Signal Polarization Selection for Aircraft Radar Control: Models and Methods

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Signal Polarization Selection for Aircraft Radar Control: Models and Methods

Table of contents :
Introduction......Page 6
Contents......Page 10
Basic Agreed Notations......Page 12
Abbreviations......Page 14
1.1 Nature of System Analysis......Page 15
1.2 System Analysis Procedure......Page 16
1.3 Objectives of a System Analysis of the Radar Detection Control Processes......Page 17
References......Page 19
2.1 Problem Formulation......Page 21
2.2 Analysis of Radar Channel Model......Page 24
2.3.1 Analysis of the Current Status of the Issue......Page 33
2.3.2 Scattering Characteristics of Large-Scale Measuring of Randomly Uneven Surface......Page 37
2.4 Methods of Noise Influence Diminishing Based on Different Polarization Structure of Target Signal and Noise Signal......Page 49
2.5.1 Research Problem Formulation......Page 60
2.5.2 Analysis of Methods for Antenna System Polarization Diagram Readjustment......Page 64
2.6 Synthesis of Signal Selection System in the Background of Passive Noise Formed by Reflections from Separation Surface......Page 69
References......Page 85
3 Primary Ways of Technical Implementation of Developed Selection System and Methods of Device’s Error Minimizing......Page 87
3.1 Estimation of Influence of Cross-Polarization Degree in Antenna System Radiation on the Accuracy of Radar Measurements. Methods of Its Diminishing......Page 89
3.2 Mirror Antenna Scattering Matrix......Page 94
3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna......Page 102
3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment......Page 109
3.5 Derivation of the Formula for Parabolic Antenna Radiation Components with Fundamental and Cross-Polarizations......Page 117
3.6 Solution for Synthesis of Line Feed with Given Amplitude Directional Diagram......Page 121
3.7.1 Structural Diagram of Conceptual Design of Radar with Synthesized Aperture......Page 129
3.7.2 Model of Matrix Joint Correlation Function for Probing and Reflected Vector Signals......Page 130
References......Page 136
4.1 Research of Scattering Polarization Characteristics of the Signals Reflected in Specular Direction from the Real Separation Surface......Page 138
4.2 Synthesis of One-Mirror Antenna Linear Feed Eliminating Cross-Polarized Radiation......Page 147
References......Page 153

Citation preview

Springer Aerospace Technology

Nikolay Kondratyevich Yurkov Alexey Yevgenyevich Bukharov Dmitry Alexandrovich Zatuchny

Signal Polarization Selection for Aircraft Radar Control Models and Methods

Springer Aerospace Technology Series Editors Sergio De Rosa, DII, University of Naples Federico II, NAPOLI, Italy Yao Zheng, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, Zhejiang, China

The series explores the technology and the science related to the aircraft and spacecraft including concept, design, assembly, control and maintenance. The topics cover aircraft, missiles, space vehicles, aircraft engines and propulsion units. The volumes of the series present the fundamentals, the applications and the advances in all the fields related to aerospace engineering, including: • • • • • • • • • • • •

structural analysis, aerodynamics, aeroelasticity, aeroacoustics, flight mechanics and dynamics orbital maneuvers, avionics, systems design, materials technology, launch technology, payload and satellite technology, space industry, medicine and biology.

The series’ scope includes monographs, professional books, advanced textbooks, as well as selected contributions from specialized conferences and workshops. The volumes of the series are single-blind peer-reviewed. To submit a proposal or request further information, please contact: Mr. Pierpaolo Riva at [email protected] (Europe and Americas) Mr. Mengchu Huang at [email protected] (China) The series is indexed in Scopus and Compendex

More information about this series at http://www.springer.com/series/8613

Nikolay Kondratyevich Yurkov · Alexey Yevgenyevich Bukharov · Dmitry Alexandrovich Zatuchny

Signal Polarization Selection for Aircraft Radar Control Models and Methods Translator: Kudriashova Anna

Nikolay Kondratyevich Yurkov Penza, Russia

Alexey Yevgenyevich Bukharov Kamensk-Uralsky, Russia

Dmitry Alexandrovich Zatuchny Technical Sciences Moscow, Russia

ISSN 1869-1730 ISSN 1869-1749 (electronic) Springer Aerospace Technology ISBN 978-981-33-4963-6 ISBN 978-981-33-4964-3 (eBook) https://doi.org/10.1007/978-981-33-4964-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Introduction

Radar Stations (RS) represent one of the most fundamental components of the ground-based flight operation system for Civil Aviation (CA). Different types of radars, such as en-route radar facilities, airfield surveillance radars, meteo radars etc., are currently used in the industry. Exact aircraft positioning and airspace control—the main functions of radio location in CA—constitute the essential conditions of Air Traffic Management (ATM) system for effective operation. Modern tendencies of CNS/ATM development that assume transition to ADS-B and RNAV do not disallow using ordinary radar stations. It is caused, first of all, by the need of validation of obtained navigation information and other data. The problem of aircraft radar positioning at the low elevation angles when influence of multibeam EMW propagation on proper radar operation that happens to be especially strong is still remaining unsolved. In this case we observe strong backward radar signal scattering from the separation surface and local objects situated around. Immediacy of the problem, which this book is devoted to, is confirmed by all available information on the current status of the issue of low-flight operation support. In the circumstances of strong masking noise formed by reflections from the space-distributed objects like sea or terrain surface, acquisition of reliable goniometric information becomes a serious problem for modern airspace control radar aids. If terrain surface produces mirror reflections of radar signal on its way to the target, we receive double radar image of the object (“antipode” phenomenon) due to target’s phase center shift. Synthesis of radio technical aids was previously based on the assumption that signal and noise were both fully polarized, and their polarizations coincide with each other and with polarization of antenna itself. However, in real circumstances polarization of the scattered wave depends on many factors and, generally, differs from antenna polarization. Besides, part of useful signal energy is being lost due to polarization selectivity of antenna system. Therefore, supplement of the known selection methods with a device which can analyze polarization structure of radar signal seems to be very valuable, because it provides significant enhancement of radar system noise immunity. v

vi

Introduction

However, currently there are no quite reliable and technically simple radio systems capable of selection by means of polarization readjustment. Several attempts to create an adaptive polarization selection system were made recently. But, in our opinion, of special interest is the question of passive noise protection with the system where polarization selection methods are applied for the primary radar signal processing. Solution to this problem represents definite scientific and unquestionable practical interest. This book deals with the questions on synthesis of polarization selection system in the background of passive noise formed by reflections from space-distributed targets. This synthesis is fulfilled as close as possible to its ideal configuration in terms of maximal signal-to-noise ratio for the matched load of radar station antenna system. As was pointed out in the book herein, adjustment of emitted EMW polarization conditions provides average target RCS increment, what, in turn, leads to the rise of signal-to-noise ratio for the antenna system load. The problem of useful signal selection is discussed during analysis of scattering polarization characteristics of large-scale impedance on a randomly uneven surface. This analysis is based on the introduction of Kotler correction to the vector form of Kirchhoff integral that is used for calculation of scattered field parameters. Application of this correction is caused by the need to take into account finite dimensions of the scattering object. From the result of the analysis we receive improved formulas for calculation of polarization conditions for EMW scattered by the large-scale target. Analysis of passive noise impact diminishing methods helped to substantiate (based on specified expressions for the field scattered by underlying surface) the methods for signal-to-noise ratio increment due to additional loss of mirror channel noise power for a given polarization of the signal, as well as due to using of selection system where processing of the signal scattered backward from the separation surface, and the useful signal from the target is carried out with the consideration of time or frequency separation of these signals and their polarization structure differences. Synthesis of a device that implements the suggested selection method was carried out on the basis of requirements to the system of useful signal selection with the passive noise background. System design was based on the method of UHF device synthesis by its conductivity matrix. The possible ways of developed system technical implementation are also analyzed in the monograph. On the basis of derived SM of mirror antenna, we ran synthesis of uniform PD. Synthesis problem solution was reduced to specifying the linear feed modification in accordance with such DP on cross-polarization that would provide cross-polarization suppression and practical implementation of the feed for a given field distribution. This approach to designing of antenna system with uniform PD is substantiated as the most universal compensation method for synthesis of systems with almost ideal characteristics due to mutual suppression of the errors caused by two or more parasites present in the system. Thus, making the combined effect of two negative phenomena close to zero, we receive the required PD by compensation method, very much alike how we struggle with aberrations in optics.

Introduction

vii

The book contains data on experimental dependences of polarization SM elements’ values for real underlying surface that were received by two position measuring radar system operating in quasicontinuous mode at the wavelength of 3.2 cm with pre-defined polarization of probing and response radiation. Comparison between theoretical and experimental dependences of these elements from view point elevation for the given griding angles  and azimuth φ of the view point is made. We also determined the optimal radiation polarization for minimization of the mirror channel signal level, which is responsible for the major part of errors in positioning of a low-flying aircraft. We deliver the results of experimental operability testing for synthesized parabolic antenna linear radiator in terms of cross-polarization suppression in the radiation zone of this mirror antenna. Synthesis method for uniform PD of two-mirror antenna is offered by synthesis of both radiator and modified subdish profile. One of the most advantageous ways is specified for design of radiator with pre-determined characteristics. The correctness of UHF device analysis by matrix methods is experimentally confirmed in the monograph, which allows giving recommendations on synthesized selection system adjustment as well as provides applicability of the proposed computing program to analysis of UHF devices that can be described by equivalent octopole. Nowadays, we can see an increasing interest to designing of multifunctional airborne radar systems (ARS) due to enhancement of hardware components and soaring computing capacities. In the meantime, ARS capacities became much higher after equipping it with the function of thermal radio radiation measurement. Inclusion of radiometer in ARS configuration and combining active radar channel with passive radiometric one is justified by significant improvement of weight dimensions characteristics, reduction of aperture area, enhancement of emission security, and obtaining of additional information. In this setting the most properly sized elements of the passive radiometric channel (antenna system and waveguide) will serve for both active and passive channels. Volume of radiometric receiver designed on the modern hardware elements will not exceed 1% of the active channel sender–receiver volume. Besides, many ARSs are equipped with monopulse antenna systems (MAS) that have two commutated channels. Therefore, the objective of radiometer design that would consider MAS capacities for radiometric measurements within ARS seems to be very relevant. It would allow improved thermal radio contrast resolution along the angular coordinates and thus offers better solution to the problem of target search and tracking. The task of design for radiometric receiver model with high resolution and improved accuracy is related to the class of inverse tasks, where we have to determine the reason of any event by its consequences. The task of reconstruction of radar signals and images received from different measuring equipment also refers to this type of problems. The general solution to this problem does not exist, because solution happens to be unstable, especially in the presence of fluctuation noise. The specifics of inverse problem’s solution of thermal radio image reconstruction consists in the necessity for determination of true distribution of contrast temperature

viii

Introduction

by the measured distribution of antenna temperature. The last is in fact the resultant of true contrast distribution with antenna system’s DP. Operation like this leads to the smoothing of real distribution of microwave radiation. Despite the numerous publications devoted to solution of incorrect inverse problems, including tasks of signal and image reconstruction, there are almost no works on the synthesis of models for radio heat image measuring processes in the context of computing resources and high processing rate limitations that make them critical considerations for multifunctional ARS. Besides, there are no mathematical models for radiometric measurers that allow taking into account MAS capacities for radiometric measurements within ARS. Thus, development of such mathematical model of MAS radiometer within ARS as well as design of the model for object’s thermal radio contrast measurement processes represents important and challenging issue. Successful solution to this problem will provide ARS multifunctionality and will drastically improve its performance characteristics.

Contents

1 General Principles of System Analysis of the Problems of Radar Contrast Increment Control Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Nature of System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 System Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives of a System Analysis of the Radar Detection Control Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Synthesis of Signal’s Polarization Selection System with the Background of Passive Noise Formed by Reflections from Distributed Targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis of Radar Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Analysis of Scattering Polarization Characteristics of Large-Scale Impedance of Randomly Uneven Surface . . . . . . . . . 2.3.1 Analysis of the Current Status of the Issue . . . . . . . . . . . . . . . 2.3.2 Scattering Characteristics of Large-Scale Measuring of Randomly Uneven Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Methods of Noise Influence Diminishing Based on Different Polarization Structure of Target Signal and Noise Signal . . . . . . . . . 2.5 Analysis of Current Methods and Devices for Polarization Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Research Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Analysis of Methods for Antenna System Polarization Diagram Readjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Synthesis of Signal Selection System in the Background of Passive Noise Formed by Reflections from Separation Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5

7 7 10 19 19 23 35 46 46 50

55 71

ix

x

Contents

3 Primary Ways of Technical Implementation of Developed Selection System and Methods of Device’s Error Minimizing . . . . . . . 3.1 Estimation of Influence of Cross-Polarization Degree in Antenna System Radiation on the Accuracy of Radar Measurements. Methods of Its Diminishing . . . . . . . . . . . . . . . . . . . . 3.2 Mirror Antenna Scattering Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Derivation of the Formula for Parabolic Antenna Radiation Components with Fundamental and Cross-Polarizations . . . . . . . . . . 3.6 Solution for Synthesis of Line Feed with Given Amplitude Directional Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Model of Matrix Joint Correlation Function of Probing and Reflected Vector Signals for Conceptual Design of Aerial Radar with Synthesized Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Structural Diagram of Conceptual Design of Radar with Synthesized Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Model of Matrix Joint Correlation Function for Probing and Reflected Vector Signals . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Experimental Testing of Theoretical Results . . . . . . . . . . . . . . . . . . . . . . 4.1 Research of Scattering Polarization Characteristics of the Signals Reflected in Specular Direction from the Real Separation Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Synthesis of One-Mirror Antenna Linear Feed Eliminating Cross-Polarized Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

75 80 88 95 103 107

115 115 116 122 125

125 134 140

Basic Agreed Notations

θ, ϕ

→ → → i x, i y, i z → → → i θ , i φ, r o

ψ ηuϕ x, y, z t K λ F(ro ) F(θ, ϕ) Fθ,ϕ (θ, ϕ) J (t) I (t) →



E, H GX , GY P = EE21 θo , ϕo PS , PN η r m β ER , EL α = arctg r θi Pt , Pr M [S]∗ [S]T

angles of spherical coordinate system, unit axes of rectangular coordinate system, unit axes of spherical coordinate system, grazing angle, elevation and azimuth scattering angles, present coordinates of curvilinear radiator, present non-dimensional coordinate of curvilinear radiator, wave number, wavelength, vector directional pattern, complex pattern, directional pattern, amplitude-phase distribution of electrical current, amplitude distribution of electrical current, components of electromagnetic field, dispersions of orthogonal components of electromagnetic field, phasor, direction of antenna radiation maximum, power of signal and noise, antenna system polarization efficiency, ellipticity ratio, degree of polarization, orienting angle of polarization ellipse, right and left-polarized circular components of electromagnetic wave, ellipticity angle, incident angle, polarization characteristics of transmitting and receiving antennas, expectation sign, complex conjugation sign, matrix transposition sign, xi

xii

[S]+  ⊗ det[S] Sp [S] inf[X ] Sup[X ] ∇ ∇2 = ∇ · ∇ Sgn X ...

Basic Agreed Notations

Hermitian conjugation matrix sign, sign of element membership, Kronecker matrix multiplication sign, matrix determinant, matrix spur, greatest lower bound, least upper bound, (nabla operator) vector differential operator, Laplace operator, signum function, assembly value averaging.

Abbreviations

AC ADS-B AP ARS AS CA CNS/ATM DC DG DP EMW EP EMW PB PD PS RA RCS RI RS RSA XMIT RSA SDP SM UHF V

Aircraft Automatic Dependent Surveillance-Broadcast Aerial platform Airborne radar system Air space Civil aviation Communication, navigation, surveillance/air traffic management Directional coupler Directive gain Directional pattern Electromagnetic wave Elliptically polarized electromagnetic wave Polarization basis Polarization diagram Phase shifter Receiving antenna Radar cross-section Radar image Radar station Synthesized aperture radar transmitter Radar with synthesized aperture Signal digital processing Scattering matrix Ultra-high frequency Valve

xiii

Chapter 1

General Principles of System Analysis of the Problems of Radar Contrast Increment Control Processes

1.1 Nature of System Analysis System analysis is a relatively new science. It is based on a comprehensive (systemic) approach to solving problems, which in the case of complex large-scale systems (including radar systems) is the most rational way of making decisions. System analysis has proven to be effective and efficient in solving complex problems in various fields of human activity, including economic practice, transport, MIS design, etc. The complexity of control tasks during radar detection necessitates the application of the system analysis in this area [1]. The essence of the system analysis is distilled to the fact that the solution of a particular problem depends on the solution of general problems in the system at large. Consider the key features of the system analysis. 1.

2.

3.

System analysis is a comprehensive methodology for solving research and practical problems. It is implemented when a research problem (objective) appears, or for finding solutions to research problems, or for organizing a research process. The methodology of the system analysis requires that a problem is stated from a system point of view. Hence, a systemic research considers the subject as a system to the extent that it is dictated by the real research objectives. The system analysis requires a generalized understanding of the subject of the research, which is defined as a system. Consideration of parts of a problem can be attempted only after the main connections have been established between them. The system analysis involves a dynamic understanding of the subject. It is assumed that we are dealing with an evolutionary system which changes its state, structure, and behavior in the course of its development.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 N. K. Yurkov et al., Signal Polarization Selection for Aircraft Radar Control, Springer Aerospace Technology, https://doi.org/10.1007/978-981-33-4964-3_1

1

2

1 General Principles of System Analysis of the Problems …

The essence of the system analysis is the methodology which can be briefly defined as “a methodology for studying a situation as a system and enabling a complex solution to the problem” [2]. At the core of the concept of the system analysis are systems. A system is a grouping of components linked together by inputs and outputs to achieve some common cause or objective. This grouping is considered as a single whole and is determined by the conversion of inputs into outputs, as a set of subsystems or a component of a larger system. The second important concept of the system analysis is the concept of organization, including hierarchical structure. Organization is the arrangement of interdependent subsystems in a single system. From the hypothesis of a hierarchical structure of a system, an important principle follows—relationships between the subsystems are not equally strong, and a system is arranged only by the most interdependent subsystems, while the rest have a relatively low impact on their behavior. One of the important management tasks is to recognize the need for specialization of functions, where one subsystem (the control object) only creates a fundamental possibility of solving a problem, while the other ensures the effectiveness of a solution. This entails the most important principle of the system analysis—a thorough and comprehensive study of a problem (task). Connected with this, another principle of the system analysis has it that whenever possible, the statement of a problem and its solution methods should be clearly distinguished. A problem is considered to be formulated correctly only when a change in its solution does not require a revision of its statement [3].

1.2 System Analysis Procedure The system analysis procedure is a sequence of basic actions that must be performed in order to solve a problem. The key and relatively independent steps of the system analysis are: statement of a problem (objective); as-is analysis; system design; analysis of the to-be system; preparation for implementation and operation; implementation; and evaluation of the results. Problem-statement can be considered as a subject of study which evolves in the process of a system analysis. Therefore, it may happen that the ultimate statement of the problem will differ significantly from the original. This is very characteristic of the system analysis. It is equally important to take into account the possible deviation of the as-is system (the control object) from its desired state. The deviation may even exceed the permissible limits and hamper the achievement of the objective. From a procedure point of view, analysis of a problem involves the study of its root causes and alternative solutions. Chronologically, the first logical step is to analyze the as-is situation. A situation means a set of different states of the system and its environment at the same

1.2 System Analysis Procedure

3

point in time characterized by a certain number of connections and requiring certain actions. Given the practical orientation of the system analysis, it is necessary to study the as-is situation, i.e., to analyze the existing system and situation. The purpose of this analysis is to identify patterns of system functioning and determine its characteristics and properties that are important for solving the problem [4]. The analysis reveals structural elements and their relationships in the system: functions performed by individual components, methods and approaches used, main flows in the system (material, financial, information), scope, type, and form of resources [5–7]. The purpose of the as-is analysis is to determine the shortcomings of a system along with the key factors and root causes of the problem to be solved. Logically, this step transforms into the system design. Therefore, the as-is analysis step is similar to project analysis. Project analysis consists in finding errors and deficiencies in the system design and eliminating them so that the system meets the requirements. At this step, it is analyzed whether the selected structure can provide the required system behavior. The main objective of the implementation step is to verify the effectiveness of a system in achieving its targets [5, 8].

1.3 Objectives of a System Analysis of the Radar Detection Control Processes Before we proceed to the description of the objectives of a system analysis of the radar detection control processes, we would like to emphasize that this paper considers control technology as an information process. Proceeding from this, the objectives of the system analysis of the radar detection control problems include the study, analysis, and design of information systems that ensure the most efficient control of the radar detection operations [7–9]. In recent years, a comprehensive risk analysis has attracted much attention in the development of the theory and methods of analysis and control in radar detection. Risk analysis involves a comprehensive approach covering the problems stemming from various risks and hazards. Risk analysis allows us to identify the relationships between the risk sources and evaluate their possible impacts on the accuracy of the air traffic control systems [10–12]. A radar detection control system is a complex hierarchical system which includes a regional control system (higher level), city control systems, and district control systems, each being composed of its components and as such representing an independent system. Analysis of the radar detection control system is fraught with certain difficulties due to its specifics. We have singled out some of them that should be taken into account when studying the processes of controlling radar detection.

4

1.

2.

3. 4.

1 General Principles of System Analysis of the Problems …

A radar detection control system is a complex, dynamic, hierarchical system. It consists of weakly interconnected elements interacting with each other mainly during a radar detection of targets, including low-flying targets. This system is characterized by relative stability of the target and functional purpose of its components. However, the ultimate and functional purpose of the system is not defined, since it depends on the objectives and strategies used by its components for one or another kind of radar detection, and on the interdependence arrangement in each case. As an object of study, a radar detection control system is unique in the sense that it possesses a wide variety of factors, relationships, and processes. Most of the processes occurring in the system cannot be directly quantified, and many processes in this system (e.g., forecasting for some radar detection processes) are difficult to formalize. Radar detection control cannot be fully algorithmized and includes heuristic procedures in some of its blocks. Since it is impossible to completely formalize the management processes of this complex system and, even more so, to build a single formalized model, it is necessary to take into account the possibility of creating an interconnected complex of formalized models and informal representations.

Specific features mentioned in 3 and 4 necessitate some caution during the study and synthesis of the system. The above features of a radar detection control system as an object of study impose certain requirements on the system analysis methodology. Thus, its methodology should be based on logic and research methods that allow efficient research on the problem. In the study and analysis of the existing system, attention should be paid to the following [13–16]: • The key tasks performed by the system at all levels and among all of its elements; • Whether the existing information system adequately accomplishes its management objectives; • Whether there is a need for improving the procedure and/or organization of radar detection control processes; and • Whether there is a need for improving the interaction between the components of the system. System analysis includes the following steps [17–19]: • Analysis of the as-is radar detection control process at various levels of its organization; • Development of standard designs for the radar detection control subsystems and their elements; • Analysis of the efficiency of the designed systems during their trial operation; and • Linking the designed systems to other elements of the system. Table 1.1 shows the sequence of steps.

1.3 Objectives of a System Analysis of the Radar Detection Control Processes

5

Table 1.1 The key steps of the system analysis Step number Description

Summary of objectives and deliverables

1

Statement of the purpose

List of goals and objectives of the system analysis. Definition of the object of study, and formulation of the outcomes of the system analysis

2

Description of the object of study

System description of the object under study. Description of the goals, criteria and limitations of the system, and its main functions. Design of a system model

3

Specification of the purpose

A list of specific goals and objectives of the study which are necessary and adequate to achieve the overall purpose of the study

4

Analysis of the object of study

Quality evaluation of the management functions. Finding alternatives to accomplish the objectives

5

Evaluation of alternatives and selection Analysis and evaluation of alternatives. of a draft solution to the problem The choice of subsystems and the general control system

The objectives of the system analysis of the radar detection control system split into two general groups: 1. 2.

Analysis objectives related to the study of the properties and characteristics of the system; and Synthesis objectives, which are reduced to the choice of the organizational and functional structure of the system and its constituent subsystems.

References 1. Drachev AN, Farafonov VG, Balashov VM (2014) Control procedure for geometricallycomplex surfaces. Issues Radioelectron 1(1):91–99 2. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Lukashov KG, Mironov OS, Panfilov PS, Parusov VA, Raisky VL, Sarychev VA (2017) Radio physical support of ultrashort-pulse radar systems. In collected volume “Problems of remote sensing, radio-waves diffraction and propagation” Compendium of lectures. RAS Scientific Council on radio-wave propagation. Murom institute (subsidiary) of Vladimir State university n.a. Stoletov A.G and Stoletov N.G., pp 5–21 3. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Mironov OS, Panfilov PS, Parusov VA, Sarychev VA (2016) Technologies of ultra-short-pulse environmental radio location with high range resolution. Meteorol Bull 8(3):17–22

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1 General Principles of System Analysis of the Problems …

4. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Lukashov KG, Mironov OS, Panfilov PS, Parusov VA, Raisky VL, Sarychev VA (2016) Experimental research of ultrashort-pulse radar system’s characteristics. In collected volume “Radiophysical methods in environmental remote sensing”. In: VII Russia-wide scientific conference proceedings. Murom institute (subsidiary) of Vladimir State university n.a. Stoletov A.G and Stoletov N.G., pp 196–202 5. Yershov GA, Zavyalov VA, Sinitsyn VA (2019) Improvement of radar observability parameters for aircrafts by methods of multi-position radio location. In collected volume “Innovation technologies and technical facilities of special purpose”. In: XI-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2, pp 21–24 6. Balashov VM, Drachev AN, Michurin SV (2019) Methodology of reflectors’ control for mirror antennas. In: Metrological support for innovation technologies. International forum: Abstracts, pp 44–46 7. Balashov VM, Drachev AN, Smirnov AO (2019) Methods of coordinate measurements during geometrically-complex surfaces control. In: Metrological support for innovation technologies. International forum: Abstracts, pp 41–43 8. Zatuchny DA (2017) Wave reflection features analysis during data transmission from the aircraft flying over urban terrain. Inf Commun 2:7–9 9. Ivanov YV, Petukhov SG, Sinitsyn VA (2017) Application of radar landing system for aircraft automatic landing support. In collected volume “Innovation technologies and technical facilities of special purpose”. IX-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol. 2. Ministry of science and education of Russian Federation, Baltic State Technical University “Voenmech” n.a. Ustinov D.F., pp 326–330 10. Zatuchny DA, Kozlov AI, Trushin AV (2018) Discernment of the objects situated within irradiated area. Inf Commun 5:12–21 11. Kozlov AI, Logvin AI, Sarychev VA (2007) Radar polarimetry. Radar signals polarization structure. Radiotechnics 640p 12. Myasnikov SA, Sinitsyn VA (2019) Design features of new landing radar. In collected volume “Innovation technologies and special purpose technical facilities”. In: XI-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2, pp 79–84 13. Rassadin AE (2010) Possible RSA system based on CMS MATLAB. In: Proceedings of XIVth radiophysics scientific conference. N. Novgorod: NNSU n.a. Lobachevsky N.I. Publishing house, pp 173–174 14. Proshin AA, Goryachev NV, Yurkov NK (2018) Calculation of radio-wave attenuation. PC program registration certificate RUS 2019612561, 05 December 2018 15. Proshin AA, Goryachev NV, Yurkov NK (2018) Calculation of dew point. PC program registration certificate RUS 2019612562, 05 December 2018 16. Rassadin AE (2010) R-function and atomic functions tools as mathematical foundation for design of aerial radar system with synthesized aperture. In: Pupkov KA (ed) Intellectual systems: IX international symposium proceedings. RUSAKI, Moscow, pp 224–228 17. Kozlov AI, Amninov EV, Varenitsa YI, Rumyantsev VL (2016) Polarimetric algorithms of radar target detection in the active noise background. Tula State Univ Bull Tech Sci 12–1:179–187 18. Sinitsyn VA, Sinitsyn EA, Strakhov SY, Matveev SA (2016) Methods of signals forming and processing in primary radar stations. St. Petersburg 19. Yurkov NK, Kuatov BG, Yeskibayev ET (2019) Algorithm of parametric and time control of aircraft movement management parameters. In: International symposium “Reliability and quality” proceedings, vol 1, pp 219–221

Chapter 2

Synthesis of Signal’s Polarization Selection System with the Background of Passive Noise Formed by Reflections from Distributed Targets

2.1 Problem Formulation Exact aircraft space and time positioning is of great importance for civil aviation radar systems. Parasite signal from the separation surface has the main impact on the positioning accuracy. The detection capacities of radar station are sometimes worsened by specular reflected signal that may dualize radar image (appearance of “antipode”). Implementation of new principles of selection systems design based, for example, on the difference between polarization coefficients for electromagnetic waves (EMW) reflected from the target and from the background is of our main interest [1–3]. In our country, as well as abroad, significant attention is devoted to improving the resolution capabilities of angle measurement devices and their energetic potential increment. However, issues of using EMW polarization characteristics for the purposes of radar station performance improvement are still understudied and are of limited application. Solution to this problem can be found in the field of polarization selection methods. Under polarization selection we shall mean herein the increment of signal-to-noise ratio due to different polarization structures of the target response and the noise. As known, the main parameter of a radar target which radio location deals with is its radar cross-section (RCS). Target with higher RCS has higher signal-to-noise ratio. RCS is defined as a following limit: G = lim

 

 

Sr e f , 4π R02   S f all 

(2.1)

  where  Sr ef  is the  power flow density of reflected radiation at the distance R0 from the target;  S f all  is the power flow density of falling (radiating) plane EMW of given polarization near the target. The limit transition is fulfilled in order to exclude RCS dependence on the distance R0 . © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 N. K. Yurkov et al., Signal Polarization Selection for Aircraft Radar Control, Springer Aerospace Technology, https://doi.org/10.1007/978-981-33-4964-3_2

7

8

2 Synthesis of Signal’s Polarization Selection System …

For linear polarized plane wave Umov-pointing vector is defined as [4]: |S| =

|E|2 , 2Z 0

(2.2)

where Z 0 is the space characteristic wave impedance. With this in view we can represent target’s RCS in a simpler form: G = lim

 2   2 Er e f 4π R0    E f all 2

= lim

   Hr e f 2  .  H f all 2

4π R02 

(2.3)

In accordance with (2.3), to find target’s RCS we have to determine the value of reflected field and its polarization at the receiving point. Average RCS of the fartet can be determined as: G t = 4π R02 PtT [P]Pt ,

(2.4)

where power scattering matrix [P] is defined as [P] = M{[S]∗ |S|} Thus, target RCS depends on both scattering matrix (SM) and polarization of falling radiation, which in turn depends on vector Pt . Modern radar stations process signals with linear or circular polarizations. During processing the observed target RCS decreases due to change of EMW polarization shape after reflection. The resulting scattered wave is recepted by radar station antenna with the following efficiency:    0 0 ∗ 1 + p s p  a   . η=  1 + | pa |2 1 + | ps |2

(2.5)

Polarization efficiency of radar signal reception depends on polarization param0

0

0

eters of antenna pa and signal Ps, and the higher it is, the closer the P s is to 0 pa .

2.1 Problem Formulation

9

Thus, higher the level of signal depolarization after reflection, the more is the loss in RCS measurement due to less polarization efficiency of receiving antenna. High operation efficiency of radar station, in general, is provided only by the full polarization reception of radar signal from the target (the numerator of fraction in Eq. 2.5 increases). It is worth to notice that for every target there is a polarization basis where the target scattering matrix (SM) has a diagonal form. It means that if a signal is polarized according to this basis, depolarization does not happen, i.e., these polarizations provide maximal polarization efficiency. The polarization of the signal is reflected from the target most efficiently and we shall refer to herein as the first natural polarization of the target. In real circumstances the RCS defined in (2.1) depends on radar antenna system quality. Correctness of this statement is based on the well-known fact that arbitrary antenna system with aperture (the most common one in radio location) radiates not only signal with principal polarization but also signal with the polarization which is orthogonal to the principal one. This phenomenon, called cross-polarization, is well studied and widely represented in scientific literature. This parasite phenomenon leads to RCS decrease due to energy redistribution through the two orthogonally polarized channels, decreasing the numerator of the fraction in (2.1). Radar channel in the broadest strokes can be represented as shown in Fig. 2.1. Here a radar channel is modeled by equivalent multipoles that describe radiation, propagation, reflection, and reception of EMW. Considering that in the great majority of practically important cases all radar channel elements are representable by linear passive multipoint circuits, or multipoles, and neglecting the effect of propagation

α 1in 1 1

3 Radar

VP Radar target

3 4

antenna system 2 2

4

5 6

HP

[S]

[M] 5 6

Hor. Pol.

Ver. Pol. 9 10 9 10 7 8 7 8

12

12

14

14

11

11

13

13

Separation surface [N] N

[]S]

Fig. 2.1 Representation of antenna system in an arbitrary polarization basis

10

2 Synthesis of Signal’s Polarization Selection System …

medium properties, we can describe radar channel by connection 12-pole [M] with two octopoles [S] and [N] (see Fig. 2.1) that define scattering properties of the target and separation surface accordingly. In an arbitrary polarization basis, the antenna system operates with two orthogonal polarizations oriented along the orts of this basis. Therefore, the antenna system of one-position radar can be represented by equivalent 12-pole [M] with two input channels (1 and 2). Spatial separation of the target and separation surface can be modeled by two pairs of antenna output channels (3–5 and 7–9), one of them connected to the [S] and the other with [N] that determines scattering parameters of underlying surface. Channels 3 and 5 (and accordingly 7 and 9) are spatially linked but formed by orthogonal polarizations and directed at the target and underlying surface, accordingly. In Fig. 2.2 we can see the diagrams of mirror target image (“antipode”) formation and the formation of noise from reversal reflections (Fig. 2.2b). The suggested radar channel model comprises both cases shown in Fig. 2.2, while in the second case channel lines 11–12 and 13–14 (see Fig. 2.1) are disconnected. In general, radar target is radiated by the signals from different angular directions (see Fig. 2.2a); therefore, its equivalent multipole contains two pairs of orthogonally polarized channels (for directions to the radar station and to the underlying surface). In this case low-flying object can be represented by its scattering matrix [S] of the fourth order. Similarly, separation surface is represented by its scattering   matrix [N]. Thus, radar antenna system is loaded by equivalent octopole S (see Fig. 2.1) formed by cascade connection of two octopoles [S] and [N]. In this configuration the antenna output channels 3, 5, 7, and 9 are  connected with mismatched loads represented by diagonal elements of SM S (see Fig. 2.3). These elements  S 44 , S 66 , S 88 , S 1010 depend on reflection characteristics of both separation surface and target. 











2.2 Analysis of Radar Channel Model Maxwell equations’ linearity allows application of matrix methods to analysis of UHF devices characteristics [5–7]. Based on matrix analysis methods, let’s find reflection index α1in from system input (see Fig. 2.1) that defines amplitude of the power received from the two angular directions. We shall assume that input channels 3, 5, 7, and 9 of [M] octopole are loaded by mismatched loads S 44 , S 66 , S 88 , S 1010 . The input channel (see Fig. 2.1) is fully matched with the feeding generator, and the second feeding channel is matched and loaded on the radar receiver. Further, for the sake of simplicity of the following derivations, we shall consider an idealized model of antenna system; namely, we assume full matching between feeding channels and antenna (m 11 = m 22 = m 33 = m 44 = m 55 = m 66 = 0), as 







2.2 Analysis of Radar Channel Model

11

[S] [M]

[S]

l1 l3

l2

( l1 =l2 +l3)

a)

[S] [M]

l1

[S]

l2

[N] (l1 =l2)

b) Fig. 2.2 a Specular image (“antipode”) formation diagram; b introduction of backward reflections

well as absence of any mutual exchange between these channels (m 12 = m 21 = m 34 = m 43 = m 56 = m 65 = m 45 = m 54 = m 35 = m 53 = m 63 = m 36 = m 46 = m 64 = 0). Elements m 14 , m 16 , m 23 , m 25 and their equivalent (for mutual antenna) m 41 , m 61 , m 32 , m 52 values determine the interdependency between input and output channels that operate on orthogonal polarizations. These elements are conditioned

12

2 Synthesis of Signal’s Polarization Selection System …

4

[S] =

8

10

S44 S46

6

S48

S410

4

S64 S84

S68 S88

S610 S810

6 8 10

S66 S86

S104 S106 S108 S1010

S11 0

4 4

12

S13 0

0 S22 0 S24 S31 S33 0 0 S42 0 S44

6 6

11

n11 n12 n13 n14

12 0 14

11 13

n21 n22 n23 n24 n31 n31 n33 n34

14

13

n41 n42 n43 n44

10 10 8 8

[S] Fig. 2.3 Radar channel model for a low-flying target

by cross-polarization phenomenon in aperture antennas. For the cases when we can neglect cross-polarization effect, these elements are equal to zero. As an example of cross-polarization analysis, let’s assume that only m14 and m41 differ from zero. With all these considerations in mind, we can represent the scattering matrix [M] (see Fig. 2.1) as follows: ⎡

m 11 ⎢m ⎢ 21 ⎢ ⎢m [M] = ⎢ 31 ⎢ m 41 ⎢ ⎣ m 51 m 61

m 12 m 22 m 32 m 42 m 52 m 62

m 13 m 23 m 33 m 43 m 53 m 63

m 14 m 24 m 34 m 44 m 54 m 64

m 15 m 25 m 35 m 45 m 55 m 65

⎤ ⎡ 0 m 16 ⎢ ⎥ m 26 ⎥ ⎢ 0 ⎥ ⎢ m 36 ⎥ ⎢ m 13 ⎥=⎢ m 46 ⎥ ⎢ m 14 ⎥ ⎢ m 56 ⎦ ⎣ m 15 m 66 0

0 0 0 m 24 0 m 26

m 13 0 0 0 0 0

m 14 m 24 0 0 0 0

m 15 0 0 0 0 0

⎤ 0 m 26 ⎥ ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎦ 0

As a rule, the radar antenna system does not contain any non-mutual elements, therefore we can consider it a mutual system. Then we can represent M] in the following way: ⎡

0 ⎢0 ⎢ ⎢ ⎢m [M] = ⎢ 13 ⎢ m 14 ⎢ ⎣ m 15 0

0 0 0 m 24 0 m 26

m 13 0 0 0 0 0

m 14 m 24 0 0 0 0

m 15 0 0 0 0 0

⎤ 0 m 26 ⎥ ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎦ 0

(2.2.1)

2.2 Analysis of Radar Channel Model

13

Reflection index from the input of antenna system represented by equivalent multipole [M] in Fig. 2.1 can be determined by the following formula:

(2.2.2)

Here matrix [M] is given by the formula (2.2.1), while matrix [B] (after connection of mismatched loads S 44 , S 66 , S 88 , S 1010 to the outputs 3, 5, 7, 9 of multipole [M], accordingly, and loading input channel 2 by the matched load) is defined as: 









0 ⎢0 ⎢ ⎢0 ⎢ [B] = ⎢ ⎢0 ⎢ ⎣0 0

0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 S44 0 S166 0 0 0 S188 0 0 0

0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(2.2.3)

1 S1010

According to (2.2.2) we receive: ⎡

0 ⎢ 0 ⎢ ⎢ m 13 ⎢ [M] − [B] = ⎢ ⎢ m 14 ⎢ ⎣ m 15 0

0 m 13 m 14 m 15 0 0 0 m 24 0 m 26 0 − S144 0 0 0 1 m 24 0 − S66 0 0 0 0 0 − S188 0 1 m 26 0 0 0 − S1010

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

(2.2.4)

Let’s determine the numerator of (2.2.2) as follows: |[M] −

[B]|2,3,5,7,9 1,1

1 = S44 S66 S88 S1010

 2   m 15 S88 + m 214 S66 + m 213 S44 .  2  2 m 26 S1010 + m 224 S66 − m 214 S66 m 224 (2.2.5)

And denominator |[M] −

[B]|2,3,5,7,9 1,1

 2  m 26 S1010 1 2 = + m 24 . S44 S66 S88 S1010 S66

(2.2.6)

After several transformations we finally receive the expression for reflection index from the first channel input of antenna system (see Fig. 2.1):

14

2 Synthesis of Signal’s Polarization Selection System …

(2.2.7)

Similar expression can be obtained for reflection index from the second input. Expression (2.2.7) determines the value of power that goes to the input of radar station receiver loaded by radar channel represented by its model in Fig. 2.1. The first summand determines the value of noise-free power, the second describes the separation surface influence, and the third summand appears due to cross-polarization phenomenon defined by element m 14 of SM [M] of antenna system (see (2.2.2)). For the case when system operates on the polarization in a given basis and we succeed in reducing to zero element S88 of SM [S] of equivalent [N] and [S] multipole, i.e., when we manage to avoid multipath energy propagation, then the existing cross-polarized antenna radiation transmits the noise signal into radar information processing channel with the value defined by the third summand in the expression. Thus, the problem of underlying surface impact minimization should be solved by reducing to zero both elements S88 and m 14 of antenna system SM, i.e., by creation of the uniform PD of aperture antenna [8–10]. Multipole [S] that represents radar channel without antenna system is connected to antenna system by two spatially divided pairs of its channels operating on orthogonal polarizations. As can be seen from Fig. 2.1, separation surface impact minimization consists in minimization of reflection indices for channels 8 and 10 of multipole [S]. If any increase of reflection indices S44 and S66 of channels 4 and 6 takes place, then significant increase of signal-to-noise ratio can be observed on the matched load of antenna system. Thus, our main interest is concentrated on the multipole [S] analysis, namely, in minimization of S88 and S1010 and maximization of elements S44 and S66 of its SM. Consider multipole [S] in Fig. 2.3. It consists of multipole [S] that defines scattering parameters of radar target and octopole [N] that determines scattering polarization parameters of underlying surface. For comprehension easing assume that target’s SM [S] is defined as follows: ⎡

S11 ⎢0 [S] = ⎢ ⎣ S31 0

0 S22 0 S42

S13 0 S33 0

⎤ 0 S24 ⎥ ⎥. 0 ⎦ S44

(2.2.8)

We also assume absence of interconnection between the input (4 and 6) and the output (12 and 14) channels (S12 = S21 = S34 = S43 = 0) and also absence of interconnection between the input and the output channels operating on orthogonal polarizations (S14 = S41 = S23 = S32 = 0). The latter presumption expels depolarization phenomenon from consideration, which is well confirmed by experimental results. Since separation surface is of main interest for our further study, let’s define its SM in the most general form:

2.2 Analysis of Radar Channel Model

15



n 11 n 12 ⎢ n 21 n 22 [N ] = ⎢ ⎣ 31 n 32 n 41 n 42

n 13 n 23 n 33 n 43

⎤ n 14 n 24 ⎥ ⎥. n 34 ⎦ n 44

(2.2.9)

Coefficients n11 and n44 here are responsible for backward energy reflection from the target on orthogonal polarizations; and coefficients n11 , n22 define back scattering of orthogonally polarized signals directed from radar station to separation surface. Coefficients n12 and n21 define signal’s depolarization during its scattering on separation surface, and elements n34 , n43 are depolarization of the signal from the target (along the l3 path in Fig. 2.2). Matrices of back scattering from the surface for different irradiating directions are defined as follows: – for l2 path (see Fig. 2.2):

(2.2.10)

– for l3 path (see Fig. 2.2):

(2.2.11)

Other elements of SM (2.2.9) are determined for underlying surface in mirroring direction along the l2 and l3 paths, respectively: (2.2.12) And for the path l3 − l2 : (2.2.13)

Since reversal of propagation direction corresponds to SM transposition, then SM of separation surface (2.2.9) can be modified as follows:

16

2 Synthesis of Signal’s Polarization Selection System …



n 11 ⎢ n 12 [N ] = ⎢ ⎣ n 13 n 14

n 12 n 22 n 23 n 24

n 13 n 23 n 33 n 34

⎤ n 14 n 24 ⎥ ⎥. n 34 ⎦ n 44

(2.2.14)

After clarification of physical significance of separation surface SM [N] elements, let’s find formulas for elements S44 , S66 , S88 , S1010 of SM [S] that drastically have impact on the value of power registered by radar station. We shall define these elements by tools of matrix analysis methods. First, compose matrix [R] that contains elements of [S] and [N] scattering matrices is as follows: ⎡

S11 ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 [R] = ⎢ ⎢0 ⎢ ⎢ S31 ⎢ ⎣0 0

0 S22 0 0 0 0 0 S42

0 0 n 11 n 12 n 13 0 n 14 0

0 0 n 12 n 22 n 23 0 n 24 0

0 0 n 13 n 23 n 33 0 n 34 0

S13 0 0 0 0 S33 0 0

0 0 n 14 n 24 n 34 0 n 44 0

⎤ 0 S24 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ S44

(2.2.15)

Since we neglect the propagation effects, i.e., we admit direct connection of multipoles [S] and [N] (see Fig. 2.3), then matrix [B] elements 11–12, 12–11, 13–14, and 14–13 are equal to 1. All other elements of this matrix are equal to zero. According to (2.2.2) this will help us to find expressions for S44 , S66 , S88 and S1010 elements of SM [S] and we have: ⎡

n 33 ⎢ −1 [N ] = |[R] − [B]| = ⎢ ⎣ n 34 0

−1 S33 0 0

n 34 0 n 44 −1

⎤ 0 0 ⎥ ⎥ = (n 33 S33 − 1)(n 44 S44 − 1) − n 2 S33 S44 . 34 −1 ⎦ S44 (2.2.16)



According to (2.2.2), element S 44 is defined as: ⎡

n 33

⎢ −1 ⎢ [N ] = |[R] − [B]| = ⎢ ⎣ n 34 0

−1

n 34

0



S33 0 0 ⎥ ⎥ ⎥ 0 n 44 − 1 ⎦ 0 −1 S44

= (n 33 S33 − 1) (n 44 S44 − 1) − n 234 S33 S44 . 

The respective element S 66 of SM [S] is given by:

2.2 Analysis of Radar Channel Model

17

⎤ S22 0 0 0 S24 ⎢ 0 n −1 n 0 ⎥ ⎥ 33 34 1 1⎢ ⎥ ⎢ 11,12,13,14 = |[R] − [B]|2,2 = ⎢ 0 −1 S33 0 0 ⎥ ⎥ N N⎢ ⎣ 0 n 34 0 n 44 −1 ⎦ S42 0 0 −1 S44   1  2 = n 34 S33 + n 33 S33 n 44 (S44 S22 − S24 S42 ) + n 33 S22 S33 . N ⎡



S 66

(2.2.17)



For S 88 we have 1 |[R] − [B]|11,12,13,14 3,3 N   2 S + n 2 S _n (n S + n S ) + n S S (n n − S n ) + n n 1 11 44 44 11 33 44 33 44 34 34 13 33 14 44 11 33 33 = . N +n 13 S33 S44 (2n 14 n 34 − n 13 n 44 )



S 88 =

(2.2.18) And, finally: 1 |[R] − [B]|11,12,13,14 4,4 N   2 1 n 24 S44 + n 24 S44 S33 (n 23 n 34 − n 24 n 33 ) + n 23 S33 S44 (n 34 n 24 − n 23 n 44 ) + n 22 n 33 S44 = .   N n 33 n 44 − n 234 − n 22 n 44 S44



S 1010 =

So, in front of us, there is a problem of minimization of underlying surface influence on the radar station operation. Meanwhile, in a given polarization basis defined by polarization mode of radiation-reception, as we can see from (2.2.7), noise reduction should be fulfilled by the methods that will decrease the level of the signal reflected back from the separation surface, i.e., we have to minimize coefficient S 88 of SM [S] (see Fig. 2.3). Besides, we have to decrease the level of cross-polarization, namely, to minimize element m 14 of antenna system SM [S] [11, 12]. Let’s consider expression (2.2.15) that defines coefficient S 88 . We represent it in the following way: 





S 88

  1 n 11 (1 + S33 S44 n 33 n 44 − n 34 S33 S34 S44 − n 33 S33 − n 44 S44 )+ . = N n 13 (n 13 S33 + 2n 14 n 34 S33 S44 − n 13 n 44 S33 S44 ) + n 214 S44 (2.2.19)

The first summand reduces to zero when n 11 = 0. Since the first summand is determined by the first element in separation surface in reversal direction (2.2.10), then selection of polarization that minimizes this element will allow decreasing of parasite power that inflow through antenna in the information processing system. The second summand is mainly determined by coefficient n 13 that represents an element of underlying surface SM in the mirror direction. This second summand defines the portion of radar station energy re-reflected to the target (2.2.12). Choice of

18

2 Synthesis of Signal’s Polarization Selection System …

radiation polarization that decreases this coefficient allows receiving positive effect in the form of higher signal-to-noise ratio amidst strong passive noise formed by reflections from underlying surface. Elements of the matrix (2.2.17) that determine radar target parameters depend on radiating EMW polarization structure. Society of elements  [S]t =

S11 S12 S21 S22



that prescribes the target SM in the reversal direction, for the zero-loss target (unitary target SM) has the form: ⎤ 2 1 − S21 S12 ⎦.  [S]t1 = ⎣ 2 S21 1 − S12 ⎡

Presence of energy loss during reflection from the target doesn’t rule out interconnection of elements in (2.2.19). Expression (2.2.19) obviously demonstrates that when S12 , S21 = 0 (what corresponds to radiation with natural polarization of the target) S11 and S22 reach their maximal values. As follows from (2.2.13) and above-mentioned considerations, increase of element S11 improves the useful signal reception. Thus, analysis of suggested radar channel model (see Fig. 2.1) allows specifying main directions to solution to the problem of reducing passive noise formed during EMW reflection from the underlying surface. The main directions are: – decrease of the wave power reflected backward from the separation surface (minimization of S88 element by methods of polarization adjustment through decrease of n 11 element of underlying surface SM [N] (2.2.17)); – minimization of the noise level from the parasite direction (l3 in Fig. 2.2) due to multibeam energy propagation through decrease of n 13 element of SM [N] that determines radar station power re-reflected in the mirror direction; – increase of the useful reflected signal power due to polarization adjustment in order to minimize S11 element of the target SM [N] (radiation of the signal with first natural polarization); – exclusion of parasite cross-polarization phenomenon influence on radiation of aperture radar antenna through minimization of m 14 element of radar station SM [M] (2.2.4). Solution to the problem will be incomplete unless we take into account the influence of UHF system of primary information processing on the operation of the radar angular measurement system as a whole. The methodology of UHF systems matrix analysis suggested in the book provides an implementable solution to this problem along with practical recommendations on equipment adjustment [13, 14].

2.3 Analysis of Scattering Polarization Characteristics …

19

2.3 Analysis of Scattering Polarization Characteristics of Large-Scale Impedance of Randomly Uneven Surface 2.3.1 Analysis of the Current Status of the Issue Electromagnetic field registered by radar antenna from the low-flying target is the result of interference between the direct wave and the wave reflected from the underlying surface. For elevation angles that meet the following condition: l sin θn

≤ λ/4,

(2.3.1)

the reflection will be more mirroring than scattering which is typical for the angles that don’t satisfy (2.3.1). Here, l is the peak-to-trough height of the terrain surface; θn is the incident angle. The noise registered by radar antenna can be divided into two major groups: additive noise and multiplicative noise. In the first approximation we can consider additive noise as noise formed by backward scattering from underlying surface, while multiplicative as noise formed by mirror reflection in the radiating direction. We shall consider spectrum of the useful signal reflected from the target quite narrow and symmetrical around the signal frequency which well corresponds with experimental research. It follows from the fact that fluctuation spectrum width due to DP rotation of antenna system does not exceed several dozens of Hertz. The fluctuation spectrum width of the target varies in the same limits. Similar conclusion can be made on the noise scattered signal, with the difference; however, that noise input signal registered by the radar system can be represented as a sum of large number of partial random signals reflected from large number of elementary radiators. After that, if amplitudes and phases are statistically independent, then the resulting signal can be represented by the sum of stationary signal and random normally distributed signal with the zero mean value. The target signal covering the path II (see Fig. 2.4) can be represented by multiplicative mixture of the signal and the noise: X 1 (t) = S1 (t)ν,

(2.3.2)

where S1 (t) is the probing signal, ν is the stationary random process that characterizes multiplicative noise. Multiplicative noise belongs to the class of slow noises whose frequency spectrum is concentrated in the quite narrow band around Doppler frequency. Therefore, selecting it by means of Doppler filter is troublesome, or sometimes even impossible. The noise spectrum width is proportional to characteristic size of undulances on the underlying surface. For real targets an approximation of slow amplitudes and phases fluctuations of orthogonally polarized signal components seems to be justifiable. Correlation time for the functions that represent the effect of multiplicative noise is

20

2 Synthesis of Signal’s Polarization Selection System …

I

n II ha n

n

ha

=2ha sin

n

Fig. 2.4 Specular reflection formation diagram for radar signal

much longer than measurement time, which allows consideration of only additive mixture of signal and noise. Numerous experiments and theoretical research confirmed that separation surface is the primary source of aircraft positioning errors that appear during object’s elevation angle measurements in specific conditions of radio waves propagation. Other factors, such as atmospheric refraction or turbulent processes in troposphere, are of secondary importance at the distance to the target less than 10 km. Terrain surface may cause significant boresight errors even for comparatively low reflection indices. In order to decrease these errors, special screens are used in the form of semi-planes with horizontal boundary situated at some distance from the antenna, along with wavelength shortening that leads to lower reflection index due to increasing influence of underlying surface unevenness. However, increasing the level of diffuse scattering negatively impacts the number and types of boresight errors. There are several research works that proposed original method for positioning accuracy by increasing using of antenna system with roaming base center. Application of this antenna type allows reducing of the problem of underlying surface influence on the radar operation to the case of additive noise only, but doesn’t provide optimal signal selection in the presence of reflections from terrain surface. Thus, existing selection methods can solve the problem of useful signal selection from the multiplicative mixture of signal and noise by reducing this problem to selection of the signal from the additive mixture. For the object with elevation angle θ antenna axis will form this angle with the horizon plane and signals re-reflected from the scattering surface will be registered by radar at the angle 2θn from the antenna equisignal axis (see Fig. 2.4). When the target

2.3 Analysis of Scattering Polarization Characteristics …

21

is situated at the angles equal to antenna’s DP width or higher, signal re-reflected from the scattering surface will not significantly affect the amplitude and phase of the direct signal and will not seriously impact radar operation. However, appearance of this signal in the difference channel leads to the error that will cause the change in antenna orientation and reception of the signal from the false direction. Therefore, increment of the power ratio of direct and re-reflected signals is a very important issue. This ratio is defined as: G1 G se S = 2 = 2 , I ρ G4 ρ

(2.3.3)

G1 is the ratio of antenna gains in the main and side lobes of DP for the angle G se = G 4 2θn down from the equisignal direction; ρ is the surface reflection index. All values in (2.3.3) are averaged within the limits of elevation angles where reflection takes place (the so-called angles of shiny surface visibility ϕd ). Antenna gain integration within ϕd angles additionally confirms the fact that design of the systems with very narrow DP and full suppression of side radiation still remains the most effective method of multipath EMW propagation extinguishing. The most common solution to this problem in practice is using of additional constructions that do not allow signals from “parasite” directions to antenna reception as if decreasing antenna gain in the side lobe. However, these methods of terrain surface impact control have limited application due to engineering constructions inconvenience. There are opportunities in design of needle-type DP in a given wavelength range. Another known method of passive noise impact control is increasing of signal carrier frequency. Other things being equal, it leads to increasing diffuse scattering that can be easily excluded by any known selection system for radar signal. However, increasing of carrier frequency is limited by worsening of ultrashort wave’s propagation in precipitation, scattering on dust particles in the air along with side effects of turbulence in troposphere, and atmospheric refraction [15]. Therefore, the need appears in the design of signal selection system operating in the presence of the passive noise background and based on new principles. The problem of signal polarization selection for a low-flying target amidst reflections from separation surface can be solved based on analysis of polarization scattering characteristics of underlying surface. Since the great majority of radar stations operate in S-band (cm wavelength band) and underlying surface is randomly uneven, then distributed object (separation surface) makes the main influence on parasite radar signal forming. This separation surface has finite conductivity and random topographic inequalities with characteristic size much longer than EMW wavelength. Besides, size range of scattering areas varies widely. For example, during scattering from the sea surface, desert, or steppe, reflecting area is defined by cross-section of antenna DP spatial body by separation surface and can reach quite high values. For terrain with forest or other vegetation, scattering area is smaller, being defined mainly by the sizes of meadows, flood-planes, or forest aisles. The intensity of scattering from these areas will be determined by their size, electrical characteristics,

22

2 Synthesis of Signal’s Polarization Selection System …

Fig. 2.5 Graphic chart of the average RCS from EMW incidence angle for low vegetation areas

peak-to-valley value, and their spatial orientation toward radar station. Besides, the scattering properties vary with azimuth angle and radiation polarization. All the above-mentioned considerations significantly complicate theoretical research of scattering properties, putting forward experimental methods. From the works devoted to experimental research of scattering properties, we can conclude that for uneven surfaces correlation between reflection index and polarization is weak. It was shown that for small θn angles, average RCS deviation of underlying surface areas is insignificant for horizontal, vertical, or circular polarizations. In Fig. 2.5 you can see an experimental dependence of average RCS for areas with low vegetation (grass, shrub) from the radiation incident angle. Degree of reflected signal depolarization is always below 10 dB, i.e., level of depolarized component is always lower than the level of components with initial polarization. For the vertical polarization this degree is 8 ÷ 15 dB, and for the horizontal it is 2 ÷ 12 dB. Depolarization degree is mainly determined by reflections from the sharp edges and weakly correlated with wavelength. However, as was shown in some works, depolarization increases for vertical polarization with wavelength shortening. Experimental research of the sea surface scattering properties allowed for conclusion that for low gliding angles the following dependence is observed for linear polarized RCS components: G 2,2 ≥ G 2,1 = G 1,2 ≥ G 1,1 ,

(2.3.4)

2.3 Analysis of Scattering Polarization Characteristics …

23

where G 2,2 and G 1,1 are RCS for emitted/received waves with horizontal and vertical polarization, respectively; G 2,1 and G 1,2 are RCS for emitted/received orthogonal linear polarized waves. However, with increasing sea disturbance, when the shadowing phenomenon appears, we have: G 2,2 ≥ G 1,1 = G 2,1 ≥ G 1,2 . In the same work, it is shown that correlation coefficients between G i, j (i, j = 1, 2) components are significantly below the unit. The maximal value of correlation coefficient observed between G 2,2 and G 1,1 can reach 0.25 (others are below 0.15). These results allow affirming that scattering polarization characteristics of distributed targets depend on both polarization conditions of incident wave and incident angle. RCS dependence on incident angle is especially strong. Analysis of modern experimental research on separation surface scattering properties has shown that for every distributed target a polarization condition exists which after reflection fully transforms into a polarization orthogonal to the initial and cannot be detected by antenna system (the so-called polarization of zero signal). This fact provides an opportunity to decrease the power level of radiation reflected from the local objects by means of antenna PD readjustment.

2.3.2 Scattering Characteristics of Large-Scale Measuring of Randomly Uneven Surface Nowadays, we witness persistent attempts to create the theory of scattering from a spatially distributed target. Theory of reflection from underlying surface is substantiated in terms of the so-called resonance mechanism. However, not all experimental results can be properly explained in terms of this theory [16–19]. These unexplained phenomena are: – exceeding of the scattered EMW intensity with horizontal polarization over the intensity of EMW with vertical polarization under increasing sea disturbance; – deeply disperse pattern of the scattered field space-and-time distribution during radiation by horizontally polarized EMW under relatively weak wind; with increasing sea disturbance this deeply disperse pattern appears for vertically polarized EMW too; – quite wide spectrum of the signal scattered by the sea surface. Since radiation wavelength in radar applications is much less than typical sizes of reflecting surface and even its random undulations, then the problem of calculation of scattering polarization characteristics for large-scale impedance surface can be solved in terms of Kirchhoff approximation.

24

2 Synthesis of Signal’s Polarization Selection System …

Since the reflecting properties of separation surface (assumed as an arbitrary spatially distributed target) depend mainly on the properties of open areas, we should solve this problem for the surface with finite size and with its arbitrary orientation toward the source of EMW of arbitrary polarization. Exactly this model of the real surface will be used hereinafter. The scattered field is determined by the vector form of Kirchhoff integral: E=

1 4π

 (ik[n, HS ]ϕ + [[n, E S ]∇ϕ] + (n, E S )∇ϕ)d S,

(2.3.5)

S

where n is the unit vector of internal normal to the surface, E s , HS are values of electrical and magnetic fields on the reflecting surface for the case of finite impedance relief, ϕ is the Green function that for Fraunhofer approximation has the following form: ϕ∼ = R −1 exp{i(k R − x, r1 ) ,

(2.3.6)

ϕ∼ = R −1 − i x R −1 exp{i(k R − x, r1 ) . Here r1 is the radius-vector of the current point on the surface, x is vector of scattering, with |X | = |K | = wc , and R is the distance to the viewpoint. Other notations are clear from Fig. 2.6. Equation (2.3.5) defines the field in the semi-sphere as a function of equivalent sources distributed on the infinite surface [S] surrounding this semi-sphere. As can be easily shown, for R -> ∞, E vector does not meet radiation principle, because vector E is not orthogonal to the X in the far zone. Putting limitations on the scattering surface size as an area with finite dimensions leads to the discontinuities on the perimeter γ surrounding this area. Compliance with the equation of charge and current continuity is provided by introducing some linear charge density distributed along this contour: G=

1 (τ, HS ), ik

(2.3.7)

where τ is the unit vector tangent to the contour γ . Equation (2.3.7) is used in the expression for the scattered field that is represented by the sum of surface integral and volume integral degenerating to the contour integral: 1 4π



1 ρ∇ϕd V → 4πik

V

For R -> ∞, (2.3.5) will take the form:

 (τ, HS )∇ϕd S. γ

(2.3.8)

2.3 Analysis of Scattering Polarization Characteristics …

25

0`

0`` R

K

R-r 01 R+r 01 R0

n

r1 S1 P

t

Fig. 2.6 Coordinate system for scattering performance measurement from impedance randomly uneven surface with finite dimensions

E=

1 4πik

 (r, HS )∇ϕdl + γ

1 4π

 {ik[n, HS ]ϕ + [[n, HS ]∇ϕ] + (n, E S )∇ϕ}d S. S1

(2.3.9) Conversion of the contour integral (2.3.9) into a surface one according to Stokes theorem results in Kirchhoff–Kotler formula for the scattered field that relates scattered field E S with fields E S u HS defined on the limited surface S1 :    1 E= ik[[n, E S ]∇ϕ] − ([n, HS ]∇)∇ϕ − k 2 [n, HS ]ϕ d S, (2.3.10) 4πik S1

where ([n, HS ]∇ϕ) =

 j=1

i −1 [n, HS ]∇ ∂∂ϕ j . Xj

This scattered field satisfies the radiation principle, since vector E is orthogonal to X for R → ∞. Thus, the main feature of the following analysis of scattering properties is derivation of expressions for scattered field based on Kirchhoff vector form with Kotler amendment.

26

2 Synthesis of Signal’s Polarization Selection System …

An accepted approximation of large-scale undulations on the reflection surface allows for application of physical optics methods. Fields E s and H s will be represented by the sums of incident wave and the wave reflected from the plane tangent to the surface at a given point, according to reflection law of geometrical optics. We shall consider incident wave front to be a plane fully determined by the vector K. Compliance with optics laws is provided by observance of the following equation: (n, K 0 ) = −(n, X 0 ).

(2.3.11)

For the following analysis let’s introduce local orthonormal basis ( p, t, n) on the random surface with normal distribution of undulances heights. This basis is related with the local normal n by the following expressions: P=

[[n,k0 ]n] [n,k0 ] t −0 = |[n,k |[[n,k0 ]n]| ; 0] −1 −1 −1 −∇ X ξ i +∇Y ξ j +k

n= √

;

,

(2.3.12)

1+(∇ X ξ )2 +(∇Y ξ )2

where ξ (x, y) is the stationary random differentiable function that describes the heights of all points on the surface, ∇m is the projection of differential operator ∇ on the appropriate axis. We can represent the components of the incident field corresponding to the horizontal and vertical polarization in the following way: E oh = E ot t; HO H = HO P p + Hon n HO V = Hot t; E O V = E op p + E on n.

(2.3.13)

Scattered fields are determined by Fresnel reflection coefficients that are given for the surface by the expressions: R H = (cos θ −

ε1 − sin2 θ )(cos θ +

RV = (ε1 cos θ −

ε1 − sin2 θ)−1 ;

ε1 − sin2 θ )(ε1 cos θ +

(2.3.14)

ε1 − sin2 θ )−1 ;

where ε1 = ε − j (4π G/ω) is dielectric permittivity; G is separation surface conductivity. For the sake of further reasoning convenience, we represent these coefficients as the sum of the real and imaginary parts defined as follows: Re R H =

cos2 θ − μ2 ; (cos θ + μ cos ϕ)2 + (μ sin ϕ)2

Im R H = −

2μ sin ϕ cos θ ; (cos θ + μ cos ϕ)2 + (μ sin ϕ)2

2.3 Analysis of Scattering Polarization Characteristics …

Re R V =

ε2 cos2 θ − μ2 + ε12 cos2 θ ; (2 cos θ + μ cos ϕ)2 + (μ sin ϕ − ε1 cos θ )2

Im R V = − where θ =

1 2

27

(2.3.15)

2μ cos θ (ε sin ϕ + ε1 cos ϕ) ; (2 cos θ + μ cos ϕ)2 + (μ sin ϕ − ε1 cos θ )2

arccos(sin ψ sin η − cos ψ cos η cos ϕ); μ=

 4

(ε − sin2 θ )2 + ε12 ;

ε1 1 . ϕ = − ar ctg 2 ε − sin2 θ Transition from the non-local basis (i, j, k) is fulfilled by multi  to the local basis plication of the field components in the basis i −0 , j −0 , k −1 by corresponding directional cosines of the local basis that are fully determined by the direction of the random normal n and the wave vector of the incident field. Therefore, Hs expansion in the local basis will depend on the slope angles of the tangent plane in every point, i.e., reflection coefficients will be random functions of the angles. Since characteristic undulances are significantly bigger than the wavelength, then we can apply method of stationary phase for scattered field evaluation. For the finite integration area, asymptotic value of the integral is determined by integrand behavior in stationary phase points and by the contour integral along the boundary of integration area. It was shown that contour integral can be neglected in comparison to share of stationary phase points within the area S1 . Assuming the stationary phase points to be situated far from the boundary γ and far from each other (due to the fact that undulation characteristic size is much longer than wavelength), the expression for the scattered field according to the Kirchhoff–Kotler formula can be written as follows: ik ei{k(R+R0 ) } 4π R R0   ⎡ ⎧ ⎧ (n, k0 ) + n , x0 }− ⎪ ⎪ ⎪ ⎪  ⎧      ⎢ ⎨ ⎨ ⎨ x0 , H0 + xo , H0 } + 2H1 n , x0 − k0 (n , H ) ⎢x H ⎣ 0 ⎪ 0 ⎪ −n  ⎪ ⎩ ⎪ ⎩ ⎩ −x −0 n −0 , H −1 S

E=

0

⎤ ⎥ i{(k −x )r } ⎥e ds, ⎦

1

where q = k − x; kq0 = k(k0 − x0 ). Modifying exponent factor as %  & (q0 r ) , exp{ik(R + R0 + (q0 , r ))} = exp ik(R + R0 ) 1 + R + R0 it is easy to see that stationary phase points should satisfy the following condition:

28

2 Synthesis of Signal’s Polarization Selection System …

 ∇ 1+

q0 r R + R0

 = 0.

(2.3.16)

And corresponding fixed values of the tangent planes slopes in the stationary phase points are equal to (∇ X ξ )0 = −

qX ; qZ

(∇Y ξ )0 = −

qY . qX

(2.3.17)

Here dS =



1 + (∇ X ξ )2 + (∇Y ξ )2 d xd y =

1 qZ

 q Z2 + q X2 + qY2 d xd y.

(2.3.18)

Scattered field can be finally represented as:   ik E −0 = exp i k(R + R0 ) }F −1 (∇ X ξ )0 4π R R0      1 exp i q −1 r −0 } × q Z2 + qY2 + q X2 d xd y. qZ

(2.3.19)

S

Vector multiplier F determines polarization characteristics. Here we won’t consider its multiplicands in (2.3.19) that are determined by statistical and electrical terrain parameters. Multiplier F can be represented as  F = [x0 {2H1 (n, x0 ) − n{(x0 , H0 ) + (x0 , H1 )} − k0 (n, H0 ) − x0 (n, H1 )} ](∇x ξ )0 . (2.3.20) Or  F = [x0 {2H1 (n, x0 ) − n{(x0 , HS ) − k0 (n, H0 )} ](∇x ξ )0 ,

(2.3.21)

where values H0 , H1 and HS should be taken in the local basis ( p, t, n). Assuming the incident wave having arbitrary polarization, i.e., having all three components in the non-local basis (i, j, k), and assuming scattering vector having arbitrary orientation, we can receive the expression for vector function F. For this purpose we have to determine incident field wave vector and scattering vector through the gliding angle ψ, elevation angle η, and azimuth angle ϕ. Besides, we also have to determine incident angle θ between vector k0−1 and tangent plane to the surface in the reflection point. The value of θ, along with electric parameters ε and G, fully determines the Fresnel reflection coefficients on the fixed frequency ω of the incident radiation of both polarizations.   For the fixed position of local basis p −0 , t −0 , n −0 and incident radiation wave vector belonging to yz plane, and with

2.3 Analysis of Scattering Polarization Characteristics …

k0 = cos ψ j − sin ψ k;

29

|k0 | = 1,

x0 = − cos η sin ϕi + cos η cos ϕ j + sin η k; q0 =

|x0 | = 1,

 1  q X i + qY j + q Z k , |q|

(2.3.22) (2.3.23) (2.3.24)

where q X = k X − x X = cos η sin ϕ, qY = cos ψ − cos η cos ϕ, q Z = −(sin ψ + sin η), √ 1 |q| = 2 (1 − cos ψ cos η cos ϕ + sin ψ sin η) 2 according to (2.3.12), in the stationary phase points, where condition (2.3.17) is satisfied, normal n is determined as: n0 =

 1  q X i + qY j + q Z k , |q|

(2.3.25)

and vectors (t and p) have the following projections: t0 =

 1 t X i + tY j + t Z k , |t|

(2.3.26)

where t X = k0 Y x0Z − k0Z x0Y = cos ψ sin η + sin ψ cos η cos ϕ, tY = k0Z x0X = sin ψ cos η sin ϕ, t Z = −k0Y x0X = cos ψ cos η sin ϕ; |t| =



cos2 η sin2 ϕ + (cos ψ sin η + sin ψ cos η cos ϕ)2 ,

where  1  P0 =  −0  p X i + p y j + p Z k , p  p X = − cos η sin ϕ, pY = cos ψ + cos η cos ϕ,

(2.3.27)

30

2 Synthesis of Signal’s Polarization Selection System …

p Z = sin η − sin ψ | p| =



1

2(1 + cos ψ cos η cos ϕ − sin η sin ψ) 2 .

Substituting in (2.3.24) the necessary values and after several transformations we finally receive: − → − → F = Re F + Im F ,

(2.3.28)

where − → Re F = Re FX i + Re FY j + Re FZ k; Im F −1 = Im FX i + Im FY j + Im FZ k.

(2.3.29)

Since we want to know the value of the scattered field at the point determined by elevation angle η and azimuth angle ϕ, then vector multiplier F, defined in (2.3.28), should be represented in the basis (

x , y , z ) tied with reception antenna (see Fig. 2.6). to We shall assume that in this basis axis y is directed along x 0 , axis z is orthogonal  → x 0 and belongs to the plane ( z , x0 ), axis z is orthogonal to the plane z , − x0 , i.e. i 0 =

[x0 , k] ; |[x0 , k]|

j0 = x0 ; k0 = −

[x0 [k, x0 ]] . |[x0 [k, x0 ]]|

(2.3.30)

From which   i 0 = cos ϕ i + sin ϕ j, i 0  = 1

 

k 0  = 1 k0 = sin η sin ϕ i − sin η cos ϕ j + cos k;

(2.3.31)

is determined by elevation angle. Expression (2.3.28) is correct for any scattering vector x 0 direction and for arbitrary polarization of incident radiation. Specifying of radiation polarization significantly simplifies the above expressions, because, for example, components H oy and H oz are absent for vertical polarization, and component H ox is absent for horizontal polarization. − → − → Since orths i 0 and k0 define horizontally and vertically polarized components of scattering vector, then in order to get scattering matrix of the large-scale impedance − → − → on randomly uneven surface, we have to receive expressions for i 0 and k0 compo nents of F vector for horizontal (Hox = 0) and vertical Hoy = Hoz = 0 radiation − → polarizations. Orth i 0 component will be the main for horizontal polarization; orth − → k0 component is for vertical polarization. Components’ phases of the scattered field will be defined as:

2.3 Analysis of Scattering Polarization Characteristics …

ϕ2 = ar ctg

31

Im F2 Im F1 ; ϕ1 = ar ctg . Re F2 Re F1

(2.3.32)

Polarization parameters of scattered wave orthogonal expanding are defined by the following expressions: Re2 F2 + Im2 F2

p=

Re2 F1 + Im2 F1

; δ = ϕ2 − ϕ1 ,

(2.3.33)

where p is polarization coefficient, δ is phase parameter, indices 1 and 2 refer to vertically and horizontally polarized components of the field, respectively. Ellipticity coefficient K and orientation angle γ of polarization ellipse are defined through p and δ as follows: ' K =

p 2 cos2 γ cos2 δ + sin2 γ − p sin2 γ cos δ , p 2 sin2 γ cos2 δ + cos2 γ + p sin2 γ cos δ   1 2p γ = ar ctg cos δ . 2 1 − p2

(2.3.34) (2.3.35)

As was shown, target RCS is determined by the value of scattered field pointing vector. It was also shown that power scattering matrix is a comfortable tool for RCS calculation. Power scattering matrix is defined as follows: ⎡ − →− →∗ ⎢ ⎢ p = F F = ⎢ ⎣



∗ ∗ F22 F22 + F21 F21



 ∗

∗ F12 F22 + F11 F12

∗ ∗ F22 F21 + F21 F11

⎤

⎥ ⎥ ⎥, ⎦  ∗

(2.3.36)

∗ F12 F21 + F11 F11

where the first index symbol defines radiation polarization, and the second symbol is polarization of reception antenna during measurement of matrix components. In Figs. 2.7 and 2.8 the results of polarization parameters are given for the wave scattered by 2D statistically uneven sloping surface with finite conductivity. These results have been obtained by two methods: according to Kirchhoff formulas without taking into account the finite size of underlying surface and according to the method proposed in this book. The comparison of results shows that application of Kirchhoff– Kotler formulas slightly modifies the curves’ form, however, the difference grows with gliding angle decrement. Figures 2.9 and 2.10 present the curves of polarization coefficients for zero azimuth angle, fixed electrical parameters of the surface, and different gliding angles. Figures 2.11 and 2.12 give an idea of polarization parameters change in the plane defined by different azimuth angles ϕ for vision angles corresponding to gliding radio wave propagation.

32

2 Synthesis of Signal’s Polarization Selection System … о

=0О

=0О

`=2-j1,62 4,0

О

=70

3,0

=30О =6 0О[72]

=30О 2,0

1,0 =70О 0

0 100

90

110

120

90

о

100

110

120

о

Fig. 2.7 Results of polarization characteristics of the wave scattered by two-dimension statistically uneven shallow surface with finite conductivity K

о

=0О

=0О

`=2-j1,62 0,12

=70

36

=30

0,09

О

=30

О

О

27

0,06

18

0,03

9 =70О

`=2-j1,62 0

0 90

100

110

120

о

90

100

110

120

о

Fig. 2.8 Results of polarization characteristics of the wave scattered by two-dimensional statistically uneven shallow surface with finite conductivity

The received results specify the dependence of polarization parameters on surface electrical properties, on radiation and observation angles, as well as give an idea of similar dependencies for small ϕ and η angles during energy propagation in radiation direction (η < 90º).

2.3 Analysis of Scattering Polarization Characteristics … P

33

о

=0О

=0О

О

=4

=4О

1,0

8,0 `=2-j1,62

0,75

6,0

`=2-j1,62

`=67-j23

0,5

4,0 `=80-j0,06 `=67-j23

0,25

2,0

`=80-j0,06 0

10

20

30

о

0

10

20

30

о

Fig. 2.9 Calculated curves of polarization coefficients for different grazing angles and assigned surface electrical parameters (azimuth angle is equal to zero) K

о

=0О

=0О

=4О

=4О `=2-j1,62

0,12

36

`=67-j23 0,09

27 `=2-j1,62 `=80-j0,06

0,06

18 `=67-j23

0,03

9 `=80-j0,06 0

0 90

100

110

120

о

90

100

110

120

о

Fig. 2.10 Calculated curves of polarization coefficients for different grazing angles and assigned surface electrical parameters (azimuth angle is equal to zero)

Presented dependencies provide valuable information on electrical properties of scattering surface for given ϕ and η angles.

34

2 Synthesis of Signal’s Polarization Selection System … P

о

`=2-j1,62

`=2-j1,62 =4О

О

=4

1,0

О

=15

8,0

О

10

О

5

0,75

6,0

0,5

4,0 О

=5

О

0,25

10

2,0

О

15

0

10

20

30

о

0

10

20

30

о

Fig. 2.11 Change of polarization parameters in the plane defined by different azimuths ϕ for viewing angles at the scattering object that correspond to grazing radio wave propagation о

K

`=2-j1,62

`=2-j1,62 =4О

=4О

О

=5

1,0

36 =10О

0,75

27 =5 О

=15О

0,5

18 =10О

0,25

9 =15О

0

10

20

30

о

0

10

20

30

о

Fig. 2.12 Change of polarization parameters in the plane defined by different azimuths ϕ for viewing angles that correspond to grazing radio wave propagation from the scattering object

2.3 Analysis of Scattering Polarization Characteristics …

35

Thus, based on the received expressions for the components of electromagnetic field scattered by a large-scale impedance on randomly uneven surface, we can calculate all polarization parameters of electromagnetic wave. Comparison of calculated results with existing theoretical curves allows for conclusion on the correctness of polarization parameters calculation approach and even for adjustment of existing theoretical curves. However, this comparison does not eliminate the need for calculation results experimental testing for a large-scale underlying surface with given values of dielectrical permittivity and conductivity.

2.4 Methods of Noise Influence Diminishing Based on Different Polarization Structure of Target Signal and Noise Signal Until recently, synthesis of radio technical aids purposed for useful signal filtering in the background of noise was based on the assumption that both signal and noise are fully polarized, their polarization coincides with reception antenna polarization, and corresponds with radiation polarization as well. However, in reality scattered wave polarization depends on many factors and generally can differ from antenna polarization. In these circumstances the partial loss of useful signal power occurs due to decrease of polarization detection efficiency for polarized signal. Therefore, signal processing by technical aids that do not take into account signal’s polarization conditions is not optimal. Under optimal processing hereinafter we shall mean achievement of maximal signal-to-noise ratio [20]. One of the most effective methods of passive noise elimination that still has not been applied widely is polarization selection. All previous systems were based on the assumption of scalar nature of radiation, dealing with integral characteristics of reflected signals. Consideration of vector nature of signal field allows for achievement of some advantage in signal-to-noise ratio. Some works of Russian authors and authors from other countries were devoted to solution to the problem of waves’ selection by their polarization parameters. All existing selection methods are based on radar PD modification according to certain laws and can be combined with statistical methods of radar signal processing. Application of an arbitrary polarization selection system should be evaluated by the following issues: – increased magnitude of the reflected target signal for selected polarization compared to the systems with circular or linear polarization; – improvement of signal-to-noise ratio for the matched load on the antenna operating on the selected polarization and receiving the mixture of target signal and noise, comparing to the systems with circular or linear polarization; – ability to apply different methods of UHF signal processing.

36

2 Synthesis of Signal’s Polarization Selection System …

Let’s consider performance potential of some methods of polarization selection based on the above-mentioned requirements. During reception of two independent signals with polarization condition defined by the vectors p˙ 1 and p˙ 2 , respectively (assuming the need of full suppression of the wave p˙ 1 and full reception ( of wave p˙ 2 ) antenna will suppress wave p˙ 1 with polarization efficiency η = 1 p˙ 1∗ and receive wave p˙ 2 with efficiency defined by (2.5). When parameters of polarization ellipse for the wave and antenna system coincide, including the sign that determines the direction of field vectors rotation, the antenna is called-matched with input signal polarization. If we maximize expression (2.5) by p˙ a , we can find optimal polarization that provides the highest possible signal-to-noise ratio. Due to the numerous reflections from fluctuating targets and local objects, radar signal ceases to be fully polarized and becomes partially depolarized. According to Stokes theorem, any arbitrary partially polarized wave can be represented by the sum of two components, one of which is fully polarized, and the other is not polarized. Since methods of polarization selection are effective for polarized signals only, hereinafter we will consider only combinations of fully polarized signal and non-polarized noise that can represent all cases conceivable in radar surveillance. Nowadays different areas of technical practice demonstrate the need in using the systems of optimal reception. The concept of optimality is defined by the adapted criterion of optimality, and the system which is optimal for one criterion may, of course, not be optimal for another [21]. As a feature of a given system optimality some quality indicator is usually taken. As a rule, optimization task consists of searching for the minimum of some functional (e.g., total cost or penalties), while sometimes searching for the maximum value of another functional (bottom line result, effect or benefit). In our case the optimality criterion is maximal signal-to-noise ratio. As a quality indicator of designed system, we will choose achievement of maximum value of the ratio: K =

PS , PN + PS N

(2.4.1)

where PS , PN , PS N are power values for useful signal, external noise, and equivalent input receiver self-noise, respectively; the last value can be represented as an additional unpolarized noise that is summed with the component of external noise, according to Stokes theorem. In the presence of fully unpolarized noise, the maximal value of (2.4.1) can be reached when antenna system is matched with the target signal, because numerator is maximal for this case while denominator value is constant for unpolarized noise. Nowadays, only the simplest optimal systems can be designed with simple components; more complicated systems require significant computation capacities [22, 23]. Mathematical setting of the design task is also quite complicated. Therefore under the term “optimal system” we shall mean that the system provides maximal value of (2.4.1) ratio. However, the design of this system is beyond the scope of this book. In this book we shall consider the design of the system that is quite close to the

2.4 Methods of Noise Influence Diminishing …

37

Fig. 2.13 Dependence of polarization selection methods from the noise signal polarization

A1

c n

c

n

A n

optimal in the sense of (2.4.1) ratio, but simultaneously quite simple in its technical realization. Hereinafter we shall call this system quasioptimal. As known, polarization selection methods are most effective when the noise is fully polarized. Assuming both signal and noise to be partially polarized, we can modify (2.4.1) in the following way: K =

A PS [0, 5(1 − m S ) + m S cos2 δ S /2] , A PN [0, 5(1 − m N ) + m N cos2 δ N /2 + PS N ]

(2.4.2)

where A is antenna aperture, PS(N ) is power flow density of the signal (noise), m S(N ) is polarization degree of the signal (noise), δ S(N ) is angular distance on Poincare sphere between polarization points of the signal (noise) and the antenna (see Fig. 2.13). Assume that PS N in (2.4.2) has some external unpolarized noise. Then we have: K =

PS 1 + m S cos δ S · , PN1 1 + m N1 cos δ N

(2.4.3)

where m N1 =

PN A PS N . ; PN1 = PN + 2 PN A + 2PS N A

(2.4.4)

Considering the spherical triangle ACn (see Fig. 2.13), we can see that the maximal value of the second compound in (2.4.3) corresponds to ϕ = π , i.e., the point that displays optimal antenna polarization is situated on a large circle passing through the points C and P (e.g., point A in Fig. 2.13). Thus, δ N = δ S + ς and K =

1 + m S cos δ S . m N1 cos(δ S + ς )

(2.4.5)

38

2 Synthesis of Signal’s Polarization Selection System …

From the expression (2.4.5) it follows that for ς = π δ Sopt = 0; when ς = 0 δ Sopt = π for m S < m N1 and δ Sopt = 0 for m S > m N1 . The more the value of m N1 , the closer is optimal polarization to the polarization orthogonal to the noise, while with m N1 decrease and m S increase optimal polarization becomes closer to the signal polarization. The benefit from the optimal polarization of receiving antenna is equal to η=

1 + m S cos δ Sopt 1 + m N1 cos ς K max = = , K0 1 + m N1 cos(δ Sopt + ς ) 1 + mS

(2.4.6)

where K 0 is the K value for the case when antenna is matched with signal polarization. The efficiency of signal polarization selection was estimated for the noise with random uniformly alternating slope of polarization ellipse and with ellipticity angle ϕT = ϕ N i = const (i = 1, 2, 3, . . .) for three different cases: 1 2 3

when polarization directions of the noise and selector coincide, when they are opposite, for the sign-variable direction of the noise field rotation.

For this purpose the ellipticity angles of polarization selector have been determined for which noise power on the matched antenna load γ N was minimal. The curves for γ N min dependencies from β angle are given in Fig. 2.14 for all three cases. Fig. 2.14 Graphic charts of γPmin depending on β

п наим

1,0

0,8 2

1 0,6

q=0,5

4

0,4 3

0,2

0

10

о

20 о

30о

40о

2.4 Methods of Noise Influence Diminishing …

39

Knowing γ N min , we can estimate polarization selection efficiency, i.e., the degree of polarized noise suppression, using the following expression [21]: η = γ S /γ N min ,

(2.4.7)

where γ S is the polarization coefficient of signal reception equal to the signal power on the antenna load. The curves shown in Fig. 2.14 correspond to the following values:

γ1N min

⎧ 1 + sin 2ϕ π π N ⎪ , npu ϕ S ≤ , 0 ≤ ϕ N ≤ ⎨ 2 4 4 = π π 1 + cos 2ϕ ⎪ N ⎩ , npu ϕ S = 0, ≤ ϕN ≤ 2 8 4

(2.4.8a)

for the first case (curves 1 and 2); γ2N min =

1 − sin 2ϕ N π π , npu ϕ S = − , 0 ≤ ϕ N ≤ 2 4 4

(2.4.8b)

for the second case (curve 3); and γ3N min = Φγ1N min + (1 − q)γ2N min , npu ϕ S =

π 4

(2.4.8c)

for equiprobable direction of noise polarization q = 0, 5 (curve 4). So, while for unpolarized noise there is no talk about polarization selection, for polarized noise, the polarization selection methods can significantly improve signalto-noise ratio of the matched antenna load. When we know beforehand the polarization structure of the target signal and noise, we can solve the problem of optimal signal reception by the device with adjustable PD of the antenna. However, in real circumstances we may not have these data, because, as was mentioned above, scattering polarization characteristics depend on many random factors. Let’s consider the problem of signal selection in the circumstances when the phenomenon of multipath EMW propagation is present (see Fig. 2.4). This problem solution is of great practical importance, because elevation angle track on target by modern radio technical aids in these circumstances is inconsistent; sometimes even impossible. Since radar signal reflected from the separation surface can be represented by the random permanently bound normal process with the zero mean value, the corresponding SM of this surface can be described as:  [S(t)]n =

 S11n (t) S12n (t) , S21n (t) S22n (t)

(2.4.9)

40

2 Synthesis of Signal’s Polarization Selection System …

where Si j (t) are random processes. In the most general terms target SM can similarly be described as:   S11t (t) S12t (t) . (2.4.10) [S(t)]t = S21t (t) S22t (t) The task of polarization selection consists in the most appropriate (in the sense of criterion (2.4.1)) mapping of matrix (2.4.10) in polarization basis of radar antenna in the background of matrix (2.4.9). It was shown that average power received by radar antenna is defined as follows: Pr ec =

λD1 D2 PZ G t ηN , (4π )3 R12 R22

(2.4.11)

where D1 , D2 are directive gains of transmitter and receiver antennas, PZ is radiation power, R1 and R2 are distances from transmitter to the target and from the target to receiver, respectively; G t is average target RCS: G t = 4π R22 p tT ∗ [P] p t , .

(2.4.11a)

η N is radar polarization efficiency defined in (2.4.7); according to (2.3.36), power scattering matrix [P] is tied with the target scattering matrix [S]t in the following way:   [P] = M [S]∗ [S] ,

(2.4.12)

where M is expectation sign. Expression (2.4.11a) shows that target RCS depends not only on scattering polarization properties but also on the kind of radiation polarization. This dependence allows for application of polarization selection methods for solution to the problem of useful signal acquisition on the background of passive noise. For the monostatic radar let’s assume that radiating wave vector is uniquely associated with receiving antenna vector, i.e., p t and p r . Expression (2.4.11) defines the value of resultant signal reflected from the target along with noise background that falls on the antenna. In order to provide better signal-to-noise ratio due to polarization difference between the target signal and noise signal, the following methods are proposed for passive noise minimizing. First, as has already been told, at least two radiation polarizations exist that don’t suffer from any transformations after reflection, i.e., they don’t demonstrate depolarization, and the reflection coefficient for the first natural polarization is maximal. Therefore, target radiation by the signal with this polarization enhances the reflected signal power.

2.4 Methods of Noise Influence Diminishing …

41

In order to determine natural polarization, we have to find the matrix of unitary transformation [U ] that diagonalizes power scattering matrix [P]. For this purpose we have to solve the following system of matrix equations: [U ][P][U ]−1 = [λi δi k ]2 [U ][U ]∗ = 1,

(2.4.13)

where λi (i = 1, 2) are characteristic numbers of matrix [P]. − → The required vector P 0t of radiating wave (the target first natural polarization) provides the maximal value of the square form: − → − →0 T ∗ P t [P] P 0t = ,

(2.4.14)

The numerator of fraction (2.4.1) takes the maximal value for monostatic radar when Pt = Pr . Let’s consider the following argumentation in order to estimate the benefit of the useful signal power increment because of the result of using target natural polarization. Assume power scattering matrix (SM) in an arbitrary polarization basis having a form: ⎤ ⎡ 0 0 p 11 p 12 (2.4.15) [P] = ⎣ 0 0 ⎦. p 21 p 22   The first natural polarization in this basis can be defined as E 1T = E 1 , E 2 e j ψ . The second natural polarization (orthogonal to the first one) will be defined by the   vector E 2T = E 2 , −E 1 e j ψ . Transition from any given polarization basis to the natural basis is fulfilled by the matrix [Q]:  q11 q12 . [Q] = q21 q22 

(2.4.16)

Taking into account the correctness of the following expressions: 

    1 E1 and = E2 e j ψ 0     1 q12 E2 . = q22 −E 1 e j ψ 0

q11 q12 q21 q22



q11 q21

We can conclude that matrix [Q] has the form:

(2.4.17)

(2.4.18)

42

2 Synthesis of Signal’s Polarization Selection System …

 [Q] =

 E2 E1 . E 2 e j ψ −E 1 e j ψ

(2.4.19)

This matrix should provide the transition from any given polarization basis to the natural one without total power change, i.e., it must be unitary. From the unitary requirement (2.4.19) it follows that ⎡

 ⎤   E1 1 − E 12 A ⎦ = E1 1 . [Q] = ⎣  Ae j ψ −e j ψ 1 − E 12 e j ψ −E 1 e j ψ The reversal matrix has the form: ⎡  ⎤   2 −j ψ −j ψ E 1 − E e 1 1 ⎦ = E 1 1 Ae [Q]−1 = ⎣  , A −e− j ψ 1 − E 12 −E 1 e− j ψ

(2.4.20)

(2.4.21)

 E2 where A = 1 − E11 . In accordance with (2.4.13), power SM (2.4.11), determined in an arbitrary basis, after transformation [Q]−1 [P][Q] that defines transition to the natural basis transforms to the diagonal form as follows: 

λ1S 0



0 λ2S

 =

p11 + p12 Ae jψ + p21 Ae− jψ + A2 p22

p11 A − p12 e jψ + p21 A2 e− jψ − p22 A

p11 A + p12 A2 e jψ − p21 e− jψ − Ap22

p11 A2 − p12 Ae jψ − p21 Ae− jψ + p22 A

 .

(2.4.22) From here we receive expressions that define yet unknown coefficient A and phase multiplicand e jψ : %

p11 Ae jψ2 − p12 e j(ψ+ϕ12 ) + p21 A2 e j(ϕ21 −ψ) − p22 Ae jϕ22 = 0 . ϕ11 + ϕ21 − ψ = ψ + ϕ12 + ϕ22

(2.4.23)

General solution of the equations system (2.4.23) is quite difficult, but it is massively simplified for the case when power SM (2.4.15) parameters are known in the initial basis. After obtaining the A and e jψ values from (2.4.23), it is not difficult to determine the benefit of using power increment after transition from initial basis to the natural one. We should just substitute A e jψ in the expression p11 e jϕ2 + p12 Ae j(ψ+ϕ12 ) + p21 Ae j(ϕ21 −ψ) + p22 A2 e jϕ22

(2.4.24)

2.4 Methods of Noise Influence Diminishing …

43

that defines the power of the signal received by antenna tuned on the first natural polarization of the target. The ratio of (2.4.24) to the SM (2.4.15) p22 element’s module determines the required benefit. On average, this benefit of using natural polarization can reach 5–6 dB compared to the linear polarization. For bi-static radar stations, the reception polarization p r that would maximize the numerator of (2.4.1) should be chosen in a way that provides maximal value of registered useful power. For this purpose, as was shown, the polarization of receiving antenna should be matched with fully polarized part of reflected target signal. For these parameters the denominator of (2.4.1) stays the same for unpolarized noise or slightly decreases for the partially polarized noise due to change of polarization selectivity of receiving antenna defined by (2.4). The second method of minimizing passive noise can be based on the mirror channel signal suppression through selection of arrival direction of the signal. During mirror reflection, the main part of the noise is produced by the signals reflected from the separation surface only once, since every reflection significantly reduces signal’s power. Therefore, signals multireflected from the separation surface do not contribute much in the level of parasite radiation and can be neglected. Assume power scattering matrix [PS ] defined in some basis for the separation surface according to (2.4) and some vector characterizing the radiation polarization that coincides with one of the orths of the basis chosen. In this case the signal reflected by the terrain surface in the mirror direction is defined as: 

P11S P12S P21S P22S

    1 P11S . = 0 0

(2.4.25)

The signal scattered by the radar target in the direction of the path II (see Fig. 2.4) is being reflected from the same area of terrain surface but with modified directions of radiation-reception that correspond to transpositioned power SM. Therefore, the received signal has polarization structure defined by the following expression: 

P11S P21S P12S P22S



p1 p2



 =

 P11S p1 P21S p2 . P12S p1 P22S p2

(2.4.26)

The radiating wave here has two components in the polarization basis chosen, because, generally, some depolarization of radiating wave happens during its reflection from the target. Radar antenna fully (without suppression) receives reflected wave polarized along the first orth of basis chosen. Therefore, the first component will produce the noise determined by the P11S element of distributed target SM, and the second component will produce the noise determined by the sum P11S + P. Thus, we can conclude that in order to minimize the mirror channel influence on radar operation, we have to choose the polarization basis that would provide minimal values of P11S and P21S elements of distributed target SM.

44

2 Synthesis of Signal’s Polarization Selection System …

As known, the transition from a linear polarization basis to another scattering matrix is determined by the following matrix: 

 cos α cos cos α sin β − j sin α cos β [Q] = , − cos α cos cos α cos β

(2.4.27)

where ( α =(ar ctg r , r is ellipticity coefficient, β is orientation angle, and α ∈ [−π 4, π 4], β ∈ [0, 2π ]. Transformation of the initial polarization scattering matrix can be written as: [s] = [Q]−1 [s]lb [Q].

(2.4.28)

According to (2.4.28), we can choose the basis where sum of squares of elements P11S and P21S in the power scattering matrix for distributed target will be minimal. This work will be done in paragraph 2.6 during calculation of scattering polarization characteristics of large-scale impedance on randomly uneven surface. Graphic dependencies of α and β of these optimal bases from scattering angle η are given in Fig. 2.15. The calculated theoretical curves allow for conclusion on the type of the most preferable radiation polarizations that provide minimal squared values of the noise formed due to the multipath EMW propagation. Another method of minimizing passive noise is based on the combination of polarization selection with other known methods of noise tuning out. The proposed method of radar noise-immunity improvement is applicable for elimination of separation surface back-scattered signals. In contrast with useful radar signal, the noise signal is characterized by time and Doppler frequency separation. The noise signal which we want to tune out is constantly registered by antenna system unlike the signal from the target situated at some distance from radar station. Since reflection from the separation surface is locally concentrated, i.e., range of elevation angles for noise background is quite narrow, we can consider polarization characteristics of back-scattering to be constant within the range of observation angles. So, we have to readjust the polarization diagram of the antenna in such a way that would minimize noise signal in the absence of the target due to polarization selectivity of antenna system. For this purpose we have to tune the antenna system in operation with the zero signal polarization for distributed target. Since noise signal is a slowly varying function, we can separate it from the fast varying target signal which will be registered with the polarization efficiency defined in 2.5. The more is the difference between polarization conditions of the target signal and the noise signal, the more effectively will this system increase the signal-to-noise ratio. Thus, along with already developed selection methods (directional diagram narrowing and compensation of side radiation, taking into account time and frequency particularities of the noise and target signals), the problem of elimination of passive noise influence on the radar operation can be solved based on polarization selection,

2.4 Methods of Noise Influence Diminishing …

45

О

О

`=2-j1,62 О

=6 100

5

=4О

80

4

60

3

40

2

20

1

0

10

20

30

40

о

Fig. 2.15 Graphic charts of the optimal polarization bases angle values depending on the scattering angle

fulfilled by adjustment of polarization basis the radar is operating on, in order to achieve the following: First, the minimal values of non-diagonal elements of power SM of the target (P12t = P21t = 0), i.e., operation on the first natural polarization of the target; Second, the minimal values of P11S and P21S elements of power SM of separation surface in the mirror direction, which would allow for minimizing multipath EMW propagation; Third, the minimal value of the first element of power back-scattering matrix of separation surface using time or frequency particularities of the target signal and the noise (zero polarization of the noise background). The substantiation of the method chosen for polarization selection of the target signal in the background of passive noise is given in the book along with the design of the device for tuning out of the noise and technique for its engineering implementation.

46

2 Synthesis of Signal’s Polarization Selection System …

2.5 Analysis of Current Methods and Devices for Polarization Selection 2.5.1 Research Problem Formulation Nowadays, polarization selection methods are widely applied for prevention from influence of precipitation on the radar operation based on the change of polarization type, from linear to circular during the rain. Here we use the difference in reflecting properties of the target and hydrometeors which allows for decreasing of parasite signal level in the processing channel [16]. This kind of polarization is widely and fruitfully used, but other methods of tuning out of the noise have also been developed. For example, there is a polarization selection device based on the modifiable PD of the receiving antenna according to the noise polarization type. In this device antenna polarization type changes either randomly or according to computer program control. If for some kind of noise polarization the amplitude of the noise exceeds some threshold value, the computer program is modified in order to use this kind of polarization less frequently [12]. However, this way of selection is difficult to implement and, besides, continuous PD variation brings in additional amplitude modulation in the signal channel due to the change in antenna polarization efficiency. Simple target signal selection method in the background of reflections from the sea surface was proposed based on a priori information about polarization structure of the signal scattered by the sea surface for the relatively narrow range of elevation angles and wind speed. This method is based on the fast rotation of linearly polarized antenna around its axis with the following registering of received signals’ amplitudes in different moments of time (for the different antenna rotation angles). Serious limitations of this method are decrement of space scanning speed due to data accumulation during polarization separation and lack of a priori information on polarization parameters of the signal scattered by the separation surface for many cases of radar surveillance. Recently, some attempts were made to theoretically substantiate design of polarization selection adaptive systems based on optimal control of radar antenna polarization type. It was shown that such structure can be designed on the basis of system’s parameters controlling according to modified optimality criterion. However, design of these systems turned out to be very complicated. Besides, UHF signals processing methods do not allow avoiding signal transformation and its lowfrequency processing. Since every signal transformation necessitates additional losses, adaptive systems failed to find wide application, besides, due to their very complex processing systems. Analysis of literature source that was conducted showed the absence of noncomplicated polarization system that doesn’t operate in UHF range and require any transformations, although optimal in the sense of (2.4.1) criterion. In this book an attempt is made, based on considerations given in Sect. 2.4, to synthesize a quasioptimal system of polarization selection represented by UHF device capable to adjust

2.5 Analysis of Current Methods and Devices …

47

antenna polarization in such a way that antenna PD satisfies the condition of maximal value for ratio (2.4.1). As known, all methods of polarization parameters measurement are based on determined functional dependencies of voltage induced on the matched load of receiving antenna from polarization parameters of radiation analyzed. Majority of indirect methods of polarization parameters measurement are implemented in the devices that contain polarization analyzer with mechanical rotation. Polarimeter (Fig. 2.17a) consists of rotatable waveguide I connected on one end with unipolarization antenna and on the other end with polarization detector 2 through rotatable connector 3. Detector 2 is positioned for reception of vertically oriented field components. Figure 2.16 demonstrates the view of polarimeter in axonometric projection. In this device the following dependence of linearly polarized wave components’ amplitudes and phases from the basis axes rotation angle is used:  E 1 = a cos2 β + r 2 sin2 β  E 2 = a sin2 β + r 2 cos2 β 2r   = ϕ1 − ϕ2 = ar ctg  . 1 − r 2 sin2 β

(2.5.1)

Mechanical rotation of analyzer impacts system’s processing rate, therefore application of devices of this type is limited. Polarimeters that analyze two components of the signal decomposed in a given polarization basis became the most perspective. The functional diagram of this polarization analyzer is given in Fig. 2.17b. Waveguide polarization analyzer consists of wave splitter made in the form of rectangular waveguide II with the round hole or cruciform slot 2. Output of wave splitter is connected by waveguide Sect. 2.3 to the input of amplitude and phase measurer 4 (see Fig. 2.17b) assembled as T-joint equipped with detector 10, phase shifter 7, and attenuators 6 and 8. This device operation requires several sequential adjustments of attenuators and phase shifter which reduces system response speed and makes it inapplicable for real reflected target signals. The design of proposed polarization parameters measurer is free from the above-mentioned drawbacks while having all advantages of above described models. Mechanical rotation is avoided by using polarization rotator 12 (Fig. 2.18). According to (2.5.1), when large axis of polarization ellipse coincides with the x-axis of decomposition basis (β = 0), phase shift between two linearly polarized wave components is equal to 90°. This circumstance is used in processing of the signals formed by polarization splitter 2 and consisting of two orthogonally polarized linear components of radiation analyzed. Linearly polarized components of EMW enter to the slot bridge hybrid 5 that has the following scattering matrix:

48

2 Synthesis of Signal’s Polarization Selection System …

2 3 1

4

1

2

Antenna

Rotating Polarization connection detector

Control system

Generator

Mixer

Generator

Phase detector

Fig. 2.16 Polarimeter view in axonometric projection

Amplifier

Detector

Videoamplifier

2.5 Analysis of Current Methods and Devices …

49

4 6 3

2

7

5

3

8

9

10 а)

11

b)

Fig. 2.17 a Polarimeter; b operation diagram of polarization analyzer

E1

1

12

E3

2

10

11

10

11

5

E2

EY

Fig. 2.18 Polarization rotator



[S]sm

0 1 ⎢ 1 =√ ⎢ ⎣ − j 2 0

1 0 0 j

−j 0 0 1

⎤ 0 j⎥ ⎥. 1⎦

(2.5.2)

0

On its outputs we will have E 3 = √a2 E 1 − j √a2 E 2 e jφ , E 4 = − j √a2 E 1 + √a2 E 2 e jφ

(2.5.3)

where E 1 , E 2 are wave components defined in (2.5.1),  = π2 . Polarization rotator 12 with ideal performance characteristics will change orientation angle β of polarization ellipse only. For p = 0, (2.5.3) takes the form: E 3 = √a2 + √a2 r , E 4 = − j √a2 + j √a2 r

50

2 Synthesis of Signal’s Polarization Selection System …

where r = EEYX . Thus, on one output of the slot bridge hybrid we have sum signal, on the other the differential signal. The output extreme corresponds to β = 0. For circular polarization (r = 1) one output is empty and for reversal rotation direction outputs swap. There are no direct phase measurements and mechanical rotation in the proposed device. System’s processing rate is determined by speed of polarization plane rotation. Modern ferrite rotators are quite reliable at the operation frequencies about 10 kHz, which makes UHF polarimeter applicable for analysis of arbitrary radar signals. Original method of EMW polarization condition analyzing based on using multimode round waveguide was proposed in [11, 12], where separate energy output of different types of waves is fulfilled in order to determine EMW polarization characteristics. Analysis of the devices for field polarization characteristics measurements has shown that basic elements of automatic measurement systems are module of receiving antenna PD modification and module that commands the fixation of measured parameters. Since the role of antenna systems’ PD modification is quite important on all stages of application of radar signals polarization characteristics, let’s dwell on this issue.

2.5.2 Analysis of Methods for Antenna System Polarization Diagram Readjustment EMW of arbitrary polarization can be represented by the sum of two waves with polarizations oriented along the orths of a chosen decomposition basis. This statement stands as a foundation for the great majority of the methods for design of antenna with arbitrary polarization and readjustable polarization diagram (PD). The first appeared and still widely applied are devices for antenna PD modification that introduce some selective phase delays immediately in the wave front of the components of the signal radiated by antenna, not in its wave drive. Different polarization gratings may serve as an example of these devices. The requirements imposed on these devices include low energy loss of the wave transformed, reasonable bandwidth, parameter’s stability over time, producibility, and operational convenience. Dynamic systems are additionally required for high processing rate, wide dynamic range, modulation linearity, energy response acceptability etc. Polarizers based on polarization gratings don’t meet several, if not the majority, of above-mentioned requirements; therefore, they failed to find wide use in radio location, though they are successfully applied in radio astronomy as polarization selectors. The simplest method of antenna PD modification is change of phase relationships between the signals with orthogonal polarizations by dielectric plates. Rotation of the

2.5 Analysis of Current Methods and Devices …

51

plate around its axis creates polarization modulation of antenna PD. However, hardly these systems can provide arbitrary polarization, because the range of polarization modification is limited. Application of these systems is also confined by their low processing rate inherent to all mechanical systems, as well as by large signal strength loss during reflection from and absorption by dielectric plate. There are limitations on power capacity for the waveguide with dielectric loading. Existing models of similar devices allow for some parameters improvement but don’t solve the problem in general. Despite the drawbacks related to mechanical systems of antenna PD control, they found their place among analyzers of slowly varying signals. There is an antenna system which, in our opinion, has all advantageous features of mechanical devices for antenna PD modification. This system consists of fixed conical horn, rotatable waveguide section made in the form of round waveguide smoothly converting to rectangular waveguide, and fixed section of rectangular waveguide connected with transmitter (see Fig. 2.19). Sections’ rotation relative to each other modifies signal’s polarization from linear to circular, while simultaneous rotation of both sections corresponds to rotation of polarization plane. Here we use the difference between propagation constants for two orthogonal waves oriented along main ellipse axes. Similar principle is put in foundation of this device operation. The simplest device for arbitrary polarization of unipolarized antenna has the form of square section cone with two exciters on the orthogonal walls of feeding square waveguide. For proper operation of this device we have to find appropriate amplitude and phase dependencies for voltages that feed exciters. Technical implementation of the change of polarization plane orientation angle is realized by introducing phase mismatch 1 into signals of orthogonal circularly polarized components of radiating wave (see Fig. 2.20).

3

1

2

2

3

1 Motor

1-1

Motor

2-2

E

3-3

E

E

1

E E1

E2

E

E2

E1

Fig. 2.19 Mechanical system of antenna polarization diagram control

52

2 Synthesis of Signal’s Polarization Selection System …

Orthogonal channel

Input Parallel

Hybrid connection

Hybrid connection

Output

channel

Phase shifter

Fig. 2.20 Principal technical diagram of the device for the change of orientation angle of wave polarization plane

In the linear decomposition basis the ratio of the amplitudes of two orthogonally polarized signals is proportional to tg 22 , where 2 is phase shift between signals defined in (2.5.17). Phase shift 2 is introduced by phase shifter 2 (see Fig. 2.21). Design of an arbitrary polarization control system is based on the diagrams presented in Figs. 2.20 and 2.21 combined. The resulting device (see Fig. 2.22) contains two double T-type bridges as hybrid connections. Phase shifters that introduce phase shifts 1 modify phase relation between the currents inducing orthogonal linearly polarized components of the cone radiation wave, while phase shifters with phase shift 2 modify their amplitude relation. System’s operation is clearly demonstrated by vector diagram of signal forming in E- and H-ports of double bridge presented in Fig. 2.23. Application of non-mechanical phase shifters allows to gain the required processing rate of the system. So, independently modifying the values of phase shifts 1 and 2 , we can receive any given radiation polarization. System’s technical implementation is quite simple and allows not only independent radiation of a signal with arbitrary polarization but also simultaneous reception of a signal with another polarization, which is provided by arresters and isolators that are included. System’s response speed is limited by the changing speed of phase shifts 1 and 2 .

Orthogonal channel Input Parallel channel

А

Hybrid connection

Polarization

splitter

Phase shifter

Fig. 2.21 Phase shift produced by phase shifter

2.5 Analysis of Current Methods and Devices …

53

Detector II

А II

Р

V

PS

PS

Р

Р

V

I V

Р

PS

PS

V

Р

Р

Detector I

V Transmitter

Е2 Ф/2 Е1

Fig. 2.22 Control system for arbitrary polarization (PS—phase shifter, V—valve, P—phasor)

Е1

Phase shifter I

А

Е`3

Hybrid connection I

Polarization. splitter

Е2

Е3

Е`1

Е`2

Parallel channel

DC Hybrid connection

ЕY

Phase shifter II

Е`Y

II

DC

Orthogonal channel

/2 Phase discriminator

Servomotor

Servomotor

Phase dicriminator

Error signal

Error signal

Fig. 2.23 Functional diagram of device for automatic EMW polarization monitoring

Thus, existing methods of antenna PD adjustment allow for modification of radiation polarization in a wide range. The most perspective methods are based on introducing phase mismatch in the channels of orthogonally polarized components of the signal. The main limitation for system’s response speed is response delay of the device that controls the phase of UHF signal. Accuracy of required PD adjustment depends on the parameters of UHF elements involved in the system, as well as on the value of these elements’ mismatch.

54

2 Synthesis of Signal’s Polarization Selection System …

As was shown, polarization characteristics of the target are functions of time. Therefore we have to have the device with the adjustment of antenna PD that would provide permanent match with signal polarization. In other words, we need useful signal polarization tracking system. The functional diagram of the device for EMW polarization automatic tracking is given in Fig. 2.23. This device tunes on the polarization of the incoming signal automatically. This system’s antenna comprises a set of orthogonal vibrators or unipolarization antenna with polarization splitters that select linearly polarized components of incoming EMW. Hybrid connections are intended for forming two orthogonal channels. EMW polarization tracking device is designed according to the same functional diagram as presented in Fig. 2.21. The distinctive feature of this device is existence of the feedback that allows for tuning of antenna system polarization to the polarization of received signal. The feedback is realized through partial power take-off from the signal by directional coupler and forwarding it to the phase  shifters control. Phase shifters register drive signals proportional to the values j and  produced by phase discriminators. Servomotors track incoming error signals and correct phase shifts induced by phase shifters. By doing so the system stays in tune with incoming wave polarization. However, just like all systems with mechanical controlling elements, this system intrinsically owns all the limitations discussed above. Thus, creation of tunable systems is possible for systems with feedback only. Moreover, this feedback should operate on the frequency close to the signal frequency in order to keep up with quickly changing signal’s functions of time. However, such systems have not yet been developed. During analysis of above-mentioned systems we assumed that all involved elements have ideal performance characteristics. However, as experimental research proved, this is not always true. Main polarization measurements errors appear due to cascade antenna connection with polarization splitter which is usually an indispensable part of great majority of systems that exist. Analysis of cascade connection between two main parts of polarization selection system based on transfer function matrix gave the limit requirements for partially polarized signal that would keep its polarization structure without slipping. This condition boils down to unitarity of transfer functions for the device comprising unipolarization antenna and two-channel polarization splitter. The review conducted on polarization selection systems has shown that unwearying efforts are being undertaken for polarization selection systems design, but still there is no appropriate UHF system capable of solution to the selection problem. In this book we don’t consider systems for adaptive reception fulfilled by multichannel low-frequency signal processing as well as issues of antenna arrays application that often help to solve some selection problems, because these issues are of great separate interest. It is worth noting that design of an optimal polarization selection system for the real radar signal in general case is a very complicated understudied problem.

2.5 Analysis of Current Methods and Devices …

55

Therefore, it makes sense to discuss the design of selection system that is close to the ideal system’s signal-to-noise ratio, i.e., the so-called quasioptimal system for polarization selection. The next chapter is devoted to the set of problems of such system’s design.

2.6 Synthesis of Signal Selection System in the Background of Passive Noise Formed by Reflections from Separation Surface Based on all knowledge that we have on polarization selection methods and devices for their implementation, we are going to design a system that allows for technical realization for one of the proposed methods of polarization tuning out of passive noise formed by reflections from separation surface. As calculations fulfilled for mirror channel signal’s influence minimizing have shown (the case P11 = P21 = 0), we have to operate in a definite polarization basis that determines radar system’s polarization for emission/reception. This selection method is very much alike the recommendation to switch from linear polarization to circular one during precipitations. Our theoretical conclusions have recommendative nature on application of certain polarization during tracking of low-flying targets for narrow range of elevation angles. The first proposed method—tuning on the first natural polarization—has many unsolved issues when passive scatterers are present in the radar surveillance area, though providing significant benefit in the signal-to-noise ratio [8]. For proper operation of the system that provides tuning on the first natural polarization we need either to know a priori information about polarization structure of reflected target signal or be able to discern target signal from the noise. Since target signal is always mixed with the noise during target surveillance in the background of multiple reflections from separation surface, implementation of this method is complicated. However, the differences in Doppler frequency shift between the signals of target and noise allow for signal’s separation from the noise and antenna system’s adjustment on the first natural polarization of the target. The system capable of permanent registering target signal with the first natural polarization provides stable operation in the natural basis of the target. In this case object’s SM and radiation vector have the following form:    1 λ11 0 − → , pt = . [S]t = 0 λ12 0 

(2.6.1)

It is evident that coherence matrix for reflected signals in this basis will be equal to:

56

2 Synthesis of Signal’s Polarization Selection System …



 |λ11 |2 0 , [ρ1 ]t = 0 0

(2.6.2)

where power of received signal will be determined by the value |λ11 |2 . Since polarization of reflected noise generally doesn’t coincide with useful signal’s polarization, the noise SM doesn’t have zero elements:  [S] N =

S11 S12 S21 S22

 (2.6.3)

Coherence matrix in the target’s natural basis has the form: 

pi j



 N

=

 ∗ |S11 |2 S11 S12 . ∗ S11 S12 |S12 |2

(2.6.4)

Since signals reflected from underlying surface are partially polarized, matrix (2.6.4) can be modified as: 

pi j



 N

=

  2  τ2 0 k1 k2 e jβ k1 + . k1 k2 e− jβ k22 0 τ2

(2.6.5)

The first summand here determines unpolarized component, while the second summand the fully polarized component of the parasite signal. In accordance with (2.6.4) and (2.6.5) we can write ) ∗ |S11 |2 = τ 2 + k12 , S11 S12 = k1 k2 e jβ ; ∗ |S12 |2 = τ 2 + k22 , S11 S12 = k1 k2 e− jβ

.

(2.6.6)

Receiving antenna is tuned on the first natural polarization of the target; therefore, p r = [1, 0]T , and the noise power output on the matched load is equal to: WΠ = τ 2 + k12 .

(2.6.7)

The total intensity of reflection from the target is determined by the expression: P0 =

 1 1  E 1 (t)E 1∗ (t) + E 2 (t)E 2∗ (t) , E(t)E(t) = 2Z 0 2Z 0

(2.6.8)

where Z 0 is free space characteristic impedance. Considering (2.6.6), the total intensity of the noise will be equal to: PO N = |S11 |2 + |S12 |2 = 2τ 2 + k12 + k22 .

(2.6.9)

2.6 Synthesis of Signal Selection System in the Background …

57

Then signal-to-noise ratio on the matched load of the receiving antenna will take the form: 2τ 2 + k12 + k22 PO N = . WN τ 2 + k12

(2.6.10)

√ √ Here τ = (1 − m), m = 1 − 4( p11 p22 − p12 p21 ). It was shown that antenna system tuned on the first natural polarization of the target on its matched load has the following signal-to-noise ratio: |s12 |2 Pon K2 = 1 + 22 = 1 + . Wn K1 |s11 |2

(2.6.11)

Thus, for antenna system tuned on the first natural polarization of the target, the increasing degree of signal depolarization after its reflection from separation surface (increment of element |s12 |2 ) increases the noise suppression level in antenna system. Target radiation by the wave with polarization that minimizes values of P11N and P21N of separation surface SM (what leads to less influence of mirror reflection) corresponds to decrement of s12 element’s value in the SM of distributed target in backward direction, i.e., decreases the suppression level of backward scattering. Thus, we have to find a compromise solution to two problems: on one hand, radiation of the wave with the first natural polarization of the target, as a method of backscattering minimization, requires increment of ρ21N value; on the other hand, the proposed method of tuning out of mirror channel signal is based on minimization of this element’s value. Practical solution to this dilemma lies, in our opinion, in initial radiation of signals with polarization that is recommended for minimization of “antipode” phenomenon influence with the following adjustment on the first natural polarization of the target. The main system’s module that provides optimal polarizable reception of the useful signal is the module for antenna system PD adjustment. These modules, based on two-channel system with separate parameters’ regulation for each channel, provide all the required range of polarization and have the potential for system’s operational speed improvement. Radar antenna system should be tuned on radiation of signal with the first natural polarization of the target, because this signal doesn’t suffer depolarization and will be fully received by antenna, thus providing maximal value to the numerator of (2.4.1). However, due to permanent change of target aspect angle and target motion relative to radar, first natural polarization of the target also fluctuates. It makes necessary the automatic tracking after parameters of the first natural polarization of the target. It was shown that polarization transformation during reflection from the stable radar target is equivalent to some rotation of Poincare sphere (see Fig. 2.13). The axis and angles of rotation are determined by target’s polarization features and radiating wave polarization type. In particular case of unitary SM (what is true for the majority of

58

2 Synthesis of Signal’s Polarization Selection System …

radar objects) tg γ0 = 1, and sphere rotation will be the same for any polarization of radiating wave.   → −  − → − → T Assume − q ,→ q as natural polarization basis of the target, and t = t , t 1

2

i

1

2

as some radiating wave polarization represented in this basis. Reflected wave’s polarization condition can be defined as:  Es =

    E 1s E 1i λs1 0 = . 0 λs2 E 2s E 2i

(2.6.12)

Experimental research of real targets’ reflective properties has shown that moduli of reflection coefficients for signals with natural polarization significantly differ for real targets. Let us agree to signify the bigger coefficient as λs1 . Then matrix (2.6.4) can be written as:   jϕ     Ae 1 0 λs1 0 jϕ1 1 0 ( = = Ae , (2.6.13) 0 λs2 0 Be jϕ2 0 B Ae j (ϕ2 −ϕ1 ) ( where B A < 1, because λs1 is the bigger reflection coefficient. In accordance with (2.6.4) and (2.6.5), reflected wave is described by the following matrix-column:   E . (2.6.14) Ae jϕ1 B j1i(ϕ2 −ϕ1 ) e E 2i e jϕi A If we radiate the signal with polarization parameters defined in (2.6.6), then reflected signal will be defined as follows:  Ae jϕ1

 E 1i  B 2 j2(ϕ −ϕ ) jϕ , e 2 1 E 2i e i A

(2.6.15)

where ϕi = ϕ2i − ϕ1i . After n cycles of this operation, i.e., after multiple correction of radiated wave according to received wave polarization, antenna system ( will radiate a signal with the first natural polarization of the target, since lim (B A)n = 0. Polarization n→∞ error, i.e., the difference between the calculated polarization and the true first natural polarization will be insignificant due to finite number of cycles [14, 15]. Experimental data from several works have shown that values λs1 and λs2 for real targets are separated by 5–7 dB in average and required polarization is reached after just 2–3 adjustment cycles. We should adjust receiving antenna polarization in such a way that would exclude the long noise signal from the system processor unit.

2.6 Synthesis of Signal Selection System in the Background …

59

The simplest method (in the sense of its technical implementation) of polarized passive noise minimization is, in our opinion, selection method based on minimization of the first element of power back-scattering matrix for the separation surface by means of antenna PD adjustment on the polarization that corresponds to zero signal from the distributed target. As was shown, the underlying surface polarization parameters of scattering can be described by stationary and stationary-bound random process with zero mean value. Therefore, we may consider an assumption on the noise constant polarization for every given moment of time (that we entered during analysis of characteristics of signal reception from distributed target) to be valid. If polarization of the signal reflected from distributed target changes over time in a way that during some time T0 all Poincare sphere is covered, then for infinite period of time Stokes parameters will be equal to zero, i.e., the wave will be fully unpolarized. Thus, wave’s polarization degree depends on observation time. If the observation time is much less than T0 , for every moment we will have almost fully polarized signal scattered by underlying surface. For this case, as was shown above, optimal polarization of receiving antenna will be close to the polarization which is orthogonal to the noise polarization. Our task consists in the most appropriate (in the sense of (2.4.1) criterion) mapping of matrix (2.4.10) in polarization basis of radar antenna in the background of matrix (2.4.9). Thus, assumption of fully polarized noise signal and adopted approach on its slow fluctuations lead us to the design of polarization selection system, and the more accurately are these conditions accomplished, the closer will be our system to the optimal. We shall consider this system as quasioptimal. Experimental results confirm the correctness of the above-mentioned approach to the really observed processes of EMW scattering. This proposed subdivision method of tuning on the first natural polarization of the target can be successfully applied for implementation of noise influence minimiza( tion. Since ratio λs1 λs2 is quite significant (appr. 5–7 dB), this system will rapidly tune on the reception of the signal with the first natural polarization of distributed target. If we simultaneously tune the receiving antenna on the polarization which is orthogonal to the first natural polarization of the separation surface, the noise signal will not go into the input of processing module, and (2.4.1) ratio will have the maximal value. Let’s run the synthesis of the system that tunes on the natural polarization of the target for the most general case. Elliptically polarized harmonic wave can be presented in the linear orthogonal decomposition basis in the complex form as: − → − → − → E (t) = ( i x E x e jψx + i y E y e jψ y )e j (wt−kz) ,

(2.6.16)

− → where E x , E y are amplitudes; ψx , ψ y are phases of harmonic vector E projections on the axis OX and OY, accordingly. Hereinafter, we will omit the multiplicand exp( j (wt − kz)), assuming harmonic dependence of the wave over time.

60

2 Synthesis of Signal’s Polarization Selection System …

3

1 1

[S] = 2

S11 S12 S13 S14 S21 S22 S23 S24 S31 S32 S33 S34 S41 S42 S43 S44

2

3 4 4

Fig. 2.24 Device for polarization adjustment that forms elliptically polarized signal

For the wave (2.6.8) electromagnetic field of unit intensity (E x2 + E y2 = 1) expression (2.6.16) can be modified in the form: − → − → − → E (t) = ( i x cos ϕ + i y sin ϕe− j )e jψx ,

(2.6.17)

where  = ψ y − ψx ; cos ϕ = E x , sin ϕ = E y . As known, for  = π/2 expression (2.6.9) defines elliptically polarized wave with polarization diagram in the form of ellipse with ellipticity angle ϕ and tilt angle β = 0. Value ψx is the phase of this wave. Expression (2.6.9) for  = π/2 and ellipticity angle β = 0 on the complex plane takes the form: E(t) = (cos ϕ + i( j sin ϕe− jθ ))e jψ .

(2.6.18)

After registering a signal of unit amplitude on input 1, polarization adjustment device that forms elliptically polarized signal under the influence of particular factors must form on its outputs 2 and 4 the signals proportional to cos ϕ and j sin ϕe jθ accordingly (see Fig. 2.24). Therefore, the first column of this system’s SM must consist of the following elements: 0, cos ϕ, 0, j sin ϕe jθ . Assuming full matching of both channels in this system along with their ideal decoupling, we will have the following zero elements of SM [S] (see Fig. 2.24): S11 , S22 , S33 , S44 , as well as S13 , S31 , S24 , S42 . Idealized polarization adjustment module doesn’t have ohmic losses, i.e., its SM is unitary. SM unitary, in turn, provides correctness of the following relationships between the SM elements: n *

|Sik |2 = 1, k = 1, 2, . . . , n,

(2.6.19)

i=1 n * i=1

∗ Si p Siq = 0,

p, q = 1, 2, . . . , n.

(2.6.20)

2.6 Synthesis of Signal Selection System in the Background …

61

After registering a signal of unit amplitude on input 3 (see Fig. 2.24) of the module, on its output the signal must be formed that is polarized orthogonally to the wave (2.6.10). Orthogonality of elliptically polarized waves consists in 180° shift and additional phase delay between orthogonally polarized components of the wave and in the reversal value of the wave phase modulus. Thus, if for the signal of unit intensity on the input 1 we receive on the spatially orthogonal outputs 2 and 4 (see Fig. 2.24) the signal defined as vector-column E 1T = (cos ϕ + j sin ϕe− jθ ),

(2.6.21)

then for the unit signal on the input 3 we should receive the signal determined as: E 2T = ( j sin ϕe− jθ , cos ϕ),

(2.6.22)

   −1 E  E  ϕ for which  E x2y2  = cos =  E x1y1  and phase delay ϕ2 = 0−(−θ + π/2) = θ −π/2 sin ϕ differs from ϕ1 = θ + π/2 exactly by 180◦ . According to the above-mentioned considerations, the third row of the designed device’ SM should consist of the following elements: S31 = 0; S32 = j sin ϕe− jθ ; S33 = 0; S31 = cos ϕ. Let us agree to consider our device as mutual, then its SM will be symmetrical: ⎤ 0 cos ϕ 0 j sin ϕe jθ ⎥ ⎢ cos ϕ 0 j sin ϕe− jθ 0 ⎥. [S] = ⎢ − jθ ⎣ 0 cos ϕ ⎦ 0 j sin ϕe 0 cos ϕ 0 j sin ϕe jθ ⎡

(2.6.23)

Corresponding to (2.6.15) transmission matrix has the form: ⎡

− jθ

e − j sin ϕ

0

⎢ e jθ ⎢ − j sin 0 ϕ ⎢ [T ] = ⎢ jθ 0 ⎣ − j cos ϕe − jθ 0 − j cossinϕeϕ

− jθ

j cossinϕeϕ 0 0 − jθ

e j sin ϕ



0 − j cossinϕeϕ e jθ j sin ϕ



⎥ ⎥ ⎥ ⎥ ⎦

(2.6.24)

0

It is easy to see that matrix (2.6.16) is in fact a multiplication of two transmission matrices, one of which ⎡

0

⎢ − j ⎢ ϕ [T ] H0 = ⎢ sin ϕ ⎣ − j cos sin ϕ 0

ϕ − sinj ϕ j cos 0 sin ϕ ϕ 0 0 − j cos sin ϕ 1 0 0 j sin ϕ cos ϕ 1 − j sin ϕ j sin ϕ 0

⎤ ⎥ ⎥ ⎥ ⎦

(2.6.25)

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2 Synthesis of Signal’s Polarization Selection System …

represents the transmission matrix of directional coupler with variable division factor, while the other ⎡

0 e jθ 0 ⎢ 0 e− jθ 0 =⎢ ⎣ 0 0 e− jθ 0 0 0

[T ] P S

⎤ 0 0 ⎥ ⎥ 0 ⎦ e jθ

(2.6.26)

determines some phase-shifting octopole with the SM ⎡

[S] P S

0 0 e− jθ ⎢ 0 0 0 =⎢ ⎣ e jθ 0 0 0 e− jθ 0

⎤ 0 e jθ ⎥ ⎥. 0 ⎦ 0

(2.6.27)

Taking the common factor e jθ (2.6.19) out of the matrix, we receive ⎡

[S] P S

0 01 ⎢ 0 00 = e jθ ⎢ ⎣ e jψ 0 0 0 10

0



e jψ ⎥ ⎥, 0 ⎦ 0

(2.6.28)

where ψ = 2θ . This SM describes the device whose channels contain non-reciprocal phase shifters (1–3, 2–4), where phase shift ψ is present in the channel 1–3 during the reverse wave propagation and in the channel 2–4 during direct wave propagation. Thus, during emission we have to introduce phase shift ψ into the signal that appears in the fourth pole of directional coupler (2.6.17), while the signal phase relation for the second pole stays the same. During signal reception the reversal operation takes place. It’s easy to notice that phase-shifting octopole [S] P S (Fig. 2.25) by its operation is equivalent to operation of non-mutual phase shifter with phase shift ψ introduced into the 1 − 2 leg of octopole [S] P S . Scattering matrix of directional coupler (2.6.17) has the form: ⎡

[S] DC

⎤ 0 cos ϕ 0 j sin ϕ ⎢ cos ϕ 0 j sin ϕ 0 ⎥ ⎥. =⎢ ⎣ 0 j sin ϕ 0 cos ϕ ⎦ j sin ϕ 0 cos ϕ 0

(2.6.29)

Expanding goniometric functions according to Euler’s formulas, moving refer( ence planes in the direction from the device on the ecsmetric length equal to −π 4 and taking out the common factor e− jϕ , we receive:

2.6 Synthesis of Signal Selection System in the Background …

63

Fig. 2.25 Phase-shifting octopole

⎤ 0 1 + e− j x 0 1 − e− j x − jx ⎥ e 0 1 − e− j x 0 ⎥ ⎢1 + e = −j − jx − j x ⎦. ⎣ 0 1+e 0 1−e 2 1 − e− j x 0 1 + e− j x 0 ⎡

[S] DC

− jϕ ⎢

(2.6.30)

It’s easy to show that SM (2.6.22) corresponds to conductivity matrix [Y] of the device. Matrix (2.6.22) is the sum of two matrices Y1 and Y2 : ⎡

0 ⎢1 Y1 = j ⎢ ⎣0 1

1 0 1 0

0 1 0 1

⎡ ⎤ 0 1 1 ⎢ 0⎥ ⎥; Y2 = je− j x ⎢ 1 0 ⎣ 0 −1 1⎦ −1 0 0

0 −1 0 1

⎤ −1 0 ⎥ ⎥. 1 ⎦

(2.6.31)

0

Using synthesis methodology, we will receive the synthesis diagram of [Y1 ] device presented in the Fig. 2.26. Synthesis diagram of [Y2 ] device (2.6.23) is presented in the Fig. 2.27. Multiplication of matrix [Y2 ] by exponential multiplicand corresponds to cascade connection of the device from Fig. 2.26 with the phase shifter having the following transmission matrix ⎤ 0 0 e− jψ 0 ⎢ 0 e− jψ 0 0 ⎥ ⎥ [T ] = ⎢ ⎣ 0 0 e− jψ 0 ⎦ 0 0 0 e− jψ ⎡

(2.6.32)

that determines octopole with non-reciprocal phase shifters in the ports 1–2 and 3–4 (see Fig. 2.27). Thus, matrix (2.6.21) can be technically realized by parallel connection of (2.6.23) devices. Considering that for design of antenna polarization adjustment device (2.6.15) we have to add to one of the ports a non-reciprocal phase shifter with the

64

2 Synthesis of Signal’s Polarization Selection System …

/4 YO

42

YOA=2Y0

YOA

/4

YO

32

12

YO

YOA

YOA

/4

22

YO

11

YO

/4 Fig. 2.26 Diagram of device [Y1 ] synthesis

3 /4 YO

41

YOA=2Y0

YOA

/4

YO

31

YOA

YOA

21

/4

YO

3 /4 Fig. 2.27 Diagram of device [Y2 ] synthesis

shift θ, the final synthesis diagram of radar antenna polarization adjustment device is given in Fig. 2.28. Technical realization of this diagram in coaxial or strip lines doesn’t cause any difficulties. In Fig. 2.29 you can see a realization of the device in a strip line, whereby connective ports have normalized characteristic conductivity equal to two; from this the width of strip waveguide is chosen. Waveguide arrangement of synthesized

2.6 Synthesis of Signal Selection System in the Background …

65

Fig. 2.28 Diagram of synthesis of the device for radar antenna system polarization readjustment

Fig. 2.29 Strip line accomplishment of the device synthesized

66

2 Synthesis of Signal’s Polarization Selection System …

Fig. 2.30 Functional diagram of the system based on the synthesized antenna polarization readjustment system and time separation of target/passive noise signals

device is similar to cascade coupling of two hybrid connections (e.g., double T-type waveguide bridges) connected to the matrix (2.6.19) by the octopole. Synthesized device for antenna system PD adjustment can be used as a foundation for useful signal selection system in the background of passive noise. Since signal of additive noise induced by reflections from the distributed target (separation surface) has some specific features in time of arrival and Doppler frequency shift comparing to the target signal, we can, using, for example, difference in arrival time for the target signal and noise, create a polarization selection system that would suppress additive noise and increment the (2.4.1) ratio. Operation of this system, whose functional diagram is given in Fig. 2.30, is based on the synthesized antenna polarization adjustment system and on time separation of target signal and the noise [19]. First, the emitted radar signal goes through the system without its polarization structure distortion, i.e., emitted signal has linear polarization. Signal reflected from the underlying surface at the moment t1 reaches the antenna (see Fig. 2.31). At this moment feedback circuits are activated by the control signal from the modulator. Feedback circuits are formed by reflected signal’s take-off in directional couplers, its detecting and selection of differential signal which through the delay line goes to the control of phase shift value in variable phase shifter 11. Phase shifter forms the shift between the signals of orthogonally polarized components of the received EMW that are selected by polarization splitter. In this case, in the sum port of double waveguide T-junction we receive doubled voltage of the noise signal, while in the difference port this signal is absent. Adjustment of zero signal in the difference port of T1 junction is possible due to feedback circuit. In the moment t 2 the system is ready to register the target signal in the sense that system “remembers” its polarization condition that minimizes influence of polarized signal scattered by the distributed target which is no more admitted to the processing channel connected with the difference port of T1 -junction. Since polarization state

2.6 Synthesis of Signal Selection System in the Background …

67

U мод

t0 t1

0

t5 t6

t

R кл

Rmax

0

t1

t2

t1

t2

t6

t

Uпр

0

t3

t4 t5 t6

t

Рис. 1.4.8

Fig. 2.31 Epures of signal reflected from underlying terrain

of useful target signal differs from the noise polarization, the signal proportional to the difference of values of target polarization components presented in natural noise basis enters to the difference arm of the t-junction. At the moment t 2 , the feeder locks, and all signal difference goes into radar channel for further processing. As we noticed earlier, polarization characteristics of the signal scattered by the distributed target cannot suffer significant change during the observation time t3 − t4 ; therefore, noise signal is almost absent in the difference arm of the T1 -junction, while useful radar signal (though a little bit faded due to polarization selectivity of elliptically polarized radar antenna) goes to the reception channel for processing. Thus, the system presented in Fig. 2.30 is capable of tuning the useful signal off the noise formed by the back-scattering from distributed target due to difference in polarization structure of these signals and their arrival time separation. A similar system can be designed on the basis of the difference in Doppler frequencies shifts for the target signal and noise signal reflected from the separation surface [21]. However, this system would be more complicated. Time period t1 − t2 intended for system tuning on signal polarization reception, on one hand, is limited by system’s response rate of readjustment and, on the other hand, should be chosen based on minimal radar shadow area.

68

2 Synthesis of Signal’s Polarization Selection System …

Fig. 2.32 Inclusion of additional phase shifter with 180º phase shift

The proposed selection system (see Fig. 2.30) fulfills UHF processing of signal polarization, incurring low loss level and insignificant distortion of received signal’s polarization structure. Similar system can be used for radar system adjustment for reception and emission of signal with the first natural polarization of the target provided that parasite signal is absent or a system discerning useful signal from the noise is present. In this case, as was shown, we have increment of useful signal energy in reception channel by 5–6 dB in average. Since target’s reflection coefficient for the second natural polarization is lower than the first one, we can achieve additional noise decrement by tuning the system (see Fig. 2.30) on the operation with the second natural polarization of separation surface. We can reach this goal by introducing additional phase shifter with the shift 180º, as it is shown in Fig. 2.32. The main obstacle on the way of polarization selection systems’ wide application is the absence of rapid-action phase control system for UHF signal; therefore the problem of UHF signal phase adjustment is very important. Existing systems for UHF signal phase delaying are based on introducing discontinuities to the wave propagation path with the purpose to modify UHF signal’s propagation conditions. These can be dielectrical or ferromagnetic plates, p-i-h-diodes, ferrite magnetized rods, ordinary waveguide narrowing, etc. However, all these hardware components introduce significant distortions resulting in power loss, throughput mismatching, significant inertia (in mechanical systems), and insufficient range of phase modification. There is an effect of UHF signal’s phase change based on the interaction of the ordinary wave with the higher-order wave in the multiwave waveguide during ordinary wave transformation into higher-order wave on the irregularities of the real waveguide. We can consider different waves in multiwave waveguide as power lines situated in one space. In ideal waveguide these lines are independent, but for real waveguides this assumption is incorrect.

2.6 Synthesis of Signal Selection System in the Background …

69

The following factors determine the degree of working wave transformation into parasite wave [17]: (1) (2)

association between the waves (linear interdependence), degeneration of the waves, i.e., equality of their phase speeds. Degenerated waves can fully exchange their energy, even for small association.

The working wave activated at the path inlet propagates undistorted up to some waveguide cross section Z = P1 (to be more exact, up to the section (p1 , p1 + dp)) where it partially transforms into different parasite waves. From the section (p1 , p1 + dp) to some next section (p2 , p2 + dp) appeared parasite waves propagate inalterably along with the main wave. In the section (p2 , p2 + dp) the reverse parasite waves’ transformation happens into main working wave which propagates further unchanged. After these transformations, due to working wave’s interactions with higher-order waves, the former wave gains some phase shift: ϕ1 (z) = p1 z +

N *

ϕ1 j (z),

(2.6.33)

j=z

where z  t ϕ1 j (z) = − 0

C j (t)C j (s)e− j j1 (t−s) sin β j1 (t − s)dsdt

(2.6.34)

0

is the working wave phase distortion formed due to relation of working wave with jth parasite wave. ϕ1 j (t) value can be both positive or negative. Expression (2.6.33) contains association coefficient between two waves C j (a). The general expression for this coefficient has the form: C jl =

1 2k(β j − βl )



 dl  l j E n E n − Hzl Hzj − Hsl Hsj ds, dz

(2.6.35)

where integral should be taken along the cross-section outline. E n , Hs are projections of electrical to the normal n and magnetic vector and to the tangent s to metal surface, respectively; normal n is directed toward the metal surface, orth s is directed along the metal surface so that vectors n , s , and z form right-handed coordinate system. Expression (2.6.35) indicates that in multiwave waveguide waves’ association coefficient depends on amplitudes of electrical and magnetic components of parasite wave. Wave’s phase characteristics influence the wave’s projections on the orths n

and s , changing hereby the value of phase mismatch. This effect can be used for design of device for UHF signal’s phase control based on higher-order waves interaction in the waveguide. For this purpose we have to change parasite wave’s amplitude or phase characteristics.

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2 Synthesis of Signal’s Polarization Selection System …

Fig. 2.33 Effect of phase distortion value control for UHF signal in UHF phase shifter

The proposed UHF phase shifter is presented in Fig. 2.33. Here multiwave waveguide Sect. 2.1 is feeded by working wave emitter 2. By exciter 3 we generate in the section control signal of the same frequency but for parasite wave type. Changing control signal’s amplitude and phase, we effectively govern the value of UHF signal phase distortion determined by coefficient of higher-order waves’ association in the waveguide. Figure 2.34 presents one of possible technical realizations of this device, where control signal enters the multiwave guide 1 from diffraction transformer 3. The proposed device allows for solution to the problem of UHF rapid-action polarization selection system’s design, because we receive an opportunity to adjust antenna system PD by means of phase control in radiated signal in accordance with the phase condition of received signal, simultaneously excluding wave transformation loss and reaching the necessary response time. Phase shifter does not introduce any limitations on power throughput, since it doesn’t contain any foreign objects. Thus, the main requirements to polarization selection systems are as follows: – quick response antenna PD adjustment system; – achievement of maximal independence between orthogonally polarized channels and minimal mismatch between its UHF hardware components; 1 2 ТЕ11

ТЕ11,

3

4

Fig. 2.34 Design of UHF phase shifter

5

6

2.6 Synthesis of Signal Selection System in the Background …

71

– loss minimization along with sufficient bandwidth; – stability of parameters, technological effectiveness, and operational convenience; – minimization of radiation polarization adjustment errors in radar antenna due to cross-polarization phenomenon in aperture antennas.

References 1. Golovanov DV, Pozdyshev VY, Sarychev VA (2000) Resolution of polarization-modulated signals by range and velocity. Transp Probl 4:291–297 2. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Lukashov KG, Mironov OS, Panfilov PS, Parusov VA, Raisky VL, Sarychev VA (2016) Experimental research of ultrashort-pulse radar system’s characteristics. In collected volume “Radiophysical methods in environmental remote sensing”. In: VII Russia-wide scientific conference proceedings. Murom institute (subsidiary) of Vladimir State university n.a. Stoletov A.G and Stoletov N.G., pp 196–202 3. Balashov VM, Drachev AN, Michurin SV (2019) Methodology of reflectors’ control for mirror antennas. In: Metrological support for innovation technologies. International forum: Abstracts, pp 44–46 4. Akinshin NS, Fomichev MY, Tsybin SM (2017) Equipment package methods of measurements prompt analysis based on application of random sequencies generator. Electron Inf Syst 4(15):47–56 5. Akinshin NS, Bystrov RP, Menshikov VL (2017) Mathematical models for directional patterns and radar cross sections of antenna systems in ground based radars. Antennas 8(240):53–66 6. Akinshin NS, Varenitsa YI, Khomyakov KA (2016) Joint assessment of coordinate and polarization parameters of radar objects; TulaSU Review, “Technical sciences” series, issue 2. TulaSU, Tula, pp 3–14 7. Akinshin NS, Rumyantsev VL (2014) Evaluation of polarization selection efficiency for ground based targets. TulaSU Review, issue 12, Part 2, “Technical sciences” series. TulaSU, Tula, pp 173–182 8. Balashov VM, Drachev AN, Smirnov AO (2019) Methods of coordinate measurements during geometrically-complex surfaces control. In: Metrological support for innovation technologies. International forum: Abstracts, pp 41–43 9. Alyoshkin AP, Artyushkin AB, Balashov VM (2015) Optimization procedures application for enhancement of two-beam over-the-horizon radio location method efficiency. Radio Ind 2:23–30 10. Andreev PG, Yurkov NK, Grishko AK, Kochegarov II, Zhumabayeva AS (2019) Research on the influence of dielectric material on UHF-signal propagation at elevated temperatures. In the bulletin “Wave electronics and information communication systems”. In: Proceedings of XXII-nd international scientific conference: in two volumes. Saint-Petersburg, pp 6–11 11. Golovachev MV, Kochetov AV, Mironov OS, Panfilov PS, Sarychev VA, Khomyakov IM (2014) UHF-band ultra-short-pulse radar system. In collected volume “Radiophysical methods in environmental remote sensing”. RAS Scientific Council on radio waves propagation, pp 255–260 12. Grishko AK, Nefedyev DI, Yurkov NK (2019) Design decisions structural optimization in conditions of multicriteriality and uncertainty. In: Proceedings of the international symposium “reliability and quality”, vol 2, pp 319–322 13. Zatuchny DA, Kozlov AI, Trushin AV (2018) Discernment of the objects situated within irradiated area. J Inf Commun 5:12–21 14. Ivanov YV, Sinitsyn VA, Smirnov VV, Strakhov SA (2017) Radar signals’ adaptive processing in the background of noise reflections based on method of maximal entropy. In collected

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15.

16.

17. 18.

19.

20. 21. 22.

23.

2 Synthesis of Signal’s Polarization Selection System … volume “Innovation technologies and technical facilities of special purpose”. In: XI-th Russiawide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2. Ministry of science and education of Russian Federation, Baltic State Technical University “Voenmech” n.a. Ustinov D.F., pp 330–333 Ivanov YV, Petukhov SG, Sinitsyn VA (2017) Application of radar landing system for aircraft automatic landing support. In collected volume “Innovation technologies and technical facilities of special purpose”. IX-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol. 2. Ministry of science and education of Russian Federation, Baltic State Technical University “Voenmech” n.a. Ustinov D.F., pp 326–330 Yershov GA, Zavyalov VA, Sinitsyn VA (2019) Improvement of radar observability parameters for aircrafts by methods of multi-position radio location. In collected volume “Innovation technologies and technical facilities of special purpose”. In: XI-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2, pp 21–24 Kozlov AI, Logvin AI, Sarychev VA (2007) Radar polarimetry. Radar signals polarization structure. Radiotechnics 640p Kozlov AI, Maslov VY (2018) Solution to inverse scattering problem and object’s shape reconstruction based on reflected electromagnetic wave’s field structure. Sci Bull Moscow State Tech Univ Civ Aviat 21(3):160–168 Myasnikov SA, Sinitsyn VA (2019) Design features of new landing radar. In collected volume “Innovation technologies and special purpose technical facilities”. In: XI-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2, pp 79–84 Kozlov AI, Amninov EV, Varenitsa YI, Rumyantsev VL (2016) Polarimetric algorithms of radar target detection on the active noise background. Tula State Univ Bull Tech Sci 12–1:179–187 Sinitsyn VA, Sinitsyn EA, Strakhov SY, Matveev SA (2016) Methods of signals forming and processing in primary radar stations. St. Petersburg Myasnikov SA, Sinitsyn VA (2019) Design features of adaptive two-components’ noise rejection filter in primary coherent-pulse radar systems. In collected volume “Innovation technologies and technical facilities of special purpose. In: IX-th Russia-wide research to practice conference proceedings. “Library of “Voenmech. BSTU Annals” series, vol 2, pp 76–79 Rassadin AE (2010) R-function and atomic functions tools as mathematical foundation for design of aerial radar system with synthesized aperture. In: Pupkov KA (ed) Intellectual systems: IX international symposium proceedings. RUSAKI, Moscow, pp 224–2288

Chapter 3

Primary Ways of Technical Implementation of Developed Selection System and Methods of Device’s Error Minimizing

The principal cause of polarization parameters measurement errors in polarization selection system is non-ideality of antenna feeder system and UHF guides characteristics in the radar. Leaving aside consideration of audio-frequency section of useful signal processing system and useful signal’s influence on measurements, we will assess the influence of antenna system and UHF path on the selection of device operation. For this purpose we will consider influence of cross-polarization mismatch between mirror antenna system and main module of antenna polarization adjustment device, as well as influence of irregularities in the values of bridges’ output ratios, electrical lengths of joining waveguides sections, and coefficients of real loads reflection on the adjustment accuracy of the required polarization [1]. From the result of this analysis we will formulate requirements to the characteristics and parameters of designed system’s real elements. Polarization structure of mirror antenna systems’ radiation is influenced by the following factors: • • • • • • • •

polarization structure of radiator’s field in the far zone; radiator’s inclination relative to focal axis in non-symmetrical antenna; reflector depth; antenna directivity (D/λ); radiator near zone; local currents interaction on the mirror due to finite reflector curvature; edge waves from reflector border; radiation into back semisphere.

During application of long-focus (Z /D > 0.25) and narrow-beam (D/λ >> 1) antennas we can neglect the last four factors and use aperture method for analysis. For radar antennas with main lobe to side lobe ratio 20–25 dB, the first condition is satisfied, while the second condition is satisfied automatically. The unit vector of the main linear polarization can be represented in spherical coordinates as © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 N. K. Yurkov et al., Signal Polarization Selection for Aircraft Radar Control, Springer Aerospace Technology, https://doi.org/10.1007/978-981-33-4964-3_3

73

74

3 Primary Ways of Technical Implementation …

x

x`

i

r i

r

F

y

z

y`

Fig. 3.1 Fundamental linear polarization

− → − → − → − → → r + cos θ sin i θ + cos ϕ i ϕ , i OC H = i x = sin θ sin −

(3.1)

and cross-polarization field as → − → − → −−→ − → r + cos θ cos ϕ i θ − sin i ϕ ). i OC H = i x = sin θ cos −

(3.2)

The notations adopted are clarified by Fig. 3.1. The main reason of cross-polarized radiation is the presence of y th component of mirror surface current, whereby the x th component causes cross-polarization 52 dB lower, and the z th component to be 23 dB lower than the y th component for the angle θ < 4◦ [2–4]. As will be shown in this chapter, cross-polarization is the main reason for many measurement errors; therefore, this issue is of great concern nowadays. The picture clarifies the correct coordinate system choice in order to provide one-to-one correspondence between radar efficiency indicators. As we have already noticed, non-ideality of UHF devices’ parameters is also a source of polarization adjustment errors in radar systems. We will analyze this phenomenon using matrix methods. Advisability of matrix tools application for theory of waveguides is conditioned not only by matrices exceptional convenience (despite their seeming complexity) but also by the fact that polarization parameters

3 Primary Ways of Technical Implementation …

75

of the analyzed radiation can be expressed in terms of scattering matrices, which will allow thereafter for these results application in analysis of device’s parameters in general. Due to narrowness of reflected signal spectrum, hereinafter we will not consider any frequency dependence. It allows for application of idealized scattering matrices.

3.1 Estimation of Influence of Cross-Polarization Degree in Antenna System Radiation on the Accuracy of Radar Measurements. Methods of Its Diminishing As known, mirror antenna radiation field has some cross-polarization degree even in the direction of the main lobe of directional pattern. Theoretical substantiation of this fact can be found in the works of B.E. Kinber and other authors. To avoid misunderstandings due to different interpretations of terminology, we will give the definition of cross-polarization. Consider Fig. 3.2 that presents spherical wave front with the components of the main and cross-radiation in linear, circular, and elliptical bases. Thus, in orthogonal coordinates one unit vector determines the direction of main polarization, and orthogonal to it is the direction of cross-polarization. In spherical coordinates orthogonal I

II Z

X

X

Z

III Z

Y

X

Y

X

Z

Z

Y

X

Y

X

Z

Y

Y

Fig. 3.2 Spherical wave front with the components of fundamental and cross-radiation in linear, circular, and elliptic bases

76

3 Primary Ways of Technical Implementation …

Fig. 3.3 Chart of amplitude-phase distribution of perturbated field in the aperture, normalized to dispersion, depending on error correlation radius

0,8 0,6 0,4 0,2

0

1

2

cm

components of radiation are determined by the vectors tangent to the surface of the sphere with unit radius. The same definition is applicable for the most general case shown in Fig. 3.2. So, by the cross-component of an arbitrary antenna radiation we will mean the component is caused by appearance of field distribution asymmetry on the aperture. This component is not registered by antenna system. Mirror antenna’s cross-polarized lobe is situated by its maximum in the first minimum of directional pattern (DP), and its level is comparable with the level of the DP first side lobe on the main polarization. This statement is correct for the plane which contains the cross-polarized field intensity maximum. As a consequence of parasite polarization, the accuracy of angle measuring radars worsens, because radiation is received from the wider space angle than it would be in the case of uniform PD [5, 6]. Antenna systems’ cross-polarization errors also lead to additional decrement of the average directive gain. Figure 3.3 presents the chart of amplitude-phase distribution (normalized to dispersion) of perturbed field in the aperture depending on error correlation radius:  = G2 −

G 2 I 2 (Cm , 0, 0) , 8

(3.1.1)

where G a2 = G 2Φ = G 2 ; Cm is the spatial correlation radius in relative units. The dashed line illustrates decrement of antenna-directive gain on the main polarization. The higher the antenna radiation level on cross-polarization, the more is decrement in antenna directive gain. For quite large spatial correlation radii this decrement can reach significant value [7]. One of the possible error sources in positioning of the target with unknown radiation polarization by the method of instant amplitude

3.1 Estimation of Influence of Cross-Polarization Degree …

77

comparison of signals is as follows. In general case, antenna system directionalfinding characteristics depend on ellipticity coefficient, polarization ellipse slope, and antenna beam width which are all functions of direction, and also depend on received wave’s polarization characteristics [8]. Cross-polarization distorts antenna system DP. As known, in mono-pulse system, the sum and the difference in directional patterns of the antenna take the form as shown in Fig. 3.4a. The corresponding directional patterns on the cross-polarization are presented in Fig. 3.4b. These figure curves obviously demonstrate that the phenomenon of crosspolarization in antenna system radiation significantly worsens systems’ angle measurement accuracy by introducing additional errors. Enhancement of information capacity and productivity of artificial satellites communication channels becomes very important nowadays. One of the most perspective methods for the frequency band utilization efficiency improvement is forming of two orthogonally polarized beams by one reflector. Efficiency of such two-beam reception depends on achievement of required level in polarization isolation [9, 10]. Isolation value is determined as difference of power registered on the Xth and Yth radiator’s outputs during reception of the signal with a given polarization:  2  2     Icp = 10 log  E xˆ  − 10 log E yˆ  .

(3.1.2)

An original system is proposed for two-beam satellite communication systems that allows for increased isolation between orthogonal channels. This device is used for polarization correction on the reception end of the system that transmits signals on two orthogonal polarizations. An additional pilot signal is transmitted on one polarization. On the other hand, polarization of this signal is detected as unwanted and is transferred to polarization shifter that corrects polarization to minimize the error. Pilot signal serves as an indicator of direction and amplitude of parasite signal appearance in the orthogonally polarized channel. When radar station operates on the ground with fully polarized target signal, this signal is mixed with unpolarized noise signal of high intensity. In the crosspolarization lobe of antenna system DP this noise signal acquires polarization and penetrates into processing channel. This phenomenon impacts measurement accuracy, leading sometimes to the total information loss. Many original works are devoted to the problem of cross-polarization suppression in mirror antennas radiation. Methods of parasite polarization suppression by means of special compensation antennas, polarization filters, and sloped radiators were described in several works. The principle of cross-polarization suppression by compensation antennas in operational angular range is similar to side lobes suppression. In some special cases application of filtering wire grids installed into antenna aperture turned to be an effective tool for cross-polarization suppression. Method of parasite polarization rejection by the lattice of curved wires with its experimental

78

3 Primary Ways of Technical Implementation … E

a

I

-

1

+

0

1

E II

-

1

0

+

1

+

1

E

-

1

0

b Generator of pilot-signal

Generator of A-polarity signal

Phase shifter

Rec

Detector of C-polarized signal

Rec Detector of B-polarized signal

Detector Detector of D-polarized signall

Fig. 3.4 a Sum and tracking patterns for mono-impulse system. b Corresponding patterns for cross-polarization

proof was developed for the large radio telescope in Pulkovo. However, this method is very complicated and practically applicable for particular antenna systems only. The sum and difference of DP correction method exists based on phase-sensitive device situated between the radiator and reflector of mirror antenna. Phase-sensitive device operation depends on polarization type and impacts the parasite polarization,

3.1 Estimation of Influence of Cross-Polarization Degree …

79

leaving unchanged the wave with working polarization. Such device can be realized in the form of dielectrical grids situated in two quadrants of the mirror. Many recent works comprise all possible designs of radiators with complementary elements that distort amplitude-phase distribution of electromagnetic field in the mirror antenna aperture or in the immediate vicinity of the mirror. However, all these devices are not theoretically substantiated, and all achieved results were mainly based on meticulous, often intuitive, experimental efforts of research workers. All the above-mentioned methods allow for cross-polarization rejection by 8– 10 dB in average, but the problem of uniform PD design for mirror antenna has not yet been solved for general case. With the help of aperture method Jones E.M. showed that if we use elementary magnetic radiator as antenna feed, an appropriate combination of electric and magnetic dipoles can completely remove cross-polarization. With that in mind, it can be shown that for cone radiator whose initial field is determined by equivalent electrical and magnetic currents in the aperture, the phenomenon of cross-polarization will be faded in comparison with dipole feed. To push down the cross-polarization level, Cassegrain antenna is proposed with its hyperbolic counter reflector. In the majority of works devoted to this problem, main attention was paid to design of radiator with circularly symmetrical directional pattern (the so-called scalar radiators with huge dimensions have similar DP). However, radiator’s directional pattern equality in the main planes does not fully eliminate cross-polarized radiation, because antenna system’s scattering properties were not taken into account. Since two antenna systems with identical DPs can differ by their influence on the falling wave, causing difference in the scattered power, the neglection by scattering phenomenon does not seem to be justifiable. It is worth noticing that in some cases it is more preferable to achieve the desired combined effect from two or more negative phenomena instead of struggle with each phenomenon separately. This way was chosen, for example, in optics for struggle with aberrations. Nowadays, several works have appeared which had intuitively applied this principle. For example, it is proposed to overcome the negative phenomenon of DP asymmetry in circularly polarized parabolic antenna with offset mono-pulse radiator (that causes cross-polarization effect) by exciting an additional difference DP which is equivalent to exciting cross-polarized signal that balances negative effect of initial cross-polarization [11–13]. The mirror antenna is needed here for radiation. In this book we synthesize antenna system with uniform PD where the radiating field with parasite polarization is induced in order to eliminate cross-polarization. We define the conditions that provide the absence of cross-polarization in the resulting field radiated by this antenna. Thus, we substantiate an opportunity of antenna radiation cross-polarization elimination by means of exciting parasite polarization of opposite direction.

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3 Primary Ways of Technical Implementation …

3.2 Mirror Antenna Scattering Matrix In the background of noise, target signal’s polarization selection efficiency can be seriously impacted by the presence of parasite matching between two orthogonally polarized antenna operation channels due to cross-polarization phenomenon. Therefore, uniformity of antenna PD may serve as a criterion of its optimality [14]. Nowadays, cross-polarization phenomena in mirror antennas are given pride of place. However, traditional description of antenna features through directional pattern and polarization characteristics cannot be complete, since antennas with similar amplitude and phase characteristics may differ by the mode and degree of their influence on the incident wave, i.e., by their scattering properties. Therefore, analysis of scattering properties of antenna system allows generalizing the conclusions that were made. In modern scientific literature there are two general approaches in the analysis of antenna-feeder device scattering properties [15–18] based on representation of antenna’s DP as a combination of partial diagrams and on representation of antenna system by equivalent waveguide multipole where every channel operates on its own polarization. First attempts of scattering matrix (SM) application to antenna system analysis were undertaken by American researchers in the middle of twentieth century. The principle idea was in antenna radiation field expanding by spherical harmonics with the consequence that resulting SM had infinite order what made it uncomfortable, if not useless, for practical application. In this book SM is introduced on the basis of finite number of antenna system’s directional patterns defined for a given design configuration. Based on the result, under the assumption of absence of ohmic losses in the system, we can calculate all SM elements andthen, fulfilling the task of  → → e2 the strict theoretantenna system optimal design, apply in an arbitrary basis − e1 , − ical methods that link amplitudes E1 and E2 of orthogonal field’s components with that correspond to electrical field similar amplitudes E 1T and E 2T in the waveguide  → → e2 T : intensity components in the basis − e1 T , − 

E 1T E 2T



     E1 j11 j12 E1 ˙ = [J] = . E2 j21 j22 E2

(3.2.1)

Thus, matrix [ j] takes into account not only field distortion by antenna system but also components modification caused by transformation of the basis. Here j11 and j22 are transmission factors of orthogonal components of analyzed radiation in corresponding channel. In principle, any antenna with matrix [ j] of (3.2.1) type can be used for polarization measurements, provided that reciprocal matrix [ j]−1 exists. In this case polarization of analyzed radiation will be determined as: 

E1 E2



 = [J ]

−1

E 1T E 2T

 .

(3.2.2)

3.2 Mirror Antenna Scattering Matrix

81

It means that criterion of antenna’s worthiness for polarization measurements is unequality of antenna’s Jones matrix to zero. The value of det [J ] determines the measurements accuracy. The most preferable det[J] value is 1. In the direction of antenna system’s DP main lobe this condition is met. However, angular shifts of radiation source in relation to antenna axis may lead to the errors due to different dependencies of matrix elements from angular coordinates θ and ϕ: ji j = αi j f i j (θ, ϕ), where αi j is constant complex multiplicand; f i j (θ, ϕ) is complex angular function with its maximal value equal to unit. Functions f i j are characteristics of antenna directivity for orthogonal components ( f 11 and f 22 ) and for cross-polarized radiation ( f 12 and f 21 ). With that in mind, matrix (3.2.1) in a linear orthonormalized basis can be represented as:  fx x fx y . [J ] = α f yx f yy 

(3.2.3)

Using Jones matrix of (3.2.3) type for antenna feeder device, after several simple transformations we can receive antenna’s Mueller matrix which binds Stokes radiation parameters on antenna input with radiation parameters on antenna output. In this case antenna can be represented by equivalent octopole. The transition operator from Jones matrix to Mueller matrix has the form: ⎡

1 1 ⎢ 1 [T ] = √ ⎢ ⎣ 0 2 0

0 0 1 −j

0 0 1 j

⎤ 1 −1 ⎥ ⎥ 0 ⎦ 0

(3.2.4)

whereby the transition itself is being fulfilled according to the following expression:   [M] = [T ] [J ] ⊗ [J ]∗ [T ]−1

(3.2.5)

where ⊗ represents sign of Kronecker matrix product. Derived from (3.2.5) expressions for Mueller matrix elements can be used for the analysis of influence of received radiation from antenna on Stokes parameters. Antenna which is capable of reception of radiation with arbitrary polarization can be represented as a two-channel system with orthogonally polarized channels [19–21]. This antenna system is equivalent to an octopole presented in Fig. 3.5, where input arms 1-1 and 2-2 of octopole [S] are situated in the focal plane of mirror antenna, while output arms 3-3 and 4-4 are situated in the far radiation zone.

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3 Primary Ways of Technical Implementation …

Fig. 3.5 Antenna system representation by the equivalent octopole

Channels 1-3 and 2-4 refer to orthogonally polarized beams of two principal polarizations, while 1-2 and 2-3 refer to cross-polarization of respective channels. Let us represent incident and reflected waves in the waveguide and in the far radiation zone by corresponding matrices-columns:  [a2 ] =

a1 a2

 ; [b2 ] =



b1 b2



  ; ap =



a3 a4



  ; bβ =



b3



b4

Here all elements are normalized so that their squares were equal to the power of respective wave. Operation of two-channel (unipolarized) antenna is described by the following matrix expression: ⎤ ⎤   ⎤⎡ ⎡ [aα ] [bα ] [Sαα ] Sαβ ⎥ ⎥ ⎢ ⎢ [b] = ⎣ − − −⎦ = [S] [a] = ⎣ − − − − − − ⎦ ⎣ − − −⎦,         Sββ Sβα bβ aβ ⎡

(3.2.6)

where [S] is antenna system SM. Antenna without ohmic losses retains power without change ([a]T [a] = [b]T [b]), therefore its SM is unitary. The physical meaning of (3.2.6) matrix coefficients is clarified during consideration of antenna operation in reception and transmission modes.   Assume, for example, that radiation is transmitted into free space, i.e., column aβ is equal to zero. Then expression (3.2.6) takes the form: [bα ] = [Sαα ] [aα ];

    bβ = Sβα [aα ]

(3.2.7)

3.2 Mirror Antenna Scattering Matrix

83

For reception antenna (when [aα ] is equal to zero) similar expressions look as follows:         [bα ] = Sαβ aβ ; bβ = Sββ aβ

(3.2.8)

From (3.2.7) it follows that block [Sαα ] of [S] matrix determines the reflection   coefficients from antenna input at the point of its connection with the feeder. Sβα determines transmission   antenna’s radiation characteristics. Expressions (3.2.8) features setting power show that block Sαβ determines antenna receptional   transmission coefficients into waveguide, while Sββ describes antenna system’s scattering parameters.     Blocks Sαβ , Sβα of (3.2.6) matrix are nothing but Jones matrix for two-channel antenna system. System’s mutuality is determined by equation: 

  T Sαβ = Sβα .

We will consider SM of idealized antenna system that is fully matched with the feeder by both channels and without mutual dependency between input signals in orthogonal channels. These limitations don’t impact our consideration’s generality, but significantly simplify the following derivations, since block [Sαα ] of (3.2.6) matrix is equal to zero. Using unitarity of antenna SM and symmetry of mutual antenna’s   matrix, it is not difficult to obtain systems of equations that determine block Sββ elements: ⎫ ∗ ∗ S23 + S34 S24 = 0; S33 ⎬ ∗ ∗ S23 + S44 S24 = 0; S34 ⎭ ∗ ∗ ∗ ∗ S13 + S24 S23 + S34 S33 + S44 S34 = 0; S14 ⎫ ∗ ∗ S13 + S14 S14 = 1; S13 ⎪ ⎪ ⎬ ∗ ∗ S23 + S24 S24 = 1; S23 ∗ ∗ ∗ ∗ S13 + S23 S23 + S33 S33 + S34 S34 = 1; ⎪ S13 ⎪ ⎭ ∗ ∗ ∗ ∗ S44 = 1. S14 S14 + S24 S24 + S34 S34 + S44

(3.2.9)

(3.2.10)

Finally from these expressions for such antenna’s SM we have: ⎡

0 ⎢ 0 ⎢ ⎢ −− [S] = ⎢ ⎢ ⎢ fx x ⎣ fx y

0 | 0 | −− −− f yx f yy

fx x f yx −− | f x y |2 | fx x | K

fx y f yy −−



⎥ ⎥ ⎥ ⎥, ⎥ | fx x | fx y | − fx x K ⎥ ⎦ |f |f | f |2 f | − xfxx x x y K | f xxxy| f ∗ x x 22 K xy

(3.2.11)

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3 Primary Ways of Technical Implementation …

→ During reception of the wave E i = − x E i ei (wt−γ R) amplitudes of scattered waves are determined by the following matrix expression: ⎡

⎤ ( f x − j f x y )E i ⎡ ⎤ i ⎢ ⎥ 0 ⎢  ( f x y − j f y )E ⎥ ⎢ 0 ⎥ ⎢ ⎥ | fx | fx y | f x y |2 ⎢ ⎢ ⎥ Ei K ⎥ ⎢ ⎥ = [S]⎣ E i ⎦. | fx | + j fx ⎢ ⎥  ⎣ ⎦ |f |f − j Ei | f |2 f 2 − xf x x y − j | fxxy| f ∗ 2x K E i

(3.2.12)

xy

Release power on the matched loads of both channels is equal to:  2  2 Pa = |b1 |2 + |b2 |2 = ( f x − j f x y )E i  + ( f yx − j f y )E i  .

(3.2.13)

Antenna’s scattered field is characterized by the power of the waves with the amplitudes equal to the difference between the total and incident fields, i.e. (b3 − a3 ) and (b4 − a4 ). The power reflected from antenna is equal to: Ps = |b3 − a3 |2 + |b4 − a4 |2 .

(3.2.14)

Taking into account (3.2.11) we have:   2    2         f x y 2 f 2  f x y 2 | fx | fx y | fx | fx y    x K + j Ei  . +j K − 1}E i  +  − Ps =  −j  ∗ 2     | fx | fx fx | fx | fx y

(3.2.15) Expression (3.2.15) can be used for the optimization of scattering properties of antenna system, namely for optimization of scattered power. It can be achieved due to the presence of degree of freedom in choosable exponential factor ψ, while functions f i j are fixed and strictly determined for a given antenna configuration. As an example, let us consider method of scattering properties optimization for simplest unipolarized antenna made as an open-end square waveguide with cross section Q × Q operating on two spatially orthogonal waves of TEiq type. For this antenna we have: f x = f y : f x y = f yx , E θ = f θ (θ, ϕ) = −

(3.2.16)

γH cos u sin ϑ πa W0 (1 + i0 cos θ )  2 sin ϕ; 2u λR k −1 ϑ

(3.2.17)

γH cos u sin ϑ πa W0 (cos θ + i0 )  2 cos ϕ; 2u λR k −1 ϑ

(3.2.18)

τ

E ϕ = f ϕ (θ, ϕ) = −

τ

3.2 Mirror Antenna Scattering Matrix

85

λ is the wavelength for free space; R is the distance to observation point; Z is sin θ cos ϕ; ϑ = ka sin θ sin ϕ. characteristic impedance of free space and u = ka 2 2 For the components f x,y, f x y,yx we have: fx = f y = −

E θ sin ϕ cos θ + E ϕ cos ϕ  ; 1 − sin2 ϕ sin2 θ

f x y = f yx =

E θ cos ϕ − E ϕ sin ϕ cos θ  . 1 − sin2 ϕ sin2 θ

(3.2.19)

From expressions (3.2.16), (3.2.19), and (3.2.13) we can determine the power released on the matched load: Pa =E i2 N 2 [(tg 2 ϕ cos2 θ − 4 sin2 ϕ sin2 ϕ/2) − j2tgϕ(tg 2 ϕ cos2 θ − tgϕ sin ϕ − tgϕ cos θ + 0, 5)]; Here N =

πQ Z λR o

cos u

(3.2.20)

sin ϑ

( 2nπ ) −1 ϑ For square waveguide in E-plane, where ϕ = 0, the scattered power, according to (3.2.14), is equal to: 2

  Ps = (N − 1)N 2 e j2ψ − 2N 1 − N 2 e jϕ − 2N 1 − N 2 e jψ .

(3.2.21)

Ps minimum will be observed for the following ψ:  ψ = 0, 5 arg(2N 1 − N 2 eiψ ). An arbitrary one-mirror antenna system with radiator operating on the fixed polarization that coincides with one of two basis orths can be represented as combination of two multipoles. For general case, mirror antenna radiator is described by equivalent six-pole having one input and two output channels (for main and cross-polarizations) [22, 23]. Reflector in this case performs two-channel signal processing and can be represented by equivalent octopole. Hereinafter we take into account partial transformation of energy with main polarization into cross-polarized channel due to cross polarization effects. Since radiator’s output signals are simultaneously reflector’s input signals, onemirror system can be represented by cascade connection of above-mentioned multipoles. Reflector’s equivalent octopole is determined by SM (3.2.9). With similar restrictions that we placed before on the antenna systems under consideration, we can by the same token derive the radiator’s SM that has the form

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3 Primary Ways of Technical Implementation …

⎤ ⎤ ⎡ f 0x y 0 f 0x 0 m 12 m 13 ⎢ ∗ ∗ jτ iτ ⎥ − f 0x [M] = ⎣ m 12 m 22 m 23 ⎦ = ⎣ f 0x f 0x y f 0x y e ⎦ ye f ∗ ∗ 0x y jτ f 0x y − f 0x f 0x y e jτ f 0x e m 13 m 23 m 33 | f0x y | ⎡

(3.2.22)

Here f ox and f ox y are radiator’s DP on the main and cross-polarizations, respectively. The τ value depends on radiator’s design features, and its appropriate choice allows to minimize antenna scattering power. Cascade connection of multipoles (3.2.9) and (3.2.22) creates a new multipole with SM [N ] presented in Fig. 3.6. To determine elements of this matrix, we will use method of oriented graph. Oriented graph of (3.2.9) and (3.2.19) multipoles’ cascade connection is presented in Fig. 3.7. According to non-intersecting loop rule, elements n 12 and n 13 that determine characteristics of energy transmission into channels of the main and crosspolarization can be easily obtained in the following form:

[N ] 1

[S ]

[M ]

1

2 2 3 3

Fig. 3.6 Cascade multipoles connection α4

S44

b`4

a`4 S34

S14 m12

b1

1` Eα

1

1 1

m23

m13

αГ

a2

S13

a`1 S14

m22

b2

m13 m33

S24

S33

S13

m12 a1

S24

b`3

b1 `

m23

b3

a`3 S23

S23 b`2

Fig. 3.7 Oriented graph of cascade multipoles connection

α3

3.2 Mirror Antenna Scattering Matrix

n 12 =

n 13 =

87

EG 2 [m 12 s13 (1 − s44 α4 − m 33 s24 α4 ) + m 12 s14 α4 s34 B 2 +m 13 s24 (1 − s44 α4 − m 22 s14 α4 ) + m 13 s24 α4 s34 ];

(3.2.23)

EG 2 α3 ) + m 12 s13 α3 s34 [m 12 s14 (1 − s33 α3 − m 33 s23 B 2 +m 13 s24 (1 − s33 α3 − m 22 s13 α3 ) + m 13 s23 α3 s34 ];

(3.2.24)

2 2 B = 1 − s44 α4 − s33 α3 − m 33 s24 α4 − m 22 s13 α3 2 2 2 − m 22 s14 α4 − m 33 s23 α3 + α4 s44 s33 α3 + m 22 s44 α4 s13 α3 2 2 2 + s33 α3 α4 s24 m 33 + m 33 s24 α4 m 22 s13 α3 2 2 + m 22 s14 α4 α3 s33 + m 33 s23 α3 α4 s44 ;

E G = I is the generator’s e.m.f.; α3 and α4 are reflection coefficients from ambient space. Zero value of n13 coefficient determined by (3.2.24) provides optimal antenna system’s PD. Setting n13 equal to zero, we obtain the most general condition for synthesis of mirror antenna uniform PD as follows: 3 f ox y = P f ox y + Q

P= Q=

(3.2.25)

f y − f x y K α3 − f yx f x y K α3 ; f y f yx2 F3 eiτ

2 f 0x f x y − f 0x f x2y K α3 / f x − f 0x2 f x y f yx α3 eiτ − f 0x f yx f x y kα3

f y f yx2 α3 eiτ

(3.2.26) ;

(3.2.27)

Solving (3.2.25) for f ox y , we can easily obtain such radiator’s DP for the crosscomponent that is needed for total elimination of cross-polarized radiation for a given mismatch. Assuming antenna’s full matching with ambient space, i.e., α3 = α4 = 0, we can significantly simplify Eqs. (3.2.23) and (3.2.24): n 12 = m 12 s13 + m 13 s 23 ;

(3.2.28)

n 19 = m 12 s14 + m 13 s24 .

(3.2.29)

From the last expression we derive n 13 = f ox f x y + f ox y f y = 0 ,

(3.2.30)

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3 Primary Ways of Technical Implementation …

from where it follows that for cross-polarization elimination in this antenna we have to synthesize radiator with such DP for the cross-component that must meet the following condition: f 0x y = − f 0x f x y / f y .

(3.2.31)

In accordance with (3.2.16) we have: Eψ = EΦ =

A(θ, ϕ) cos ϕ + sin ϕ cos ψ , cos ϕ − A(θ, ϕ) sin ϕ cos ψ

(3.2.32)

A(θ , ϕ) = f x y (θ , 4)/ f y (θ , ϕ). Thus, one possible way of cross-polarized radiation elimination is forming of such amplitude-phase distribution in radiator that would provide DP on the crosspolarization in accordance with (3.2.25) or (3.2.27). The other most obvious way is modification of mirror profile in order to achieve a similar effect. These issues are considered in the theory of antenna synthesis [24, 25]. The next subdivision of this chapter is devoted to the problem of radiator synthesis for antenna system with uniform PD, where we also will substantiate the applicability of uniform PD synthesis conditions for the case of multimirror antennas [26].

3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna During consideration of antenna systems’ polarization characteristics, it is comfortable to use such coordinates for the far zone field that decomposition basis for output field would coincide with decomposition basis for input field. For narrow-beamed antennas in this case we can consider this requirement to be fulfilled within the area of DP we interested in the most: the main lobe and the first side lobes are situated near the mirror axis. Therefore synthesis condition (3.2.32) can be represented by dependence in spherical coordinates system. As follows from (3.2.31), the task of uniform PD synthesis reduces to finding of radiator cross-polarized component Fy (, ϕ) from the known antenna DP. Let us consider one-mirror antenna system in the form of parabolic mirror radiated by an open-end rectangular waveguide with cross-section a × b where main wave propagates. The computation program for DP calculation was written according to mathematical expressions derived. The calculation results are represented in Fig. 3.8, where dash-line represents radiator DP on the main polarization.

3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna

1,0

89

F

f0X 0,8

fY

0,6

0,4 fXY 0,2

degrees

0

1

2

3

4

5

Fig. 3.8 Results of directional pattern calculation

According to (3.2.30), radiator’s DP on cross-polarization was calculated in order to suppress mirror antenna radiation on cross-polarization. The resulting radiator’s DP is given in Fig. 3.9. The main theme of this subdivision is synthesis of amplitude-phase distribution of radiator’s current that would form DP on parasite polarization in accordance with (3.2.30). Since the maximum of cross-polarized field intensity of mirror antenna radiation is observed in the plane  = π/4, we are interested the most in the synthesis of uniform PD exactly in this plane. During consideration of the above-mentioned problem solution we will put the following restrictions on the synthesized system [27–29]: 1. 2. 3.

The system is linear. Coefficients in differential equations that describe all processes are constant. All processes are stationary.

This synthesis task can be represented as synthesis task for a cammed radiator with the fixed profile and known requirements to field distribution in the antenna that provide its technical realization. Scalar DP is a dependence of one of cross components from angular coordinates:

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3 Primary Ways of Technical Implementation …

1,0

0,8

0,6

f0XY

0,4

0,2

degrees

0

1

2

3

4

5

Fig. 3.9 Resulting directional pattern of primary feed

 F3a∂ (, ) =

Jy (ρ)  exp[ik(ρ0 r0 )]ds,

(3.3.1)

δ

where ρ is the radius-vector of integration point in the s area; r0 is unit orth of − → − →  is spherical coordinate system ( r0 ,  ,  ) directed onto integration point; Jy (ρ) vector of electrical current density and we assume that currents in the s area depend on time according to the exponential law exp iωt. Dependence on time and frequency in DP expressions is usually omitted provided that we talk about monochromatic signal. Equation (3.3.1) determines the dependence between antenna directional pattern F and aperture function χ. In operator form this expression can be written as Aχ = F

(3.3.2)

Linear, additive, and quite continuous operator A is determined by Eq. (3.3.2). Its specific form depends on area v and characteristics of aperture function χ. Operator A is widely called as direct operator of antenna synthesis. The task of radiating system synthesis consists in finding aperture function χ from the operator Eq. (3.3.2) of the first type, based on arbitrarily assigned directional pattern F. Due to arbitrary form of F function, Eq. (3.3.2) may have no solution in general case. Besides, the solution

3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna

91

χ = A−1 F can be unstable, because reciprocal operator A−1 (A−1 A = 1) is not continuous. Tasks of this type belong to the class of incorrect tasks. They require especially careful consideration during their study. For practical purposes, it’s enough to obtain an approximate solution when required DP is calculated with certain accuracy. Therefore hereinafter we will find an approximate solution (quasi-solution) of Eq. (3.3.2) that provides minimal mean square deviation between the left and right parts of Eq. (3.3.2). Thus, the solution is reduced to minimization of square functional. We will consider the required complex function χ and assign directional pattern F as elements of abstract Hilbert spaces x ∈ X,

F ∈ X,

For both spaces we introduce scalar product:  (x j , xk ) =  (F j , Fk ) =

x j (ρ)  x¯k (ρ)dv, 

(3.3.3)

F j (, ) F¯k (, )ds,

(3.3.4)

v

S1

where S1 is sphere of unit radius. The overline means complex conjugation. Now, from the initial task of antenna synthesis we can formulate the task of minimization of the following functional: (x) = Ax − F 2

(3.3.5)

The formal solution to synthesis task sometimes is represented by amplitude-phase distribution which may happen extremely difficult to realize. Technically realizable solution should provide insignificant variations of calculated current distribution for small variations of directional pattern parameters. This condition provides stability to task solution [30]. While strict or approximate solution is being found as an element of Hilbert space L2 defined in the interval [−1, + 1], during technical implementation we can encounter some difficulties due to fast oscillations of distribution function. Fast oscillations assume that only small fraction of complex input power can be radiated, since two similar closely situated elementary radiators will operate with very large phase difference. In these circumstances antenna will be ineffective. This phenomenon takes place when aperture length is small and required DP is no more a slowly changing function. As a parameter that regulates solution’s “complexity” we will choose a coefficient of high-directivity which is proportional to antenna radiation power. Since calculated solutions will insignificantly differ from each other, their

92

3 Primary Ways of Technical Implementation …

active powers will also be close to each other. Therefore, from all diagrams that differ from the assigned one less than by δ1, we will choose the one that has minimal coefficient of high-directivity, whereby δ1 is evaluated in square metrics, i.e. ⎡ ⎣

2π

⎤0,5 |F(, 0 ) − F C (, 0 )|2 d ⎦

= δ1 ,

(3.3.6)

0

where Fs(, 0 ) is antenna’s synthesized DP. On the other hand, since antenna system directivity is inversely proportional to power of ohmic loss, then minimizing of ohmic loss will enhance antenna system gain. The task of finding χ (solution) from the F set by “input data” from the set v, χ = R(u) is called correctly formulated on the sets (F, v) that are metrical spaces with distances ρ F (x1 , x2 ) and ρu (u 1 , u 2 ), where x1 , x2 ∈ F, u 1 , u 2 ∈ V , if the following requirements are fulfilled: 1. 2. 3.

For every element u ∈ V , a solution x exists from F set; Solution is uniquely determined; Solution must continuously depend on input data, i.e., for any ε > 0 such δ(ε) exists that if ρu (u 1 , u 2 ) ≤ δ and x1 = R(u 1 ), x2 = R(u 2 ), then ρ F (x1 , x2 ) ≤ ε. This requirement is also called the feature of task stability. In coordinate system 0˜ X˜ Y˜ Z˜ for the left part of (3.3.1) equation, we have  Fy (, ) =

Jy (ρ)  exp[ jk(ρ r0 )]d s˜

(3.3.7)



Let’s introduce dimensionless coordinates x = k x, ˜

J = k Y˜ , z = k z˜ ,

as well as u = sin  cos , V = sin  sin ,

(3.3.8) (3.3.9)

that are directional cosines of the orth r0 . Then expression (3.3.7) can be represented in the form:  Fy (, ) =

1

−1

J (t)l(t) exp{i[n(t) sin  cos  + r (t) sin  sin s(t) cos ]}dt l(t) =

 2  2  2 1/2 n (t) + n (t) + s (t) .

(3.3.10)

Here J (t) = Jy (t) is complex amplitude of J (t) vector. Cammed radiator L is described by the system of differential parametric functions:

3.3 Synthesis of the Uniform Polarization Diagram for Mirror Antenna

x˜ = n(t), ˜

J˜ = r˜ (t), z˜ = s˜ (t); −1 ≤ t ≤ 1.

93

(3.3.11)

Notice that the following functions were introduced in (3.3.10): n(t) = k n(t), ˜ r (t) = k r˜ (t), s(t) = k s˜ (t), where k is wave number that characterizes electrical dimensions of radiator. For the linear radiators (and radiator in the form of open-end rectangular waveguide belongs to this class) Eq. (3.3.11) will take the form n(t) = G t; r (t) = s(t) = 0.

(3.3.12)

whereby function u(, π/4, t) is defined as 20π 2 exp u(, π/4, t) = 3 cos3 πt G



 √ 2 2 πt πt 2tg sin  + tg cos  , G 2 G

(3.3.13)

And the core K (ζ, t) is determined by the expression: K (ζ, t) =

200π 5 9 cos3

πt G

cos3

πζ G

,

(3.3.14)

where corresponding coefficients c(t) are equal to:  ⎧ ⎪ ⎪ c = F(, 0 )(u(, π/4, tm )dt; m1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

S

cm2 = −(u(, π/4, tm );

(3.3.15)

cm3 = −(u (, /4, tm ).

Let’s assign the following form of the weight function: ⎧ ⎨ 1, Ψ ∈ [00 , 20 ]; ρ(Ψ ) = ρ1 , Ψ ∈ [20 , 100 ]; ⎩ ρ2 , Ψ ∈ [100 , 900 ].

(3.3.16)

With an additional condition, the level of side lobes in the angular interval 2◦ ≤  ≤ 10◦ should not exceed −23 dB; in the interval 10◦ ≤  ≤ 90◦ it should not exceed −55 dB. Let’s also agree on limitation that synthesized DP should not deviate from assigned Fy (, 0 ) values more than 5%. In this case,

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3 Primary Ways of Technical Implementation …

⎡ δ = ρ()(Ax − y) = ⎣

 π/2 0

⎤1/2 1   ρ 2 () J (t)u(, π/4, t)dt − F(, π/4)|2 d ⎦ .  −1

(3.3.17) A solution of (3.3.2) equation was found for several ρ1 and ρ2 (ρ2 > ρ1 ) values by the method of mesh functions (n = 20, p1 (t) = 0, p2 (t) = 1); parameters β1 , β2 , β3 were determined, and then the current distribution was obtained that provides the required DP of radiator on cross-polarization. During the calculation of result, the ρ1 and ρ2 values equal to 2 and 4, respectively, were chosen, with α ∈ [10−7 , 10−8 ]. Similar calculations have been made for α = 5 · 10−8 with their result taken as a required solution, since although mesh functions method defines field distribution values for the definite number of pre-determined points and does not provide enough information on field’s behavior in between these points, due to the small size of the radiator (< 30 α), this information is of little worth. Results of parabolic antenna’s linear radiator synthesis can be approximated as shown in Fig. 3.10, where dash-line represents phase distribution along the radiator. Thus, Fig. 3.10 represents the synthesized amplitude-phase distribution of cross current in linear radiator that should be induced near parabolic mirror’ radiator in order to obtain uniform PD of mirror antenna in the plane situated between the two antenna focal planes ( = π/4).

0

-8

-16

A 10 -2 8

-24

6

-32

4

-40

2

-48

0 -10

-8

-6

-4

-2

0

2

4

Fig. 3.10 Results of synthesis of parabolic antenna linear primary feed

6

8

10

t

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment

95

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment Let’s determine SM for the device represented in Fig. 3.11 and signify output terminals for every block. In accordance with adopted notations we have the following SM for elements that comprise the device: 13 57 ⎡ ⎤ 00 cc ⎢ 0 0 c −c ⎥ ⎢ ⎥ ⎣c c 0 0 ⎦ c −c 0 0

1 3 ; 5 7

9 11 13 15 ⎡ ⎤ 00 c c ⎢ 0 0 −c c ⎥ ⎢ ⎥ ⎣ c −c 0 0 ⎦ cc 0 0

9 11 ; 13 15

(3.4.1)

where C = √12 For quadrupoles: 6 10   0 e− jφ 6 ; e jφ 0 10

14 18   0 e− jψ 14 ; e jψ 0 18

8 12   01 8 ; 1 0 12

16 20   0 1 16 ; 1 0 20

(3.4.2)

that is, first two quadrupoles are ideal phase shifters with non-reciprocal phase shifts ϕ and ψ, while second two quadrupoles represent immediate connection of UHF devices. For polarization shifter, ⎡

0 ⎢0 ⎢ ⎣c 0

2 4 17 19 ⎤ 0c0 2 0 0 c⎥ ⎥4 ; 0 0 0 ⎦ 17 c 0 0 19

(3.4.3)

Using expressions (3.4.1)–(3.4.3), we compose octopole’s SM that is included in the following expression:

Fig. 3.11 Functional diagram of device

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3 Primary Ways of Technical Implementation …



⎤ ⎡ ⎤ b1 ⎢ b ⎥ ⎡ [s ] | [s ] ⎤ a1 ⎢ ⎥ 2 1 2 ⎢ a2 ⎥ ⎢ ⎥ ⎥ ⎢ − − − ⎥ = ⎣ −− −− −− ⎦⎢ ⎣ a3 ⎦. ⎢ ⎥ ⎣ b3 ⎦ [s3 ] | [s4 ] a4 b4

(3.4.4)

The required octopole’s matrix is determined by the expression: [s] = [s1 ] − [s2 ]([s4 ] − [F])−1 [s3 ],

(3.4.5)

where [F] is square matrix for coupled terminals that consists of nulls and units (units refer to the terminals that should be coupled, and other coefficients are equal to zero). Since matrix [S1 ] in (3.4.3) consists of zeroes, (3.4.5) takes the form [s] = [s2 ](−[E] + [s4 ])−1 [s3 ].

(3.4.6)

After several simple, although cumbersome transformations of (3.4.6) we finally receive system’s SM in the form ⎡

0

jϕ ⎢ ⎢ −ce jψ e 2+1 [S] = ⎢ ⎣ 0 jϕ −c e 2−1

⎤ jϕ jϕ −ce jψ e 2+1 0 −c e 2−1 jϕ ⎥ 0 −ce jψ e 2+1 0 ⎥ ⎥; jϕ jϕ 0 −c e 2−1 ⎦ −ce jψ e 2+1 jϕ 0 −c e 2−1 0

(3.4.7)

The required matrix after transformations has the form e jϕ + 1 = 2e jϕ/2 cos ϕ/2; e jϕ − 1 = 2 je jϕ/2 sin ϕ/2; e− jϕ + 1 = 2e− jϕ/2 cos ϕ/2; e− jϕ − 1 = −2 je− jϕ/2 sin ϕ/2.

(3.4.8)

⎤ 0 e jϕ e jξ cos ξ 0 − je jξ sin ξ ⎥ 1 ⎢ −e− jϕ e− jξ cos ξ 0 −e− jϕ e− jξ sin ξ 0 ⎥; [s] = √ ⎢ jϕ jξ jξ ⎣ 0 e cos ξ ⎦ 0 je e sin ξ 2 0 e− jξ cos ξ 0 je− jξ sin ξ (3.4.9) ⎡

where ξ = ϕ/2. Taking out a common factor e jξ , we receive SM of ideal device for antenna polarization adjustment

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment

97

⎤ 0 e jψ cos ξ 0 − j sin ξ ⎥ e jξ ⎢ −e− jψ e−2ξ cos ξ 0 − je− jψ e− j2ξ sin ξ 0 ⎥. (3.4.10) [s] = √ ⎢ jψ ⎣ 0 cos ξ ⎦ 0 je sin ξ 2 0 e− j2ξ cos ξ 0 je−2ξ sin ξ ⎡

The corresponding transmission matrix ⎡

0

− j/ sin ξ

j cos ξ sin ξ

0

⎢ − jψ − jψ − j2ξ ξ ⎢ − j e sine ξ 0 0 j e sincos ξ [T ] = ⎢ ⎢ − j cos ξ 0 0 − j sin1 ξ ⎣ sin ξ − jψ − jψ − j2ξ ξ 0 j e sincos − j e sine ξ 0 ξ

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(3.4.11)

As we have already noticed, one of the main reasons of field polarization structure measurement errors is mismatching between elements that constitute the device considered. Therefore, assessment of mismatchings in system’s operation is a mandatory condition of our analysis. During derivation of (3.4.10) matrix we applied matrix theory of waveguide networks. Advisability of this theory mathematical tools’ application is conditioned not only by their exceptional convenience (despite seeming complexity), but also by the fact that parameters of the analyzed device can be expressed in terms of SM elements, what allows thereafter for these results application in analysis of device’s main parameters in general. Hereinafter we will not consider an issue of SM elements frequency dependence. This assumption is justifiable because radars transmit in the quire narrow frequency band where hybrid waveguide networks, being the main source of different losses, keep their parameters unchanged within 5% frequency fluctuations [31, 32]. Using obtained SM, we will run analysis of network elements’ influence (namely, their non-ideality) on the accuracy of antenna system PD adjustment device. For this purpose we have to determine the complex values of transmission coefficients from system’s input to its output and specify the type of these coefficients dependence on system’s parameters. In the final form analysis issues can be formulated as follows [24]: – Assessment of influence of channels reactive loads on the input reflection coefficient of ideal system having SM (3.4.10); – Analysis of influence of non-ideality of UHF elements’ properties on adjusted polarization parameters. This question includes study on effect of variations in coefficients of UHF power splitting by waveguide bridges and polarization splitters, as well as mismatching coefficients for separate UHF elements, and values that determine parasite interconnection of orthogonal channels.

98

3 Primary Ways of Technical Implementation …

Fig. 3.12 Functional diagram of radar system receive path

Solution of the first question consists in derivation of expression that would determine the value of input reflection coefficient for the device (see Fig. 3.11) provided that its output terminals 2, 3, and 4 are connected with mismatched loads having reflection coefficients equal to antenna system’s coefficients of mismatch between orthogonally polarized channels and to the reflection coefficient of radar receive path (Fig. 3.12). Write down the SM for the system’s components (Fig. 3.11) in general form: ⎡

1

3

5

Γ111

7

Γ112

c11

⎢ [s]T 1 = ⎢ Γ122 c12 ⎢ Γ112 ⎢ ⎢ c11 c Γ 133 ⎣" " 12 2 2 1 − c11 1 − c12 Γ134 ⎡ [s]T 1

9 Γ211

11

13 Γ212

15 c21

"

" ⎤ 2 1 − c11 ⎥1 " 2 ⎥3 1 − c12 ⎥ ⎥ , 5 Γ134 ⎥ ⎦ 7 Γ144

2 1 − c21

(3.4.12)



⎢ ⎥ 9 " ⎢ ⎥ 2 = ⎢ Γ212 Γ222 −c22 1 − c22 ⎥ 11 ⎢ ⎥ , ⎢ c21 ⎥ 13 −c Γ Γ 22 233 234 ⎣" ⎦ " 15 2 2 1 − c21 1 − c22 Γ234 Γ244

(3.4.13)

Here matrices [S]TI and [S]TI I that determine parameters of waveguide T-type phase shifters in general form can be represented as

[s]ϕ1 =



6 10 14 18    Γφ11 e− jϕ 6 [s]ϕ2 = Γφ21 e− jψ 14 , . e+ jϕ Γφ12 10 e+ jψ Γφ22 18

(3.4.14)

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment

99

According to (3.4.5), it is not difficult to determine values of all elements in the SM of generalized octopole (3.4.1) for assigned values of elements in SMs (3.4.12)– (3.4.15). Analysis of the influence of UHF elements’ non-ideal properties on the coefficient values in the generalized SM of polarization adjustment system is run by tools of simulation modeling. For this purpose a computing program was developed where properties of unitary SM had been used. In Figs. 3.13, 3.14, 3.15, 3.16, 3.17 and 3.18 the charts are presented for the SM elements S12 , S14 , S23 , and S34 of the generalized octopole in dependence from waveguide T-junctions Ci j and polarization Fig. 3.13 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

|S|

S12 S14

0,6

0,4 S23 S34 0,2

0 0,2571

Fig. 3.14 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

0,3471

0,4371

0,5271

0,6171

C11

0,6171

C12

|S|

S23

0,6

S34

0,4 S12 S14 0,2

0 0,2571

0,3471

0,4371

0,5271

100 Fig. 3.15 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

3 Primary Ways of Technical Implementation … |S|

S12 S23

0,6

0,4 S14 S34 0,2

0 0,2571

Fig. 3.16 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

0,3471

0,4371

0,5271

0,6171

|C21|

0,6171

|C22|

|S| 1

S14 S34

0,6

0,4 S12

S23

0,2

0 0,2571

0,3471

0,4371

0,5271

splitter C pj ratio values. These ratio values varied from 0.25710678 to 0.70710678 with the increment 0.015. As it follows from the presented charts, the above-mentioned elements are influenced the most by polarization splitter ratio value. As for waveguide T-junctions, dependence of SM elements from Ci j is inverse proportional, which allows to achieve an ideal adjustment for the system as a whole. For double T-type waveguide bridges these coefficients have the following values: α111 = α211 = 0.11; α112 = α212 = 0.1; α112 = α222 = 0.15; α133 = α233 = 0.14; α134 = α234 = 0.11; α144 = α244 = 0.14.

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment Fig. 3.17 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

101

|S|

0,6

0,4

0,2

S12 S14 S23 S34

0 0,2571

Fig. 3.18 Graphic charts for scattering matrix elements of generalized octopole depending from ratio values in waveguide T-junctions and polarization splitter

0,3471

0,4371

0,5271

0,6171

|CP1|

0,6171

|CP2|

|S|

0,6

0,4

0,2

S12 S14 S23 S34

0 0,2571

0,3471

0,4371

0,5271

With corresponding ratio values, C11 = C21 = 0.7, C12 = C22 = 0.68. All other reflection coefficients can be neglected. Polarization splitter has coefficients C S1 = C S2 = 0, 7. The data given in this paragraph allow for taking into account the influence of non-ideality of UHF elements that comprise the system, on its operation. Since we know which octopole’s parameters are influenced by accuracy of this or that element, process of system’s adjustment is significantly facilitated. This adjustment should also take into account the different degree of every element’s influence on the general transmission coefficients of the whole system [14].

102

3 Primary Ways of Technical Implementation …

Fig. 3.19 Oriented graph of generalized octopole

For analysis of influence of UHF elements’ mismatch and variation coefficients of power ratio for waveguide T-junctions and polarization splitters on the system’s operation, it is suitable to represent a generalized octopole shown in Fig. 3.11 by its oriented graph which for reciprocal device connected with mismatched loads has the form shown in Fig. 3.19. The terminal 1 is connected to the generator with reflection coefficient α1 , and terminals 2 and 4 are connected to the antenna system with reflection coefficients α2 and α4 for respective channels. Terminal 3 is loaded by reception device with mismatch determined by reflection coefficient α3 . Under the assumption of full reception system’s matching α3 = 0, the ratio is defined as:    S12 (1 − S44 α4 ) + S14 S24 α4 2 1 − |α2 |2 P2   = (3.4.15) P4 S14 (1 − S22 α2 ) + S12 S24 α2  1 − |α4 |2 To determine the input power for the second terminal when the load is mismatched, we have to multiply the power for the matched load by the following correction coefficient Qtoby opredelit mownost, postupawu na nagruzku vtorogo pleqa pri rassoglasovanii, neobhodimo znaqenie mownosti, opredelenno bez uqeta rassoglasovani, umnoit na popravoqny kofficient, opredelemy sleduwim obrazom:

3.4 Analysis of Scattering Matrix of Ideal Device for Polarization Readjustment

  (1 − S33 α3 ) + S13 S23 α3 κ1 = κ1 |S14 | =  S14 S (1 − S α ) + S S α 2

14

33 3

13 34 3

2   

103

(3.4.16)

If α3 = 0 then P2 1 = and K 1 = 1 P4 |S14 |2

(3.4.17)

On the other hand, if the device has ideal characteristics, i.e., S11 = S22 = S33 = S44 = S13 = S24 = 0 and, besides, we can assume |S12 = 1| then Eq. (3.4.17) transforms into equation P2 1 1 − |α2 |2 = P4 |S14 |2 1 − |α4 |2

(3.4.18)

Consequently, κ1 =

1 − |α2 |2 r2 = 2 r4 1 − |α4 |



r4 + 1 r2 + 1

2 (3.4.19)

where r2 and r4 are VSWR of antenna system orthogonal channels. Value of mismatch between orthogonal channels is determined as   1 C = 20 log10  S

12

   

(3.4.20)

Thus, using expressions (3.4.16) and (3.4.21), we can determine the influence of UHF elements’ mismatch for the system shown in Fig. 3.11 on the power supplying the antenna, which, in turn, influences the accuracy of required polarization adjustment as well as mismatch between orthogonal channels. Hereinafter, we consider the phase shift introduced by phase shifter PS1 (Fig. 3.11) equal to zero.

3.5 Derivation of the Formula for Parabolic Antenna Radiation Components with Fundamental and Cross-Polarizations The density of the surface current induced on the surface of parabolic reflector is determined by well-known expression →



J = 2 n × H,

(3.5.1)

104

3 Primary Ways of Technical Implementation … →

where n is vector of the unit normal to the mirror surface with the following projections: nx =

sin ψ sin ψ − cos ψ ; ny = ; nz = 2 cos φ 2 sin ϕ 2

(3.5.2)



Magnetic field vector H can be determined based on the following vector equation: →

#

H=

→% ε $→ u × E μ

(3.5.3) →

Here ε and μ are propagation space parameters, E is vector of the incident → electrical field, u is unit vector that determines energy propagation direction such that u x = sin ψ cos φ; u y = sin ψ sin φ; u z = cos ψ

(3.5.4)

From (3.5.1)–(3.5.3) it follows that #



J =2

ε μ

$→ →% n × u ×E



(3.5.5)

For the parabolic antenna with radiator in the form of open-end rectangular waveguide with cross-section a ×b and main type of the wave in the waveguide, projections of directional pattern (DP) are equal to E p = 0;

⎫ μ ⎪ ⎪ ;⎪ E ψ = (1 + cos ψ) F(ψ, φ) ⎬ ε ; # μ⎪ 1 ⎪ H ⎪ Eφ = (cos ψ + γ10 ) F(ψ, φ) ⎭ tgφ ε #

γ10H

where & γ10H

=

 1−

λ 2a

2 ;

% $ % $ $ π %2 e−ikp cos π a sinλψ cos φ sin π b sinλψ sin φ $ . F(ψ, φ) = − %2    2 p π a sin ψ cos φ π 2 sin ψ sin φ − λ 2 Expression (3.5.6) can be represented as

(3.5.6)

3.5 Derivation of the Formula for Parabolic Antenna …

105

E p = 0;

  E ψ = Φ(ψ, φ) sin φ + γ10H cos ψ sin φ ;   ; E φ = Φ(ψ, φ) cos ψ cos φ + γ10H cos φ ;

(3.5.7)

where     # $ % −ikp cos πλa sin ψ cos φ sin πλb sin ψ sin φ μ π 2e ' . Φ(ψ, φ) = − 2  2 ( πa ε 2 p sin ψ sin φ sin ψ cos φ − π λ

2

Assume (R, θ, ϕ) are coordinates of the observation point. Then the field reflected from the mirror will be determined by the following expressions: ω μ e−ik R E = −j 4π R



 

   → → → u u eik p l u R ds, J− J ·



R

R

(3.5.8)

S →

where u is unit vector in the direction of the observation point; l is unit vector to the R

point M; p is distance to the point M; S is reflector surface. ds = 2 p 2

sin ψ dψ dφ; 2

p=

f cos2 ψ 2

(3.5.9)

;

(3.5.10)

u Rx = sin θ cos ϕ; u Ry = sin θ sin ϕ; u Rz = cos θ f is mirror focal distance. →

(3.5.11)



To determine vector E , we have to receive expressions for J components based on (3.5.5)–(3.5.7). From these equations and also from ⎫ E x = E ψ cos ψ cos φ − E φ sin φ;⎪ ⎬ E y = E ψ cos ψ sin φ + E φ cos φ; ⎪ ⎭ E z = −E ψ sin ψ

(3.5.12)

we can easily obtain for the components of surface current: # Jx (ψ, φ) =

  ψ ε Φ(ψ, φ) sin φ cos φ sin ψ 1 − cos ψ cos 2ψ + sin γH μ 2 10

106

3 Primary Ways of Technical Implementation …

 ⎤ cos ψ + cos2 ψ ⎢ cos ψ cos φ ⎥ 2 ⎢ ⎥ # ⎢ ⎥ ε ψ H⎥ Φ(ψ, φ) ⎢ Jy (ψ, φ) = 2 − sin cos φ sin ψ cos ψ γ10 ⎥ ⎢ μ 2 ⎢ ⎥ ⎣ ⎦ ψ 2 H sin φ + cos ψ γ10 + cos 2 &    ψ ε Jz (ψ, φ) = 2 Φ(ψ, φ) sin sin φ 1 + γ10H cos ψ 1 + cos2 φ sin2 ψ μ 2 (3.5.13) ⎡

2

According to (3.5.8), electrical field in the observation point is formed by the following components of the surface current:   Jx| = −Jx (ψ, φ) 1 − sin2 θ cos2 ϕ + Jy (ψ, φ) sin2 θ sin ϕ cos ϕ + Jz (ψ, φ) sin θ cos θ cos ϕ;   | Jy = Jy (ψ, φ) 1 − sin2 θ sin2 ϕ + Jx (ψ, φ) sin2 θ sin ϕ cos ϕ + Jz (ψ, φ) sin θ cos θ sin ϕ

(3.5.14)

Electrical field with main polarization, according to (3.5.8), is determined as   FOC H (θ, ϕ) = 1 − sin2 θ sin2 ϕ  − sin2 θ sin ϕ cos ϕ  − sin θ cos θ sin ϕ



Jy| (ψ, φ)e jγ (ψ,φ,θ,ϕ) dψ dφ

S

Jx| (ψ, φ)e jγ (ψ,φ,θ,ϕ) dψ dφ

S

Jz| (ψ, φ)e jγ (ψ,φ,θ,ϕ) dψ dφ

(3.5.15)

S

while cross-polarized radiation is % ⎤ ⎡$ 1 − sin2 θ cos2 θ ⎥ ⎢  ⎥ ⎢ ⎥ ⎢ ⎢ Jx| (ψ, φ) · e jγ (ψ,φ,θ,ϕ) dψ dφ − sin2 θ sin ϕ cos ϕ J y| (ψ, φ) e jγ (ψ,φ,θ,ϕ) ⎥ 1 ⎥ ⎢ E K pocc (θ, ϕ) = ⎥ ⎢S S ⎥ E OC H (θ, ϕ) ⎢  ⎥ ⎢ ⎥ ⎢ | jγ (ψ,φ,θ,ϕ) ⎦ ⎣ dψ dφ − cos θ sin θ cos ϕ Jz (ψ, φ) e dψ dφ S

(3.5.16) where we have γ (ψ, φ, θ, ϕ) =

2π f [sin ψ sin θ cos(ϕ − φ) + cos ψ cos θ − 1] λ cos2 ψ12

(3.5.17)

3.5 Derivation of the Formula for Parabolic Antenna …

107

In the Eqs. (3.5.15) and (3.5.16), the following components of the surface current are used:   ψ 1 | | γ H J ; J y (ψ, φ) Jx (ψ, φ) = Q(ψ, φ) sin φ cos φ sin ψ 1 − cos ψ cos2 ψ + sin 2 2 10   $ % ψ H , =Q(ψ, φ) cos ψ cos2 φ cos ψ + cos2 ψ − sin × cos2 φ sin ψ cos ψ − γ10 2 % ψ$ 2 sin φ + cos2 φ cos2 ψ + γ cos ψ Jz + cos 2 $ %( ' ψ H cos ψ 1 + cos2 φ sin2 ψ =Q(ψ, φ) sin sin φ 1 + γ10 (3.5.18) 2

where # Q(ψ, φ) =

ψ ε 4 f 2 sin 2 p Φ(ψ, φ) ψ −ikp 4 μ cos 2 e

(3.5.19)

3.6 Solution for Synthesis of Line Feed with Given Amplitude Directional Diagram Expression (3.3.10) has the form 1 J (t)l(t) exp{n (t) sin ψ cos φ + r (t) sin ψ sin φ + S(t) cos ψ}dt

Fy (ψ, φ) = −1

For the sake of simplification of the following derivations, we represent (3.3.10) as: 1 Fy (ψ, φ) =

Jy (t) μ(ψ, φ, t)dt.

(3.6.1)

−1

Expression (3.6.1) is a linear Fredholm integral equation of the first kind. Similar equations can be easily obtained for arbitrary radiating surface, with the only difference in the type of the core that depends on antenna geometry and on polarization of the current (field) vector in its aperture. For a linear Fredholm integral equation of the first kind with no matter how smooth kernel K (t, y) (even analytical)

108

3 Primary Ways of Technical Implementation …

b K (t, y) x(y)dy = u(t),

c≤t ≤d

(3.6.2)

a

solution is being found in the class of continuous functions F[G]. We will evaluate deviation of the right part of u(t) in L 2 metrics (in Hilbert space), and x(y) in C metrics (in space of continuous functions), i.e.

pu (u 1 , u 2 ) =

⎧ d ⎨ ⎩

|u 1 (t) − u 2 (t)|2 dt

) 21

;

(3.6.3)

c

p F (x1 , x2 ) = sup|x1 (y) − x2 (y)| y ∈ [a, b]

(3.6.4)

Assume that for some right part u = u 1 (t) function x1 (y) is a solution to (3.6.2) equation. If instead of u 1 (t) we know just some of its approximation that slightly differs from u 1 (t) in L 2 metrics, then we can talk only about finding the solution to (3.6.2) equation that is close to x1 (y). The right part may not exhibit enough smoothness wherein Eq. (3.6.2) has no solution, because kernel K (t, y) is a smooth function. Consequently, exact solution to (3.6.2) equation with just approximately known right part u(t) = u 1 (t) cannot be taken as a solution approximated to x1 (y). In these cases the first requirement for correctly formulated task is not fulfilled. Besides, task (3.6.2) does not exhibit stability feature, i.e., requirement 3 for correctly formulated task is not fulfilled. Really, function x2 (y) = x1 (y) + B sin ωy will be a solution to (3.6.2) equation with the right part b K (t, y) sin ωy dy

u 2 (t) = u 1 (t) + B

(3.6.5)

a

Obviously, for any B, deviation ⎧ ⎤2 ⎫ 21 ⎡ ⎪ ⎪ ⎪ ⎬ ⎨d b ⎦ ⎣ pu (u 1 , u 2 ) = |B| K (t, y) sin ωy dy dt ⎪ ⎪ ⎭ ⎪ ⎩c a

(3.6.6)

can be made arbitrarily small for large enough ω, whereas for the corresponding solutions x1 (y) and x2 (y) p F (x1 , x2 ) = sup|B sin ωy| = |B| y ∈ [a, b]

(3.6.7)

3.6 Solution for Synthesis of Line Feed …

109

Thus, task (3.6.2), as well as task (3.6.1), is resolved by try and guess method. However, Tikhonov A.N. developed a new approach for these tasks solution based on the fundamental concept of regularizing operator.   According to Tikhonov, element xα = R u γ , α , generated from regularizing operator R(u, α), where α = α(δ) is reconciled with the source data accuracy u δ and pu (u 1 , u δ ) ≤ δ, should be taken as an approximate solution to (3.6.2) equation. This solution is called a regularized solution. Numeric variable α is called regularization parameter. If we know that pu (u 1 , u δ ) ≤ δ, then, according to definition of regularizing operator, we can choose such value of regularization parameter α = α(δ) that for δ → 0 regularized solution xα (δ) = R(u δ , α(δ)) approaches to the initial exact solution x E , i.e., p F x1 , xα(δ) → 0 (in L 2 metrics). It justifies a proposal to take regularized solution as an approximate solution to (3.6.2) equation. Method of regularizing operators R(u, α) design is based on variation principle and consists in the following. Suppose [x] is some non-negative functional defined on the subset L 1 of the set L 2 and for every number d > 0, set L d of elements x for which [x] ≤ d is compact within L 2 . It is not difficult to see that choice of [x] functional is nonunique. Assume we know, besides, that x1 ∈ L 1 and deviation of the right part u δ from the exact value u 1 does not exceed δ, i.e., pu (u 1 , u δ ) ≤ δ. Then approximate solution should be sought in the Q δ class of the x elements that satisfy the condition pu (Rx, u δ ) ≤ δ 2 . But the set Q δ is not compact, it is too wide. Hereafter we will consider only those elements of Q δ set for which functional [x] is defined, i.e., we will consider elements of the set L 2 = Q δ ∩ L 1 only. It is shown that instead of minimization of [x] functional on L 1 set, we can solve the task of functional [x] minimization on the L 1 set provided that for the sought element x the requirement pu (Ax, u δ ) ≤ δ is fulfilled. This is a constrained extremum problem. We will solve it by Lagrange’s method of undetermined multipliers, i.e., we will find the minimum for the functional M α [u δ , x] = pu2 (Ax, u δ ) + α [x],

(3.6.8)

where numeric parameter α is determined by the condition pu (Axα , u δ ) = δ, where xα is element for which the functional M α [u δ , x] reaches its lowest value. Element xα can be considered as a result of application of some operator R1 that depends on parameter α, to the right part u = u δ of (3.6.2) equation, i.e. xα = R1 (u δ , α)

(3.6.9)

Tikhonov A.N. shown that R1 (u δ , α) is a regularizing operator. It’s worth to notice that although the initial task (3.6.2) does not exhibit stability feature, the task of functional M α [u, x] minimization is stable against slight change in the source data u. Therefore functional M α [u, x] is called smoothing functional [27]. This feature was achieved due to consideration of the function [x] that stabilizes

110

3 Primary Ways of Technical Implementation …

the solution. Regularizing   parameter α can be determined by deficiency, i.e., from the condition pu Axαδ , u δ = δ. Let’s apply regularization method to Fredholm integral equation of the first kind, which the problem of mirror antenna’s uniform PD synthesis is reduced to. Suppose A is integral operator with the kernel K (t, s). Then Eq. (3.6.2) takes the form b K (t, s) × (t)dt = u(s), c ≤ s ≤ d.

(3.6.10)

a

It is easy to show that A is a linear operator. Let’s take [x] in the form of the functional: b  [x] =



dx p1 (t) dt



2

+ p(t)x 2 dt,

(3.6.11)

a

where p1 (t) and p(t) are assigned non-negative functions. In this case the equality condition for the first variation of the functional M α [u, x] has the form: ⎞b    . b  d x d ⎟ ⎜ p1 (t) − p(x)x(t) + K (s, ξ )x(ξ )d(ξ ) − B(t)dt + αp1 (t)x(t)δx(t)⎠ = 0. ⎝α  dt dt  a a a

b



-

(3.6.12)

Here δ x(t) is an arbitrary variation of x(t) function such that both x(t) and x(t) + δ x(t) belong to the class of admissible functions: d K (s, ξ ) =

d K (t, s)K (t, ξ )ds;

B(t) =

c

K (t, s)u(s)ds = B(t),

(3.6.13)

c

Condition (3.6.12) is fulfilled provided that ⎧ . b  ⎨ d dx −α p (t) − p(t)x(t) + K (t, ξ )x(ξ )dξ = B(t), ⎩ dt 1 dt

(3.6.14)

a

and b p1 (t)x | (t)δ x(t)a = 0.

(3.6.15)

So, if we know the values of required solution x(t) to (3.6.12) equation on one or both ends of [a, b] interval, then in searching the minimum of M α [u, x] functional

3.6 Solution for Synthesis of Line Feed …

111

only those functions can be considered as admissible that take the assigned values at the end(s) of the interval. Besides, they should have a generalized derivative integrable with its square. In this case functions δ x(t) become equal to zero on the end(s) and condition (3.6.15) is fulfilled. Thus, the problem of finding the regularized solution xα (t) is reduced to finding the solution of integro-differential Eq. (3.6.14) that satisfies the conditions x(a) = x1 ,

x(b) = x2

(3.6.16)

If values of required solution x(t) on the ends S = a and S = b are unknown, then condition (3.6.15) can be satisfied by assuming x | (a) = x | (b) = 0,

(3.6.17)

where x1 , x2 are known numbers, and solution to (3.6.14) that satisfies condition (3.6.16) should be taken as a regularized solution to (3.6.10). In order to synthesize normalized DPs with maximum in the assigned point, we propose to reduce synthesis problem to minimization of the functional M(x) = [x] + β1 Re A0 x + β2 Im A0 x + β3 Re A30 x,

(3.6.18)

where β1 , β2 , β3 are yet unknown Lagrange multipliers. Normalization requirement for synthesized DP in the direction (ψ0 , φ0 ) can be written as Re A0 x = 1,

Im A0 x = 0,

Re A(1) 0 x = 0,

(3.6.19)

here Fc (ψ0 , φ0 , t) = A0 x = (x, μ(ψ0 , φ0 , t)). The equation for calculation of current distribution in radiating system for [x] will be as follows: −α

  b d dx p12 (t) + α p 2 (t)x(t) + k(s, ξ )x(ξ )dξ dt ds a

d −

K (t, s)u(s)ds + (β1 + β2 ) μ(ψ0 , φ0 , t) + β3 μ−1 (ψ0 , φ0 , t) = 0

(3.6.20)

c

x(a) = c1 , x(b) = C2 , βm = 0, 5βm , m = 1,2,3; the line means the operation of complex conjugation.

112

3 Primary Ways of Technical Implementation …

In order to solve the Eq. (3.6.20), we will write down the following equations:  d 2 p1 (t)x | + α p 2 (t)x + A∗1 A1 x − A∗1 y1 = 0; dt x(a) = c1 , x(b) = c2

(3.6.21)

 d 2 p1 (t)x | + α p 2 (t)x + A∗1 A1 x + μ(ψ0 , φ0 , t) = 0; dt x(a) = x(b) = 0

(3.6.22)

 d 2 p1 (t)x | + α p 2 (t)x + A∗1 A1 x + μ−1 (ψ0 , φ0 , t) = 0; dt x(a) = x(b) = 0

(3.6.23)

−α

−α

−α

A non-negative weight function p(ψ) is introduced here so that A1 = p(ψ)A,

y1 = p(ψ)y.

(3.6.24)

This function allows for taking into account the assigned requirements to the shape of the main lobe for narrow-beamed antenna system and to the level of side lobes. Unfortunately, optimal weight function cannot be specified by above-mentioned requirements only. Invariable ρ allows for regulation of mean square approximation value of the main lobe in the synthesized DP to the assigned function F(ψ0 , φ0 ), while ρ1 allows for level of the side lobes, whereby integral level of side lobes is a constantly decreasing function of ρ1 parameter. For the majority of practical cases weight function ρ(ψ) is chosen equal to the unit in the main lobe area. Suppose that parameter α is known. Let’s multiply Eqs. (3.6.21) and (3.6.22) by β1 + iβ2 each, and (3.6.23) by β3 ; then sum them all term-by-term. After elementary transformations we will receive d  2 2 (α) ) p (t) x + αp 2 (t)x (α) + A∗1 A1 x (α) dt 1 −A∗1 y1 + (β1 + iβ2 )μ(ψ0 , φ0 , t) + β3 μ−1 (ψ0 , φ0 , t) = 0; −α

(3.6.25)

where x α = x (o) + (β1 + iβ2 )x (1) + β3 x (2)

(3.6.26)

Comparing (3.6.20) and (3.6.25) we can see that x (α) is linear combination of x , x (1) , x (2) distributions. Parameters { β} can be easily found by substituting DP normalizations into (3.6.19) condition. Really, using features of scalar multiplication, after several not very complicated transformations we receive the following set of linear equations with respect to parameters { β} : (0)

3.6 Solution for Synthesis of Line Feed …

113

 (1)   (1)   (2)   (0)  ⎧ ⎪ ⎨ β1 Re x , μ − β2 Im x , μ + β3 Re x , μ = 1 − Re x ,μ ; β1 Im x (1) , μ + β2 Re x (1) , μ + β3 Im x (2) , μ = −Im x (0) , μ ; ⎪         ⎩ β1 Re x (1) , μ−1 − β2 Im x (1) , μ−1 + β3 Re x (2) , μ−1 = −Re x (0) , μ−1 , (3.6.27) where μ = μ(ψ0 , φ0 , t). Thus, to find the field distribution in the antenna, it is sufficient to solve (for the known α) Eqs. (3.6.21)–(3.6.23) and the set (3.6.27). Every equation in (3.6.21)–(3.6.23) can be written in the form:   1 d J (t) d 2 2 −α p (t) + α p (t)J (t) + k(ξ, t)J (ξ )dξ − c(t) = 0 dt 1 dt

(3.6.28)

−1

J (−1) = b − 1;

J (1) = b1 ;

where x = J (t); kernel K (ξ, t) is determined by (3.6.13) and for every (3.6.21)–(3.6.23) equations is equal. Accordingly, A∗1 y1 , −μ(ψ0 , φ0 , t) , −μ−1 (ψ0 , φ0 , t) ; b−1 = c1 and b1 = c2 for (3.6.21); b−1 = b1 = 0 for (3.6.22) and (3.6.23). In the interval [−1,1] let’s choose the points tm = −1+mh, where m = 1,2…,h = 2n. After application of the known difference scheme Eq. (3.6.28) can be substituted by the set of linear equations with respect to current value distributions in tm points. To be definitive, let’s choose the following difference scheme:    d J (t)  = h −1 J (tm ) − J (tm−1 ) ;  dt t=tm  2       p J − J (t ) (t ) (t ) d d J (t) m m m−1 1  p 2 (t) = h −2   dt 1 dt t=tm − p12 (tm−1 ) J (tm−1 ) − J (tm−2 )

(3.6.29)

(3.6.30)

We expand the integrator in the left part of (3.6.28) using, for example, trapezoid rule on the system of tm points: 1 −1





0, 5K (ξ0 , tm )J (ξ0 ) + K (ξ1 , tm )J (ξ1 )

⎢ K (ξ, tm )J (ξ )dξ = h ⎣ +... + K (ξl , tm )J (ξl ) + ...+

⎥ ⎦, l = 1, 2, ..., n − 1

K (ξn−1 , tm )J (ξn−1 ) + 0, 5K (ξn , tm )J (ξn )

(3.6.31) Substituting in (3.6.28) its terms for t = tm according to (3.6.28)–(3.6.28) formulas and after simple transformations we receive the linear equation with regard to current values J (ξe ). When variable t runs all points tm (m = 0,1,2,…,n), integral Eq. (3.6.28) transforms into the following set of linear equations:

114

3 Primary Ways of Technical Implementation … n 3

aml Je = Cm ;

J0 = b − 1,

Jn = b, m = 2, 3 . . . n,

(3.6.32)

l=0

where

aml =

⎧ k(tm , ξl ); l = m − 2, m − 1, m; ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ + αpm2 , l = m; k(tm , ξl ) − αh −2 p1,m ⎪ ⎪ ⎪   ⎪ ⎨ k(tm , ξl ) + αh −2 p 2 + p 2 , l = m − 1; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2 p1,m = p12 (tm );

k(tm , ξl ) −

1,m 1,m−1 −2 2 αh p1 , m − 1,



k(tm , ξl ) =

l = m − 2;

(3.6.33)

0, 5h K (tm , ξl ), l = 0, n; h K (tm , ξl ), l = 0, n;

pm2 = p 2 (tm ); cm = c(tm );

Jl = J (tl ); ξl = tl .

Set (3.6.22) is a set of n + 1 linear equations with n + 1 unknown Jl s. For p1 (t) = 0 boundary conditions J0 = b − 1, Jn = b should be excluded from the system (3.6.32), but index m must run all values from 0 to n in this case. Set (3.6.32) should be substituted by the set of equations with real coefficients, since computer cannot operate with complex variables. For this purpose we write functions J (t) and c(t) in the form: J (t) = I1 (t) + j I2 (t); C(t) = c1 (t) + jc2 (t) ; where I1 (t) = Re J (t); I2 (t) = Im J (t); C1 (t) = Re c(t); C2 (t) = Im C(t)

(3.6.34)

Substituting (3.6.34) into (3.6.32) we come to the conclusion that set (3.6.34) is equivalent to the following set with real coefficients: n ' ( 3 (1) (2) aml I1,l − am,l I2,l = c1,m ; m = 2, 3, . . . n; l=0 n ' ( 3 (2) (1) aml I1,l + aml I2,l = c2,m ; m = 2, 3, . . . n; l=0

I1,0 = Re b−1 ;

Il,0 = Im b−1 ;

I1,n = Re b1 ;

I2,n = Im b1 ;

(3.6.35)

(1) (2) where aml and aml are real and imaginary parts of coefficient aml , accordingly; I1,l = I1 (te ); I2,e = I2 (te ).

3.7 Model of Matrix Joint Correlation Function …

115

3.7 Model of Matrix Joint Correlation Function of Probing and Reflected Vector Signals for Conceptual Design of Aerial Radar with Synthesized Aperture For the full-scale simulation of the project design development, including virtual testing of radar with synthesized aperture (RSA) on the aerial platform (AP) in the model environment, we need conceptual design of RSA on AP. For implementation of virtual testing capacities for RSA on AP we will use mathematical modeling environment MATLAB that has elaborated set of tools for digital signal processing (DSP). To fulfill computer modeling for radar systems with full polarization probing, we have to design a model for matrix joint correlation function of probing and reflected vector signals.

3.7.1 Structural Diagram of Conceptual Design of Radar with Synthesized Aperture The main goal of conceptual design of radar with synthesized aperture (RSA) on the aerial platform (AP) is creation of virtual RSA on AP for simulation of the full-scale process of project design on computer models, including virtual RSA on AP testing in model environments [9]. Structural conceptual design diagram for RSA on AP developed on the basis of exact subject knowledge is presented in Fig. 3.20 (digital and literal numbers of the units in Fig. 3.20 represent mathematical models for RSA subsystems).

Fig. 3.20 Structural conceptual diagram for RSA design

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3 Primary Ways of Technical Implementation …

Here 1 = RSA antenna; 2 = receiver (RCV) and transmitter (TRM) of the RSA; 3 = unit of signal digital processing (SDP) which forms radar image (RI) for transmission on the ground; 4 = TRM of RI; 5 = antenna of TRM; 6 = receiving antenna (RCV); 7 = RI receiver model; 8 = unit of the secondary RI processing. The underlying surface, atmosphere and AP constitute the environment for RSA on AP. Presented in Fig. 3.20, the RSA structure allows taking into account the influence of AP trajectory irregularities and construction oscillations on RI quality not only in the 1–2–3 route, but also in 4–5–6–7 route. RSA simulation testing is provided by the choice of mathematical models for underlying surface features (block A in Fig. 3.20), atmosphere properties on the RSA’s working frequency (block B in Fig. 3.20), atmosphere properties as a wideband radio channel (block C in Fig. 3.20), and trajectory irregularities of RSA on AP (block 3 in Fig. 3.20).

3.7.2 Model of Matrix Joint Correlation Function for Probing and Reflected Vector Signals To fulfill computer modeling of radar systems with full polarization probing, we need to design the model of matrix joint correlation function for probing and reflected vector signals. For this purpose we introduce model of vector probing signal in the form of response from linear matrix filter to the delta-function. As a model of scattering object we took an assembly of independent point scatterers distributed in space and having different velocities in general case [17]. For general case, narrow-band wave group limited in time and frequency band can be represented by complex vector u(t, ω) with its orthogonal components as parametric complex functions of time  u(t, ω) =

 f˙1 (t, ω) , f˙2 (t, ω)

(3.7.1)

where ω is signal’s instant frequency, its parameter. Introduction of wave group u(t, ω) instant frequency concept in this case is appropriate due to bandlimitedness of ordinary radar probing signals. For these signals the condition ω denotes sign of averaging of A0 (t) enveloping for the interval of A0 (t) duration that determines the time span of the signals described by functions f˙1 (t, ω) and f˙2 (t, ω). In practice, two different types of scalar probing signals are used that determine time–frequency structure of orthogonally polarized components of vector probing signal [7, 8]: Signals with complex enveloping in the form of smooth function that has time derivative in every point; Signals with complex enveloping in the form of threshold function whose derivative in time is not defined for some points, while the number of these points is countable. An example of the first type is a linear frequency modulated signal. An example of the second type is phase-modulated signal, modulated in accordance with some code sequence. Signals of the first type are described by the function ins tan t f r equency

2 f˙(t, ω) = R(t) · e j (ω0 t+kt +ϕ0 )

6 74 5 = R(t) · e j[(ω0 + kt) ·t+ϕ0 )] ,

Signals of the second type are described by the function f˙(t, ω) = R(t) · e j (ω0 t+π·m{t}+ϕ0 ) ,

(3.7.2)

where R(t) is rectangular enveloping of the function, k is invariable coefficient that determines the rate of linear variation of oscillation frequency, m{t} denotes

118

3 Primary Ways of Technical Implementation …

binary (0,1) code sequence that defines the law of phase setting for oscillations with frequency ω0 . Signals of the second type are called noise-like signals when their autocorrelation function has a single “narrow” peak in the point of zero shifts in time and frequency. It’s clear that considered scalar signals can be generated in the form of response from linear filters excited by delta-function. Thus, model of considered vector probing signals u0 (t, ω) forming can be represented by response of two-channel filter shifted on the carrier frequency ω0 , as is shown in Fig. 3.21. In computer modeling two-channel filter is implemented in the form of two enquired memory registers that contain preliminary stored response functions g˙ 1 (t), g˙ 2 (t) that correspond to the chosen signal type. Using the model of radar one-position channel, we can generally represent reflected signal in the form u p (t, ω) =

N 3 i=1

u pi (t, ω) =

N 3 i=1

4

gi (τi , i ) ∗u0 (t, ω) = G (τ, ) ∗ u0 (t, ω), 56

G (τ,)

7 (3.7.3)

where gi (τi , i ) is matrix response function of ith point scatterer; τi and i are N : delay time and frequency shift for ith scatterer; G (τ, ) = gi (τi , i ) is matrix i=1

response function for spatially distributed radar object. Thus, reflected signal (3.7.3) is the sum of elementary signals whose shape repeats the shape of emitted signal, but their amplitude, phase, and polarization are determined by coordinate, velocity, and polarization parameters of elementary scatterers that constitute spatially distributed object. Algorithm and results of computer modeling for spatially distributed objects have been given above. From the physical point of view, process of probing electromagnetic signal reflection from complicated

Fig. 3.21 Narrow-band vector signal shaper

3.7 Model of Matrix Joint Correlation Function …

119

spatially distributed object is described by the expression ur (t) =

N 3

uri (t) =

i=0

=

N 3

N 3

Si (t) · ai (t) · exp(−i · k · ri (t)) · ur (t − t0i ) =

i=0

gi (t) ·ur (t − t0i ) = G (t) ·

i=0

N 3

ur (t − t0i ) = G (τ ) ⊗ ur (t),

i=0

4 56 7 G (t)

is the wave number; λ is wavelength; t−t0i = τ ; matrix function G (τ ) where k = 2π λ is the sum of matrix response functions for all objects that form radar environment, and represents the result of vector signal u0 (t, ω) filtration by vector filter with structural diagram shown in Fig. 3.22 (ur (t, ω) is reflected vector signal). Every leg of vector filter with impulse characteristics G (τ, ), in turn, represents linear filter for one of components of vector probing signal ( f˙1(2) (t, ω)). Impulse characteristics of this filter G i j (τ, ) (i, j = 1, 2) are formed by the sum of all N impulse responses gi j (τ, ) from every elementary reflector described by matrix response function  g˙ (τ, ) = n

 n n (τ, ) g˙ 12 (τ, ) g˙ 11 , n = 1, N . n n (τ, ) g˙ 22 (τ, ) g˙ 21

This is shown schematically in Fig. 3.23. Vector linear filter transforms input vector into some output vector that belongs to the same set as input vector does. Physically it means that reflected electromagnetic

Fig. 3.22 Filtration of probing vector signal during reflection

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3 Primary Ways of Technical Implementation …

Fig. 3.23 Filtration of orthogonally polarized components of probing signal u0 (t, ω) during its reflection from spatially distributed object

wave has similar nature as incident wave has, differing from it by parameters of amplitude, phase, frequency, and polarization.  p T p Reflected vector signal ur (t, ω) = f 1 (t, ω) f 2 (t, ω) observed in the output of antenna with full polarization reception bears all available information on coordinate and polarization parameters of separate radar objects that form radar environment in terms of active narrow-band radio location. Complete appliance of electromagnetic field’s vector characteristics assumes correct estimation of response matrix function G (τ, ) for the whole group of elementary reflectors forming radar environment, according to the results of reflected vector signal ur (t, ω) observation. The correct estimation of matrix response function G (τ, ) is a procedure of matrix bi-dimensional transvection of received vector signal ur (t, ω) with radiated signal u0 (t, ω), used as a reference signal, by shift parameters τ and . For every given value of frequency shift  = i matrix transvection can be realized by reflected signals processing through matrix filter matched with radiated signal. Compensation of received signals’ Doppler frequency shift i is fulfilled by reflected signals’ frequency shift by −i , while processing system becomes multichannel by the range and Doppler frequency. Structural diagram of received signal filtration is shown in Fig. 3.24. Impulse characteristics g˙ 1∗ (τ, ) i g˙ 2∗ (τ, ) are coupled with impulse characteristics of the filters in the probing vector signal shaper shown in Fig. 3.21. The total set of output responses J˙i j (t, ω) (i, j = 1, 2) from matched filters represents the matrix joint correlation function for probing and reflected vector signals J(t, ω). This function is an estimation of response function G (τ, ) for radar object. Thus, after taking into consideration the above-mentioned models of forming probing vector signal and matrix response function for distributed radar object, the model of matrix joint correlation function for radiated and reflected signals can be represented by structural diagram shown in Fig. 3.25. The input parameters for this model are:

3.7 Model of Matrix Joint Correlation Function …

121

p Fig. 3.24 Filtration of orthogonally polarized components f˙1(2) (t, ω) of reflected vector signal ur (t, ω) during its reception (algorithm of initial processing)

Fig. 3.25 Model of matrix joint correlation function for radiated and reflected vector signals as an operation of sequential delta-function filtration

Shape of the orthogonal (by their polarization and time-frequency structure) components f˙1 (t, ω) and f˙2 (t, ω) of vector probing signal that determine impulse characteristics g˙ 1 (t) and g˙ 2 (t); Matrix response function G (τ, ) for spatially distributed radar object determined by statistics of polarization and coordinate parameters of elementary reflectors that constitute the radar object. In the output of the model, an estimation of matrix response function for spatially distributed radar object is formed. The presented structural diagram may serve as a foundation for development of algorithm and computer program for modeling of the process of signal initial processing in radar station with full polarization probing.

122

3 Primary Ways of Technical Implementation …

References 1. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Lukashov KG, Mironov OS, Panfilov PS, Parusov VA, Raisky VL, Sarychev VA (2017) Radio physical support of ultrashort-pulse radar systems. In collected volume “Problems of remote sensing, radio-waves diffraction and propagation” Compendium of lectures. RAS Scientific Council on radio-wave propagation. Murom institute (subsidiary) of Vladimir State university n.a. Stoletov A.G and Stoletov N.G. 5-21 2. Akinshin NS, Peteshov AV, Bystrov RP, Rumyantsev VL (2018) Statistical characteristics of periodically nonstationary radar signals. Advances of modern radio electronics. 5:10-14 3. Akinshin NS, Bystrov RP, Rumyantsev VL, Peteshov AV (2018) Sequential rank procedure for radar signals detection. Advances of modern radio electronics. 5:3-9 4. Akinshin NS, Peteshov AV, Sigitov BB (2018) Modeling algorithm for radar image forming by polarimetric radar station with synthesized aperture. TulaSU Review, “Technical sciences” series, issue 2, Tula: TulaSU 2:32-40 5. Akinshin NS, Varenitsa YI, Khomyakov KA (2016) Joint assessment of coordinate and polarization parameters of radar objects// TulaSU Review, “Technical sciences” series, issue 2. TulaSU, Tula, pp 3–14 6. Akinshin NS, Fomichev MY, Tsybin SM (2017) Equipment package methods of measurements prompt analysis based on application of random sequencies generator. Electronic information systems 4(15):47-56 7. Akinshin NS, Bystrov RP, Menshikov VL (2017) Mathematical models for directional patterns and radar cross sections of antenna systems in ground based radars. Antennas 8(240):53-66 8. Akinshin NS, Rumyantsev VL (2014) Evaluation of polarization selection efficiency for ground based targets//TulaSU Review, issue 12, Part 2, “Technical sciences” series.- Tula: TulaSU, p 173-182 9. Akinshin RN, Yesikov OV, Zatuchny DA, Peteshov AV (2019) Model of the matrix joint correlation function for probing and reflected vector signals for conceptual design of the radar with synthesized aperture on the aerial platform.- Scientific bulletin of Moscow State Technical University of Civil Aviation 22(02):86-95 10. Alyoshkin AP, Artyushkin AB, Balashov VM (2015) Optimization procedures application for enhancement of two-beam over-the-horizon radio location method efficiency. Radio industry 2:23–30 11. Andreev PG, Yurkov NK, Grishko AK, Kochegarov II, Zhumabayeva AS (2019) Research on the influence of dielectric material on UHF-signal propagation at elevated temperatures. In the bulletin “Wave electronics and information communication systems”. In: Proceedings of XXII-nd International scientific conference: in two volumes. Saint-Petersburg, pp 6-11 12. Balashov VM, Drachev AN, Michurin SV (2019) Methodology of reflectors’ control for mirror antennas. In the book “Metrological support for innovation technologies”. International forum: Abstracts, pp 44-46 13. Balashov VM, Drachev AN, Smirnov AO 2019) Methods of coordinate measurements during geometrically-complex surfaces control. In the book “Metrological support for innovation technologies”. International forum: Abstracts. pp 41-43 14. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Mironov OS, Panfilov PS, Parusov VA, Sarychev VA (2016) Technologies of ultra-short-pulse environmental radio location with high range resolution. – Meteorological bulletin. 8(3):17-22 15. Golovachev MV, Kochetov AV, Mironov OS, Panfilov PS, Sarychev VA, Khomyakov IM (2014) UHF-band ultra-short-pulse radar system. In collected volume “Radiophysical methods in environmental remote sensing”. RAS Scientific Council on radio waves propagation. pp 255–260 16. Antsev GV, Bondarenko AV, Golovachev MV, Kochetov AV, Lukashov KG, Mironov OS, Panfilov PS, Parusov VA, Raisky VL, Sarychev VA (2016) Experimental research of ultrashort-pulse radar system’s characteristics. In collected volume “Radiophysical methods in

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environmental remote sensing”. VII Russia-wide scientific conference proceedings. Murom institute (subsidiary) of Vladimir State university n.a. Stoletov AG and Stoletov NG pp 196–202 Brikerights SG, Zatuchny DA, Ilyin EM, Polubekhin AI (2017) Methodological approach to non-traditional radar methods’ positioning in perspective radio location application. Bulletin of Saint-Petersburg State University of Civil aviation, 3(16):72–84 Grishko AK, Nefedyev DI, Yurkov NK (2019) Design decisions structural optimization in conditions of multicriteriality and uncertainty. In: Proceedings of the International symposium “Reliability and quality”. 2:319–322 Yershov GA, Zavyalov VA, Sinitsyn VA (2019) Improvement of radar observability parameters for aircrafts by methods of multi-position radio location. In collected volume “Innovation technologies and technical facilities of special purpose”. XI-th Russia-wide research to practice conference proceedings. vol. 2 “Library of “Voenmech. BSTU Annals” series. pp 21–24 Drachev AN, Balashov VM (2018) Checklist analysis of reflector production accuracy for mirror antenna. Issues of radioelectronics. 1:105–109 Drachev AN, Farafonov VG, Balashov VM (2014) Control procedure for geometricallycomplex surfaces. Issues of radioelectronics. 1(1):91–99 Kozlov AI, Maslov VY (2018) Solution to inverse scattering problem and object’s shape reconstruction based on reflected electromagnetic wave’s field structure. Scientific bulletin of Moscow State Technical University of Civil Aviation. 21(3):160–168 Myasnikov SA, Sinitsyn VA (2019) Design features of new landing radar. In collected volume “Innovation technologies and special purpose technical facilities”. XI-th Russia-wide research to practice conference proceedings. In 2 vol. “Library of “Voenmech. BSTU Annals” series. pp 79–84 Myasnikov S.A., Sinitsyn V.A. Design features of adaptive two-components’ noise rejection filter in primary coherent-pulse radar systems. In collected volume “Innovation technologies and technical facilities of special purpose”. IX-th Russia-wide research to practice conference proceedings. In 2 vol. “Library of “Voenmech. BSTU Annals” series. 2019. p. 76–79 Kozlov AI, Amninov EV, Varenitsa YI, Rumyantsev VL (2016) Polarimetric algorithms of radar target detection in the active noise background. Tula State university bulletin. Technical sciences. 12–1:179–187 Kochegarov II, Lysenko AV, Zhikharev KV, Yurkov NK (2019) Trusov AV Program for adjustment and control of multichannel generator module in vibration testing machine. PC program registration certificate RUS 2019615254 10.04.2019 Rassadin AE (2010) Possible RSA system based on CMS MATLAB // Proceedings of XIVth radiophysics scientific conference. Novgorod N, NNSU n.a. Lobachevsky N.I. Publishing house, pp 173–174 Sinitsyn VA, Sinitsyn EA, Strakhov SY, Matveev SA (2016) Methods of signals forming and processing in primary radar stations. St Petersburg Sogomonyan KE, Yurkov NK (2019) Effect of ultra-short electromagnetic impulses on unmanned aerial vehicles. International symposium “Reliability and quality” proceedings. 2:315–317 Proshin AA, Goryachev NV, Yurkov NK (2018) Calculation of radio-wave attenuation. PC program registration certificate RUS 2019612561 of 05.12.2018 Proshin A.A., Goryachev N.V., Yurkov N.K. Calculation of dew point. PC program registration certificate RUS 2019612562 of 05.12.2018 Yurkov NK, Kuatov BG, Yeskibayev ET (2019) Algorithm of parametric and time control of aircraft movement management parameters. International symposium “Reliability and quality” proceedings 1:219–221

Chapter 4

Experimental Testing of Theoretical Results

4.1 Research of Scattering Polarization Characteristics of the Signals Reflected in Specular Direction from the Real Separation Surface In the third chapter of this book the main directions have been outlined for experimental testing of primary theoretical results received during research on the problem of elimination of ground surface influence on radar operation. It was underlined that the main emphasis should be put on the correctness of the chosen approach to analysis of polarization scattering characteristics of large-scale impedance on randomly uneven surface, i.e., getting experimental dependencies of distributed target’s SM elements values from the given feed angle and azimuth. Therefore, the first subdivision of this chapter is devoted to description of experiment on getting these characteristics and their comparison with theoretical values based on introduction of Kotler amendment to vector form of Kirchhoff integral. This comparison allows for conclusion on the usefulness of proposed method for calculation of distributed targets’ polarization scattering characteristics. The second chapter of this book is devoted to synthesis of the uniform PD for one-mirror antenna system by creation of linear feed that minimizes this antenna’s cross-polarized radiation. Achieved amplitude-phase distribution of cross current in the synthesized radiator is simulated by additional dipole of the ordinary waveguide– dipole feed. Experimental proof of synthesized radiator operation and its influence on the level of cross-polarization are considered in the second subdivision of this experimental chapter. The results proved the correctness of theoretical derivations. Experimental testing of theoretical expressions for assessment of SM elements’ values for large-scale impedance on randomly uneven surface in the linear polarization basis pegged with two-position radar receiving antenna was run on the base of created device for measuring scattering characteristics of arbitrary radar targets. This device was performed as a two-position measuring radar station with transmitting unit radiating signals of 3.2 cm wavelength and arbitrary polarization. General view of transmitting part of measuring polarization system is presented in Fig. 4.1. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 N. K. Yurkov et al., Signal Polarization Selection for Aircraft Radar Control, Springer Aerospace Technology, https://doi.org/10.1007/978-981-33-4964-3_4

125

126

4 Experimental Testing of Theoretical Results

Fig. 4.1 General view of transmitting part of measuring polarization system

Radiation of the wave with given polarization is fulfilled by two-mirror antenna system I with DP width equal to 1.5º on the half-power level (Fig. 4.2). Separation of orthogonally polarized channels in the system is not worse than 25 dB [1, 2]. UHF transmitting part of measuring radar station is performed according to the two-channel schedule that allows radiating signals with any given polarization due to change of signals’ phase by phase shifter 3 (Fig. 4.2). Control of polarization type is fulfilled by oscillograph 2 (Fig. 4.2) that registers transformed low-frequency signals from both channels. Separation of orthogonally polarized components of the signal radiated in the linear basis is fulfilled by polarization splitter joined with radiator’s round waveguide that provides radiation of the waves with arbitrary polarization.

4.1 Research of Scattering Polarization Characteristics of the Signals ...

127

Fig. 4.2 Two-mirror antenna system

During the field tests transmitter was installed at the height of 2 m above the surface of compacted snow with the depth 0.5 m and maximal peak-to-trough value 0.25–0.3 m [3, 4]. Receiving unit of polarization scattering characteristics measurer shown in Fig. 4.3 provides reception of unipolarized reflected signal by two-mirror antenna system I (Fig. 4.4) with the same parameters as for transmitting antenna. Polarization splitter separates orthogonal linear-polarized components of the received signal that go

128

4 Experimental Testing of Theoretical Results

Fig. 4.3 General view of receiving device of polarization scattering characteristics measurer

further to the two-channel receiving device 2. Separation of receiving channels is not worse than 25 dB. Fast-response recorder HZ27 was used as terminal equipment registering the value of reflected power received by two orthogonal channels for a given polarization of radar target radiation. Calibration of the receiving unit was run not far from the transmitter. The distance was chosen so that to provide antenna system’s far zone, on one hand, and to avoid the influence of underlying surface on the measured parameters, on the other. The

4.1 Research of Scattering Polarization Characteristics of the Signals ...

129

Fig. 4.4 View of two-mirror antenna system for the receiving device

transmitter was adjusted for radiation with circular polarization and with ellipticity coefficient not less than 0.95. The transmitting unit of polarization measuring device had been adjusted with linear-polarized polarization indicator 3 (Fig. 4.4) fulfilled in the form of linearpolarized antenna system rotatable around the axis that determines the direction to the transmitter. Pointer situated near the polarization measurer served as a power indicator.

130

4 Experimental Testing of Theoretical Results

Installation of two-position radar system receiving unit on the moveable platform (Fig. 4.3) at the height 2 m from the separation surface allowed to vary radiation and reception angles in a narrow range due to horizontal movements of receiving unit [5]. During the experiment on measuring polarization scattering characteristics of a large-scale impedance on randomly uneven surface in the mirror direction, grazing angle ψ and reflection angle η had been varied within 4°÷ 8° by alongside movements of receiving unit (Fig. 4.5). The reflecting surface area was chosen in such a way that movement of maximal reflection areas due to change of radiation angle would not influence significantly the measurements results. Figure 4.6 presents the dependencies of power SM elements P11 and P22 for the fixed grazing angle ψ = 4° for different electrical parameters / ε = ε + jε1 / of the

x

Fig. 4.5 Design of an experiment on registering of scattering polarization characteristics in mirror direction P22

P 11

=4 О =0 О

=4 О =0 О 0,8

0,8

0,6

0,6

0,4

0,4

`=2-j1,62

`=67-j23

`=80-j0,06

`=2-j1,62 0,2

0,2

`=67-j23

`=80-j0,06 0

10

20

30

о

0

10

20

30

о

Fig. 4.6 Graphic charts of scattering matrix elements P11 and P22 for a given grazing angles and different reflecting surface electric parameters (azimuth angle is equal to zero), calculated in accordance with suggested methodology

4.1 Research of Scattering Polarization Characteristics of the Signals ... P 11

131

P22 `=2-j1,62

`=2-j1,62

=4 О

0,8

0,8

0,6

0,6

=4 О

=5 О 0,4

0,4

10 О 15 О

=15 О

0,2

0,2

10О 5О 0

10

20

30

о

0

10

20

30

о

Fig. 4.7 Graphic charts of P11 and P22 elements that define energetic parameters of scattering for the waves with vertical and horizontal polarization from the sea surface for different azimuth angles (λ = 3.2 cm)

reflecting surface for zero azimuth angle φ = 0◦ of observation point calculated in accordance with proposed method. The analysis of received curves indicated the similarity between energetic parameters of the scattered field and dependencies of Fresnel reflection coefficients’ values [6]. Figure 4.7 provides dependencies of elements P11 and P22 values that determine energetic parameters of scattering from the sea surface / ε = 67 − j23 for the waves with vertical or horizontal polarization, wavelength 3.2 cm, and different radiation angles. Dependencies presented in Figs. 4.6 and 4.7 demonstrate that the power of the reflected wave with vertical polarization is less than power of the wave with horizontal polarization for the whole range of scattering angles η [7, 8]. Figure 4.8 provides dependencies of cross elements P12 and P21 of power SM calculated for different azimuth angles and given electrical parameters of separation surface. Figures 4.9 and 4.10 give an idea of the change of SM elements in dependence from radiation angle for separation surface with ε’ = 2- j·1.62 that corresponds to the snow surface. Experimental values of power SM [P] elements received by above described measuring device are given in the same figures. As it follows from presented dependencies, experimental results conform with theoretical results calculated on the base of Kotler amendment into vector form of Kirchhoff integral within 25% measurement accuracy [9]. Experimental results are given for ψ = 3°.

132

4 Experimental Testing of Theoretical Results P12

P21 `=67-j23

`=67-j23

=4О

0,2

0,2

0,15

0,15

0,1

0,1

=15 О

=10 О

=15 О

0,05

=4О

0,05

10О 5О 0

10

20

=5О о

30

0

10

20

о

30

Fig. 4.8 Graphic charts of scattering matrix cross elements P12 and P21 , calculated for different azimuth angles and given separation surface electric parameters P 11

P22

=0 О

=0 О

`=2-j1,62 0,8

0,8 =4 О

0,6

0,6

=6 О

=4 О 0,4

0,4 =6

О

0,2

0,2 `=2-j1,62 0

4

8

12

о

0

4

8

12

о

Fig. 4.9 Graphic charts of scattering matrix elements for different radiation angles for the case of snow separation surface

4.1 Research of Scattering Polarization Characteristics of the Signals ... P 12

P21

=5О

133 =5О

`=2-j1,62

`=2-j1,62

0,2

0,2

0,15

0,15

0,1

0,1

=4 О

0,05

=4О

=6О

0,05

=6О 0

4

8

12

о

0

4

8

12

о

Fig. 4.10 Graphic charts of scattering matrix elements for different radiation angles for the case of snow separation surface

Measurement errors occur due to error of calibration attenuator’s readings; variations of scattering area size; error of recorder unit readings; change of dielectrical permittivity due to different weather conditions [10, 11]. Figures 4.9 and 4.10 show points of experimental readings received by statistical methods of measurement results processing. Sections of recorder tape were divided into intervals with the length defined by the rate of fluctuations caused by the wind. From the value of recording pen deviation with consideration of calibration attenuator fading, the average Pi value was determined for each interval. As a result of measurement, the average value of all Pi was taken. The performed experiment proved one more time the correctness of theoretical derivations. Thus, the results of the analysis proved to be useful for assessment of scattered power value for the wave of given polarization and known parameters of ground surface dielectrical permittivity. This knowledge is especially useful for research of multipath radio wave propagation over uniform underlying surfaces with relatively smooth relief, as well as for analysis of these conditions’ influence on radar operation during aircraft surveillance [12].

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4 Experimental Testing of Theoretical Results

4.2 Synthesis of One-Mirror Antenna Linear Feed Eliminating Cross-Polarized Radiation As was mentioned during the synthesis of uniform PD for mirror antenna, linear feed that reduces cross-polarized component in mirror antenna radiation to the minimum should have cross-current amplitude-phase distribution similar to presented in Fig. 3.2. For experimental proof of this statement, it is necessary to create linear feed with required distribution and prove experimentally the effect of cross-polarization suppression provided by the feed. One-mirror parabolic antenna system was made with mirror diameter 1 m and focal distance 0.4 m. The feed of this antenna was of waveguide–dipole type, excited by rectangular waveguide on the wavelength 3.2 cm. On the main polarization this linear feed has DP similar to DP of rectangular open end waveguide. Antenna mirror with the general view shown in Fig. 4.11 was made of dielectric material covered by conducting layer of tinfoil. Dielectrical permittivity of mirror base material is close to that of free space (except that in the center of the mirror a metal ring with diameter 0.3 m is situated as a holder for mirror accessories). The mirror design is presented in Fig. 4.12. For initial waveguide–dipole feed shown in the right upper corner of Fig. 4.13, the experimental DPs were taken on the main and cross-polarizations in the plane slanted 45° toward antenna system’s main planes. In this plane, as known, the intensity of cross-polarized radiation is maximal. Received DPs are presented in Fig. 4.14. The dashed lines represent calculated DPs of mirror antenna for the main and cross-polarizations. Difference between calculated and experimental curves can be accounted for by imperfection of measuring equipment and mirror design. As it follows from experimental DPs of mirror antenna, cross-polarization radiation level is situated 22 dB below the level of the main radiation, which is well within the results of experimental research of similar antenna systems [12]. As one of the possible feed types that have required distribution of cross current, the waveguide–dipole feed presented in Fig. 4.15 was chosen. For different lengths of additional feed dipole l experimental DPs of parabolic antenna were taken for the main and cross-polarizations. For every length the levels of cross-polarized radiation were measured. The bestresults in terms of minimization of cross-polarized radiation were received for l = λ 2 when cross -polarization level decreased to 34 dB (see curve 2 in Fig. 4.15). As a redundant check of the performed experiment validity, the radiation characteristics were taken experimentally for parabolic antenna with the feed having cross-polarization on the current distribution similar to the distribution of the main current, i.e., of cos (x) type. It was achieved by rotation of additional dipole of waveguide–dipole feed by 90º. As experiments have shown, for this complicated feed the level of cross-polarized radiation significantly exceeds the level of parasite polarization in radiation of parabolic mirror with initial feed.

4.2 Synthesis of One-Mirror Antenna Linear Feed ...

135

Fig. 4.11 General view of antenna mirror

As known, cross-polarization level for the cut mirror is usually 50–55 dB below the level of the main radiation [8]. The received level of cross-polarization 40 dB should be accounted for by imperfection of measuring equipment introducing systematic error. In order to exclude this error, we will compare the cross-polarization level suppression achieved due to feed modification by synthesized amplitude-phase distribution with the level of cross-polarization for the cut mirror considering the latter as corresponding to antenna system with uniform DP. As can be seen from Fig. 4.15, where DPs are presented for the initial mirror with waveguide–dipole feed (1), for mirror with synthesized feed (2) and for cut

136

4 Experimental Testing of Theoretical Results

Fig. 4.12 The mirror design

mirror radiated by the initial waveguide–dipole feed (3), feed with amplitude-phase distribution synthesized in accordance with the requirement of cross-polarized radiation minimization produces DP that is very close to uniform one. It confirms the assertion made in this book on required amplitude-phase distribution that eliminates cross-polarized component in mirror antenna radiation. Some divergence from ideally uniform DP can be explained by disbalance between quite complicated amplitude-phase distribution and its very simple implementation. In presented antenna system suppression of the field with parasite polarization by 12–15 dB is achieved by relatively small modification of linear waveguide–dipole feed. Similar cross-polarization suppression in antenna systems with the mirrors of cut type is much more expensive, difficult to implement, and sometimes even impossible to realize due to insurmountable technical obstacles. In our case a minor feed modification allowed to achieve cross-polarization level similar to that in the systems with uniform DPs and polarization diagrams. It’s worth to notice that horn feeds, although theoretically having the required characteristics for cross-polarization suppression, in practice do not provide the required level of cross-polarization suppression. It is explained by the fact that during formulation of conditions required for mirror antennas’ feeds (which horn feeds meet quite well) antenna system’s scattering properties had not been taken into

4.2 Synthesis of One-Mirror Antenna Linear Feed ...

137

Fig. 4.13 The initial waveguide–dipole feed

account. Since antenna systems, having different scattering properties, can differently influence the incident wave, this approach suffers from potential drawbacks, neglecting some important properties of antenna systems. Therefore, approach to the problem of uniform PD synthesis for mirror antenna proposed in this book and based on consideration of scattering properties is more complete and, as experiments have proved, more correct in terms of elimination of cross-polarization influence on antenna system radiation. The so-called scalar feeds, while having satisfactory polarization properties, are difficult to implement due to their large dimensions fraught with practical difficulties. An open end of a round multimode waveguide can be used as mirror antenna feed that implements synthesized amplitude-phase distribution of cross current. The required distribution is created by exciting the superposition of the waves of TE21, TE01, TM01 types by, for example, double-waveguide T-type bridges. For two-mirror antenna systems the described method of synthesis can be applied assuming that six-pole [M] corresponds to the system “feed–counter-reflector”. Synthesis of required distribution is run as follows. Since DPs f ox and f ox y of the feed (which is counter-reflector in this case) are determined not only by properties of the feed itself but also by the shape of reflecting surface of counter-reflector, we have an opportunity to synthesize a uniform PD for two-mirror antenna system by means of not only feed synthesis by calculated distribution of cross-current for a given shape of

138

4 Experimental Testing of Theoretical Results

E

0,8

0,6

0,4

0,2

-4

-3

-2

-1

0

1

2

3

4

о

Fig. 4.14 Experimental DPs on the main and cross-polarizations in the plane slanted 45° toward antenna system’s main planes

4.2 Synthesis of One-Mirror Antenna Linear Feed ...

139

l 20 log |D|

D 0,08

-22

0,07 -24 0,06 1

0,05

0,04

-28

0,03 -30 0,02 -34 2 0,01

0

-40

3

1

2

3

(град)

Fig. 4.15 Experimental DPs of parabolic antenna on the main and cross-polarizations

counter-reflector surface, but also by means of counter-reflector profile modification which provides additional features for synthesis of required DPs f ox and f ox y . The performed experiment on synthesis of linear feed with assigned amplitudephase distribution of cross-current has shown that theoretical derivations given in this book have found their experimental proof in synthesis of linear feed that minimizes cross-polarized radiation of mirror antenna. Any parasite phenomena that would

140

4 Experimental Testing of Theoretical Results

worsen DP shape or other antenna characteristics have not been noticed during experiments. In order to create the required DP for counter-reflector on the main and crosspolarizations, we receive an opportunity for synthesis of uniform PD of two-mirror antenna system by both synthesis of the feed and synthesis of counter-reflector’s surface profile.

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