RADIO WAVE PROPAGATION FUNDAMENTALS [2 ed.] 9781630818456, 1630818453

Table of contents :
Radio Wave Propagation Fundamentals Second Edition
Contents
Preface to the Second Edition
Preface to the First Edition
Chapter 1
Introduction
1.1 Brief Historical Overview
1.2 Classification of Radio Waves by Frequency Bands
1.3 The Earth’s Atmosphere and Its Structure
1.4 Classification of Radio Waves by Its Propagation Mechanisms
1.5 Interferences in RF Transmission Links
References
Chapter 2 Basics of Electromagnetic Waves Theory
2.1 Electromagnetic Process
2.1.1 Maxwell’s Equations of Electrodynamiscs
2.1.2 Boundary Conditions of Electrodynamics
2.1.3 Time-Harmonic Electromagnetic Process: Classification of Media by Conductivity
2.2 Free Propagation of Uniform Plane Radio Waves
2.2.1 Uniform Plane Wave in Lossless Medium
2.2.2 Uniform Plane Wave in Lossy Medium
2.3 Polarization of the Radio Waves
2.3.1 Basic Relationships
2.3.2 Linear Polarization (LP)
2.3.3 Circular Polarization
2.3.4 Elliptical Polarization
2.4 Reflection and Refraction of Plane Radio Wave from the Boundary
of Two Media
2.4.1 Introductory Remarks
2.4.2 Normal Incidence on a Plane Boundary
2.4.3 Oblique Incidence on a Plane Boundary
2.4.4 Power Reflection and Transmission
2.4.5 Reflection of the Radio Wave from the Boundary of Nonideal Dielectric Medium
2.5 Radiation from Infinitesimal Electric Current Source: Spherical
Waves
2.6 Spatial Area Significant for Radio Waves Propagation
2.6.1 Huygens’ Principle
2.6.2 Fresnel Zones
2.6.3 Knife-Edge Diffraction
2.6.4 Some Practical Applications of the Fresnel Zones Concept
References
Appendix 2A Useful Mathematical Relations
2A.1 Trigonometric Equalities
2A.2 Vector Analysis
Appendix 2B Basic Relations in Infinitesimal Electric Current Source
Radiation Analysis
2B.1 Helmholtz Equation for Vector Potential
2B.2 Radiation from the Electric Current Point Source
Appendix 2C Fresnel’s Integrals
Chapter 3
Basics of Antennas for RF Links
3.1 Brief Introduction
3.2 Basic Parameters of Antennas
3.2.1 Radiation Pattern and Directivity
3.2.2 Radiation Resistance, Loss Resistance, and Antenna Gain
3.2.3 Antenna Effective Length
3.2.4 Antenna Effective Area of the Aperture
3.3 General Relations in Radio Wave Propagation Theory
References
Appendix 3A Definition of the Antenna’s Far Field Zone
Chapter 4 Impact of the Earth Surface on Propagation of Ground Waves
4.1 Propagation Between Antennas Elevated Above the Earth’s Surface: Ray-Trace Approach
4.1.1 Flat Earth Approximation Case Study
4.1.2 Propagation over the Spherical Earth Surface
4.1.3 Specifics of Propagation over a Rough and Hilly Terrain
4.1.4 Optimal Path Clearance and Choice of the Antenna Elevations
4.1.5 Propagation Prediction Models in Urban, Suburban, and Rural Areas
4.2 Propagation Between Ground-Based Antennas over the Flat Earth
4.2.1 Antennas over the Infinite, Perfect Ground Plane
4.2.2 Leontovich Approximate Boundary Conditions and Structure of RadioWaves Near the Earth’s Surface
4.2.3 Propagation over the Real Homogeneous Flat Earth
4.2.4 Propagation Along the Real Inhomogeneous Flat Earth: Coastal Refraction
4.3 Asymptotic Diffraction Theory of Propagation over the Spherical Earth Surface
4.3.1 Basic Concepts
4.3.2 Propagation Between Ground-Based Antennas
4.3.3 Propagation Between Elevated Antennas
4.3.4 Specifics of Propagation Estimates in Penumbra Zone
References
Appendix 4A Input Impedance of the Radiating Current Element Above the PEC Ground Plane
Appendix 4B Diffraction of Radio Wave Around Earth’s Surface: Basic Theory
4B.1 General Solution of the Wave Equation Relevant to Propagation Factor
Appendix 4C Specifics of Ground Reflection Calculations Between Vertically Polarized Antennas
Appendix 4D Table of Roots (k) of Equation (4.56) for the Given Combination of p and x Parameters
Chapter 5 Atmospheric Effects in Radio Wave Propagation
5.1 Dielectric Permittivity and Conductivity of the Ionized Gas
5.2 Regular Refraction of the Radio Waves in the Atmosphere
5.3 Standard Atmosphere and Tropospheric Refraction
5.4 Reflection and Refraction of the Sky Waves in the Ionosphere
5.5 The Impact of the Earth’s Magnetic Field on Propagation of the Radio Waves in the Ionosphere
5.5.1 Propagation Along Geomagnetic Field Lines
5.5.2 Propagation Perpendicular to Geomagnetic Field Lines
5.5.3 Propagation of the Radio Wave Arbitrary Oriented Relative to the Earth’s Magnetic Field
5.5.4 Reflection and Refraction of the Radio Waves in the Magneto-Active Ionosphere
5.6 Specifics of Ionospheric Propagation of ELF and VLF in the Earth-Ionosphere Waveguide
5.6.1 General Remarks
5.6.2 Propagation of ELF Signals
5.6.3 Propagation of LF and VLF Signals: High-Order Modes
5.7 Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism
5.7.1 Secondary Tropospheric Radio Links
5.7.2 Analytical Approaches in Description of the Random Tropospheric Scatterings
5.7.3 Physical Interpretation of Tropospheric Scatterings
5.7.4 Effective Scattering Cross-Section of the Turbulent Troposphere
5.7.5 Statistical Models of Tropospheric Turbulences
5.7.6 Propagation Factor on Secondary Tropospheric Radio Links
5.7.7 The Specifics of the Secondary Tropospheric Radio Links Performance
5.8 Attenuation of the Radio Waves in the Atmosphere
5.8.1 Attenuations in Troposphere
5.8.2 Attenuations in Ionosphere
References
Appendix 5A Volumetric Spectrum for Autocorrelation Function of
Statistically Homogeneous and Isotropic Random Field
Appendix 5B Some Theoretical Aspects of the Ionospheric Layers Generation
5B.1 Ionospheric Gaseous Composition and Physical Processes Related to Ionospheric Propagation
5B.2 Chapman Model and Structure of the Real Ionosphere
Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere
5C.1 General Relations
5C.2 Propagation Along Geomagnetic Field Lines
5C.3 Propagation Across Geomagnetic Field Lines
Chapter 6
Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception
6.1 Multiplicative Interferences (Signal Fades)
6.1.1 Fluctuation Processes and Stability of Radio Links
6.1.2 Fast Fading Statistical Distributions
6.1.3 Slow Fading Statistical Distribution
6.1.4 Combined Distribution of Fast and Slow Fades
6.2 Additive Interferences (Noises)
6.2.1 Internal Noises of One- and Two-Port Networks
6.2.2 Noise Figure and Noise Temperature of the Cascaded Two-Port Networks
6.2.3 Noise Figure of the Passive Two-Port Network
6.2.4 Antenna Noise Temperature
6.2.5 Environmental (External) Noise
6.2.6 Basics of RF Link Performance Stability Analysis
6.3 Methods of Improvement of RF Systems Performance
6.3.1 Use of Noise Suppressing Modems for the Analog RF Links Performance Improvement
Problems
References
Selected Bibliography
Acronyms
List of Symbols
About the Author

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Radio Wave Propagation Fundamentals Second Edition

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Radio Wave Propagation Fundamentals Second Edition Artem Saakian

artechhouse.com

Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library.

ISBN-13: 978-1-63081-844-9 Cover design by Andy Meaden meadencreative.com © 2021 ARTECH HOUSE 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, includ­ ing photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this informa­ tion. Use of a term in this book should not be regarded as affecting the validity of any trade­ mark or service mark.

10 9 8 7 6 5 4 3 2 1

Dedicated to the memory of my mother, Victoria Badalyan

Contents Preface to the Second Edition Preface to the First Edition

xi xiii

CHAPTER 1 Introduction1 1.1  1.2  1.3  1.4  1.5 

Brief Historical Overview 1 Classification of Radio Waves by Frequency Bands 4 The Earth’s Atmosphere and Its Structure 6 Classification of Radio Waves by Its Propagation Mechanisms 11 Interferences in RF Transmission Links 15 References18

CHAPTER 2 Basics of Electromagnetic Waves Theory

21

2.1  Electromagnetic Process 21 2.1.1 Maxwell’s Equations of Electrodynamics 21 2.1.2 Boundary Conditions of Electrodynamics 25 2.1.3 Time-Harmonic Electromagnetic Process: Classification of Media by Conductivity 31 2.2 Free Propagation of Uniform Plane Radio Waves 35 2.2.1  Uniform Plane Wave in Lossless Medium40 2.2.2  Uniform Plane Wave in Lossy Medium42 2.3 Polarization of the Radio Waves 47 2.3.1 Basic Relationships 48 2.3.2 Linear Polarization (LP) 49 2.3.3 Circular Polarization 50 2.3.4 Elliptical Polarization 52 2.4 Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 63 2.4.1 Introductory Remarks 63 2.4.2 Normal Incidence on a Plane Boundary 65 2.4.3 Oblique Incidence on a Plane Boundary 67 2.4.4 Power Reflection and Transmission 74 2.4.5 Reflection of the Radio Wave from the Boundary of Nonideal Dielectric Medium 76 2.5 Radiation from Infinitesimal Electric Current Source: Spherical Waves 76 2.6 Spatial Area Significant for Radio Waves Propagation 79 2.6.1  Huygens’ Principle 79 vii

viii Contents

2.6.2 Fresnel Zones 81 2.6.3 Knife-Edge Diffraction 87 2.6.4 Some Practical Applications of the Fresnel Zones Concept 92 References98 Appendix 2A Useful Mathematical Relations 98 Appendix 2B Basic Relations in Infinitesimal Electric Current Source Radiation Analysis 101 Appendix 2C Fresnel’s Integrals 109 CHAPTER 3 Basics of Antennas for RF Radio Links

113

3.1 Brief Introduction 113 3.2 Basic Parameters of Antennas 113 3.2.1 Radiation Pattern and Directivity 114 3.2.2 Radiation Resistance, Loss Resistance, and Antenna Gain 120 3.2.3 Antenna Effective Length 122 3.2.4 Antenna Effective Area of Aperture 128 3.3 General Relations in Radio Wave Propagation Theory 129 References135 Appendix 3A Definition of the Antenna’s Far Field Zone 135 CHAPTER 4 Impact of the Earth Surface on Propagation of Ground Waves 4.1 Propagation Between Antennas Elevated Above the Earth’s Surface: Ray-Trace Approach 4.1.1 Flat Earth Approximation Case Study 4.1.2 Propagation over the Spherical Earth Surface 4.1.3 Specifics of Propagation over a Rough and Hilly Terrain 4.1.4 Optimal Path Clearance and Choice of the Antenna Elevations 4.1.5 Propagation Prediction Models in Urban, Suburban, and Rural Areas 4.2 Propagation Between Ground-Based Antennas over the Flat Earth 4.2.1 Antennas over the Infinite, Perfect Ground Plane 4.2.2 Leontovich Approximate Boundary Conditions and Structure of Radio Waves Near the Earth’s Surface 4.2.3 Propagation over the Real Homogeneous Flat Earth 4.2.4 Propagation Along the Real Inhomogeneous Flat Earth: Coastal Refraction 4.3 Asymptotic Diffraction Theory of Propagation over the Spherical Earth Surface 4.3.1 Basic Concepts 4.3.2 Propagation Between Ground-Based Antennas 4.3.3 Propagation Between Elevated Antennas 4.3.4 Specifics of Propagation Estimates in Penumbra Zone

137 137 138 146 156 162 164 174 175 178 184 188 193 193 198 204 206

Contents

ix

References211 Appendix 4A Input Impedance of the Radiating Current Element Above the PEC Ground Plane 212 Appendix 4B Diffraction of Radio Wave Around Earth’s Surface: Basic Theory 215 Appendix 4C Specifics of Ground Reflection Calculations Between Vertically Polarized Antennas 225 Appendix 4D Table of Roots (k) of Equation (4.56) for the Given Combination of p and x Parameters 228 CHAPTER 5 Atmospheric Effects in Radio Wave Propagation

229

5.1 Dielectric Permittivity and Conductivity of the Ionized Gas 229 5.2 Regular Refraction of the Radio Waves in the Atmosphere 234 5.3 Standard Atmosphere and Tropospheric Refraction 236 5.4 Reflection and Refraction of the Sky Waves in the Ionosphere 241 5.5 The Impact of the Earth’s Magnetic Field on Propagation of the Radio Waves in the Ionosphere 247 249 5.5.1 Propagation Along Geomagnetic Field Lines 5.5.2 Propagation Perpendicular to Geomagnetic Field Lines 253 5.5.3 Propagation of the Radio Wave Arbitrary Oriented Relative to the Earth’s Magnetic Field 255 5.5.4 Reflection and Refraction of the Radio Waves in the Magneto-Active Ionosphere 258 5.6 Specifics of Ionospheric Propagation of ELF and VLF in the Earth-Ionosphere Waveguide 260 260 5.6.1 General Remarks 5.6.2 Propagation of ELF Signals 264 265 5.6.3 Propagation of LF and VLF Signals: High-Order Modes 5.7 Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 269 269 5.7.1 Secondary Tropospheric Radio Links 5.7.2 Analytical Approaches in Description of the Random Tropospheric Scatterings 271 5.7.3 Physical Interpretation of Tropospheric Scatterings 276 5.7.4 Effective Scattering Cross-Section of the Turbulent Troposphere 279 5.7.5 Statistical Models of Tropospheric Turbulences 280 5.7.6  Propagation Factor on Secondary Tropospheric Radio Links 286 5.7.7 The Specifics of the Secondary Tropospheric Radio Links Performance 291 5.8 Attenuation of the Radio Waves in the Atmosphere 296 5.8.1 Attenuations in Troposphere 296 5.8.2 Attenuations in Ionosphere 300 References307

x Contents

Appendix 5A Volumetric Spectrum for Autocorrelation Function of Statistically Homogeneous and Isotropic Random Field Appendix 5B Some Theoretical Aspects of the Ionospheric Layers Generation Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere CHAPTER 6 Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

308 309 315

325

6.1 Multiplicative Interferences (Signal Fades) 325 6.1.1 Fluctuation Processes and Stability of Radio Links 325 331 6.1.2 Fast Fading Statistical Distributions 341 6.1.3 Slow Fading Statistical Distribution 6.1.4 Combined Distribution of Fast and Slow Fades 346 6.2 Additive Interferences (Noises) 353 353 6.2.1 Internal Noises of One- and Two-Port Networks 6.2.2 Noise Figure and Noise Temperature of the Cascaded Two-Port Networks 356 6.2.3 Noise Figure of the Passive Two-Port Network 358 6.2.4 Antenna Noise Temperature 361 362 6.2.5 Environmental (External) Noise 6.2.6 Basics of RF Link Performance Stability Analysis 370 374 6.3 Methods of Improvement of the RF Systems Performance 6.3.1 Use of Noise Suppressing Modems for the Analog RF Links Performance Improvement 375 6.3.2 The Brief Overview of Digital RF Link Performance Improvement Approaches 376 382 6.3.3 Use of Spread-Spectrum Discrete Signals 6.3.4 Diversity Reception Technique 386 References394 Acronyms 397 List of Symbols

399

About the Author

403

Index 405

Preface to the Second Edition The preparation of the second edition of this book for publication was initiated on the basis of further pedagogical experience and engineering practice of the author at NAVAIR. As well, it was stimulated by new advances in the theory and technology of constructing RF links that took place after the publication of the first edition. Among the important changes and additions made by the author in the second edition, the following should be noted: •

•

• •

• • •

•

New materials related to decomposition of the arbitrary (elliptically) polarized radio wave into two cross-orthogonal waves of linear polarization, and/ or into two circularly polarized waves of the opposite sense (right and left); these changes are presented in Section 2.3. A new, clear understanding of the features of the reflection of a radio wave with polarization parallel to the plane of incidence is presented. This idea seems to be especially important when analyzing the signal reception by the vertically polarized antenna raised above the ground (see Section 2.4 and Appendix 4C). Physical explanation of the term antenna’s far field zone (Appendix 3A). Specifics of ionospheric propagation of radio waves from extremely low frequencies (ELF) to low (LF) frequencies in the Earth-ionosphere waveguide (Section 5.6). Physical proof of the appearance of ionospheric layers using the example of the theoretical (Chapman) model (Appendix 5B.2). Detailed theoretical evaluations of the radio wave propagation specifics in ionospheric magnetoactive plasma (Appendix 5C). A new, statistical approach in the description of the fades of the received signal due to random interference between direct-to-reflected waves (Section 6.1.2). Several new examples and end-of-chapter problems.

The author expresses the hope that the book, in its new edition, will be interesting and useful to a wide audience of students and professionals.

xi

Preface to the First Edition The main goal of this text is to satisfy the growing demand in propagation study materials for students and specialists. The core materials of the proposed text are developed from lecture notes that have been offered by the author for graduate students at Patuxent Graduate Center of Florida Institute of Technology. The materials included into the text have been extended beyond the needs of the single-semester course and may be used for continuous self-study. The objective of this text is to support senior-level undergraduate and graduate electrical engineering (EE) students with an introduction of the basic principles of electromagnetic waves propagation of radio frequencies (RFs) in real conditions relevant, but not limited, to communications and radar systems. It is also to emphasize the primary role of the antenna-to-antenna propagation path in overall performance of those systems. Some of the practicing engineers who need a quick reference to the basics of propagation mechanisms and principles of engineering estimates and designs may use this text in their everyday routine. It may be useful not just for the students and specialists in the area of radar and communication technologies, but also for the students, scientists, and engineers of the adjacent areas of science and engineering/technology, such as antenna engineering, astrophysics, geomagnetism, and aeronomy. Chapter 1 is an introductory chapter, which outlines the definitions and classifications that are commonly used and are adopted by international organizations such as IEEE and ITU. A brief survey of the structure of the Earth’s atmosphere is considered in this chapter in support of atmospheric propagation phenomena that are covered in Chapter 5. Chapter 2 covers the basics of electromagnetic waves theory with the emphasis on those specific for the RF propagation, such as polarization of radio waves, their reflections and transmission at the interface of two media, as well as diffraction on the knife-edge obstacles. As a subject of special interest, the Fresnel’s zones are analyzed based on Huygens-Kirchhoff’s principles of electromagnetics. This is to clarify the concept of ray forming (i.e., the concept of the spatial area actively involved in canalization of energy of the radio waves). Some engineering applications are presented in this chapter to demonstrate the variety of application of that concept. Chapter 3 is a brief journey to the basic antenna parameters that is needed to evaluate and analyze antenna-to-antenna propagation path. The last section presents the main relations, such as the Friis formula and link budget equation, as well as introduces propagation factor, which is an important measure of the impact of real conditions on the propagation of radio waves. Those main relations are of importance to evaluate the radar equation, as well as for radio link power budget analysis in communication systems.

xiii

xiv

Preface to the First Edition

Chapter 4 is devoted to propagation of ground radio waves (i.e., the waves that propagate in vicinity of the Earth’s surface) being affected by that interface. It appears to be methodically reasonable not to involve any atmospheric effects into consideration within the scope of this chapter, in order not to confuse the reader by mixing ground effects and atmospheric effects. The methods of propagation factor calculations are first considered here for the flat-Earth approximation, based on Leontovich’s boundary conditions with further extension for the cases of convexity of the Earth’s ground. Propagation features over the inhomogeneous paths are presented within Mandelshtam’s takeoff-landing concept that is qualitative, rather that quantitative: it allows readers to better understand the behavior of the waves that propagate along mixed type paths. Some quantitative estimates are given to support the engineering estimates. The coastal refraction effect is analyzed as a particular case along with numerical estimates discussed. As a case of the propagation in mixed, rough terrains, the propagation in urban, suburban, and rural areas is considered in Section 4.1.5. Section 4.3 introduces the asymptotic diffraction theory (V. A. Fock)—the theory of diffraction of radio waves over the spherical Earth’s surface. Engineering approaches for practical applications are demonstrated by using sample examples. Chapter 5 is dedicated to the effects of the atmosphere on propagation of radio waves. Smooth refraction in the troposphere and reflections from the ionospheric layers are analyzed in conjunction with the regular inhomogeneities of the refraction index in those atmospheric regions. Scattering of the radio waves of UHF and higher frequency bands from the random variations of the tropospheric refraction index (from air turbulences) are considered here by using the principles of statistical radio-physics. The results are brought to the level of engineering applications and design of the over-the-horizon troposcatter communication links. Troposcatter radio-links are not widely used currently, although the understanding of the physical mechanisms of the scatterings in the troposphere may become a background for the understanding of scattering phenomenon in general. Thus, there’s no need to discuss them one by one for other scatterings, such as from ionospheric meteor trails, raindrops, and so on. The chapter concludes with an analysis of absorptions in atmosphere. Both absorptions in tropospheric gases and hydrometeors as well as in ionospheric layers are introduced to support signal attenuation estimates on a variety of RF radio links. Chapter 6 is devoted to the reception of radio waves, which is of the highest importance for practical applications in radio links design. Two types of interferences, namely multiplicative (fading) and additive (noise), are analyzed in conjunction with the signal-to-noise ratio (SNR) and communication stability. Statistical distributions of fast and slow fades are considered, as well as a combined distribution to predict a long-term stability of the communication system performance. This analysis results in an engineering method for power margin calculation, which is to ensure that the objective of the communication stability is met. Two components of the additive noise are considered: internal noise (receiver noise) and external noise (environmental noise) received from the surrounding areas. The sensitivity of the receiver is discussed in order to define a threshold of the received signal level. The chapter concludes with a section that outlines the basic methods of improvement of the radio wave reception: (1) the use of noise-resistive signals, such as analog FM,

Preface to the First Edition

xv

(2) the use of the spread-spectrum signals, and (3) the diversity reception technique. The main goal of this chapter is to build a bridge between RF system structure and propagation conditions and mechanisms (i.e., to show a direct coupling between them). The scope of the book includes a wide variety of aspects of radiophysics; therefore, the proposed text does not attempt to cover all details of the subject, but rather it encourages a creative approach among readers. A background in the mathematics and electromagnetics required in engineering and physics curricula is assumed. Some of the unique mathematical techniques and evaluations are incorporated within appropriate chapters. Appendix 2A provides some useful mathematical relations. The numerous calculation examples are to support better understanding of the core materials of the text. The problems at the end of the chapters may become highly supportive for the study process. The problems solution manual is available for teachers and instructors on the Artech House Web site to support the teaching process. This text may be used in senior elective or entry-level graduate courses. The list of symbols contains only generalized notations for the quantities (e.g., f for frequency) in base or derived units. Subscripts are used within the text to denote specific application of the particular notation (e.g., fc for critical frequency of the ionospheric plasma in hertz); for the multiple and fractional quantities within the text, the units are indicated in symbols by the proper subscripts (e.g., fc, MHz for the ionospheric plasma frequency in megahertz). First, second, and third derivatives are notated with prime, double prime, and triple prime, respectively. Complex numbers and complex vectors are notated with the dot above the symbol. I wish to express my deepest gratitude to my reviewers, especially to Dr. D. K. Barton, whose valuable comments and recommendations helped improve the text significantly. I’d like to thank Prof. V. E. Arustamyan from the State Engineering University of Armenia for his revision and constructive criticism of very first draft of the manuscript in Armenian, as well as Ms. Svetlana Avanian for help with typing. I’d also like to express my appreciation to Ms. Catherine Wood for her help with grammatical and syntax corrections of most of the chapters. I’m also thankful to my colleague, Mr. Frederick Werrell, for his review of Chapter 3, as well as to my students for critical remarks during and after the classroom studies. My appreciation is forwarded to Mr. Norm Chlosta, the director of the Patuxent Graduate Center of the Florida Institute of Technology, for his support and for providing me with the opportunities to test the course with smart audience of graduate students. Finally, I’d like to express my thankfulness to my wife Arous: without her understanding, patience, and support, this work would not be possible.

CHAPTER 1

Introduction 1.1  Brief Historical Overview The beginning of the era of using electrical signals to transmit information over a distance is considered to be 1844, when the first telegraph signal was transmitted via cable as a guiding system between distant points. The next significant success in the development of communication systems is considered to be the construction of the first trans-­Atlantic cable line, which officially launched in the summer of 1858, when a congratulatory telegram was sent from Queen Victoria to U.S. President James Buchanan. However, a revolutionary event in communication technology occurred in May 1895 when the Italian-­born American engineer G. M. Marconi and the Russian physicist A. S. Popov (Figure 1.1) carried out the transmission of a communication signal between spatially separated points using radio waves—­ electromagnetic waves that freely propagate in the earth’s atmosphere. Thus began the era of wireless communications. This initial effort was particularly notable, as it was conducted without the support of the traditional wire guiding line first used by Marconi in 1844 for his wire telegraphy efforts. Instead, a spark-­gap was implemented as a transmitting source for the electromagnetic radiation and a coherer was utilized as a receiving device. It is difficult to overestimate the importance of that invention for society, and for the advancements it heralded for wireless communications. Yet, that invention might not have been possible without prior theoretical hypotheses of the existence of free propagating electromagnetic waves, which were made in 1864 by the Scottish mathematician and theoretical physicist J. C. Maxwell [1]. Maxwell’s greatest merit was a theoretical prediction of the displacement currents in dielectrics and vacuums, which had generalized the concept of current continuity in Ampere’s law. The fundamental equations of electromagnetism (known as Maxwell’s equations) were later updated to achieve complete and symmetric form by the introduction of magnetic currents. The introduction of displacement electric and magnetic currents in dielectrics, and in free space, made possible the comprehension of the nature of electromagnetic waves capable of propagating for long distances independent of any physical guidance such as wires or waveguides. Those electromagnetic waves are identified as radio waves when relevant to scientific and commercial applications. Experimental verification of the existence of electromagnetic waves was achieved by H. Hertz in the 1880s, when he demonstrated a “propagation” of the spark from a transmitting Leyden jar to the terminals of remote receiving antenna. The revolutionary role of Maxwell’s equations, which are based on experimental investigations of M. Faraday and A. M. Ampere, is hard to overemphasize for the immense progress they have allowed science, engineering, and associated technologies. 1

2 Introduction

Figure 1.1  Guillermo Marconi (left) and Alexander Popov (right).

Soon after Marconi’s and Popov’s experiments with the transmission of telegraph signals over the distance of several miles in 1895, Marconi was able to greatly extend the range of propagation from the U.K. to Canada, over the Atlantic Ocean, in December 1901. This accomplishment was possible by the use of a sinusoidal carrier, a resonant LC filter at the receiver’s input, and a vertical grounded radiator (antenna), which thereby demonstrated the advantage of vertical polarization for the frequency ranges of tens and hundreds of kilohertz. The immense success of Marconi’s long-­range signal transmission motivated engineers and research scientists to find a reasonable explanation for existence of propagation mechanisms [2]. Ionospheric propagation is a mechanism that was initially assumed to be overall predominant. It was introduced in 1902 after A. E. Kennelly, in the United States, and O. Heaviside in the United Kingdom, postulated the existence of the ionized region in the upper atmosphere, which seemed to reflect the radio waves and thereby support their long-­range propagation. The second mechanism of propagation was based on the assumption of the existence of surface waves, which may occur only at the interface of two media, such as the boundary of atmosphere and Earth’s ground. A detailed theoretical analysis of surface propagation was conducted by the German physicist A. Sommerfeld in 1909 [3], an analysis that was based on Maxwell’s equations for vacuum to perfect electric conductor flat boundary. A specific approach for the solution near the boundary of two ideal media was achieved and was repeated 10 years later by H. Weyl in a more explicit form [4]. It was shown that two propagating modes exist: (1) a regular TEM mode of a spherical phase front that is specific to free space propagation; for this mode the field strength is in inverse proportion to propagation distance, and (2) a surface mode of a cylindrical phase front, which is tightly bound to the interface of two media, and which may exist only in the vicinity of that interface, whose field strength is in inverse proportion to the square root of the propagation distance. For moderate and large distances, the second component becomes predominant. In the late 1930s the conceptualization of surface waves, relevant to the real Earth’s ground electromagnetic constants, was further developed by J. Zenneck and K. A. Norton [5]. The third mechanism of propagation was based on the assumption of the diffraction of radio waves around the Earth’s convex surface. These assumptions were proposed by several mathematicians and scientists prior to and during World War I. They were first presented by G. N. Watson in 1919 in the form of estimates of the field at the receiving point for the ideal conductive Earth surface, as an attempt

1.1  Brief Historical Overview 3

of direct solution of Maxwell’s equations. Later, in 1937, Van-­Der-­Paul and Bremmer adopted Watson’s approach by applying arbitrary ground constants. Based on those developments, C. R. Burrows created an engineering method that allowed representation of the results in a form that was convenient for the applications. Directly following World War-­II, in 1945-­46, Soviet physicists V. A. Fock and M. A. Leontovich introduced a Maxwell’s equations solution in parabolic form relevant to the diffraction problem [8]. The complete theory was developed by taking into account the properties of the Earth ground in a wide frequency range. It was broadly used, until quite recently, for the over-­the-­horizon propagation analysis (diffracted field analysis) of radio waves associated with frequencies up to tens of megahertz. Both surface wave and diffracted wave approaches result in a single conclusion: the lower a radio wave’s frequency, the more favorable the propagation conditions are. Prior to and after World War II, the need to further increase the volume of transmitted information within a single communication link, as well as the need to advance radar system performance, motivated an increase in carrier frequencies. Thereby higher and higher frequencies were employed for the support of communication and radar systems design, along with newly developed antenna radiating systems and advances radio frequency (RF) components, together providing increased performance and effectiveness. Radio waves implementing frequencies of hundreds of megahertz and up were considered to be employed. For those higher frequency ranges, propagation analysis is based most substantially on geometric optical approaches. The first empirical expression for the intensity of the current induced in a remote receiving antenna for distances within line-­of-­sight (LOS) was developed by L. W. Austin in 1911, in the United States. That expression was fairly close to the formula developed analytically, in former USSR, by B. A. Vvedensky in the late 1920s [9]. Vvedensky’s formula was based on a ray-­tracing (geometrical optics) approach. For higher frequency bands, the atmospheric effects became considerable and had to be taken into account. Thus, after World War II, a significant amount of attention was focused on issues such as attenuation in atmospheric gases, as well as refraction, reflection, and scatter of the radio waves in the lower and upper atmosphere. One of those effects, namely the tropospheric scatter of microwaves by atmospheric turbulence, was discovered in the late 1940s and was later theoretically validated by H. G. Booker and W. E. Gordon in 1950 [10]. Some limited military and commercial applications of the tropospheric scattering effect take place currently, in nominated troposcatter radio links within the United States and several European countries. A new era of ionospheric propagation research and investigation initiated after the launch of the first satellite (“Sputnik”), by the USSR in 1957, to the Earth’s orbit. This culminated in the current large network of specialized ionospheric ground-­ based and satellite-­based radars, sounding stations (ionosondes), and satellite systems worldwide that allow obtainment of a complete set of data for long-­term predictions of the status of ionospheric layers. Those predictions are widely used for the radio links design and for deployment in high-­frequency (HF) and higher frequency bands, which are directly affected by the ionosphere. In modern times, the radio wave propagation theory and applications still remain subjects of extremely high interest to the science and engineering technological

4 Introduction

world and are consistently in further development and expansion via numerous worldwide programs.

1.2  Classification of Radio Waves by Frequency Bands A radio wave is defined by the Institute of Electrical and Electronics Engineers (IEEE) as “an electromagnetic wave of radio frequency” [11]. Each particular radio frequency belongs to the radio spectrum, which is a wide range of frequencies from several hertz up to 3 THz. The entire radio spectrum is divided into frequency bands, shown in Table 1.1, that are based on decimal division. This standard classification is accepted by the International Telecommunications Union (ITU), which is comprised of 189 member-­countries. A selection of applications pertinent to the radio waves used in engineering, science and technological efforts includes, but is not limited to the following: • • •

Wireless communication systems, including satellite communication systems and, wireless local area networks (WLAN); Radar systems; Telemetry, radio-­remote control, radio-­navigation and radio astronomy.

Adhering to ITU-­R1 recommendations, the range for each frequency band extends from 0.3 × 10N to 3 × 10N Hz,2 where N is a band number given in the first column of Table 1.1. There are many subdivisions within each band, reliant upon allocations to services and the inhabitant worldwide regions [12, 13]3. Terminology noted in the last two columns of Table 1.1 is commonly used, but not officially accepted. A subdivision of microwave bands, shown in Table 1.2, is widely used in radar and satellite applications [14]. Both the upper and lower limits of the radio spectrum are outlined conventionally and rely entirely on progressions made in science and technology. For instance, until the mid-­1930s radio communications designs were based on technologies that allowed only utilization of frequencies lower than 100 MHz, as that represented the upper limit of radio frequencies at that time. In the 1930s and 1940s, overarching progress in the design of a new generation of the radar systems invoked a claim of involving higher frequencies. Invention of multiple new types of devices, including magnetrons, klystrons, traveling-­wave tubes, and others, allowed the expansion of the upper limit of radio frequencies to approximately 10 GHz and higher. In the mid-­1950s to early 1960s, when the new generation of quantum electronic devices, such as MASERs, were developed by C. H. Townes, (United States), N. G. Basov, and A. M. Prokhorov (USSR), they were based on achievements in quantum radio-­spectroscopy. Further broadening of the upper 1. The Radio Communications section of the ITU. 2. Conventionally, the upper limit is included into the band, and the lower limit is excluded. 3. The US frequency allocations to variety of services may be found in National Telecommunications and Information Administration’s (NTIA) website under the following URL: https://www.ntia.doc.gov/files/ ntia/publications/2003-­allochrt.pdf.

1.2  Classification of Radio Waves by Frequency Bands 5 Table 1.1  ITU Classification of the Radio Waves by Frequencies Frequency Band Name by ITU-R

Acronym

Frequency Range, Hz

Wavelength in Meters

Descriptive Name

Acronym

—

Extremely low frequency Very low frequency Low frequency Medium frequency High frequency Very high frequency Ultra high frequency Super high frequency Extremely high frequency —

ELF

< 3 ´ 103

> 105

–

–

VLF

(3 to 30) ´ 103

104 to 105

–

LF

(30 to 300) ´ 103

103 to 104

MF

(0.3 to 3) ´ 106

102 to 103

HF

(3 to 30) ´ 106

10 to 102

VHF

(30 to 300) ´ 106

1 to 10

Miriameter waves Kilometer waves Hectometer waves Decameter waves Meter waves

UHF

(0.3 to 3) ´ 109

10-1 to 1

SHF

(3 to 30) ´ 109

10-2 to 10-1

EHF

(30 to 300) ´ 109

10-3 to 10-2

—

(0.3 to 3) ´ 1012

10-4 to 10-3

4 5 6 7 8 9 10 11 12

Microwaves

Band Number, N

Decimeter waves Centimeter waves Millimeter waves Submillimeter waves

– – – MW DMW CMW MMW SMMW

Table 1.2  Microwave Bands Subdivision for Radar and Satellite Applications Band Name

L

S

C

X

Ku

Frequency range, GHz Band name Frequency range, GHz

1–2 K 18–27

2–4 Ka 27–40

4–8 V 40–75

8–12 W 75–110

12–18 mm band 110–300

Table 1.3  Classification of Optical Wave Optical Band Name

Wavelength in Meters (µm)

Frequency Range, in Hz

Far infrared (IR) band Medium IR band Near IR band Visible light Ultraviolet rays

2 ´ 10–­5 m to 10–­4 m (20 µm to 100 µm) 1.5 ´ 10–­6 m to 2 ´ 10–­5 m (1.5 µm to 20 µm) 7 ´ 10–­7 m to 1.5 ´ 10–­6 m (0.7 µm to 1.5 µm) 4 ´ 10–­7 m to 7 ´ 10–­7 m (0.4 µm to 0.7 µm) 10–­8 m to 4 ´ 10–­7 m (0.01 µm to 0.4 µm)

(3 to 15) ´ 1012 (15 to 200) ´ 1012 2 ´ 1014 to 4.3 ´ 1014 4.3 ´ 1014 to 7.3 ´ 1014 7.5 ´ 1014 to 3 ´ 1016

limit of radio frequencies became possible, not just by including CMW, MMW and SMMW, but also by instituting coherent optical waves to develop lasers. Implementation of these new types of devices made possible the amplification and generation of coherent radiation even in an optical domain, and thereby traditional radio technologies and principles became applicable to the optical frequency domain as well. Optical waves that are used for wireless information transmission and processing are sometimes called optical radio waves [15]. Table 1.3 shows the classification of optical waves by frequency bands, which is considered a nonofficial classification and is widely used by specialists in different areas of engineering and science.

6 Introduction

In tandem with other items contributing to expansion of the lower limit of the radio spectrum, there are also rationales based on the needs of global military communication services and navigation, as well as scientific research in the areas such as geophysics, atmosphere science, and radio astronomy.

1.3  The Earth’s Atmosphere and Its Structure The real Earth’s atmosphere has a complex structure, which significantly impacts radio wave propagation, causing effects such as smooth refraction, scatter, and energy absorption of the radio wave. Variations of the electromagnetic parameters of the atmospheric air are highly dependent on its gaseous composition, pressure, humidity and ionization. The vertical profile4 of distribution of the main composites of atmospheric air is shown in Figure 1.2 [15]. As one may notice from the diagram, for heights of up to approximately 90 km, the gaseous composition of the atmosphere is homogeneous as a result of the continuous mixture caused by ascending, descending, and horizontal air streams that permanently exist in that area. At that range of height, the atmospheric air is composed of about 78% molecular nitrogen and about 20% to 21% molecular oxygen, despite the fact that they have different molecular weights. The remainder is a mix of carbon dioxide, argon, ozone, and other gases. This atmospheric area of homogeneous distribution of gases is conventionally divided into two regions, the troposphere and the stratosphere. The troposphere is the lowest portion of the Earth’s atmosphere. It contains approximately 80%of the atmosphere’s mass, and 99% of its water vapor and aerosols. The average ceiling of the troposphere is approximately 17 km in its middle latitudes. It is deeper in tropical regions (up to 20 km and more), and is shallower near the Earth’s poles (about 7 km in summer and an indistinct measurement in the winter). The remaining portion of the homogeneous atmospheric area is known as stratosphere. One of the specific features of the troposphere, which distinguishes these two regions, is a rapid decrease of the concentration of water vapors reliant on elevation. In fact, the humidity level is highly dependent on weather conditions. The main characteristics of the troposphere are air pressure, usually measured in millibars, and absolute humidity, which is also measured in millibars. Based on numerous observations and collaborative measurements carried out worldwide, in 1925 the International Commission for Aeronavigation introduced the international standard atmosphere, which was later renamed, and is called now the standard troposphere. It represents a hypothetical troposphere with characteristics that are averaged from the measurements to portray all locations and seasonal influences. Those characteristics are noted here [15]:

4. The graph of the elevation dependence of any parameter of the atmosphere is called a vertical profile of that particular parameter. Another example of vertical profile is free electrons’ distribution, which is shown in Figure 1.3, or elevation dependence of the tropospheric air refraction index (refractivity) which is shown in Figure 5.4.

1.3  The Earth’s Atmosphere and Its Structure 7

Figure 1.2  Diagram of the gaseous composition of atmospheric air (by percentage).

• • • • •

A sea-­level air pressure of 1,013 millibars; A constant vertical gradient of the air pressure of negative 120 millibars per kilometer; A sea-­level temperature of 290K; A constant vertical gradient of the air temperature equal negative 5.5K per kilometer; Relative humidity of 60%, which is assumed to remain elevation independent.

The average vertical profile of the temperature is:

T (h) = T0 - 5.5 × h (1.1) Here T0 = 290K and elevation h in kilometers. Equation (1.1) may be understood based on the following rationale: the tropospheric air is transparent to solar thermal radiation; thereby, it does not cumulate the heat directly from solar radiation. The bulk of that thermal radiation penetrates through the troposphere freely and reaches the Earth’s surface, where it is absorbed. The air layers adjacent to the Earth’s surface become heated due to heat transfer and air convection processes. The higher the elevation, the less the effect of these processes is, resulting in linear decay of the temperature given by expression (1.1). A similar linear vertical profile is specific for averaged tropospheric air pressure5. As noted later in Chapter 5, from the viewpoint of atmospheric propagation problems analysis, the most important parameter is the dielectric permittivity e of the atmospheric air. It closely relates to the refraction index, defined as n = ε . The mean value of tropospheric air refraction index, being tightened to atmospheric

5. As noticed from numerous observations, for the altitudes higher than 10 km the linearity of the vertical profile of mean temperature and air pressure becomes significantly destroyed. However, the atmospheric air at those high altitudes is extremely sparse; therefore, those distortions do not play a significant role in propagation mechanisms specifically on radar and communication paths.

8 Introduction

air characteristics, appears as a smooth, linearly decaying function of elevation (see Figure 5.4). At the same time, a large number of globally noted experiments and measurements, undertaken over many decades, have demonstrated existence of seasonal and random fluctuations of atmospheric air in all atmospheric regions. In the troposphere, the main mechanism of their generation is stipulated by the horizontal and vertical movements of air masses. Under proper conditions, those movements become turbulent (i.e., the air masses of different refraction indexes are mixed randomly in space and time, resulting in random space-­time fluctuations of the refraction index). These turbulent movements may be observed in the visible region of the spectrum of electromagnetic radiation in multiple ways to include the twinkling of stars, the wavering appearance of objects seen over the Earth’s surface that is heated by sun, and the conversion trails left by the exhaust gases of aircraft jet engines. The same processes take place in radio frequency bands, and all of these examples demonstrate that the air in the troposphere is present in a random, erratic flow. The stratosphere does not alter the propagation of radio waves significantly, as it is the atmospheric region containing fairly constant gases, whose composition is of a very low density. The ionosphere is the upper part of the Earth’s atmosphere, which extends from 60 kilometers upwards. At these elevations, the atmospheric air becomes ionized (i.e., the neutral atoms and molecules split into positively charged ions and free electrons). This state of matter is called plasma. Regarding the latest data obtained from the ionospheric research, the upper border of the ionosphere is above 20,000 km. Ionization of atmospheric air is caused by intensive radiation flow, emanating from the outer space that is sometimes referred to as cosmic rays. Cosmic rays are the intensive flow of a variety of elementary particles and photons6 composed of a wide range of energies. A major contribution to the total intensity of this radiation comes from the sun. Physics courses will teach that in order to ionize the gas cloud (i.e., to tear off an electron from an atom or molecule), a quantum of energy greater than a work function, We is to be applied. This amount of energy must be expended to break the bond between an electron and atom (or electron and molecule), and may be acquired from the composite part of cosmic ray (i.e., from this radiation that comes from outer space). Forms of this radiation may include the following: • •

Elementary particles (protons, neutrons, electrons, and so on); Photons of electromagnetic character, such as ultraviolet, x-­rays and g-­ radiation.

Hence, two types of ionization are to be distinguished: (1) strike ionization, caused by particles, and (2) photo ionization, caused by photons. For photo ionization the

6. Acting as the envelopes of electromagnetic waves, photons are often considered particles, due to some properties that are specific to the elementary particles. For instance, these particles are able to eject the electrons from the atom’s boundary by bombarding them, or bouncing from each other like the balls in billiard game.

1.3  The Earth’s Atmosphere and Its Structure 9

energy carried by photon must be greater than or equal to the work function; that is,

h f ³ We, (1.2) where: h = 6.626 × 10- 34 J × s is a Planck’s constant, and f is the frequency of the photon. The maximum wavelength (threshold wavelength) of the radiation, which is able to cause the ionization, may be found from (1.2) as



λ £ λ max =

ch , (1.3) We, min

where c = 3 · 108 m/s is the speed of light in free space7, and We, min represents the minimal energy in joules that a single particle or photon in a cosmic ray may have. That portion of energy may also be expressed in electron-­volts (eV), if the relation 1eV = 1.6 · 10-19 J is applied. Among the composite gases of atmospheric air, nitrogen oxide has lowest value of the work function, We = We, min = 1.48 · 10-18 J. Thus, the maximum wavelength of radiation that is able to ionize this gas is found from (1.3) as l max = 0.134 m. From (1.3) one also may realize that only ultraviolet radiation, as well as the radiation of the shorter wavelengths such as x-­rays and g-­radiations, is able to cause the ionization of atmospheric air. From numerous observations and measurements, it has been concluded that the photo-­ionization in the real atmosphere is caused by radiation in the range of wavelengths from 0.03 to 0.14 µm. The ionization degree of the atmospheric air may be expressed by the number of free electrons per unit volume (predominating per cubic centimeter), which is known as the concentration of electrons (or plasma concentration), Ne. An experimental graph of the vertical profile of ionospheric plasma concentration is shown in Figure 1.3. The ionization of atmospheric air starts from the height of about 60 km on upwards. It is shown analytically [15] that for the hypothetical homogeneous gaseous composition of the atmosphere and for the exponential model of the vertical profile of air pressure, the single-­layered vertical profile of the plasma concentration will be obtained when the ionizing radiation is monochromatic. However, in real conditions for a complex structure of air composition, as well as for the complex mixture of ionizing radiation (multiparticle and multiphoton cosmic rays), the vertical profile of the ionospheric plasma concentration becomes multilayered (stratified) as shown in Figure 1.3. As may be noted from that figure, four layers exist during the day: D layer (60 to 90 km), E layer (90 to 120 km), F1 layer (180 to 230 km), and F2 layer (230 km and up). A sporadic ES layer of fairly high plasma concentration may appear and disappear randomly. Starting from elevations of about 400 km and higher, there is no stratification but only a smooth decrease of the concentration of ionospheric plasma.

7.

Note that the ionization process itself does not depend on the intensity of radiation but on the wavelength of the ionizing radiation.

10 Introduction

Figure 1.3  Vertical profile of plasma concentration in the real ionosphere.

Aside from ionization, there are also recombination processes, where randomly moving free electrons may collide with positively charged atoms and molecules, resulting in the recovery of neutral particles. It thereby becomes clear that the rate of the recombination process is as great as the number of free electrons and positive charged particles (i.e., as great as the corresponding plasma concentration). Consider the limited spatial volume of the initially neutral atmospheric air is being ionized. Then, after the ionization is complete (i.e., plasma is generated), the considered volume will remain neutral overall, due to the number of generated negative free electrons remaining equivalent to the number of positively charged atoms, or to the molecules with an equal amount of total charge. Initially, when ionizing radiation is applied to neutral gas, the rate of generation of charged particles is nearly constant, causing an increase of plasma concentration. When plasma concentration increases, after a certain period of time a balance between ionization and recombination will be achieved, so far as the rate of recombination is proportional to the ionospheric plasma concentration Ne. Thus, the plasma’s concentration stabilizes after the transition period ends. Typically the ionization and recombination processes are in balance at noon and at midnight; whereas during sunrise (or sunset), when the radiation coming from the sun appears (or disappears), this balance is destroyed, and smooth changes of the ionization concentration (increasing in the morning hours and decreasing in the evening hours) may be observed. This phenomenon results in the disappearance of two layers during night time hours: D layer and the F1 layer. Only two of the overall layers, namely the E layer and F2 layer, will remain during night-­time hours, with a plasma concentration that is much less than it is during the day (Figure 1.3). Finally, it must be noted that the profile of ionospheric plasma concentration shown in that figure is simply a graph of the averaged values of Ne. In reality, there are some random fluctuations of Ne around each point of the graph, similar to those occurring in a refraction coefficient of troposphere. Several factors that cause the fluctuations of Ne are:

1.4  Classification of Radio Waves by Its Propagation Mechanisms 11

• • •

•

Random fluctuations of intensity of the ionizing radiation emanating from outer space; Turbulent movements caused by horizontal and vertical drafts of the ionospheric plasma; Fast invasion of micrometeors and cosmic dust, acting as an additional source of the ionization, causing highly ionized and randomly distributed prolonged paths of ionizations; Magnetohydrodynamic waves originated by the influence of the Earth’s magnetic field in the presence of mobile masses of ionized air.

These random fluctuations of ionospheric plasma concentration result in random scatterings (not reflections) of radio waves, which are most intensively observed in HF and VHF bands.

1.4  Classification of Radio Waves by Its Propagation Mechanisms Two types of radio waves propagation are: (1) guided propagation, and free (unguided) propagation. Free (unguided) propagation of radio waves occurs between corresponding antennas in the Earth’s atmosphere, under-­water, or in free space;8 this is in contrast to guided propagation, which occurs in manmade guiding systems, such as wire-­lines, coaxial cables, waveguides, and optical fibers. However, only free propagating radio waves are subjects for detailed consideration in this textbook. The following terms are introduced for classification of radio waves by propagation mechanisms between transmitting and receiving antennas. A direct radio wave (or simply direct wave) is a radio wave that propagates from a transmitting to a receiving point over “an unobstructed ray path” [11] (i.e., over the trajectory that is either a straight line or close to it). One example of a direct radio wave is one that propagates via an Earth-­to-­space (uplink), space-­to-­ space, or space-­to-­Earth (downlink) path of a satellite communication system [see Figure 1.4(a)]. A reflected radio wave (or a reflected wave) is a wave that travels to the receiving point via a reflection from a boundary of two media, where the boundary is of a size that is much larger than a wavelength and is relatively close to the flat surface,9 The examples portraying radio waves traveling to the receiving point via reflections are reflections from the Earth’s surface or via structures such as landscapes, metallic bodies placed into orbits, and so on. Near ideal reflection occurs via the ionized layers in the ionosphere, when the radio wave of low frequencies (up to 30 MHz) propagates between corresponding points A and B, as shown in Figure 1.4(b). A scattered (or secondary) radio wave is one that appears when the scatterings take place during propagation. Scatterings may be observed when the radio 8. 9.

Free space is defined as “space that is free of obstructions and that is characterized by the constitutive parameters of a vacuum” [11]. The IEEE standard definition [11] is as follows: “for two media, separated by a planar interface, that part of the incident wave that is returned to the first medium.”

12 Introduction

Figure 1.4  (a) Direct radio wave, (b) reflected radio wave, (c) scattered radio wave, and (d) diffracted radio wave.

wave stochastically reflects from a rough, random surface with the average size of the roughness comparable or less than the wavelength, or during propagation of the radio wave through a medium that contains randomly shaped or space-­ time-­distributed irregularities. Typically, these volumetric scatterings are observable when the dimensions of scatterers (or random globules) are comparable or less than the wavelength itself. Each globule then plays the role of a secondary (virtual) radiator of the random origin. A superposition of the multitude of secondary waves that arrive to the receiving point B produces the resultant field. Radio waves scattered from the small-­scale irregularities of the refractive index of tropospheric air may be considered as an example of a secondary radio wave [Figure 1.4(c)]. These random irregularities exist in the lower portion of the atmosphere (troposphere) even in clear atmospheric air as turbulences caused by the horizontal and vertical movements of atmospheric air masses. The random volumes of the irregularities of atmospheric air are able to scatter the microwaves effectively within a wide range of angles. That effect is the main mechanism of long-­range propagation of DMW and CMW, which are able to propagate over the horizon for distances of many hundreds of kilometers. This phenomenon is known as far (over-­the-­horizon) tropospheric scatter propagation of microwaves, or as troposcatter propagation. The scattered propagation caused by the irregularities of ionization in the ionosphere is another example of propagation by the mechanism of secondary radio waves. Those irregularities are mainly generated by small-­scale particles (micrometeors) and dust coming from the outer space. These particles result in highly ionized footprints with an average length of several meters. Therefore, the phenomenon of scatter propagation through the irregularities of the ionosphere takes place mainly with radio waves within the VHF frequency band. Note that both micrometeors

1.4  Classification of Radio Waves by Its Propagation Mechanisms 13

and space dust are present in the ionosphere permanently, so these types of secondary waves may be observed all day long, regardless of the season. A diffracted radio wave (or simply a diffracted wave) is defined as “an electromagnetic wave that has been modified by an obstacle or spatial inhomogeneity in the medium by means other than a reflection or refraction” [11].10 As known from a typical college physics course, any material body placed across a propagation path may be considered an obstacle only if its linear dimensions are comparable or greater than the wavelength. Otherwise, the wave will spill over that material body (i.e., will diffract on it) and will easily arrive at the observation point placed behind the obstacle. For a rough estimate of propagation distance, one may take into account that the diffraction will take place when h ≤ l, where h is shown in Figure 1.4(d). The following approximate geometric relations may be written based on the expansion of the cos(Q/2) into Taylor’s series for small Q angles. In fact, for the real conditions the propagation distances are much smaller than the Earth’s radius, a = 6370 km. Thus, h = a - a cos



é æ 1 Q2 ö ù Q2 a Q (1.4) » a ê1 - ç 1 ÷ú = ç 8 2 2 4 ÷ø úû êë è

where

Q = R/a (1.5) represents a geo-­central angle between corresponding points A and B, and R indicates a curvilinear distance (arc) between those points. Taking into account the relation h ≤ l, as well as (1.4) and (1.5), we may define a maximum distance of the propagation of a diffracted as



R»

8 a λ , (1.6)

or, by expressing R in kilometers and l in meters, we may obtain

Rkilometers » 7

λ meters . (1.7)

The expression (1.7) portrays a rough estimate of the limits of propagation distances through diffraction mechanism for the radio waves of various frequency bands. From (1.7), it may be noted that the greater the wavelength, longer is the propagation distance, and the easier the diffraction can occur. This mechanism creates more favorable conditions for the propagation of the radio waves at frequencies less than 30 kHz, at which the propagation distances may reach up to 1000 km. On the other hand, for higher frequencies such as HF, the diffraction mechanism of propagation may not be considered as an essential propagation mechanism when the maximum distances of HF propagation, caused by the diffraction, 10. Later we will use the term refraction to identify the bending of the propagation path in the stratified troposphere, or ionosphere, and the term refracted wave to identify the wave that penetrates from one medium into the second through their interface.

14 Introduction

Figure 1.5  Illustration of the ionospheric propagation mechanism of HF radio waves (sketch is not to scale).

become almost equal or even less than the line-­of-­sight (LOS)11 distance. For VHF and higher frequencies, the real observations portray the distances as much greater than those noted from (1.7). That mechanism is shown in Figure 1.4(c) and given in Section 5.6. The set of terms, direct, reflected, scattered, and diffracted relate to the mechanisms specifying how the radio wave arrives the observation (receiving) point. Now we present another set of terms, which allow the classification the radio waves based on the spatial area that the propagation paths are traveling through. A sky wave (or ionospheric wave) is “a radio wave that propagates obliquely toward, and is then returned from the ionosphere” [11]. This type of radio wave is localized in the spatial region between the ionosphere and the Earth’s surface and is shown in Figure 1.4(b). It is thereby evident that it may also be called a reflected wave, if we intend to specify the mechanism of propagation. Note that the long-­range propagation distances of HF radio waves are stimulated by the mechanism of subsequent reflections of sky waves from the ionosphere and the Earth’s surface. These results in propagation distances of thousands of kilometers (see Figure 1.5). A ground wave is a radio wave that propagates “from a source in the vicinity of the surface of the Earth (i.e., a wave that would exist in the vicinity of the Earth’s surface in the absence of the ionosphere)” [11]. Two ground wave modes may be considered independently existing: •

•

A surface wave is non-­TEM mode that propagates along the Earth’s surface and is guided by the air-­ground boundary; this type of wave is specific to radio waves generated by so called low-­elevated antennas.12 A space wave is a superposition of direct and ground-­reflected TEM waves in the vicinity of the Earth’s surface; this type of wave is specific to radio waves

11. Line-­of-­sight is a term that is common to the propagation paths in frequency ranges VHF and higher. The higher a frequency, the closer propagation properties are to those of the optical waves. 12. Antenna elevation above the Earth’s surface that is close to, or less than a wavelength.

1.5  Interferences in RF Transmission Links 15

generated by elevated antennas,13 mainly in frequency ranges of VHF and higher. Per these definitions, one may conclude that the contribution of each component into the ground wave depends strictly on the radiating antenna height above the Earth’s ground surface. For an antenna with an elevation of several more wavelengths above ground level (high-­elevated antenna) the space wave component of the ground wave is predominant. Otherwise, for a ground-­based antenna (i.e., for an antenna with an elevation that is comparable or less than the wavelength), the surface wave component will become predominant. The issues will be discussed further in Chapter 4.

1.5  Interferences in RF Transmission Links The quality of information transmission via a radio transmission link between corresponding points, as well as the quality of a radars performance, is highly impacted by the presence of disturbances to the desired signal. The term interference is commonly used in communications engineering practices to manifest those disturbances of a desired signal. Meanwhile, in physics and in electromagnetic theory, the same term is used to convey the superposition of the considered electromagnetic wave with other electromagnetic wave(s), either of different origin or of the same origin, but arriving at the observation point from different propagation directions. In this section, we will use the first meaning of the term interference. The second meaning will be examined in the following chapters. Interferences typically destructively impact the content of information received, as opposed to the information actually transmitted. Considered destructive random process that occur in the receiving mode, these interferences may come into view in two different forms: •

They may be in the form of random fluctuations of the parameters of the desired signal. For instance, when a monochromatic signal14 passes through the propagation path, then both the amplitude and the phase of the signal will randomly fluctuate. The rate of these random fluctuations is much less than the rate of the oscillations of the signal’s carrier, as typically the quasiperiod of those random fluctuations is hundreds of milliseconds and up. Thereby, the fluctuations may be simply interpreted as multiplications of the amplitude of the signal by slow random variable(s). This is similar to passing the signal through a linear two-­port network with a randomly fluctuating transmission coefficient, where the input voltage (or current) is multiplied by the transmission coefficient of a random character. Hence, this type of

13. Antenna elevation above the Earth’s surface that is greater than several wavelengths. In order to be considered as an elevated antenna, the feeding line must not be radiating. 14. To present a strict approach, the quasi-­monochromatic signals are to be discussed simply because any information-­carrying, modulated signal is never purely monochromatic. However, in practice, RF signals that are transmitted through propagation paths are narrowbanded in most cases; thus, they may be considered monochromatic for analytical purposes.

16 Introduction

•

interference is referred to as multiplicative interference, or simply fading. This form of interference is originated in the propagation medium (i.e., along the propagation path). Another form of interference appears in both the propagation medium and in conjunction with the receiver (including a receiving antenna). It occurs independently and simultaneously with the desired signal, superimposing (overlaying) to the desired signal. It is therefore known as additive interference, or noise. In contrast with fading, this type of interference may be specific to an extremely wide spectrum, affecting all applicable RF frequency bands.

Based on these definitions, the output signal of the receiver may be written in time domain as

s (t) = κ (t) × s (t) + n(t). (1.8) Here s(t) is the desired signal, k(t) is the multiplicative interference, and n(t) represents the additive interference. Both k(t) and n(t) indicate random processes. To present an example of multiplicative interference, the fading of the voice volume of an HF-­broadcast audio signal may be considered. Another example is the deep fades encountered with a receiving antenna output signal, which is indicated by troposcatter, associated with over-­the-­horizon microwave radio links. Two types of multiplicative interference, the slow and fast fading of the signal level,15 are presented in the classification chart shown in Figure 1.6. Additionally, a display of seasonal variations of a received signal is included (conventionally) to indicate a multiplicative interference. The uniform hum of an HF-­broadcasting receiver’s audio output may be considered an example of additive interference(s) that permanently exists, regardless of the existence of a desired signal within a broadcasting channel. In Figure 1.6, the reader will find the classification of additive interference (noise). Receiver noise (or internal noise) is a noise type that appears in different areas of the predetection (linear) section of receivers, including antenna, feeders, RF and IF amplifiers, and filters. The nature of inner noise is stipulated by several physical phenomena to include the following: •

Random thermal movements of free electrons in resistors, conducting wires, antennas, and associated elements: The long-­term average summation of vector velocities for electrons is zero. Therefore, if no other forces are applied, there is no draft of the electrons cloud within the circuit. For short time periods the chaotic motions of electrons result in jumps (short pulses) of current/voltage, and the overlay of a large number of these short and overlapping pulses appears as a steady hum, which is known as thermal noise. The spectrum of thermal noise covers the entire RF range, and the higher the temperature, the more intense the movements are and bigger the power

15. Occasionally the seasonal variations of signal levels may also be considered as an example of multiplicative interference. However, they may be excluded if the previously known character of those variations is accounted for.

1.5  Interferences in RF Transmission Links 17

Figure 1.6  The classification of interferences associated with RF channels.

•

spectral density of thermal noise is. However, thermal noise may exist even without a current flowing through the associated element. The discrete character of associated particles (i.e., electrons and vacancies) when flowing through an active electronic component, such as a diode, transistor, or running wave tube, may cause a random sequence of splashes within in the circuits. This phenomenon is known as shot effect. However, it will not occur if the current is significantly large, as the natural averaging effect of a larger current will flatten off the fluctuations of the current. It is only for small currents flowing through a component that the shot effect becomes significant. This is particularly specific to the first conductive stage of RF signal amplifiers (typically placed right at the receiving antenna output, or before the feed line associated with the receiver).

External noise usually penetrates into the receiver from the propagation medium. The three types of external noises are described as follows: •

Man-­made noise: Examples of man-­made noise include the noise generated by the ignition systems of automobiles, powerful electric motors, power transmission lines, and high-­power distribution equipment as well as by

18 Introduction

•

•

other residential and commercial systems. In these cases, electromagnetic energy will be radiated into our frequency bands of interest. Atmospheric noise, which is generated by two sources, and therefore may appear within two types: lightening discharges within the troposphere, and a steady noise background generated by collisions between atoms and molecules within the tropospheric air layer. The spectral density of the first type of atmospheric noise will typically be concentrated in the lower part of the RF spectrum (i.e., VHF, UHF and higher) the intensities of this first type of noise may be ignored, as they are not comparable to the effects of the second type of noise. The second type of the noise is characterized by noise spectral power density, which increases in direct proportion to the square of the frequency and, therefore, reigns significantly within the microwave frequency bands. Cosmic noise signifies the RF radiation emanating along the observation direction. The intensity of cosmic noise is reliant upon the location of where the receiving antenna is directed/aimed. Note that the ionosphere is impenetrable by the radio waves with frequencies of less than 30 MHz, so this type of noise is applicable only to radio links at operating frequencies higher than VHF – most typically, those whose receiving antenna(s) is directed skywards (i.e., for satellite downlinks). The RF thermal radiation of the Earth’s surface may also be considered cosmic relative to these radio links. Numerous radio-­ astrophysical observations have indicated that the most intense cosmic noise comes from the sun, the center of our associated galaxy (the Milky Way), or from several other constellations within the universe.

Additional details about interferences and their estimates may be found in Chapter 6.

References   [1] Maxwell, J. C., “A Dynamical Theory of the Electromagnetic Field,” Proc. Roy. Soc., London, Vol. 13, 1864, pp. 531–536.   [2] Burrows, C. R., “The History of Radio Wave Propagation Up to the End of WW-­I,” Proceedings of the IRE, Vol. 50, 1962.   [3] Sommerfeld, A., Annalen der Physics, Vol. 28, 1909, p. 665.   [4] Weyl, H., Annalen der Physics, Vol. 60, 1919, p. 481.   [5] Norton, K. A., “The Propagation of Radio Waves Over the Surface of the Earth and in the Upper Atmosphere, Part-­I,” Proceedings of the IRE, Vol. 24, No. 101936, pp. 1367–1387, and “Part-­II,” Proceedings of the IRE, Vol. 25, No. 9, 1937, pp. 1203–1236.   [6] Watson, G. N., Proceedings of the Royal Society, Vol. 95, 1918, p. 83.   [7] Van Der Pol, B., and H. Bremmer, Philosophical Magazine, Vol. 24, 1937, pp. 141 and 825, Vol. 25, 1938, p. 817, Vol. 27, 1939, p. 216.  [8] Фок, B. A. Диффракция радиоволн вокруг земной поверхности. Изд-­во AH UCCP, Москва, 1946. (Fock, V. A., Diffraction of the Radio Waves Around Earth’s Surface, USSR Academy of Sciences Publishing Press, Moscow, 1946, in Russian.)  [9] Bведенский, Б. A. Bестник теоретической и экспериментальной электротехники. No.12, стр. 439-­446, Москва, 1928. (Vvedensky, B. A., Messenger of the Theoretical and Experimental Electrotechnics, No.12, Moscow, 1928, pp. 439–446, in Russian.)

1.5  Interferences in RF Transmission Links 19 [10] Booker, H. G., and W. E. Gordon, “Radio Scattering in the Troposphere,” Proc. IRE, Vol. 38, April 1950, pp. 401–421. [11] IEEE Standard Definitions of Terms for Radio Wave Propagation. IEEE Std 211-­1997. [12] Withers, D., Radio Spectrum Management, IEE Telecommunication Series 45, 1999. [13] Derek, M. K., Ah Yo, and Emrick, R., “Frequency Bands for Military and Commercial Applications,” Ch.2 in Antenna Engineering Handbook, Fourth Edition, McGraw-­Hill Co., 2007. [14] IEEE Standard 521-­2002 (Revision of IEEE Standard 521-­1984), IEEE Standard Letter Designations for Radar-­Frequency Bands, 2002. [15] Dolukhanov, M. P., Propagation of Radio Waves, Moscow, USSR, Mir Publishers, 1971.

CHAPTER 2

Basics of Electromagnetic Waves Theory 2.1  Electromagnetic Process 2.1.1  Maxwell’s Equations of Electrodynamiscs1

Dynamic electromagnetic process is considered a unity of two processes, identified as time-varying electric and magnetic. Analytically, the unification of these processes is expressed by a system of Maxwell’s equations, as follows:    ∂D  ∇×H= + Jtot   (Ampere’s law) (2.1) ∂t





   ∂B ∇×E= −   (Faraday’s law) ∂t

(2.2)



  ∇ ⋅ D = ρtot   (Gauss’ law)

(2.3)

  ∇ ⋅ B = 0   (Law of continuity of magnetic field lines) (2.4)   In this system of equations, E and   H represent electric and magnetic field strengths, respectively, whereas D and B represent electric and magnetic induction vectors (or electric and magnetic flux density vectors) for particular points of space. These vectors are further coupled by the constitutive parameters of the medium. εˆ , e0, µˆ , and μ0.     D = ε oεˆ E, and B = µ0 µˆ H (2.5)





Here, absolute dielectric permittivity and absolute magnetic permeability of free space (vacuum) are

εo =

1 10−9 F / m (Farads per meter), and µ0 = 4π ⋅ 10−7 H /m (Henries per meter)  (2.6) 36π

1. The term electrodynamics is utilized here to describe the area of “electromagnetism” relative to time-­ varying electromagnetic processes, in contrast to “electrostatics” and “magnetostatics,” which are to represent to areas of study relevant to time-constant electric and magnetic fields, respectively.

21

22

Basics of Electromagnetic Waves Theory

respectively, whereas εˆ and µˆ signify relative dielectric permittivity and relative magnetic permeability specific to the particular medium.2 All media herein are categorized based on εˆ and µˆ as follows: • • • •

Constant and parametric media representing media time-dependence; Homogeneous and inhomogeneous media presenting spatial dependence; Specific character such as isotropic (for scalar εˆ = ε , and µˆ = µ ) and anisotropic (for tensor εˆ , and/or µˆ )3 media; Linear, if εˆ and/or µˆ are field intensity independent, and non-linear otherwise.

If the relations in equations (2.5) are taken into account, then equations (2.1) and (2.2) may be rewritten as:

    ∂ ∇ × H = ε0 εˆ × E + Jtot (2.7a) ∂t



   ∂ ∇ × E = µ0 µˆ × H (2.7b) ∂t

(

)

(

)

Per expressions  (2.7a),  (2.7b), and (2.3), one may realize that electric and magnetic field vectors E and H  interrelate and are coupled to the volumetric  total conducting electric current Jtot and volumetric total electric charge rtot. Jtot and rtot are defined by the same electric charges (electrons, ions) that are able to move freely within the considered spatial area.4 Therefore, it is not surprising that they are coupled by the law of current continuity:



  ∂ρ ∇ ⋅ J = − tot (2.8) ∂t The movements of free electric charges may be stipulated either by external force, or by internal (secondary) electromagnetic field, once it’s generated so far by externally forced movements of charges. Thus, the total conducting current and charge may be presented as sums:    Jtot = JExt + J (2.9a)

ρtot = ρExt + ρ (2.9b)  Here JExt and rExt represent the  components of the current and charge that are stipulated by external source(s); J and r induced by the electromagnetic field within the medium that contains free moving charges. It’s well known that Coulomb force, applied to the charged particle, is in direct proportion to the amount of charge and

2. In general cases these parameters of the medium are tensor quantities. 3. A particular case of anisotropic medium, namely magnetoactive ionospheric plasma, and propagation in that medium will be considered in Chapter 5. 4. Here defined in contrast to the bonded charges, such as those bonded to the crystalline lattice of solid matter.

2.1  Electromagnetic Process 23







 the electric field intensity. Therefore, current J is expected to be in direct proportion to the electric field as well:   J = σ E (2.10) The coefficient s is called a conductivity of the medium and is proportional to the concentration of free charges (i.e., proportional to volumetric density of free electric charge) in considering point of space. If (2.9a) and (2.10) are substituted, then (2.1) may be rewritten as      ∂E ∇ × H = ε 0ε + σ E + JExt . (2.11) ∂t The physical meaning of the first term, on the right-hand side, translates to a volumetric current density that exists in space, regardless of the existence of free charges; it is stipulated in dielectric medium (or vacuum) by the time variations of the electric field. This term is conventionally known as a volumetric displacement  electric current, or displacement electric current volumetric density, Jdis. By analogy, the right-hand side term in (2.2) is known as a displacement magnetic current volumetric density. The importance of these two displacement currents is difficult to overestimate. Indeed, only these currents are accountable for keeping the electromagnetic process running, as they permit continuous energy exchange between electric and magnetic fields within the united electromagnetic process. The balance of energy exchange within the spatial area containing the electromagnetic process may be assessed as follows:   First, we multiply both sides of equations (2.11) and (2.2) by E and H , respectively, and subtract (2.11) from (2.2). Next, we will refer to identity (2A.22), given in Appendix 2A, which may be applied to the left-hand side to result in the following equation:       H ⋅ ∇ × E − E ⋅ ∇ × H = ∇ ⋅ ( E × H ) (2.12)

(

)

(

)

Now use the transforms that are applied to thsse right-hand side:



   ∂H  ∂E     − µ0 µ H − ε 0ε E − σ E ⋅ E − E ⋅ JExt = ∂t ∂t   ∂  µ µ H 2 ε oε E 2  2 =−  0 +  − σ E − E ⋅ JExt ∂t  2 2 

(2.13)

Here, on the right-hand side the expression in parenthesis represent the energy per unit volume (volume density of energy) cumulated by electric and magnetic fields, respectively. The second term represents the power loss per unit volume, due to finite conductivity of the medium. The higher the conductivity of medium, the greater the rate of collisions of charge-carrying free particles within the matter is, and therefore the higher the rate of transformation of energy of the electric field

24

Basics of Electromagnetic Waves Theory

into heat. The last term in (2.13) represents solely the power of the external source implemented into the electromagnetic field. Now we may integrate both right- and left-hand sides presented by (2.12) and  (2.13), respectively, within the volume V that contains an external source JExt .   ∇ ⋅ ( E × H )  dV = ...   ∫

V



   ε ε E2 ∂  µ µ H2 dV  − ∫ σ E2 dV − ∫ E ⋅ JExt dV ... = −  ∫ 0 dV + ∫ o 2 2 ∂t V  V V V

(2.14)

Here Wm = (1/ 2) ∫ µ0 µ H2 dV and We = (1/ 2) ∫ ε0ε E2 dV denote total energies cuV

V

mulated within volume V by magnetic and electric fields, PL =

∫V σ E

2

dV denotes

the total thermal   loss of power of the electromagnetic field within volume V, and PExt = − ∫ E ⋅ JExt dV the total power given to the electromagnetic field by the exV

ternal source.5 The identity (2A.17) that is known as Gauss theorem, given in Appendix 2A, may be applied to the left-hand side of (2.14), namely, as: 





 

∫V ∇ ⋅ ( E × H )] dV  = ∫ ( E × H ) d S (2.15)



S

Here the volume integral is replaced by the integration along the closed surface S, which surrounds volume V. Note that d S illustrates a vector surface element directed outward of the volume, normally to the surface at the considering point. Therefore, (2.14) may finally be rewritten as: PExt =



   ∂ ( E × H ) d S . (2.16) (Wm + We ) + PL +  ∫ S ∂t

This is a mathematical formulation of the balance of energy of the electromagnetic field known as Poynting theorem: the amount of power given to the electromagnetic field by the external power source within a limited spatial area: • • •

Is partially consumed to increase the energy stored by electric and magnetic fields in that spatial area; Is partially dissipated within the volume as a thermal power loss; Partially flows away from the volume as a radiated power.

The surface integral on the right-hand side of equation (2.16) allows introduction of a vector of power flow density known as Poynting vector, which shows the

5.

The negative sign indicates a power that is inserted into the electromagnetic field, in contrast to the positive power that is subtracted from the field.

2.1  Electromagnetic Process 25

Figure 2.1  Sketch of the main constituents of the electromagnetic process.



amount of power passed through the unit surface that is placed orthogonal6 to the direction of flow in particular point of space.    Π = E × H . (2.17) It can be seen from (2.17) that the unit for the Poynting vector – that is, the  unit for the power flux density (magnitude of vector Π) is: (V/m) · (A/m) = W/m2. Figure 2.1 shows the transformations between time-varying electric and magnetic fields, within the electromagnetic process, as it follows from Maxwell’s equations.   The external source of electric current JExt generates the initial magnetic field H. The time-variant magnetic field, at any  arbitrary point A, appears as a source of displacement electric current µ µ ·∂ H ∂t, which forces   the generation of the 0 electric field E . Consequently, a time-varying electric  field E, at point B, appears as a source of displacement magnetic current ε0ε ·∂E ∂t that initiates the secondary  magnetic field H ′ , and so on. This process may remain infinitely long in time, if there are no losses in considering spatial area (i.e., if the conductivity of the medium is equal to zero). 2.1.2  Boundary Conditions of Electrodynamics

Maxwell’s equations noted in (2.1) – (2.4) represent differential equations in partial derivatives. In a spatial area free of sources, the electromagnetic field is to be 6.

 If the unit surface is not orthogonal to vector Π from (2.17), then a scalar product is appropriate. Namely  in (2.16), the arbitrary elementary surface dS is represented by the surface element vector dS , so  oriented  the scalar product Π ⋅ d S represents an infinitesimal amount of power that flows through that element.

26

Basics of Electromagnetic Waves Theory

considered stand-alone, as an independently existing form of matter. The first two Maxwell’s equations are utilized here to describe the interrelations between electric and magnetic fields, and may be rewritten as:    ∂E  ∇ × H = ε 0ε + J, (2.18) ∂t    ∂H , (2.19) ∇ × E = − µ0 µ ∂t





 where J depicts a conductive current from Ohm’s law, defined by (2.10). The set (2.18) – (2.19) is a system  of two first-order linear equations with two unknowns being depicted as E and H. As evident in collegiate mathematics, a general solution of the system contains arbitrary (undefined) integration constants, creating a multivalued (ambiguous) solution. For real conditions, when configuration of boundaries between media is known, the initial conditions may be set up to calculate those integration constants. In electrodynamics applications, the initial conditions are conventionally called boundary conditions, because they represent the act of “bonding” (restraining) the values of electric and magnetic fields to those boundaries. In other words, these boundary conditions allow transformation of a general solution of Maxwell’s equations into a particular solution that is specific for the given configuration of the boundaries in which the electromagnetic field is being defined. That particular solution is known as a boundary value problem. For more consistency we will now consider those conditions in detail. First, we take volume integrals of (2.3) and (2.4) within a volume V:

 

 

∫ (∇ ⋅ D) dV = ∫ D ⋅ d S = ∫ ρtot dV , (2.20)

V



S









∫ (∇ × B ) dV = ∫ B × d S = 0, (2.21)

V



V

S

For these equations, the divergence theorem is utilized [see (2A.17) in Appendix 2A]. Here S denotes a closed surface representing the boundary of the volume V.  The vector surface element dS is always directed outbound to volume V. This volume encompasses a part of the boundary between two media via constitutive parameters (e1, μ1, s1) and (e2, μ2, s2) as shown in Figure 2.2. Additionally, n indicates a unit vector normal to the boundary of media and directed from medium 2 towards  medium  1. As D and B vectors are specific to any particular point of space, we may choose to minimize volume V enough to allow uniformity of the field distribution within that volume as well as on its boundary. Taking that fact into account, we may write the following expression    D ⋅ dS = (D1 × n) d S = Dn1d S (2.22a) for the top base of cylinder, and

2.1  Electromagnetic Process 27

Figure 2.2  Integration area in (2.20) and (2.21) integrals.





   D ⋅ dS = −(D2 ⋅ n) d S = −Dn 2 d S (2.22b)   for the bottom base of cylinder, wherein D1 , Dn1 and D2 , Dn2 represent the electric field inductions vectors and their normal components of the first and second media, respectively. These vectors and their components are considered constant along the top and bottom  surfaces ΔS, as mentioned earlier. Hence, the left-hand side of (2.20) indicating D-vector flow through the closed surface may be rewritten as    D ⋅ dS = [(D 1 − D2 ) ⋅ n ]∆S = (Dn1 − Dn 2 ) ∆S + ΞD (2.23) ∫ S



Here, the first term on the right-hand side shows a D-vector flow through both bases of the cylinder, and ΞD represents a flow through the side surface of the cylinder. Hence it is evident that if we shrink the cylinder vertically towards the boundary of media (i.e., take Δh → 0), then ΞD will disappear. The right-hand side of (2.20) represents a total charge enclosed within the volume V. If rtot is considered a volumetric charge density, then obviously for Δh → 0 the right-hand side of (2.20) will also disappear. However, for a wide range of electromagnetic problems, free charge is allocated within a tiny layer directly atop the boundary surface of two media. With that, it is appropriate to consider the surface charge rS that is distributed in a minute layer of infinitesimal thickness per unit surface atop the boundary.7 Thus, the right-hand side of (2.20) may be represented as rSΔS, and finally that expression may be transformed into the following equation:   (D1 − D2 ) ⋅ n = Dn1 − Dn 2 = ρS (2.24)



Similarly, expression (2.21) may be transformed into the following:   (B1 − B2 ) ⋅ n = B n1 − B n 2 = 0 (2.25) Equations (2.24) and (2.25) represent boundary conditions for the components of electric and magnetic fields that are normal to the boundary of two media. As one may conclude, the normal component of magnetic field flux density consistently

7.

Such charges do not exist in nature; thus, this solution simply represents a mathematical abstraction.

28



Basics of Electromagnetic Waves Theory

remains continuous across the boundary between media, whereas the normal components of the electric flux density may have a discontinuity if a free surface charge exists on the boundary (i.e., exists within the infinitesimal layer that surrounds the boundary of two media). Now, the boundary conditions may be evaluated for the tangential components of the electric and magnetic fields if (2.18) and (2.19) are integrated within a surface S (ABCD) residing across the boundary and orthogonal to them, as shown in Figure 2.3.       ∂E  ∫ ∇ × H ⋅ d S = ∫ ε0ε ∂ t × d S + ∫ J ⋅ d S (2.26) S S S

(

)

∫(



S

    ∂H  ∇ × E ⋅ d S = − ∫ µ0 µ ⋅ d S (2.27) ∂t S

)

 Here d S = s0 d S displays a surface vector-element that is directed orthogonal to the surface ABCD, with s0 as a unit vector orthogonal to that surface. Now we apply Stoke’s theorem [see (2A.16) from the Appendix 2A] to the left-hand side of (2.26):



 

∫ (∇ × H ) ⋅ d S = S

∫

  H ⋅ d S (2.28)

Contour ABCD

It is apparent from Figure 2.3 that the integration path along the rectangular contour ABCD may be expressed as

∫

      H ⋅ dS = H1 ⋅ AB + H2 ⋅ CD + Int (2.29)

Contour ABCD D

where Int represents a line-integral along sides BC and DA of height Δh. That integral will disappear for the vanishing height of the rectangle (i.e., for Δh →

Figure 2.3  Integration area in surface integrals (2.26) and (2.27).

2.1  Electromagnetic Process 29

0). If the following three unit cross-orthogonal vectors s0, n, and l0 are positioned as shown in Figure 2.3, then the vectors AB and CD may be replaced by AB = l0 ⋅ L = s0 × n ∆L , and CD = − l0 ∆L = − ( s0 × n ) ∆L. Then (2.29) may be rewritten after simple transformations as:

(

lim∆h→0

)

    H ⋅ dS = [H1 ⋅ ( s0 × n ) − H2 ⋅ s0 × n ] ∆L =

∫

(

Contour ABCD

  = [ n × (H1 − H2 )] ⋅ (s0 ∆L)

)

(2.30)

The first term on the right-hand side of (2.26) will vanish to zero when Δh →  0, as well as will a second term, except in the case when the conducting current J has a surface instead of spatial distribution (i.e., flows into a layer of infinitesimal thick ness along the boundary of media, as shown in Figure 2.3). Hence if JS represents a surface conducting current, then  the second term on the right-hand side of (2.26) may be replaced by the flow of JS through the line segment MN as shown:

∫



  JS ⋅ (s0 ⋅ dL) = (JS ⋅ s0 ) ∆L (2.31)

MN



Making (2.30) and (2.31) equal will result in    [ n × (H1 − H2 )] = JS (2.32) or in scalar form,



H1τ − H2τ = JS (2.33)



This procedure may be repeated for (2.27) and will result in   [ n × (E1 − E2 )] = 0 (2.34) or, in scalar form:



E1τ − E2τ = 0 (2.35) Physical meaning of (2.32) and (2.34) is defined as follows: tangential components of the electric and magnetic field strengths remain continuous across the boundary of two media, unless there exists surface current flowing on the boundary that results in a jump of the tangential component of the magnetic field strength. That jump is equal to the value of that surface current density. In some applications in antennas and microwave theory, a boundary with perfect electrical conductor (PEC) is of interest and is identified for its infinite conductivity (i.e., for s = ∞). Based on expression (2.7a), along with (2.10), for the  spatial areas free of sources (Jext = 0) , it becomes evident that the electric field must

30

Basics of Electromagnetic Waves Theory

be assumed to be equal to zero because the induced conducting current σ E may not be infinitely large. The absence of the electric field results in the absence of the magnetic field, thereby indicating an absence of the entire electromagnetic process within the PEC medium. Hence, the boundary conditions (2.24), (2.25), (2.32), and (2.34) may be rewritten for the PEC boundary via the following formulations:  ρ E ⋅ n = E n = S . (2.36) ε 0ε  H ⋅ n = H n = 0. (2.37)   n × H = JS , or Hτ = JS (2.38)  n × E = 0 or Eτ = 0 (2.39)



From formulas (2.36) through (2.39) one may conclude that the tangential component of the electric field and the normal component of the magnetic field on the boundary of PEC are always equal to zero (i.e., the electric field lines are always perpendicular to PEC boundary, whereas the magnetic field lines are always tangential to it). Example 2.1

The slope angle of electric field lines changes when passing from one medium into another, as shown in Figure 2.4. Find the angle a2 if e1 = 5, e2 = 1, angle a1 = 60°, and there are no free surface charges distributed along the boundary of those media. Solution

tan α1 = • •

Eτ 1 En1

,   tan α 2 =

Eτ 2 En 2

.

Based on (2.35) E1t = E2t, thus En1 ⋅ tana1 = En2 ⋅ tan a2. From (2.24) for rS = 0, we may assume: e1En1 = e2En2. By substitution into the previous expression we may derive finally:

Figure 2.4  Sketch of the electric field lines on the border of two media

2.1  Electromagnetic Process 31



1   ε α 2 = tan−1  2 tan α1  = tan−1  3  = 19.1°.(Answer) 5    ε1

2.1.3 Time-Harmonic Electromagnetic Process: Classification of Media by Conductivity



If time variations of electric and magnetic fields in dynamic electromagnetic processes are assumed to be harmonic (sinusoidal), which is of predominant interest in science and technology, then significant simplifications in Maxwell’s equations may be achieved by applying a complex variables analysis. If, for example,  time-harmonic oscillations of the electric field are presented analytically as E cos(ω t + ϕ ),  then transformation into the complex form Eexp[ i (ω t + ϕ )] allows representation as follows:8    E cos(ω t + ϕ ) = Re Eexp[i (ω t + ϕ ] = Re Eexp[iω t ] exp[iϕ ] (2.40)

{

}

{

}

The most attractive feature of complex analysis is the fact that time derivatives (or time integrals) may simply be replaced by multiplication (or division) by the factor iw, which transforms differential equation into algebraic. Indeed,

 ∂  Eexp[i (ω t + ϕ )] = i ω Eexp[i (ω t + ϕ )] (2.41) ∂t

{

}

{

}



Even a time-harmonic multiplier, exp(iwt), may be cancelled out of Maxwell’s equations. Then Maxwell’s equations are written not for the timevarying electric and magnetic fields but for their vector phasors, such as:  E = x0 E X exp(iϕ X ) + y0 EY exp(iϕY ) + z0 E Z exp(iϕ Z ), where EX, EY, EZ and jX, jY, jZ represent a coordinate-dependent amplitudes and initial phases of oscillation of the electric field vector in Cartesian coordinates. A phasor for the magnetic field vector or any other scalar or vector variable may be introduced similarly. Taking these statements into account, Maxwell’s equations for the source-free spatial region may be expressed in complex form as follows:     ∇ × H = iω ε0ε E + σ E (2.42)



   ∇ × E = −iω µ0 µ H (2.43)



8.

  ρ ∇ × E = tot (2.44) ε ε0   ∇ × H = 0 (2.45)

Here the Euler’s formula exp(ia) = cosa + isina is used, where i = −1 .

32

Basics of Electromagnetic Waves Theory

Then expression (2.42) may be transformed to:9



       σ   ∇ × H = Jtot = Jdis + J = iω ε0ε E + σ E = iωε0  ε − i E (2.46) ωεε0    Jtot is shown as a sum of displacement, Here the total volumetric current density   Jdis and conducting, J , current densities. The same expression (2.46) for the lossless medium (s = 0) can be rewritten as:     ∇ × H = Jdis = iω ε0ε E (2.47) If (2.46) is compared with (2.47), then a complex relative permittivity of the considering medium (or material) may formally be introduced as



ε = ε − i

σ = ε − i 60λ 0σ (2.48) ωε0

In (2.48) the following substitution is used:w = 2pc/l0, where l0 represents the wavelength in free space, and c = 3.108m/s represents the speed of light. For further analysis we’ll assume both e and s scalar quantities (i.e., the media under consideration being isotropic).10 Now, without limitations to generality, consider for simplicity a single-component electric field (e.g., a field that is directed along x-axis) of zero initial phase shift.  Then in (2.46) we may replace E by scalar E. Thus, for the lossy medium, (2.46) may be rewritten for the same component in phasor form as

(

)

J tot = iω ε0ε E = iωε0 ε − i60λ 0σ E = Jdis − i J (2.49) As one may notice from (2.49), the total volumetric current density is a complex quantity with displacement current as a real part and conducting current as its imaginary part. The ratio between imaginary and real parts of the complex dielectric constant (2.48) is the same as the ratio of conducting and displacement current densities. This ratio may be used to specify the rate of losses in considering medium or material. In complex plane it represents tangent of the slope angle de, identified as loss angle (Figure 2.5).



9.

tan δ ε =

J Jdis

=

60 λ 0 σ σ = (2.50) ω ε 0ε ε

Here and later we’ll assume the absence of the magnetic losses, which is specific for the propagation problems. Moreover, for most of the cases of RF propagation in the Earth’s atmosphere, along its surface, and in space, the propagation media are assumed to be nonmagnetic (i.e., μ = 1). 10. Anisotropic properties of ionospheric plasma interacting with the Earth’s magnetic field and its impact on radio waves propagation is considered briefly in Chapter 5. All other propagation mechanisms relate to propagation in isotropic media only.

2.1  Electromagnetic Process 33

Figure 2.5  Conducting, displacement, and total current densities in a complex plane.

Generally speaking, both e and s are frequency-dependent as shown in the graph presented in Figure 2.6. However, in some frequency ranges they behave almost constant (see Table 2.1). Based on the values of tan de, classification of media by conductivities is introduced conventionally as follows: • • •

tan δ ε < 10−1   – for dielectrics; −1 10 ≤ tanδ ε ≤ 10   – for semiconductors; tanδ ε > 10   – for conductor.

From (2.50) one may notice that the value of tande is dependent on frequency w. The same medium may exhibit different behavior in different frequency ranges. For several media listed in Table 2.1, frequency dependencies are presented in Figure 2.7. As one may notice from the figure, marble and mica may be considered ideal dielectrics for the entire RF spectrum. Quartz, paraffin, glass, atmospheric air, polyethylene, and so on may also be considered ideal dielectrics for the entire radio spectrum. Nearly all metals, including copper, iron, aluminum, mercury, behave as ideal conductors in the entire RF spectrum up to 1010 Hz. Several media, such as wet and dry soil, seawater and freshwater, and ice, may display variant properties in different frequency bands. For example, a dry soil behaves as a conductor in the LF frequency band, as a semiconductor in MF and HF frequency bands, and as a lossy dielectric in higher frequency bands. Example 2.2

Estimate the frequency ranges for seawater (e = 80, s = 1 S/m) to be considered as conductor, semiconductor, and dielectric. Assume e and s are frequency independent. Solution •

First transform (2.50) to obtain the frequency: f =

•

60 σ c (2.51) ε ⋅ tan δ ε

Here c = 3⋅108 m/s is speed of light in free space. The results of calculations based on (2.51) are shown in Table 2.2.

34

Basics of Electromagnetic Waves Theory

Figure 2.6  Frequency dependence of permittivity and conductivity for different propagation media provided by the International Telecommunication Union [1] for the following media: A = seawater, B = wet soil, C = freshwater, D = medium dry soil, E = very dry soil, F = pure water, G = ice.

Table 2.1  Electromagnetic Properties of Different Media Reference, (from Figure 2.6)

Medium

e

s, S/m

Applicable Frequency Range (Hz)

A B C E H K

Seawater Wet soil Freshwater Dry soil Marble Mica

~80 10–30 ~80 3–6 ~8 ~7

1–5 3.10– 3–3 · 10–2 10–3– 2.4 · 10–2 1.1 · 10–5– 2 · 10–4 10–7–10–9 10–11–10–15

0–109 0–108 0–108 0–109 103–108 103–108

2.2  Free Propagation of Uniform Plane Radio Waves 35

Figure 2.7  Frequency dependence of tande for variant media (see Table 2.1 for details by proper references).

Table 2.2  Calculation Result for Example 2.2 tan de < 10-1 10-1 < tan de < 10 tan de > 10

f > 2.25 GHz 22.5 MHz ≤ f ≤ 2.25 GHz f < 22.5 MHz

Dielectric Semiconductor Conductor

2.2  Free Propagation of Uniform Plane Radio Waves



As an electromagnetic process, we’ll analyze the propagation of the radio waves based on Maxwell’s equations that are specifically conditioned to free propagation in the media, such as the Earth’s atmosphere, other space, water of seas and ponds, and Earth’s ground. Consider a time-harmonic field11 that is initially generated by an external source. The first two of Maxwell’s equations in a region, which do not contain a source (a source-free region), may be expressed in complex form as:    ∇ × H = iωε0ε E (2.52)    ∇ × E = −iωµ0 µ H (2.53) To solve this system of two linear equations in partial derivatives,  differential  we exclude one of the unknowns E or H to bring this system to one equation of

(

)

11. Known also as monochromatic. The term comes from optic and originated from the Greek phrase single colored. Single colored radiation in optics means a radiation of a single frequency. The same meaning is adopted for the RF spectrum.

36



Basics of Electromagnetic Waves Theory

one unknown. With  that in mind, we will take a curl from both sides of (2.53), and then substitute ∇ × H from (2.52):       ∇ × ∇ × E = −iωµ0 µ ⋅ ∇ × H = k 2 E (2.54) where



k = ω

ε0ε µ0 µ = β − iα (2.55)

represents a complex propagation constant with a physical meaning that we will define later in this text. If the equality (2A.18) from Appendix 2Ais applied to the  left-hand side of (2.54), and (2.3) is taken into account, that is, ∇ × E = 0 for the medium free of electric charges, then (2.54) may be rewritten as

  ∇2 E + k 2 E = 0 (2.56)  Here ∇2 indicates a Laplacian operator applied to E vector. In Cartesian coordinates, ∇2 is expressed as:



∇2 =

∂2 ∂x2

+

∂2 ∂y2

+

∂2 ∂z 2

(2.57)

 Thus, for the vector phasor E(x, y, z) = x0 E x (x, y, z) + y0 E y (x, y, z) + z0 E z (x, y, z) the Laplacian is

 ∇2 E = x0∇2 E X + y0∇2 E Y + z0∇2 E Z (2.58) where x0 , y0 , and z0 are the unit vectors along X, Y, and Z coordinates. Equation (2.56) displays a particular form of well-known wave equation called Helmholtz equation, which is a wave equation written for the time-harmonic process. The function given by (2.59) may be considered one of the simplest solutions of Helmholtz equation:



 E = x0 E m X exp[i(ω t − k z)] (2.59) In this expression, E mX is a complex amplitude (phasor) that includes the real amplitude EmX and the initial phase shift j0 (i.e., E m X = E m X e iϕ0). It’s easy to show by substitution, that (2.59) satisfies (2.56). Regarding (2.59), the electric field contains a harmonic time and z-dependence only, with an amplitude value of E m X. In other words, all planes parallel to the XOY are the planes of constant values of the field for the fixed time instance. As electric and magnetic fields are coupled to each other in every point of space, it is reasonable to expect that field components of both electric and magnetic fields are portrayed as:

2.2  Free Propagation of Uniform Plane Radio Waves 37

    ∂E ∂E ∂H ∂H = = = = 0 (2.60) ∂x ∂y ∂x ∂y



 The value of the magnetic field coupled to (2.59) may be found by substituting E into (2.52). In Cartesian coordinates, a curl on the left-hand side of (2.52) is conveniently presented in form of determinant. Thus (2.52) may be written as

  ∇×H=



x0

y0

z0

∂ ∂x  H

∂ ∂y  H

∂ ∂z  H

X

Y

 = iωε0ε ⋅ E (2.61)

Z

Taking into account (2.60), and observing the fact  that the right-hand side of (2.61) does not contain the Z- and Y components of E field, it may be rewritten as: −



 dH Y  )] (2.62) = iωε0ε E m,X exp[ i (ω t − kz dz

Here the partial derivative is replaced by the regular derivative due to the single-coordinate dependence. If we integrate (2.62), then the result will be:    H = y0 H m,Y exp[ i (ω t − kz)] (2.63)

where



ω ε0ε   H E m,X (2.64) m ,Y = k   Both vector phasors E and H contain the total phase ϕ = ω t − β z (2.65) where b is a real part of the complex number defined by (2.55). Note that it affects a phase and is called phase coefficient, whereas the imaginary part, a, affects the amplitude and is called attenuation coefficient, or attenuation constant (see Section 2.2.2 for details). Surfaces represented by j = const in space, namely, the surfaces of constant values of the phase, are called wave fronts. Consider the value of j that remains unchanged (i.e., we’ll try to find the velocity of movement of the wave front along the Z-axis). Then, any increase in time ∆t is to be related to the change in coordinate ∆z (Figure 2.8), so thus



ϕ 1 = ω t − β z = ϕ 2 = ω (t + ∆t) − β (z + ∆z) = const (2.66)

38

Basics of Electromagnetic Waves Theory

Figure 2.8  Movement of the plane wave front. Only a square fragment of the infinity wave front is represented here.

From (2.66) it is logical to define the phase velocity, which is the velocity of the movement of the plane wave front along the Z-axis: v=



∆z ω = (2.67) ∆t β

If, in particular, ∆z represents a distance between two adjacent maximums of the wave pattern (i.e., ∆z = l), then the time interval that is needed to move from one maximum to the next is ∆t = T = 1/f = 2p/w, where T is a period of the oscillations, f is the linear frequency, and w is angular frequency. From (2.67) the phase coefficient may be expressed in terms of wavelength in considering medium as

β=

2π (2.68) λ

Based on these considerations, one may conclude that the electromagnetic process described by the solutions (2.59) and (2.63) represents a wave that propagates along the Z-axis with the constant velocity v and contains plane infinite wave fronts, supporting the uniform distribution of the electric and magnetic fields along that plane. For the particular case of propagation along the Z-direction, wave fronts are flat surfaces parallel to the XOY plane. These types of waves are known as uniform plane waves because of the constant distribution of the amplitudes and phases of electric and magnetic fields along those planes. They do not exist in nature in their pure form and are introduced as a mathematical abstraction to simplify understanding of the physical concepts and make easier the formal mathematical evaluations. In reality, the distances covered by the radio waves are much greater than the geometric size of source. It will be shown later that for those long distances, this abstraction is a good foundation for the radio wave propagation theory and is capable of outlining all propagation problems qualitatively and quantitatively. Note to the Reader

Expression (2.54) describes a plane radio wave that propagates along the Z-axis  For propagation in homogeneous medium of the complex propagation constant k.

2.2  Free Propagation of Uniform Plane Radio Waves 39

along an arbitrary direction defined by a distance-vector r , (2.59) is presented in its generic form

   E(r ) = E m exp(ω t − k ⋅ r )

(2.59a)

    where k = β − iα is defined as a complex wave vector, which combines β , phase    vector and α , attenuation vector. Generically both β and α may be orientation-dependent that is specific for the anisotropic media. The dot product in (2.59a) parentheses allows calculation of both the phase β ⋅ r of the wave and its amplitude  decay α ⋅ r . Note that β is perpendicular to the phase front at the observation point, and it shows the direction of propagation of the phase front (unlike the direction of the Poynting vector that shows the direction of the canalization of energy). The   imaginary part of k , the complex wave vector α , shows the direction of the fastest decay of the amplitude. From (2.64) one may realize that if a radio wave contains only an X-component of the electric field (i.e., that is directed along positive X-axis) then the magnetic field may have only a Y-component that is directed along the positive Y-axis. From the same equation, it may be confirmed that for the propagation of a time-harmonic radio wave in lossless media (s = 0), both ε and k become real numbers, and therefore, the Poynting vector (2.17) also becomes real if the cross product is defined as



   Π = E × H* (2.69)



  where the phasor H* represents a conjugate of the complex vector H. In a lossy      medium Π is a complex vector. Disposition of E and H satisfies the right-hand rule as shown in Figure 2.9. An expression similar to (2.69) may be written for the amplitude phasor of the Poynting vector as    Πm = E m × H*m (2.70) For practical applications, it is more convenient to introduce a scalar that is derived from the complex Poynting vector as an RMS value of the instant power flow density (i.e., power flow averaged within a period of oscillation):

   Figure 2.9  Disposition of vectors E, H, and Π in a free space according the right-hand rule.

40

Basics of Electromagnetic Waves Theory

(

)

   1 Π = Re E m × H*m (2.71) 2



The factor 1/2 is due to ratio between amplitude and RMS values of any time-harmonic oscillations. It’s analogous to the relation between voltage amplitude, current amplitude, and the power consumption within an electric circuit element. As one may note from (2.64), if the parameters of propagation medium ε and μ are constant in time and space, then the ratio between electric and magnetic fields is time and space independent. That ratio may be used to identify a property of the propagation medium called intrinsic impedance of medium and may be defined from (2.64), if (2.55) is taken into account, as well as μ = 1 is assumed:   = E mX = W  H mY



W µ0 = 0 , Ohm (2.72) ε0 ε ε

Here W0 = 120p ≈ 377 Ω is a characteristic impedance of free space, and the values for e0 and μ0 are utilized from (2.6). 2.2.1  Uniform Plane Wave in Lossless Medium

 the value of relative magnetic permeability μ may be In (2.55) for parameter k, assumed to be μ = 1, due to the nonmagnetic types of most of mediums the RF propagation theory deals with, such as salt and sweet water, ground soil, and atmospheric air. Another assumption is s = 0, due to lossless propagation conditions in the atmospheric air. Under those two conditions k becomes a real number (i.e., k = β ). If this value of k is substituted into (2.67) then the result for propagation phase velocity will become 1

v=



ε0ε µ0

=

c

ε

(2.73)

where

c=

1

ε0 µ0

=

1 −9

−7

(10 /36π ) ⋅ (4π ⋅ 10 )

= 3 ⋅ 108 m/s (2.74)

represents speed of light in vacuum. Expression (2.73) demonstrates that propagation phase velocity in any real medium is always less than in free space. An exception is the case of propagation within ionospheric plasma medium, which will be discussed in Chapter 5. Recall that the wavelength is a distance that is covered by the radio wave within one period of oscillation (i.e., within time interval T = 1/f = 2p/w), as follows:

λ = v ⋅T =

1 c 1 2π c λ 0 = = (2.75) ε ε f ε ω

2.2  Free Propagation of Uniform Plane Radio Waves 41

Here,

λ0 =



c 2π c = (2.76) ω f

denotes a wavelength of the radio wave in free space (i.e., in vacuum).12 Now consider the propagation constant k defined by (2.55). For this particular case of propagation in lossless medium,

k=ω

ε0 µ0 ⋅ ε =

ω c

ε=

2π λ0

ε=

2π rad/m (2.77) λ

Parameter k is usually referred to as a wave number. To be more specific, we have to note that in physics the wave number is defined as a reciprocal of the wavelength, 1/l, showing the number of the wavelengths on a unit distance along the propagation path, whereas the quantity 2p/l is referred to as angular wave number. However, most of the authors in electromagnetics are using the term wave number to specify 2p/l, so we’ll follow the same in this text. To summarize (2.77), we emphasize that the higher the oscillation frequency, the greater the wave number is. As mentioned earlier, in the case of the absence of losses, s = 0 regarding (2.50),  and W become real numbers with (2.55), and (2.72) three complex quantities, ε , k, the following values:



k=β =



W=

2π

λ0 / ε

120 π

ε

≈

=

2π rad/m (2.78) λ

377

ε

Ohms (2.79)

  Thus, as indicated in (2.64), vectors E and H, being cross-perpendicular to each other, are not phase-shifted (i.e., maximums and minimums of these vectors are allocated in the same points of space). As follows from (2.59) for this case of lossless medium (k is a real number), the amplitude of the wave is Z-independent (i.e., it remains  unchanged along the entire propagation path, the Z-axis). If the conjugate of H mY is substituted from (2.72) into (2.71), then the effective (RMS) power flow may be found as:

Π=

1  E*  E 2 Re  E m × m  = m (2.80) 2  W  2 W

12. Do not confuse the term free propagation with propagation in free space. Free propagation refers to unguided propagation that may occur in any medium such as air, water, soil, whereas propagation in free space means propagation in ideal conditions (vacuum) that is free of any material substance.

42

Basics of Electromagnetic Waves Theory

Figure 2.10  Structure of the radio wave in lossless medium.

This expression allows us to reach the conclusion that the Poynting vector also becomes real, time and space independent. The distribution of the electric and magnetic fields along the propagation path is shown in Figure 2.10. 2.2.2  Uniform Plane Wave in Lossy Medium

In this case of non-zero conductivity s  0, the complex relative permittivity, intrinsic impedance, and wave number are complex, and the expressions (2.48) and (2.72) may be rewritten in following forms:

ε = ε − i



σ = ε ′ − i ⋅ ε ′′ (2.81a) ωε0

where



ε ′′ =

σ (2.81b) ω ε0

 = W ⋅ ε i ΦW (2.82) W In this case, (2.55) is complex and is rewritten here:



k = β − iα (2.83) For the plane uniform radio wave that propagates along the Z-direction, (2.59) may be presented in the following form, after (2.83) is substituted into it:



  E = x0 E m X e i(ω t − k z) = x0 E m X e −α z ⋅ e i (ω t − β z) (2.84)



Similarly, based on (2.63), the expression for the magnetic field may be presented as:  −α z i (ω t − k z)   H = y0 H = y0 H ⋅ e i(ω t − β z) (2.85) mY e mY e

2.2  Free Propagation of Uniform Plane Radio Waves 43

where:  H mY =



E m X E m X − iΦ W (2.86)  = W e W

It is obvious from (2.84) and (2.85) that the amplitudes of electric and magnetic components of the radio wave decay exponentially as e–az. Coefficient a introduced in (2.55) as the attenuation coefficient and b as the phase coefficient, which is a part of the total phase wt –bz. The expression (2.86) demonstrates the fact that there is an initial (permanent) phase shift, ΦW, between electric and magnetic fields oscillations. This phase shift may be transformed into the distance shift ∆Z = ΦW /b between electric and magnetic waves, as shown in Figure 2.11. The decrease of the intensity of radio wave during propagation may also become evident if the expression for the effective (RMS) value of the Poynting vector is recalled. That expression may be derived if (2.84) and the complex conjugate of  = H mY from (2.86) are used along with (2.71). Π=



2  Em E * 1  ,X −2α z e Re  E m,X m,X e iΦW  = cosΦ W (2.87) 2  W W 2 

From (2.87) it is evident that in case of propagation in lossy medium, the average power density flow not just decays exponentially along the propagation path. In addition, its initial value becomes smaller than that for propagation in lossless medium, by the value of the so-called power factor, which is equal to cosΦW. The important parameters of propagation, namely, attenuation coefficient a and phase coefficient b, are considered composite parts of the complex propagation constant k . If in (2.55) we assume μ = 1, then taking into account (2.81) will result in

κ = β − iα = ω ε0 µ0



... =

ω c

... =

2π λ0

ε =

ω c

ε −i

σ = ... ωε0

 σ  2π ε exp  −iarctan n = ωε0ε  λ 0 

ε exp ( −iδ ε ) = ... (2.88)

 δ δ  ε  cos ε − i ⋅ sin ε  2 2 

Here, 2



ε =

 σ  ε ε2 +  = (2.89) cosδ ε  ωε0 

is the magnitude of complex dielectric permittivity, expressed by its real part, e, and by angle of losses, de, given by (2.50) and shown in Figure 2.5. Thus (2.88) may be rewritten as:

44

Basics of Electromagnetic Waves Theory

Figure 2.11  Structure of the uniform plane wave in lossy (semiconducting) medium.



β − iα =

2π λ0

ε cosδ ε

 δε  δε  cos 2 − i ⋅ sin 2  (2.90)  

Thereby, the attenuation and phase coefficients may be derived by taking real and imaginary parts in both sides equal to each other:

α=

2π λ0

δ ε sin ε (2.91a) cosδ ε 2



β=

2π λ0

δ ε cos ε (2.91b) cosδ ε 2

Here l0 is a wavelength in free space. Expressions (2.91a) and (2.91b) may be simplified for following two extreme cases. 2.2.2.1  Low-Loss Dielectric Medium

δε 1 ≈ tan δ ε , and therefore 2 2 expressions (2.91a) and (2.91b) may be simplified as follows: In this case tan de 1 (or δ ε ≈ ); therefore, 2 δ δ 1 cos ε ≈ sin ε ≈ . Thus, 2 2 2

ε = cos δ ε

ε ≈

σ , and ωε0

2.2  Free Propagation of Uniform Plane Radio Waves 45

σ ω µ0 (2.94) 2

α ≈β≈



In all other cases of semi-conducting media, which are more specific for real propagation conditions, only the original (2.91a) and (2.91b) formulas are applicable. The attenuation constant is specified by a Neper-per-meter (Np/m) unit. The physical meaning of that unit may become clear if (2.84) and (2.85) are recalled. From any of these equations, one may conclude that if a = 1 Np/m, then the amplitude of the radio wave, which passes a distance of one meter, will decrease e = 2.71…13 times. Indeed, from (2.84), the attenuation coefficient is found as

α Np/ m =



1 E mX (z = 0) ln (2.95) E mX (z) z

Then, for the ratio of the amplitudes equal e = 2.71…, and for the propagation distance of z = 1m, the value of the right-hand side in (2.95) is equal to unity. Another unit, decibel-per-meter (dB/m), which is widely used in engineering applications, is defined as

α dB/ m =

 E m X (z = 0)  1 ⋅ 20log   (2.96) z  E m X (z) 

and is more commonly annotated. The relation between those two units is

α dB/ m = 8.68 ⋅ α Nep/ m (2.97)



Attenuation of the radio wave may now be introduced as A = α ⋅ z (AdB = α dB / m ⋅ z) (2.98)



that displays a total decrease of the radio wave amplitude along the entire homogeneous propagation path of the length z. In inhomogeneous media, where e and/ or s are varying in space from point to point, a also becomes variable. Suppose the entire inhomogeneous propagation track consists on several discrete segments of homogeneous paths, as shown in Figure 2.12. Then it’s apparent that after passing the entire z distance, the amplitude of the radio wave will become

E m (z) = E m ⋅ e −α1⋅∆z1 ⋅ e −α 2 ⋅∆z2 ⋅⋅⋅⋅⋅ e −α n ⋅∆zn = E m ⋅ e

− ∑ α k ⋅∆zk k

(2.99)

for the total distance of

z = ∑ ∆zk (2.100) k

13. This is the base of the natural logarithm.

46

Basics of Electromagnetic Waves Theory

Figure 2.12  Radio wave propagation pattern in a discretely inhomogeneous medium.

If a is a continuous function of the distance, it is apparent that (2.99) may be transformed into integral form as follows: − α (z ′)⋅dz ′ E m (z) = E m0 ⋅ e ∫ (2.101)



Here Em0 is the initial amplitude. Thus, the expression for total attenuation along the entire propagation path may be rewritten in an integral from as z

A = ∫ α (z ′) ⋅ dz ′ (2.102)



0

Example 2.3

Calculate attenuation coefficient and phase coefficient for the radio wave propagating in dry soil (e = 4, and s = 2·10-4 S/m). Calculations are to be performed for the following three frequencies: f1 = 6 kHz, f2 = 600 kHz, f3 = 6 GHz. Solution •

For the frequency f1 = 6 kHz, the wavelength in free space is l01 = 50000 m. Then, regarding (2.50), tan δ ε =



60 λ 01 σ

ε

= 150 >> 1. (2.103)

Hence, expression (2.94) is applicable for both the attenuation coefficient and phase coefficient; that is,



α≈

σ ω µ0 = 2

2 ⋅ 10−4 ⋅ 2π ⋅ 6 ⋅ 103 ⋅ 4π ⋅ 10−7 = 2.177 ⋅ 10−3 Np/m(Answer) 2

β ≈ 2.177 ⋅ 10−3 rad/m

•



(Answer)

For the frequency f2 = 600 kHz, the wavelength in free space is l02 = 500 m. Then, tan δ ε =

60 λ 02 σ

ε

= 1.5 (de = 0.9828 rad)

(2.104)

2.3  Polarization of the Radio Waves 47

Hence, neither of the approximate approaches given in Sections 2.2.2.1 and 2.2.2.2 is applicable. The general expressions (2.91a) and (2.91b) must be used for the attenuation coefficient and phase coefficient, respectively.

α=

2π λ 02

β= •

2π λ 02

 0.9828  4 = 0.0159 Np/m (Answer) sin  cos 0.9828  2 

δ 2π ε ⋅ sin ε = 2 500 cos δ ε

 0.9828  4 = 0.03 rad/m  (Answer) cos  cos 0.9828  2 

δ ε 2π ⋅ cos ε = 2 500 cos δ ε

For the frequency f3 = 6 GHz, the wavelength in free space is l03 = 0.05 m. Then,



tan δ ε =

60 λ 03 σ

ε

= 1.5 ⋅ 10−4 0 means the Y-component lags behind the X-component, whereas if d < 0 there is an opposite relation between components. Now we separate cross-orthogonal normalized components of (2.121) and present them in the form of a system of equations:



X = cos ω t Y = M ⋅ cos(ω t − δ )

  (2.122) 

It may be seen that (2.122) is a parametric form of arbitrary-oriented ellipse with the center at the origin of the coordinates, as shown in Figure 2.17(a). t is a tilt relative to the horizontal axis. For further processing, we will rotate the XOY coordinate system by angle t counterclockwise, as shown in Figure 2.17(b). This allows to make ellipse symmetric relative to OX′ and OY′ axes in a new X′OY′ coordinate system. The linear transform from XOY coordinates to X′OY′ may be presented as

X = X ′cosτ − Y ′sin τ Y = X ′sin τ + Y ′cos τ

  (2.123) 

Now we solve (2.123) for X′ and Y  by finding the inverse transform of the matrix of coefficients on the right-hand side; that is,

X ′ = X cos τ + Y sin τ Y ′ = − X sin τ + Y cos τ

  (2.124) 

54

Basics of Electromagnetic Waves Theory

Figure 2.17  (a) Polarization ellipse in Cartesian coordinates, and (b) rotation of the coordinate system. (Note: the OZ-axis is directed from picture to viewer; that is, the direction of wave movement.)

Next, we substitute X and Y from (2.122) into (2.124) and use the following notation wt – z.



X ′ = cosζ cos τ + M cos(ζ − δ )sin τ =

(2.125)

= cosζ cos τ + M cosζ cosδ sin τ + M sinζ sinδ sinτ ,

Y ′ = − cosζ sin τ + M cosζ cos δ cos τ + M sinζ sin δ cos τ (2.126) On the other hand, the parametric equation for ellipse in the new X′OY′ coordinate system may be written as



X ′ = a cos (ζ + ∆) Y ′ = ± b sin (ζ + ∆)

  , (2.127) 

where a and b are the major and minor semi-axes of the ellipse, respectively (see Figure 2.18), and ∆ is the arbitrary initial phase shift. Note from Figure 2.18 that the plus sign in (2.127) results in counterclockwise rotation of the total E vector, which signifies a right-hand elliptical polarization

Figure 2.18  Polarization ellipses for right-hand elliptical polarization (RHEP) and left-hand elliptical polarization (LHEP).

2.3  Polarization of the Radio Waves 55

 (RHEP), whereas a minus sign represents a clockwise rotation of the total E vector; that is, a left-hand elliptical polarization (LHEP).17 The following two polarization parameters are introduced to identify the orientation of the ellipse and its shape: •

The angle of ellipticity, e e, which shows how stretched the ellipse is:  b b tan ε e = ± , then ε e = tan−1  ±  (2.128)  a a

•



The axial ratio of polarization ellipse: AR = ±

a = cot ε e , b

ARdB = 20log

a   (logarithmic format) b

(2.129)

From this definition it is obvious that if the magnitude |AR| changes from 1 to ∞ (from 0 to ∞ in decibels), then the sense of polarization will change from circular (CP) to linear (LP). For those changes, the values of the angle ee may vary in range –45° ≤ ee ≤ +45°. The system of (2.125)–(2.126) is the same as (2.127). Therefore, we may take proper parts of those equations equal to each other, as follows:



a cosζ cos ∆ − a sinζ sin ∆ = ... ... = cosζ cos τ + M cosζ cos δ sin τ + M sinζ sin δ sin τ ±b sinζ cos ∆ ± b cosζ sin ∆ = ... ... = − cosζ sin τ + M cosζ cos δ cos τ + M sinζ sin δ cos τ

(2.130)

(2.131)

Now we take like terms in (2.130) and (2.131) equal:

a cos ∆ = cos τ + M cos δ sin τ (2.132)



a sin ∆ = − M sin δ sin τ (2.133)



±b cos ∆ = M sin δ cos τ (2.134)



±b sin ∆ = − sin τ + M cos δ cos τ (2.135) Next, we take the sum of squares of all four expressions (2.132) – (2.135). After simplifications the result is

17. Here the Z-axis is directed from the picture towards the observer, in contrast to that in Figure 2.13(b). Therefore, we ended up with the counterclockwise rotation of the resultant vector for RHEP, in contrast to the clockwise rotation, when the Z-axis has the opposite direction (i.e., directed from the observer towards the picture), as shown in Figure 2.13(b).

56

Basics of Electromagnetic Waves Theory

a2 + b2 = 1 + M 2 (2.136)



which stands for a conservation of the power while switching from XOY to X′OY′ coordinates. Now we divide (2.134) by (2.132), and (2.135) by (2.133):

±

M sinδ cosτ M sin δ − sin τ + M cos δ cos τ b = = = (2.137) a cosτ + M cosδ sinτ 1 + M cos δ tan τ − M sin δ sin τ

The following transforms may be applied to (2.137): M sin δ = 1 + M cos δ tan τ



M cos δ tan τ tan τ − M cos δ (2.138) = M sin δ M tan τ sin δ

1−

M 2 sin2δ tan τ = (1 + M cos δ tan τ ) (tan τ − M cos δ ) (2.139)



Now we open the parenthesis, combine like terms, and substitute M = Em,Y/ Em,X = tanΨ: tan τ



2

1 − tan τ

=

M 1− M

2

cosδ =

tan Ψ 1 − tan2 Ψ

cos δ (2.140)

Finally,18 tan2τ = tan2Ψ cosδ (2.141)



If Ψ and d are known a priori, then t may be found by solving (2.141); that is,

τ=

1 π tan−1 [tan2Ψ cos δ ] ± n ,  n = 0, 1, 2, … 2 2

(2.142)

Each value of n in (2.142) may be related to one position of the polarization ellipse in space. In other words, we have an aggregate of ellipses. But only two of those positions are different. Those are defined by the values n = 0 and n = 1. All other positions (for all other values of n) will repeat those two. Hence, it is reasonable to consider only two values of t:

τ1 =





τ2 =

18. Here 2 tan x/(1 – tan2 x) = tan 2x.

1 tan−1 [tan2Ψ cosδ ] (2.143) 2

1 tan−1 [ tan2Ψ cos δ ] + 90° (2.144) 2

2.3  Polarization of the Radio Waves 57

It’s easy to show that (2.143) associates to the values of tan Ψ < 1 (Ψ < 45°), whereas (2.144) associates to tan Ψ > 1 (Ψ > 45°). If tan Ψ = 1 (Ψ = 45°), then t = ± 45°. In order to obtain ee, first we find the product ± ab by multiplying (2.132) by (2.134), and (2.133) by (2.135), and adding the results.

(

)

± ab = cosτ + M cosδ sinτ M sinδ cosτ + ...



... + (− M sinδ sinτ )(−sinτ + M cosδ cosτ )

(2.145)

After simplifications the following expression may be obtained: ± ab = M sinδ . (2.146)



Now we use (2.128) to evaluate the ratio.19 ±



2ab 2

a +b

2

=±

2b /a 2

1 + (b /a)

=

2tanε e 1 + tan2ε e

= sin2ε e (2.147)

On the other hand, we may modify the same expression by using (2.136) and (2.146):

±

2ab 2

a +b

2

=

2 M sin δ 1+ M

2

=

2tanΨsin δ 1 + tan2 Ψ

= sin2Ψ sin δ (2.148)

A combination of (2.147) and (2.148) results in

sin2ε e = sin2Ψsin δ (2.149) Expressions (2.141) and (2.149) allow us to obtain the direct relations between initial parameters of the linear cross-polarized components (M = tan Ψand d ) and parameters of polarization (t and ee). Axial ratio (AR) may further be calculated from (2.129). To find the inverse relations, one may obtain cos d from (2.149), substitute into (2.141), and solve for cos 2Ψ. The result is



cos2Ψ = cos2τ cos2 ε e (2.150) which may be used to obtain Ψ. If (2.149) is divided by (2.141), then



tan δ =

1 sin2 ε e (2.151) cos2 Ψ tan2τ

The final expression may be developed if cos 2Ψ is substituted from (2.150). Then the result

19. Trigonometric relation, 2 tan x/(1 + tan2 x) = sin 2x is used.

58

Basics of Electromagnetic Waves Theory

tan δ =



tan2 ε e (2.152) sin2τ

allows us to find the phase shift d between initial LP components of the elliptically polarized wave. Example 2.4

Find polarization parameters of the radio wave (i.e., sense of polarization); that is, angle of ellipticity, ee, axial ratio, AR, and tilt angle t of the polarization ellipse, if parameters of cross-polarized LP-components (vertical and horizontal) are • • •

Amplitude of V-pol component: Ey,m = 8mV/m, Amplitude of H-pol component: EX,m = 3mV/m, Phase shift between components: d = 60°.

Solution • •



M = tanΨ = EY,m/EX,m = 3/8 =2.67, thus Ψ = 69.4°, 2Ψ = 138.9° We use (2.144) because of the condition M = tan Ψm >1 t = 0.5 tan–1[tan(2Ψ)⋅cos δ]+90° = 0.5 tan–1[tan(138.9°)cos(60°)] +90° = 78.2°

•

•

(Answer)

Now using expression (2.149) to obtain e e sin 2ee = sin 2Ψ⋅sind = sin 41.1°⋅sin (60°) = 0.5693, therefore ee = 17.35° (Answer) Axial ratio AR = cot(ee) = cot(17.35°) = +3.2 (10 dB). (Answer) The sense of polarization is RHEP for positive e e (see Figure 2.19).

Example 2.5

Find parameter M = tan Ψ (the ration of the amplitudes of initial cross-polarized components) of the radio wave and the phase shift d if the following parameters of polarization ellipse are provided: angle of ellipticity, ee = –15° (LHEP), and tilt angle, t = 75°. Solution •

Using the expression (2.150)

Figure 2.19  Polarization ellipse sketch for the calculated parameters in Example 2.4.

2.3  Polarization of the Radio Waves 59

cos 2Ψ = cos 2t ⋅ cos 2ee = cos(2⋅75°)⋅cos(2⋅(–15)) = –0.75 thus Ψ = 69.3°, M = tan Ψ = 2.65.



•

Using the expression (2.152) tan δ =



(Answer)

tan 2 ε e tan ( −30° ) = = −1.1547 , thus δ = − 49.1° .(Answer) sin 2τ sin ( 150° )

2.3.4.2 Decomposition of the Elliptically Polarized Radio Wave into Two CrossPolarized CP Radio Waves

Decomposition of elliptically polarized radio wave into two circularly polarized waves is often helpful in RF propagation analysis. For further processing, the basic CP unit vectors are to be introduced [4]. As shown in Section 2.3.3, an ideal circularly polarized (CP) radio wave may be obtained if two LP-components of equal amplitudes Em are superimposed with initial phases that are shifted by ± 90 degrees; that is,



   π  ECP =  x0 E m + y0 E m exp  ± i   = E CP u . (2.153)  2   Here x0 and y0 are unit vectors in Cartesian coordinates, u is a rotating unit vector, and E CP is a generic notation for the complex amplitude (phasor) of the CP radio wave. Recall from Section 2.3.3 (cases 3 and 4) that positive sign relates to LHCP wave, whereas the negative sign – to RHCP wave. Based on principle of conservation of power one may assume that the sum of squares of LP amplitudes [i.e., the sum of squares of terms on the left-hand side of (2.153)] is equal to the square of the amplitude of the resultant CP wave presented on its right-hand side; that is, ECP2 = 2Em2 (or E m = (1/ 2)ECP ). Hence (2.153) may be rewritten in following two forms if normalized to the amplitude of LP component (i.e., to Em)20 r=

l =

1 2 1 2

x0 − i

x0 + i

1 2 1 2

y0 – RHCP unit vector,

(2.154a)

y0 – LHCP unit vector,

(2.154b)

 If assumed that electric field E of the radio wave of any arbitrary elliptic polarization is presented as a combination of RHCP and LHCP components with the  amplitude phasors E R and E L , then amplitude phasor E of that EP radio wave may be presented as

20. It may easily be proven that the amplitude of each unit vector, (2.154a) and (2.154b) is equal to a unity.

60

Basics of Electromagnetic Waves Theory

 E = E R r + E L l (2.155)



It’s easy to prove that the following dot products of each unit vector and its conjugate is equal to a unity. r ⋅ r* = 1 l ⋅l*=1



  

21

(2.156)

Then each CP component, E R or E L, may be extracted from total field (2.155) by using the following procedure(s):  E R = E ⋅ r *  E L = E ⋅ l *



   . (2.157)  

Figure 2.20 demonstrates graphically three examples of how the combination  of the E R and E L components end up with the locus of the tip of the resultant E vector of the arbitrary elliptic polarization as a combination of RHCP and LHCP waves. In that figure we assumed, for simplicity, that the component phasors, E R and E L , are real numbers (no initial phase shifts), which results in polarization ellipses to not be tilted. In the general case of non-zero phase shift between CP components, the resultant polarization ellipse will become tilted as shown in Figure 2.17(a). To evaluate this generic case, we substitute (2.154a) and (2.154b) into (2.155).



 E − E R E E E + E R E = E R r + E L l = R (x0 − i y0 ) + L (x0 + i y0 ) = L x0 + i L y0 (2.158) 2 2 2 2 This expression demonstrates the conformity between circular components of the expanded elliptically polarized wave and its linear components. It follows from (2.158) that if E R = − E L (out-of-phase oscillations), the polarization ellipse degenerates into LP along the Y-axis; whereas if E R = E L (in-phase oscillations), the resultant LP wave is polarized along X-axis, as shown in Figure 2.21. Similar to decomposition of the EP wave into two cross-polarized LPs (Section 2.3.4.1), we can affirm the following statement: The parameters of the polarization ellipse do not depend on the absolute amplitudes and initial phases E R and E L, but rather on their relative values. Therefore, let’s normalize (2.158) similar to (2.121)—a result of normalization of (2.122).



x0 + i

1 − K  y (2.159) y0 = x0 + i M 0 1 + K

21. Asterisks in r * and l * depict complex conjugate of proper unit vectors. The orthogonality of r and l may easily be proven if the dot product r ⋅ l * is assessed, which is equal to zero.

2.3  Polarization of the Radio Waves 61

Figure 2.20  Forming of the polarization ellipse as an overlay of RHCP and LHCP component paths (propagation is directed towards observer).

where

  = 1 − K (2.160) M 1 + K In (2.160) we have the following notation:



E E K =  R = R exp(i χ) = K exp(i χ) (2.161) EL EL which is the ratio of the amplitudes of CP-components, and c is a phase shift between them. Recall (2.121) from Section 2.3.4.1:



  = E m,Y = E m,Y exp(−iδ ) = tanΨexp(−iδ ) = M exp (−iδ ) (2.162) M E m,X E m,X

Figure 2.21  Combining of two CP radio waves of opposite sense and equal amplitudes (RHCP and LHCP) into an LP radio wave.

62

Basics of Electromagnetic Waves Theory

This is a ratio of complex amplitudes of LP components with d – a phase shift between them, and M = tan Ψ, a ratio of real amplitudes of LP components. With (2.160) in mind, proper solutions for M and d may be defined as follows: 1 − 2K cosχ + K 2

)=M= abs(M



1 + 2K cosχ + K 2

= tanΨ (2.163)

22    K sinχ  −1  K sinχ     −  tan−1  1 − K cosχ  + tan  1 + K cosχ   − π if Re(1 − K) < 0     arg(M) = δ =  . (2.164)  −1  K sinχ   −1  K sinχ    − − + tan ≥ 0 K) tan if Re(1   1 − K cosχ   1 + K cosχ      

An inverse relation may be found from (2.160) in symmetric form:  1− M (2.165) K =  1+ M



From (2.165) on, we may finally define the expressions for magnitude and phase as abs(K ) = K =



1 − 2M cosδ + M 2 1 + 2M cosδ + M 2

(2.166)

  − M sinδ  −1  M sinδ    tan−1   − tan  1 + M cosδ  − π if Re(1 − M) < 0 1 cos δ − M   arg(K ) = χ =  (2.167)  tan−1  − M sinδ  − tan−1  M sinδ   if Re(1 − M) ≥ 0  1 + M cosδ   1 − M cosδ   

Example 2.6

Find components of CP decomposition of elliptically polarized wave considered in Example 2.4: • •

Phase shift between LP components: d = 60°. Ratio of the amplitudes of LP components: M = tanΨ = 8/3 = 2.67.

22. For any complex number, the z = x + iy argument must be determined numerically as arg(z) = atan2(y,x), which covers the angles in range [–p, +p] However, the regular inverse tangent is also usable; namely, the expression arg(z) = atan(y/x) is used for x ≥ 0, or arg(z) = atan(y/x) – p for x < 0.

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 63

Solution •

Ratio of amplitudes of CP components is found from (2.166): K=

•

1 − 2 ⋅ 2.67 ⋅ cos 60° + 2.67 2 1 + 2 ⋅ 2.67 ⋅ cos 60° + 2.67 2

= 0.7 759.(Answer)

Phase shift between CP components is found (2.167):

 −2.67 ⋅ sin 60°   2.67 7 ⋅ sin 60°  − tan−1  − π = −142.96°. (Answer) χ = tan−1    1 − 2.67 ⋅ cos 60°   1 + 2.67 ⋅ cos 60° 

2.4 Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 2.4.1 Introductory Remarks

Consider an electromagnetic wave of the plane wave front that falls onto the flat boundary XOY of two media, with the angle in incidence qi. The incident ray path is positioned in YOZ-plane, as shown in Figure 2.22(a). The reflected ray path is always positioned in the same YOZ-plane, and the following two conditions are satisfied at the boundary of media: phase matching condition and boundary condition. The phase matching condition is expressed as

β 1sinθi = β 2sinθt (2.168)



wherein qi and qt represent angles of incidence and refraction, respectively, and phase coefficients b1 and b2 are defined by one of the formulas (2.91b), (2.93), or (2.94). Particularly for low-loss or lossless (clear) media, when attenuation is ignorable, (2.168) may be transformed into

ε1 ⋅ sinθi =



ε 2 ⋅ sinθt (2.169a)

which is known from optics as Snell’s law, written for the refraction indexes ε1 = n1 and ε 2 = n2 of both media. Reflection and refraction effects for these types of low-loss or lossless media are considered below. Notes

For a generalized approach, quantity k given by (2.83) is considered as a   a complex   complex wave vector k = β − iα , so for the arbitrary oriented distance-vector r expression of the wave field may be written in generalized form     (2.59) for the intensity as E = E m X exp[i(ωt − k ⋅ r )] . The dot product in parentheses allows the calculation    of both the phase β ⋅ r and the decay of the amplitude (i.e., the attenuation) α ⋅ r. It  must be noted here that the real part of the complex wave vector β is perpendicular

64

Basics of Electromagnetic Waves Theory

Figure 2.22  (a) Disposition of incident, reflected, and refracted waves at the boundary of two media, and (b) vertical pattern of the incident and refracted wave fronts at the boundary of two media (1-incident wave, 2-reflected wave, 3-refracted wave).



to the phase front at the observation point, and it shows the direction of propagation of the phase front (unlike the direction of the Poynting vector, which shows the direction of canalization of energy).  The imaginary part of the complex wave vector α shows the direction of the fastest decay of the amplitude. In isotropic media those vectors support the same direction (i.e., the amplitude decay takes place in the same direction as a direction of the movement of the phase front). At the boundary of two media shown in Figure 2.22(a) the general form of (2.168) is     k1 ⋅ z0 = k z1 = k2 ⋅ z0 = k z 2 (2.169b) If shown separately for the real and imaginary parts, then (2.169b) may be rewritten as



β 1sinθi = β 2sinθt (2.170a)



α 1sinθi = α 2sinθt (2.170b) See [10] for more details. The physical interpretation of the phase-matching condition may be clarified if b1 and b2 in (2.168) are replaced based on (2.78) or (2.93):

λ1

sinθi

=

λ0 ε 1 sinθi

=

λ2 sinθt

=

λ0 ε 2 sinθt

(2.171)

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 65

The meaning of (2.171) is as follows: as shown in Figure 2.22(b) on the vertical XOZ-cut, the phase fronts along the interface of two media supposed to be matched (i.e., points of maximums and minimums supposed to be stitched to each other on that interface).  The boundary conditions for the tangential components of the total electric E  and total magnetic H vectors must also be taken into account. Regarding (2.33) and (2.35), in the absence of conducting currents, the tangential components of the electric and magnetic fields strengths are continuous across the boundary (interface) between media. That is,

Eτ (1) = Eτ (2), (2.172)



Hτ (1) = Hτ (2) . (2.173) where the superscript in parenthesis indicates medium under consideration. Another assumption that is important for further analysis is that the radio wave with the arbitrary polarization may be decomposed into two waves of cross-orthogonal polarizations of initially chosen orientations. Note that in the particular case of propagation over the Earth’s ground those initial polarizations will be vertical (V-pol) and horizontal (H-pol). In other words, the consideration of reflection and refraction phenomenon at the boundary of two media may be limited for just two cases: for the linear polarization that is (1) parallel to the boundary, and (2) polarization that is perpendicular to it. In order to retrieve the resultant wave, after the separate analysis is complete, it will be followed by the superposition of the proper components of reflected (or refracted) waves.

2.4.2  Normal Incidence on a Plane Boundary

In this particular case of normal incidence, the plane wave travels in a direction perpendicular to the flat interface of two ideal dielectric media (qi = 0). Part of the total energy of the wave reflects from the boundary, and the remaining part penetrates (refracts) through the boundary into the second medium. The relative positioning of electric, magnetic, and Poynting vectors are interrelated by the right-hand rule (as shown in Figure 2.23a) for all three waves: incident, reflected, and refracted. Conventionally, the ratio of the reflected electric field strength and the incident field is called the electric field reflection coefficient, which generically is presented as a complex number:



E i⋅Φ E Γ E =  r = Γ E ⋅ e Γ (2.174) Ei where E r and E i are electric fields of reflected and incident waves, respectively. Similarly, a magnetic field reflection coefficient is defined as



 H i⋅Φ H Γ H =  r = Γ H ⋅ e Γ (2.175) Hi

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Basics of Electromagnetic Waves Theory

Figure 2.23  (a) The pattern of a normally incident wave onto the boundary of two media, and (b) transmission line analogy.

The magnitudes of both reflection coefficients are always less than one, which means that the intensity of reflected wave is always less than the intensity of the incident wave. This statement is called the reflection intensity loss or simply reflection loss. ΦΓE and ΦΓH are reflection phases for electric and magnetic field reflection coefficients, respectively. They represent the phase shift between incident and reflected waves. The initial phases ΦΓE and ΦΓH in (2.174) and (2.175) are always non-positive, which denotes the fact that the initial phase of the reflected wave always lags behind the initial phase of the incident wave. Therefore, the reflection phase is called sometimes a reflection phase loss. In other words, the phase of the reflected wave must never pass ahead of the phase of incident wave, as it would contradict the cause-effect principle. For this particular case the boundary conditions (2.172) and (2.173) may be written as:

E i + E r = E t (2.176)



 −H  =H  (2.177) H i r t and based on (2.72), written for either incident or reflected wave, the expression (2.177) may be transformed into



(E i − E r ) ε1 = E t

ε 2 (2.178)

By solving the system of equations (2.176) and (2.178) for the electric field reflection coefficient (2.174) for the normally incident wave, one may obtain Γ E norm =

ε1 −

ε 2

ε1 +

ε 2

(2.179)

As seen from this formula, the reflection coefficient ΓEnorm for the interface of lossless media becomes a real number, either positive (if e 1 > e 2) or negative (if

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 67

e 1 < e 2). Real positive ΓEnorm shows the decrease of the amplitude of the reflected wave relative to incident, while keeping the phase unchanged, (ΦΓE = 0). Real negative ΓEnorm shows the decrease of the amplitude and inversion of the initial phase (ΦΓE = 180°). A formula for ΓHnorm may be developed similarly as Γ H norm = − Γ E norm = −

ε1 −

ε 2

ε1 +

ε 2

(2.180)

Expressions (2.179) and (2.180) are similar to those in the transmission lines shown in Figure 2.23(b), where the voltage reflection coefficient Γ V relates to Γ Enorm , and the current reflection coefficient Γ I relates to Γ Hnorm ; that is, ΓV =

Z0 − ZL Z − ZL = −Γ I = − 0 (2.181) Z0 + ZL Z0 + ZL

If reflecting surface is a perfect electric conductor (PEC) then, by assuming e 2 → ∞ one may obtain form (2.179) and (2.180) Γ E norm = −1, and Γ Hnorm = 1 , which means that after reflection the electric field remains the same in magnitude, whereas the orientation becomes inverted. The magnetic field keeps both magnitude and orientation the same as for the incident wave.

2.4.3  Oblique Incidence on a Plane Boundary 2.4.3.1  Introductory Notes

As mentioned above, the arbitrary polarized radio wave may be decomposed into two LP waves of cross-orthogonal polarizations. After analysis of reflection of each component, the final result may be obtained by superimposing them to define the resultant wave structure. To analyze reflection in the case of oblique incidence, we use q and j variables of the spherical coordinate system (see Section 2A.2.1 in Appendix 2A).  We start with the normal incidence (qi = 0, considered in the previous section; Ei vector oriented along the Y-axis) and then increase qi continuously in two  cross-perpendicular planes: in the YOZ-plane of incidence (j = 90°), where the Ei vector of incident wave always remains parallel to that plane  as qi variates [Figure 2.24(a)], and in the XOZ-plane of incidence (j = 0), where Ei vector of incident wave always remains perpendicular to that plane [Figure 2.24(b)]. Two types of reflection coefficients are introduced below: •

•

 Γ || , for E vector parallel to plane of incidence [i.e., colinear with unit vector θ0 (see Figure 2A.1(b)];  Γ ⊥ , for E vector perpendicular to plane of incidence (i.e., colinear with unit

vector ϕ 0 ).

For further analysis, it is important to understand how to determine the sign of the reflection coefficient for both polarizations: the reflection coefficient for oblique

68

Basics of Electromagnetic Waves Theory

Figure 2.24  Graphical presentation of E-vector orientation in two principal planes, and for variable angles of incidence: (a) YOZ-plane, where the E-vector remains parallel to the plane of incidence, and (b) XOZ-plane, where the E-vector remains perpendicular to the plane of incidence.

incidence is considered positive, if, for the angle of incidence tending to zero (qi = qr → 0), the E-vectors of the incident and reflected rays become codirected, or negative, if the E-vectors become counterdirected. Examples of vector orientations of electric fields in a case of oblique incidence for positive and negative reflection coefficients are given in Figure 2.25. 2.4.3.2  Oblique Incidence of Parallel Polarized Radio Wave

  In this case electric  field vectors of all three rays ( Ei for the incident ray, E r for the reflected ray, Et for the refracted ray) will lie in the same YOZ-plane [(Figure 2.24(a)]. The cross positioning of all three vectors are depicted from Figure 2.25(a) or 2.25(c) in such a way, that it is relevant to positive Γ || . As shown below, the reflection coefficient(s) are dependent on the angle of incident, and may change the sign while the angle of incidence is swiping in 0 ≤ qi ≤ 90° range. Then, a reorientation of the reflected E-vector is to be arranged properly, based on Figure 2.25. The boundary conditions (2.172) and (2.173) in this case may be written as (see Figure 2.26):

E τ (1) = E i,y + E r,y = E i cosθi + E r cosθ r = E t ,τ = E t cosθt = E τ(2), (2.182)



 (1) = H  −H        (2) H τ i,x r,x = H t,x = H i − H r = H t = H t,x = Hτ . (2.183)

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 69

Figure 2.25  Examples of E-field vector orientations for positive and negative reflection coefficients and for different polarizations of the incident field.   E-vector directed to the observer, ⊗  E-vector directed from the observer.

For further evaluations, we must define cosy, which is found from (2.169a) as follows:

cos θt =

1−

ε1 2 sin θi (2.184) ε2

Figure 2.26  Positions of E-vectors for the parallel polarized electromagnetic wave of oblique inci   dence to the flat reflection boundary (Πi , Πr , and Πt are proper Poynting vectors).

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Basics of Electromagnetic Waves Theory

Now we may rewrite (2.182) and (2.183) with relations (2.79), (2.86), and (2.184) in mind.



(E i + E r ) cosθi = E t

1−

ε1 2 sin θi (2.185) ε2

(E i − E r ) ε1 = E t ε 2 (2.186) If this system is solved for reflection coefficient (2.174), then the result for the considering case of parallel polarization becomes



ε cosθi − ε1 ε 2 − ε1 sin2 θi Γ ||E = − 2 . (2.187) ε 2 cosθi + ε1 ε 2 − ε1 sin2 θi Consider the following two particular cases of reflection of the parallel polarized waves. 2.4.3.2.1 Radio Wave Incident from Sparse Medium onto the Border with Dense Medium (e1 < e2)

E For the small angles of incidence (qi), the value of Γ || in (2.187) is a real, negative number. The increase of the angle of incidence (qi) from zero up results in E E decrease of the magnitude Γ || = | Γ || |, which, as may be seen from (2.187) by taking the numerator equal to zero, tends to zero for the angle of incidence defined as



sinθi,0 =

ε2 (2.188) ε1 + ε 2

There is no occurrence of reflection at the angle qi,0; thus, it is called the angle of total refraction, or Brewster’s angle. Further increase of qi results in real positive values of Γ E|| within the range of the angles q i,0 >q i > 0. Hence the reflection phase ΦEΓ, jumps from 180° to 0° at qi,0. The patterns of angular dependencies of the magnitude and phase of reflection coefficient for this particular case are shown in Figure 2.27 (dotted lines). 2.4.3.2.2 Radio Wave Incident from Dense Medium onto the Border with Sparse Medium (e1 > e2)

If qi = 0 in this particular case, then the reflection coefficient is positive, and therefore the reflection phase is equal to zero. The increase of q i from zero up will result in same total refraction phenomenon at the same Brewster’s angle qi,0, defined by expression (2.188), as with the previous case. When the angle of incidence passes E through the value of qi,0, then Γ || turns the sign from positive to negative (i.e., reflection phase jumps from zero to 180°). Further increase of the angle of incidence (qi > qi,0) will result in another phenomenon that occurs at so-called critical angle, qi = qi,cr, when the expression under the square root in (2.187) becomes equal to zero. Then qi,cr is defined as

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 71

Figure 2.27  Magnitude and phase of the electric field reflection coefficients at 400 MHz and for the boundary of two ideal dielectrics with parameters e 1 = 1, e 2 = 3. [qi,0 = 60° from (2.188)].



sinθi,cr =

ε2 (2.189) ε1

For the values of angle of incidence qi > qi,cr, the expression under the square root sign becomes negative; thus, the numerator and denominator of (2.187) become complex conjugate relative to each other. E Therefore, for all q i > q i,cr, the magnitude of Γ || will remain equal to unity, whereas the reflection phase ΦEΓ, will smoothly decrease from 180° to 0°, as shown in Figure 2.28 (dotted lines). Hence qi.cr is called the total reflection angle.

72

Basics of Electromagnetic Waves Theory

Figure 2.28  Magnitude and phase of the electric field reflection coefficients at 400 MHz and for the boundary of two ideal dielectrics with parameters e 1 = 3, e 2 = 1. [qi,0 = 30°,qi,cr = 35°].

2.4.3.3  Oblique Incidence of Perpendicular Polarized Radio Wave

In this case, electric field E-vectors of all three rays lie in the same XOY-plane, which is perpendicular to the plane of incidence XOZ as shown in Figure 2.29. The orientations of electric field vectors are taken the way that relates to those presented in Figure 2.25 for the presumably positive reflection coefficient. The boundary conditions (2.172) and (2.173) may be rewritten in the following form:

 (2)  (1) = H  −H      H τ i,x r,x = H i cosθ i − H r cosθ r = H t ,τ = H t cosθ t = Hτ (2.190)



E τ (1) = E i,x + E r,x = E t,x = E i + E r = E t = E τ (2) (2.191)

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 73

Figure 2.29  Positions of vectors for the perpendicular polarized electromagnetic wave of oblique incidence to the flat reflection boundary.

Simplifications similar to those provided for the parallel polarized incident waves are applicable to (2.190) and (2.191) as well: • •

qt may be expressed in terms of qi using Snell’s law (2.169a); Replace magnetic field strengths by electric, using (2.72) in the following form: H = x0 ε1 /W0 E .

(

)

After combining (2.190) and (2.191) the ratio (2.174) for electric field reflection coefficient may be found as

Γ E⊥ =

ε1 cosθi − ε 2 − ε1 sin2 θi ε1 cosθi + ε 2 − ε1 sin2 θi

(2.192)

From (2.192) it may be seen that unlike the previous case of parallel polarization, the perpendicularly polarized radio wave can never satisfy the total refraction condition (i.e., the perpendicularly polarized radio wave will never be totally refracted (penetrate) into the second medium). Next we consider two particular cases of reflection of perpendicular polarized waves. 2.4.3.3.1 Radio Wave Incident from Sparse Medium onto the Border with Dense Medium: (e1 < e2)

E For all angles of incidence qi the value of Γ ⊥ represents a real and negative fraction, which urges to minus one, when qi becomes close to 90°. That is, the magnitude of Γ E⊥ smoothly increases, while the reflection phase remains constant and equal 180° for all values of angle of incidence qi as shown in Figure 2.27 (solid lines).

2.4.3.3.2 Radio Wave Incident from Dense Medium onto the Border with Sparse Medium: (e1 > e2)

In this case, the total reflection occurs at the same condition (2.189). Hence, for E the angles of incidence between 0 and qi,cr, the reflection coefficient Γ ⊥ remains

74

Basics of Electromagnetic Waves Theory

a positive fraction and tends to 1 when qi becomes close to qi,cr. The magnitude of Γ E⊥ tends smoothly to one, while the reflection phase remains equal to zero. For E the values of qi between qi,cr and 90° the magnitude of Γ ⊥ remains equal to one, while the value of reflection phase increases smoothly from 0° to 180°, as shown in Figure 2.28 (solid lines). Note that both expressions (2.187) and (2.192) will convert to (2.179) regardless of the sense of polarization if the normal incidence (qi = 0) is considered. 2.4.4  Power Reflection and Transmission

Electric field reflection coefficients for parallel and perpendicular polarized radio waves incident to the border of two dielectric media are given by (2.189) and (2.192). Magnetic field reflection coefficients may be derived similarly, based on the systems of equations represented by (2.185) – (2.186), and (2.190) – (2.191). If in those equations electric fields are replaced by magnetic fields using W1/2 = W0 / ε1/2 [see (2.79)], and then further evaluated (2.175) for both polarizations, it may simply be shown that Γ H = − Γ E (2.193)



For the transmission coefficients, the definitions are based on the following expressions for the electric and magnetic fields, respectively:



E i⋅Φ T E =  t = T E ⋅ e T E (2.194a) Ei



H i⋅Φ T H =  t = T H ⋅ e T H (2.194b) Hi Evaluation results for Γ E , Γ H , T E, and T H are provided in Table 2.3. To best evaluate reflection and transmission of the power of radio waves, we have to keep in mind that the power flow must be considered across the boundary in a direction normal to the border. The normal components of the incident, re  flected, and transmitted (refracted) waves (Πi Norm , Πr Norm and Πt Norm ) shown in Figure 2.30 must be taken into account. Power flow density at the border of lossless media may be defined from (2.87). So, for the origin of the coordinate system that is placed on the border z = 0, the power flow is defined as Π = Em2/2W (a = 0 and ΦW = 0), and an expression for the power balance right at the interface may be written as



ΠiNorm − Πr Norm = Πt Norm (2.195) Then (2.195) is rewritten as



Ei 2 E2 E2 cosθi − r cosθ r = t cosθt (2.196) 2 W1 2 W1 2 W2

2.4  Reflection and Refraction of Plane Radio Wave from the Boundary of Two Media 75

Electric Magnetic Electric Magnetic

Parallel polarization Perpendicular polarization

For electric and magnetic fields

Table 2.3  Complete Set of Reflection Coefficient Formulas

For power reflection and transmission for either parallel or perpendicular polarizations

Reflection coefficient

ε cosθi − ε1 ε 2 − ε1 sin2θi Γ E|| = − 2 ε 2 cosθi + ε1 ε 2 − ε1 sin2θi

Transmission coefficient

T E|| =

Reflection coefficient

ε cosθi − ε1 ε 2 − ε1 sin2θi Γ H|| = 2 ε 2 cosθi + ε1 ε 2 − ε1 sin2θi

Transmission coefficient

T H|| =

Reflection coefficient

Γ E⊥ =

Transmission coefficient

T E⊥ =

Reflection coefficient

ε cosθi − ε 2 − ε1 sin2θi Γ E⊥ = − 1 ε1 cosθi + ε 2 − ε1 sin2θi

Transmission coefficient

T H⊥ =

Reflection coefficient

Γ P = Γ E

Transmission coefficient

T P = T E

2 ε1 ε 2 cosθi

ε 2 cosθi + ε1 ε 2 − ε1 sin2θi

2 ε 2 cosθi

ε 2 cosθi + ε1 ε 2 − ε1 sin2θi ε1 cosθi − ε 2 − ε1 sin2θi ε1 cosθi + ε 2 − ε1 sin2θi 2 ε1 cosθi

ε1 cosθi + ε 2 − ε1 sin2θi

2 ε 2 cosθi

ε1 cosθi + ε 2 − ε1 sin2θi 2

2

ε 2 − ε1 sin2θi ε1 cosθi

Figure 2.30  Power flow directions for the oblique incidence of the radio wave onto the border of dielectric media.

76

Basics of Electromagnetic Waves Theory

where W1 and W2 indicate the intrinsic impedances of first and second media, respectively. Now we may define the power reflection and transmission coefficients as ΓP =





TP =

Norm t Norm Π i

Π

=

Er2 E i2

ΠNormr Norm Π i

=

E r2 Ei

2

2

= Γ E (2.197)

ε 2 cosθt = TE ε1 cosθi

2

ε 2 − ε1 sin2θi ε1 cosθi

(2.198)

Hence, taking into account (2.197) and (2.198), the expression (2.196) may be rewritten as

1− Γ P = T P (2.199) Note that (2.199) represents the power balance that may be easily verified by direct substitution of ΓE and TE for either parallel or perpendicular polarizations into (2.197) and (2.198), and further into (2.199). Table 2.3 summarizes analytical expressions that relate to these cases. 2.4.5 Reflection of the Radio Wave from the Boundary of Nonideal Dielectric Medium

This case is specific for radio waves propagation conditions. For the RF frequencies, the soil, freshwater, and seawater, as well as ionosphere, behave as semiconductors; therefore, in (2.189) and (2.192) one or both ε1 and ε 2 must be considered as complex number defined by (2.48). It may be seen that for nonideal dielectric boundaries a phenomenon such as total reflection or total refraction will not purely appear due to the presence of the power losses. An example of the angular dependence of the reflection coefficients for the real conditions is shown in Figure 2.31, obtained from (2.189) and (2.192) by using numerical evaluation.

2.5 Radiation from Infinitesimal Electric Current Source: Spherical Waves For the spatial area that contains time-harmonic source, the first two Maxwell’s equations may be written as

   ∇ × H = iωε0ε ⋅ E + JExt , (2.200)



  ∇ × E = −iωµ0 µ ⋅ H , (2.201)  where electric current density, JExt is induced by the external source, and is distributed within a limited volume V. Regarding Poynting theorem (2.16), the energy of

2.5  Radiation from Infinitesimal Electric Current Source: Spherical Waves 77

Figure 2.31  The angular dependencies of magnitude and phase of reflection coefficient for the reflection of parallel and perpendicularly polarized radio wave of frequency of 400 MHz from soil with the following parameters: e = 3, s 2 = 0.03 S/m

the source is spent not only to increase the energy stored by electric and magnetic fields, and dissipated in form of a loss within the area V, but also generates an outgoing electromagnetic radiation. This radiation may exist if last integral in (2.16) is non-zero.23 Solution of the systems (2.200) – (2.201) for the radiated field may not be obtained directly, meaning if we try to turn the two first-order linear differential equations for two unknowns into a single differential equation of the second order   *  23. The integral in complex form  ∫S E × H d S may become equal to zero because of boundary conditions on the closed surface S. If, for instance, the volume V is surrounded by the PEC, then regarding (2.39) the tangential component of the electric field becomes equal to zero, which results in zero value of the above integral.

(

)

78

Basics of Electromagnetic Waves Theory

for one unknown, then generically it becomes unsolvable. The main idea of the commonly used indirect approach for obtaining the result is to introduce so called  auxiliary magnetic potentials (i.e., vector potential A, and scalar potential Φ).24 Introduction of these auxiliary functions allows for obtaining solutions for the fair number of the radiation problem. As an example, the radiation from the linear electric current of the infinitesimal size is considered in Appendix 2B. Results of the solution are presented by expressions (2B.25) and (2B.30). If time-harmonic multiplier exp(iwt) is included into those analytical results, then (2B.25) and (2B.30) may be rewritten as follows: •

For the magnetic field:  )]  Il  exp[i (ω t − kr H= ik sinθ ⋅ ϕ 0 (2.202) 4π r

•

For the electric field:  )]  Il exp[i (ω t − kr E= iω µ0 µ sinθ ⋅ θ0 (2.203) 4π r



In these two expressions, we assume that the electric current filament, I with uniform current distribution is placed symmetrically at the origin of the spherical coordinates and directed along the z-axis as shown in Figure 2.32. The length of the current filament is assumed to be infinitesimal (i.e., l L. More detailed analysis of the ring-segment diffractor may be found in [8]. 2.6.4.3 Effective Area of the Radio Wave Reflection from Flat Boundary

In Section 2.4, the reflection phenomenon was considered based on the ray concept of propagation (i.e., the reflection was assumed occurring from an infinitely small point C). In reality, as mentioned previously, a finite spatial volume, limited at least by the first Fresnel zone, is involved in propagation; thereby, the reflection must be considered not from the C spot, but from the surrounding flat area, defined as an intersection of the Fresnel zone with a flat boundary of two media (see the shaded area in Figure 2.44). If Figure 2.36 and the expression (2.219) are recalled, then we may realize that to enable the field intensity of the reflected wave so is the same as for the infinite reflecting boundary, the cross-sectional (transversal) size b of the intersection between Fresnel’s ellipsoid and the reflecting boundary (Figure 2.44) must be taken as

b ≥ R1 / 3 =

λ r10 r20 (2.247) 3 r0

2.6  Spatial Area Significant for Radio Waves Propagation 97

Figure 2.44  Configuration of the area effective for the reflection from a flat boundary of the Earth.

The longitudinal size of the effective reflection area may approximately be evaluated from the geometrical sketch, shown in Figure 2.44 as a≈



b r0 b ≈ (2.248) sinγ h1 + h2

if it is assumed, that the elevation angle is small enough (i.e., g 1 then z n, q » z n, q =¥ + (4.164) q z n

As follows from the right-hand-side of (4.164), if q > 30, then for practical calculations the value of z n may be chosen approximately the same as for q = ¥ (i.e., z n, q » z n, ¥). Based on data provided in Table 2.1 (see Chapter 2), we may confirm by simple calculations that for any type of real Earth’s ground the statement qhoriz = ¥ is always true for the horizontally polarized wave of any RF frequency. On the other hand, if |q| < 0.5, then we may choose the value of z n,q » z n,0, which directly relates to q = 0. It is thereby evident from (4.160) and (4.163) that this case may occur for vertically polarized waves only. Figure 4.32 demonstrates the dependence of parameter q on frequency for two polarizations and for radio waves propagating along different types of Earth surfaces. As one may conclude from these graphs, for horizontally polarized radio waves the values of z n may always be chosen from the second column of Table 4.7, regard­ less of the frequency and type of the Earth’s ground along the propagation path (i.e., z n = z n, q =¥). For the vertically polarized radio wave, some caution is needed, as: • •

The assumption z n = z n, q =¥ is applicable for frequencies higher than 500 MHz. In the case of sea water and wet soil, the assumption z n = z n, q = 0 (last column of Table 4.7) is applicable for frequencies less than 50 kHz.

4.3 Asymptotic Diffraction Theory of Propagation over the Spherical Earth Surface 197

Figure 4.32  Frequency dependence of parameter q for propagation along seawater (e = 80, s = 3 S/m), wet soil (e = 30, s = 0.03 S/m), and dry soil (e = 3, s = 10-5 S/m).

•

For intermediate frequencies, (4.163) and (4.164) must be used for z n estimates.

Within the illuminated zone 1 (see Figure 4.31), the series (4.158) converges very slowly; thus, the calculation of the propagation factor becomes either highly time/effort consuming or even impossible. Therefore, it is meaningless to use it instead of the interference formulas given in Sections 4.1 to 4.3. However, in the shadow zone it converges fairly fast; thus, any term higher than the first one may be ignored and a single-term (sometimes called single-mode) formula may be adopted as follows:



F =2 πX

exp(i X z1) z1 - q 2

w (z1 - y1) w (z1)

w (z1 - y2 ) w (z1)

(4.165)

This expression represents a main achievement of Fock’s approach known as the outcome of the asymptotic diffraction theory (ADT). Expression (4.165) contains three separate terms. The first term,



U (x) = 2 π X

exp(i X z1) z1 - q 2

(4.166)

is known as an attenuation factor. It depends only on normalized distance X. The second and third terms are symmetric and may be written in the general form of (4.167). They depend only on the transmitting and receiving antennas’ elevations y1 and y2, respectively; that is,



V (y1, 2 ) =

w(z1 - y1, 2 ) w(z1)

(4.167)

Those terms are called height-gain function. So, in general, for the given value of q, the single-term propagation factor may be expressed as a product:

198



Impact of the Earth Surface on Propagation of Ground Waves

F (X, y1, y2 , q) = U (X, q) V (y1) V (y2 ) (4.168) The fact that both height-gain functions are counted in (4.168) symmetrically comes from the principle of reciprocity in electromagnetics and confirms the cor­ rectness of all statements of the Fock’s asymptotic diffraction theory. 4.3.2  Propagation Between Ground-Based Antennas

As noted previously, the term ground-based antenna signifies y1,2 f02, where f01 and f02 are derived from (5.60) by taking it equal to zero: 2



f  f f01 =  H  + fc2 − H 2 2  

(5.62)

2



f02

f  f =  H  + fc2 + H 2 2  



(5.63)

Those expressions are the same as (5.55) and (5.56).

Figure 5.17  Frequency dependence of a dielectric constant for ordinary and extraordinary rays for transverse-propagation in magneto-active ionospheric plasma.

5.5  The Impact of the Earth’s Magnetic Field on Propagation of the Radio Waves 255

Some properties of transverse-propagation in magneto-active plasma are as follows: 1. During propagation in magneto-active plasma, the extraordinary ray generates an additional electric field component Ez along the propagation direction, which, according to some estimates provided in Appendix 5C.3, as well as in [4–8], is 90° phase shifted relative to Ey and is much smaller by magnitude than the initial component Ey. If the extraordinary ray is the only one that propagates in considered ionospheric plasma-medium, then the superposition of those two electric field components, Ey and Ez, will result in elliptic polarization of the wave, with polarization ellipse lying in Y0Z plane. 2. Similar to the previous case of longitudinal propagation, in this case of transverse propagation, magneto-active plasma exposes different properties (eord and eext) for ordinary and extraordinary rays. Therefore, each of those two components of linearly polarized radio waves will propagate with different velocities, so at the end of propagation distance, an extra phase shift between them will occur: ∆Φ =



ω r c

(

ε ord − ε ext

)

(5.64)

The total field strength, as a superposition of those two components: E = x0 Ex + y0 Ey



(5.65)

may have different polarization for different values of DF (see Section 2.3). While moving along the medium, one polarization type may smoothly be transformed into another, depending on distance covered. It must also be noted that the sense of polarization may become even more complicated if the Ez component is taken into account in (5.65). 3. When the frequency of the radio wave becomes close to resonant, f ≈ f∞, then similarly to the case of longitudinal propagation one of the components of the arbitrary polarized wave, namely the extraordinary component, will experience an intensive attenuation and may disappear almost completely, being absorbed by the medium. In other words, the magneto-active plasma medium may be considered a polarization filter at that particular resonant frequency. 5.5.3 Propagation of the Radio Wave Arbitrary Oriented Relative to the Earth’s Magnetic Field

Consider the general case when the Pointing vector of the radio wave has an arbitrary g angle with the Earth’s magnetic field H0. Let’s decompose the magnetic field vector into longitudinal

HL = H0 cos γ

(5.66)

HT = H0 sin γ

(5.67)

and transversal

components, as shown in Figure 5.18.

256

Atmospheric Effects in Radio Wave Propagation

Figure 5.18  Disposition of Earth’s magnetic field into transversal and longitudinal components for an arbitrary direction of propagation in magneto-active ionospheric plasma.

For the previous case of transverse propagation, the radio wave splits into two linearly polarized waves: ordinary (Ex) and extraordinary (Ey). Without restrictions to the generality, we may assume that the magnetic field H0 lies in XOZ plane; thus, for the linearly polarized plane wave propagating along Z axes, the Ex component is concurrent with the direction of HT , and Ey is perpendicular to it. Thus, Ex occurs as an ordinary component of the radio wave, and Ey, as an extraordinary component. Regarding to longitudinal HL field, the primary wave may be decomposed into RHCP (E+) and LHCP (E-) circularly polarized components. Superposition of one of the linear components (Ex, for instance) with one of the circular components (E-, for instance) will result in a left-hand elliptically polarized wave E1, propagating along 0Z axes8 and with the main axes of the polarization ellipse coinciding with HT . Similarly, the superposition of Ey with the other circular component E+ will result in a right-hand elliptically polarized wave, E2 , with the main axes of the polarization ellipse orthogonal to HT and with same propagation direction (see Figure 5.19). Electric field vector of each elliptically polarized wave may be represented as E1 = Eord + E−   E2 = Eext + E+  



(5.68)

For this general case the dielectric constant of magneto-active plasma is defined by the following formula [2–5, 8]: 2fc2

ε1,2 = 1 − 2f 2 −

1−

fT2 fc2 /

  fT4 ±  f 2  1− f 2 / f 2 c 

(

1/ 2

)

2

 2 2 + 4 f fL   

(5.69)

Here e1 and the “+” sign in front of the brackets relate to the propagation of the E1 wave, while e2 and the “–” sign relate to the propagation of the E2 wave. Two other notations in (5.69) are

8.

fT =

e µ0 H 2π me T

It is assumed that in general the magnitudes of components are not equal to each other.

(5.70)

5.5  The Impact of the Earth’s Magnetic Field on Propagation of the Radio Waves 257

Figure 5.19  The pattern of two elliptically-polarized waves, generated in magneto-active plasma by primary wave. In this general case the Earth’s magnetic field is arbitrarily oriented relative to the direction of propagation of the primary radio wave.

and fL =



e µ0 H 2π me L

(5.71)

Expression (5.69) may be reduced considerably in two extreme cases. •

Case 1: fT4 fc2  2

 1 − f  



•

2

(5.72)

which is satisfied under the condition f ≈ fc ; it usually takes place in HF or MF bands. Then (5.69) may be simplified down to (5.59) and (5.60), so thus, the propagation conditions become the same as for transverse propa­ gation, regardless of the value of the g angle. In other words, the ray will split into two cross-orthogonal linearly polarized rays, which propagate independently (ordinary and extraordinary). Therefore, this case is often called quasi-transverse propagation. Case 2: fT4



>> 4 f 2 fL2

fc2  2

 1 − f  

2

> fc ; this is typical for UHF and higher frequencies. Then (5.69) may simplified down to (5.53) and (5.54), so thus, the propagation conditions become the same as for longitudinal propagation, regardless of the value of the g angle; the ray will split into two circularly polarized components, which propagate independently. This case is called quasi-longitudinal propagation. Note, finally, that after transformations of (5.69), the values fL or fT must be kept the same as from (5.70) and (5.71), respectively, instead of fH. 5.5.4 Reflection and Refraction of the Radio Waves in the Magneto-Active Ionosphere

The presence of Earth’s magnetic field modifies the patterns of reflection and refraction of the radio waves in ionosphere, due to dependence of the parameters of the medium on propagation direction and polarization of the radio wave. This is a typical case of propagation in an anisotropic medium. Consider the oblique incidence of the radio wave on the boundary of magneto-active ionosphere. The incident wave splits into two independently propagating waves, as discussed previously (see Figure 5.20). The reflections from the ionosphere are generally used for HF long-distance propagation (HF long-range radio links), so the radio wave frequency is usually close to fc. Based on (5.72) for those links, the reflection may be considered for the quasi-transverse propagation condition. At the input of the ionospheric layer, the incident wave, radiated from point A, splits into two independent linearly polarized cross-orthogonal waves: ordinary and extraordinary. The reflection condition (5.37) for the ordinary ray may be rewritten here as

f ≤ fmax = sec ϕ 80.8 ⋅ Ne, ordinary

(5.74)

where Ne,ordinary is the ionospheric plasma concentration needed to satisfy reflection condition for the ordinary ray. Now rewrite the reflection condition (5.37) for the extraordinary ray, which may appear hypothetically in two cases:

f ≤ f0,1 sec ϕ

(5.75)

Figure 5.20  Split of the radio wave in magneto-active ionosphere: (a) reflection pattern for the extraordinary (1), and ordinary (2) rays, and (b) refraction pattern for LHCP (1) and RHCP (2) rays.

5.5  The Impact of the Earth’s Magnetic Field on Propagation of the Radio Waves 259

and

f ≤ f0,2 sec ϕ

(5.76)

where f0,1 and f0,2 are defined by (5.62) and (5.63), respectively. Due to the relation f0,1 < fc < f0,2, the expression (5.76) is easier to satisfy, so we may ignore condition (5.75). Then the MUF for the extraordinary ray may be found from (5.76) and (5.63) as



fmax = sec ϕ  80.8Ne,extraord + (fH / 2)2 + fH / 2   

(5.77)

The same signal frequency and the same angle of incidence are considered for both ordinary and extraordinary rays. Hence, taking (5.74) and (5.77) equal to each other, the following expression may be written here:

80.8Ne,ordinary = 80.8Ne,extraord + (fH / 2)2 + fH / 2

(5.78)

The expression (5.78) shows clearly the following relation between ionospheric plasma concentration needed to satisfy the reflection conditions for the ordinary and extraordinary rays:

Ne,extraord

< Ne,ordinary

(5.79)

The reflection heights of both rays may be found based on a vertical profile of the ionospheric plasma concentration Ne(h): either on experimental profile shown in Figure 1.2, or on widely used parabolic model shown in Figure 5.21. It may be seen from the Figure 5.21 that the reflection point for an ordinary ray is higher than for an extraordinary (i.e., h2 > h1). It must be noted that the absorp­ tion rate as well as the dispersion rate for the extraordinary ray is much higher than that for ordinary ray. The structure of a geomagnetic field is such that for the low and medium latitudes, the Earth’s magnetic field lines are close to horizontal

Figure 5.21  Reflection heights definition for ordinary and extraordinary rays, based on a vertical profile of the parabolic model of ionospheric plasma concentration.

260

Atmospheric Effects in Radio Wave Propagation

and directed from north to south. That’s the reason the ordinary ray has a horizontal polarization; therefore, the horizontally polarized waves are preferable for HF communication lines design. At the higher latitudes, namely, for the Earth’s polar zones, HF communications are rarely used due to the high absorption rate (abnormal absorptions) in those areas. Considering the refraction of the ray path in the ionosphere (e.g., on Earth-tospace or space-to-Earth links), it must be emphasized that for frequencies higher than VHF, the radio wave penetrates through the ionosphere, and is shifted from its initial propagation direction due to refraction, as shown in Figure 5.20(b). From the general expression (5.69) for dielectric constant of magneto-active ionospheric plasma, it is clear that the impact of the medium will be lower if the difference between the currier frequency of radio wave and the cutoff frequency is greater; that is, f >> fc



(5.80)

Under this condition: 1. The dielectric constant becomes close to one, so the ray path will become close to the straight line. 2. Ionospheric absorptions will decrease because according to (5.9b) the conductivity of the medium will decrease. 3. The polarization disturbances and signal dispersions will also decrease. While designing the Earth-to-space (or space-to-Earth) radio links, the condition (5.80) is to be met as much as possible. As mentioned previously, the frequencies VHF and higher are preferably to be used for Earth-to-space or space-to-Earth propagation paths, which relates to the quasi-longitudinal propagation [see condition (5.73)]. If the linearly polarized radio wave propagates vertically through the ionosphere, then the Faraday’s rotation may be expected. The random rotations in the polarization plane of the LP signal may result in polarization losses at the reception point due to polarization mismatches between Tx and Rx antennas. If the propagation track is sloped, then most likely just one of the circularly polarized (CP) components of the linearly polarized (LP) radio wave will arrive at the destination due to different paths for the CP components, as shown in Figure 5.20(b). Hence the usage of LP radio wave may result in loss of 50% energy (3 dB). To avoid those losses, a CP radio wave is considered more preferable for satellite radio links such as satellite communication, navigation, and broadcast.

5.6 Specifics of Ionospheric Propagation of ELF and VLF in the Earth-Ionosphere Waveguide 5.6.1 General Remarks

The lowest ionized layers, D-layer, which exist in daytime only, and E-layer have significant contribution in propagation of the radio waves from ELF (less that 3 kHz) to VLF (up to 30 kHz). The propagation mechanism is due to the reflective properties of the lower ionospheric layers and the earth’s surface for waves in these

5.6 Specifics of Ionospheric Propagation of ELF and VLF in the Earth-Ionosphere Waveguide261

frequency ranges; the area between them (between ionosphere and the earth), serves for channeling of the energy of radio waves, and contributes to guided propagation over significant distances. Recall expression (5.9a/b) for permittivity and conductivity of nonmagnetoactive ionospheric plasma:9

ε = 1−

σ = ε0

ω c2 ω 2 + ξ2 ω c2ξ ω 2 + ξ2

    , (5.81)   

Here e = 1.6 ⋅ 10–19C, me = 9.1 ⋅ 10–31 kg, e0 = (1/36p) ⋅ 10–9F/m, and

ωc =

Ne e 2 (5.82) ε 0 me

is a plasma (Langmuir) frequency. Based on data provided in Table 5.1, it’s easy to confirm that for ELF and VLF the expressions (5.81) may be reduced to



ω  ε = 1−  c   ξ 

ω2 σ = ε0 c ξ

2

    , S/m   

(5.83)

for both, D- and E-layers. Calculations are shown below, in Table 5.4. It may be seen from Table 5.4 that the D-layer of the ionosphere acts as nonreflecting (permittivity is close to unity), whereas the E-layer reflects those radio waves with reflection losses that are comparable to Earth’s ground reflection losses. In other words, the area between E-layer and the Earth’s ground serves as a spherical waveguide for the wavelengths that are comparable with the height of the layer or even greater. Geometry of this waveguide is shown in Figure 5.22. Note that it is easy to assess the depth of penetration (skin depth) of the radio wave into the reflecting layer as well as into ground by using the formula inverse to (2.94), namely δ c = 1 α = 2 ωµ0σ . If typical values are substituted, then one may end up with penetration depth of 20 to 500 meters. In other words, the surfaces may be considered as lossy reflectors, and the area between ionosphere and Earth’s surface may be considered as a propagation region between reflecting spherically shaped concentric surfaces. We’ll start our analysis by considering the ideal case, when the surfaces are made of PEC, and there are no losses due to atmospheric absorptions between those surfaces. If assumed that the equivalent isotropic radiated power EIRP = PTx GTx (PTx is radiated power, and GTx is antenna gain) is uniformly distributed along conical surface of median radius r shown in Figure 5.22(b), then the value of the Poynting vector (power per unit surface) may be defined as

9.

The impact of the Earth’s magnetic field on the properties of the Earth-ionosphere waveguide is negligible.

262

Atmospheric Effects in Radio Wave Propagation

Table 5.4  Conductivity and Permittivity of Ionospheric D- and E-Layers Ionospheric Layer D (day only) E (day) E (night)

h, height [km]

x [1/s] 7

60 to 90

10

100 to 140

105



wc [rad/s]

s [S/m]

5

6

5.6 · 10 – 1.8 · 10 1.8 · 106 – 3.78 · 107 4 · 106 – 5.6 · 106

Π=

e –7

–6

2.77 · 10 – 2.86 · 10 2.86 · 10–4 – 1.26 · 10–1 1.4 · 10–3 – 2.77 · 10–3

0.97 – 0.99 Negative Negative

P TxGTx (5.84) S

where S is the area of conical surface in Figure 5.22(b), and may approximately be found from the geometry as   h S ≈ 2π h  a +  sinΘ  (5.85) 2  



Then, per definition (3.40), the propagation factor may be calculated from

Π=

P TxGTx P G = Π0 F 2 = Tx 2Tx ⋅ F 2 (5.86)   h 4π r 2π h  a +  sin Θ    2   F=



2r  h a + 

h sinΘ 2 

(5.87)

In (5.86) P0 is magnitude of the Poynting vector in free space, and r is the horizontal curvilinear distance to observation point shown in Figure 5.22(a). Taking into account (3.39), the expression for electric field may be written as

Eideal = E0 F =

30PTxGTx F= r

60PTxGTx  h h  a +  sin Θ 2 

V ,   (5.88) m

Figure 5.22  (a) Earth-ionosphere waveguide, general view (not to scale), and (b) 3D image of the conical surface of the height CD = AB = h, which is normal to the Poynting vector at the fixed distance r from transmit point.

5.6 Specifics of Ionospheric Propagation of ELF and VLF in the Earth-Ionosphere Waveguide263

Figure 5.23  Calculated RMS ELF electric field dependence on curvilinear horizontal distance along the great circle for the following propagation conditions: f = 300 Hz; current at the base of vertical monopole, Ieff = 40 A; height of the monopole; hant = 50 m; height of the reflecting E-layer (nighttime) = 100 km, conductivities, sion = 2 ⋅ 10–3 S/m, sg = 10–3 S/m for ionosphere and Earth’s ground, respectively.

where E0 is free space (reference) electric field. Distance dependence of the calculated electric field for this ideal case is presented in Figure 5.23 (dotted line). In (5.88) the expression Θ = r/(a + h/2) is used to transform geocentric angle Θ to distance. For the ELF and VLF the height of the monopole antenna, hant (as well as its effective length) is always much less than the wavelength, leff ≈ 2hant fc,n = c/lc,n, and c is the speed of light. The expressions for field components may be written as  nπ x  − ikz z E x,n = E x,max ,n cos  e  h 

 nπ x  − ikz z E z,n = E z,max ,n sin  e  h   nπ x  − i kz z H y,n = H y,max ,n cos  e  h 

     (5.101)    

It is to be noted that (5.98) – (5.100) are applicable to TEn modes as well, whereas the field components are  nπ x  − ikz z E y,n = E y,max,n sin  e  h   nπ x  − ikz z H x,n = H x,max,n sin  e  h 

 nπ x  − i kz z H z,n = H z,max,n cos  e  h 

     (5.102)    

It is important to emphasize that the case n = 0 is applicable to transverse-magnetic modes only, resulting in non-zero and x-independent components Ex and Hy whereas all other components disappear. As follows from (5.99), the cutoff wavelength and frequency are lc,TEM = ∞, and fc,TEM = 0, respectively, which is typical for TEM waves in any other guiding system and in free space. It’s obvious that among transverse-electric modes the case n = 0 is meaningless because all transverse components (5.102) disappear, resulting in the disappearance of the field itself. Figure 5.25 shows the frequency ranges where the TMn and TEn modes are able to propagate, if the height of ionospheric reflecting layer is h = 100 km. Attenuation coefficients may be calculated based on the following expressions [6, 17]:

  k   1 ωε0  k  α TM,n = 2  0  ⋅  ⋅ Σ  = 2  0  ⋅ α TEM (5.103)   kz,n   2h 2  kz,n 



 k2   1 ω ε  k2   0 α TE,n = 2  c  ⋅  ⋅ Σ  = 2  c  ⋅ α TEM (5.104) 2   k0kz,n   2h  k0kz,n  where Σ is defined by (5.94),



kz,n k0 = cos γ n =

nλ 0 2h

(5.105)

268

Atmospheric Effects in Radio Wave Propagation

Figure 5.25  Frequency ranges supporting propagation of different modes of ELF and VLF in Earth-ionosphere waveguide for the height of reflecting layer, h = 100 km.

2 kc,n

2

2h  kc,n  = (5.106) kz,n k0 n λ 0  k0 



kc,n = 2p/lc,n = kx,n is the cutoff wave number, k0 > kc,n is the free space wave number shown in Figure 5.24, and kz,n = k0 cos gn is phase coefficient. Note that attenuation coefficient for TEM mode may be derived from (5.97), if taken n = 0 and therefore kz,0 = k0: 1 1 ω ε0 α TEM = α TM,0 = ⋅ Σ (5.107) 2 2h 2



which is the same as that from (5.93). It’s important to note that phase coefficient for TEM mode, b = kz,0 = k0, is taken to be equal to that in free space because the losses are ignored; expression (5.92) more precisely accounts for the losses in ionosphere and Earth’s ground. Finally, the effective (RMS) electric field of either TMn or TEn mode may be calculated if the proper value for ideal case (5.88) of nonabsorptive propagation is multiplied by exponential term, e–ar; that is

E=

60PTxGTx  h a + 

h sin Θ 2 

exp(−α r) =

170π Ieff hant  h λ h  a +  sin Θ 2 

V exp(−α r)   (5.108) m

Here a is defined by one of expressions (5.103), (5.104), and r is a distance along the great circle between corresponding points. It’s easy to realize that (5.108) takes into account the divergence of the energy of the radio wave while propagation within the first quadrant of the great circle, and its convergence within the second quadrant, all the way to the antipode distance. Unfortunately, the expression (5.108) may not be used for precise calculations of the electric field, but for qualitative, rough estimates only. That is because, as mentioned above, the analysis was based on the plane wave propagation concept and limitations caused by geometrical optics approach. In the LF (and partially in VLF) frequency band the following empirical formulas for electric field RMS values for the distances greater than 2,000 km are developed by Austin and Cohen; these developments are based on series of experiments carried out for the former U.S. National Bureau of Standards (currently NIST) [8]:

5.7 Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 269

Figure 5.26  Calculated distance dependencies of the cumulative RMS electric field strength of LF and VLF frequency bands for the height of reflecting layer, h = 100 km.



Erms =



Erms =

300 PTx,kW GTx rkm 120π hant,m Ieff,A

λm rkm

Θ  0.0014 rkm   mV  exp  − ,  (5.109) 0.6  sin Θ  m  λ  km  Θ   mV   0.0014 rkm ,  exp  −  (5.110) 0 .6 sin Θ λkm   m  

Here the units are shown in subscripts of the variables. Note that (5.109) and (5.110) count for the cumulative, total electric field, which is composed of the variety of different propagation modes typical for LF and partially for VLF. Examples of these dependences are shown in Figure 5.26. The graphs in Figure 5.26 represent median signal levels. In reality deep fluctuations (fades) of the signal are being observed. This is caused by random interference between composite modal component of the cumulative signal, as well as by fluctuations of the height of reflecting layer, especially when between the observation points along the arc of the great circle the time of day changes from day to night (or vice versa).

5.7 Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 5.7.1  Secondary Tropospheric Radio Links

Although the asymptotic diffraction theory examined in Section 4.3 appears to be a powerful tool for radio wave propagation predictions in the shadow region behind

270

Atmospheric Effects in Radio Wave Propagation

the horizon line, the only way of counting the impact of the atmosphere within ADT concepts is the introduction of the Earth’s equivalent radius, which takes into account only a smooth standard atmospheric refraction. ADT prediction results are in fairly good agreement with measurements in relatively low frequency bands up to HF. For VHF frequencies and up, ADT predicts the abrupt decrease of field intensity for over-the-horizon distances (i.e., in the diffraction zone), whereas numerous observations performed after World War II have shown much higher field intensities than those predicted by ADT. Figure 5.22 demonstrates qualitatively the dependence of the field intensity on wide range of distances that are predicted by ADT as well as observed in real conditions at VHF and higher frequencies: in fact, the higher the frequency is, the sharper the predicted field decays in the shadow zone. In the mid-1930s, several pioneering measurements were carried by G. Marconi and other investigators, which have shown a considerable difference (up to 100 decibels) compared to diffraction theory predictions. One of the features of the received field is very deep fast fading of the signal level at the reception point. In the late 1940s, a formal explanation as well as fairly good quantitative description of this phenomenon was given by H. Booker and W. Gordon [9] with the later updates by V. Tatarskii and S. Rytov [10]. The basic explanation of the nature of fairly intensive field at the receiving point for over-the-horizon distances is as follows: •

•

In the presence of real atmosphere, the existence of the field in shadow area is a result of scattering of the primary radio wave from random irregularities of the tropospheric air known as turbulences. Effects such as twinkling stars and the wavering appearance of objects observed over the Earth’s surface heated by the sun testify that the atmospheric

Figure 5.27  Distance-dependence of field strength for VHF/microwave propagation (a qualitative pattern). R0 = horizon distance, 1 = theoretical prediction from ADT, 2 = real measurements (averaged curve).

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 271

•

•

•

•

•

air is in random, erratic movements that exist permanently, regardless of weather condition, geographic region, and season. Statistically, those irregularities appear as spherical volumes (i.e., globules, eddies) with the refraction index slightly different from the surroundings; those globules exist permanently in troposphere, even in clean air conditions. Typical dimensions of those globules that are most intensively involved in scattering of the UHF and higher band frequencies vary from several millimeters to hundreds of centimeters; thus, maximum effectiveness of those reradiations must be expected in a range of the wavelengths comparable to the range of those dimensions, which typically belong to microwave frequency bands. Although a scattering from one single globe may never be sensed, the superposition of the huge amount of those scattered fields, called random ensemble or Rayleigh ensemble of waves, causes fairly sensible field strength at the observation point. A number of those scattering globules is unpredictably large and is defined by the common volume between transmitting and receiving antenna beams; the common volume is located at heights from about 1 to 8 km above the surface. Deep fading of the received signal, which is specific for the troposcatter propagation, is a result of random interference of the scattered (elementary) fields at the reception point (i.e., is a result of constructive and destructive interference of those elementary fields due to random phase shifts between them).

5.7.2 Analytical Approaches in Description of the Random Tropospheric Scatterings

Assume that the dielectric constant e of the troposphere is a variable that randomly fluctuates in time and space. The time variations of e are much slower than the rate of oscillations of a propagating radio wave. Thus, it may be assumed that within several periods of oscillations of the radio wave, the random spatial distribution e remains unchanged (i.e., the randomness occurs only as a function of coordinates). In other words, only the random spatial fluctuations around the spatial average, 〈e〉, may be accounted to simplify the analysis; that is,

ε (r ) = ε + ∆ε (r )

(5.111)

where De is a randomly fluctuated part of dielectric constant, and r = x0 x + y0 y + z0 z is the radial vector-coordinate of the observation point. This time-independent model of frozen turbulences is quite convenient for analytical evaluations instead of a time-dynamic approach, and, as follows from the numerous of experiments, the final results are close enough to those observed. For further simplifications, we may assume 〈e〉 = 1, which is acceptable for the tropospheric scatterings analysis. Recall (2.5), which constitutes a relation between vectors of the electric field strength (E ) and its induction (D):

D = ε0ε E = ε0 (1 + ∆ε )E = ε0 E + P

(5.112a)

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Atmospheric Effects in Radio Wave Propagation

Here P, by its classical definitions, is a microscopic vector of polarization of the medium at a particular point of space within a unit volume. From (5.112a), P = ε0∆ε E



(5.112b)

If referred to Figure 5.28, we realize that time-harmonic polarization P of the medium at point C is forced by electric field E incident from transmission point A. In other words, the field, scattered from point C (secondary radiation), may hypothetically be presented as a primary field radiated by the dipole with polarization moment: dp = ε0∆ε EC dVsc (5.112c)



Here dp is a resultant dipole moment from all polarized particles (i.e., atoms, molecules) induced by the incident electric field EC within elementary volume dVsc. The following vector quantities are introduced to support further analysis (see Figure 5.28): • •

k0 the wave vector of the incident wave, which is directed orthogonal to in­­ cident wavefront; its magnitude is equal 2p /l0. r1 is the radius-vector of the elementary volume dVsc. It is directed from the transmitting point A to the point where the elementary volume dVsc is located

Taking into account the previously introduced quantities, the expression (4.1) for the amplitude of the incident field at point C may be rewritten as



(

)

60PTxGTx exp i ω t − k0 ⋅ r1  E C =   r1

(5.113)

Figure 5.28  Positioning of vectors for tropospheric scattering analysis. (Origin of the coordinate system is located at point A.)

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 273

For the elementary dipole with the dipole moment dp , placed in point C, the field strength in reception point B at the distance r2 from point C in complex scalar form is defined by (2B.38) from Appendix 2B, which may here be rewritten as



dE B =

k02 dp sinθ exp  −iksc ⋅ r2  4πε0ε r2

(5.114)

where ksc is the wave-vector of the scattered wave, shown in Figure 5.28. Note that the multiplier sinq in (5.84) is the normalized radiation pattern of the elementary dipole as expected from its physical meaning. In our case, the dipole axis is collinear to the exciting field E C ; thus, q must be counted from that axes (see Figure 5.28). After combining (5.112c), (5.113), and (5.114), the following result may be obtained for the scalar complex field strength at the observation point:



k2 ∆ε (r 1) 60PTxGTx dE B = 0 (5.115) sinθ exp i(ω t − k0 ⋅ r1 − ksc ⋅ r2 )  dVsc r1r2 4π ε Here it is assumed that the vector of the scattered field strength will keep the same direction as the original vector E C . As one can see from (5.115), the field dE B is proportional to xe = De /e. Hence, both the amplitude and phase of dE B will randomly fluctuate as the position of the scattering point C varies within the scattering volume Vsc. In order to define the power flow density of the scattered radio wave at the receiving point (the average magnitude of Pointing vector), consider scattered field strengths dE B′ and dE B′′ arriving at observation point B from elementary scattering volumes dV′sc and dV″sc positioned at point C’ and C”, respectively, as shown in Figure 5.29.

Figure 5.29  Superposition of two scattered rays at the reception point from different volumetric elements of the scattering volume.

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Atmospheric Effects in Radio Wave Propagation

Taking into account (2.71), the total average power flow intensity at the receiving point B may be expressed in the following form of the volumetric integral: ΠB, ave =



 1  1  ′′ )  Re  ∫ (dE B′ dE B 2 Vsc W0  

(5.116)

 ′′ is a complex where W0 = 120p is the wave impedance of the free space, and dE B    ′′ into (5.116): conjugate of dEB′′ . Using (5.115) we may substitute both dEB′ and dE B   ∆ε ′(r1)  k04 60PTxGTx 1  θ sin ΠB, ave = Re  ∫∫ sin θ ′′ ′   ⋅....... 2 Vsc (4π )2W0 r1 r2 r1 + ρ r2 − ρ  ε    ε ′′(r1 + ρ)   .....  ′ ⋅ r2 + ik0′′ ⋅ (r1 + ρ) + iksc ′′ ⋅ (r2 + ρ)  dVsc′ dVsc′′   exp  −ik0′ ⋅ r1 − iksc ε    (5.117) where the vector coordinates, shown in Figure 5.29, are related to each other as follows: r1′′ = r1 , r1′ = r1′′+ ρ = r1 + ρ , r2′′ = r2 , r2′ = r2′′ − ρ = r2 − ρ





(5.118)

The heavy expression (5.117) may be considerably simplified for real conditions when both antenna beams, Tx and Rx, are fairly narrow; thus, the distances r1 and r2 are much greater than the linear dimensions of the common scattering volume Vsc. This volume has an elevation above the Earth’s surface much less than the horizontal distances. Hence, k0′ ≈ k0′′ , ksc ′ ≈ ksc ′′ , θ ′ ≈ θ ′′ , r1 + ρ ≈ r1 , r2 − ρ ≈ r2



(5.119)

For conditions (5.119), as well as W0 = 120p ohms, the expression (5.117) may be simplified down to: ΠB, ave ≈

k04 PTxGTx sin2 θ (4π )3 r12 r22

 ∆ε ′(r ) ∆ε ′′(r + ρ)  1 1   exp − i ( k − k ) ⋅ ρ dV  sc  dVsc ∫  sc 0  ∫ ε ε Vsc Vsc 

(5.120)

Evidently, the integral in braces in (5.120) is a the spatial autocorrelation function of the random scalar field, ξε (r ) = ∆ε (r ) / ε , which we denote here as



ψ ε (r1, ρ) =

1 ∫ ξε′ (r1 + ρ) ξ ′′ε (r1) dVsc Vsc Vsc

(5.121)

In general, ye depends on r1 , as well as on the distance ρ of two points within the common volume Vsc. Fortunately, the fluctuations of tropospheric refraction index are statistically homogeneous (i.e., they are independent on position r1 )

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 275

and are only dependent on distance ρ between two points. Thus, (5.121) may be written as

ψ ε = ψ ε (ρ)



(5.122)

and (5.120) is represented as ΠB, ave ≈



k04 PTxGTxVsc sin2 θ (4π )3 r12 r22

∫ ψ ε (ρ) ⋅ exp[−iq ⋅ ρ ]dVsc

Vsc

(5.123)

where q = ksc − k0



(5.124)

is called a scattering vector. ψ ε (ρ) is considered a deterministic measure of the random field ∆ε (r ) . Then the integral in (5.123) may be expressed in terms of volumetric Fourier-spectrum of ψ ε (ρ) field:11



Sε (q) =

1

∫ ψ ε (ρ)exp(−iq ⋅ ρ)dVsc

(2π )3 Vsc

(5.125)

with the inverse transform



ψ ε (ρ) =

∫

Sε (q)exp(iq ⋅ ρ) dVsc

Vsc

(5.126)



The physical meaning of the (5.126) transform is that the autocorrelation function ψ ε (ρ) is represented as superposition of the continuum of deterministic plane waves Sε (q)exp(iq ⋅ ρ) , each of which has an amplitude of Sε (q) as a function of the spatial frequency q (scattering vector). Note that in the theory of random processes (theory of time-domain random functions), the transform (5.125) is known as WienerKhinchin transform, whereas in theory of random fields (theory of space-domain random functions), it’s known as Shannon-Whittaker transform. If (5.125) is substituted into (5.123), then the result is



ΠB, ave ≈

k04 PTxGTx sin2 θ Sε (q) 8r12 r22

Vsc

(5.127a)

Finally we use the replacement q = p /2 + qs illustrated in Figure 5.28. Then (5.127a) will be rewritten as



ΠB, ave ≈

k04 PTxGTx cos2 θ sc Sε (q) 8r12 r22

Vsc

(5.127b)

Here qsc is known as a scattering angle. 11. As seen later, the volumetric function ψ s (ρ) is symmetric. Thus, (5.125) becomes a real function. Therefore, when transforming (5.127) into (5.120) the sign Re was omitted.

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Atmospheric Effects in Radio Wave Propagation

Figure 5.30  (a) Flat projections of the autocorrelation function of the random scalar field, and (b) spectrum of the spatial harmonics of autocorrelation function of the random field De.

5.7.3  Physical Interpretation of Tropospheric Scatterings

As shown earlier, spatial distribution of the random field ξε (r ) may be expanded into the superposition of a continuum of elementary plane waves. To be more specific, note that the expansion is applied not to the random field ξε (r ) itself, but to one of its deterministic characteristic such as autocorrelation function ψ ε (ρ). It 2 2 contains the main statistical parameters, such as variance, ξε = (∆ε / ε ) , and the average dimension of random irregularities Le [i.e., the average sizes of the globules of turbulences (see Figure 5.30)]. One of the properties of the Shannon-Whittaker transform is similar to property of the Fourier transform,12 namely, the narrower the initial function ψ ε (ρ) is, the wider the spectrum becomes, and vice versa. There’s a following approximate relation between the width of the main lobe of the correlation function (the average dimension of the irregularities), Le, and the width of spatial spectrum of irregularities: qmax ~ 1/Le. To understand the mechanism of how the random medium impacts the scattering process, let’s consider a single Sε (q) component of 3-D spectrum of irregularities. If, for instance, q is directed vertically down, then the 3-D shape of that component of the spectrum of autocorrelation function may be represented as a stratified medium with the harmonically changing intensity of e in the vertical direc­­ tion (Figure 5.31) with the amplitude Se and the spatial period of Λ = 2π / q . Loci of the maximums of spatial densities are the parallel planes separated by L, as shown with dashed lines in Figure 5.31. 12. As it was mentioned previously, the Shannon-Whittaker transform is the same as a Fourier transform, which is applied to spatial correlation function ψ s (ρ) (i.e., it allows us to find the Fourier spectrum of the spatial harmonics of the given function ψ s (ρ) ).

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 277

Figure 5.31  Scattering of the radio wave from a single spatial harmonic of the turbulent fluctuation of dielectric constant in the troposphere.

As seen from Figure 5.31, all rays reflected toward the receiver will superimpose at the receiving point with the same phase (constructive interference) if the phase shift between two adjacent rays is equal to 2p (or is a multiple of 2p). If we take into account that the difference in distances between two adjacent rays is equal to 2BC, then from Figure 5.31 it’s easy to derive geometrically k0 (2BC) =



θ  2π  2Λ sin sc  = 2π  2  λ0 

(5.128)

Then the spatial period of the considering harmonic may be found from (5.128) as Λ=



λ 2 sin(θ sc / 2)

(5.129)

The same expression may be found from Figure 5.31 based upon disposition of three vectors, k0 , ksc , and q. Indeed, taking into account the equal magnitudes k0 = ksc =



2π λ

(5.130)

and



q =

2π Λ

(5.131)

278

Atmospheric Effects in Radio Wave Propagation

the following relation may be developed:

ksc − k0 = 2 k0 sin

θ sc =q 2

(5.132)

After substituting (5.130) and (5.131) into (5.132), the expression (5.129) may be verified, which is known in optics as a Bragg’s diffraction condition with the vector form of (5.124). It allows us to find relations between l, L and qsc (i.e., the relations between directions of incident and diffracted waves) if the diffraction takes place on one of the spatial harmonics of the 3-D correlation function of random field of the dielectric constant. Now if (5.127b) is recalled, then one may notice that the average power flow density at the receiving point is proportional to the amplitude of spatial harmonic. Hence, the intensity of the scattered field, PB,ave, will vanish for scattering angles greater than qmax when Se becomes very small (see Figure 5.30). The relation between qmax and qmax is found from (5.132) as

 λ (5.133) θ max = 2 sin−1  0 qmax    4π As mentioned previously, the upper limit of spatial spectrum of xe fluctuations, namely, qmax, is in inverse proportion to the average dimension of irregularities (qmax µ 1/Le); therefore, the greater the average dimensions of irregularities is, the narrower the spatial spectrum is. Thus, the angular aperture of the scattered rays is narrower, as demonstrated in Figure 5.32.

Figure 5.32  Patterns of scatterings of radio waves on random irregularities of the tropospheric turbulences and their vector diagrams (2qmax = angular aperture of scatterings): (a) scatterings on small-scale irregularities, and (b) scatterings on large-scale irregularities.

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 279

5.7.4  Effective Scattering Cross-Section of the Turbulent Troposphere

For engineering applications, it’s useful to introduce the effective scattering cross-section (ESCS) of the troposphere. Consider a unit volume (e.g., 1 m3) that surrounds point C located within the scattering volume Vsc. Now we replace hypothetically the unit volume of scatterers that surround point C by a flat PEC surface ssc, called ESCS, which produces at the receiving point B the same amount of received power as produced by the primary irregularities located within that unit volume. Similar to (3.41), we may obtain the incident power flow density at the scattering point C produced by the transmitter. That is, ΠC =



PTxGTx

(5.134)

4π r12

where r1 is the direct distance from transmitting point A to the scattering point C (Figure 5.28). Then, according to the definition for ESCS, the total power flow of the wave scattered by the unit volume that surrounds point C may be found as PC = ΠC σ sc =

PTxGTx 4π r12

σ sc

(5.135)

The power flow density at the receiving point B is



Π′B =

PC 4π r22

=

PTxGTxσ sc

(5.136)

(4π )2 r12 r22

where PC is substituted from (5.135), and r2 is the direct distance from scattering point C to receiving point B. Finally the total power flow density at B point, induced by the entire scattering volume, may be defined if we multiply (5.136) by Vsc; that is,13



ΠB = Π′BVsc =

PTxGTxσ scVsc (4π )2 r12 r22

(5.137)

If (5.137) is compared with (5.127b), then ESCS may be found in the following form:

σ sc = 2π 2k04 cos2 θ sc Sε (q)

(5.138)

Expression (5.138) demonstrates ssc(qsc) dependence, not only due to the presence of the cos2 qsc factor, but also because of the presence of the spectral compo­­­ nent Sε (q), which is highly dependent on q vector, namely, on both of its magnitude 13. For higher accuracy, the product sscVsc must be replaced by the volume integral ∫Vsc ssc ⋅ dVsc. However, for practical engineering applications, ssc is assumed to be uniformly distributed within scattering volume Vsc, which results in simple product sscVsc instead of the volume integral.

280

Atmospheric Effects in Radio Wave Propagation

and orientation qsc. It’s easy to realize that the maximum value of ESCS will appear at qsc = 0 [i.e., ssc,max = ssc(0)]. The graph of the normalized function Σ sc (θ ) =



σ sc (θ sc ) σ sc,max

(5.139)

is called a scattering diagram and is considered next for particular troposcatter models. 5.7.5  Statistical Models of Tropospheric Turbulences 5.7.5.1  Gaussian Model

In statistical radiophysics, the autocorrelation function ψ ε (ρ) is most commonly presented in the form of a Gaussian function. It is assumed that this function doesn’t depend on the direction of the spatial coordinate ρ , but only on its magnitude. This kind of statistically homogeneous random medium is called a statistically isotropic random field.



 ρ2  ψ ε (ρ) = ξε2 exp  − 2   2Lε 

(5.140)

where Le is the average radius of the irregularities of the tropospheric air that are spherically shaped by statistical means, and

ξε2



2

 ∆ε  =    ε 

(5.141)

is standard deviation of fluctuations of the relative dielectric permittivity of the clear tropospheric air that usually belongs to the range from 1.5 ⋅ 10-7 to 3.3 ⋅ 10-6. The lower limit corresponds to the winter season, and the upper limit, to the summer season. It is independent on the elevations above the Earth’s surface (at least for the elevations of up to 5 km). Spatial spectrum (5.125) of statistically isotropic field presented particularly by (5.140) may be introduced in the following form after transformations provided in Appendix 5A.



Sε (q) =

1

∞

∫ ψ ε (ρ)sin(qρ)ρ d ρ

2π 2q 0

(5.142)

If (5.140) is substituted into (5.142), then the expression for the spatial spectrum may be derived as follows [11]:



Sε (q) =

 q 2 L2  ε exp −  3 /2 2 (2π )  

ξε2Lε3

(5.143)

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 281

Now substitute (5.143) into (5.138) and (5.139) and take into account (5.132) to determine final expressions for ESCS and for the scattering diagram, respectively:



σ sc (θ sc ) =

 θ  π 4 2 3 k0 ξε Lε cos2 θ sc exp  −2k02L2ε sin2 sc  2 2  

 θ  Σ sc (θ sc ) = cos2 θ sc exp  −2k02L2ε sin2 sc  2  

(5.144) (5.145)

The 2-D scattering diagrams that are expressed in (5.145) analytic form in polar coordinates are illustrated in Figure 5.33 for different Le /l0 ratios obtained by using a MATLAB routine. 5.7.5.2  Kolmogorov-Obukhov Model

Although the previously considered Gaussian model (5.140) has various practical applications, in the case of tropospheric turbulences it is applied notionally and is not based on real aerodynamic processes related to generation of the tropospheric clear-air inhomogenieties of the dielectric constant (eddies). More accurate quantitative results may be obtained if scattering phenomenon analysis is linked to a

Figure 5.33  Scattering diagrams of the turbulent troposphere plotted in polar coordinates for the Gaussian model of turbulences (verifies the concept demonstrated in Figure 5.27).

282

Atmospheric Effects in Radio Wave Propagation

model that has been developed based on the real aero-hydrodynamic processes of generation of those inhomogenieties. Regarding to the principles of hydrodynamics, there are two types of movement of any gaseous or liquid masses: •

•

Laminar: A particular type of streamline flow where the gas (or fluid) in thin, parallel layers tends to maintain uniform velocity with constant magnitudes and directions [Figure 5.34(a)]. Turbulent: When, under certain conditions, a laminar flow of gas (liquid) turns into stochastically distributed eddies. That is, when the magnitudes and directions of velocities of the composite atoms and molecules turn to be randomly distributed in time and space [see Figure 5.34(b)].

The type of movement of gas or liquid is usually specified by Reynold’s number:



Re =

ϑ vlflow ν

(5.146)

isflow a density, v is the magnitude of velocity vector v, lflow is the cross-sectional Here ϑ vl Re = dimension  ν of the flow, and ν is the viscosity of the medium. Each gas or liquid has the specific critical value of Reynolds number, Recr. So if for a particular medium Re > Recr, then the movement of gas or liquid becomes turbulent; otherwise, it is laminar. The values of variables in (5.146) for the real troposphere are such that the movement of the air masses is almost always turbulent, which results in a permanent presence of the fluctuations of its dielectric constant. Figure 5.35 demonstrates the spectrum of spatial harmonics of the fluctuations of the tropospheric dielectric

Figure 5.34  Movements of tropospheric air masses: (a) laminar, and (b) turbulent.

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 283

Figure 5.35  Spatial spectrum of random fluctuations of dielectric constant of the troposphere: 1 = range of eddies formation, 2 = energy conservation range (inertial range), 3 = energy dissipation range.

constant as a function of the mechanical spatial wave numbers. As seen from the figure, the spectrum is divided into three ranges: 1. Range of formation of the eddies (globules) of largest sizes, L0, called the outer scale of turbulences, which may vary from 100m to 1,000m, and even larger. 2. Inertial range, where kinetic energy of the movements remains unchanged. Eddies become smaller in size gradually, with gradually increasing velocity of spinning of the air masses inside each eddy. The range of sizes is l0 < l < L0 (2p /L0 < q < 2p /l0), where l0 is its lower limit, called the inner scale of turbulences, which varies in range of dimensions approximately from 1 cm to 1 mm. 3. Dissipation range, where the lower limit l0 of the eddy’s size is reached; thus, the maximum spinning velocity results in destruction of the eddies followed by the dissipation of their kinetic energy (i.e., irreversible transformation of that energy into heat while the globules disappear). Regarding the theoretical investigations by A. N. Kolmogorov and A. M. Obukhov, the spatial spectrum of correlation function for random fluctuations of dielectric constant of the troposphere in the area of inertial transforms is expressed in the following analytical form [10]:



Sε (q) = 0.033 Cε2q

−

11 3



(5.147)

In (5.147) C e2 is in m–2/3 and called a structural constant that is dependent on the season of the year and on elevation of the scattering point. Sometimes the values of C e2 are given in cm–2/3 units. To transform C e2 from one unit to the other, recall that

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Atmospheric Effects in Radio Wave Propagation

(1 cm)-2/3 = (0.01m)-2/3 = 21.54(1m)-2/3, which means the value of C e2 in cm-2/3 must be multiplied by 21.54 to get it in m–2/3 units. Seasonal variations of the structural constant near the Earth’s surface are in the range from 10-14 cm-2/3 (summer) to 10-16 cm-2/3 (winter). The average vertical profile of C e2 may be expressed as



 Cε2 (h) = Cε2 (h0 ) 

h  h0 

−α

(5.148)



where C e2(h0) is defined for the elevation of h0 = 30m. a is a constant, which has the following values, defined empirically: a = 2/3 for the winter season and a = 4/3 for the summer season (see Figure 5.36). Note that expressions (5.138) and (5.139) become meaningless if qsc → 0, because (5.137) is valid only in inertial range shown in Figure 5.35; that is, it is valid for the spatial harmonics q starting with their minimum value of



q = 2k0 sin

θ sc 2π ≥ qmin = 2 L0



(5.149)

up to the maximum qmax. Indeed, for the scattering angle qsc = 0, the KolmogorovObukhov model violates the physical picture: it results in infinite scattering intensity, while limited value is expected. This is the shortcoming of the model. On the other hand, if that is compared with the Gaussian model then one may realize that the Kolmogorov-Obukhov model considers an existence of multiscale irregularities, which is typical for the troposphere, while the Gaussian model considers only monoscale irregularities, with the dimensions dispersed (spread) around its mean value, Le. Despite this fact, the use of Gaussian model is still applicable if the angles qsc are close to zero.

Figure 5.36  Vertical profiles of the structural constant Cs2 for the turbulent troposphere.

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 285

In MMW and optical bands, when the values of scattering vector become large enough (i.e., when the interaction between incident waves with the turbulent medium takes place in dissipation range of the spectrum), the following relation:



q = 2k0 sin

θ sc 4π θ 2π = sin sc ≥ = qmax 2 2 l0 λ

(5.150)

dictates the usage of an additional exponential multiplier to be inserted into (5.147), which allows us to take into account an abrupt decrease of Se(q) seen from Figure 5.33:



 q  Sε (q) = 0.033 Cε2q−11/3 exp  − q  max 

(5.151)

For the small values of the magnitude of scattering vector (i.e., in range of formation of the tropospheric turbulences), where



q = 2k0 sin

θ sc 2π ≤ = qmin 2 L0

(5.152)

the data obtained from the Kolmogorov-Obukhov model considerably distinguishes from real observed values of Se(q). Later an update of the model was introduced by transforming (5.151) into his final form, pertinent for the practical applications



 q  Sε (q) = 0.033 Cε2 (q2 + qmin2 )−11/6 exp  −  qmax 

(5.153)

It was shown theoretically by Kolmogorov and Obukhov that the outer scale of atmospheric turbulences, L0, is linked to the mean-square value of fluctuations of the dielectric constant of the troposphere ξε2 and to the structural constant C e2 by the following expression [10]:



Cε2 =

2 ξε2

(5.154)

L02 /3

If the decaying tendency of the vertical profile C e2(h) is taken into account from (5.148) or from Figure 5.36, then it may be realized that the higher the elevation is, the greater the outer scale of the turbulences is (i.e., greater the dimensions of the newborn eddies are). Now we substitute (5.153) into (5.138) and (5.139) to define ESCS and scattering diagram in analytical form for this model:



2   L0 θ sc   π 0.033 k04 Cε2L011/3 2  σ sc (θ ) = 3 θ 1 + 2 i n cos s sc 4 2     λ0 (2π )2  

−11/6

 l θ  exp  −2 0 sin sc  2   λ0

(5.155)

286



Atmospheric Effects in Radio Wave Propagation

2   θ   L Σ sc (θ sc ) ≈ cos2 θ sc 1 +  2 0 sin sc   2     λ0  

−11/6

 l θ  exp  −2 0 sin sc  2   λ0

(5.156)

Examples of scattering diagrams (5.156) for the Kolmogorov-Obukhov model of the tropospheric turbulences are shown in Figure 5.37. 5.7.6  Propagation Factor on Secondary Tropospheric Radio Links

In order to obtain the total power received at point B, the power flow density PB form (5.137) must be multiplied by antenna’s effective aperture form (3.36); that is,



PB =

PTxGTxGRx λ02σ sc (θ sc )Vsc (4π r1r2 )2 4π



(5.157)

If this expression is compared with (3.44), then the propagation factor may be found as



F=

σ sc (θ sc )Vsc 4π

r r1r2

(5.158a)

where r1 and r2 are the distances from scattering point C to transmitting A and receiving B points, respectively, and r is the distance between transmitting and receiving points. Expression (5.158a) may be rewritten for the total horizontal distance R between transmission and reception points along the great circle if assumed r1 ≈ r2 ≈ R/2: F=

2 R π

σ scVsc



(5.158b)

Figure 5.37  Scattering diagrams of the turbulent troposphere plotted in Cartesian coordinates, based on (5.156) model for the fixed values of the inner (l0) and outer (L0) scales, and for different wavelengths (verifies the concept demonstrated in Figure 5.32).

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 287

The scattering volume Vsc is defined as the intersection of main beams of the transmitting and receiving antennas, as shown in Figure 5.38. For simplicity we’ll assume both diagrams are symmetric relative to the main radiation direction; that is, half power beam widths (HPBW) are equal in both E- and H-planes for each antenna:

γ 1E = γ 1H = γ 1

(5.159)



γ 2E = γ 2H = γ 2

(5.160)

Then by definition (3.16) given in Chapter 3 antenna gains may be expressed in terms of beam widths as GTx ≈



4π

and GRx ≈

4π

(5.161) γ 22 For maximum efficiency of the secondary tropospheric radio links, the angles of elevation of both antennas are chosen close to zero; that is, the main beams of both antennas are directed tangential to the Earth’s surface (i.e., directed horizontally). That allows having a common scattering volume Vsc as close to Earth’s ground as possible. The reason is because the lower the elevation of the scattering volume is, then smaller is the scattering angle so the higher the intensity of scatterings are. From Figure 5.38(b),

γ 12

Vs ≅ AreaMNLP ⋅ MM ′



(5.162)

where

AreaMNLP ≅

b d , and d = MM sinθ

(5.163)

Figure 5.38  The definition of the scattering volume of troposphere: (a) the intercept of the radiation diagrams of Tx and Rx antennas, and (b) the detailed sketch of the scattering volume (note that the vertical scale is highly exaggerated for clarity).

288

Atmospheric Effects in Radio Wave Propagation

If C is the center point of the scattering volume, then from Figure 5.38(a) the following approximate expressions may be written: d ≈ AC ⋅ γ 1 = r1γ 1   b ≈ BC ⋅ γ 2 = r2γ 2  



(5.164)

It’s easy to realize that if the elevations of the antennas’ main beams are close to horizontal, then the scattering angle qsc is the same as the geo-central angle. For real conditions, the following approximation may be applied: sinθ sc ≈ θ sc ≈



R ae

(5.165)

where ae = 8,500 km is the Earth’s equivalent radius, and R is the horizontal distance between corresponding points. If g1 and g2 are found from (5.161), then, by combining (5.162)–(5.165), we find Vsc ≈



R2 ae (4π )3 /2 8 GTx GRx

(5.166)

It was assumed GTx > GRx, because the width d of the volume VS was defined based on the beam width of the antenna with higher gain (transmitting antenna in this case). It’s easy to show that for GTx < GRx, the denominator in (5.166) will become GRx GTx . In real conditions the mean-geometric of those two values may be used instead: 3 /4

3 /4

GTx GRx GRx GTx = GTx GRx



(5.167)

Thus, (5.166) may be rewritten as Vsc ≈



R2 ae (4π )3 /2 8 G3 /4G3 /4 Tx Rx

(5.168a)

This expression may easily be transformed to Vsc ≈



(Rγ )3 R3 (4π )3 /2 ≈ 8θ sc 8θ scG3 /2

(5.168b)

where (5.165) is taken into account as well as the similarity between corresponding antennas; that is, G = GTx ≈ GRx ≈ 4p /g  2. If (5.168) is substituted into (5.158a), then the following formula may be found for the power propagation factor: F 2 ≈ 4 π ae



σ sc (θ sc ) ∆G

(5.169)

where

3 /4

3 /4

∆G = GTx GRx

(5.170)

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 289

Now before we refer to ssc(qsc) to make expression (5.169) applicable for practical estimates, we have to take into account the following statements: •

•

The path length of the troposcatter links is practically limited from 200 km (for relatively wideband systems with the bandwidth up to several megahertz) to 1,000 km (for narrowband systems with the bandwidth no more than tens of kHz). The carrying frequencies are limited from 200 MHz (due to antenna size limitations) to 5 GHz (due to increasing of the atmospheric absorptions).

Based on the above statements it’s easy to estimate that for the given distances the geo-central angle qsc = R/ae along the great circle is limited to 0.0235 < qsc < 0.1177 radians. Then the magnitude of scattering vector from (5.132), q = q = 2k0 sin(θ sc / 2), will be ranged within 0.0986 m-1 < q < 12.3 m-1 limits. If those limits are compared with the limits (qmin ≤ 0.063 m-1 qmax ≥ 628 m-1) of the energy conservation range (range 2 in Figure 5.35), then it is easy to realize that mostly those irregularities are responsible for the scattering, which are located in range 2 only. Therefore, for further evaluation of (5.169) for the power propagation factor, the original version (5.147) of the spectrum of turbulences may be employed, without counting on corrections made to adjust both ends of the spectrum. After combining (5.138), (5.147), and (5.169), we obtain F2 ≈



=



4 π ae 2 4 2π k0 cos2 θ sc 0.033Cε2q−11/3 ∆G

 θ  39.26 ⋅ 106 4 k0 cos2 θ scCε2  2k0 sin sc  2  ∆G 

−11/3

(5.171)

The following notations are used in (5.171): • • • • •

DG is defined by (5.170) as unitless. k0 is a free space wave number in 1/m. ae = 8.5 ⋅ 106 m is equivalent Earth’s radius. qsc is a scattering angle defined by (5.165). Ce2 is a structural constant for atmospheric turbulences in m-2/3.

Example 5.2

Estimate the propagation factor on the tropospheric radio link for the frequency 1 GHz (free space wavelength, l0 = 30 cm, wave number, k0 = 2p /l = 20.944, 1/m), and horizontal distance of R = 300 km between transmitting and receiving antennas with the gains GTx = GRx = 50 dB (105 unitless). The value of the structural constant in vicinity of the Earth’s surface, at the height h0 = 30m, may approximately be taken: Ce2 = 10-14 cm-2/3 for summer season and Ce2 = 10-16 cm-2/3 for winter season (see Figure 5.36). Find the power received, if the radiated power is PTx = 2 kW. Solution •

The scattering angle is calculated by taking it equal to geo-central angle, q = r/aeq = 300/8,500 = 0.053 rad.

290

Atmospheric Effects in Radio Wave Propagation

•

The height of the scattering volume is found from triangle AOC on Figure 5.39.     1 1 hC = ae  − 1 = 8, 500  − 1 ≈ 5.3 km  cos(θ /2)   cos(0.0353 /2) 

•

•

We define the structural constant at the elevation hC from (5.148) (or roughly estimated from Figure 5.36): • Ce2(5.3 km) = 10-16(5,300/30)-2/3 = 3.18 × 10-18 cm-2/3 = 6.85 × 10-17 m-2/3 (for winter season). • Ce2(5.3 km) = 10-14(5,300/30)-4/3 = 10-17 cm-2/3 = 2.15 × 10-16 m-2/3 (for summer season). • Note that here we ignore seasonal variations of the smooth tropospheric refraction that may result in variations of the common volume elevation above the ground; standard tropospheric refraction is assumed here. Power propagation factor is calculated from (5.171): • For winter seasons: F2 =

39.26 ⋅ 106 (105)3 /2

20.9444 cos2 0.0353 ⋅ 6.85 ⋅ 10−17

 0.0353  ×  2 ⋅ 20.944 ⋅ sin 2  

−11/3

= ...

… = 4.95 ⋅ 10-11(-103 dB) •

(Answer)

For summer seasons: F2 =

39.26 ⋅ 106 5 3 /2

(10 )

20.9444 cos2 0.0353 ⋅ 2.15 ⋅ 10−16

 0.0353  ×  2 ⋅ 20.944 ⋅ sin 2  

−11/3

= ...

… = 1.55 ⋅ 10-10(-98.1 dB)

Figure 5.39  Definition of the scattering volume elevation above the ground.

(Answer)

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 291

•

Power received on reference propagation path (free space) is found from (3.42): PRx,0 =

•

(4π R)2

=

2, 000 ⋅ 105 ⋅ 105 ⋅ 0.32 (4π ⋅ 300, 000)2

= 0.1267 W (126.7 mW)

Power received on real propagation path is found as PR = PR,0 ⋅ F2: • For winter seasons: PRx = 6.27 ⋅ 10-12 W (-112 dBW)

•



PTxGTxGRx λ02

(Answer)

For summer seasons: PRx = 1.96 ⋅ 10-11 W (-107 dBW)

(Answer)

These considerations of the propagation factor and power received on troposcatter radio links allow us to make only the rough assessments of their median values. Unfortunately, more accurate calculations may not be performed due to the lack of complete and precise set of data about the parameters of the atmospheric turbulences: we do mean the data of structural constant global distributions and its spatial and time variations. For more accurate results, the reader may be referred to ITU-R document [12], which provides an empirical calculation method based on numerous of observations on existing troposcatter radio links, but it is not based on analysis of physical mechanisms of propagation. 5.7.7  The Specifics of the Secondary Tropospheric Radio Links Performance 5.7.7.1  Antennas Gain Effect on Link Performance

As seen from (5.171) the propagation factor on secondary tropospheric radio links is highly dependent on antenna gains GTx and GRx. The propagation factor is getting lower by DG, when either GTx or GRx (or both) increase unlike other types of radio links, where propagation factor is independent on antennas. The cause of this phenomenon is quite clear: the increase of the antenna gains will result in a decrease of the beam widths g1 and g2; according to (5.168a) and (5.168b) this results in a decrease of the scattering volume VS, as demonstrated in Figure 5.38. If (5.170) is expressed in decibels,



∆GdB =

3 + GRx,dB ) (G 4 Tx,dB

(5.172)

then, as follows from the plot of (5.172) shown in Figure 5.40, the loss of the antennas’ gains (in decibels) as a function of total antenna gain is linearly increasing with the slope of 3/4.

292

Atmospheric Effects in Radio Wave Propagation

Figure 5.40  Antennas’ total gain losses on secondary tropospheric radio link (theoretical prediction).

In reality, DG becomes considerable when the total gain exceeds 65–70 dB, as may be seen from Figure 5.40. Another reasonable explanation of this phenomenon is as follows: due to turbulences, scattered field is a subject of intensive amplitude and phase fluctuations at the reception point along the antenna’s aperture; correlation distance of those fluctuations across the propagation path is comparable to the antennas dimensions. If the antenna’s dimension is less than that correlation distance, then within the antenna’s aperture the received field is correlated (i.e., its structure becomes close to the structure of plane wave). If the antenna’s dimension is greater than correlation distance, then uncorrelated fluctuation along the receiving aperture (especially phase fluctuations) results in destructive integration of the field values, resulting in destruction of the antenna gain. It is clear that when transmitting and receiving antennas are not identical, then the antenna with larger dimension suffers more than that of smaller dimension. Total antenna gain of GTx + GRx = 90–100 dB seems to be the limit, which is meaningless to exceed; further investments are not paid off. Hence, the value for each antenna gain is usually not taken larger than 45–50 dB. 5.7.7.2  Signal Level Fluctuations at the Receiving Point (Fading)

One of the specifics of the troposcatter radio link is the presence of intensive random fluctuations of the signal level at the receiving point, which may be classified into three independent categories: • • •

Fast fading; Slow fading; Seasonal variations.

As noted previously, the cause of fast fading is a random interference of the multitude of secondary waves, interfering at the receiving point.14 On this type of 14. Detailed analysis of the random interference (multipath interference) is given in Chapter 6, along with the references for terms and definitions.

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 293

radio link, the multitude of secondary, interfering partial waves may be considered almost an ideal Rayleigh ensemble. The fast fading is considerably deep and has the Rayleigh-type probability distribution of the field strength (see Chapter 6 for details). Slow fading on troposcatter radio links is described by lognormal probability distribution with fair accuracy (see Section 6.1.3.2 for more details). Standard deviation of the median signal level, sy, depends on propagation path length and season of the year, as shown in Figure 5.41. To understand the reason for the decrease in intensity in slow fades distancedependence, recall that the greater the distance is, higher the elevation of the scattering volume is VS, and the more stable the atmospheric conditions are. Those conditions are less dependent on seasonal variations at high elevations than at low elevations. The seasonal dependence of those variations may become clear if taken into account the fact that during winter season the Earth’s surface is heated less. Therefore, the vertical gradient of the atmospheric refractivity is less, dN/dh; according to (5.25) the curvature of ray trace due to regular smooth tropospheric refraction is less, and the position of the scattering volume VS is higher. The higher the elevation of the scattering volume, the less the impact of seasonal variations. Although the intensity of slow fades is less during the winter season, the short-term median signal at the reception point is always less than in summer. Therefore, it is common practice to design such radio links for the winter season to guarantee performance all year long. On the other hand, in terms of stability of the signal level at the receiving point, the worst conditions arise during the summer season due to an increasing of the deepness of fading (both fast and slow). Therefore, the estimates for the stability of those radio links is preferably carried out for the summer seasons, based on approaches given in Chapter 6, and the values of sy obtained from Figure 5.41. One of the specific features of troposcatter communication systems is the use of diversity receiving principles that allows struggling against very deep fading. Note that the previous approaches allow us to estimate only the long-term median received power. If the sensitivity of the receiver is taken equal to that median power level, then apparently the designer may not expect the overall performance quality (steadiness) of the system to be greater than 50% by definition. To reach the required performance quality, an increase of the radiated power (i.e., the embedding

Figure 5.41  Dependence of the standard deviation of slow fading of the received field level on distance for troposcatter radio link.

294

Atmospheric Effects in Radio Wave Propagation

of power margin) is not the solution for such systems because of the limitations in power budgets. The only means of achieving the required performance steadiness is a combination of the power margin with diversity reception techniques that are widely used in troposcatter communication lines. Those techniques are considered in Section 6.3.3 in detail. 5.7.7.3  Limitations to Signal Transmission Bandwidth

Consider an amplitude-modulated signal with a carrier frequency f0 and doublesideband sinusoidal modulation. The waveform has a periodical shape with the amplitude envelope of the sinusoidal form. If Fmod is modulating frequency, then the frequency bandwidth is Df = 2Fmod. Thus, the spectrum is located in frequency range from fmin = f0 - Fmod to fmax = f0 + Fmod. To analyze transmission of this signal through the troposcatter propagation path, we refer to Figure 5.38 in order to define the phase shifts between spectral components fmin and fmax. Two extreme rays scattered from the highest and lowest points of the scattering volume VS, ANB and APB, have difference in distances, which may roughly be estimated as



∆r = APB − ANB ≈ Rγθ sc =

R2γ ae

(5.173)

for g1 = g2 = g and equal distances r1 ≈ r2 = R from transmission/reception point to the center of the common volume Vsc along the great circle. Expression (5.173) may be rewritten, taking into account (5.161): ∆r ≈

R2 4π /G ae

(5.174)

where ae = 8,500 km is the Earth’s equivalent radius. The phase shift between two rays passed through different extreme paths for the lowest spectral harmonic is



∆ϕ min =

2π (f0 − Fmod ) 2π fmin ⋅ ∆r = ⋅ ∆r c c

(5.175)

whereas for the highest spectral harmonic it is equal to



∆ϕ max =

2π (f0 + Fmod ) 2π fmin ⋅ ∆r = ⋅ ∆r c c

(5.176)

where c = 3 ⋅ 108m/s is the speed of light in free space. As noted in Section 4.1.3, the phase distortions of the signal may be neglected if the maximum difference in phase shifts between the extreme rays and extreme spectral harmonics is less than p/2; that is,



δϕ = ∆ϕ max − ∆ϕ min = 2

2π Fmod π ∆r ≤ c 2

(5.177)

5.7  Over-the-Horizon Propagation of the Radio Waves by the Tropospheric Scattering Mechanism 295

Then the limit for signal transmission bandwidth may be defined by substitution of (5.174) into (5.177): ∆f = 2Fmod ≤



ca c ≈ e2 4∆r 4R

G 4π

(5.178)

Example 5.3 demonstrates, in particular, how the signal transmission frequency band may depend on troposcatter propagation path length. Example 5.3

Calculate and plot the distance dependence of the maximum signal transmission bandwidth for the range of distances from 200 to 1,000 km. Assume identical antennas of the gain G = 47 dB (G = 50,000) for both transmitting and receiving stations. Solution

Based on (5.178) the calculation results are presented in Table 5.5 and plotted in Figure 5.42. From the considered example, it may be seen that the bandwidth of the secondary tropospheric transmitting system is strictly limited and cannot exceed several megahertz. From (5.148) it may be concluded that some improvement of the bandwidth can be achieved by increasing the antennas’ gains. However, the gain limitations are to be applied. Unfortunately, the restrictions do not allow transmission of the video signals or high-speed digital data.

Table 5.5  Example 5.3: Calculation Results R, km Df, kHz

100 4,021.2

200 1,005.3

300 446.8

400 251.3

500 160.8

600 111.7

700 82.1

800 62.8

900 49.6

1,000 40.2

Figure 5.42  Example of the distance dependence of maximum bandwidth for troposcatter signal transmission.

296

Atmospheric Effects in Radio Wave Propagation

5.8  Attenuation of the Radio Waves in the Atmosphere 5.8.1  Attenuations in Troposphere

Propagation of the radio wave through the atmospheric air is accompanied by losses due to transfer of the energy of the wave into thermal movements of atoms and molecules of the composites (gases and hydrometeors). Those composites are considered to be uniformly distributed along the propagation path, so the attenuation coefficient in (2.96) is counted as a sum:

α = α g + αh



(5.179)

and is usually expressed in dB/km. In (5.179) ag and ah are the attenuations per unit distance (attenuation coefficients15) in atmospheric gases, and hydrometeors, respectively. The most significant contributions into the gaseous component of the attenuation coefficient, namely, ag, occurs in molecular oxygen (O2), called dry air attenuation coefficient, and in water vapors (H2O). That is, the total attenuation coefficient due to losses in atmospheric gases is

α g ≈ αO2 + α H2O





(5.180)

Frequency dependencies of aO2 and aH2O are shown on graphs in Figure 5.43; it’s seen that both terms in (5.180) are highly frequency-sensitive and are called frequency-selective absorptions [13]. From the given figure, one may realize that the attenuation of energy of radio waves becomes considerable for the frequencies higher than 5 GHz. For the frequencies less than 5 GHz, those attenuations can be neglected for any type of radio links including terrestrial, as well as Earth-to-space and space-to-Earth. Considering the frequency range of f ≥ 5 GHz, the value of ag increases with the significant jumps of attenuations caused by resonances in molecular structures of O2 and H2O, as seen from Figure 5.43. Apparently, the use of those frequencies is meaningless. From the same figure it may be seen that there are also transparency windows with relatively low attenuations; only those windows are acceptable for the radio links design. In particular, a window of frequencies 25 to 52 GHz is allocated for new, 5G cellular networks operations. Total attenuation (losses in decibels) on terrestrial radio links due to absorptions in atmospheric gases may easily be found as Lg = α gR, dB



(5.181)

where R is a horizontal distance between corresponding antennas in kilometers. Note that for the slant propagation paths, such as those for Earth-to-space and space-to-Earth, as well as for the VHF, UHF, and microwave, communication links between ground-based and airborne stations expression (5.181) is not valid any more. The reason is uneven vertical distribution of the atmospheric air tempera-

15. Cited in [13] as specific attenuation.

5.8  Attenuation of the Radio Waves in the Atmosphere 297

Figure 5.43  Attenuation coefficient of sea-level atmosphere due to molecular oxygen and water vapor for standard atmosphere: air pressure = 1,013 mbar, temperature = 15°C, water vapor density = 7.5 g/m3 (for aH O curve only). 2

ture, pressure, and density of the water vapors. The last two factors have the most significant impact on the values of aO2 and aH2O. For the standard atmospheric conditions, they both decrease exponentially. To obtain the total attenuation on the slant propagation path, the product (5.181) is to be replaced by the (2.102) integral. If Dh is the difference in altitudes of the communicating antennas, the equivalent height Dheq < Dh is introduced [13] for both dry air attenuations, Dheq,O2, and water vapors attenuations, Dheq,H2O. It must be emphasized that the air pressure decreases much slower than water vapor density; therefore, Dheq,H2O < Dheq,O2. Then for the zenith angle q of the slant propagation path, the value of the total attenuation may be calculated based on secant law; that is,

Lg = (αO2 ∆heq,O2 + α H2O∆heq,H2O )sec θ



(5.182)

The values of aO2 and aH2O are taken at the height of the lowest antenna. Figure 5.43 may be used only for rough graphical estimates of ag. For precise (line-byline calculation method) and approximate analytical calculations of ag, as well as calculation of Dheq,O2 and Dheq,H2O, the ITU document [13] may be used. The second term in (5.179), namely, the attenuation coefficient due to hydrometeors, has two composite parts:

α h = α prec + α fog

(5.183)

298

Atmospheric Effects in Radio Wave Propagation

where aprec is a component caused by precipitations (rain, snow, hail), and afog is a component caused by clouds and fog. Note that each hydrometeor may be considered a small particle (raindrops, ice balls, or snowflakes) with semi-conducting properties. The propagating radio wave excites a displacement and/or conducting currents in that particle. Regarding (2.47), the intensity of displacement currents is in direct proportion to dielectric constant of propagation medium. The density of these currents within each raindrop is significant because the dielectric constant of water is 80 times greater than in air. On the other hand, they are proportional to the frequency; thus, they become heavier at a higher frequency. From the same expression (2.47), one may realize that the displacement currents do not result in dissipation of energy of the radio wave into heat within the particle, because they’re shifted by 90° relative to electric field. However, they result in decay of energy due to scatterings similar to scatterings on tropospheric turbulences: the incident radio wave turns each particle (raindrop, snowflake) into an elementary radiator with a wide radiation pattern. Thus, the energy of directed radiation transforms into the radiation widely spread into surrounding space. This scattering mechanism may have a significant impact on loss of energy, especially if the wavelength is comparable to the size of particles. Those attenuations may be neglected for the frequencies less than 1 GHz, whereas they become considerable in microwave bands. An additional attenuation in hydrometeors is caused by conducting currents, which irreversibly turn part of the energy of radio wave into heat that dissipates within the particle. The frequency dependencies of composite parts of ah are shown in Figure 5.44(a) [8].

Figure 5.44  (a) Frequency dependence of the attenuation coefficient, ah, due to hydrometeors (rain and fog): (1) drizzling rain (0.25 mm/hr); (2) light rain (1 mm/hr); (3) moderate rain (4 mm/hr); (4) heavy rain (15 mm/hr); (5) light fog (0.03 g/m3–600m visibility); (6) medium fog (0.3 g/m3– 120m visibility); (7) dense fog (2.3 g/m3–30m visibility); and (b) propagation path range within precipitation area for Earth-to-space communication link.

5.8  Attenuation of the Radio Waves in the Atmosphere 299

The graphs in Figure 5.44 may be used for rough, approximate references only. For precise calculations the reader may be referred to [14, 15]. Note that for the dry snow and hail aprec becomes almost 25 times less than that for the same intensity of the rain at the same frequency. This statement may be taken into account when similar calculations are performed for snow and hail based on the data from Figure 5.44(a). However, wet snow results in almost the same aprec as rain of the same intensity. Total loss due to precipitations on Earth-to-space communication path of length Dprec depends on the zenith angle q [see Figure 5.44(b)] as follows: Lprec = α prec ∆ prec = α prec



hclouds cosθ

(5.184)

More complete data including geographical and seasonal statistical distributions may be found in [14]. Example 5.4

Calculate approximate diameter of parabolic dish antenna onboard geostationary satellite for the space-to-Earth communication link for initial data given below. Assume receiving station positioned in under-the-satellite point (q = 0). • • • • • • • • • •

The wavelength: l0 = 7.5 cm (f = 4 GHz). Power radiated: PTx = 30W (14.8 dB/W). Minimum received power required: PRx = 1.6 ⋅ 10-11 W = 16 pW (-108 dB/W). Receiving antenna gain: 50 dB. The distance:  r = 36,000 km, (distance to geostationary orbit). Intensity of the rain: 15 mm/hr. Visibility in foggy condition: 120m. Height of the precipitation ceiling: 1 km. Losses in antenna feed lines, and total losses in ionosphere ignored. Consider for simplicity the equivalent path lengths:



D heq,O2 = 7 km, and D heq,H2O = 1.5 km Solution

1. The specific attenuation (attenuation coefficient) is found from Figure 5.43 approximately as: aO2 = 0.007 dB/km, aH2O = 0.001 dB/km 2. Total attenuation loss due to atmospheric gases found from (5.182): Lg = (0.007 ⋅ 7 + 0.001 ⋅ 1.5) sec0 = 0.0505 dB

300

Atmospheric Effects in Radio Wave Propagation

3. Attenuation coefficient due to rain and fog defined from Figure 5.44(a): for rain: aprec ≈ 0.015 dB/km for fog: afog ≈ 0.02 dB/km 4. As a worst-case scenario, we consider the same path length for both rain and fog. Then the total loss due to attenuations in hydrometeors: Lh ≈ (0.015 + 0.02) ⋅ 1/cos0 = 0.035 dB 5. Total losses due to attenuations in troposphere (dry air and hydrometeors): LA = Lg + Lh = 0.0505 + 0.035 ≈ 0.1 dB 6. From the expression (3.45), the satellite-based transmitting antennas gain: GTx = PRx - PTx - GRx + L0 + LA =

= -108 - 14.8 - 50 + 0.1 + 195.6 = 22.9 dB (≈195 unitless) 2

2

 4π r   4π ⋅ 36, 000, 000  = 10 log  where L0 = 10 log   = 195.6 dB is a free 0.075   λ0  space loss. 7. Now we refer to (3.36) and (3.37) and take n ≈ 0.6. Then the geometrical aperture of the satellite transmitting antenna may be calculated as S=

GTx λ02 195.6 ⋅ 0.0752 π D2 = ≈ 0.146m2 = 4πν 4 ⋅ 3.14 ⋅ 0.6 4

where D is the diameter of the satellite transmitting antenna. 8. Finally, form the previous expression



D=

4S = π

4 ⋅ 0.146 = 0.43m 3.14

(Answer)

5.8.2  Attenuations in Ionosphere

The ionosphere is considered a low-loss dielectric medium with the attenuation coefficient defined by (2.92) for tand > x 2. Then (5.9b) may be rewritten as

σ≈



2.8ξ Nelec /cm3 2 (2π )2 fHz

⋅ 10−2 ≈ 7.1

ξ Nelec /cm3 2 fMHz

⋅ 10−16, S/m



(5.187)

where Nelec/cm3 is a number of free electrons per cubic centimeter, and x is a number of free electrons collisions per second. The substitution of (5.187) into (5.186) will result in

α ≈ 1.34



ξ Nelec /cm3 fMHz2

10−13, Np/m

(5.188)

For the simple estimates we may assume the values x and Ne remain constant within the absorbing layer. Particularly for the E layer, the number of collisions is taken form Table 5.1 as x ≈ 1051/sec, then (5.188) can be expressed as

α E ≈ 1.66



fcE,MHz fMHz 2

2

⋅ 10−4 , Np/m

(5.189)

where

fcE,MHz = 80.0Nelec /cm3 ⋅ 10−3



(5.190)

is the plasma frequency (Langmuir frequency) of the ionospheric E layer in megahertz. Because the value of aE is taken approximately constant within the absorbing layer, the total losses in that particular layer can be estimated based on (2.98) [instead of (2.102)]. The distance z covered by the radio wave within the absorbing layer may be estimated based on geometric sketch given in Figure 5.45(a).

z = ∆hE sec ϕ E

(5.191)

where DhE is a thickness of the E layer, and jE is the angle of radio wave incidence on E layer. By substituting (5.189) and (5.191) as well as the average thickness of the E layer DhE = 30 km into (2.98), a formula for total attenuation of the radio wave in the E layer may be presented as



AE =

AE′ fMHz2

(5.192)

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Atmospheric Effects in Radio Wave Propagation

Figure 5.45  (a, b) The reflection pattern of the radio wave from the F2 ionospheric layer in the presence of the absorbing layers D, E, and F1.

where AE′ ≈ 5fc2,E,MHz sec ϕ E





(5.193)

is the attenuation for the carrier frequency of 1 MHz (unit is: Np·MHz2). The expressions (5.192)–(5.193) may be generalized for the other absorbing layers such as D and F1 layer. To be more accurate, it must be recalled that, for simplicity, those expressions have been developed without taking into account the following factors: •

•

Within each layer the value of a doesn’t remain constant due to variations of x(h) and Ne(h) (see Section 1.3). It results in changes of a from point to point along the ray path; thus, (2.98) may not provide enough accuracy and (2.102) must be used instead The Earth’s magnetic field has a proper impact on the value of attenuation; therefore, it must be taken into account as well.

These exact evaluations provide the following expression for the total attenuation, consolidating all three absorbing layers (if the F2 layer is considered as reflecting): AΣ =



AΣ′

(f + fL )2

(5.194)

where

AΣ′ = AD ′ + AE′ + AF′ 1 = fc2, E, MHz (3 sec ϕ D + 2.5 sec ϕ E + 0.4 sec ϕ F1)

(5.195)

is for the daytime propagation, and fL is defined from (5.52) and (5.71) as

fL = fH cos γ

(5.196)

5.8  Attenuation of the Radio Waves in the Atmosphere 303

where fH ≈ 1.4 MHz and g is the angle between the direction of propagation and the geomagnetic meridian (see Figure 5.18), which is approximately the same as the direction of the Earth’s magnetic field, especially for the low and middle latitudes. The fact that AD ′  , AE′ , and A′F1 are defined by only one parameter, fc,E,MHz, is because the critical frequencies of all three ionospheric layers, fc,D, fc,E, and fc,F1 are dependent on just one single factor—the solar activity. In order to simplify the engineering approaches it’s reasonable to use just one of those three parameters (e.g., fc,E), keeping in mind that all three attenuations are bounded originally with just one source. The family of curves for the total attenuation A′S(fc,E) as a function of the length of a single-hop propagation distance (horizontal distance along the Earth’s surface), Rh, is presented in Figure 5.46. As seen from (5.195) the increase of single-hop distance results in increase of A′S due to increase of the angles of incident jD, jE, and jF1. Besides the nonreflecting attenuation, some attenuation takes place during reflection from the F2 layer, which may be calculated by using the following formula:

2 AF 2 = BF 2 fMHz

(5.197)

where BF2 may be defined from Figure 5.47 for the given distance and effective height of the F2 layer, heff, shown in Figure 5.45(b). The decaying shape of the family of attenuation curves on Figure 5.47 is due to the increase of the angle of incidence, jF2, when the horizontal distance Rh is increased, and as a result the altitude of the reflection point C′ lowers, shown in Figure 5.45(b). The lower the height of the reflection point is, the smaller the distance MC′N will be covered by the propagation path, and the less attenuation

Figure 5.46  Total nonreflecting attenuation of the radio wave of the frequency 1 MHz in ionosphere as a function of the E layer’s plasma frequency fcE,MHz.

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Atmospheric Effects in Radio Wave Propagation

Figure 5.47  Attenuation of the radio wave in reflecting F2 ionospheric layer for the frequency of 1 MHz as a function of single-hop distance, Rh, and the effective reflection height, heff.

will occur. The height heff of the vertex C′ of the track depends on the ratio of the carrier frequency and the critical frequency of the reflecting layer, namely, f/fc,F2, provided in ionospheric MUF predictions. Otherwise, it may be taken roughly heff ≈ 300 to 350 km. Those advance monthly predictions have been provided in the United States for several decades by Central Radio Propagation Laboratory (CRPL) of the National Bureau of Standards, currently known as the National Institute of Standards and Technology (NIST). Similar governmental services exist in other countries. Ionospheric predictions are in strong correlation with the index of solar activity measured in Wolf numbers (also known as Zürich numbers), which are highly predictable for many years in advance, taking into account the 11-year solar activity cycle. The total attenuations from all layers, nonreflecting and reflecting, is defined for the daytime as A = AΣ + AF 2



(5.198)

Note finally that sometimes the reflections from the layers lower than F2, such as E or F1, are possible (see Figure 5.11). The specific approaches must be applied in each particular case. For instance, if reflection takes place from the E layer, then the total attenuation may be found as follows



A≈

AD ′ 2

(f + fL )

+ BE f

f f + fL

(5.199)

5.8  Attenuation of the Radio Waves in the Atmosphere 305

where BE ≈



4 fc, E, MHz

cos2 ϕ E



(5.200)

is a semi-empiric expression, which represents the reflecting attenuation at the frequency of 1 MHz. The first term in (5.199) represents the nonreflective absorption from the D layer only, whereas the second term represents the reflective absorption within the E layer. The median value of the field strength at the receiving point for the multihop propagation path may be found from the following empirical expression: EB = E0



 n −1 1+ Γ  exp(− A) Γ 2

(5.201)

where •

• •

•

E0 is the free-space field strength (3.40) for the radio wave, which covers a distance counted along the Earth’s surface between radiation point A and reception point B. n is a number of hops. A is total attenuation in ionospheric layers, in nepers (Np), including both reflective and nonreflective absorptions. They’re defined either by (5.198) or (5.199).  )/ 2 takes into account the impact of the Earth’s surface The multiplier (1 + Γ  as the average reflection coef­­ at the transmitting and receiving points for Γ ficient at those points. For practical calculations, it may be taken approximately equal to 0.8.

Finally, the approaches considered in this section are mostly applicable to HF communication lines assessments and designs. Those lines are still in use by military and commercial users despite high competition from the satellite communication (satcom) systems. HF communication systems are still used as substitutes or backups for those highly developed systems. Example 5.5

Determine which ionospheric layer reflects the radio wave on a single-hop HF communication line (n = 1) at the frequency f = 28 MHz, and estimate the RMS field strength at the reception point if the horizontal propagation distance is R = 1,800 km. The angle between the direction of propagation and geomagnetic meridian is g = 45°. Power radiated by transmitter is PTx = 200W with the antenna gain of GTx = 20 dB (100 unitless). For ionospheric layers parameters, assume the following average values taken from Table 5.1 for a daytime: • •

D layer: Ne = 5 ⋅ 103 electrons/cm3, hmax = 75 km (nonreflecting layer). E layer: Ne = 2.8 ⋅ 105 electrons/cm3, hmax = 120 km.

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Atmospheric Effects in Radio Wave Propagation

• •

F1 layer: Ne = 3.25 ⋅ 105 electrons/cm3, hmax = 210 km. F2 layer: Ne = 2 ⋅ 106 electrons/cm3, hmax = 315 km.

Solution

1. Check the reflection conditions for each E, F1, and F2 layer: (a) find the reflection angles from (5.43), (b) calculate plasma frequencies from (5.10b), (c) calculate MUFs for the given distance from (5.47), (d) check if the condition f < MUF is satisfied. • For the E layer: (a) j = 1.369 rad (78.44°), (b) fc = 4.76 MHz, (c) MUF-E1800 = 23.7 MHz, (d) f > MUF (this is nonreflecting layer). • For the F1 layer: (a) j = 1.275 rad (73.04°), (b) fc = 5.12 MHz, (c) MUF-F1 1800 = 17.56 MHz, (d) f > MUF (this is nonreflecting layer). • For the F2 layer: (a) j = 1.1715 rad (67.12°), (b) fc = 12.71 MHz, (c) MUFF2 1800 = 32.7 MHz (d) f < MUF (this is reflecting layer). 2. Find the angle of incidence on absorbing D, E, and F1 layer, considering F2 a reflecting layer. For the geo-central angle between communicating points along the great circle q = R/a = 1800/6370 = 0.2826 rad (16.19°), y = p - (j + q/2) = 1.8288 rad (104.78°; see Figure 5.48). In triangle ADO, DO = a + habs. Then, based on the law of sines,   sin ϕ Abs sinψ a = , hence ϕ Abs = sin−1  sinψ  a + hAbs a  a + hAbs 



(5.202)

Angles of incidence for proper layers are calculated from (5.202) as follows: • • •

D layer: jD = 1.27186 rad (72.872°). E layer: jE = 1.25 rad (71.63°). F1 layer: jF1 = 1.21 rad (69.4°).

Figure 5.48  Reflecting F2 and absorbing layers of ionosphere: habs is the height of the absorbing layer; jabs is the angle of incidence of the radio wave onto absorbing layer.

5.8  Attenuation of the Radio Waves in the Atmosphere 307

1. Calculating fL from (5.196): fL = fH cosg = 1.4 cos 45° = 0.99 MHz. 2. AS′ for nonreflecting layers is defined from (5.195) A′S = 4.762(3/cos1.272 + 2.5/cos1.25 + 0.4/cos1.21) = 436.22. 3. Total nonreflecting absorption in D, E, and F1 layers is defined from (5.194): AΣ =

AΣ′

2

(f + fL )

=

436.22 (28 + 0.99)2

= 0.52 Np (4.51 dB)

4. The reflective absorption in the F2 layer is defined by (5.197) for BF2 = 7 ⋅ 10-4 and is approximately obtained from Figure 5.47: 2 AF2 = BF2f  MHz = 7 ⋅ 10-4 ⋅ 282 = 0.55 Np (4.76 dB)



5. The total ionospheric absorptions from (5.168) are A = AS + AF2 = 0.52 + 0.55 = 1.07 Np (9.29 dB)



6. RMS value of the electric field strength in free space (reference conditions) 30PT GT 30 ⋅ 200 ⋅ 100 = = 4.4 ⋅ 10− 4 V/m = is defined by (3.40) as E0 = R 1800 ⋅ 103 0.44 mV/m.  = 0.8 , and 7. The final RMS value of the electric field strength is defined for Γ n = 1 by (5.201) as: E = E0

 n −1 1+ Γ  exp(− A) = 0.44 ⋅ 1 + 0.8 ⋅ exp(−1.07) = 0.1 Γ 136 mV/m (Answer) 2 2

References  [1] Levis, C. A., J. T. Johnson, and F. L. Teixeira, Radiowave Propagation: Physics and Applications, New York: John Wiley & Sons, 2010.   [2] Davies, K., Ionospheric Radio Propagation, Washington, D.C.: NBS, 1965.  [3] Rawer, K., Wave Propagation in the Ionosphere. Theory and Applications, Vol. 5, New York: Springer-Verlag, 1993.  [4] Al’pert, Y. L., Radio Wave Propagation and Ionosphere, “Vol. I, The Ionosphere,” New York: Consultants Bureau, 1974.   [5] Budden, K. G., The Propagation of Radio Waves: The Theory of Radio Waves of Low Power in the Ionosphere and Magnetosphere, New York: Cambridge University Press, 1988.   [6] Collin, R. E., Antennas and Radio Wave Propagation, New York: McGraw-Hill, 1985.   [7] Staelin, D. H., A. W. Morgenthaler, and J. A. Kong, Electromagnetic Waves, Englewood Cliffs, NJ: Prentice-Hall, 1994.  [8] Dolukhanov, M. P., Propagation of Radio Waves, Moscow: Mir Publishers, 1971.   [9] Booker, H. G., and W. E. Gordon, “Radio Scattering in the Troposphere,” Proc. IRE, Vol. 38, April 1950, pp. 401–421. [10] Rytov, S. M., Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, New York: Springer-Verlag, 1989.

308

Atmospheric Effects in Radio Wave Propagation [11] Gradshtein, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed., New York: Elsevier, 2007. [12] ITU-R Recommendation P.617-1, “Propagation Prediction Techniques and Data Required for the Design of Trans-Horizon Radio-Relay Systems,” International Telecommunication Union, 1992. [13] ITU-R Recommendation P.676-7, “Attenuation by Atmospheric Gases,” International Telecommunication Union, 2007. [14] Crane, R. K., Electromagnetic Wave Propagation Through Rain, New York: John Wiley & Sons, 1996. [15] ITU-R Recommendation P.838-3, “Specific Attenuation Model for Rain for Use in Prediction Methods,” International Telecommunication Union, 2003. [16] Chapman, S., “The Absorption and Dissociative or Ionizing Effect of Monochromatic Radiation in an Atmosphere on a Rotating Earth,” Proc. Phys. Soc., Vol. 43, pp. 26–45, 1931. [17] Pozar, D. M., Microwave Engineering, Fourth Edition, John Wiley & Sons, Inc., 2012.

Appendix 5A Volumetric Spectrum for Autocorrelation Function of Statistically Homogeneous and Isotropic Random Field For the statistically isotropic random field ∆ε (r ) the volumetric autocorrelation function depends only on the magnitude of the distance between two correlating points, ρ = r2 − r1 ; that is,

ψ ε (r1, r2 ) = ψ ε (ρ) = ψ ε (ρ)



(5A.1)

Assume the function ye(r) is given in spherical coordinates (r, z, j). First we define the elementary volume dV in spherical coordinates as [see Figure 5A.1(b)]: dV = ρ 2 sinζ ⋅ d ρ ⋅ dϕ ⋅ dζ



(5A.2)

Vector q of the spatial frequency is conventionally taken directed along negative Z-axes as shown in Figure 5A.1(a). Then the scalar product q ⋅ ρ is needed for (5.125): q ⋅ ρ = q ⋅ ρ ⋅ cos(π − ζ ) = −q ⋅ ρ ⋅ cosζ



(5A.3)

Now substitute (5A.2) and (5A.3) into (5.125), taking into account statistical isotropy (5A.1). Sε (q) = Sε (q) =

1 3

(2π )

⋅ ∫ ψ ε (ρ) ⋅ exp(i q ρ cosζ ) ⋅ ρ 2 sinζ ⋅ d ρ ⋅ dϕ ⋅ dζ V

(5A.4)

Variables in (5A.4) may be separated and volumetric integral may be written as

Sε (q) =

1 (2π )3

⋅

2π

∫

0

 π dϕ ⋅ ∫ ψ ε (ρ) ⋅  ∫ exp( i qρ ⋅ cosζ ) ⋅ sinζ ⋅ dζ  ⋅ ρ 2 d ρ  0  0 ∞

(5A.5)

Appendix 5B  Some Theoretical Aspects of the Ionospheric Layers Generation309

Figure 5A.1  (a) Positions of vectors q– and r-, and (b) the pattern of elementary volume dV in spheri­­ cal coordinates.

Now consider the integral within the brackets in the last expression:



π

π

0

0

∫ exp(i qρ ⋅ cosζ ) ⋅ sinζ ⋅ dζ = − ∫ exp(i qρ ⋅ cosζ ) ⋅ d(cosζ ) =

π 2 1 2 exp(i qρ) − exp(− i qρ) =− ⋅ exp(i qρ ⋅ cosζ ) = ⋅ = ⋅ sin(q ρ) 0 qρ 2i i qρ qρ

(5A.6)

2p

If ∫ dj = 2p is substituted into (5A.5) along with (5A.6), then final expression for 0

the spatial spectrum of the isotropic, homogeneous field ye(r) may be obtained as Sε (q) =

1 2π 2q

∞

⋅ ∫ ψ ε (ρ) ⋅ sin(qρ) ⋅ ρ d ρ 0

(5A.7)

which is ready to be used with different ye (r) models.

Appendix 5B Some Theoretical Aspects of the Ionospheric Layers Generation 5B.1 Ionospheric Gaseous Composition and Physical Processes Related to Ionospheric Propagation

Gaseous composition of the Earth’s atmosphere is shown in Figure 1.2 in Chapter 1. As seen from that diagram from the ground level and up to 100 km, that includes

310

Atmospheric Effects in Radio Wave Propagation

troposphere and stratosphere regions; the atmospheric air composition remains almost unchanged, containing mostly molecular oxygen and molecular nitrogen in the proportion of about 22%/78%. This sounds reasonable because of continuous vertical and horizontal mass movements in those regions that result in continuous air mixing and keeping that gaseous composition unchanged. Starting from elevations of about 60 km (which is the lower border of the ionosphere) and up to new gaseous components, such as atomic oxygen and nitrogen, as well as atomic helium and other minor components, are being involved. As pointed out in Chapter 1, the air gases here are ionized due to the impact of the cosmic radiations (cosmic rays) coming predominantly from the sun (called solar wind), as well as from the center of our galaxy (Milky Way) and several constellations. Cosmic rays represent a stream of elementary particles and electromagnetic radiations containing photons of very short wavelengths and very high energies: UV, X-rays, and gamma radiations. As mentioned in Chapter 1, when colliding with neutral atoms/molecules, these photons are able to knock an electron out of an atom or molecule, resulting in generation of pairs of a free electron and positively charged atom/molecule: this process is called strike ionization. Expression (1.3) in Chapter 1 allows defining a photon’s wavelength, and therefore the type of radiation needed to ionize particular gaseous component of the ionosphere:

λ < λmax =



c ⋅ c ⋅ = 6.25 ⋅ 1018 (5B.1) We,min, joules We,min,eV

Here c = 3 · 108 m/s is a speed of light, ħ = 6.626 · 10–34 J·s is a Plank’s constant, We,min,joules and We,min,eV are minimum required energy (ionization work function) needed to tear off the electron from atom/molecule, in joules and electron-volts (eV), respectively. Table 5B.1 represents the ionization work functions and proper wavelengths for the particular gaseous component of the ionosphere to be ionized [8]. The recombination process is opposite from ionization that is a reunion of the particles of opposite charge signs—electrons with positively charged atoms/molecules, resulting in generation of the neutral particles. The intensity of the recombination is in direct proportion to the ionospheric plasma concentration.

Table 5B.1  Work Functions and Threshold Wavelengths for the Ionospheric Composite Gases Composite Gas

Ionization Work Function, We,min,eV

λmax, nm

O2 – molecular oxygen O – atomic oxygen N2 – molecular nitrogen N – atomic nitrogen He – helium H2 – molecular hydrogen H – atomic hydrogen NO – nitrogen monoxide

12.2 13.61 15.51 14.53 24.58 15.42 13.6 9.25

101.83 91.28 80.10 85.50 50.54 80.57 91.35 134.31

Note: As seen from the table all threshold wavelengths lmax relate to UV radiation.

Appendix 5B  Some Theoretical Aspects of the Ionospheric Layers Generation311

5B.2 Chapman Model and Structure of the Real Ionosphere

Let us consider the nature of the origin of ionospheric layers using an example of a simple model by following [8, 16] with comments. First, we assume the ionosphere is composed of a homogeneous, single-component gaseous medium, with the air pressure described by a barometric vertical profile:

p(h) = p0 exp(–h/H) = p0 exp(–bh), (5B.2) where p0 is the air pressure at the Earth’s ground surface, and H = 1/b is a scale of heights, which is dependent on the type of gas and its temperature. For further evaluations there is no need to define that dependence at this point. Assume, in general, that the monochrome ionizing radiation is incident with angle of c as shown in Figure 5B.1. Then the path length AB of the radiation within that layer apparently is dl =



dh (5B.3) cos χ

Let’s introduce ionization intensity as a number of free electrons generated in a unit volume per second. Consider the ionizing radiation with the value of power flow density (Poynting vector) Πe that is incident to the ionospheric layer of infinitesimal thickness dh and constant pressure within that layer. It’s easy to show that taking into account (5B.3) the ionization intensity is expressed as



Ie (h) =

dNe (t , h) 1 d Πe 1 d Πe = = cos χ dt [ν ] dl [ν ] dh

(5B.2)

Here Ne(t, h) is ionospheric plasma concentration as a function of elevation and time, [ħn] is the quantum portion of energy of the radiation, and n is the photon’s frequency. Part of the radiation power lost in tiny region shown in Figure 5B.1 is apparently in direct proportion with the path length dl, the gas pressure p(h), and radiation intensity itself; that is

d Πe = B ⋅ Πe ⋅ dl ⋅ p0 exp(−bh) = B′ ⋅ Πe ⋅ exp(−bh) ⋅

dh (5B.4) cos χ

Figure 5B.1  Ionizing radiation passing through the infinitesimal layer of ionosphere.

312

Atmospheric Effects in Radio Wave Propagation

where B is a coefficient of proportionality and B′ = B·p0. Now we integrate this differential equation. After separation of variables and assuming that the initial value of ionizing power flow density that comes from infinity is Πe,∞: Π e ,∞

∞

d Πe B′ = ⋅ exp(−bh) dh (5B.5) Πe (h) cos χ ∫h

∫



Πe

Integration results in ln



Π e ,∞ B′ = exp(−bh) (5B.6) Πe (h) b ⋅ cos χ

Then from (5B.6)   B′ Πe (h) = Πe,∞ exp  − exp(−bh)  (5B.7)  b ⋅ cos χ 



Now if we differentiate (5B.7)   B′   d Πe (h) B′ = Π e ,∞ exp −  exp(−bh) + bh   (5B.8) dh cos χ     b ⋅ cos χ



and substitute the result into (5B.2). Then the following final expression may be obtained:



Ie (h) =

Πe,∞ B′    eletrons    B′ exp −  exp(−bh) + bh    3  (5B.9) [ν ]   b ⋅ cos χ    m ⋅ s 

To find the elevation of maximum of the ionization intensity, we need to take the first derivative of (5B.9) equal to zero, and solve that equation for h = hmax. dIe (h) dh

h = hmax

=

  B′ Πe,∞ B′ b    B′  exp −  exp(−bhmax ) + bhmax   ⋅  exp(−bhmax ) − 1 = ⋅ b [ν ] b cos χ cos χ ⋅      

  B′ Πe,∞ B′ b    B′  exp −  exp(−bhmax ) + bhmax   ⋅  exp(−bhmax ) − 1 = 0 [ν ]    b ⋅ cos χ    b ⋅ cos χ



(5B.10)

Only the last bracket becomes equal to zero. Then the solution for hmax results in hmax =

1  B′  ln , m b  b ⋅ cos χ 

(5B.11)

Appendix 5B  Some Theoretical Aspects of the Ionospheric Layers Generation313

For vertically incident ionizing radiation c = 0

hmax,0 =

1  B′  ln   , m b  b

(5B.11a)

Maximum ionization value may be defined by substitution of (5B.11) into (5B.9):

Ie (hmax ) = Ie,max =

Πe,∞ b cos χ = Ie,max,0 cos χ e ⋅ [ν ]

 electrons    (5B.12)  m3 ⋅ s 

Π e ,∞ b for vertical Here e = 2.7182… is a base of natural logarithm, and Ie,max,0 = e ⋅ [ν ] incidence. It is convenient to convert (5B.9) into the form of relative ionization intensity, if (5B.12) is taken into account; that is Ie (h)

Ie,max,0

=

  B′   B′ e− bh + bh  + 1 (5B.13) exp −  b     b ⋅ cos χ

Figure 5B.2 displays the ionization intensity normalized to its maximum value at c = 0 versus heights, normalized and centered to hmax,0 value. The existence of the ionized layer may be explained qualitatively as follows: at sufficiently high altitudes above the ground, where the ionizing radiation is intensive, the atmospheric

Figure 5B.2  Normalized vertical profiles of the ionospheric ionization intensities for the Chapman model for the homogeneous gaseous composition, for several values of angle of incidence, c of the monochrome ionizing radiation.

314

Atmospheric Effects in Radio Wave Propagation

gas density is low, and free electrons generation is not intensive due to very small number of the atoms/molecules to be ionized. Further penetration of the radiation into atmosphere is accompanied with the barometric increase of the pressure of gases (higher density of the gas particles). This results in higher intensity of ionization, and more losses of power by ionizing radiation; when it penetrates more into atmosphere the radiation won’t have enough power for keep the same ionization intensity despite the larger density of the atmospheric gases. This in turn results in a decrease of ionization intensity. In other words, there’s a maximum of the ionization intensity located between the areas of low and high densities of the atoms/ molecules of the atmospheric air. As seen from this figure: (1) most intensive ionization may be observed around noontime when ionizing radiation is vertically incident, and (2) the extreme point of the ionization shifts up when changing from noon to evening hours. As pointed out in Section 1.3, the ionization process is accompanied by the inverse process of recombination of the free electrons with positively charged atoms/ molecules. Assume we have a case of one single electron and one single positively charged particle (atom/molecule) in a unit volume (1m3). Let the probability of its recombination within that unit volume be ae; then the time interval needed for the recombination is apparently proportional to 1/ae. If the number of free electrons is increased to Ne (ionospheric plasma concentration) along with the number of positively charged particles, then probability of the single electron recombination will apparently increase. This jump of probability will follow the rule of Ne2ae [8]. Therefore, the change of the resultant number of free electrons per unit volume (plasma concentration) in the presence of both ionization and recombination will take place with the rate of

dNe = Ie − Ne2α e dt

 electrons    (5B.14)  m3 ⋅ s 

It is obvious that for the balanced condition between ionization and recombination, when plasma concentration remains unchanged, dNe/dt = 0, the value of Ne may be defined from (5B.14) as

Ne,max = Ie α e

 electrons    . (5B.15)  m3 

This expression is typical for midnight and noontime, when all transitional processes due to sunrise and sunset are over. After sunset, when the ionizing radiation disappears Ie = 0, the expression (5B.14) transforms into



dNe = − Ne2 ⋅ α e (5B.16) dt with the following solution for the initial condition Ne (t)

t =0

= Ne,max

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere315

Ne (t) =



Ne,max 1 + Ne,max ⋅ α e ⋅ t

 electrons    (5B.17)  m3 

In other words, the ionospheric plasma concentration decays hyperbolically after sunset. Note that this approximate formula (5B.17) doesn’t count for the very low reluctant ionizing radiation intensity during the night hours. Finally, it is to be pointed out that in the real ionosphere the conditions related to ionization and recombination are much more complex, and therefore the real measured profiles of plasma concentration are shown in Figure 1.3 for day and night times. The reasons are as follows: • • •

•

The ionizing radiation is not monochrome, but rather includes a wide spectrum; The real ionosphere is a multicomponent, layered structure of the inhomogeneous distribution as shown in Figure 1.2; The ionization process is caused not just by electromagnetic radiations, but also by those caused by a stream of particles (mostly by protons and neutrons), as a composite part of solar wind; An additional source of the ionization is a cosmic dust and micrometeors that are arriving into Earth’s atmosphere with high enough speed/energy.

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere 5C.1  General Relations

A general expression for a plane wave in vector form is similar to (2.59) and may be written for both electric and magnetic fields:

     E(r) = E m e − ik⋅r (5C.1)



     H(r) = H m e − ik⋅r (5C.2)

   where E m and H m are initial amplitude phasors, and k is the phasor of the prop agation constant—the subject for evaluation, r, is a vector distance. Note that a time-harmonic multiplier is omitted for simplicity. If (5C.1) and (5C.2) are substituted into (2.52) and (2.53), then we can rewrite them as

   -ik × E m = −ωµ0 µ H m (5C.3)    −ik × H m = iωε0εˆ E m (5C.4) As we demonstrate below, the permittivity of the medium under consideration is assumed to be a second rank tensor quantity, which is a 3 × 3 matrix. A differential nab-

316

Atmospheric Effects in Radio Wave Propagation

  la-operator ( ∇) is replaced by −ik, an exponent multiplier; this replacement is specific to the regular differentiation procedure when dealing withexponential functions.  Now if we combine (5C.3) and (5C.4) by excluding H m and assuming that the medium is nonmagnetic; that is, µ = 1. Then we will end up with the following second order differential equation for the given vectors:

(

)

    k × k × E m = −k02εˆ E m (5C.5)



Here k0 = ω ε0 µ0 = 2π / λ0 is a free space wave number (see Section 2.2), and εˆ is a tensor of the relative dielectric permittivity. Expression (5C.15) is known as Appleton-Hartree equation in vector form for  evaluation of k. For further analysis let’s express the left-hand side through the components; that is  x 0      k × k × E m = k ×  k x   E m, x 

(

)

y0 k

z0 k

E m, y

E m, z

y

z

  x0   = k x     k y E m, z − k z E m, y  

y0 k

z0 k

k z E m, x − k x E m, z

k x E m, y − k y E m, z

y

z

  16  (5C.6)   

Then (5C.5) may be rewritten if εˆ is expressed in matrix form:  x0   k x   k y E m, z − k z E m, y 

y0 k

z0 k

k z E m, x − k x E m, z

k x E m, y − k y E m, z

y

z

     = −k02      

ε11 ε 21

ε12 ε 22

ε31

ε32

 ε13   x0 E m, x  ε 23   y0 E m, y   ε33  ⋅  z0 E m, z 

    (5C.7)   

Two particular cases are considered below that are typical for radio wave propagation analysis. 5C.2 Propagation Along Geomagnetic Field Lines 5C.2.1  Tensor of Dielectric Permittivity

In Section 5.1 we evaluated dielectric permittivity of the ionospheric plasma when the impact of the geomagnetic field was ignored. The expression (5.3) shows the sum of two colinear forces, namely inertial and friction forces, which are balanced by the force applied from the electric field of the external monochrome radiation. Recall (5.3), which is Newton’s third law of motion in vector form:



    dv  eE = Finertial + Ffriction = me + ξ me v (5C.8) dt

16. Here and below the brackets are used to distinguish the notation of the determinant of the matrix, from the notation of the matrix itself.

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere317

Here me = 9.1 ⋅ 10–31 kg is a mass of electron, x is a number of collisions of the   electron with neutral atoms/molecules per second, v and dv / dt are the electron’s movement velocity and its acceleration colinear to each other.  In the presence of DC magnetic field H0 there’s a Lorentz force (5.48) applied to the moving electron, so (5C.8) should be rewritten as     dv    eE = Finertial + Ffriction = me + ξ me v − e µ0[v × H0 ] (5C.9) dt



 To be consistent with the approaches adopted in Section 5.5 let H0 be directed along the Z-axis; that is, H0 = H0 ⋅ z0. Then (5C.9) may be rewritten for the frequency domain as e    E = (iω + ξ)v − ω H [v × z0 ] (5C.10) me



where wH is a Larmor (cyclotron) frequency defined by (5.50).   Next, let’s express (5C.10) in vector form by substituting v = (1 / eNe )J from (5.6). After simple transforms we may obtain    ω c2ε o E = (iω + ξ)J − ω H [J × z0 ] (5C.11)

Here

ωc =



Ne e 2  rad  (5C.12) ε0 me  s 

which is a plasma frequency (Langmuir frequency) in radians per second, in compliance with (5.8). If we extend the cross product on the left-hand side and present in Cartesian coordinates, then (5C.11) may rewritten as

(

)

(

ω c2ε0 E x x0 + E y y0 + E z z0 = (i ω + ξ) J x x0 + J y y0 + J z z0

)

 x  0 − ω H  J x   0

y0 J y

0

z0   J z  (5C.13)  1 

The right-hand side of this expression may be written as

(

)

 iω + ξ x0 + ω H y0  J x +  −ω H x0 + (iω + ξ) y0  J y + (iω + ξ)z0 J z (5C.14)  

318

Atmospheric Effects in Radio Wave Propagation

If (5C.13) is taken into account, then (5C.14) will turn into matrix form:

ω c2ε0



E x E

=

y

E z

iω + ξ

ωH

0

−ω H

iω + ξ

0

0

0

iω + ξ

⋅

J x J y

J z

(5C.15)

For further evaluations we invert (5C.15) to solve for matrix ||J||, and express it in terms of matrix ||E||: J x J y

=

J z

ω c2ε0 2 − ω 2 + ξ 2 + 2iωξ ωH

⋅

iω + ξ

−ω H

0

ωH

iω + ξ

0

0

iω + ξ +

0

⋅

E x E y

E z

2 ωH

iω + ξ

 (5C.16)

If (5C.16) is compared with Ohm’s law in differential form, (2.10), then it’s easy to realize that the expression in front of the ||E||-matrix is just a tensor of conductivity σˆ . Then the tensor of the dielectric constant may be found if (2.48) is recalled in matrix form written for the vacuum filled with free electrons (i.e., for the spatial area with conducting currents only):

εˆ =

ε11 ε 21

ε12 ε 22

ε13 ε 23

ε31

ε 22

ε33



= I3 − i

σˆ = ... ωε

2

... = I3 −

ω c ε0

i ⋅ ωε0 ω H2 − ω 2 + ξ 2 + 2iωξ

iω + ξ

−ω H

0

ωH

iω + ξ

0

0

0

iω + ξ +

(5C.17)

ω H2 iω + ξ

Here 1 0 0

I3 =

0 1 0

0 0 1

(5C.18)

is a unit matrix of 3 × 3 size. Then εˆ matrix may be evaluated as

εˆ =

ε1 iε 21

−iε 2 ε1

0

0

0 0 ε3

(5C.19)

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere319

with the elements

ε1 = 1 −

        (5C.20)       

 ξ ω c2  1 − i   ω

ω 2 − ω H2 − ξ 2 − 2iωξ ω ω c2  H

  ω ε 2 = 2 ω − ω H2 − ξ 2 − 2iωξ



 ω H2  ξ   ω c2  1 − i  ⋅ 1 − 2  ω   ω − 2iωξ − ξ 2    ε3 = 1 − 2 2 2 ω − ω H − ξ − 2iωξ

In the case of ideal, lossless medium (x = 0), expressions (5C.19) and (5C.20) will turn to

1−

εˆ =

ε1

− iε 2

0

iε 2

ε1

0

0

0

ε3

=

i

ω c2 ω 2 − ω H2

−i

ω c2 ωH ω ω 2 − ω H2

ω c2 ωH ω ω 2 − ω H2

1−

0

0

ω c2

 (5C.21)

0

ω 2 − ω H2 1−

0

ω c2 ω2

5C.2b  Plane Wave Propagation

Let’s recall (5C.7) and apply the following assumptions for the considered case:  k = kz z0 , k x = k y = 0. Then (5C.7) may be reduced to  x0     k × k × Em =  0    −kz E m, y

(

)

z0    k z  = −k02εˆE m = −k02εˆ ⋅  0  

y0 0 k E z m, x

E m, x x0 E y m, y 0

E m, z x0

(5C.22)

or −k z2 ⋅

x0

y0

z0

⋅

E m, x E m, y

0

= −k02 ⋅

ε1 iε 2 0

−iε 2 ε1 0

0 0 ε3

⋅

x0

y0

z0

⋅

E m, x E m, y

 (5C.23)

E m, z

Note that as follows from (5C.23) E m,z = 0, which is in agreement with the initial statement. (5C.23) may now be replaced by the system of two linear equations for E m,x and E m,y .

320



Atmospheric Effects in Radio Wave Propagation

(kz2 − k02ε1 ) E m,x − ik02ε2E m,y = 0 ik02ε 2 E m, x + ( k z2 − k02ε1 ) E m, y = 0

   (5C.24)  

The nontrivial solution of this (5C.24) exists if, and only if, the determinant of this system is equal to zero; then the solution for k z may be found as

k z1,2 = k0 ε1 ± ε 2 (5C.25) As seen from (5C.25), we ended up with two independent solutions. After substitution of e1 and e2 from (5C.21) for the lossless plasma (x = 0), one may obtain the following formulas for k z solutions:





k z1 = k0 ε1 + ε 2 = kz1 = k0 1 −

ω c2 (5C.26) ω (ω − ω H )

k z 2 = k0 ε1 − ε 2 = kz 2 = k0 1 −

ω c2 (5C.27) ω (ω + ω H )

These solutions may easily be confirmed if (5C.26/27) are substituted into (5C.24). To clarify the physical meaning of the above solutions, let’s consider the solutions for the electric field by plugging in k z1 into one of (5C.24). The equality is achieved if E m,y = −iE m,x . This means that for the first solution k z1 we have the electric field expressed as

 E m = E m,x ( x0 − iy0 ) (5C.28) which is an amplitude phasor for the RHCP wave [compare with (2.154a)]. For this RHCP component of the radio wave the dielectric constant of this anisotropic medium may be obtained from (5C.26), if we recall that k z12 = k02ε + = k02 ( ε1 + ε 2 ) and use e1 and e2 from (5C.21):



ε + = ε + = 1 −

ω c2 ω (ω − ω H )

(5C.29)

We may apply a similar procedure to the second solution, k z2; that results in E m,y = iE m,x . A proper electric field component may be expressed as

 E m = E m, x ( x0 + iy0 ) (5C.30) which relates to an LHCP component of the radio wave [see (2.154b)], and the dielectric constant of the medium for this mode may be found as



ε − = ε − = 1 −

ω c2 ω (ω + ω H )

(5C.31)

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere321

Note that in the absence of the electrons’ collision with other ionospheric particles (atoms, ions) both, e+ and e– are real numbers in limited frequency range, which confirms that no thermal losses are present in this medium. 5C.3  Propagation Across Geomagnetic Field Lines 5C.3a  Tensor of Dielectric Permittivity

In this case  we assume the direction of the external DC magnetic field is along the X-axis, H0 = H  0 ⋅ x0 shown in Figure 5.16, and propagation direction is perpendicular to it, k = k z z0 (k x = k y = 0). Then expression (5C.13) may be modified as follows:

(

)

(

ω c2ε0 E x x0 + E y y0 + E z z0 = (iω + ξ) J x x0 + J y y0 + J z z0

)

 x  0 − ω H  J x   1

z0   J z  (5C.32)  0 

y0 J y

0

which turns (5C.8) and (5C.9) into the following matrix equations:

ω c2 ε 0

E x E y

iω + ξ

0

0

0

iω + ξ

−ω H

0

ωH

iω + ξ

=

E z



J x J

=

y

J z

ω H2

iω + ξ +

2

ω c ε0 ω H2 − ω 2 + ξ 2 + 2iωξ

J x J

⋅

0

0

0

iω + ξ

ωH

0

−iω H

iω + ξ

iω + ξ

(5C.33)

y

J z

⋅

E x E y

E z

 (5C.34)

Now let’s skip the intermediate steps relating to expressions (5C.10) to (5C.18), and write the final result for tensor of dielectric permittivity in lossless medium (x = 0); that is 1−

εˆ =

ε3

0

0

ε1

iε 2

0

−iε 2

ε1

0 =

ω c2

0 0



0

ω2 1− −i

0 v

ω 2 − ω H2

ω c2 ωH ω ω 2 − ω H2

i

ω c2 ωH ω ω 2 − ω H2 1−

 (5C.35)

ω c2 ω 2 − ω H2

which, as seen from comparison with (5C.21), is simply its transposed conjugate.

322

Atmospheric Effects in Radio Wave Propagation

5C.3b  Plane Wave Propagation

 Now the equation (5C.7) is written, with (5C.35) in mind, as well as k = k z z0; that is

k z2 ⋅

E m,x E

= k02 ⋅

m,y

0 0

0



ε3

0 ε1 −iε 2

0 iε 2

E m,x E

⋅

m,y

(5C.36)

E m,z

ε1

or k02ε3E m,x

k z2E m,x k z2E m,y

k02ε1E m,y + ik02ε 2 E m,z

=

−ik02ε 2 E m,y

0



(5C.37)

+ ik02ε1E m,z

As in previous case of longitudinal propagation, two independent solutions may be obtained from this matrix equation: •

(k

)

− k02ε3 E m,x = 0. This means that for the finite value of E m,x the following k 2 − k 2ε = 0 is to be requested, which, after taking into account 2 z,1

0

z,1

(5C.35), results in

3

2

ε ord



•

if k z = k0 ε is kept in mind. This solution relates to so called ordinary wave, which behaves exactly the same way as if no magnetic field exists in the plasma medium. The physical meaning behind it is clear: as seen from Figure 5.16 vector E m, x is perpendicular to planes of rotation of each electron, therefore those circular movements are not affecting propagation of the radio wave; it behaves as it would behave in nonmagnetoactive plasma. A second solution may be found if the second and third rows of the matrix equation (5C.37) are presented as a system of equations

(k

2 z



ω  = ε3 = 1 −  c  (5C.38) ω 

)

− k02ε1 E m,y − ik02ε 2 E m,x = 0 ik02ε 2 E m,y − k02ε1E m,z = 0

   (5C.39)  

A nontrivial solution of this system is obtainable if the determinant is taken to equal zero; then the result is found as follows:

Appendix 5C Plane Wave Propagation in Homogeneous Magnetoactive Plasma of Ionosphere323

ε ext =

ω  ε12 − ε 22 = 1−  c  ω  ε1

2

1 1−



ωH   ω ω 1−  c

2

 ω

(5C.40) 2

This solution relates to the extraordinary wave, which propagates under strong impact from the magnetic field. An interesting property of this propagation mode is that both components of the radio wave exist (i.e., E m,y ≠ 0 and E m,z ≠ 0 ), being related as E m,z = i ( ε 2 ε1 ) E m,y . The conclusion here is that the resultant electric field vector resides in the YOZ-plane and is elliptically polarized with the polarization ellipse tilted towards the Z-direction. Finally, note that those two waves—namely ordinary and extraordinary—exist independently, so any wave that is arbitrary polarized in the XOY-plane may be decomposed into two components, E m,y ≠ 0 and E m,z ≠ 0, further analyzed separately.

CHAPTER 6

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception 6.1  Multiplicative Interferences (Signal Fades) 6.1.1  Fluctuation Processes and Stability of Radio Links

The field strength of the radio wave at the reception point in real conditions is counted based on propagation factor F in (3.43), which is introduced in Section 3.2 to accommodate the reference path (ideal propagation track) to real conditions. In other words, the propagation factor allows us to take into account the impact of the Earth’s surface and the atmospheric effects. The electromagnetic parameters of the Earth’s surface and the atmospheric propagation medium are not constant, but randomly fluctuating in time and space; therefore, generally the magnitude and initial phase of the complex propagation factor F must be considered random variables in the time domain at any particular receiving point. Those random time-variations/ fluctuations may be expressed as F = F(t)exp[i ΦF (t)]



(6.1)

where F(t) is a random magnitude of the propagation factor, and FF(t) is its random phase. Based on (3.43) for the RMS (effective) value of the electric field strength, the expression for complex time-harmonic field strength may be written in the following form:

E (t) = E m, 0 F exp(iω t) = Em, 0 exp{i [ω t + ΦE (t)]} F

(6.2)

E m, 0 = Em, 0 exp[i ΦE (t)]

(6.3)

where

is the amplitude phasor of the signal in free space with the magnitude of



Em,0 = E0 2 =

60 PTxGTx r

(6.4)

and E0 is the effective value of the reference field strength of the radio wave [compare with (3.43)]. The randomly fluctuating amplitude of the field strength may be defined from (6.1) and (6.2) as a product

Em (t) = Em, 0 F(t)

(6.5)

325

326

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.1  Random variations of effective field strength of the signal at the receiving point in the presence of multiplicative interference (fading).

From (6.5) one may notice that in real conditions the field amplitude at the reception point randomly fluctuates because of random variations of F(t). The value of amplitude Em,0 for the ideal propagation conditions is multiplied by a random factor that represents the fluctuations on real propagation path. This type of interference is called multiplicative interference or fading. Therefore, fading is considered fluctuation of the wanted signal caused by random changes of the signal propagation conditions.1 Methods of the probability theory and statistics are employed for the quantitative description of fading statistics. For numerical estimates, the probability of any value E of the continuous random variable E′ is formally defined as a ratio of the sum of the time periods, DTn, when E′ faded below E (E′ ≤ E) to the total observation time tTot (see Figure 6.1); that is,



P(E) =

∑ ∆Tn n

tTot

=

∆T1 + ∆T2 + ∆T3 + ∆T4 tTot



(6.6)

As one may see from Figure 6.1, the greater the value of P(E), the larger the probability P(E) is. This functional dependence is called the cumulative distribution function (CDF). Another important statistical measure of the random variable is probability density function (PDF). Suppose DP(E) is the probability of the field strength E 1. Taking into account (6.1) and (6.2), the random total phase of the radio wave at the reception point is FE(t) + FF(t). For AM systems, the initial phase FE(t) = F0 is constant (i.e., doesn’t carry information). In FM and PM systems FE(t) carries the information. Thus, at the output of demodulator the component FF(t) appears as an additive interference (noise) (see Section 6.2). Fortunately, the rate of fluctuations of FF(t) is usually much slower than the rate of time variations of FE(t) due to the informative signal. Hence, the power spectrum of this additive noise is limited to less than 10 Hz and may be easily cleared off with the highpass filter.

6.1  Multiplicative Interferences (Signal Fades) 327

ranged between E and E + DE; that is, in expression (6.6) the quantity

∑ ∆Tn is n

defined not as the cumulative time interval, when the random field strength remains below the value E, but the cumulative time interval, when it remains within E and E + DE limits. Then PDF is defined as:



∆P dP = = P′(E) ∆E→0 ∆E dE

w(E) = lim

(6.7)

which is the probability of the random variable E′(t) remaining within the unit interval that surrounds the particular value E. From (6.7) it’s easy to see that the relation between CDF and PDF is E



P(E) = ∫ w(E)dE 0

(6.8)

Per definition of the cumulative probability, CDF may not exceed the unity; therefore, w(E) must satisfy the normalization condition: ∞



∫ w(E)dE = 1

0

(6.9)

One of the useful statistical measures of the random E′(t) process that is widely used in radio communications and radar engineering applications is the median value, Emed, defined from the following equation:

P(E med ) = 0.5

(6.10a)

As follows from (6.10a), Emed has a meaning of deterministic parameter of the random variable E′(t) that represents the value of E being exceeded with the probability of 0.5 (i.e., being exceeded within 50% of the total observation time). Another important parameter of the random variable is the mean value E, called average value or statistical expectation defined from the following averaging integral: E =

1 tTot

tTot

∫

E′(t)dt

0

(6.10b)

The definitions (6.10a) and (6.10b) are considerably different, so the quantities 〈E〉 and Emed are different in general. However, for statistics those describe fading of the radio waves, the values 〈E〉 and Emed are fairly close to each other, so the dif­­­ ference may often be ignored. It’s easy to realize that neither 〈E〉 nor Emed allow us to specify the depth of fades. It is customary to obtain the fade depth based on deciles E0.1 and E0.9, which show the levels of observing signal (usually in dB) that are exceeded within 10% and 90% of total observation time, respectively.2 Therefore, from the practical 2.

Note that Emed = E0.5. Thus, Emed may be considered a decile.

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

point as a measure of the fades depth the use of the difference E0.1 - E0.9 (in dB) in RF propagation is more convenient than the standard deviation that is widely used in other probability and statistics applications. In engineering practice the term channel reliability is introduced as a measure of quality of the service: it is defined as percentage of the total serving time while the performance of the communication system is not less than its minimum (threshold);3 this threshold is usually specified by industry standards. Therefore, in practical applications another statistical measure, namely, complementary cumulative distribution function (CCDF) is introduced similar to that given in (4.139) for the complementary error function. CCDF is more commonly used in communications and radars, rather than cumulative distribution function: CCDF indicates the percentage of total observation time while random variable E′ is greater than or equal to a considered fixed value E. Hence, taking into account (6.8) and (6.9), the expression for CCDF may be written as ∞



CCDF = W (E) = 1 − P(E) = ∫ w(E) dE E

(6.11)

Within time intervals Dt1, Dt2, and Dt3 of the total observation time, tTot, shown in Figure 6.2, the fluctuating field E may fade down the minimum allowable value Emin that causes a communication outage (break). Emin is the lowest electric field that the receiver is able to sense in the presence of additive noises in terms of its ability to extract the modulating signal from the carrier. In order to estimate channel reliability, one may refer to Figure 6.2, where the short-term4 pattern of the field strength at the receiving point is presented. From this definition, the quantitative value of channel reliability may be found based on CCDF as (see Figure 6.2):

W0 = W (E min ) (6.12) It is equal to the probability that the fluctuating field at the reception point is above its minimum value (i.e., it is equal to W(E) for the value of E = Emin). It may be realized that higher the margin between Emed and Emin is, the greater the channel reliability W0 is. Therefore, for any particular radio link, the greater the ratio Emed/Emin, the higher the channel reliability. This ratio is called fade margin and in most cases is expressed in decibels, as follows:



fade margin = Emed , dB − Emin, dB

3. In some references [1], the term channel reliability = 100 ⋅ (1 - t0 /ttot) = ta /ttot is used: t0 is the total outage (break) time, ttot is total observation time, and ta is total availability time. 4. Short term is introduced to emphasize that during the total observation time period tTot, the median value of the electric field remains constant (i.e., this random process remains stationary). Therefore, we call it interval of stationarity, tStat. For different types of radio links, tStat has different values. For instance, for HF links, tStat is equal approximately several to tens of minutes.

6.1  Multiplicative Interferences (Signal Fades) 329

Figure 6.2  Sketch of the complementary cumulative distribution function W(E) of the fading field of the radio wave.

On the other hand, the probability of a communication break is apparently equal to

T0 = 1 − W0

(6.13)

Generally, these statements are applicable not just to randomly fluctuating E(t), but also to signal-to-noise ratio (SNR), which actually behaves as a random variable with the same statistical properties as for E(t), if assumed the noise intensity is unchanged within the observation time interval. Channel reliability standards for most of the service types are defined in terms of SNR rather than in terms of E(t) (i.e., the quantities such as SNRmed, SNRmin, and fade margin for SNR are commonly used in radar and communications engineering). If, instead of short-term observation, the long-term observations (hours, days, months) are conducted, then in predominant propagation cases the median field strength, as well as other statistical parameters, do not remain constant, but also are changing randomly, as shown in Figure 6.3. This means that generally

Figure 6.3  Random variations of the effective field strength at the receiving point for long-term observation interval.

330

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

fading is a complex, nonstationary random process (i.e., the process has nonstable statistical parameters, such as average, median, and standard deviation, which are also randomly fluctuating). It was noticed from numerous observations that in real conditions, any long-time interval may be divided into a several short-time intervals so that the statistical parameters within each of those short-time intervals may be considered to be almost constant. That is, each of those short-time intervals may practically be considered intervals of stationarity, tstat (see Figure 6.3). From these considerations, one may conclude that two types of fading persist in radio links: fast and slow. Fast fading appears as random variations within the stationarity intervals, whereas slow fading is considered long-term, slow variations of the median value of the field strength, which usually appears together with variations of other statistical parameters, such as standard deviation and mean value. The natures of fast and slow fades are totally different and independent. The cause of fast fades relates to multipath propagation. A typical example is tropospheric scattered propagation considered in Section 5.7. For multipath propagation, different independent rays approach the reception point by passing different paths with random propagation distances (Dr), causing random interference. Those fluctuations of Dr are the result of fluctuating positions of scattering irregularities of the tropospheric “clear air” on tropo-scatter radio links [Figure 6.4(a)], or by random changes of the reflection points and reflection heights on HF-ionospheric radio-links [Figure 6.4(b)]. Hence, the phase shifts between different rays DF = (2p/l)Dr becomes a random variable. It’s sensitive to both fluctuations of path differences Dr and the wavelength. The smaller the wavelength l is, the larger the fluctuations of phase shifts are, and therefore, the higher the fast fading rate is. When arriving at destination point B with random phase shifts within a wide range [0, 2p], different rays are superimposing differently. The extreme cases are equal-phase superposition, which causes constructive interference, and opposite-phase superposition, which causes destructive interference. The random interference is typical for most of RF links. The greater the number of superimposing rays, the deeper the fading is, and the more considerable its impact on communication stability is. Slow fading of the radio waves is caused by random changes of attenuation along the radio propagation path, which are considered a result of slow changes in the electromagnetic parameters of Earth’s surface and atmosphere, due to the random variations of temperature, humidity, air pressure, ionization, and so on. One may realize that fast and slow fading are two independent random processes acting simultaneously within the antenna-medium-antenna propagation path. There-

Figure 6.4  The multipath propagation mechanisms on (a) tropospheric and (b) ionospheric radio links.

6.1  Multiplicative Interferences (Signal Fades) 331

fore, statistical distributions are expected to be composite functions, combining fast and slow fade statistics. This issue will be considered in Section 6.1.4. 6.1.2  Fast Fading Statistical Distributions

Here we will start with simple case of two-ray random interference, continue to the case of multiray superposition of the independent wavelets (Rayleigh ensemble), and then further consider more complex cases in Section 6.1.2.4 to provide generalized approaches. 6.1.2.1  Two-Ray Random Interference

Consider a random interference of two rays; that is, a superposition of two sine waves with the constant amplitudes of the electric field Em1 and Em2, and same spatial orientation. We assume that the phase shift between them is a randomly time-dependent, Φ(t), which is evenly distributed within the range [−p; + p]. This case is typical, for example, for the interference between direct and reflected waves when antennas are placed on mobile platforms above the Earth’s ground; then the relation between electric fields at the reflection point is expressed as E m2 = Em1 ⋅ Γ E exp i ΦΓ  , where ΦΓ is reflection phase (see Section 2.4.3). Let’s keep in mind that the phase shift between those two waves is not only due to ΦΓ, but also due to the difference in distances covered by direct and reflected waves, which is a random variable of the platforms that are randomly moving, such as in the case of cars or aircraft. Generally speaking, the relation between amplitudes may be expressed as E m2 = Em1 ⋅ R exp i Φ ( t )  ; here R = Em2/Em1 is the ratio between amplitudes, and Em1 is assumed to be a real number without limitations to further analysis. Note also that Φ(t) includes both above phase shifts. The resultant random amplitude may be expressed as E m (t) = Em1 {1 + R ⋅ exp[ jΦ(t)]} (6.14)

with the magnitude of

Em (t) = Em1 1 + R2 + 2R ⋅ cos Φ(t) (6.14a)



As seen from (6.14a), the radical represents propagation factor, per its standard definition (3.43); it represents the electric field’s amplitude/rms normalized to its free space value Em1:

χ (t) =

Em (t) = 1 + R2 + 2R ⋅ cos Φ(t) (6.14b) Em1

Let’s find cosΦ(t) from (6.14b) and the transform into sinΦ(t); that is,



cos Φ =

1  2 ( 2   χ − 1 + R ) 2R

sin Φ =

2 1 − χ 4 + 2 χ 2 (1 + R2 ) − (1 − R2 ) 2R

   (6.15)   

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Now we differentiate (6.14b) and substitute cosΦ(t) and sinΦ(t) from (6.15). After simple transforms one may end up with −



1 2π

dχ2 − χ 4 + 2 χ 2 (1 + R2 ) − (1 − R2 )

2

=

1 dΦ (6.16) 2π

As pointed out above, Φ(t) is assumed to be a random variable evenly distributed within [−p; +p]. Thus, the right-hand side represents the probability of Φ being allocated within [Φ; Φ + dΦ] interval, which is equal 1/2p (see Figure 6.5). With that in mind we may conclude that the left-hand side represents the probability of the random variable c2 being allocated within [c2; c2 + dc2] interval. The following two facts are to be taken into account for further evaluations: • •

The negative sign on the left-hand side of (6.16) doesn’t have a physical meaning and may be dropped; As follows from (6.14b), two symmetric values of phase shift; namely +Φ and −Φ result in the same value of the normalized amplitude. Hence, the probability of c 2 being allocated within the range of c2 ∈ [c2, c2 + dc2] is to be doubled.

Based on these assumptions the expression for the differential distribution function (DDF) for c2 may be written as

p(χ 2 ) =

1 π

1 − χ 4 + 2 χ 2 (1 + R2 ) − (1 − R2 )2

(6.17)

For further evaluations, as well as in terms of practical applications, it is reasonable to switch to power units, namely to make the replacement c2 = x. This means that the results of further evaluations will equally be applicable to both variables, as well as to cdB = 20logc and xdB = 10logx in logarithmic format. It’s easy to realize from (6.14b) that the variable x may vary within the limits (1−R)2 = xmin < x < xmax = (1 + R)2. The cumulative distribution function, CDF(x), may be expressed as

P(x) = ∫ p(x) dx =

dx 1 + C (6.18a) ∫ π − x2 + 2x(1 + R2 ) − (1 − R2 )2

Figure 6.5.  Probability density distribution of the resultant phase for randomly superimposing two independent rays.

6.1  Multiplicative Interferences (Signal Fades) 333

The integral in (6.18a) may be found as [5, p.81, 2.261]

∫



dx − x2 + 2x(1 + R2 ) − (1 − R2 )2

= − sin−1

1 + R2 − x (6.18b) 2R

The integration constant C in (6.18a) may be found from the calibration condition P(xmin) = 0 as C = 1/2. Then (6.18a) is rearranged here if we substitute C; that is,

P(x) =

1 1 + R2 − x (6.18c) cos−1 π 2R

Note that the other calibration condition, P(xmax) = 1 is satisfied automatically. According to (6.11) the complementary cumulative distribution function is

W (x) = 1 − P(x) = 1 −

1 1 + R2 − x (6.18d) cos−1 π 2R

for the same variation limits (1 –R)2 = xmin < x < xmax = (1 +R)2. While designing RF communication links, calculation of the signal’s electric field at the reception point, or the power received from expressions (3.43) and (3.44), provide median value on the carrier frequency; that is, the value that is exceeded during 50% of the total observation time in the presence of the multiplicative interferences. This means that for the receiver sensitivity to be high enough, when the lowest signal level that the receiver is able to sense is less than median, then, as mentioned above, the difference represents the signal level margin that allows increasing the link availability. Therefore, to generalize (6.18c) and (6.18d) it is reasonable to normalize the signal level to is median value, which is easy to define from either expression by taking P(xmed) = W(xmed) = 0.5. This equation results in 2

xmed



 E = 1 + R =  med  (6.19) E  m1  2

Change of variable in (6.18c) and (6.18d) to y = x / xmed results in transformation to P(y) =



and

W (y) = 1 −

 1 + R2  1 cos−1  (1 − y)    for CDF π  2R 

 1 + R2  1 cos−1  (1 − y)    for CCDF π  2R 

(6.20)

(6.21)

with the variation limits that are easy to define:



1−

2R 2

1+ R

= ymin < y < ymax = 1 +

2R 1 + R2

(6.22)

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.6.  CCDF-s for selected values of the parameter R.

The expression (6.21) for the CCDF is shown graphically in Figure 6.6 as a function of the ratio y = E/Emed. 6.1.2.2  Random Interference of the Large Number of Independent Wavelets

In most applications, a large number of partial, independent waves of equal amplitudes and random phases, evenly distributed in the [–p, p] range, are interfering at the reception point. The intensity of each wavelet is vanishing. However, the number of those partial wavelets tends to infinity resulting in a finite overall power. This group of partial rays, called Rayleigh ensemble, is a theoretical model that closely presents the real conditions that are specific for scatterings from tropospheric turbulences or scatterings from the ionospheric irregularities due to micro-meteor ionized trails [see Figure 6.4(a)]. Each wavelet may be represented in the complex plane as a phasor with a constant amplitude and random phase, as shown in Figure 6.7(a). As noted earlier, the number of independent partial waves is extremely large (i.e., N → ∞), with vanishing intensities of each wave. Then each of them may be decomposed into real and imaginary parts, either positive or negative, depending on random initial phase shift F of the single partial wavelet. Note that we silently assume that all those wavelets propagate parallel to each other, resulting in collinear electric fields. Hence, one may replace a vector sum by algebraic summation in complex plane, as has been done in previous cases of random interference; that is, E =

N

N

N

n =1

n =1

n =1

∑ E n = ∑ Re E n + i ⋅ ∑ Im E n = X + iY

(6.23)

Regarding the central limit theorem, if N is large enough, then both terms in right-hand side of (6.23) have a Gaussian probability distribution, regardless of N

what the distribution of each component of the sum

∑ E n

n =1

is; that is,

6.1  Multiplicative Interferences (Signal Fades) 335

Figure 6.7  (a) Partial waves (E˙n and E˙k) presentation in complex plane, and (b) probability density distribution of the components of the resultant E magnitude of Rayleigh ensemble.





w(X) =

 X2  exp  − 2  2π σ  2σ 

(6.24)

w(Y ) =

 Y2  exp − 2  2π σ  2σ 

(6.25)

1

1

where s is a standard deviation of X and Y random variables that are the resultant components of the total vector E in complex plane.5 In order to evaluate the probability distribution of the E = E magnitude of the electric field, let’s first find the probability of the tip of vector E to be located in the area dS that surrounds point A with the coordinates (X, Y), as shown in Figure 6.7(b). The joint probability density function, w(X, Y) of the events X ∈[ X, X + dX ] and Y ∈[Y , Y + dY ] , which appear simultaneously, may be counted as a product of the proper distribution functions due to independency of X and Y random variables; that is, w (X, Y ) = w (X) ⋅ w (Y ) =

 X2 + Y 2  exp −  2π σ 2 2σ 2   1

(6.26)

5. Note that the standard deviation s is taken the same for both X and Y because it’s invariant to the rotations of complex coordinates.

336

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception 2 It may be seen from Figure 6.7(b) that E2 = E = X 2 + Y 2 ; therefore, (6.26) may also be considered as the joint probability density of the magnitude and phase of E = E exp(iΦ) phasor:

w(X, Y ) = w(E, Φ) =

 E2  exp − 2  2π σ 2  2σ  1

(6.27)

Now, based on probability density distribution function w(E, F), we may find the probability of the tip of vector E to be located in the area dS = E dE dF [see Figure 6.7(b)] that is expressed in polar coordinates (E, F). dW  E ∈(E, E + dE), Φ ∈(Φ, Φ + d Φ)  = = w (E, Φ) dS =

 E2  1 exp  − 2  ⋅ E dE d Φ 2π σ  2σ 

(6.28)

In order to find the probability distribution of the magnitude E, the phase F to be excluded from (6.28). In other words, (6.28) is to be integrated for F from –p to p, which allows us to find the probability for the tip of E to be located within the ring E ∈ (E, E + DE) shown in Figure 6.7(b).

dW  E ∈(E, E + dE)  = =

π

∫

−π

π

∫ dW  E ∈(E, E + dE), Φ ∈(Φ, Φ + d Φ) d Φ =

−π

 E2   E2  E exp − E dE d Φ = exp    − 2  dE σ2 2π σ 2  2σ 2   2σ  1

(6.29)

From (6.29) the probability density distribution function for the Rayleigh ensemble, or so-called differential Rayleigh distribution function, may be found as w(E) =



 E2  dW E = 2 exp  − 2  dE σ  2σ 

(6.30)

The graph of (6.30) is shown in Figure 6.8(a). The complementary cumulative Rayleigh distribution function may be obtained by substitution of (6.30) into (6.11). ∞



W (E) = ∫ w(y) dy = E

∞

 y2   E2  exp − y dy = exp   − 2  ∫ 2  2σ 2   2σ  Eσ 1

(6.31)

Similar to the previous case, the parameter s is likely be replaced by the median value, Emed. By definition



 E2  W (Emed ) = 0.5 = exp  − med   2σ 2 

(6.32)

6.1  Multiplicative Interferences (Signal Fades) 337

Figure 6.8  Rayleigh distribution: (a) probability density function; and (b) cumulative probability function.

Then s may be found by solving (6.32):



2σ 2 =

2 Emed E2 = med ln 2 0.693

(6.33)

Final expression for CCDF as a function of E/Emed may be found if (6.33) is substituted into (6.30):



2   E    W (E) = exp  −0.693    Emed    

(6.34)

The graph of probability density distribution function (6.34) is presented in Figure 6.6(a) in the linear scale and cumulative probability distribution function is shown in Figure 6.8(b) in the Rayleighian scale, which ends up with the straight line, as shown.

6.1.2.3  Further Generalization of the Fast Fading Statistics

In its pure form, the Rayleigh distribution is not quite common for fast fading statistical description in practical applications in radars and communications practices. Its main restriction is the lack of flexibility that is needed to support a variety of propagation mechanisms—hence the variety of statistical properties, particularly the fading depths and shapes of probability distribution functions. A variety of types of radio links and propagation mechanisms leads to the need for more complex analytical expressions for fading statistics that are able to:

338

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.9  Superposition of a strong, stand-alone monochromatic component with the continuum of single-scattered uncorrelated wavelets (Rayleigh ensemble).

1. Specify the physical processes in radio waves propagation; 2. Adjust to statistical properties of the particular propagation path of interest. Further updates of the fast fading statistics that involve more detailed propagation mechanisms are outlined in this section without going into the mathematical evaluation procedures. 1. R  ice distribution (known as generalized Rayleigh distribution) appears as a result of superposition of Rayleigh ensemble with a strong monochromatic (deterministic) component of E0 field strength (see Figure 6.9). This approach was originally presented in [2], where the probability density distribution function was evaluated in the following form:



w(E) =

 E2 + E2   E E0  0 exp ⋅I − 2 2  0  σ 2  σ 2σ   E

(6.35)

The graph of (6.35) is shown in Figure 6.10(a). Here I0 is the modified Bessel function of the first kind, zero order. If E0 = 0, then the distribution reduces to a regular Rayleigh distribution. The adjustment parameter b = E0/s is to adopt (6.35) to the real conditions for the given ratio between the deterministic component E0 and the total intensity of the Rayleigh ensemble. This distribution is specific for the terrestrial LOS or Earth-to-space/ space-to-Earth communication links at UHF and higher frequencies. CCDF is calculated numerically and is shown in Figure 6.10(b) in the Rayleighian scale. 2. m-distribution of Nakagami was originally presented in [3]. It is one of the most generalized descriptions of random interference for the superimposing wavelets of arbitrary amplitude and phases distributions. The probability density function is presented by



wm (E) =

2m m E2m −1 Γ(m) σ 2m

 mE2  exp − 2   σ 

(6.36)

6.1  Multiplicative Interferences (Signal Fades) 339

Figure 6.10  Rice distribution: (a) probability density distribution, and (b) CCDF.

where m and s are the adjustment parameters. This expression is applicable to any type of radio links as a fast fading statistic (see Figure 6.11). Based on (6.11), and (6.36), the CCDF for m-distribution may be defined as Wm (E) =

2m m Γ(m)σ 2m

∞

∫z

E

2m −1

∞

 m  1 exp  − 2 z 2  dz = t m −1 exp(−t)dt (6.37) ∫ Γ ( ) m   σ B 

where



B=

mE2

σ2

Figure 6.11  Probability density of m-distribution Nakagami.

(6.38)

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

The integral in (6.37) may be expressed in closed form through the incomplete gamma function [5] as follows:   m γ  m, 2 E2    σ Wm (E) = 1 − Γ (m)



(6.39)

It may be shown that the second-order mathematical moment of the random variable E(t) is defined from (6.39) as E2 = σ 2



(6.40)

and it is constant and independent on m. Median values may be found from Wm(Emed) = 0.5 by using (6.39) in the following form:  Emed 2  γ  m, m   σ 2  = 0.5 Γ (m)



(6.41)

The ratio Emed 2 /σ 2 is m-dependent. If we denote p(m) = Emed 2 /σ 2



(6.42)

The values of p(m) are calculated and given in Table 6.1. Taking into account (6.42), the expression (6.39) may be rewritten as Wm (x) = 1 −



(

γ m, m ⋅ p(m) ⋅ χ 2 Γ (m)

)

(6.43)



where

χ=



E Emed

(6.44)

is E-level, relative to median value. The sketch of CCDF for m-distribution is shown in Figure 6.12(b) in Rayleighian scale, so for m = 1 (for Rayleigh distribution) the graph becomes a straight line. 3. n-distribution (Quasi-Rayleigh distribution) is a statistical model that was verified based on numerous observations of fast fading on HF radio links and was originally presented in [6]. Parameter n is introduced to modify the Rayleigh distribution and make it adjustable to the observed fade statistics Table 6.1  The Values of the p (m) Parameter of m-Distribution m p(m)

0.5 0.456

0.75 0.606

1.0 0.693

1.5 0.789

2.0 0.839

3.0 0.891

4.0 0.918

5.0 0.922

6.0 0.945

6.1  Multiplicative Interferences (Signal Fades) 341

Figure 6.12  (a) Graph of the p(m) parameter, and (b) CCDF of m-distribution.

on real propagation paths. The idea relates to diversity reception in auto-selection mode that results in (6.154). Similar expression is adopted here for CCDF of quasi-Rayleigh distribution as follows: n

2     E      Wn (E) = 1 −  1 − exp − q (n) ⋅     E   med    



where q (n) =

Emed 2 2σ 2

 1  = ln  1  1 − 0.5 n

(6.45)

  (6.46)  

is an n-dependent parameter. The calculated values of q(n) are given in Table 6.2 and are presented graphically in Figure 6.13(a). CCDF of the n-distribution (quasi-Rayleigh distribution) is shown in Figure 6.13(b) with a Rayleighian scale for horizontal axis. 6.1.3  Slow Fading Statistical Distribution

As noted earlier, the cause of slow fading (slow random variations of the short-term median field at the reception point) are the random changes in attenuations along the propagation path. Those attenuations depend on frequencies and propagation path configurations. They may arise in different atmospheric layers and along the

Table 6.2  The Values of a q (n) Parameter of n-Distribution n q(n)

0.25 0.0645

0.5 0.2877

1 0.6931

2 1.2279

3 1.5784

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.13  Quasi-Rayleigh distribution: (a) graph of the q(n) parameter, and (b) CCDF.

Earth’s surface, being impacted by random changes in temperature, humidity, pressure, ionization intensity, or other physical conditions. Numerous observations on various radio links exhibited the logarithmic-normal (lognormal) probability distribution of slow fades considered in Section 6.1.3.2. 6.1.3.1  Normal (Gaussian) Distribution of the Random Variable

First, consider the normally distributed (Gaussian distribution) random variable z. This type of random variable is described by the following probability density distribution function: w(z) =

 (z − z )2  exp  −  2σ z2  2π σ z  1

Figure 6.14  Normal (Gaussian) probability density distribution function for z– = 1.

(6.47)

6.1  Multiplicative Interferences (Signal Fades) 343

The graph is shown in Figure 6.14, where z is a mean value of random variable z (statistical expectation), and σ z = (z − z )2 is its standard deviation. Based on (6.10a), for this case, the value of zmed may be found by solving the following equation: ∞

∫

zmed



 (z − z )2  exp  −  dz = 0.5 2 2π σ z  2σ z  1

(6.48)

The result is zmed = z



(6.49)

because of the symmetry of the function (6.47) about z = z line. 6.1.3.2  Lognormal Distribution of the Random Variable

The term lognormal means the normal (Gaussian) distribution of the logarithm of random variable E; that is, z = ln E



(6.50)

Practically it’s more convenient to rewrite (6.47) as a function of E-variable, taking into account (6.50): dW ′ [ z ] = w (z) d z = w [ z(E)] ⋅



dz ⋅ d E = dW (E) dE

(6.51)

Probability density function may be found from (6.51) as w(E) = w [ z(E)] ⋅

 (ln E − ln E)2  dz 1 = exp  −  d E E 2π σ z 2σ z2  

(6.52)

which is shown graphically in Figure 6.15. Based on (6.49) and (6.50), the expressions for statistical expectation and standard deviation may be rewritten as ln E = ln Emed



σ z = (z − z )2 ≈



 E  2  ln E  = y = σ y med

(6.53a) (6.53b)

where

y = ln

E Emed

, [Np]

is a relative level of field strength expressed in nepers.

(6.54)

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.15  Lognormal probability density distribution [simplified version of (6.52)].

The complementary cumulative distribution function for this case may be evaluated if (6.52) is substituted into (6.11) as follows: ∞

∞

W (E) = ∫ w (E) dE =



 (ln E − ln E)2  dE exp  −  2σ z2 2π σ z   E 1

∫

E

E

(6.55)

If replacements (6.53) and (6.54) are applied to (6.55), then the result is 1

W (y) =

2π σ y



∞



y′2 

y



y

∫ exp  − 2 σ 2  d y′ 

(6.56)

that is, the slow variations of the relative level (in dB or Np) of medians of the electric field strength are normally distributed. In order to standardize this expression, the replacement x=



y

(6.57)

2σy

is used, which results in 1

W (y) = W (x) =

π



∞

∫ exp(− x′ x

2

1 ) dx′ = [1 − erf (x)] 2

(6.58)

where erf (x) =

2

π

x

∫ exp(− x′

0

2

) dx′ = − erf (− x)

(6.59)

is error function, given in numerous math references (e.g., [5]).

6.1  Multiplicative Interferences (Signal Fades) 345

Figure 6.16  Lognormal cumulative probability distribution function.

Examples of lognormal cumulative distributions are shown in Figure 6.16, where nonlinear scale was adopted for the horizontal axes (Gaussian scale), which allows us to sketch the graphs as straight lines. The graphs in Figure 6.16 are drawn based on the following procedure: • • •

If x = 1/ 2 (i.e., y = sy), then from (6.58) W = 0.16 (16%). If x = −1/ 2 (i.e., y = -sy), then from (6.58) W = 0.84 (84%). Note also the fact that for any sy the graph will cross the point (0, 50%), which comes from the definition of the median value.

Hence, in a Gaussian coordinate any graph of CCDF is a straight line that goes through the points (-sy, 84), (0, 50), and (sy, 16), as seen from Figure 6.16. Lognormal distribution is appropriate as fast fading statistics for atmosphericoptical communication lines only. That’s because of multiscatter propagation of each partial wavelet coming to the reception point along with the strong deterministic component [4], as shown in Figure 6.17.

Figure 6.17  Mechanism of the superposition of strong deterministic component with an ensemble of partial, multiscattered wavelets.

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

In (6.53b) and (6.56) sy is an independent parameter, which specifies the deepness of fading and may be used to adopt the graph to real data observed. Note again the analytical expressions, given in Section 6.1.2, describing fast fading also include that kind of adopting parameters (e.g., parameter E0 / ( 2 σ ) in Ricedistribution, parameter m in Nakagami-distribution, or parameter n in quasi-Rayleigh distribution). 6.1.4  Combined Distribution of Fast and Slow Fades 6.1.4.1  Signal Stability in Long-Term Observations

Fast and slow fades always act simultaneously. That is, within the short-term stationarity intervals of observations, the median value of the field strength remains almost constant, whereas during long-term observation periods, the random variations of the median Emed(t) become observable and are considered slow fading (see Figure 6.3). The following evaluation is based on the assumption that the fast fading may be counted by introduction of the normalized random variable kfast(t) with the median value equal to 1. Thus, within the interval of stationarity (short-term observation), the field strength at the reception point may be written in the following form: E(t) = κ fast (t) ⋅ Emed



(6.60)

If the observation period is prolonged, becoming greater than the interval of stationarity, then in order to count slow random variations of the median field Emed(t), we may introduce another variable kslow(t), similar to that for the fast fad­­ ing, so the fast and slow fades may be taken into account simultaneously by the following expression: E(t) = κ fast (t) ⋅κ slow (t) ⋅ Emed , slow



(6.61)

where Emed, slow is an overall median value of the field strength for a long-term observation period (i.e., the median of the slow fading). Now we introduce the level of signal as a unitless quantity referenced to the long-term median Emed, slow:

Z(t) = ln

E(t)

(6.62) Emed , slow Then, taking the logarithm from both sides, the expression (6.61) may be rewritten as Z(t) = X (t) + Y (t)



(6.63)

where X(t) = lnκ slow (t)

Y (t) = lnκ fast (t)

  

(6.64)

In (6.64) X(t) and Y(t) represent slow and fast fading components in Np, respectively. They easily may be converted into decibels by using the relation (2.97).

6.1  Multiplicative Interferences (Signal Fades) 347

From (6.63) one may notice that the randomly fluctuating signal level Z(t) is displayed in form of the sum of two random variables, X(t) and Y(t), that are representing slow and fast fades, respectively. It’s known from the probability theory that if w(X, Y) is the joint probability density distribution function of X and Y random variables, then the double integral W=

∫∫ w(X, Y ) dX dY S

(6.65)

represents the probability that the pair of variables (X, Y ) belongs to the surface S on and XOY plane. In our case of (6.63), the complementary cumulative probability function W is counted for the surface S, which is part of the XOY plane above the line Y = Z – X, as shown in Figure 6.18. If X and Y random variables are independent, then the joint probability density distribution function may be simplified to the product

w(X, Y ) = w slow (X) ⋅ w fast (Y )

(6.66)

Hence, taking into account (6.63), the integration limits in (6.65) may be set as follows: +∞   +∞ W (Z) = ∫ w slow (X)  ∫ w fast (Y ) dY  dX =  Z − X  −∞ =

  +∞   w ( Y ) w ( X ) dX dY fast slow ∫ ∫  Z −Y  −∞ +∞

Figure 6.18  Area S of integration in (6.65).

(6.67)

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

The convolution integral (6.67) in probability theory is called composition of probabilities, and a special symbol, “⊗” is used to express (6.67) in compact presentation form. Taking into account (6.11), we may rewrite the convolutions (6.67) as W (Z) = Wslow (X) ⊗ Wfast (Y ) = ... ... =

+∞

+∞

−∞

−∞

∫ wslow ( X ) ⋅ Wfast (Z − X) dX = ∫ wfast (Y ) ⋅ Wslow (Z − Y ) dY

(6.68a)

The expression (6.67) for the cumulative probability distribution function in form of double integral may be transformed to the single-integral form, written for the probability density distribution functions of both fast and slow fades. That is, (6.67) may be rewritten as follows:6 w (Z) =

∞

∫

w slow (X) ⋅ w fast (Z − X) dX =

−∞

∞

∫ wfast (Y ) ⋅ wslow (Z − Y ) dY

−∞

(6.68b)

It may be shown that if one of the functions, wfast(Y) or wslow(X), is even (symmetric about vertical axis), then (6.68b) may be rewritten as

w(Z) =

∞

∫

w slow (X) ⋅ w fast (Z + X) dX =

−∞

∞

∫ wfast (Y ) ⋅ wslow (Z + Y ) dY

−∞

(6.68c)

which means that in this case of symmetry both expressions (6.68b) and (6.68c) may be applied to either sum of two random variables, Z = X + Y, or to their difference, Z = X - Y. Note that (6.68b) type convolutions may be more easily calculated based on the fact that the Fourier transforms of both functions, wfast(Y) or wslow(X), are just to be multiplied. Example 6.1

Consider a combined distribution of fast and slow fades if both are lognormally distributed with s1 and s2, standard deviations, respectively. Note that this is a typical case for the atmospheric optical communication links (not for RF links). Find the joint probability distribution function as well as its standard deviation. Solution

The expressions for probability density functions (PDF) for the logarithmic variables of fast and slow fades are presented the in the first column of Table 6.3, with ∞

6. Expressions

∫

Z −Y

w slow (X) dX = Wslow (Z − X) or

∞

∫

Z−X

w fast (Y ) dY = Wfast (Z − X) are employed here.

6.1  Multiplicative Interferences (Signal Fades) 349 Table 6.3  Example 6.1: Fourier Transforms of PDDFs Probability density distributions

Fourier cosine transforms [5]

w 1(X) =

 X2  exp  − 2 2π σ 1  2 σ1 

F1(ξ) =

 ξ 2σ 2  1 exp  −  2  2π 

w 2 (X) =

 X2  exp  − 2 2π σ 2  2σ 2 

F2 (ξ) =

 ξ 2σ 2  2 exp  −  2  2π 

1

1

1

1

proper Fourier transforms in the second column. The expression for the Fourier transform of the joint distribution density is a product: F(ξ) = F1(ξ) ⋅ F2 (ξ) =

(

)

 ξ2 2  1 exp  − σ 1 + σ 22  2π  2 

(6.69)

As seen from (6.69) the inverse transform is also expected to be a lognormal and may be easily obtained from [5] with same form as for w1(X), or w2(Y), and with the standard deviation of

σ 2 = σ 12 + σ 22 , dB

(6.70)

The expression (6.68a) allows to determine the long-term probability distribution by combining lognormal distribution for slow fades and one of the distribution models given in Section 6.1.2 for fast fades. Unfortunately, for RF links any combinations between fast and slow fading models may not be evaluated in closed form, and, therefore, only numerical evaluations are applicable. The graphs of several combined distributions are shown in Figures 6.19–6.21. To generalize these statements, one may consider three, four, and more independently acting fading mechanisms instead of two: the problem is to obtain the

Figure 6.19  Combination of lognormal and m-distributions.

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.20  Combination of lognormal and n-distributions.

resultant distribution function if the cumulative distribution functions (W1, W2, …, WN) due to different independent mechanisms of fades are known. The following approach may be proposed, based on (6.68a): first W1 must be composed with W2, then the result with W3, and so on. That is,

{

}

W = (W1 ⊗W2 ) ⊗ W3  ........ ⊗WN

(6.71)

The approaches presented in this section may be considered a background for the assessments of the RF link stability.

Figure 6.21  Combination of lognormal and Rayleigh distributions.

6.1  Multiplicative Interferences (Signal Fades) 351

Example 6.2

Assume E(t) = Eth = 4.3 mV/m is a minimum value of the electric field the receiver is able to sense (threshold value) for the conditions described in Example 5.5. Fast fading of the signal has Nakagami statistics with parameter m = 1.5, and slow fad­ ing is lognormally distributed with parameter s = 8 dB. Estimate the stability of communication link. Solution

If the output power of the transmitter is chosen to induce the long-term median electric field at the reception point equal to threshold, Emed, slow = Eth, then apparently from (6.66) follows Z = 0, which, according to Figure 6.19 results in 50% of stability. For the considered case Emed, slow = 136 mV/m, therefore, Z = 20log(E/Emed, slow) -30 dB. From Figure 6.19 for the given parameter m = 1.5, we may determine approximately W = 99.9%



(Answer)

As seen, the power margin of 30 dB results in fairly high communication link stability. Now consider the case when the power margin is such that received signal stability is high enough, as demonstrated in Example 6.2. It means only a very deep fades may result in signal outage when it drops below the threshold level. The reason is a significant difference between threshold and long-term median fields. Now rewrite (6.68a) for the combined probability distribution function: +∞

W (Z) =

∫ wslow (X) ⋅ Wfast (Z − X) dX

−∞

(6.72a)

As seen, in the case of deep fades, or for large negative values of Z, the cumulative distribution Wfast (Z - X) is fairly close to unity (100%) in a wide range of variations of the argument X, where the contribution of wslow(X) is significant. Thus, the value of Wfast may approximately be considered a constant and taken out of the integral. Then the expression (6.72a) may be rewritten approximately as

W (Z) ≈ Wslow (X) ⋅ Wfast (Y )

(6.72b)

In other words, the convolution (6.72a) may be replaced by the product (6.72b) of CDFs of slow and fast fades, with acceptable accuracy. The same approach may be applied to the case when more than two independent fading mechanisms remain in force. From (6.11), which shows the relation between CDF and CCDF, one may obtain:

P = 1 - W

(6.73)

P represents the probability of dropping down the fixed value of the signal (e.g., down the threshold value). Based on (6.72b), CDF for two independent fades may be rewritten as

Wresult ≈ W1 ⋅ W2 = (1 − P1) ⋅ (1 − P2 ) = 1 − (P1 + P2 ) + P1 ⋅ P2 = 1 − Presult

(6.74)

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

hence Presult = P1 + P2 − P1 ⋅ P2



(6.75)

The last term is the probability of simultaneous appearance of two drop-downs, caused by two different independent mechanisms. Under the assumptions P1 1 is a modulation index; α = 2 ⋅ (1 + mFM + mFM ) = ∆f /F ; ∆fdev is a maximum deviation of carrying frequency; F is an initial, modulating frequency; ∆f is the width of FM signal spectrum

Comparison of the modulation types shown in Table 6.5 may emphasize that three linear modulation types (AM, DSB-SC, and SSB) result in SNR gain at most equals 3 dB (2 unitless). At the same time the nonlinear type of modulation such as FM may result in much higher value of SNR gain by proper choice of modulation index. It appears that any value of SNR gain may be obtained by choosing the value of modulation index mFM for FM. In other words, it seems like even in a case when the input signal is buried within the additive noise, the output signal may spring up from the noise after demodulation, if mFM is chosen large enough. This statement is not true because of the existence of the threshold that is specific for FM systems: the linear relation c = SNRDet,Out/SNRDet,In between input and output SNRs is not in effect for the small SNRDet,In; the output SNR goes down abruptly when the input SNR is close or less than the its threshold, as shown in Figure 6.35. The improvement in FM systems is achieved at the expense of the signal bandwidth, 2∆fmax = aF, which becomes much wider than that for linear types of modulation such as AM, DSB-SC, and SSB. If SNRDet,In for the detector’s output as well as the allowable width of signal spectrum is preassigned, then SNRDet,In for the detector’s input becomes linked to those preassigned values. That allows to define the receiver’s sensitivity, based on developments given in Section 6.2.5 [expression (6.113)]. Modems such as AM, DSB-SC, SSB, and FM are just examples of how detection of the signal may affect both Tx-Rx structure design, and the RF link power budget calculations. Complete analysis of all types of analog modems is beyond the scope of this text and may be a subject for additional reading. 6.3.2 Brief Overview of Digital RF Link Performance Improvement Approaches

A simplified block diagram of the digital signal transmission system is shown in Figure 3.36. The initial signal, called baseband signal, is being transformed into digital forms on the transmit side by analog-to-digital converter (ADC): first the continuous analog signal is being discretized (quantized), and then each discrete sample

6.3 Methods of Improvement of RF Systems Performance 377

Figure 6.35  Demonstration of SNR threshold for FM signal (both axes are in logarithmic scale). Note that if the input SNR is less than threshold then FM becomes unacceptable.

is transformed by an encoder into a digit (predominantly binary). The number of single bits within the code is directly related to the number of quantization levels as M = 2m, where m is the number of bits in code. As a result, a pulse code modulated (PCM) signal is formed at the encoder’s output; the PCM signal is a sequence of bits; that is, the “1”-s (high voltage) and “0”-s (low voltage). This process of analog-to-PCM conversion is called a source coding. The transmitter allows digital modulation: a signal is mounted on the carrier and is amplified to the output power needed for transmission from antenna-to-antenna through the propagation medium. There are a variety of modulated signal structures (waveforms) being implemented by a variety of modulation schemes [16–19]. During the modulation process additional bits may be implemented into PCM signal—a redundancy, which is needed to overcome the destructive impact of the noises. This process of redundancy implementation within digital modulation is called channel coding. Except for external (environmental) and receiver (internal) noises, a significant amount of the noise is generated during the baseband signal’s sampling and encoding process in the ADC; this noise is known as quantization noise. Note that this type of noise exists regardless of type of transmission channel—either wire channel or radio channel. In other words, this portion of the overall noise power is expected to be the same even if the PCM signal goes from ADC to DAC directly, bypassing transmitter, receiver, and propagation path (see Figure 6.36). 6.3.2.1

As a first approach in overall signal-to-noise ratio reduction efforts, let’s consider the way of mitigation of the quantization noise. The peak output signal-to-quantization noise ratio is defined by the following expression [23]:

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.36  Simplified block diagram of the digital RF transmission link. Here, ADC stands for analog-to-digital converter, and DAC stands for digital-to-analog converter. Note: Modulation and demodulation are carried out in the transmitter and receiver, respectively, whereas coding and decoding are carried out in ADC and DAC, respectively.

 PS  3M 2 = (6.120)  P  N pkOut 1 + 4 ( M 2 − 1)Pe



Here, M is the number of discretization levels of the initial baseband signal: the number of bits (m) after source coding process is simply expressed as m = log2 M, which is a positive integer number. In (6.120) Pe = BER is a bit error rate (error probability of the single bit reception in presence on the noise). The values of (Ps/ PN)pkOut versus BER for the selected number of discretization (quantization) levels, M, are shown in Figure 6.37. As seen from the graphs in order to achieve high SNR for low probability of bit error, the number of discretization levels is to be selected as large as possible, which results in less quantization noise, and therefore in higher SNR. This example is to showcase the impact of a specific type of noise; namely quantization noise, and a way of its mitigation, and therefore, increasing the SNROut. 6.3.2.2

There are several other types of noises described in previous sections, which may not be lessened by the proper choice of the number of discretization levels while forming a digital signal. Predominantly, those are the noises implemented into the RF channel along the propagation path, as well as at the reception side; namely, on the signal’s way from antenna to detector’s input. A detailed evaluation of the overall noise destructive impact on a digital signal’s reception credibility is outside the scope of this text, and may be found in [16–19]. Here, in Table 6.6 we’ll just present properties of the selected digital modems (modulation-demodulation schemes) that

6.3 Methods of Improvement of RF Systems Performance 379

Figure 6.37  (Ps/PN)pkOut is a peak SNR versus BER for the selected values of the number of quantization levels.

relate to signal-to-noise statistics. As examples, Figure 6.38 shows the graphs of BER versus Eb/No for the selected modulation schemes. As seen from Figure 6.38 proper choice of modem may allow achieving some SNR reduction for the prerequired value of BER. In other words, choice of modem may be considered as a second approach in overall signal-to-noise ratio reduction effort. For example, in order to achieve the value of probability BER = 10–5, the minimum value of SNR for a digital signal at the detector’s input is to be at least 9.6 dB for BPSK, 10.2 dB for DPSK, and 12.6 dB for FSK, as shown in Figure 6.38.

Table 6.6  BER vs Eb/No for Selected Digital Modems Modulation Scheme (Abbreviated)

Full Description of the Modulation Scheme

Minimum Bandwidth Required for Bandpass (Modulated) Signal to be Transmitted through RF Link (Rbit Is a Bit Rate)

BER vs Eb/No

BPSK

Binary phase shift keying

Rbit

  E  Q  2 b     No  

FSK

Frequency shift keying

2∆f + Rbit   E  b  (2∆f is a frequency jump in carrier from Q      No   bit-to-bit)

DPSK

Differential phase shift keying

Rbit

Note: Here Q(z) =

1 exp(− Eb No ) 2

∞  z   ξ2  1 1 erfc  = ∫ exp  − 2  d ξ , and erfc(z) is the complementary error function [5]. It is used for the real 2  2 π z

argument, unlike (4.139), where it was used for the imaginary argument.

380

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.38  BER versus the ratio of energy per bit to noise spectral density (Eb/No) for selected modulation schemes.

6.3.2.3

Finally, a third approach in overall signal-to-noise ratio reduction effort is a redundancy implementation into PCM signal during digital modulation, namely a channel coding. In order to assess the efficiency of the channel coding to support the detector’s output SNR reduction, let’s turn to Shannon’s theorem:

C = Fch·log2 (1 + SNR) bits/s

(6.121)

which sets a theoretical limit C to the channel capacity for the designer to strive to achieve in practical communication system design. In (6.121) Fch is channel bandwidth. If the initial signal rate is equal or less than the C limit, then even in the presence of the noises the bit error probability in ideal channel will turn to zero for Eb/ No = –1.59 dB [23]. In other words, if the optimum coding/decoding is applied then one may end up with no error reception. Figure 6.39 demonstrates, as an example, how much gain is achievable if a Golay correction coding is applied to BPSK signal to achieve BER = 10–3 and BER = 10–5. It should be noted that rather high values of the coding gain can be achieved if a large redundancy is implemented, by paying a price either in the form of a decrease in the transmission rate, or in form of increasing the channel bandwidth. Example 6.6

Assuming that the carrier frequency can be modulated by 10% to carry information, make a plot of channel bandwidth vs carrier frequency from 1 kHz to 100 GHz. Plot also the ideal Shannon channel capacity (theoretical limit) vs carrier frequency if assumed SNR = –1.59 dB (SNR = 0.693 linear). Compare with a home Wi-Fi data transfer rate at 2.4 GHz (typically~600 Mbps) and at 5 GHz (typically~1.3 Gbps).

6.3 Methods of Improvement of RF Systems Performance 381

Figure 6.39  Demonstration of achieving of the coding gain by comparing required Eb/No-s for the given BER for the BPSK modulation scheme with and without channel coding [24].

Solution • •

The channel bandwidth, F = 0.1⋅fc is shown in the figure below (fc is carrier frequency). Channel capacity (data rate) for the Shannon ideal channel is defined from (6.121) C = F⋅log2(1 + 0.693) = F⋅0.76. It is presented in figure below.

Note: Keep in mind that the required channel SNR for the real Wi-Fi units are notably higher than those for the ideal channel, and may be of order ≈10–20 dB. Meanwhile, a large redundancy (use of correction coding) is needed, which may significantly drop the data rate.

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Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

6.3.3  Use of Spread-Spectrum Discrete Signals

For the special class of discrete signals called spread-spectrum signals a significant improvement in SNR may be achieved when a matched-filtering reception20 is used. The idea of the matched-filtering was first introduced by D. O. North in 1943 [19], and further evaluated in the late 1940s and early 1950s [20]. What follows is a brief, heuristic outline of the basics of the matched-filtering. Consider a pulse signal s(t) of duration T and of the arbitrary waveform that is applied to the input of the linear two-port network resulting in output response sout(t) of duration Tout. The goal of the matched filter (MF)21 is to “squeeze” (compress) the signal from input to output with maximum possible compression, as shown in Figure 6.40. Then the envelope of the output pulse presumably is to be sin(t)/t-shaped (or close to it) with duration (6.122)

Tout ≈ 1/∆f



where Df is a bandwidth of the input signal. For the proper reason Df remains the same from input to output of MF. Considering MF is lossless network, the total energy of the signal remains unchanged from input to output; that is, E =

∞

∫s

2

2

(t) dt = T ⋅ s =

−∞

∞

∫ sout

2

(t) dt = Tout ⋅ sout2

−∞

(6.123)

where s2 and sout2 are the values of mean power22 of input and output signals, respectively. Then, taking into account (6.122), the following relation between input and output RMS voltages may be written: sout , rms

srms

=

sout2 s

2

=

T = T ∆f = B Tout

(6.124)

where the time-bandwidth product B = TDf is called pulse compression ratio when having to deal with discrete signal. It shows how many times the pulse duration decreases from input to output of the matched filter, and as a result, how many times it increases the peak power of the input signal (not the energy of the signal). In the presence of white Gaussian uncorrelated noise, which equally affects both input and output signals, the ratio (6.124) is nothing but the ratio of SNR from input to output (i.e., it is a SNR gain for this type of transform). The values of that gain may become significant, especially for the radar applications, where the value of B may achieve 1,000 and even more. In order to develop a structure of the matched (optimal) receiver that maximizes SNR, consider a signal s(t) of the complex spectrum 20. Optimum filtering or correlation detection terms are also in use. 21. Not to be confused with abbreviation for medium frequency (see Chapter 1). 22. Actually, s2 and sout2 are the mean squares of the voltages within the pulse durations. They turn to become mean powers if the voltage drops are considered on unit resistances of 1 ohm. This assumption does not put any restrictions on further considerations.

6.3 Methods of Improvement of RF Systems Performance 383

Figure 6.40  Signal waveforms at input/output of the matched filter.



S ( jω ) = S(ω ) ⋅ exp[ jΦS (ω )]

(6.125)

that passes through MF along with the Gaussian white noise n(t) of the spectral density N0. In (6.125) S(w) is the amplitude and FS(w) is a phase spectrum of the input signal, respectively. Let MF have a transmission function in complex from:

H ( jω ) = H (ω ) ⋅ exp[ jΦH (ω )]

(6.126)

where H(w) and FH(w) are amplitude and phase transmission coefficients of MF. If one needs to superimpose all spectral components of a signal at the output with the same phase (constructive superposition) at any time moment t0, then the signal’s phase variations versus frequency FS(w) must be compensated by MF in order to obtain the phase spectrum of output signal in the form of (-wt0). Hence ΦS, out = ΦS (ω ) + ΦH (ω ) = −ω t0 , therefore

ΦH (ω ) = − ΦS (ω ) − ω t0

(6.127)

Indeed, the output response of MF will appear at the time moment of dFS,out /dw = -t0 (i.e., it is shifted from start-point of the input signal by t0). As stated, the condition (6.127) allows maximizing the output signal by superimposing all spectral components of the initial signal at the time moment t0. To minimize the noise intensity, we may assume that MF is matched to amplitude spectrum of signal; that is,

H (ω ) = a ⋅ S(ω )

(6.128)

where a is a real constant. By combining (6.127) and (6.128), the transmission function of MF may be expressed as a complex conjugate Sˆ ( jω ) of the spectrum of primary signal; that is,

H ( jω ) = a ⋅ S (− jω ) ⋅ exp(− jω t0 ) = a ⋅ Sˆ ( jω ) ⋅ exp(− jω t0 )

(6.129)

For further evaluations, the following statements from the theory of Fourier transforms must be recalled:

384

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

•

The Fourier transform of convolution of two functions is a product of the Fourier transforms of proper functions; that is, ∞

∫ s(t ′) ⋅ h(t − t ′) ⋅ dt ′ ⇔ S( jω ) ⋅ H( jω )



−∞ •

If in expression (6.130) the replacement H ( jω ) = Sˆ ( jω ) is applied, then S ( jω ) ⋅ Sˆ ( jω ) ⇔





∞

∫ s(t ′) ⋅ s(t ′ − t) ⋅ dt ′

= ψ (t)

(6.131)

−∞

•

(6.130)



it becomes an autocorrelation function for s(t) (initial, input signal). Time shift property: exp(− jω t0 ) ⋅ S ( jω ) ⇔ s(t − t0 )

(6.132)

Based on (6.130)–(6.132) the expression for output signal may be written as Sout ( jω ) = S ( jω ) ⋅ H ( jω ) ⇔ ⇔ sout (t) = a ⋅

∞

∫ s(t ′) ⋅ s[t ′ − (t − t0)] ⋅ dt ′ = a ⋅ ψ (t − t0)

−∞

(6.133)

where y(t - t0) is the autocorrelation function of the input signal. It may be seen that for the time moment t = t0, the expression (6.133) becomes ∞



sout (t0 ) = aψ (0) = a ∫ s2 (t ′) dt ′ = a E

(6.134)

−∞

that is, the maximum value of the output signal at the time moment t = t0 is directly proportional to the total energy of input signal (see Figure 6.41). Finally let’s derive the SNR gain based on these analyses. For the input SNR, the signal power may be defined from (6.123) as PS,in = E /T = s2 . The input white Gaussian noise spectral density N0 is evenly distributed in frequency range Df = fmax - fmin along the entire spectrum of the signal. Thus, the input noise power is PN,in = N0Df. Then taking into account B = TDf, we may write



SNRin =

s2 = E/(N0 B) N0 ∆f



(6.135)

For the output SNR the noise power spectral density may be defined if N0 is multiplied by power transmission function of the MF from (6.128): Nout (ω ) = N0 a2 ⋅ S2 (ω ) Nout (ω ) = N0 a2 ⋅ S2 (ω ) . Then the total output noise power may be defined by integrating the output noise power density within the effective frequency band:

6.3 Methods of Improvement of RF Systems Performance 385

Figure 6.41  Optimum reception of spread-spectrum signal: (a) block-diagram of the signal correla­ tion processing in matched (optimal) receiver, (b) input signal, and (c) output signal.

PN , out =



1 2π

f max

∫

Nout (ω ) dω =

f min

1 2π

f max

∫

N0 a2 S2 (ω ) dω =N0 a2 E

f min

(6.136)

Taking into account (6.134) and (6.136) the output SNR may be represented as SNRout =



PS, out PN , out

= E/N0



(6.137)

Hence, the SNR gain is found as

χ=



SNRout =B SNRin

(6.138)

An important conclusion from (6.138): matched (optimal) reception of the spread-spectrum signals allows achieving significant gain in SNR by increasing the phase compression ratio (base) of the signal. Thus, the proper gain will allow detecting the signals, even if they are completely buried within the noise. The following two types of signals with the large pulse compression ratio B are widely used in modern radars and communication systems: • •

Colored pulses, with linear frequency modulation (FM) waveform [Figure 6.42(a)]; those signals are sometimes called chirp signals; Phase-coded pulses, with several jumps of carrier’s phase between 0° and 180° [Figure 6.42(b)] within signal’s duration.

In those two types of signals, the energy is spread along a wide frequency range. The reader may obtain the details about practical utilization of the compression for those types of signals in [20, 21].

386

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.42  Examples of spread-spectrum signals: (a) colored pulse of linear FM waveform (chirp pulse), and (b) four-element phase-coded pulse.

6.3.4  Diversity Reception Technique

An alternative for improvement of the RF link stability is diversity reception technique (DRT). The use of DRT allows decreasing of the fast fade depth by altering its statistics. In other words, the same communication stability may be achieved for less power fade margin for particular propagation conditions compared to that for non-DRT reception. The idea of DRT is based on the use of two or more copies of the same signal (signal 1, signal 2, and so forth) sent to the reception point through different statistically independent tracks, followed by merging those signals in the receiver to decrease fading depth. Those independent tracks are called diversity branches. The independent (uncorrelated) copies of the signal are merged in receiver by using different techniques to achieve less communication breaks. Statistically independent copies of the signal have different fluctuation patterns, so if one of them falls below

Figure 6.43  Illustration of the fade depth decrease while combining two statistically independent copies of the signal by using the DRT technique.

6.3 Methods of Improvement of RF Systems Performance 387

the desired SNR level, the others most likely will remain above that level. Thus, the proper combination of independent copies of signal may result in a decrease of probability of outages and therefore in an increase of transmission channel stability, as illustrated in Figure 6.43. The methods of achieving statistically independent diversity branches are as follows: •

•

•

Space diversity, when different spatially separated receiving antennas are used: the separation between antennas [that is called diversity base, L; see Figure 6.44(a)] must be greater than the spatial correlation distance, which is usually taken as L ≥ (5 ÷ 10)l. Polarization diversity, when two orthogonally polarized radio waves of the same frequency and propagation path are used to carry different copies of the same signal; this type of diversity branches are fairly effective for HF radio links [see Figure 6.44(b)]. Angle diversity, when different angles of arrivals of the radio waves are used to form the independent copies of the signal; for instance, if a dish antenna is used, then multiple diversity branches may be formed by using multiple feeds, shifted off the focal point perpendicular to the axial line of dish reflector [see Figure 6.44(c)].

Figure 6.44  Simplified block-diagrams of basic DRTs (twofold DRT cases): (a) space diversity, (b) polarization diversity, (c) angle diversity, and (d) frequency diversity.

388

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

•

•

•

Frequency diversity, when different carrying frequencies are used: the separa­ tion between carrying frequencies must be large enough in order the copies to become uncorrelated [see Figure 6.44(d)]. Time diversity, when the signal is repeated two or more times with the time shift that is greater than fading correlation interval in time domain; the gain in reception quality is achieved by sufficient decrease in transmission rate. Rake technique, when the pulse-signal arrives the reception point being carried by the radio waves propagating along different paths, then results in time-series of the same single pulse at the reception point. Those series are the result of different time delays between different propagation paths; shift-summation (discrete convolution) of those series may allow overlaying those pulses at a specific time moment, and therefore, improve SNR at the receiver’s output.

The last two listed approaches are used mostly for digital data transmissions links, when the discrete signals are transmitted; these two approaches are considered in detail in [16]. An important issue is how to combine the copies of signal at the reception point in order to achieve maximum efficiency in SNR, or maximum communications stability. Consider two separate diversity branches with u1(t) and u2(t) signal voltages mixed with the additive noise voltages uN1(t) and uN2(t) in each channel; that is, First branch

Second branch

u1(t) + uN1(t) u2 (t) + uN 2 (t)

  

(6.139)

Assume that the mixes coming from diversity branches are combined by adding to each other with weighting coefficients a1 and a2, respectively. That is, the resultant output voltage is

Uout (t) = uout (t) + uN , out (t) = a1 [ u1(t) + uN1(t)] + a2 [ u2 (t) + uN 2 (t)]

(6.140)

The power carried by the total wanted signal at the output is in direct proportion to the following mean-square:

uout2 (t) = [ a1 ⋅ u1(t) + a2 ⋅ u2 (t)] 2

(6.141a)

whereas the power carried by total unwanted noise at the output is in direct proportion to

uN , out (t)2 = a12 ⋅ uN12 (t) + a22 ⋅ uN 22 (t)

(6.141b)

Here uN12(t) and uN22 (t) are the mean-square noise voltages in channel 1 and channel 2, respectively. The difference in presentations (6.141a) and (6.141b) comes from the fact that u1(t) and u2(t) are strongly correlated in contrast to uN1(t) and uN2(t), which are uncorrelated processes. Therefore, their mean product becomes equal to zero.

6.3 Methods of Improvement of RF Systems Performance 389

Then for the output SNR the following expression may be written: SNRout =



uout2

=

uN ,out2

[ a1 ⋅ u1(t) + a2 ⋅ u2 (t)]2

(6.142)

a12 ⋅ uN21(t) + a22 ⋅ uN 22 (t)

Now we assign signal-to-noise ratios: SNR1 =

u12 (t) uN21(t)

, and SNR2 =

u22 (t)

(6.143)

uN 22(t)

to each channel before we combine them. Then, taking into account (6.143), the expression (6.142) may be rewritten as

SNRout =

a12 ⋅ SNR1 ⋅ uN21 + 2 a1 a2 ⋅ u1(t) ⋅ u2 (t) + a22 ⋅ SNR2 ⋅ uN 22 a12 ⋅ uN12 + a22 ⋅ uN 22



(6.144a)

If a12 ⋅ SNR2 ⋅ uN21 − 2 a1 a2 ⋅ u1(t) ⋅ u2 (t) + a22 ⋅ SNR1 ⋅ uN 22 is added and subtracted in the numerator, then after proper simplifications the expression (6.144) may be rewritten as  2 2  a1 ⋅ SNR2 ⋅ uN1 − a2 ⋅ SNR1 ⋅ uN 2 

SNRout = SNR1 + SNR2 −

a12 ⋅ uN12 + a22 ⋅ uN 22



2

(6.144b)

In (6.144a) the replacement

u1(t) ⋅ u2 (t) = u12 ⋅ u22 = SNR1 ⋅ uN12 ⋅ SNR2 ⋅ uN 22

(6.145)

is used based on fact that both signals, u1(t) and u2(t), are exact scaled copies of each other. An important conclusion can be made from (6.144b), namely, maximum SNRout after combining

SNRout

max

= SNR1 + SNR2

(6.146)



may be achieved if the expression in parentheses in (6.144b) is equal to zero; that is, if a1 uN12

SNR1

=

a2 uN 22 SNR2

= C = const



(6.147)

390

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

If (6.143) is taken into account, then (6.147) may be rewritten as a1 ⋅ uN12 u12



=

a2 ⋅ uN 22 u22

=C

(6.148)

From (6.148) the unknown weighting coefficients a1, a2 . . . an may be found for the general case of n-folded diversity as follows: a1 = C ⋅

u12 uN12

, a2 = C ⋅

u22 uN 22

, ………. an = C ⋅

un2 uNn2

(6.149)

From (6.146) and (6.148) it may be concluded that maximum SNR (maximum SNR combining technique)23 may be achieved if the receiver transfer function of each diversity branch is adjusted individually to keep the ratio of wanted signal (voltage or current) and unwanted noise power the same and equal to each other before summation. One of the appropriate block diagrams based on this algorithm is shown in Figure 6.45(a). Each diversity branch has two automatic gain control (AGC) loops: one of them to control the signal voltage level [numerator in (6.149)], another one is to control the noise power level [denominator in (6.149)]. In order to make it applicable, a pilot tone is mixed up with the wanted signal at the transmission. This block diagram is used in most advanced systems, which allows obtaining of the maximum SNR. The simplified version of DRT called equal gain combining is represented in Figure 6.45(b). In this case, all weighting coefficients are taken equal to each other. Finally the selection combining is presented in Figure 6.45(c). In this case, the diversity branch that has greatest value of the SNR is switched (connected) to the detector’s input. This principle is sometimes called auto-selection DRT. As a particular case we’ll consider an analytical expression for output fast fades statistics of the auto-selection DRT to confirm the decrease of deepness of fading (see Figure 6.43 for the qualitative demonstration). Consider the case of n-folded diversity, when signal-1, signal-2, … signal-n from diversity branches are due to independently fading fields E1, E2, … En; each of those fields is assumed to have the same complementary cumulative distribution function (CCDF), W(E). Then cumulative probability distribution function is P(E) = 1 - W(E). Because all branch signals are statistically independent, the probability of combined signal being below the E-value at the selector’s output is the product of probabilities of individual branch signals; that is,

Tn (E) = T n (E) = [1 − W (E)]n

(6.150)

Thus, the probability of E-value being exceeded by all diversity branch fields simultaneously (cumulative probability distribution function) becomes

Wn (E) = 1 − Tn (E) = 1 − [1 − W (E)]n

23. This principle is also called the weighted summation technique.

(6.151)

6.3 Methods of Improvement of RF Systems Performance 391

Figure 6.45  Examples of two-folded DRT block-diagrams: (a) for maximum SNR combining, (b) for equal gain combining, (c) for switching (auto-selection combining).

If Rayleigh distribution is considered for W(E) as most common CDF that describes most deep fades of single-branch signal, then in (6.151), it may be replaced by (6.31), so n



  E2   Wn (E) = 1 − 1 − exp  − 2     2σ  

(6.152)

Expression (6.152) is referred in Section 6.1.2.3 as quasi-Rayleigh distribution, which provides statistical description of the non-Rayleigh type fades with variable depth. Expression (6.45) is nothing but a modified version of (6.152). The difference between those two expressions is that the parameter n in quasi-Rayleigh distribution may vary within the range (0, ∞) continuously, whereas parameter n in (6.152) may only belong to the discrete sequence of natural numbers. Recall the argument of the expression (6.45), and insert the noise power PN into numerator and denominator: 2



 E  E2 /PN E2 SNR = = γ = =  E  2 2 Emed Emed /PN SNRmed med

(6.153)

where the ratios are replaced by proper SNRs taking into account that signal power is in direct proportion with the square of the proper electric field strength. Then expression (6.45) may be rewritten as

392

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

Figure 6.46  SNR gain for n-folded diversity reception for following DRT combing schemes: (1) maximum SNR combining, (2) equal gain combining, and (3) auto-selection combining.



{

}n

Wn (γ ) = 1 − 1 − exp[ −q(n) ⋅ γ ]

(6.154)

where g  is SNR relative to its median per expression (6.153). The values of q(n) may be taken from Table 6.2 or from Figure 6.13(a). As mentioned earlier, the expression (6.154) is only applicable to auto-selection combining along with the graph in Figure 6.13(b). In two other considered cases, namely maximum SNR combining and equal gain combining, the CDF, Wn(g ), is expressed in terms of c2-distribution function24 (chi-square distribution) that is given in [16] in details. Graphs shown in Figure 6.42 [8] may be conveniently used for the estimates of the gain in fade margin (i.e., how much the fade margin may be decreased in comparison to nondiverse reception) as a function of number of diversity folds n and for different types of DRT combining schemes. The idea of a diversity reception has received a new qualitative development in connection with the implementation of MIMO technology in the latest and upcoming generations of LTE cellular networks. The term MIMO stands for multiple-input multiple-output, meaning multiple antennas not just on the receive side, but also on the transmit side. This technology involves the combination of diversity reception, digital signal processing, and smart antenna technology, allowing achievement of not just a significant increase in RF link stability between base station and user’s terminals (mobile phone, tablets, sensors, etc.), but also the optimization of the transmit powers, optimization of the frequency spectrum usage, as well as a significant increase of the link throughput. The idea is to split the initial digital signal of high transmission rate into several low-rate data streams; each stream of much less baseband spectrum modulates a designated subcarrier and is radiated by proper designated Tx-antenna. On the receive site inverse operations are taking place by using multiple antennas (the number of Rx antennas does not need to be equal to 24. Do not confuse with c, which represents the SNR gain.

6.3 Methods of Improvement of RF Systems Performance 393

the number of Tx antennas). The use of proper signal processing technology [direct and inverse fast Fourier transform (FFT)] allows extraction of the substreams, and then properly combining them in order to retrieve the initial signal. As a separate, independent part of MIMO technology is the use of smart antennas that allow to implement a beamforming procedure. In this way, a significant reduction in fading depth can be achieved, and thus, it is possible to carry out a significant reduction in the depth of fading, and therefore increase the stability of the radio link between base station and subscriber’s terminal. A detailed analysis of MIMO systems goes beyond the problems outlined in this book. The reader is encouraged to refer to [24], where it is clear and with sufficient detail sets out their main provisions.

Problems   P6.1 Use (6.30) to confirm it satisfies normalization condition (6.9). Express median value Emed of the Rayleigh distribution in terms of parameter s. Answer: Emed = 1.177s.   P6.2 Find the depth of the Rayleigh fading relative to its median value (E0.1/ Emed)dB - (E0.9/Emed)dB and assess the result by comparing with that found from the graph on Figure 6.8(b). Answer: 13.4 dB.   P6.3 The stability of HF communication radio link is affected by lognormally distributed slow fading with the standard deviation of s = 8 dB and by Rayleighian fast fading. The power margin on the transmitting side is to compensate both fades and to achieve communication stability of 99%. Find the value of the power margin by using two approaches: a. As a sum of margins assessed for slow and fast fades separately from Figures 6.16 and 6.12(b) (m = 1), respectively. b. From the combined distribution graphs shown in Figure 6.19 (m = 1). Which one is larger? Why? Explain in your own words. Answer: (a) 20 dB + 18.5 dB = 38.5 dB; (b) 25 dB.   P6.4 For the probability density function (PDF) of two-ray random interference that is given by (6.18), find the mean value of fluctuating electric field E expressed in terms of E1 and compare with median value given by (6.21). Hint: For any random variable x mean value is defined as x2

x=

∫ x w (x) dx ; for the considering case x1 = 0, x2 = 2E1.

x1

4E1 . π   P6.5 A three-stage passive filter has a transformation coefficients of the stages, K1, K2, and K3, respectively. Based on (6.93), (6.94), and (6.96), show that the overall noise parameters for the filter may be represented as N = 1/(K1K2K3) and T = T0(N - 1). Answer: E =

394

Fluctuation Processes, RF-Link Stability Analysis and Radio Wave Reception

  P6.6 In Example 6.3 for the case 1 scenario, calculate the minimum value of the gain for LNA if predefined (required) value of the system noise temperature for the RF unit is T = 800K. Answer: 11.47 dB.   P6.7 Confirm (6.144) by modifying (6.143).   P6.8 Estimate the increase in maximum detectable radar range if sounding monochrome pulse signal (case 1) is replaced by linear FM spread-­ spectrum signal with time-bandwidth product that equals B = TDf = 50 and optimal filtering is applied (case 2). The transmitter’s EIRP and the system’s bandwidth are assumed to remain unchanged. Hint: Use expressions (3.52), (6.106b), and (6.136). Take amplitude modulation depth for monochrome pulse equal 100% (mAM = 1). Answer: increased 2.94 times.   P6.9 For sufficient reception quality of the satellite TV broadcast FM signal, a required SNR at the detector’s output is SNRDet, out = 26.2 dB (416.9 unitless). Standard TV video signal spectrum is limited to the maximum frequency of F = 4.5 MHz. The system’s RF bandwidth is Df = 30 MHz (same as the width of FM signal spectrum). The system noise temperature is Tsys = 145K. Based on this data make the following estimates: Modulation index mFM and detector’s SNR gain c. Minimum input signal power PRx,min required to meet the value of c (6.106b). • Compare PRx,min with the power received in real conditions, PRx if signal power density at the reception point (magnitude of Poynting vector) is –110 dBW/m2 (P = 10 pW/m2). A dish antenna of diameter d = 0.6m with the aperture efficiency v = 0.67 is used at the receiver’s input. • Find the power margin DPRx = PRx /PRx,min v = 0.67 in dB. What changes will take place if the modulation index mFM is increased? Explain. Note: In real systems an additional 18.8 dB improvement in SNRDet, out is achieved due to technical modifications that are not discussed here; the resultant SNRDet, out is 26.2 + 18.8 = 45 dB. For details, refer to [22]. Answer: mFM = 4.44, c = 20 dB, PRx,min = -126 dBW, PRx = -117.2 dBW, DPRx = 8.8 dBW. • •

References   [1] Weik, M. H., Communications Standard Dictionary, New York: Van Nostrand Reinold, 1983.   [2] Rice, S. O., “Mathematical Analysis of Random Noise,” Bell System Technical Journal, Vol. 24, 1945, pp. 46–156.   [3] Nakagami, M., “The m-Distribution, a General Formula of Intensity of Rapid Fading,” in W. G. Hoffman, (ed.), Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held at the University of California, London, U.K.: Permagon Press, 1960, pp. 3–36.   [4] Rytov, S. M., Kravtsov, Y. A., and V. I. Tatarskii, Principles of Statistical Radio-Physics, New York: Springer-Verlag, 1987.

6.3 Methods of Improvement of RF Systems Performance 395   [5] Gradshtein, I. S., and I. M. Ryzhik, Tables of Integrals, Series, and Products, 7th ed., New York: Elsevier, 2007.  [6] Долyxaнов, M. П., A. C. Caaкян, и H. H. Энтинa, Квaзиpeлeeвскиe зaмирания нa кopoтких волнax. Известия AH Aрм. CCP, cepия XXVIII, Aпpeль, 1975. (Dolukhanov, M. P., A. S. Saakian, and N. N. Entina, “Quasi-Rayleighan Faded on HF,” Proceedings of the Academy of Sciences of the Rep of Armenia, Series XXVII, April 1975, in Russian.)  [7] ITU-R Recommendation P.372-7, “Radio Noise,” International Telecommunication Union, 2001.   [8] Pratt, T., and C. W. Bostian, Satellite Communications, New York: John Wiley & Sons, 1986.   [9] Uman, M. A., Understanding Lightning, Carnegie, PA: Bek Technical Publications, 1971. [10] Spaulding, A. D., and J. S. Washburn, Atmospheric Radio Noise: Worldwide Levels and Other Characteristics, NTIA Report 85-173, U.S. Dept of Commerce, 1985. [11] CCIR Report 322, “World Distribution and Characteristics of Atmospheric Radio Noise,” Xth Plenary Assembly, Geneva, 1963. [12] Smith, A. A., Radio Frequency Principles and Applications, New York: IEEE Press, 1998. [13] Balanis, C. A., Antenna Theory, Analysis and Design, New York: John Wiley & Sons, 2005. [14] Kraus, J. D., Radio Astronomy, New York: McGraw-Hill, 1966. [15] Stutzman, W. L., and G. A. Thiele, Antenna Theory and Design, New York: John Wiley & Sons, 1998. [16] Schwartz, M., W. R. Bennett, and S. Stein, Communication Systems and Techniques, New York: IEEE Press, 1996. [17] Ziemer, R. E., and W. H. Tranter, Principles of Communications, Boston, MA: Houghton Mifflin, 1990. [18] Carlson A. B., Communication Systems, New York: McGraw-Hill, 1986. [19] North, D. O., An Analysis of the Factors Which Determine Signal/Noise Discrimination in Pulsed-Carrier Systems, RCA Tech. Rept. PTR-6C, June, 1943 (reprinted, Proc. IEEE 51, No. 7, July 1963, 1016–1027; reprinted, Detection and Estimation, S. S. Haykin, (ed.), Halstad Press, 1976, pp. 10–21). [20] Cook, C. E, and M. Bernfeld, Radar Signals: An Introduction to Theory and Application, New York: Academic Press, 1967 (reprinted by Artech House, 1993). [21] Skolnik, M. I., Introduction to Radar Systems, 3rd ed., New York: McGraw-Hill, 2002. [22] Pratt, T., C. W. Bostian, and E. A. Jeremy, Satellite Communications, New York: John Wiley & Sons, 2003. [23] Couch, L. W., Digital and Analog Communication Systems, Eight Edition, Pearson Education, 2013. [24] Heath, R. W., and A. Lozano, Foundations of MIMO Communication, First Edition, Cambridge University Press, 2018.

Selected Bibliography Dolukhanov, M. P., Propagation of Radio Waves, translated from Russian by Boris. Kuznetsov, Moscow: Mir Publishers, 1971. Freeman, R. L., Reference Manual for Telecommunications Engineering, Vol. 1, New York: Wiley-Interscience Publication, 2002. O’Flinn, M., and E. Moriarty, Linear Systems, New York: John Wiley & Sons, 1987. Vaughan, R., and J. B. Andersen, Channels, Propagation and Antennas for Mobile Communications, New York: IEEE Press, 2003.

Acronyms ADC Analog-to-digital converter ADT Asymptotic diffraction theory AM Amplitude modulation BER Bit error rate CCDF Complementary cumulative distribution function CDF Cumulative distribution function CP Circular polarization CW Continuous wave DAC Digital-to-analog converter DRT Diversity reception technique EHF Extremely high frequency (30–300 GHz) EIRP Effective isotropic radiated power, W ELF Extremely low frequency (< 3 kHz) EP Elliptical Polarization ESCS Effective scattering cross-section of the tropospheric turbulences FM Frequency modulation FNBW First null beam width of the antenna radiation pattern, rad (or degrees) HF High frequency (3–30 MHz) HPBW Half-power beam width of the antenna radiation pattern, rad (or deg) IEEE Institute of Electrical and Electronics Engineers IF Intermediate frequency ITU-R International Telecommunication Union–Radio Communication Sector LF Low frequency (30–300 kHz) LHCP Left-hand circular polarization LOS Line-of-sight MF Medium frequency (0.3–3 MHz); matched filter MUF Maximum usable frequency OPN One-port network PDF Probability density function PEC Perfect electric conductor PM Phase modulation RCS Radar cross-section RF Radio frequency RHCP Right-hand circular polarization 397

398 Acronyms

RMS SHF SNR TEM TPN UHF VHF VLF WLAN

Root mean square Super high frequency (3–30 GHz) Signal-to-noise ratio Transverse electromagnetic Two-port network Ultra high frequency (0.3–3 GHz) Very high frequency (30–300 MHz) Very low frequency (3–30 kHz) Wireless local area network

List of Symbols a Attenuation coefficient, Np/m (or dB/m); angle, rad b  Phase coefficient of the radio wave, rad/m; angle, rad; parameter in generalized Rayleigh distribution function (Rice distribution) Γ E Electric field reflection coefficient Magnetic field reflection coefficient Γ H P Power reflection coefficient G g Slant angle (elevation angle) D Allowable average height of the surface roughness de Loss angle of the dielectric medium, rad . Complex relative dielectric permittivity e e0 Absolute dielectric permittivity of free space, (1/36p) × 10-9 F/m x Number of collisions in ionospheric plasma, 1/s xe Relative fluctuation of the dielectric permittivity (xe = De/e) hant Antenna efficiency Q Geocentric angle, rad q Zenith angle in spherical coordinates, rad (or degree) qi, qr, qt Angle of incidence, angle of reflection, angle of transmission (refraction) kfast Fast fading factor kslow Slow fading factor l Wavelength, m m Relative magnetic permeability m0 Absolute magnetic permeability of free space (vacuum), H/m n Antenna aperture efficiency Π Pointing vector, W/m2 Ssc(qsc) Scattering diagram r Electric charge volumetric density, C/m3 s  Conductivity of the medium, S/m; standard deviation of the random variable ssc Scattering cross-section of the turbulent troposphere (per unit volume), 1/m sRCS Scattering cross-section of the target in radar applications, m2 F Phase of the radio wave, rad; scalar (magnetic) potential, V Phase of propagation factor, rad FF Reflection phase, rad FG 399

400

List of Symbols

j Azimuth angle in spherical coordinates c Detector’s SNR gain Y Amplitude radiation pattern Spatial autocorrelation function of the random dielectric permittivity ye W Solid angle, sr w Angular frequency, rad/s Ñ Nabla vector operator 2 Laplacian operator Ñ A Diffraction loss; parameter in Okumura-Hata propagation model ¢ , AE , AE¢ , AF1 , AF¢ 1 , AF 2) A, A¢ Attenuations in ionosphere (e.g., AS , AS¢ , AD , AD → A Vector (magnetic) potential, Wb/m a Major axis of the ellipse; Earth’s average geometric radius (6,370 km) B  Time-bandwidth product for spread-spectrum signal; parameter in Okumura-Hata propagation model B Vector of the magnetic field induction, or magnetic flux density, T Specific attenuation in ionospheric reflecting layer F2 (at 1 MHz) BF2 b Minor axis of the ellipse C Parameter in Okumura-Hata propagation model Fresnel cosine integral CF Structural constant of the tropospheric turbulences Ce c Speed of light in vacuum (3 × 108 m/s) D Antenna directivity; parameter in Okumura-Hata propagation model D Vector of the electric field induction (electric flux density), C/m2 Divergence factor due to Earth’s surface convexity and due to scatterings Ddiv, Dsc  from roughness E Parameter in Okumura-Hata propagation model Median electric field strength, V/m Emed áEñ Average electric field strength, V/m E Vector of the electric field strength, V/m e Electron’s charge, 1.6 × 10-19 C erf Error function erfc Complementary error function emf Electro-motive force  F Complex propagation factor F Force, N F Modulating frequency, Hz Atmospheric noise factor (relative brightness temperature) FA f Carrier frequency, Hz Df RF signal bandwidth, Hz; receiver’s passband, Hz G Antenna gain H(0) LOS path clearance without atmospheric effect, m

List of Symbols

401

H(N)  Total path clearance on LOS link in presence of the atmospheric refraction, m Vector of the magnetic field strength, A/m H Natural unit of heights, m Hh h Height, elevation, m; Planck’s constant = 6.626 × 10-34 J × s I Current, A Radiation intensity, W/sr IS i = −1 Imaginary unit Vector of electric current spatial density, A/m2 J J S Vector of conducting current surface density, A/m K Transmission coefficient of TPN (two-port network) k Relative distance of the reflection point on LOS microwave radio link  k Propagation constant (complex), 1/m Boltzmann’s constant [1.38 × 10-23 W/(K × Hz)] kB L Total propagation path loss Propagation path loss due to atmospheric and ground effects LF L0 Free space (reference) propagation loss; outer scale of tropospheric turbulences Multiple screen diffraction loss in urban area, dB Lmsd Roof-to-street diffraction loss, dB Lrst Natural unit of distance in ADT approach, m Lr Antenna effective length, m leff Inner scale of tropospheric turbulences l0 M Parameter in Lee propagation model; large parameter in ADT approach m Parameter in m-distribution of Nakagami Mass of the electron (9.1 × 10-31 kg) me N Refractivity Plasma concentration, 1/m3 Ne NF Noise figure Noise power spectral density, W/Hz NT n  Refraction index; parameter in Quasi-Rayleigh distribution (n-distribution) n Unit vector normal to surface P Power, W Noise power, W PN Signal power, W PS P(E) Cumulative distribution function (CDF) P Polarization vector of the unit volume, C/m2 PS Radiated power, W Power transmitted (applied to antenna’s input), W PTx Power received (from the antenna’s output), W PRx p(m) Parameter in m-distribution of Nakagami

402

List of Symbols

q Earth’s ground parameter in ADT approach Scattering vector q q(n) Parameter in n-distribution R Horizontal distance between corresponding antennas along the earth surface, m Real part of the antenna input impedance, ohm Rant Antenna radiation resistance, ohm RS r Direct distance between two points, m S Area of the surface, m2 Seff Area of the antenna effective aperture, m2 SF Fresnel sine integral Se Spatial spectrum of the autocorrelation function for dielectric permittivity s Distance scale, 1/m T  Temperature (including noise temperature), K; Tesla (unit for the magnetic field induction/flux density) E  T Electric field transmission coefficient (complex) T H Magnetic field transmission coefficient (complex) Power transmission coefficient TP t Time, s U(x) Attenuation factor in ADT approach V Volume, m3 V(y) Height-gain function in ADT approach v Velocity of the radio wave, m/s W(E) Complementary cumulative distribution function (CCDF)  W Intrinsic impedance (complex), ohm Ionization energy, J; spatial density of the energy of electric field, J/m3 We Spatial density of the energy of magnetic field, J/m3 Wm w(t) Airy function w(E) Probability density function X Normalized distance in ADT approach Imaginary part of the antenna input impedance, ohm Xant x Cartesian coordinate; relative electric field strength (x = E/Emed)  x Numerical distance in W&V method for ground wave propagation xo Unit vector in Cartesian system y Normalized height in ADT approach; Cartesian coordinate yo Unit vector in Cartesian system Antenna input impedance, ohm Zant z Cartesian coordinate Unit vector in Cartesian system zo

About the Author Dr. Artem Saakian graduated with degrees in electrical engineering from the National Polytechnic University of Armenia (NPUA; the former Yerevan Polytechnic Institute), the University of Telecommunications in St. Petersburg, Russia, and the University of Massachusetts, Lowell, Massachusetts. Dr. Saakian’s professional career begun in the academic environment at SEUA; there he was an assistant professor, associate professor, the head of the chair of radio technology and communication systems, and the director of the graduate school. His engineering and scientific research experience relates to RF antennas and propagation projects funded by U.S. Navy, as well as by the former Soviet Navy and Soviet Space Administration. Dr. Saakian has served in the U.S. Navy, Naval Air Systems Command (NAVAIR), as an electronics engineer since 2005. He is currently retired and supports the Navy as an electronics engineering consultant. He is the author of more than 25 publications, including peer-reviewed papers, conference materials, and encyclopedia articles.

403

Index A Absolute curvature, 237 Absolute dielectric permittivity, 21 Absolute magnetic permittivity, 21 Additive interference. See Noise Air-Earth ground interface, 179 Airy differential equation, 222 Airy functions, 195 Ampere’s law, 21 Amplification (doubling), 85 Amplitude probability distribution (APD), 366 Amplitude radiation pattern, 116 Angle diversity, 387 Angle of coastal refraction, 192 Angle of ellipticity, 55 Angle of total refraction, 70 Angular aperture, 118 Antenna effective height, 168, 169 Antenna effective length alternate to, 128 calculation of, 124–­25 defined, 122 illustrated, 123 Antenna efficiency, 121, 361 Antennas basic parameters, 113–­29 directivity, 114–­20 effective area of the aperture, 128–­29 effective vector-length, 126 equivalent schematic representation, 121 far field zone, 135–­36 gain, 120–­22 gain effect on link performance, 291–­92 introduction to, 113 loss resistance, 120–­22 matched, 121

omnidirectional, 120 radiation beam width, 118 radiation pattern, 114–­20 radiation resistance, 120–­22 tuned, 121 Aperture efficiency, 129 Apparent heights, 147, 161, 162 Apparent reflection coefficient, 227, 228 Appleton-Hartree equation, 316 Asymptotic diffraction theory (ADT) about, 193 basic concepts, 193–­98 correctness of statements, 198 estimates in Penumbra zone, 204–­6 general solution of wave equation and, 215–­25 outcome of, 197 over spherical Earth surface, 193–­210 propagation between elevated antennas, 204–­6 propagation between ground-based antennas, 198–­204 results comparison, 200–­202 W&V approach versus, 199 Atmospheric air composition of, 6, 7 electromagnetic parameters of, 6 ionization of, 8, 9 seasonal and random fluctuations of, 8 vertical movement of masses in, 12 Atmospheric noise, 18, 362–­63, 365–­70 Atmospheric nonthermal noises, 366 Attenuation curves, 303 in hydrometeors, 298, 300 405

406 Index

Attenuation (continued) in ionosphere, 300–­307 of radio waves, 45 in reflecting ionospheric layer, 304 total, 46, 303 in troposphere, 296–­300 water vapors, 297 Attenuation coefficient defined, 37, 296 due to rain and fog, 300 frequency dependence of, 298 at sea-level atmosphere, 297 Attenuation constant, 45 Attenuation factor, 197 Attenuation loss, 299 Autocorrelation function flat projections of, 276 spatial, 274 spectrum of, 276 of statistically homogeneous and isotropic random field, 308–­9 Auto-selection DRT, 390, 392 Auxiliary magnetic potentials, 76–­79 Average value, 327 Axial ratio, 55, 57

B B&A method, 208, 209 Background sky noise, 364 Back lobes, 118 Barrier type diffractor, 94 Baseband signal, 376 Boundary conditions defined, 26 of electrodynamics, 25–­31 for field components, 224 Leontovich approximate, 178–­83 in spherical coordinates, 223 for tangential components, 65 Boundary value problem, 26 Bragg’s diffraction condition, 278 Breaking point distance, 173–­74 Brewster’s angle, 70 Brightness temperature, 362, 367

C Calibration condition, 102, 333 Central limit theorem, 334 Channel coding, 377 Channel reliability, 328 Chapman model, 311–­15 Circular polarization (CP) about, 50–­52 components, phase shift, 63 components, ratio of amplitudes, 63 Circular polarized waves components, 60 decomposition of elliptically polarized wave into, 59–­63 decomposition of LP waves into, 53 polarization losses and, 260 two, combining of, 61 Coastal refraction angle of, 192 defined, 190 pattern of, 192 Coefficient of proportionality, 312 Communication link budget, 132 Communication stability, 330, 374 Complementary cumulative distribution function (CCDF), 328, 329, 334, 341, 344 Complementary cumulative Rayleigh distribution function, 336 Complementary error function, 185 Complex analysis, 31 Complex dielectric constant, 215 Complex dielectric permittivity, 43–­44 Complex relative permittivity, 22, 32 Complex wave vector, 64 Composition of probabilities, 348 Concentration of electrons, 9, 10 Conductivity classification of media by, 31–­35 frequency dependency of, 34 non-zero, 42 Constant and parametric media, 22 Constitutive parameters, 21, 26 Cornu spiral, 111 Correction factors, 152, 153

Index 407

Cosmic dust, 11 Cosmic noise, 18, 364–­65 Cosmic rays, 8 Coulomb force, 22 Critical angle, 70 Cumulative distribution function (CDF), 326, 348, 351 Curl of vector formula, 215 Current density electrons-cloud movement and, 232 total, 33 volumetric, 32

D Deciles, 327 Dielectric constant, 234, 253, 254 Dielectric permittivity absolute, 21 complex, 43–­44 as decaying function, 234 defined, 7 of extraordinary ray, 254 of ionospheric magneto-active plasma, 250 real part of, 232 relative, 22 tensor of, 316–­19, 321 Differential distribution function (DDF), 332 Differential Rayleigh distribution function, 336 Diffracted waves, 12, 13 Diffraction loss, 90 Diffuse reflection, 157, 158 Digital Terrain Elevation Data (DTED) files, 159 Directivity defined, 116, 125 formula, 119 radiation intensity and, 118 Direct waves defined, 11 illustrated, 12 parallel, 142

Displacement electric current volumetric density, 23 Displacement magnetic current volumetric density, 23 Distance factor, 204, 205 Distance to reflection point, 148–­53 Divergence factor defined, 153 distance dependence of, 153 due to scattering, 158 graphs for, 154 Divergence of energy in radio waves, 153–­56 Divergence theorem, 26 Diversity receiving principles, 293 Diversity reception technique (DRT) auto-selection, 390, 392 block diagrams, 387 concept, 386 defined, 386 equal gain combining, 390, 392 illustrated, 386 independent diversity branches and, 387–­88 two-folded, 391 D-layer, 260–­61, 262

E Earth-ionosphere waveguide, 262, 265 Earth radius, 161 Earth’s atmosphere free propagation and, 35 structure, 6–­11 Earth’s equivalent radius, 206, 237, 238, 288 Effective (RMS) electric field strength, 156 Effective aperture, 286 Effective area of the aperture, 128, 129 Effective field strength, 329 Effective isotropic radiated power (EIRP), 132, 373 Effective power flow, 41 Effective scattering cross-section (ESCS), 279–­80, 285

408 Index

Effective vector-length, 126 E-field vector orientation, 69 E-layer, 260–­61, 262, 304–­5 Electric field phasors, 144, 145 Electric field reflection coefficients defined, 65 magnitude and phase of, 71, 72 magnitude of, 66 for parallel and perpendicular polarized waves, 74 Electric field strength calculation results comparison, 199 distance dependence of, 207 effective (RMS), 156 family of graphs of, 201 normally distributed, 344 in system of equations, 221 Electric flux density, 27 Electric induction vectors, 21 Electrodynamics boundary conditions of, 25–­31 Maxwell’s equations of, 21–­25 Electromagnetic process boundary conditions and, 25–­31 Maxwell’s equations of electrodynamics and, 21–­25 sketch of constituents of, 25 time-harmonic, 31–­35 Electromagnetic waves, 1 Electromotive force, 226 Electrons-cloud movement, 232 Elevated antennas asymptotic diffraction theory (ADT) and, 204–­6 choice of elevation, 162–­64 defined, 137 distance factor and, 204, 205 elevation factors and, 204–­5, 206 propagation between, 137–­74, 204–­6 space waves and, 15 Elevation factors, 204–­5, 206 ELF electric field dependence, calculated, 263 frequency ranges supporting propagation and, 268 height of monopole antenna and, 263 signals, propagation of, 264–­65

Elliptically polarized radio waves decomposition into CP waves, 59–­63 decomposition into LP waves, 52–­59 defined, 47–­48 excited by vertical monopole, 182 pattern of, 257 phase shift, 58 Elliptical polarization, 47, 54, 55 Emf, 131, 227 Emissivity, 364 Empirical models about, 165–­66 COST 231-Hata model, 167–­68 defined, 165 example, 168–­69 Lee model, 168 Okumura-Hata model, 166–­67 See also Propagation predication models Energy per bit to noise power density (Eb/ No) ratio, 370 Equal gain combining, 390, 392 Equivalent isotropic radiated power (EIRP), 176, 261 Error function, 344 Euler’s identity, 89 External noise, 17–­18, 362–­70 Extraordinary rays, 253–­54, 259 Extraordinary wave, 323 F Fade margin, 328, 374 Fading defined, 16, 326 fast, 292–­93, 331–­41, 346–­53 patterns, 352 slow, 293, 341–­53 See also Multiplicative interference Faraday rotation, 251, 260 Faraday’s law, 21 Far field zone, 135–­36 Fast fading appearance of, 330 combined distribution, 346–­53 defined, 292–­93 depth, 293 generalization of statistics, 337–­41

Index 409

m-distribution of Nakagami and, 338–­40 n-distribution (Quasi-Rayleigh distribution) and, 340–­41, 342 Rayleigh distribution and, 337 Rice distribution and, 338, 339 statistical distributions, 331–­41 See also Slow fading Fast Fourier Transform (FFT), 393 Field strength, as function of distance, 190 First null beam width (FNBW), 118 Flat Earth approximation, 186 approximation case study, 138–­46 propagation between ground-based antennas over, 174–­93 real homogenous, propagation over, 184–­87 real inhomogeneous, propagation along, 188–­93 Fourier transform, 383–­84 Free electrons in free space, 229 oscillating movement of, 230 trajectory in magnetic field, 248 Free propagation amplification of field and, 85–­86 defined, 11 Earth’s atmosphere and, 35 of uniform plane radio waves, 35–­47 See also Propagation Free space disposition of vectors in, 39 free electrons in, 229 impedance of, 40 line current in, 184 Poynting vector in, 262 radio wave wavelength in, 41 wave number in, 190 Free space loss, 131, 173–­74 Frequency diversity, 388 Frequency-selective absorptions, 296 Fresnel integral cosine, 89 Fresnel integral sine, 89 Fresnel’s cosine, 112 Fresnel’s integrals, 109–­12 Fresnel’s sine, 112

Fresnel zones defined, 85 practical applications of, 92–­98 solid shapes of, 86 two-dimensional shapes of, 86 Friis formula, 131, 132, 373 Frozen turbulences, 271

G Gaussian error-integral, 110 Gaussian model, 280–­81 Gauss’ law, 21 Gauss theorem, 24 Geocentric angles, 146, 148 Ground-based antennas asymptotic diffraction theory (ADT) and, 198–­204 field structure, 176 Leontovich approximate boundary conditions and, 178–­83 over infinite, perfect ground plane, 175–­78 over real homogeneous flat Earth, 184–­87 over real inhomogeneous flat Earth, 188–­932 propagation over flat Earth, 174–­93 surface wave component, 15 term usage, 175 Ground waves, 14–­15 Guided propagation, 11 Gyro-magnetic radian frequency, 248 Gyro-magnetic resonance, 252

H Half power beam width (HPBW), 118 Hankel function, 264 Height-gain function, 197–­98 Helmholtz equation defined, 36 inhomogeneous, 103, 104 simple solution to, 36 for vector potential, 101–­5 Hertzian dipole, 78, 79, 105, 109, 123

410 Index

HF propagation, 13–­14 High-gain factor, 208 High-loss conducting medium, 44–­46 Homogeneous and inhomogeneous media, 22 Horizon distance defined, 146 power flow density at, 176 real total, 150 Horizontal current element, 215 Horizontally polarized radio waves, 183, 196, 222–­25 Horizontal polarization, 195 Huygens-Kirchhoff’s principle, 156, 163 Huygens’ principle, 79–­81 Huygens’ source, 79 Hydrometeors, attenuation in, 298, 300

Ionosphere attenuations in, 300–­307 defined, 8 homogeneous magnetoactive plasma of, 315–­23 Ionospheric gaseous composition, 309–­15 Ionospheric layers generation, 309–­15 Ionospheric propagation. See Propagation

K Knife-edge diffraction, 87 Knife-edge diffraction loss, 90, 91

L I Ideal dipole, 78, 79, 105, 109, 123 Illuminated zones, 193 Incident waves, disposition of, 64 Input impedance, 120, 177, 212–­15 Integral evaluation, 82 Integration area in integrals, 27, 28 Integration constant, 222, 333 Interference additive, 16, 353–­74 classification of, 17 destructive impact of, 15 multiplicative, 16, 325–­53 random, 330, 331–­37 in RF transmission links, 15–­18 International standard atmosphere, 6 International Telecommunications Union (ITU), 4 Intrinsic impedance of medium, 39, 108, 177, 179 Ionization of atmospheric air, 9 intensity, 312, 313 photo, 8–­9 recombination and, 314 strike, 8 Ionizing radiation, 311, 313

Laminar movement, 282 Langmuir, 233 Large parameter, 194, 219 Large-scale obstacles defined, 160 reflection from, 159 Larmor (or cyclotron) frequency, 249, 317 Law of current continuity, 22 Law of Sines, 148 Left-hand circular polarization (LHCP), 51–­52, 59, 61, 252, 256, 320 Left-hand elliptical polarization (LHEP), 54, 55 Leontovich approximate boundary conditions, 180, 181 LF, propagation of, 265–­69 Linearly polarized radio waves decomposition into CP waves, 53 decomposition of elliptically polarized waves into, 52–­59 defined, 48 in magneto-active plasma, 249, 253 propagation in plasma medium, 230 Linear polarization (LP), 49–­50 Lognormal probability distribution, 293, 343–­46, 350 Longitudinal terrain profile, 160 Lorentz condition, 102

Index 411

Lorentz force, 247, 317 LOS paths, propagation predication models, 172–­74 Lossless medium, uniform plane wave in, 40–­42 Lossy medium high-loss conducting, 44–­46 low-loss dielectric, 44 uniform plane wave in, 42–­47 Low-loss dielectric medium, 44 L-shaped wire antenna, 183

M Magnetic currents, 1 Magnetic field and propagation along geomagnetic field lines, 249–­53 impact of, 247–­60 perpendicular to geomagnetic field lines, 253–­55 radio wave arbitrary oriented, 255–­58 reflection and refraction and, 258–­60 trajectory of free electrons and, 248 transversal and longitudinal components and, 256 Magnetic field reflection coefficients, 65, 66, 74 Magnetic field strength, 21 Magnetic flux density, 27 Magnetic induction vectors, 21 Magnetic permittivity, 21, 22 Magneto-active ionosphere, 258–­60 Magneto-active plasma, 250 Magnetohydrodynamic waves, 11 Man-made noise, 171, 365, 369 Marconi, G. M., 1, 2 Matched antennas, 121 Matched filter (MF) goal of, 382 output, 382 power transmission function of, 384 signal waveforms of, 383 Matched filtering, 382 Mathematical relations, 98–­101 MATLAB, 89, 159, 185, 200 Maximum detectable range, 134

Maximum SNR combining technique, 390, 392 Maximum usable frequency (MUF), 245, 246 Maxwell’s equations applying complex variables analysis and, 31 defined, 1 direct solution of, 3 electric and magnetic fields and, 26 expression, 21–­25 solution in parabolic form, 3 M-distribution of Nakagami, 338–­40 Media boundary of two, 64 classification by conductivity, 31–­35 constant and parametric, 22 electromagnetic properties of, 34 homogeneous and inhomogeneous, 22 properties in frequency bands, 33 variant, frequency dependence of, 35 Micrometeors, 11 Microwave bands, subdivision of, 5 Millington’s approximate formula, 189 MIMO (multiple-input multiple output), 392–­93 Mixed paths, 188 Multipath propagation, 330 Multiplicative interference combined distribution of fast and slow fades, 346–­53 defined, 326 fast fading statistical distributions, 331–­41 fluctuation processes and stability of radio links and, 325–­31 slow fading statistical distributions, 341–­46 See also Fading

N National Institute of Standards and Technology (NIST), 304 Natural unit of distance, 194, 206 Natural unit of heights, 195, 205, 219

412 Index

N-distribution (Quasi-Rayleigh distribution), 340–­41, 342 Noise atmospheric, 18, 362–­63, 365–­70 background sky, 364 cosmic, 18, 364–­65 defined, 16 external, 17–­18, 362–­70 man-made, 171, 365, 369 receiver (internal), 16–­17 shot effect and, 17 thermal, 16–­17, 353, 363–­64 types of, 16–­18 white, 353 Noise figure of cascaded TPNs, 356–­58 defined, 355 of passive TPN, 358–­60 Noise power, 367 Noise radiation, 362 Noise suppressing modems, 375–­76 Noise temperature antenna, 361–­62 atmospheric nonthermal noise and, 366 of cascaded TPNs, 356–­58 defined, 356 system, 371 Non-LOS paths, propagation predication models, 169–­72 No-refraction condition, 239 Normal (Gaussian) distribution of a random variable, 342–­43 Normalized distance parameter, 194, 219 Normalized height(s), 219 Norton wave, 184 Numerical distance, 184–­85 O Oblique incidence introductory notes, 67–­68 of parallel polarized radio wave, 68–­72 for perpendicular polarized radio wave, 72–­74 on plane boundary, 67–­74 power flow directions, 75 reflection coefficients for, 67–­68

Ohm’s law, 318 Okumura-Hata model, 166–­67 Omnidirectional antennas, 120 One-port network (OPN), 353, 354 Optical waves, 5 Ordinary rays, 253, 254, 259 Ordinary wave, 322 Over-the-horizon propagation interpretation of tropospheric scatterings and, 276–­78 propagation factor on secondary tropospheric radio links and, 286 random tropospheric scatterings and, 271–­76 secondary tropospheric radio links, 269–­71 secondary tropospheric radio links performance and, 291 statistical models of tropospheric turbulences and, 280–­86 turbulent troposphere and, 279 P Parallel polarized radio waves, 68–­72 Partial derivatives, 37 Partial fields, summation of, 84 Passive repeaters, ring-segment diffractors as, 94–­96 Path analysis, 164 Path clearance, 162 Pattern-propagation factor, 142 Perfect electrical conductor (PEC) air interface, 178 boundary with, 29, 30 ground, 181, 184, 215 ground plane, 175, 177, 213 plates, parallel, 265 reflecting surface as, 67 surface area, 133 Permittivity, frequency dependency of, 34 Perpendicular polarized radio waves, 72–­ 74 Phase coefficient, 37, 43 Phase matching condition, 63 Phase modulation (PM), 375 Phase pattern, 116

Index 413

Phase shift, 250, 294, 331 Photo ionization, 8–­9 Physical models defined, 165 LOS paths, 172–­74 non-LOS paths, 169–­72 See also Propagation predication models Plane boundary introductory remarks, 63–­65 normal incidence on, 65–­67 oblique incidence on, 67–­74 reflection and refraction from, 63–­76 Plane wave propagation across geomagnetic field lines, 321–­23 along geomagnetic field lines, 316–­21 general relations, 315–­16 inn homogeneous magnetoactive plasma, 319–­21, 322–­23 Plasma concentration, 9, 10, 258 defined, 8 ionospheric, 245 ionospheric concentration, 311 isotropic, into anisotropic, 249 magneto-active, 250, 255 Plasma frequency, 233–­34, 243, 301, 317 Polarization axial ratio of, 47, 55, 57 basic relationships, 48–­49 circular, 50–­52 defined, 47 elliptical, 47 horizontal, 140, 195 left-hand circular (LHCP), 51–­52, 59, 61, 252, 256, 320 linear (LP), 49–­50 mathematical evaluations and, 52–­63 mismatch, 126 of radio waves, 47–­63 right-hand circular (RHCP), 50–­51, 59, 61, 252, 256, 320 tilt angle and, 47, 49 vertical, 182, 195, 228 Polarization constant, 225 Polarization diversity, 387 Polarization ellipse axial ratio of, 55

for calculated parameters, 58 in Cartesian coordinates, 54 forming as overlay, 61 for LHEP, 54, 55 for RHEP, 54–­55 Polarization losses, 260 Polarization plane, angle of rotation, 251 Popov, A. S., 1, 2 Power factor, 43 Power flow density at horizon distance, 176 integration along spherical surface, 212 for lossless propagation medium, 115 value of, 311 for vertical radiating line current, 184 Power flow directions, 75 Power radiation pattern, 116 Poynting theorem, 24, 76–­77 Poynting vector amplitude phasor of, 39 defined, 24–­25 direction of, 39 in free space, 262 Leontovich approximate boundary conditions, 181 Probability distribution function (PDF) CDF and, 327 defined, 326 Gaussian, 342–­43 lognormal, 344 m-distribution of Nakagami and, 338–­ 39 for randomly superimposing two independent rays, 332 for Rayleigh ensemble, 336 Rice distribution, 338, 339 Propagation asymptotic diffraction theory of, 193–­ 210 atmospheric effects in, 229–­307 in discretely inhomogeneous medium, 46 of ELF and VLF, 260–­69 free, 11, 35–­47, 85–­86 guided, 11 HF, 13–­14 of HF radio waves, 14

414 Index

Propagation (continued) impact of Earth’s magnetic field on, 247–­60 ionospheric, 2, 3, 14 mechanisms of, 2–­3 multipath, 330 over-the-horizon, 269–­95 phase velocity, 40 plane wave propagation, 315–­23 quasi-longitudinal, 258 quasi-transverse, 257, 258 scatter, 12–­13 sea-to-land, 192–­93 spatial area significant for, 79–­98 stratosphere and, 8 troposcatter, 12 Propagation (between elevated antennas) about, 137 asymptotic diffraction theory (ADT), 204–­6 choice of elevation, 162–­64 flat Earth approximation case study, 138–­46 optimal path clearance, 162–­64 over spherical Earth surface, 146–­56 ray-trace approach, 137–­74 rough and hilly terrain, 156–­62 urban, suburban and rural models, 164–­74 Propagation (ground-based antennas) about, 174–­75 asymptotic diffraction theory (ADT), 198–­204 Leontovich approximate boundary conditions and, 178–­83 over infinite, perfect ground plane, 175–­78 real homogeneous flat Earth, 184–­87 real inhomogeneous flat Earth, 188–­93 Propagation constant, 36, 78, 102 Propagation distance, 13 Propagation factor ADT method and, 209–­10 B&A method and, 209 defined, 130, 325 defined by flat Earth approach, 207 dependence on magnitude, 185

distance dependence on, 141, 191 first maximum of, 141 flat Earth approximation and, 139 as function of distance, 142 general solution of wave equation and, 215–­25 in logarithmic form, 132 maximums of, 142 minimum value, 141 pattern, 142 random variations in, 130–­31 on secondary tropospheric radio links, 286–­91 Propagation loss, 132 Propagation path, 92 Propagation predication models about, 164–­65 COST 231-Hata model, 167–­68 empirical, 165–­69 Lee model, 168 LOS paths, 172–­74 non-LOS paths, 169–­72 Okumura-Hata model, 166–­67 physical, 169–­72 in urban, suburban, and rural areas, 164–­74 Wallfish-Ikegami flat-edge model, 170, 171–­72 Propagation ray patterns, 155 Propagation theory, general relations in, 129–­35 Pulse amplitude modulation (PAM), 375 Pulse code modulation (PCM), 377, 380 Pulse position modulation (PPM), 375

Q Quasi-longitudinal propagation, 258 Quasi-transverse propagation, 257, 258

R Radar cross section (RCS), 133 Radar range equation, 134 Radiation beam width, 118

Index 415

from electric current point source, 105–­9 from infinitesimal electric current source, 76–­79 intensity, 115 patterns, 118–­19 resistance, 120, 125 Radio communications, 374 Radio link design, 3 Radio waves attenuation of, 45 circular polarized, 260 classification based on spatial area, 14 classification by frequency bands, 4–­6 defined, 1, 4 diffracted, 2, 13 direct, 11, 12 divergence of energy in, 153–­56 elliptically polarized, 52–­63, 256, 257 free propagation of, 35–­47 ground, 14–­15 horizontally polarized, 183, 196, 222–­25 implementation of frequencies, 3 incident from dense medium, 70–­72, 73–­74 incident from sparse medium, 70, 73 ITU classification table, 5 linearly polarized, 48, 230, 249, 253 optical, 5 plane, 63–­76 polarization of, 47–­63 propagation theory, 3–­4, 129–­35 reflected, 11, 12 regular refraction of, 234–­36 scattered, 11–­12 sky, 14, 241–­47 space, 14–­15 spherical, 76–­79 split in magneto-active ionosphere, 258 structure in lossless medium, 42 surface, 14 vertically polarized, 182–­83, 196–­97, 222–­25 wavelength in free space, 41 Rake diversity, 388 Random fluctuations, 11, 15 Random interference defined, 330

of large number of independent wavelets, 334–­37 two-ray, 331–­34 Rayleigh criterion, 157, 159, 164 Rayleigh distribution, 337, 350 Rayleigh ensemble, 334, 336 Ray-trace approach, 137–­74 Ray traces, 83 Ray-tracing patterns, 138, 147 Receiver (internal) noise, 16–­17 Receiver sensitivity, 371 Receiving system block diagram, 370 Recombination processes, 10, 310 Rectangular-to-spherical transforms, 99 Reference conditions, 130 Reference loss, 131 Reference path, 130, 325 Reflected waves defined, 11 disposition of, 64 illustrated, 12 parallel, 142 Reflecting ionospheric layers, 246 Reflection from boundary of nonideal dielectric medium, 76 diffuse, 157, 158 effective area from flat boundary, 96–­98 from large-scale obstacle, 159 magneto-active ionosphere and, 258–­60 power, 74–­76 of sky waves, 241–­47 specular, 157 Reflection coefficients angular dependences of magnitude and phase of, 77 apparent, 158, 227, 228 electric field, 65, 66, 71, 72, 74 formulas, 75 magnetic field, 65–­66, 74 for oblique incidence, 67–­68 Reflection loss, 66 Reflection pattern, 302 Reflection phase, 66 Reflection point distance to, 148–­53 relative coordinates of, 160

416 Index

Refracted waves, disposition of, 64 Refraction magneto-active ionosphere and, 258–­60 of radio waves in the atmosphere, 234–­36 of sky waves, 241–­47 tropospheric, 236–­41 Refraction index, 7, 234 Refractivity defined, 236 elevation dependence of, 237 gradient of, 238 Relative curvature, 237 Relative magnetic permittivity, 22, 40 Relative permittivity complex, 32 dielectric, 22 magnetic, 40 Reynold’s number, 282 RF link performance improvement approaches, 376–­81 RF link performance stability analysis, 370–­74 RF link power budget, 371–­74 RF transmission links, interference in, 15–­18 Rice distribution, 338, 339 Right-hand circular polarization (RHCP), 50–­51, 59, 61, 252, 256, 320 Right-hand elliptical polarization (RHEP), 54–­55 Right-hand rule, 39, 247 Ring-segment diffractors, 94–­96 Ring-shaped antenna directors, 92–­94 Roots (k), 227 Rough surfaces propagation over, 156–­62 random, levels of, 156 reflection of radio waves from, 156 Rule of secant, 244 Rural/open areas, 167 S Scalar potentials, 78, 102 Scattered waves, 11, 12

Scattering angle, 275 Scattering coefficient, 158 Scattering diagrams defined, 280 of tropospheric turbulences, 281, 284, 286 Scattering vector, 275, 285 Scattering volume defined, 287 definition illustration, 287 elevation of, 290 height of, 290 Scatter propagation, 12–­13 Seasonal variations, 293 Sea-to-land propagation, 192–­93 Secondary tropospheric radio links antennas gain effect on performance, 291 limitations to signal transmission bandwidth, 294–­95 maximum efficiency of, 287 over-the-horizon propagation and, 269–­71 path length of, 289 performance, 291–­95 propagation factor on, 286–­91 signal level fluctuations and, 292–­94 Selection combining, 390 Separation constant, 221 Separation of the variables technique, 193–­94 Shadow areas, 79, 95 Shadow factor, 186, 208 Shadow zones, 193, 197 Shannon-Whittaker transform, 275, 276 Shot effect, 17 Side lobes, 118 Signal propagation pattern, 133 Signal-to-noise ratio (SNR), 329, 370, 371, 375–­76 Signal transmission bandwidth, limitations to, 294–­95 Single-hop propagation distance, 303 Single-term formula, 197, 206 Skip zone, 243 Sky waves defined, 14

Index 417

pattern of reflection, 242 ray patterns of, 247 reflection of, 241–­47 refraction of, 241–­47 single-hop propagation, 244 Slow fading combined distribution, 346–­53 defined, 293 distance-dependence, 293 lognormal distribution of the random variable, 343–­46 as long-term, 330 normal (Gaussian) distribution of a random variable, 342–­43 statistical distribution, 341–­46 See also Fast fading Smooth atmospheric refraction, 239 Snell’s law, 178 Source coding, 377 Space diversity, 387 Space waves, 14–­15 Spatial frequency, 275 Specular reflection, 157 Spherical coordinate system, 115 Spherical Earth surface distance to reflection point, 148–­53 divergence of energy in radio waves, 153–­56 ideal propagation model and, 165 propagation over, 146–­56 zoning, 194 Spherical-to-rectangular transforms, 99 Spherical waves, 76–­79 Spiral waves, 252 Spread-spectrum signals, 382–­86 Standard atmosphere, 236 Standard troposphere, 6 Statistical expectation, 327 Statistically homogeneous random medium, 280, 308–­9 Statistically isotropic random field, 280, 308–­9 Stoke’s theorem, 28 Stratosphere, 6, 8 Strike ionization, 8 Structured constant, 283 Suburban areas, 167

Surface waves defined, 2, 14 Norton wave, 184 Zenneck wave, 184

T Takeoff-landing concept, 189, 193 Taylor series, 146 Thermal losses, 120, 121 Thermal noise, 16–­17, 353, 363–­64 Thevenin’s voltage source, 126 Tilt angle, 47, 49, 126 Tilted DP, 126, 127 Time diversity, 388 Time-harmonic electromagnetic process, 31–­35 Time-harmonic multiplier, 31 Time-shift property, 384 Total attenuation, 46 Total path clearance, 240 Total propagation loss, 132 Total reflection angle, 71 Transmission bandwidth, limitations to, 294–­95 power factor, 74–­76 signal analysis, 294 troposcatter, maximum bandwidth for, 295 Transmission coefficients, 74 Transverse electromagnetic (TEM) waves, 264, 265, 267 Trigonometric equalities, 98–­99 Troposcatter propagation, 12 Troposphere attenuations in, 296–­300 defined, 6 dielectric constant of, 271, 277 ESCS of, 279–­80 movements of air masses, 282 Tropospheric refraction fluctuations of, 274 LOS link in presence of, 239 smooth, 239 standard atmosphere and, 236–­41 types of, 241

418 Index

Tropospheric scatterings physical interpretation of, 276–­78 positioning of vectors for analysis, 272 random, 271–­76 Tropospheric turbulences effective scattering cross-section of, 279–­80 Gaussian model of, 280–­81 inner scale of, 283 Kolmogorov-Obukhov model, 281–­86 outer scale of, 283, 285 pattern of scattering, 278 scattering diagrams of, 281, 284, 286 statistical models of, 280–­86 vertical profiles of structural content, 284 Tuned antennas, 121 Turbulent movement, 282 Turbulent movements, 8, 11 Two-port network (TPN) cascaded, 356–­58 defined, 354 module, feedline, and receiver as, 372 noise parameters, 355 passive, noise figure of, 358–­60 Two-ray random interference, 331–­34

U Uniform plane waves defined, 38 in lossless medium, 40–­42 in lossy medium, 42–­47 structure in lossy medium, 44 Unit vector transform, 99 Urban areas, 166–­67

Vector of polarization, 272 Vector potentials defined, 78, 101 Helmholtz equation for, 101–­5 magnetic, 104 Vertically polarized radio waves, 182–­83, 196–­97, 222–­25 Vertical polarization, 182, 195, 228 VLF frequency ranges supporting propagation and, 268 height of monopole antenna and, 263 signals, propagation of, 265–­69 Volumetric current density, 32

W Wallfish-Ikegami flat-edge model, 170, 171–­72 Water vapors attenuations, 297 Wave equations, 36, 222–­25 Wave fronts, 37, 38 Wave number defined, 41 in free space, 190 Weyl and Van der Pol (W&V) approach applicability of, 198–­99 calculation method, 199 data comparison, 199 results comparison, 200–­202, 203 Weyl and Van der Pol (W&VdP) formula, 184 Whistling atmospherics (whistlers), 252 White noise, 353 Wiener-Khinchin transform, 275 Wireless communications, 374 Wolf numbers, 304 Work function, 8

V Vector analysis, 99–­101 Vector differential operations, 100–­101 Vector integral relations, 99–­100 Vector of dipole moment, 109

Z Zenneck wave, 184 Zürich numbers, 304

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