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 1786305402, 9781786305404

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Digital Communication Techniques

Series Editor Guy Pujolle

Digital Communication Techniques

Christian Gontrand

First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2020 The rights of Christian Gontrand to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019953804 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-540-4

Contents

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

History Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv List of Acronyms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxxix

Chapter 1. Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Modulation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Main reasons for modulation . . . . . . . . . . . . . . . . . . 1.1.2. Main modulation schemas . . . . . . . . . . . . . . . . . . . . 1.1.3. Criteria for modulation via electronics . . . . . . . . . . . . 1.1.4. Digital modulation: why do it? . . . . . . . . . . . . . . . . . 1.2. Main technical constraints . . . . . . . . . . . . . . . . . . . . . . 1.3. Transmission of information (analog or digital) . . . . . . . . . 1.3.1. Characteristics of the signal that can be modified . . . . . . 1.3.2. Amplitude and phase representation in the complex plane. 1.4. Probabilities of error . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Bit error ratio versus signal to noise ratio . . . . . . . . . . . 1.4.2. Demodulator: intended recipient decoder . . . . . . . . . . . 1.5. Vocabulary of digital modulation . . . . . . . . . . . . . . . . . . 1.6. Principles of digital modulations . . . . . . . . . . . . . . . . . . 1.6.1. Polar display . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2. Variations of parameters: amplitude, phase, frequency. . . 1.6.3. Representation in a complex plane. . . . . . . . . . . . . . . 1.6.4. Eye diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 1 2 2 2 6 7 7 10 11 12 14 17 19 19 20 21

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1.7. Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1. Frequency multiplexing . . . . . . . . . . . . . . . . . 1.7.2. Multiplexing – time . . . . . . . . . . . . . . . . . . . 1.7.3. Multiplexing – code . . . . . . . . . . . . . . . . . . . 1.7.4. Geographical (spatial) multiplexing . . . . . . . . . . 1.8. Main formats for digital modulations . . . . . . . . . . . 1.8.1. Phase-shift keying . . . . . . . . . . . . . . . . . . . . 1.8.2. BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3. The QPSK . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. Error vector module and phase noise . . . . . . . . . . . . 1.9.1. Plot QPSK reference constellation . . . . . . . . . . . 1.9.2. Effects of phase noise on 16-QAM . . . . . . . . . . 1.9.3. Phase noise: effects of the signal spectrum . . . . . . 1.9.4. Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.5. Spectrum analyzer . . . . . . . . . . . . . . . . . . . . 1.9.6. Measures of the error vector module of a signal modulated by a noisy 16-QAM . . . . . . . . . . 1.10. Gaussian noise (AWGN) . . . . . . . . . . . . . . . . . . 1.10.1. AWGN channel . . . . . . . . . . . . . . . . . . . . . 1.10.2. Ratio between EsNo and SNR . . . . . . . . . . . . 1.10.3. Behavior for real and complex input signals . . . . 1.11. QAM modulation in an AWGN channel . . . . . . . . . 1.11.1. QAM demodulation . . . . . . . . . . . . . . . . . . . 1.11.2. Detecting phase error . . . . . . . . . . . . . . . . . . 1.12. Frequency-shift keying . . . . . . . . . . . . . . . . . . . 1.12.1. Binary FSK . . . . . . . . . . . . . . . . . . . . . . . . 1.13. Minimum-shift keying . . . . . . . . . . . . . . . . . . . 1.13.1. Bit error ratio (BER)/Gaussian channel . . . . . . . 1.13.2. Typical analytical expressions used in “berawgn” . 1.14. Amplitude-shift keying . . . . . . . . . . . . . . . . . . . 1.14.1. On–off keying . . . . . . . . . . . . . . . . . . . . . . 1.14.2. Modulation at “M states” . . . . . . . . . . . . . . . 1.15. Quadrature amplitude modulation . . . . . . . . . . . . . 1.15.1. Limits on theoretical spectral efficiency . . . . . . . 1.15.2. I/Q imbalance . . . . . . . . . . . . . . . . . . . . . . 1.15.3. QAM-M constellations . . . . . . . . . . . . . . . . . 1.16. Digital communications transmitters . . . . . . . . . . . 1.16.1. A digital communications receiver . . . . . . . . . . 1.16.2. Measures of power . . . . . . . . . . . . . . . . . . . 1.16.3. Power of the adjacent channel. . . . . . . . . . . . . 1.16.4. Frequency measures . . . . . . . . . . . . . . . . . . 1.16.5. Synchronization measures . . . . . . . . . . . . . . . 1.17. Applications. . . . . . . . . . . . . . . . . . . . . . . . . .

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81 81 83 84 85 85 89 90 93 94 95 97 98 99 99 101 104 105 106 109 117 118 120 121 121 123 129

Contents

vii

1.17.1. Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17.2. Digressions or precisions, around modulations. . . . . . . . . . . .

129 131

Chapter 2. Some Developments in Modulation Techniques . . . . . .

137

2.1. Orthogonal frequency division multiplexing . . . . . . . . . . . . . 2.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2. Multicarrier modulations . . . . . . . . . . . . . . . . . . . . . . 2.1.3. General principles . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4. How to choose N? . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5. Practical aspects. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.6. COFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7. Equalization and decoding . . . . . . . . . . . . . . . . . . . . . 2.1.8. The multiuser context . . . . . . . . . . . . . . . . . . . . . . . . 2.1.9. Code division multiple access . . . . . . . . . . . . . . . . . . . 2.1.10. Schematic ordinogram . . . . . . . . . . . . . . . . . . . . . . . 2.1.11. Data in OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.12. OFDM: advantages and disadvantages . . . . . . . . . . . . . 2.1.13. Intermediate conclusion . . . . . . . . . . . . . . . . . . . . . . 2.1.14. QPSK and OFDM with MATLAB system objects . . . . . . 2.1.15. FDM versus OFDM: difference between FDM and OFDM . 2.2. A note on orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Global System for Mobile Communications . . . . . . . . . . . . . 2.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Forming a GSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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137 137 138 143 145 145 147 149 150 150 152 155 156 157 159 162 170 174 174 175 178 178 178 182

Chapter 3. Signal Processing: Sampling . . . . . . . . . . . . . . . . . . .

183

3.1. Z-transforms . . . . . . . . . . . . . . . . . 3.1.1. Transforms . . . . . . . . . . . . . . . . 3.1.2. Inverse z-transform . . . . . . . . . . . 3.2. Basics of signal processing. . . . . . . . . 3.3. Real discretezation processing. . . . . . . 3.3.1. Real discretization comb. . . . . . . . 3.3.2. Real sampled signal . . . . . . . . . . 3.3.3. Blocked, sampled signal . . . . . . . . 3.3.4. Model of real sampled signals . . . . 3.3.5. Uniform quantifying . . . . . . . . . . 3.3.6. Signal quantification step: rounding . 3.3.7. Signal quantification step: troncature

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3.3.8. Quantification solution . . . . . . . . . . . . . . . 3.3.9. Additive white Gaussian noise (AWGN): a simple but effective model . . . . . . . . . . . . . . . 3.3.10. Quantification error and quantification noise . 3.3.11. In practice, sample and hold and CAN . . . . 3.3.12. Spectra of periodic signals . . . . . . . . . . . . 3.3.13. Non-periodic signal spectrums . . . . . . . . . 3.3.14. PSD versus delay . . . . . . . . . . . . . . . . . 3.3.15. FT of a product: the Plancherel theorem . . . 3.3.16. Periodic signal before sampling. . . . . . . . . 3.3.17. Spectrum of sampled signals . . . . . . . . . . 3.3.18. Conditions for sampling frequency . . . . . . . 3.4. Coding techniques (summary) . . . . . . . . . . . . .

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Chapter 4. A Little on Associated Hardware . . . . . . . . . . . . . . . . .

203

4.1. Voltage-controlled oscillator. . . . . . . . . . . . . . 4.2. Impulse sensitivity function . . . . . . . . . . . . . . 4.3. Phase noise . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. At passage to zero . . . . . . . . . . . . . . . . . 4.3.2. At the peaks . . . . . . . . . . . . . . . . . . . . . 4.4. Phase-locked loop . . . . . . . . . . . . . . . . . . . . 4.4.1. Study of a fundamental tool: the PLL . . . . . . 4.4.2. Schematic structure of the PLL . . . . . . . . . . 4.4.3. Operation of the loop: acquisition and locking. 4.4.4. Charge pump . . . . . . . . . . . . . . . . . . . .

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Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

233

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243

Appendix 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

291

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

Acknowledgements

Acknowledgements are owed to the non-exhaustive list below: Chafia Yahiaoui from the Ecole Supérieure d’Informatique d’Alger (Technical University of Algeria), and my telecom colleagues at INSA Lyon: Guillaume Villemaud, Jean-Marie Gorce, Hugues Benoit-Cattin, Attila Baskurt, Stéphane Frenot, Thomas Grenier, Jacques Verdier, Gérard Couturier, Patrice Kadionic, Alexandre Boyer and Carlos Belaustegui Goitia among others, for their detailed observations, as well as their helpful commentaries. Kind acknowledgements also go to Omar Gaouar, my kindly mate at INSA FES, a networker, but also a music buff. This work is supported by the UpM (Union pour la Méditerranée – Mediterranean Union). It has been accomplished at the Centre d’Intégration en Télécommunication et Intelligence Artificielle (Center of integration in telecommunications and artificial intelligence), INSA FES, UEMF.

Preface

Impressive developments in Information and Communications Technologies (ICT) have naturally led universities and technical schools to develop the electrical engineering (EI) training they provide. This is particularly true in the wireless communications sector. In fact, communications as part of the transmission of data, whether verbal or in video form, is finding more and ever more varied applications. It is becoming necessary for future graduates to understand and master problems linked to the implementation of radio links, depending on the environment, formatting and source data flow, on the power available to the antenna and on the receiver’s selectivity and sensitivity. This book only requires an introductory level of understanding in mathematics. It does not aim to suffice in and of itself, but rather to convince the reader of the wealth of this domain and its future, to provide good building blocks that will lead to fruition elsewhere. Manufacturers’ concise application notes also seem vital for any researcher/engineer. Technological innovation plays a very important role in the ICT domain. It therefore seems necessary for training courses now to provide welladapted and innovative content in teaching and associated tools, while still mastering, as well as possible, the fundamental nature of teaching, which is the only guarantee of a solid and lasting education. This book is aimed at professional diploma students and engineering and masters students. However, it could also perhaps be aimed at researchers in related domains, such as that of hardware, with, for example, phase-locked loops and their central components: voltage-controlled oscillators, and the

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famous associated phase noise. Of course, there is an entire domain linked to what is known as firmware, which must be taught, but there are also mathematical tools already in use, for relativity for example, or cryptography, indeed, older forms of coding must be revisited, such as that of Claude Shannon. Christian GONTRAND November 2019

Introduction

The word “communication” is now a catch-all in modern society; in its most basic sense, it makes it possible to share information. A department that in any French university or technical school might historically have been labeled as “humanities” (at the end of the 1960s, particularly focused on human resources or sociology); was often later reduced to “communication and humanities”, both terms having become interchangeable in the meantime. Perhaps, now devoid of a clear meaning, nothing will be left apart from the term communication? This word must not be amalgamated into others: information (transport), (en)coding. Perhaps later semantics are involved in this book, in a strict, technical sense, certainly not in any modernistic sense. I.1. Why digitize the world? For broadband communications, transmissions are limited by physical constraints, such as noise or interference, resulting from system imperfections and physical components modifying the transmission of the signal sent. Distortion of the signal over the course of the broadcast is, similarly, a concern. Hence, there is a need for a clear separation of the signals sent, so that, in practice, they remain distinct when they are received. The transmission of a set of signals undergoes data dispersion over time, leading to intersymbol interference. Signals reflected from buildings, the ground or vehicles cause this dispersion, depending on the length of the paths traveled. The significance of this phenomenon depends on the

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frequency (above all high frequency), which can vary stochastically, via, for example, the signal’s phases over time (after reflection of obstacles: echoes). They often generate signals, added destructively, or at reception. The resulting signal will therefore be very weak, or sometimes almost nonexistent. These signals can also be added constructively; the final signal will therefore be more powerful than one that arrives via a direct path. We note that multiple paths do not present only drawbacks, since they enable communication even when the transmitter and receiver are not in direct contact (for example, Transcontinental Communications). A signal is often corrupted when it crosses different paths between transmitter and receiver: data bits that reach the receiver are subject to delays. This distorted signal will be interpreted poorly by the receiver. In broadband communications, signals are limited by constraints: transmission errors are attenuated when the signal is digitized. For example, for the voice, the amplitude of the signal is typically measured 8,000 times per second and its value is coded in an 8-bit sequence (of 0s and 1s) – we refer here to sampling. The receiver decodes the sequence of the original signal, thus reconstructing the signal sent. Using only 0s and 1s leads to a low (or indeed non-existent) probability of error. The propagation channel can be modeled via an impulse response (see: linear system, Dirac comb); the signal received r(t) is therefore none other than the filtering of the signal sent x (T) through the propagation channel c (t) and can therefore be written in baseband, via a convolution to which noise is often added (see Langevin term added), modeling the system imperfections. Reference is made to frequency-selective channels when the signal transmitted x (t) occupies a [−W / 2, W / 2] frequency band, which is wider than the propagation channel’s coherence bandwidth, c (t), (propagation channel defined as the inverse of the propagation channel’s maximum delay spread Tr). In this case, the frequential components of x(t) separated from the coherence bandwidth undergo different attenuations. In broadband digital systems, symbols are often sent at a regular interval of time T, at a maximum path delay time Tr; the signal received at an instant t can be expressed as a weighted sum (affected by path attenuations) of the signal transmitted simultaneously (the propagation time for the electromagnetic waves is often neglected, as these propagate at the speed of light) and signals sent at previous instants, a multiple of the (sampling) period.

Introduction

xv

I.2. Temporal representation of a channel The coefficients of the propagation channel are given by the values taken for various multiple moments of T: [|c(0)|, |c(T)|, |c(2T)|,|c(3T)|, |c(4T)|, |c(5T)|]. If we focus on mobile radio, between buildings, at 5Ghz, T is in the order of 50 ns; Tr equates to 450 ns. Designers need to reduce interference caused by multiple reflections of the signal and extract the signal. Equalization means balancing the effects of distortions resulting from these multiple paths. To do this, it is necessary to identify the attenuation coefficients that model the effect of the propagation channel c (t). Current technologies, used in industrial applications, call on training sequences; a “chosen sequence” is sent regularly, known by the sender and the intended recipient. This method makes it possible to know the channels’ different phase shifts and delays, and gives good results in practice. On the other hand, if the sampling period is too short in relation to the delay Tr (as is the case with high flow transfers; the number of coefficients c(iT) (typically: 0 ≤ i ≤ 5) to be determined can be great, see matrix inversion). Thus, the transmission of high flows when there are several paths present can quickly increase the complexity and therefore the cost of the terminals. A channel’s selective frequency: the signal to be transmitted has frequency components attenuated differently through the propagation channel. This phenomenon is produced when the signal has a broader frequency band than the propagation channel’s consistent band. A channel’s consistent band is defined as the minimum pass band for which losses from the two channels are independent. This phenomenon is one of the main obstacles to transmission reliability: in fact, it is necessary to estimate the channel (which triggers a loss of flow in moving environments) and also to equalize it (which increases receiver complexity). Digital equalizer complexity depends on the number of the propagation channel’s paths (determined by the relationship between the duration of equalization, Tr, and the sampling period, T), but also the type of constellation transmitted – see Fresnel diagram. The bits are transmitted in the form of symbols rather than as they are. The number of bits contained in each symbol indicates the size of the constellation; the greater this size, the higher the flow. The average size of these constellations generally has a fixed threshold because of the power limits at the terminals.

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Whyy is not it possible p to increase the flow indefinitely by inncreasing constelllation size? The T transmisssion rate caan be increassed by enlarrging the constelllation. But, if i we speak of the rate as a the numbeer of bits perr second arrivingg perfectly att the receiverr, then this iss impossiblee; the greaterr the size of the constellationn (at a fixeed power, which w is alw ways normallized for mission cost), then the closer the vaalues of the symbols questionns of transm between transmittted. It is noot easy thereffore for the receiver r to discriminate d two values riddled with w errors resulting r from noise. Wee can really increase w (i.e. transm mission speeed) by increaasing the coonstellation. The rate the flow thereforre has a threeshold calledd channel cap pacity. The idea of an eerror-free transmisssion was sccarcely imagiined by scien ntists at the end of the 1950s. At this tim me, it was naatural to reduuce the probability of traansmission eerrors by reducingg binary flow w, thus defiining channeel capacity. It I was only with the work off Claude Shaannon at the start of the 1920s that encoding e em merged to solve thhis dilemma. I.3. The e need for coding

Figure I.1.. Different type es of codes

So thhat the intennded recipiennt can underrstand the message m broaadcast, it must bee as close ass possible to the initial message. m Whhatever the pprinciple

Introduction

xvii

behind the broadcast, disruption will be added to the information and will distort it. It is therefore necessary to eliminate this interference; this is the first goal of coding. I.4. Synoptic bases on information theory

Figure I.2. Shannon diagram (source: item of interest for the recipient. Channel: origin of the phenomenon of propagation but also of disruption)

We will consider a discrete channel without memory. The word “discrete” refers to the fact that the real signal has already been transformed, if it is analog, into a binary digital signal, which is no longer continuous. “Without memory” means that the noise is modeled via a conditional probability of B given that A is independent of time. From a theoretical perspective, we will approximate this channel using a white Gaussian channel, which means that all the bits have the same broadcast probability, whatever their position. H entropy: this defines the quantity of information provided by the source; it depends on the 0’s and 1’s probability of appearance. If a single message is possible, the entropy is null. The entropy makes it possible to measure the quantity of information lost after noisy transmission or encryption. I.4.1. Shannon–Hartley theory There is a quantity of maximum theoretical information that can be transmitted by the channel. For any channel, there is a coding algorithm such

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that the message sent by the source is received with an arbitrarily weak error rate. I.4.1.1. A little math A message X is a set of basic elements xi characterized by their probability of occurrence. The quantity of information it transmits is a measure of its unpredictability: the more predictable a message is, the less information it provides. If x is a basic message and p(x) its probability of transmission, then the quantity of information h(x) it transmits is defined by h(x) = −log (p(x)). We note that if this message has a probability of 1, the information broadcast, h, is null. Source entropy: this is defined by the source’s average quantity of information, which translates mathematically into the expected value of the intrinsic quantity of information from each basic message. H(X) therefore depends only on the probability of broadcast of 0 or 1. k

H(X) = E(h(xi)) = −Σ p (xi ) * log2 ( p(xi ) ) i=1

– Unit: it is shannon. We see that entropy is maximum for a uniform broadcast probability, i.e. for p(xi = 0) = p(xi = 1) = 1/2, which is the case in a symmetrical binary channel. H(X/Y) is also defined, called ambiguity or conditional entropy, which is linked directly to the probability of error of the channel’s transmission. k

H(X/Y) = E(h(xi/y)) = −Σ p(yj) * H(X/Y = yj) j=1

Introduction

xix

with k

H(X/Y = yj) = −Σ p(xi/yj) * log2 ( p(xi/yj) ) i=1

In fact, a binary variable X can take only two values: 0 or 1 (Figure I.3). In the modeling of the binary symmetrical channel, whatever its initial value, there is a probability of error p = pe, so that the bit can be changed into its opposite, and there is a therefore a probability p = 1 − pe that the bit can be transmitted.

(1 – p)

0

0

p

X

Y

p

1

(1 – p)

1

Figure I.3. Probability of error (symmetrical binary channel)

For entropy values, we therefore have: H(X/Y) = −plog2(p) − (1 − p)log2(1 − p) si 0 this is the fourth column of H; this means that the fourth bit of the word sent is erroneous. As we are dealing with a binary, it is enough to replace the 1 with a 0, and find the corresponding word (here c2). NOTE.– There are still undetectable configurations of errors. I.6. Coding techniques I.6.1. Interleaving Interleaving is a coding technique that consists of permutating a sequence of bits to distance errors from one another as much as possible. The errors are distributed all along an s sequence; the percentage of errors at each place

Introduction

xxvii

is therefore not very high and they can therefore be corrected. The bits will then be put back into order to recover the initial message. Concretely, interleaving is used, for example, over the CD: if it has a score, the errors are concentrated in the same place; they are distributed along a long sequence so that they can be detected and then corrected. This is Reed–Solomon block coding. Today, there is no rule for interleaving; different interleavers must be tested to choose the one that gives the best result. For turbocodes, the interleaver is an integral part of the code design (the interleaver is chosen depending on the code). But this technique poses a problem: an interleaver is often designed for a precise length of code; it will no longer work if the error packet extends over a great length. In turbocodes, Golden interleavers are used most as they have good spreading properties. NOTE.− Interleaving is also used for convolutive codes. I.6.2. Convolutive codes I.6.2.1. General remarks The principle behind convolutive codes was invented in 1955 by Peter Elias, a professor at MIT. Unlike block codes, which cut the message into finite blocks, we will consider here a semi-infinite sequence of information that passes through several shift registers. The number of these registers is called code memory. For example, we consider the convolutive code shown in Figure I.6.

X

Figure I.6. Convolutive coder

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D Digital Communication Techniqu ues

It haas a memory equal to 2. At A the instan nt t, we thereefore consideer bits at, at-1, at-2. At the outpuut, we will have

(moduloo addition 2)). We represent thhe code usinng a transittion diagram m (Figure I.7). This mbination off shift registters, the schema describes, for each poossible com nput bit. Eachh case in thee schema coders’ output messsage, dependding on the in s registerrs. The digitts beside thee arrows correspoonds to a sttate of the shift indicatee the input biits and the coded c bit corrresponding to t the transittion. For examplee, if the codeer, initializedd at 00, receiives the sequuence 101, thhe coded messagee leaving willl be 11 10 00 (Figure I.8 8).

Figure I.7. Transition dia agram. For a color c version of o this figure, see www.iste e.co.uk/gontra and/digital.zip

Figure I.8. Response R to message m 101

Introduction

xxix

I.6.2.2. RSC and NSC codes Two categories of convolutive codes are particularly interesting to study: – Recursive systematic convolutional (RSC) codes: a convolutive code is called systematic if the input is recovered at its output. In addition, we call a code recursive if its shift registers are “fed” by their content. For example, an RSC code is shown in Figure I.5: it is systematic, since its output X is identical to the input. It is also recursive since, at input, we find shift registers for information that is also found in these same registers (“the information also circulates from right to left”). It has been shown experimentally that only RSC codes are likely to reach Shannon’s limit.

X

Figure I.9. RSC coder

– Non-systematic convolutional (NSC) codes: these have the advantage over systematic codes of providing more information: any of the coder’s output bits is informed about a number of bits from the source message. The decoder therefore has more elements and corrects more errors. It is for this reason that NSC codes have mainly been used since the start of the 1990s. I.6.2.3. Decoding example: Viterbi algorithm The algorithm most widely used in sequential decoding was invented by Andrew Viterbi, in 1967. It makes it possible to find the most probable sequence of states that might have produced the sequence measured. Although the message is initially a semi-infinite sequence, in practice, it is

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Digital Communication Techniques

broken down into very large blocks (from 100 to 100,000 bits, or more) and the coder is initialized at 0 between each block. The principle behind the algorithm is to examine all the message’s possible paths through the state-transition diagram by gradually deleting the least probable. We code and then decode the Viterbi algorithm. For a simple explanation of the algorithm’s operation (Figure I.9), we use the trellis code representation. This diagram is a variant of the transition diagram that represents the possible successive states of the shift registers and the transitions between these states. First, we calculate, for each word possible, its word metric, i.e. its distance from the word received. Then we calculate the accumulated metrics of the different paths. We then keep only the minimum distances and continue in this way to the end of the block. In fine, we carry out the most probable journey and we recover the original sequence: the error has been corrected.

Figure I.10. Convolutive codes

Introduction

xxxi

Figure I.11. Viterbi algorithm a

Figure I.12. Basic calculation units s in the Viterbi decoder

I.6.3. Turbocodes T s I.6.3.1. Characteriistics S theoretical Turbbocodes makke it possiblee to come verry close to Shannon’s limit, seet out in the Shannon–H Hartley theoreem, which defines the m maximum quantityy of non-errooneous data per p channel.

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Diigital Communiccation Techniqu ues

We can c calculatee the capacitty of a Gausssian channell with the help of the relationship: C = ½ log2(1+2R Eb/N0), Cc ) where R represents the yield off the code an nd Eb/N0 the energy e per bbit of the input signal over the density of the chaannel’s noise. The weaaker this relationship, the moore effectivee the error-co orrecting codde, since it m makes it possiblee to recoverr the initial message with w a low probability p oof error, despite substantial noise. n c trace Cc using Eb/N0 (dB). We can On the t graph bellow are represented the theoretical t liimits in the ccase of a continuoous channel (“Shannon limit”) and in the case of a binary channel that hass been subject to binarry phase mo odulation – see below (“BPSK limit”).

Figure I..13. Gaussian n channel capa acity versus Eb/N0 (dB). Copyright © C. C C Schlegel, Trrellis and Turb bo Coding, IEE EE Press, 200 04

A turrbocode’s efffectiveness is i also defineed by: – thee bit error raate, which makes it possiible to measuure the propoortion of errors thhat the code is not able too correct; – thee flow, whichh correspondds to the codee’s running speed. s

Introduction

xxxiii

I.6.3.2. The different types of turbocodes

Figure I.14. Turbocodes. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Advantages of block turbocodes Block turbocodes (BTC) are very flexible in terms of complexity and the code’s yield. They support blocks of any size and a great variety of yield between 1/3 and 0.98 but, above all, offer excellent performances at high yields. Unlike convolutive turbocodes, BTCs have no error floor. They can be used when we want an extremely low error rate. In addition, the BTCs have decoders that can act at very high speeds. Advantages of convolutive turbocodes In addition to giving better performances for a low signal/noise ratio, it is easier to pass from one yield to another with convolutive turbocodes (CTC). Turbocoding converges more quickly with CTCs; they are therefore more effective for a low Eb/N0 rate. They are, moreover, more practical when we want to send continuous data. In theory, CTCs can support any yield, but in practice, they are used for yields of ½, 2/3, ¾. In addition, a CTC requires less memory than a BTC.

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I.6.3.3. Crossword analogy A simple analogy makes it possible to understand the principle behind turbocodes: this is a crossword with two dimensions – lines and columns. The lines, with their respective definitions, represent the first level of coding, and the columns the second. This definitions correspond to the redundancy introduced by the code. The metaphor is also valid for the decoder: the person filling in the crossword uses horizontal definitions to fill some boxes, then the vertical definitions enable them to confirm or remove their previous choices and to carry on filling in the boxes. By successive iterations, they thus try to decipher the crossword. Sometimes, they may fail and boxes may remain empty or incorrectly filled; these are the very occasional errors that are not corrected by turbocodes. This is how Claude Berrou (Telecom Rennes) described the subject: Our coding adds redundancy to the transmission. If, for example, we send “Blanc et immaculé” at the same time, and “Flanc et immaculé” is received, it will still be understood! With turbocodes, a sort of crossword grill is sent with definitions in the lines and columns. This is verified in one sense and also in the other. This is the turbo effect, or iterative coding: as in a turbo engine, where the energy of the exhaust reinforces the admission, via a compression turbine; the coding by column reinforces the coding by line. He also adds the fact that “NASA had created a less effective decoder that cost million dollars and took up the space of two marine canteens in place of our chips (microchips) which can be held in one hand”2.

2 Source: www.espace-sciences.org, www.institut-telecom.fr.

History Pages

1969 – ARPANET (Advanced Research Projects Agency Network – USA Army): the first Internet system is created. 1979 – In Japan, the first cell phone network is created.

Figure H.1. Telecom: curriculum vitae

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Figurre H.2. The pio oneers

Figure H.3. Telegraph//telephone

History Page es

Figure H.4. Begin nnings of radio o communicattions

Figure H.5. H First deplloyments

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Digital Commun nication Techniq ques

F Figure H.6. Th he emitter of Lyon – La Dou ua

List of Acronyms

ADC: Analog Digital Converter ADSL: Asymmetric Digital Subscriber Line AMS: Adaptive Modulation Scheme AP: Access Point APSK: Amplitude- and Phase-Shift Keying ASK: Amplitude-Shift Keying BER: Bit Error Rate, Binary Error Rate BPSK: Binary Phase-Shift Keying CDMA: Code Division Multiple Access COFDM: Coded Orthogonal Frequency-Division Multiplexing CP: Cyclic Prefix DAB: Digital Audio Broadcasting DAC: Digital Analog Converter DMT: Discrete Multi-Tone DPSK: Differential Phase-Shift Keying

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Digital Communication Techniques

DSP: Digital Signal Processing DVB-T: Digital Video Broadcasting – Terrestrial FEC: Forward Error Coding FFT: Fast Fourier Transform FIR: Finite Impulse Response FSK: Frequency-Shift Keying ICI: Inter-Carrier Interference IDFT: Inverse Discrete Fourier Transform IFFT: Inverse Fast Fourier Transform MC-CDMA: Multi-Carrier CDMA MCM: Multi-Carrier Modulation MMSE: Minimum Mean Square Error MT: Mobile Terminal NRZ: No Return to Zero OFDM: Orthogonal Frequency-Division Multiplexing OFDMA: Orthogonal Frequency-Division Multiple Access PAPR: Peak to Average Power Ratio PLL: Phase-Locked Loop PSD: Power Spectral Density PSK: Phase-Shift Keying QAM: Quadrature Amplitude Modulation VCO: Voltage-Controlled Oscillator

1 Modulation

1.1. Modulation? 1.1.1. Main reasons for modulation – In the modulation process, one characteristic of a high-frequency (passband) carrier signal is modified according to the instantaneous amplitude of the signal to be processed (in its baseband). – Why does modulation make sense for signal transmission (distance, etc.)? – Several signals are transmitted on the same channel. – Capacitative or inductive devices need high frequency (carrier) to operate (to ensure Stability and a good Noise Rejection). 1.1.2. Main modulation schemas – Examples of application: transmitting audio and video signals. – Mobile radio communications, such as cell phones. Basic modulation types are as follows: – amplitude modulation (AM): modifies the amplitude; – frequency modulation (FM): changes the frequency; – phase modulation (PM): changes the phase.

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

2

Digital Communication Techniques

1.1.3. Criteria for modulation via electronics Requirements for signal processing are as follows: – speed; – reliability and precision; – low energy consumption. Electronics serve to transport information at least energy cost, as quick as possible, in the greatest quantity possible, with the highest quality possible and with maximum reliability. NOTE.– In electrical engineering, the electrical signal serves to transport the energy and not the information. 1.1.4. Digital modulation: why do it? Digital modulation provides more information capacity, compatibility with digital data services, better security levels, better communication quality and wider availability than analog, for example. 1.2. Main technical constraints – available passband; – usable power; – level of system background noise. The radiofrequency (RF) spectrum is aimed at many users and communication services are only growing. Digital modulation has greater capacity to propagate large quantities of information than analog. If all the frequencies that form the signal to be transmitted are found in the passband supporting the transmission, this signal can be applied to the line directly.

Modu ulation

3

Figure 1.1 1. Transmissio on of a signal in baseband. For a color version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

If alll or some off the frequenncies forming g the signal to t be transm mitted are found outside o the baandwidth of the transmisssion medium m, a modulaation will make it possible to shift all the frequencies,, in blocks, to t higher freqquencies (see the so-called heeterodyne proocess) (Figurre 1.2).

Figure 1.2.. Transmission n of a signal via v modulation. For a color version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

Eachh frequency f of the signaal “becomes”” f + Δf. If Δff suffices, all frequenciess of the sign nal thus moduulated are trranslated within the t passbandd. Once this procedure p haas been madde by the trannsmitter,

4

Digital Communication Techniques

a modulator for translating the frequencies and a demodulator are needed at reception to regroup the frequencies, at the initial value, into blocks. As the system is generally conversational, each station is equipped with a MODEM (MODulator-DEMdulator).

Figure 1.3. Heterodyne system. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Simplicity/bandwidth compromise The need to have a broad passband limits the number of users. On the other hand, transmitters and receivers can be used with a lower passband. The increasing transition toward spectral efficiency calls for ever greater complexity in hardware; this requires sophisticated designs, it is not simple to manufacture or test. There are many compromises, depending on whether communication is aerial, wired, analog or digital. Developments in industry An important transition has occurred in recent years from analog AMs and FM/PM, towards digital modulation techniques.

Modu ulation

5

Figure 1.4. Trends in n industry

NOTE.–– The main types of multiplexing m (or “multiplle access”) aare time divisionn multiple access (TD DMA) and code division multiplee access (CDMA A). These arre two differrent ways of adding divversity, enabbling the signals to t separate from fr one anoother (see bellow). The strong trennd toward digital mod dulation, coompared to analog, munication, it i is more com mpatible provides more data capacity andd more comm with diggital data serrvices and itt provides more m data seccurity, higherr quality communnication andd faster systtems. The constraints arre as follow ws: more bandwiddth available, the noisee and powerr allowed thhe system. The RF spectrum m should be shared; more and more users are innvolved. Commuunications seervices are growing. Digital D moduulation scheemas are finder greater g possiibilities for conveying c high h quantitiees of inform mation, if we com mpare them too analog moddulation scheemas. Comprromise bettween simp plicity and bandwidth b Therre is a basic point p in com mmunication systems. A simple hardw ware can be usedd for transmittters and receeivers, but th his requires the t spectrum m to have a broadd bandwidthh, hence feewer users. On the otther hand, complex transmittters and receivers can be used to traansmit the saame informattion over a lower transmissionn bandwidthh; in contrastt, this requirees very sophhisticated

6

Digita al Communicatio on Techniques

materiall. This com mpromise exiists if comm munication occurs via aiir or via wires, iff it is analog or digital.

Figure 1.5. Digital modu ulation chain

F Figure 1.6. Fundamental compromises

1.3. Tra ansmission n of inform mation (ana alog or digital) To trransmit a siggnal in the airr, there are essentially thrree stages: 1) a pure carrier is created att the receiverr; 2) thhis carrier is modulated with w the inforrmation to bee transmittedd; 3) att the receiveer, any modiification or change c of siignal is deteccted and demoduulated. Losssless codingg (see files, binary flo ow) is appliied to the source’s propertiies to reducee the volumee of data to be transmittted; in fact, iit means eliminatting redunddancies (“ass” for comp pression). Lossy L codinng takes accountt of the receiiver’s properrties; with lossy coding, it is informaation that is a prioori useless orr uninformattive, that is deleted. d In faact, audio annd video sourrce coders use u both losssless and losssy codes via digittization of annalog data.

Modulation

7

1.3.1. Characteristics of the signal that can be modified There are therefore only three of a signal’s characteristics that can be changed over time: size, phase or frequency. However, the phase and the frequency are simply different ways of visualizing or measuring the main changes in the signal. – For AM, the amplitude of a high-frequency conveyor signal (carrier) is changed in proportion to the instantaneous amplitude of the message’s modulation. – FM is the most popular analog modulation technique used in communications systems; the amplitude as well as the modulating carrier is kept constant, while its frequency is changed by the message’s modulation signal (this frequency excursion may be even greater, the higher the central frequency). – Amplitude and phase may be modulated simultaneously and separately, but it is difficult to generate and detect. In its place, in practical systems, the signal is separated into another set of independent components: 1 (in-phase) and Q (quadrature). These components are orthogonal, so they do not interfere with one another. 1.3.2. Amplitude and phase representation in the complex plane The periodicity of the frequency response is clear; since after completing a circle (2 π: frequency variation from 0 to 1, or 0 to Fe), the vector will be found in the same position (Figure 1.7.). z = a + ib Amplitude (module): ϼ= + 2) Phase (argument): = arctang (b/a) +/- k Remember: z= ϼ ∗ z=ϼ( + .

)

Figure 1.7. Polar/rectangular (Cartesian) conversion

8

Digital Communication Techniques

Figure 1.8. Amplitude, frequency, phase and/or amplitude shift-keying

Digital transmission systems broadcast information between a source and a receiver or an intended recipient using a physical support such as a cable, optical fiber or propagation along a radioelectric channel. The signals transported may be, ab initio, digital in origin, as in data networks, or analog in origin (speech, images, etc.). Where it is analog, it is then digitized. The transmission system should therefore convey signals containing the information from the source to the receiver, with as few errors as possible; this is the domain of reliability. Below is a synoptic schema of a modulator, placed in a global digital transmission system: modulator/demodulator.

Modu ulation

9

F Figure 1.9. Ge eneral schema a of a modulattor

– Thhe source seends a digital message in the form m of a seriess of bits (binary digits). – Thhe coder can sometimes delete binary y elements, a priori insiggnificant (data coompression or source coding; c com mpression onlly being poossible if there iss redundanccy), or, on the contrarry, additionn of redunddancy to informaation with a view v to proteecting it against noise andd parasites present in the trannsmission chaannel (channnel coding). Channel codding it only possible if the fllow from thee source is less than the transmissionn channel’s capacity (the proobability off error Pe teends, in this case, tow ward 0, according to researchh by Hartley––Shannon). – Moodulation haas the role off adapting th he signal specctrum to the channel (physicaal medium) on o which it is sent. – Onn the side off the receiveer, the demodulation andd decoding ffunctions are in faact the respective inversees of the mod dulation andd coding funcctions on the sidee of the receivver. Therre is an essenntial charactteristic that makes m it possible to com mpare the differennt transmissioon techniquees with one an nother: – Thhe probabilitty of error Pe P per bit transmitted t m makes it posssible to evaluatee the quality of a transmiission system m. It dependss on the transsmission techniquue used as well w as the sysstem source//coder/modullator/channell.

10

Digital Communicattion Techniquess

1.4. Pro obabilities of error

Figurre 1.10. Proba ability of error. For a color version v of thiss figure, see www.iste.co.uk w k/gontrand/dig gital.zip

Figure 1.11. Probability of wrong deciisions. For a color c version w k/gontrand/dig gital.zip of thiss figure, see www.iste.co.uk

Modulation

1.4.1. Bit B error ratio versus signal to noise n ratio

Figure 1.12. Errror rate by bitt, for a unipola ar and antipod dal transmissio on, a according to th he signal to no oise ratio. For a color versio on of this figure e, see www.iste e.co.uk/gontra and/digital.zip

Figure 1.13. Prrobability of errror in erfc (errf complementtary). For a collor version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

11

12

Digital Communication Techniques

Figure 1.14. Probability of error by bit. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

1.4.2. Demodulator: intended recipient decoder The three essential characteristics that make it possible to compare the different techniques with one another are as follows: – The probability of error Pe per bit transmitted is an important criterion for the quality of a transmission system. It depends on the transmission technique used, as well as the channel on which the signal is transmitted. Pe is a theoretical value, for which an unbiased estimation, in the statistical sense, is the Bit Error Rate: BER. – The spectral occupation of the signal sent should be known so the passband of the transmission channel can be used effectively. The current trend is increasingly to use modulations with high spectral efficiency. – The complexity of the receiver, which is there to reform the signal sent, is the third major aspect of a transmission system.

Modulation

13

Figure 1.15 5. The transm mission chain

Figu ure 1.16. Ordiinogram of a transmission t c chain

The purpose of modulation m i therefore to adapt thee signal to bee sent to is c of “playing” “ onn one or the trannsmission chhannel. Thiss operation consists more parameters off a carrier wave w S(t) = Acos(ct + ϕc) centeredd on the channell’s frequencyy band. The modifiable parameters p arre therefore: – thee amplitude: A; – thee frequency: fc = ωc/2π; – thee phase at oriigin: ϕc.

14

Digital Communication Techniques

In binary modulation, information is transmitted with the aid of a parameter that takes only two possible values. In the modulation procedures M-ary (M = 2: binary modulation; M-ary, e.g. bi-nary), information is transmitted with the aid of a parameter that takes M values. This makes it possible to link a word of n binary digits to a modulation state. The number of states is therefore M = 2n. These n (= log2 M) digits result from breaking into packets the n digits from the bit stream that come from the coder. The types of modulation most frequently used are the following: – phase-shift keying: PSK; – differential phase-shift keying: DPSK; – quadrature amplitude modulation: QAM; – frequency-shift keying: FSK; – amplitude-shift keying: ASK. 1.5. Vocabulary of digital modulation Symbol A symbol is an element of an alphabet. If M is the size of the alphabet, the symbol is then called M-ary; important example: M = 2, the symbol is called binary. By grouping, in the form of a block, n independent binary symbols, we obtain an alphabet of M = 2n M-ary symbols. Thus, a symbol M-ary conveys the equivalent of n = log2M bits. Modulation speed R This is the number of state changes per second of one or more parameters modified simultaneously: a phase change in a carrier signal, a frequency excursion and a variation in amplitude are state changes. The “modulation speed” R = 1/Τ is expressed in “bauds” (see the work of Émile Baudot, a French telegraphy engineer (1845–1903)).

Modulation

15

Binary flow D = 1/T This is defined as the number of bits transmitted per second; it is therefore expressed in “bits per second”. It is equal to or greater than the modulation speed at which a state change will represent a bit or group of bits. For an M-ary alphabet, we have the fundamental relationship: T = nTb which is D = n R. There is an equality between source flow and modulation speed only in the case of a binary source (binary alphabet). The quality of a link is connected to the bit error rate. BER (bit error ratio) This is the number of false bits/number of bits transmitted. A priori, Pe and BER are different. In a statistical sense, we have Pe = E(BER) (expected value; BER tends toward Pe if the number of bits transmitted tends toward infinity). Spectral efficiency The spectral efficiency parameter: η = D/B.

of

a

modulation

is

defined

by

the

It is expressed in “bit/second/Hz”. D is the “binary flow” and B is the width of the band occupied by the modulated signal. For a signal using M-ary symbols, we have: η = 1/T Blog 2 M bit/sec/Hz. For given B and T, the spectral efficiency increases with the number of bits/symbol n = log2M. Hence the M-ary modulation. Spectral power density As for the power spectral density (PSD) of the modulated signal m(t), some signal theory formulae remind us that if αm(t) = xc(t) + jxs(t) represents the signal in baseband of m (t) = Re [α m.exp(j(ωct+φc) )] (see below), and if

16

Digital Communicattion Techniquess

γαm(f) is the PSD of α m.(t) off the modulaated signal m(t), m then thee PSD of the moddulated signaal will be: γm m(f) = 1/4 [γγαm (f − f c) + γαm (f + f c)]].

Figure e 1.17. Powerr spectral dens sities (low passs type)

The PSDs for the (so-called)) online codees are of the low frequenncy type. Indeed, the radio channel is RF. The problem m is solved using u an RF ccarrier. As we w will see later, l different digital mo odulations PSDs P are reppresented in dB (ddecibels) andd the frequenncies are positive. They display d a cenntral lobe at low frequencies, f and weaker secondary lo obes at highher frequencies (from 8 to 14 dB). The poower is therefore essentiially carried by the centtral lobe, hence thhe precaution of choosinng the lowpaass filter, so as not to altter, or to alter only slightly, thhis main lobee. Wheen the modullation is linear, carrying out modulation has the eeffect, in most caases, of transllating the PS SD of the modulating signnal. The PSD of the modulated signal m(t) is linked to the wave shhape g(t) w often bee rectangularr) by its Fourrier transform m (FT): G(f). (which will In faact, the PSD D is the FT of the auto o-correlationn function oof g (see temporaal convolutioons). Symbo ol clock The symbol clocck representss the exact frequency fr andd synchronizzation of s clock transmissiions, the the trannsmission off different syymbols. In symbol

Modulation

17

carrier transmitted is at the correct amplitude/phase values (I/Q) to represent a specific symbol (a specific point in the constellation). Binary flow and symbol rates The binary flow is the frequency of a bit stream of the system. For example, a series using system binary radio at 8 bits sampling at 10 kHz for the voice. The binary rate, the base radio flow rate, would be 8 bits multiplied by 10 Kbits per second or 80 Kbits per second (we will ignore, a priori, the extra bits needed for synchronization, error correction, etc.). 1.6. Principles of digital modulations The message to be transmitted is issues from a binary source. The modulating signal, obtained after coding, is a (so-called) baseband signal, which may be complex and written as: Z(t) = k Zk.g(t -kT) with Zk = (Ik + jQk). ZK (t) = Ik(t) + jQk(t). The function g(t) is a wave shape considered in the interval [0, T[ with: kT < t < (k + 1)T. In ASK, PSK and QAM modulations, the modulation transforms this signal c(t) into a modulated signal m(t) so that: m(t) = Re[κ Zk(t) ej ( ωct + φc)] The frequency fc = ωc/2π and the phase at origin (in time) ϕp characterize the sinusoidal carrier, with phase ωc + fc used for the modulation. If Zk (t) = Ik(t) + jQk(t) are real (Qk(t) = 0), the modulation is called one-dimensional and if they are complex, the modulation is called two dimensional. The modulated signal is also written more simply: m(t) = k Ik (t).cos(ωct + ϕc) − k Qk (t).sin(ωct + ϕc) or indeed: m(t) = I(t).cos(ωct + ϕc) − Q(t).sin(ωct + ϕc), positing: Ι(t) = κ Ik(t) and Q(t) = k Qk(t)

18

Digital Communication Techniques

The signal I(t) modulates the amplitude of the in-phase carrier: cos(ωct + ϕc) and the signal Q(t) modulates in amplitude the quadrature carrier: sin(ωct + ϕc). In most cases, the elementary signals Ik(t) and Qk(t) are identical almost to a coefficient and they use the same impulse form g(t) also called “train”. Ik(t) = Ik.g(t − kT) and Qk(t) = Qk.g(t − kT) The signals I(t) and Q(t) are also called “modulating trains” and are written as: I(t) =  Ik (t)g(t − kT) and Q(t) = Qk (t)g(t − kT) Symbols Ik and Qk, respectively, take their values in the alphabets (A1, A2, … AM) and (B1, B2, … BM). The theoretical schema of the modulator is represented in Figure 1.18.

Figure 1.18. General form of the modulator

The various types of modulation are defined by the alphabets described above and by the g(t) function. To each symbol sent there corresponds an elementary signal of the form: mk(t) = Ik.g(t − kT).cos(ωct + ϕc)−Qk.g(t − kT).sin(ωct + ϕc),

Modulation

19

that cann be represennted (see Figgure 1.20) in a two-dimensional spacce whose base vecctors are: g(tt − kT). cos(ω ωct + ϕc) and d − g(t − kT).sin(ωct + ϕc). 1.6.1. Polar P displa lay The amplitude and a phase are a representted together,, simply, in a polar m, and the caarrier is the reference. Just J as for frrequency and phase; diagram these arre measuredd in relationn to a refereence signal, the carrier in most communnication systtems. The am mplitude is an n absolute orr relative vallue; both are usedd in digital coommunicatioon systems. Polaar diagrams are the basis b for many m displayys used inn digital communnications, although it is usual to desscribe the siggnal vector uusing its rectanguular coordinaates: I (in-phhase) Q (quad drature). 1.6.2. Variations V of parametters: ampliitude, phas se, frequency Figuure 1.19 shoows differentt forms of polar p modullation. The phase is represennted as an anngle. AM chhanges only the t amplitudde of (modulates) the signal. The T PM channges only thee phase (the argument) of o the signal. AM and PM cann be used together. t FM M seems sim milar to PM M, although iit is the frequency that is thee command parameter, p raather than thee relative phaase.

Figurre 1.19. Variatiions: amplitud de, phase, freq quency

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Digital Communication Techniques

It is important to note the difference between the bit flow and the symbol flow. The signal bandwidth needed for a given communication channel depends on the flow of symbols, not on the binary flow. The flow of bits is the flow frequency of bits in a system. 1.6.3. Representation in a complex plane The modulated signal m(t) integrates independent information via Ik(t) and Qk(t), which are two baseband signals, respectively, called in-phase and Q. The recovery of Ik(t) and Qk(t) could be achieved only if both these signals are in a band limited at the [-B,B] with B < fp (Rayghley condition). A representation in complex plane corresponds one point to the various types of modulation. The whole group linked to the symbols is called a constellation.

Figure 1.20. Position of a symbol in complex plane (from Fresnel)

The distribution of points depends on the following criteria: – To be able to distinguish between two symbols, one should respect a minimum distance dmin between the points representing these symbols. The greater the distance, the lower the probability of error. The minimum distance between all these is: d min = Min(dij) with dij =

2

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21

– This should be compared to the definition of the Hamming distance. A CODE’S HAMMING DISTANCE.– This is the criterion that makes it possible to evaluate a code’s detecting power as well as its corrective power. The Hamming distance between two words (noted dh) = the number of positions that have distinct values, for example: (110011, 101010) = 3: the number of 1 of the exclusive OR (XOR). A code’s Hamming distance C (written Dh(C)) = the minimum distance between two words in the code. For example: Dh = ([110, 101; 011]) = 2 and DH = ([001, 0101, 1001, 0110, 1010, 11001]) = 2. To each signal sent, there corresponds an elementary signal mk(t) and by the same token, energy needed to transmit this symbol. In the constellation, the distance between a point and the origin is proportional to the square root of the energy that must be provided during the time interval [kT, (k+1)T[ to send this symbol. The average power used to transmit symbols is proportional to  |Ci| : the peak power divided by 2. The two criteria mentioned above are antagonistic; in fact, it is tempting, on the one hand, to extend the symbols to the maximum to decrease the probability of error, and, on the other hand, to make them close to the origin to minimize the energy needed for transmission. The criteria for choosing a modulation are as follows: – The constellation that, depending on the applications, highlights the low energy needed to transmit symbols or a low probability of error. – The spectral occupation of the modulated signal. – The simplicity of operation (with, possibly, a symmetry between the points in the constellation). 1.6.4. Eye diagram The eye diagram represents the values at the receiver. These are sampled repetitively and are applied to the input of the vertical deviation, while the horizontal deviation is synchronized with the signal’s flow. The name of this diagram comes from the fact that for many codings, the motif obtained resembles a series of eyes framed by two horizontal rails.

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Digital Communication Techniques

Several performance criteria can be deduced for this. Whether or not the signals are too long, too short, poorly synchronized with the system clock, at too high or low a level, have a great deal of noise, are very slow during state changes, or indeed if there are too many overtakings or too much inertia. An open eye indicates a signal carrying little distortion. This may be caused by intersymbol interferences (IES) or noise; the eye tends to close. Separate eye diagrams can be produced, one for data from channel I and the other for data from channel Q. The eye diagram displays the amplitudes of I and Q depending on time, are in indefinite persistence mode, with tracings. Transitions I and Q are shown separately and an “eye” (or eyes) are formed when the symbol is decided. The binary phase-shift keying (BPSK)/quadrature phase-shift keying (QPSK) (Figure 1.21; see below) has four distinct states of I/Q, one in each quadrant of the circle: two levels for I, 2 levels for Q. This forms a simple eye for each I and Q. Other arrangements use more levels and create more nodes in time, where the traces pass. The example below is that of a 16-QAM signal (see below) that has four levels forming distinct eyes. The eye is open at each symbol. A good signal has these crossing points wide open.

Figure 1.21. Eye diagram: I and Q

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23

Figure 1.22. Intersymbol F I interferences an nd eye diagram ms. For a colo or version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

Figure 1..23. Eye diagrrams (e.g. QPS SK; Agilent). For F a color version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

1.7. Mu ultiplexing In CDMA C systeems (see beelow), userss use in shaared time: a digital channell with very high flow, whilst w also recovering at a very highh flow a digital sequence s onn their transm mission. A different d sequuence is asssigned at each ouutput terminal so that the signals can be b discerned from one annother by correlating them wiith the superrposed sequeence. This is based on coodes that

24

Digital Communication Techniques

are shared between base and mobile stations. Because of the choice of coding, there is a limit of 64 code channels on the direct line. The inverse line has no practical limit on the number of codes available. Channel sharing The RF spectrum is a finite resource. Multiplexing or channelization is used to separate different users of the spectrum. We speak, below, of multiplexing using frequency, time, code and geography. Most communication systems are a combination of these multiplexing methods. 1.7.1. Frequency multiplexing

Figure 1.24. Multiplexing frequency

This breaks down the frequency band available into smaller channels at fixed frequency. Each transmitter or receiver uses a separate frequency. This technique has been used since the 1990s. Transmitters have a narrow or limited band. A narrow frequency band transmitter is used with a narrowband filter receiver so that it can demodulate the signal wanted and reject unwanted signals, such as signals interfering from adjacent radios.

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25

1.7.2. Multiplexin M ng – time TDM MA multipleexes severaal transmitteers or receeivers at thhe same frequency, i.e. separrates the trannsmitters over time; userrs (A, B, C, etc., see Figures 1.23 and 1.224) have theiir own time interval i (IT; time slot [TS]), thus sharing the same frequency. The T simplesst type is tiime-divisionn duplex t and the receiver on thhe same (TDD). This multipplexes the transmitters frequency. TDD is used, for example, e in a simple tw wo-directionnal radio where a button is prressed to speeak and releaased to listenn. This type of TDD, howeveer, is very slow. Digitaal wireless radios suchh as CT2 (C Cordless Telephoone) and DE ECT (digital enhanced co ordless teleccommunications) use the TDD D, but they multiplex m hunndreds of perriods per second. TDM MA is also ussed in the diggital cellularr system GSM M and also inn the US NADC--TDMA system, the publlic switched telephone t neetwork. If we are interessted in the notion n of spaace–time freqquency, thenn we can define an equivaleence betweeen a group of users sharing s a pparticular frequency band andd a case wheere each user has accesss to a fractioon of the band alll the time.

Figure 1.25. CDMA: all users on each e frequency y and users arre separated b by code. or a color verssion of this figu ure, see www w.iste.co.uk/gon ntrand/digital.zzip Fo

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Digital Communication Techniques

Figure 1.26. TDMA principle

1.7.3. Multiplexing – code CDMA is a method of access where multiple users are authorized to transmit simultaneously on the same frequency; it is a code specific to each user, which discriminates between data. Frequency-division multiplexing is therefore used, but the channel is 1.23 megahertz wide. In the case of CDMA US telephones, an additional type of channelization is added in the coding format. 1.7.4. Geographical (spatial) multiplexing Another sort of multiplexing: geographical or cellular. If two transmitter/receiver (transceiver) pairs are distant enough from one another, they can operate at the same frequency and not interfere with one another. There are few systems that do not use geographical multiplexing (see international broadcasting stations): clear-channel (in the US, in AM, very robust with regard to interference), amateur stations, and some low frequency military radios with no geographical limitations, broadcasting globally, etc. 1.8. Main formats for digital modulations Here, we will discuss the main digital formats for digital modulation, their main applications, relative spectral efficiencies and some variations in

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27

modulation types used in practical systems. Fortunately, there is a limited number of modulations that form construction blocks for absolutely any system. Table 1.1 covers different modulation formats in wireless and video communications; we will return to it. Modulation format

Applications

MSK1,GMSK

GSM, CDPD

BPSK

Deep-space telemetry, cable modems

QPSK,

Satellite, CDMA, NADC, TETRA, PHS, OODC, LMDS, DVB-S, cable (return path), cable modems, TFTS

π/4 DQPSK FSK, GFSK

DECT, Pagination, Rank mobile data, AMPS, CT2, HERMES, land mobile, public security

8, 16 VSB

North Americana IV digital (ATV), broadcasting, cable

8PSK

Satellite, aviation, telemetry points, broadband video surveillance systems

16 QAM

Digital microwave radio, modems, DVB-C, DVB-T

32 QAM

Terrestrial microwave, DVB-T

64 QAM

DVB-C, modems, set-top box (STB), MMDS

256 QAM

Modem, DVB-C (Europe) digital video (US) Table 1.1. Modulation formats and applications

This is a digital FSK at continuous phase. Like QPSK, MSK is encoded into bits alternating the moments in quadrature, the component Q is delayed by half the duration of a symbol. But, unlike the squared signals used in QPSK, MSK modulation encodes each bit on a half-sine. We are dealing with a constant module signal, reducing problems with nonlinear distortions.

1 MSK: minimum-shift keying; the specific case of FSK (Δf/D = 1/π).

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1.8.1. Phase-shift keying

Figure 1.27. Example of PSK modulations: constellation of symbols in phase modulation PSK-M

It seems obvious to increase the M (i.e. the number of bits transmitted per symbol). This presents the following advantages and drawbacks: – The spectral efficiency 1/T*B log2M bit/s/Hz increases for a given bandwidth B; but the probability of error per symbol Ps(e) then increases, and, so as not to degrade it, it will be necessary to increase the energy emitted per bit: Eb. – This type of modulation, which is simple to achieve, is scarcely used for M > 2. Its performances are not as good as those of other modulations, as it happens, in noise resistance. Phase shift modulations are also often known by the acronym PSK, for “phase-shift keying”. We return to the general expression for a digital modulation: m(t) = Re[κ Zk(t) ej ( ωc t + φc)], with: Zk(t) = Ik(t) + j. Qk(t), recalling below the allure of a “transceiver” (Figure 1.28).

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Figure 1..28. I and Q: (a) ( radio transm mitter; (b) radiio receiver

The elementary signals s Ik(t) and a Qk(t) usee the same waveform w g(tt), which is here a rectangulaar phase, of duration d T and a an ampliitude equal tto A, if t belongss to the intervval [0, T[ T and moreoover equal to 0 elsewheree. We still s have: Ik(t) ( = Ik.g(t − kT) k and Qk(tt) = Qk.g(t − kT) k so Zk(t) = (Ik + jQk).g(t − kT T) = Zk.g(t − kT) k In thhis case, thee Zk symbolls are distrib buted over a circle, hennce it is written in Eulerian form f as: Zk(t) = (Ik + jQ Qk) = ejϕk . Hencce Ik = cos ϕk and Qk = sin(ϕk) and Ik(t) = cos(ϕk).g(t ) − kT) Qk(t) = sin(ϕk).g(t ) − kT). One could build several PSK K-Ms for a single given value v of M w where the c To im mprove perfoormances symbolss are arrangeed in any waay around a circle. over nooise, the sym mbols must be b distributed regularly around the ccircle (it will thenn be easier too distinguishh between theem, on averaage). All the possible phases are a thus convveyed by thee following ex xpressions:

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Digital Communication Techniques

φk = π/M + 2kπ/M when M > 2 φk = 0 where π/ when M = 2 NOTE.– The Zk symbols take their values in an alphabet of M > 2 elements {ejφk) where ϕk is defined above with k = 0,1,… M-1. We can also say that Ik and Qk simultaneously take their values in the alphabet {cos(ϕk)} and {sin(ϕk)}. The modulated signal becomes: m(t) = Re[k ej φk .g(t-kT ej ( ωct + φc)] = Re[k .g(t-kT) ej ( ωct + φc+ φk)] Considering only the interval [kT, kT+1]: m(t) = Re[Aej ( ωct + φc+ φk)], i.e.: m(t) = A. cos(ωct + ϕc + ϕk) m(t) = A. cos(ωct + ϕc) cos(ϕk) − A.sin(ωct + ϕc) sin(ϕk) The expression above indicates that the carrier phase is modulated by the argument ϕk of each symbol, which explains the name given to PSK. It should also be noted that the carrier phase is modulated in amplitude by the signal A.cos(ϕk) and that the quadrature carrier sin(ωct + ϕc) is modulated in amplitude by signal A.sin(ϕk). In other words, a PM can be considered an AM-C modulation (amplitude without carrier) by a binary anti-polar signal. Its bandwidth and spectrum are identical to OOK modulation (see below) unless there is no line frequency fp (with OOK). The PSK expression demonstrates well that it is a constant envelope; the envelope being the module of the complex envelope. This property is useful for transmissions on nonlinear channels, which makes PSK a tool of choice for satellite transmissions. The benefit of having a constant-envelope modulated signal is that it makes it possible to use amplifiers in their area of best performance (class E, class F), often linked to a nonlinear mode of operation.

Modulation

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Thus, the arrangement of the symbols on a circle does not only translate into a constant envelope, but also by an identical energy implemented to transmit each symbol. Both aspects are of course intimately linked. 1.8.2. BPSK

Figure 1.29. BPSK

Figure 1.29 is an example of a diagram where the states are mapped using zeros and ones. The symbol flow is the binary flow divided by the number of bits that can be transmitted for each symbol. If one bit is transmitted per symbol, as in BPSK, then the symbol rate is identical to the binary flow. If two bits are transmitted per symbol, as in QPSK, then the symbol rate is half the binary flow. The symbol is sometimes called the baud rate. Note that the baud rate is not identical to the binary flow. These limits are often confused. This is why more complex modulation formats use a higher number of states defining the information with a narrower RF spectrum band. An important example of QPSK modulation is therefore BPSK2 modulation. It is a binary modulation (a single bit is transmitted per period T): n = 1, M = 2 and ϕk = 0 or π 2 BPSK: Binary Phase-Shift Keying: phase change of the carrier wave. The BPSK digital modulation technique is the simplest form of modulation by phase shift. It uses two phases that are separated by 180°.

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Digital Communicattion Techniquess

The symbol thereefore takes itts value in th he alphabet {−1, { 1} (±

).

ωct + ϕc). Heree, the modulaation only occcurs on the carrier in phhase I: cos(ω This is a one-dimeensional moddulation. Th he modulatedd signal is ttherefore written for t belongiing to the intterval [00, T[ : m(t) = ±A.cos(ωct + ϕc) The BPSK consttellation is shhown in Figu ure 1.30.

Fig gure 1.30. BP PSK. For a collor version of this t fig gure, see www w.iste.co.uk/go ontrand/digital..zip

a 0 are reppresented Binaary input data in polar foorm with thee symbol 1 and with a constant c ampplitude level. The processs of signal traansmission ccoding is followed by an imm mediate no-reeturn-to-zero (NRZ) encooder. The am mplitudes linked to 1 or 0 are (1)) andd − (0)), or the conttrary. s is theerefore: Eb. The energy per symbol me: Accoording to tim

.

Modulation

33

To demodulate d t original binary orderr of 1 and 0, the 0 the signal entering the BPS SK is passeed to a corrrelator, whiich is formeed of the m multiplier (betweeen the signal received annd all the posssible signalls) and the inntegrator via a rulle. If the resuult is > 0, thee device prod duces a “1”, otherwise, 0. BPSK is used more in satelliite communications, becaause of its siimplicity u Other addvantages of BPSK incluude the improvement and robbustness of use. in bandw width powerr, because it can only tran nsmit one biit per symbol, and so cannot be b used for high-flow h appplications. 1.8.2.1. BPSK: mo odulation an nd demodullation The modulator represented r i Figure 1.3 in 31 is formedd of a multipplier that changess the frequenncy on a digittal train codeed NRZ.

Figure 1.31. BPSK modulator m

The receiver reqquires the use u of a coh herent demoddulation (seee Figure 1.33), a simplified synoptic of a BPSK demo odulator. m m on (NRZ codes) 1.8.2.2. A small memorandum Bipolarr NRZ codess NRZ Z: there is no return of thee voltage to level 0 over a symbol’s dduration. There arre two types of NRZ codde: unipolar and a bipolar (see Figure 1.32).

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Digital Communicattion Techniquess

Figure 1.32. Unipolar U and biphase b pulses

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35

Figure e 1.33. Top: un nipolar RZ cod de (remains at a zero). Botto om: bipolar RZ Z code

Manche ester codess Codiing is achievved here via a variation in n the level off voltage: – froom −Vm to Vm: V rising eddge: codes att 1; – froom −Vm to Vm: V falling edge: e codes at a 0; – or the reverse.

Figure 1.34. Biphase e pulses

In thhe case of Manchester – or bi-phased – codees, the probblems of desynchhronizing thee receiver aree less prohib bitive, as eacch symbol iss marked by a varriation in volltage.

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Digital Communicattion Techniquess

Figure 1.35. BPSK dem modulator

If r(tt) = B.cos(ω ωct + ϕc + ϕk ), the signal without noise is receiveed by the receiverr during the time t intervall [kT, (k + 1))T[. After muultiplication with the recovereed carrier, we w obtain: S S1(t) = B.cos((ωct + ϕc + ϕk ).cos(ωct + ϕc) If, after a filteringg, to eliminnate the freq quency com mponent 2fc: S2(t) = (B/2)cos ϕk. s trannsmitted, The receiver shoould still recover the rhyythm of the symbols mple the siggnal S2(t) in the middle of each perriod. Afterwaards, the then sam symbol that emittedd –1 or 1, ϕk takes the value π or 0 and the siign S2(t) becomees negative orr positive, shhowing the binary data reeceived “0” oor “1”. 1.8.2.3. The “BPS SK” spectrum m he power speectrum of g(tt) which, The spectrum off the basebannd signal is th here, is a rectangulaar impulse (F Figure 1.36):

 sin π fT T γαm ( f ) = A T    π fT 

2

2

The spectrum off the signal modulated m is shifted s by ± fp. f

Modulation

37

Figure 1.36. Spectral efficiency of a BPSK. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

1.8.3. The QPSK It is used intensively in applications that include CDMA, wireless local radio loop, iridium (a voice/data satellite system) and local DVB-S (Digital Video Broadcasting; videotrans). The signal is shifted in 90 increments: 45–135, 45, or 135. These points are chosen as they are easy to represent, using an I/Q modulator. Only two values of Q and I are needed, giving two bits per symbol. There are four states (22). DPSK actually has much greater spectral efficiency (see bandwidth) than BPSK can have (B: binary), potentially a ratio of 2. The TDMA version of the North American Digital Cell (NADC) achieves a data rate of 48 Kbits per second on a bandwidth of 30 kHz or 1.6 bits per second per hertz. The system is based on a /4 DQPSK and transmits two bits per symbol. The theoretical efficiency should be 2 bits per second per hertz, while in practice, it is 1.6 bits per second and per hertz. NOTE.– DQPSK: Differential QPSK: the information is not carried by an absolute state, but by the transitions between states. Another example is a digital micro-wave radio using 16 QAM.

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Digital Communication Techniques

16 QAM (4 bits per symbol: 24 symbols) is more sensitive to noise and distortions than DPSK. This type of signal is often sent on a direct visibility micro-wave link (as well as an optimized high-power transceiver) or on a wire that generates little noise or interference. In the case of digital microwave radio, the bit rate is 140 Mbits per second on a broad frequency band: 52.5 MHz. The spectral efficiency is therefore 2.7 bits per second and per hertz. It is also known as frequency splatter. Very slow changes in power waste precious transmission time as the transmitter cannot send data when it is not completely filled. This can also cause error rates on bits that are high at the start of the transmission. In addition, the peak and average power levels should be well framed, since excessive power from an amplifier can lead to compression or disconnection. These phenomena distort the modulated signal and usually lead to a spectrum regrowth. The example in Figure 1.37 shows a differential quadrature phase shift (π/4 DQPSK) – direct transitions between states – as described in the standard TDMA of the NADC. In the figure, a burst of DQPSK is represented in 157 symbols.

Figure 1.37. Constellation diagram (Agilent). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

The polar diagram shows several symbols at once. This means that it indicates the instantaneous value of the carrier at any time on the continuous

Modulation

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line including the symbol times, represented as of l/Q or amplitude/phase values. The constellation diagram shows an instant of this same burst, with values displayed only at points of decision. The constellation diagram displays phase errors as well as amplitude errors at points of decision. Transitions between points of decision affect the bandwidth transmitted. This display shows the path that the carrier takes, but it does not indicate the errors at points of decision explicitly. The constellation diagrams provide specifications on variable power levels, the effects of filtering and spectrums (in video: moiré) or pixilation (aliasing). How do we remove these interferences between symbols (IBS)? By considering the Nyquist criterion, which states in essence that it is not possible to transmit without interference between symbols a signal of symbol duration T on a bandwidth channel lower than 1/(2T). Solution 1 If a signal is NRZ M-ary, the set of symbols is Si(t) = (2i – 1)Vm*h(t), with −M/2 + 1 ≤ i ≤ M/2 and h(t) being the door function (pulse); the signal will have a PSD in sinc, with limitless support. If we replace h(t) with a sinc impulse, its PSD will have a (finite) gate width: 1/T of height (M2 – 1).VmT/3. A signal with this PSD will not be altered by a low-pass channel of BP 1/2T; so, no IES, This is not the case if we replace h with a sinc; the symbols are recovered as they are of infinite duration. The IES is null as PSD is limited and is the wise choice at the moment of sampling. However, symbols of indefinite duration cannot be created in practice. Solution 2 Instead of a sinc, we use raised cosine impulses:

40

Digital Communication Techniques

 π rt   cos  T S   t    f (t ) = sin c  2  2 rt   TS    1−    TS 

NOTE.– The closer r is to 1, the more f is attenuated. Frequencies f including the module verify: (1-r)/2T  ≤ 1 + r)/2Tn. Then, the PSD is the previous multiplied by 1/4.(1 + sin(πT/r(1/r.abs(f)))2. Otherwise, the PSD is null. In this case, the PSD spreads further, the greater the size of r. For frequencies whose module is ≤ 1-r/2T, the PSD has the same value as for the previous solution, i.e. one PSD of one sinc. 1.8.3.1. More on quadrature phase-shift keying: towards some modulation and demodulation simulation programs Another extension of PSK is the digital modulation technique, of a higher order than PSK, which uses a four-level phase state to transmit 2 bits/symbol simultaneously, by selecting one of four carrier phase shifts spaced at 0, π/2, π and 3π/2, where each phase value corresponds to a distinct pair of message bits 00,01,10,11. This enables the signal to have twice the information using the same bandwidth. This means that QPSK is more effective by bandwidth than BPSK. This is therefore an AM on two levels on each of the quadrature carriers. This has been chosen for GSM, as it produces a minimum product Q.B (Q: flow; B: spectral width). In this case: n = 2, M = 4 and ϕk = π/4 + kπ/2. 1.8.3.2. QPSK: modulation and demodulation In the constellation diagram of a QPSK (Figure 1.43), there is a path from any one symbol to one of the three others. In fact, there is one chance in four that the signal’s trajectory will traverse the origin (the 0 V). In this case, I and Q change. This critical case poses problems caused by great variations in amplitude exacerbated by possible nonlinearities in amplifier circuits, as these cause distortions (and spectral regrowth) widening modulation subbands.

Modulation

41

Figure e 1.38. Conste ellation diagram m of a π/4 QP PSK: (a) possib ble states for θk when θk-1 = n π/2; (b)) possible stattes when θk-1 = n π/4; (c) all possible state es

To circumvent this probleem, other modulation m formats havve been t concept of o differentiaal modulationn. developped: one mighht consider the NOTE (D Differential PSK P or DPSK).– The infformation dooes not depennd on the absolutee value of thhe state, but on the transsition betweeen states. Thhere may be unauuthorized traansmissions. This is the case for π//4 DPSK, w where the trajectorry does not cross the orrigin (0,0) (ssee Figure 1.38(c)). 1 In aaddition, compareed to GMSK, a classicc cellular modulation, m π π/4DPSK, liinked to filteringg in raised coosine, has greeater spectrall efficiency (the ( filteringg tends to round thhe corners off the signals on the right--hand side off Figure 1.39).

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Digital Communicattion Techniquess

Figure 1.39. Positions (sta ates) in the BP PSK constellattion representt otif of specific bits (symbol) and a symboll time a mo

F Figure 1.40. QPSK: Q I/Q. Forr a color versio on of thiss figure, see www.iste.co.uk w k/gontrand/dig gital.zip

Modulation

43

A “QPS SK” constelllation The QPSK constellation is represented r in i Figure 1.441. It shows that the assignm ment of bits at a points of thhe constellation usually happens h accoording to Gray coode (modificaation of a sinngle bit at a time, t when a number is inncreased by one unit). u QPSK chronogram m SK chronogrram. It highliights the disttribution Figuure 1.41 repreesents a QPS of numbbered bits inn the incomiing bit stream m { zk }(o ik) toward biitstreams { Ik } (oor ak) and{ Qk }(or bk) as well as th he delay neeeded on the in-phase channell to achieve both bit floows. We also observe thhat the phasse of the modulatted signal m(t) can change from 0, ±π.2, ± ου π raadiants whenn passing from onne symbol to another, which w happeens easily when w we obsserve the QPSK constellation c .

Fig gure 1.41. QP PSK phase mo odulation chro onograph (Deg gauque/Kadion nik)

Modula ation The chronogram m (Figure 1..41) highligh hts the simpple relationsship that exists between the even e bit pairss and the Ik, and betweenn the odd bitss and the Qk. For a homothetyy and by designating the set of valuess of the bit sstream as { Ik } too the rhythm:

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Digital Communication Techniques

m(t) Re[ Zk(t) ej  c t + c m(t) I(t).cosct cQ(t).sinct c or only considering the time interval [kT, kT+1]. m(t) A.(1-2i2k)g(t-kT).cosctc(1-2i2k+1)g(t-kT).sinct c m(t) A.Ik.cosct cA.Qk.sinct c The incoming bit stream { Zk } (or{ ik }) is divided into a { Ik }, or { ak } bit stream, switched to the in-phase channel for the even bits, and a { Qk }, or { bk } bit stream, switched to the quadrature channel for the odd bits. The speed of bit streams { Ik } and { Qk } is therefore half the speed of the incoming bit stream { ik }. The synoptic schema of the modulator shown in Figure 1.42 shows the demultiplexing of the bit stream at the entry of the modulator in two bit streams on the in-phase and quadrature channels. Both bit streams are therefore coded in NRZ. The paths of the schema represent the relationship: m(t) I(t).cosct cQ(t).sinct c and therefore calls for two multipliers.

Figure 1.42. QPSK modulator. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Modulation

( ) = A cos(2π t +  ); 0 ≤ t ≤ T, i = 1,2,3,4 with  =

(2 − 1) 4

( ) = √ cos(  ), = √ sin( ) And so:  = tan PSD: γ =

∗ (2T ) ∗ (

(

)

A QPSK, via MATLAB, SCILAB or OCTAVE % function [s,t,I,Q] = qpsk_mod(a,fc,OF) clear all fc = 1000 fs = 20e3; OF = 8 fs = OF*fc t = 0:1/fs:0.1; w = 7.5e-3; z0 = [1,0,1,1,0,0,0,1,1,1] z = 2*z0-1 z_even = [z(2),z(4),z(6),z(8),z(10)] z_odd = [z(1),z(3),z(5),z(7),z(9)] %ak = [1,1,1,1,1,1,] N = length(z0) z = 2*z0-1 x = rectpuls(t,w); %xfutur = zeros(N,1001) %tpast = -45e-3;

45

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Digital Communication Techniques

xfutr0 = 0; Bip0 = 0; NRZ0 = 0 NRZ_even0 = 0 NRZ_odd0 = 0 %xpast = rectpuls(t-tpast,w)*(-1); N2 = N/2 for k = 1: N tfutr = 15e-3; Bip = Bip0+(rectpuls(t-(tfutr*(k-1)),w))/1; xfutr = z0(k)*(rectpuls(t-(tfutr*(k-1)),2*w))/1+xfutr0; NRZ = NRZ0+z(k)*(rectpuls(t-(tfutr*(k-1)),2*w))/1; xfutr0 = xfutr; Bip0 = Bip; NRZ0 = NRZ; end SIZE_NRZ = size(NRZ) for k = 1: N2 tfutr = 15e-3; % xfutr = z0(k)*(rectpuls(t-(tfutr*(k)),w))/1+xfutr0; NRZ_even = NRZ_even0+z_even(k)*(rectpuls(t-(tfutr*(k-1)),2*w))/1; NRZ_odd = NRZ_odd0+z_odd(k)*(rectpuls(t-(tfutr*(k-1)),2*w))/1; xfutr0 = xfutr; NRZ_odd0 = NRZ_odd; NRZ_even0 = NRZ_even; end

Modulation

Figure1 = Figure plot(t,Bip) % plot(t,x,t,Bip)* ylim([-0.2 1.2]) Figure2 = Figure plot(t,xfutr) ylim([-0.2 1.2]) Figure3 = Figure plot(t,NRZ) ylim([-1.2 1.2])

Figure 1.43. NRZ

% SEQUENCE (to be concatenated with the previous section) ===================== % Function to modulate an incoming binary flow, using a conventional QPSK % input: binary data flow (0 and 1) to modulate

47

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Digital Communication Techniques

%w = 5e-3 dim = 2001 % OF: -oversampling factor -, (multiples of fc) – takes at least 4 OF = 4 fe = OF*fc % its sampling frequency %t = -0.1:1/fs:0.1; Te = 1/fe % t = 0:w/10:10*w; t = 0:Te:(length(dim)-1)/fe; rp = rectpuls(t,w); length = size(t) lengthrp = size(rp) %s – QPSK signal modulated with carrier %t – time base for the signal modulated through the carrier %I – non-modulated Ichannel (no carrier) %Q – non- modulated channel (no carrier) L = 2*OF % number of samples in each symbol (QPSK: 2 bits per symbol) Lfe = L/fe % ak = 2*a-1 %NRZ encoding: 0- > -1, 1- > +1 Q = NRZ_even(1:1:end); I = NRZ_odd(1:1:end); % Flow of odd and even bits SIZE_I = size(I) SIZE_Q = size(Q) lengthI_initial = length(I) I = repmat(I,1,1).'; Q = repmat(Q,1,1).'; I = I(:).'; Q = Q(:).'; lengthI = length(I) %t = 0:1/fs:(length(I)-1)/fs; % time base

Modulation

n = 10 nfe = 10*fe t = 0:1/nfe:(length(I)-1)/nfe; iChannel = I.*cos(2*pi*fc*t);qChannel = -Q.*sin(2*pi*fc*t) s = iChannel + qChannel; % Base band signal modulated by QPSK Pelot = 1; Pelot = 1; %Pelot = 0, if you do not intend to see shape plots if Pelot == 1, % Wave form at the transmitter Figure;subplot(3,2,1);plot(t,I,'LineWidth',2); % Wave form in base band on the I arms % zoomed on the first symbols xlabel('t'); ylabel('I(t)- baseband’);xlim([0,0.0125]); %% % _ el('t'); ylabel('I(t)- baseband’);xlim([0,10*Lfs]); subplot(3,2,2);plot(t,-Q,'LineWidth',2); % Wave form in base band on the Q arms % zoomed on the first symbols xlabel('t'); ylabel('Q(t)baseband’);xlim([0,0.0125]); subplot(3,2,3);plot(t,iChannel,'r');%I(t) with carrier’ xlabel('t'); ylabel('I(t)- with carrier’);xlim([0,0.0125]); subplot(3,2,4);plot(t,qChannel,'r');%Q(t) with carrier’ xlabel('t'); ylabel('Q(t) & carrier’);xlim([0,0.0125]); % waveform QPSK zoomed on the first symbols %xlabel('t'); ylabel('s(t)');xlim([0,10*Lfe]); hold on; subplot(3,2,5);plot(t,s,'b');%s(t) with carrier’ % plot (t,s, 'g' ) xlabel('t'); ylabel('s(t)');xlim([0,0.0125]); subplot(3,2,6);plot(t,s,'b');%s(t) with carrier’ xlabel('t'); ylabel('s(t)');xlim([0,0.0125]); % end

49

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Digital Communicattion Techniquess

Figure 1.44. QPSK; at the e transmitter: timing t diagram ms. For a colorr ee www.iste.c co.uk/gontrand d/digital.zip version off this figure, se

% contiinuation (to be b concatenaated with the previouus MATLAB section) ====== ========== ========= === In thhe demodulattor, the signaal received iss multiplied by b a referencce signal (cf. cohherence) andd the two 2PSK 2 are in ntroduced intto the in-phhase and quadratuure channelss. The outputt multiplied from each channel c is inttegrated. The outtput of the inntegrator is compared c to a threshold value and a decision is takenn. The receeiver should also recov ver the rhythhm of the symbols transmittted. Oncee this has beeen done, thee binary sequ uences in thee ouptuts of both the in-phasee and the quuadrature chhannels are combined c viia the multipplexer to generatee the sequennce of moduulated binarry data. Thee QPSK is uused for transmisssion using satellite appplications succh as videocconferencingg, mobile telephonne systems and other digital d comm munications using u an RF F carrier. The matthematical reepresentationn of the QPSK signal is expressed e as:

Modulation

where Ts is the period of the symbol, Es is energy per symbol. QPSK demodulation In the same manner, we obtain for channel B: The receiver % function [a_cap] = qpsk_demod(r,fc,OF) % Function to demodulate a conventional QPSK signal % r – signal received upstream of the receiver % fc – carrier frequency in Hertz % OF – oversampling factor (multiples of fc) – at least 4 is better % L – upsampling actor on the in-phase and quadrature arms % a_cap – binary flow detected % OF = 8 fs = OF*fc; % sampling frequency L = 2*OF; % sampling over a duration 2Tb. r = s; t = 0:1/fs:(length(r)-1)/fs; % time base x = r.*cos(2*pi*fc*t); % x = I.*cos(2*pi*fc*t); y = -r.*sin(2*pi*fc*t); % y = -Q.*sin(2*pi*fc*t);

% Arms % Arms Q

51

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Digital Communication Techniques

x = conv(x,ones(1,L)); % convolution; integration over duration L (Tsym = 2*Tb) y = conv(y,ones(1,L)); % convolution; integration over a duration L (Tsym = 2*Tb) %x = x(L:L:end);% Arm I – sample at each instant of symbol Tsym %y = y(L:L:end);% ArmsQ – sample at each instant of symbol Tsym a_cap = zeros(1, 2*length(x)); a_cap(1:2:end) = x.' > 0; % even bits a_cap(2:2:end) = y.' > 0; % odd bits doPlot = 1; % For the receiver constellation plot % if doPlot = 1, Figure; plot(x(1:200),y(1:200),'o'); end if doPlot = 1, Figure; plot(x(1:102),y(1:102),'o'); end

Figure 1.45. A truncated constellation

Modulation

53

Coheerent demoddulation is applicable when the receiver haas exact knowleddge of the frequency and a the phasse of the caarrier. The synoptic schema of a coherennt demodulattor for QPSK K is shown inn Figure 1.466.

F Figure 1.46. Coherent C QPS SK demodulato or

The QPSK deemodulator is essentiially formeed of two BPSK demoduulators. In fact, the signaal received (aafter potentiaal bandpass ffiltering) is demoodulated in two t parallel channels by y two quadrature carrierrs. Some techniquues make itt possible too synchronizze the locall oscillator w with the carrier at transmisssion. The quuadrature sig gnal is generrated from tthe local oscillatoor and a phasse shifter of π/2. π If r(tt) = Ik.cos(ωc + ϕc) − Qk.ssin(ωct + ϕc), the non-noiisy signal is received by the receiver r in thhe time interrval [kT, (k + 1)T[. For channel A aand after multipliication with the t recovered carrier, wee obtain: S Sa1(t) = [ Ik.cos(ωct + ϕc) − Qk(ωct + ϕc)].cos(ωct + ϕc. After filtering, we w have elimiinated the freequency com mponent 2fc. fr MATLA AB 1.8.3.3. A QPSK from mQPSKTrannsmitterReceeiver.m. Th his examplee shows a digital comm communnications system using QPSK modu ulation. It uses u communnications

54

Digital Communication Techniques

system objects to simulate the QPSK transceiver. In particular, it illustrates methods using “wireless” in the world of real communications, such as carrier frequency and a phase offset, the re-establishment of synchronization and frame synchronization, and the time delay. The data transmitted from the QPSK undergo fadings that simulate the effects of wireless transmission such as the addition of added white gaussian noise (AWGN). To confront this weakening, this example provides a reference design of a digital receiver including raw compensation based on a fast Fourier transform (FFT) frequency, the end compensation using a phaselocked loop (PLL); the re-establishment of symbol synchronization is based on this PLL, the frame synchronization techniques and the phase ambiguity resolution. Three main objectives – Modeling a wireless communication system that can recover a message corrupted by various simulated channel impairments. – This example presents some blocks for QPSK system design from the MATLAB library, including the raw carrier frequency and end compensation, the synchronization recovery in closed circuit with bit consolidation and extraction. – Illustrating the creation of higher level system objects, which contain other objects so as to model larger components of the system under test. Models The runQPSKSystemUnderTest function models this communication environment. The QPSK transceiver model is divided into three main components. 1) QPSK transmitter: produces the bit stream, code, module and filter. 2) QPSK channel: models the channel with the carrier offset, the synchronization offset, and AWGN. 3) QPSK receiver: models the receiver, including the components for re-establishing the phase, synchronization recovery, decoding, demodulation, etc.

Modulation

55

Description of different components Transmitter: This component produces a message in ASCII characters, converts these into bits and adds a Barker code (oversampling by 2) for the frame synchronization sent to the receiver, plus the headers. The data are then modulated using QPSK, and filtered via a square root-raised cosine (Figure 1.52). The payload, in bits, (of the frame) is scrambled, thus guaranteeing a balanced distribution of zeros (see an improvement in data transition density) and other zeros for achieving synchronization recovery at the receiver (see estimating the frequency offset).

Figure 1.47. QPSK transmitter

Figure 1.48. Transmitter QPSK (MATLAB Inc)

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Digital Communicattion Techniquess

Beloow we indiccate some effects, e asseessed using the autocoorrelation functionn of Barker codes, c by diff fferent pulse shaping (S. Rao). R

F Figure 1.49. Barker-13 B phasse coded puls ses for differen nt pulse shape es

Figure e 1.50. Norma alized autocorrrelation functio on (lag) for a modu ulated Barker--13 impulse

NOTE.–– A Barker code c is a finiite sequence of N valuess of +1 and −1, with the ideaal autocorrellation propperty, such th hat the off-ppeak autocoorrelation coefficients (non-cyyclical) verifyy:

Modulation

57

= mall as possiible: | | ≤ 1, 1 ≤ are as sm

.

Figure 1.51 1. Filtering thro ough various raised r cosiness. For a color version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

Takiing an FFT, we w obtain thee following low l pass (Figgure 1.52).

Figure 1.5 52. Filtering an n impulse in ra aised cosine. For a color version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

Channel: ( show the eeffect of Figuures 1.53 (ssimulations) and 1.54 (measures) filteringg in raised coosine on a QP PSK.

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Digital Communicattion Techniquess

Figure 1.53. Filtering F F a QPS SK signal using g a raised cossine. For a collor version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

F Figure 1.54. Spectrum S of the modulated QPSK Q signal for f a binary flo ow of 20 Kbits/s witho out filtering strreams I and Q (Agilent). Forr a color versiion of this figure, see ww ww.iste.co.uk/g /gontrand/digittal.zip

Modulation

59

Figure 1.55. Spectrum F S of the modulated QPSK Q signal for f a binary flo ow of 20 Kbits/s with filtering (Agilent). (A For a color version n of this figure,, see www.iste e.co.uk/gontra and/digital.zip

i For a color version n Figure 1.56. Spectrum of a rectangular impulse. w k/gontrand/dig gital.zip of thiss figure, see www.iste.co.uk

Specctral conditioons (bandwidth) of the symbol rate can be observed in phase shhift modulattions in eighht states (8PS SK). This is a variation of PSK: eight poossible statess via signal transition, at any momentt. The signall’s phase can takee any of the eight valuess at any timee of the symbbol: 23 = 8: tthere are three bitts per symbool. This meanns that the sy ymbol rate iss a third of thhe binary flow.

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Digital Communication Techniques

This component simulates the effects of free-to-air transmission. It breaks down the signal transmitted with the phase and the frequency offset, a temporal drift to imitate the clock’s skew between the transmitter and the receiver, and the AWGN. Receiver:

Figure 1.57. This component recreates the original message sent. It is divided into seven subcomponents (Matlab Inc.)

1) Automatic gain control: its output power is adjusted via the inverse of the square root of the oversampling factor, so that the input amplitude from the raw frequency compensation subcomponent is stable and approximately 1. This ensures that equivalent gains from the phase and synchronization error detectors remain constant in time. The CAG is placed before the raised cosine reception filter, so that the amplitude of the signal can be measured with an oversampling factor of four. This process improves the precision of the evaluation. 2) Gross frequency compensation: uses nonlinearities and an FFT to give an approximate estimate of the frequency offset to compensate. 3) The frequency offset is estimated using a system object, *comm.PSKCoarseFrequencyEstimator*, and compensation is achieved using *comm.PhaseFrequencyOffset*. 4) Synchronization recovery: activates synchronization recovery with closed-loop scalar processing to overcome the delay effects caused by the

Modulation

61

channel, using the system object: *comm.SymbolSynchronizer*. The object implements a PLL to correct the symbol synchronization error in the signal received. A zero-crossing synchronization error detector is chosen here. The input is a frame of samples of constant length. The object’s yield is a frame of symbols whose length can change via reinforcement and extraction, depending on delays in the channel involved. 5) Preambule detection: detects the place of the Barker code at input using the object *comm.PreambleDetector*. This runs an algorithm based on cross-correlations to detect the known order of the symbols at input. 6) Frame synchronization: runs frame synchronization and converts input of symbols of variable length into outputs of constant length using *FrameSynchronizer*. 7) Data decoder: achieves resolution and demodulation of phase ambiguity. In addition, the data decoder compares the regenerated message with the transmitted one and calculates the bit error ratios (BERs). 1.8.3.4. Operation and results After using the System Under Test script and obtaining BERs values for simulated communication of a QPSK, the following MATLAB code is run. When you launch simulations, error rate data for the bits is displayed, as well as some results in graph form, respectively: – constellation: the output constellation diagram *Raised Cosine Receive Filter*; – output power spectrum *Raised Cosine Receive Filter*; – output constellation diagram *Fine Frequency Compensation* output; – (partial) synchronization error estimated from *Timing Recovery*. BER = runQPSKSystemUnderTest(prmQPSKTxRx, useScopes, printReceivedData); fprintf('Error rate = %f.\n',BER(1)); fprintf('Number of detected errors = %d.\n',BER(2)); fprintf('Total number of compared samples = %d.\n',BER(3));

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Digital Communication Techniques

Initialization The commqpsktxrx_init.m script initializes simulation parameters and generates the structure prmQPSKTxRx. prmQPSKTxRx = commqpsktxrx_init % QPSK system parameters useScopes = true; % true if scopes are to be used printReceivedData = false; % true if the received data is to be printed compileIt = false; % true if code is to be compiled useCodegen = false; % true to run the generated mex file prmQPSKTxRx = struct with fields: M: 4 Upsampling: 4 Downsampling: 2 Fs: 200000 Ts: 5.0000e-06 FrameSize: 100 BarkerLength: 13 DataLength: 174 ScramblerBase: 2 ScramblerPolynomial: [1 1 1 0 1] ScramblerInitialConditions: [0 0 0 0] sBit: [17400×1 double] RxBufferedFrames: 10 RaisedCosineFilterSpan: 10 MessageLength: 105 FrameCount: 100 PhaseOffset: 47 EbNo: 13 FrequencyOffset: 5000 DelayType: 'Triangle' CoarseCompFrequencyResolution: 25

Modulation

63

PhaseRecoveryLoopBandwidth: 0.0100 PhaseRecoveryDampingFactor: 1 TimingRecoveryLoopBandwidth: 0.0100 TimingRecoveryDampingFactor: 1 TimingErrorDetectorGain: 5.4000 ModulatedHeader: [13×1 double] Rolloff (fallout factor, band excess at: 0.5000) TransmitterFilterCoefficients: [1× 41 double] ReceiverFilterCoefficients: [1× 41 double] 1.9. Error vector module and phase noise The script below, which is MATLAB/OCTAVE/SCILAB compatible, calculates the transmission spectrum from a symbol modulated using QPSK according to Es/N0 for different root mean square (RMS) phase noise values. % Simulation script for a spectrum transmitted using QPSK % symbols altered by phase and thermal noise % ––––––––––––––––––––––––––––– % Simulation script for a spectrum transmitted using QPSK % symbols altered by phase and thermal noise % ––––––––––––––––––––––––––––– clear; close all; N = 10^5; % number of symbols os = 4; % oversampling factor %Es_N0_dB = 40; Es_N0_dB = [0:1:6]; phi_rms_deg_vec = [0:1:6]; % root-raised cosine filter t_by_Ts = [-4:1/os:4]; beta = 0.5;

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Digital Communication Techniques

ht = (sin(pi*t_by_Ts*(1-beta)) + 4*beta*t_by_Ts.*cos(pi*t_by_Ts*(1+beta)))./(pi*t_by_Ts.*(1(4*beta*t_by_Ts).^2)); ht((length(t_by_Ts)-1)/2 + 1) = 1 -beta + 4*beta/pi; ht([-os/(4*beta) os/(4*beta)] + (length(t_by_Ts)-1)/2 + 1) = beta/sqrt(2)*((1 + 2/pi)*sin(pi/(4*beta)) + (12/pi)*cos(pi/(4*beta))); ht = ht/sqrt(os); for ii = 1: length(Es_N0_dB) for jj = 1: 2: length(phi_rms_deg_vec) % Transmitter ip_re = rand(1,N) > 0.5; % % generation ds [ 0,1]: uniform probability ip_im = rand(1,N) > 0.5; % generation ds [ 0,1]: uniform probability s = 1/sqrt(2)*(2*ip_re-1 + j*(2*ip_im-1)); % QPSK modulation % impulse formatting s_os = [s ; zeros(os-1,length(s))]; s_os = s_os(:).'; s_os = conv(ht,s_os); s_os = s_os(1:os*N); % addition of thermal and phase noise n = 1/sqrt(2)*[randn(1,N*os) + j*randn(1,N*os)]; % thermal noise phi = phi_rms_deg_vec(jj)*(pi/180)*randn(1,N*os)*sqrt(os); % phase noise y = s_os.*exp(j*phi) + 10^(-Es_N0_dB(ii)/20)*n; % spectrum transmitted [Pxx1(jj,:) W2 ] = pwelch(y,[],[],1024,'twosided'); % adjusted filtering y_mf_out = conv(y,fliplr(ht)); y_mf_out = y_mf_out(length(ht):os:end);

Modulation

65

% vector error error_vec = (y_mf_out-s); evm(ii,jj) = error_vec*error_vec'; theory_evm(ii,jj) = 10^(-Es_N0_dB(ii)/10) + 2 - 2*exp(-(phi_rms_deg_vec(jj)*pi/180).^2/2); % plot(10*log10(theory_evm)) end end figure (1) plot(10*log10(theory_evm)) xlabel(' Es/N0'); ylabel(' vector error '); %axis([1 6 - 40 30]); grid on %hold on figure (2) plot([-512:511]/1024,10*log10(fftshift(Pxx1))) % fftshift can be useful for visualizing the FT with the % zero-frequency component in the middle of the spectrum. pxx = pwelc(x): NOTE.– Welch provides an estimator of the power spectral density. It reflects the evaluation of the power spectral density (PSD), pxx, of the input signal, x, found, breaking the signal down into several segments, then applying a window (see temporal data weighting, thus limiting the Gibbs phenomenon: this is a side effect, linked to drift discontinuities; window functions accord more importance to data from the center of the segment to that from the edges, which leads to a loss of information. The recovery of segments, which we will average, makes it possible to reduce this effect). When x is a vector, it is treated as a simple channel. When x is a matrix, the PSD is calculated independently for each column and stored in the corresponding column of the pxx. If x has real values, pxx is a one-sided evaluation of the PSD. If x is complex, pxx is a double-sided evaluation of PSD. By default, x is divided into the longest segments possible so as not to exceed eight segments with an overlap of 50%. Each segment is provided with a Hamming window, whose coefficients can be given by: w(n) = 0.54 – 0.46 cos (2πn/N),

0≤



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Digital Communication Techniques

Fliplr: this MATLAB function reflects A with its columns reversed from left to right (i.e. turned around a vertical axis). xlabel('frequency, kHz'); ylabel('amplitude, dB'); legend('0 deg rms', '2 deg rms', '4 deg rms'); title('Spectrum Es/N0 = 40dB, filtering in root-raised cosine; phase noise rms'); %axis([-0.5 0.5 -50 5]); grid on

Figure 1.58. Amplitude of the vector error depending on the signal/noise ratio. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 1.59. Calculation of a typical spectrum of a vector error. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Modulation

The MATLAB (Figure 1.60).

documentation

gives

this

type

of

67

oscillogram

Figure 1.60. Typical spectrum of a vector error, filtered by a raised cosine (MATLAB). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 1.61. Constellation after filtering in raised cosine. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Digital Communication Techniques

Figure 1.62. Constellation after fine frequency offsets. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 1.63. Characteristic of a phase detector (the zig-zags are not ideal). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

In Figures 1.64 and 1.65, we find simulations of QPSK constellations. % Create a QPSK modulator. mod = comm.QPSKModulator;

Modulation

§ Determine the reference constellation points. refC = constellation(mod) refC = 0.7071 -0.7071 -0.7071 0.7071

+ + -

0.7071i 0.7071i 0.7071i 0.7071i

% Plot of the constellation. constellation(mod) % Create a PSK demodulator having 0 phase offset. demod = comm.QPSKDemodulator('PhaseOffset',0); constellation(demod) 1.9.1. Plot QPSK reference constellation Create a QPSK modulator. mod = comm.QPSKModulator; Determine constellation reference points. refC = constellation(mod) refC = 0.7071 + 0.7071i -0.7071 + 0.7071i -0.7071 - 0.7071i 0.7071 - 0.7071i Of the constellation. constellation(mod) %% Plot QPSK Reference Constellation % %%

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Digital Communicattion Techniquess

% Creatte a QPSK modulator. m mod = comm.QPSK c KModulator; %% % Deterrmine the refference constellation poin nts. refC = constellation c n(mod) %% % Plot the t constellaation. constelllation(mod) %% % Creatte a PSK dem modulator %% havving 0 phase offset. demod = comm.QPS SKDemodulator('PhaseO Offset',0); % Plot its i reference constellationn. The |consttellation| method works for % both modulator annd demodulaator objects. constelllation(demodd) Resuult:

F Figure 1.64. Creating C a QPS SK constellatio on

Modulation

Figure 1.65. Another A QPSK K constellation n

Creatio on of a PSK K demodulattor, with 0 offset o demod = comm.QPS SKDemodulator('PhaseO Offset',0); constelllation(demodd) Phase noise on a QPSK sign nal % Creaation f’ of a QPSK moddulator objeect and a phase noise object. qpskMoodulator = coomm; phNoisee = comm.PhhaseNoise('L Level',55,'FreqquencyOffsett',20,'SampleeRate',1000);; % Produuces random m QPSK data. Considers the t signal via v the phase noise objectt. d = randdi([0 3],10000,1); % randoom (random m/stochastic): generation of pseudo-raandom numberrs. x = qpskkModulator((d); y = phN Noise(x);

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Digital Communication Techniques

Displays ‘below’: Figure 1.66, the constellation diagram of the noisy QPSK signal. The phase noise shows a rotation fluctuation on the constellation diagram. = comm.ConstellationDiagram; constDiagram(y): noisy in-phase constellation plot:

Figure 1.66. Noisy in-phase constellation. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Raised cosine function %function [p,t,filtDelay] = raisedCosine Function(alpha,L, Nsym) % Generate raised – cosine (RC) impulse % alpha – roll-off factor (excess band factor; beyond the Nyquist criteria), alpha = 2 % L – oversampling factor L=4 % Nsym – filter span in symbols Nsym = 4 % Returns the output impulse p(t) that spans the discrete-time

Modulation

% base -Nsym:1/L:Nsym. Also returns the filter delay when the % function is viewed as an FIR filter Tsym = 1; t = -(Nsym/2):1/L:(Nsym/2); % ± discrete-time base A = sin(pi*t/Tsym)./(pi*t/Tsym); B = cos(pi*alpha*t/Tsym); % handle singularities at p(0) and p(t = ±1/2a) p = A.*B./(1-(2*alpha*t/Tsym).^2); p(ceil(length(p)/2)) = 1; %p(0) = 1 and p(0) occurs at center temp = (alpha/2)*sin(pi/(2*alpha)); p(t == Tsym/(2*alpha)) = temp; filtDelay = (length(p)-1)/2; %FIR filter delay = (N-1)/2 plot(p) scatterplot(p) end

Figure 1.67. Delay: FIR filter (digital filters with finite-duration impulse response)

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1

0.8

0.6

0.4

0.2

0

-0.2

0

2

4

6

8

10

12

14

16

18

Figure 1.68. Probability density

%% Phase noise effects on 16-QAM (MATLAB Inc.) % Add a phase noise vector and frequency offset vector to a 16-QAM signal. % Then, plot the signal. %% % Create 16-QAM modulator having an average constellation power of 10 W. modulator = comm.RectangularQAMModulator(16,... 'NormalizationMethod','Average power','AveragePower',10); %% % Create a phase noise object (MATLAB Inc.). pnoise = comm.PhaseNoise('Level',50,'FrequencyOffset',20); %% % Generate modulated symbols. data = randi([0 15],1000,1); modData = modulator(data); %% % Apply phase noise and plot the result.

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75

y = pnoise(modData); scatterplot(y) % Scatter: dispersion, collision. Scatter plot 3 2

Quadrature

1 0 -1 -2 -3 -3

-2

-1

0

1

2

3

In-Phase

Figure 1.69. QAM: phase error

1.9.2. Effects of phase noise on 16-QAM Adds a phase noise vector and offset to the frequency vector on a 16-QAM signal. Then, it plots the signal. It creates the 16-QAM modulator with an average constellation power of 10 W. modulator = comm.RectangularQAMModulator(16,... 'NormalizationMethod','Average power', 'AveragePower',10); % Create a phase noise object. pnoise = comm.PhaseNoise 50,'FrequencyOffset',20); % Generate modulated symbols. data = randi([0 15],1000,1);

('Level',-

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modData = modulator(data); % Apply phase noise and plot the result. y = pnoise(modData); scatterplot(y) Hence:

Figure 1.70. Phase error (QAM)

1.9.3. Phase noise: effects of the signal spectrum Creation of a sinusiodal wave generator with a carrier frequency of 100 hertz, a sampling rate of 1000 hertz, and a frame size of 10.000 samples. sinewave = dsp.SineWave('Frequency',100, 'SampleRate',1000,... 'SamplesPerFrame',1e4,'ComplexOutput',true); % Creation of a phase noise object (pnoise). It indicates the level of phase noise for -40 dBc/Hz of offsets offset by 100 hertz and -70 dBc/Hz for offsets offset by 200 hertz

Modulation

pnoise = comm.PhaseNoise('Level',[-40 70],'FrequencyOffset',[100 200],...

77

-

'SampleRate',1000); % Spectrum analyzer: spectrum = dsp.SpectrumAnalyzer('SampleRate',1000,'Spectral Averages',10,'PowerUnits','dBW'); % Production of a sinusoidal wave. Adds the phase noise to the sine wave. Noisy signal spectrum plots (see Figure 1.72). x = sinewave(); y = pnoise(x);

spectrum(y) We obtain:

Figure 1.71. Carrier (spur: see line) and phase noise. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Digital Communication Techniques

The phase noise is –40 dB to ±100 Hz of the carrier. The phase noise is –70 dB under the carrier for offsets greater than 200 Hz. 1.9.4. Algorithms

Figure 1.72. Phase noise

The output of the object of the phase noise, yk, is linked to the input sequence xk, input term, by yk = xkejϕk, where ϕk is the phase noise. The phase noise is a Gaussian noise so that ϕk = f (hk), where hk is the term of the noise and f represents a filtering. If FrequencyOffset is a scalar quantity, then we use the digital infinite impulse response (IIR) filter in which the numerator coefficient, λ, is λ = G2πfoffset10 L/10. An IIR filter is characterized by a response to the values of the input signal as well as the previous values of the same response. It is also called a recursive filter. This filter is one of the two types of linear digital filter. The other possible type is the finite impulse response filter (RIF filter). Unlike the RII filter, the response of the RIF filter only depends on the values of the input signal. Consequently, the impulse response of an RIF filter is always of finite duration. foffset is the offset frequency in hertz and L is the level of phase noise in dBc/Hz. FrequencyOffset is a vector; an FIR filter is used. The phase noise is set by the introduction of a scale log10 for frequency offset on the range [df, fs/2], where the DF is the resolution of the frequency and fs is the sample rate. The frequency resolution is equal to fs/2 (= 1/NT), where N is the number of samples. The object increases the NT until the frequency resolution is less than the minimum value of the FrequencyOffset vector or a maximum value of 512 is reached. This value has been chosen to balance the

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79

contradictory conditions of a small frequency resolution and a rapid filtering. The object properties correspond to the block parameters. This object uses a draw of random numbers often based on the remainder of the recursive divisions. In addition, the block uses an initial value to initialize the generator of the random number. Each time the system that contains the block re-runs, the block produces the same series of (pseudo-) random numbers. 1.9.5. Spectrum analyzer In MATLAB, (and other simulators) it is easy to represent the spectrums (see Figure 1.73): Fs = 100e6; % Sampling frequency fSz = 5000; % Size of the frame sin1 = dsp.SineWave(1e0, 5e6,0,'SamplesPerFrame',fSz,'SampleRate',Fs); sin2 = dsp.SineWave(1e1,15e6,0,'SamplesPerFrame',fSz,'SampleRate',Fs); sin3 = dsp.SineWave(1e2,25e6,0,'SamplesPerFrame',fSz,'SampleRate',Fs); sin4 = dsp.SineWave(1e3,35e6,0,'SamplesPerFrame',fSz,'SampleRate',Fs); sin5 = dsp.SineWave(1e4,45e6,0,'SamplesPerFrame',fSz,'SampleRate',Fs); scope = dsp.SpectrumAnalyzer; scope.SampleRate = Fs; scope.SpectralAverages = 1; scope.PlotAsTwoSidedSpectrum = false; scope.RBWSource = 'Auto'; scope.PowerUnits = 'dBW'; for idx = 1:1e2 y1 = sin1(); y2 = sin2();

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y3 = sin3(); s y4 = sin4(); s y5 = sin5(); s scope((y1+y2+y3+y4+y5+0.0001*randn(fS Sz,1)); end

Figure 1.73. Simulation off a spectrum analyzer. a For a color version n w k/gontrand/dig gital.zip of thiss figure, see www.iste.co.uk

Figu ure 1.74. A 16 QAM

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81

1.9.6. Measures of the error vector module of a signal modulated by a noisy 16-QAM %comm.RectangularQAMModulator(16,...) % %Measure EVM of Noisy 16-QAM Modulated Signal(MATLAB InC.) %% % Create an EVM object. ConFigure it using name-value pairs to output % maximum EVM, 90th percentile EVM, and the symbol count. evm = comm.EVM('MaximumEVMOutputPort',true,... 'XPercentileEVMOutputPort',true, 'XPercentileValue',90,... 'SymbolCountOutputPort',true); %% % Generate random data symbols. Apply 16-QAM modulation. The modulated % signal serves as the reference for the subsequent EVM measurements. data = randi([0 15],1000,1); refSym = qammod(data,16,'UnitAveragePower',true); %% % Pass the modulated signal through an AWGN channel. rxSym = awgn(refSym,20); %% % Measure the EVM of the noisy signal. [rmsEVM,maxEVM,pctEVM,numSym] = evm(refSym,rxSym) % Measure the EVM of a Noisy16 QAM Modulated Signal Example Results: rmsEVM = 10.0838 maxEVM = 30.9821 pctEVM = 14.9950 numSym = 1000 1.10. Gaussian noise (AWGN) clear all % Create a sawtooth wave at. t = (0:0.1:20)'; %x = sawtooth(t, 0.5); x = square(2*pi*0.15*t)

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Digital Communication Techniques

% Apply white Gaussian noise (AWGN) and plot the results. %y = awgn(x,20,'measured'); plot(t,[x x], 'LineWidth',1) legend ('non-noisy signal: pulses') Consider a 101010 train.

Figure 1.75. At the transmitter: pulses: 101010. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 1.76. In the channel: pulses: 101010. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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83

Figure 1.77 7. Filtering (avverage, integra ation). For a color c version of thiss figure, see www.iste.co.uk w k/gontrand/dig gital.zip

Fiigure 1.78. Sa ampling/thresh holding (without error) (1010 010). For a co olor version off this figure, se ee www.iste.c co.uk/gontrand d/digital.zip

1.10.1. AWGN channel e usee an AWGN N (added whhite Gaussiaan noise) The following examples channell: QPSK trannsceiver andd general QAM Q modulaation in the AWGN channell.

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Digital Communication Techniques

Level of AWGN channel noise: The relative power of the noise in an AWGN channel is generally described by quantities such as: – signal-to-noise ratio (SNR) per sample. This is the real input parameter of the AWGN function; – energy to bit ratio over PSD of noise (EbNo). This quantity is used by the BERTool (from MATLAB) and the functions for evaluating this tool’s performances; – symbol energy/noise PSD (EsNo) ratio; – ratio between EsNo and EbNo. The ratio between EsNo and EbNo, both expressed in dB, is the following: Es/N0 (dB) = Eb/N0 (dB) + 10log10(k), where k is the number of data bits per symbol: fb /B; fb is the net bit rate and B (= 1/ Tsamp) is the channel BW. In a communication system, k can be influenced by the size of the modulation alphabet or by the flow of an error-checking code. For example, if a system uses a ½ rate code and a 8-PSK modulation, the number of bits of data information per symbol (k) is the product of the code rate and the number of bits coded per symbol modulated: (1/2) log2 (8) = 3/2. In such a system, 3 bits of information correspond to six coded bits, which correspond in turn to two 8-PSK symbols. 1.10.2. Ratio between EsNo and SNR The ratio between EsNo and SNR, both these quantities being expressed in dB, is, for complex input symbols, the following: Es/N0 (dB) = 10log10(Tsym/Tsam) + SNR(dB) Tsym is the period and the Tsamp of the signal’s symbol is its sampling period. For example, if a complex baseband signal is oversampled by a factor of 4, EsNo exceeds the corresponding SNR by 10 log10 (4).

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85

1.10.3. Behavior for real and d complex x input sign nals The following fiigures illustrate the difference between real and complex cases by b showing the noise PSDs P Sn(f) of o a real paassband whiite noise process and its compplex passbannd equivalentt. We need n to be aw ware that sometimes thee noise poweer is written aas N0 / 2 when negative n freqquencies andd baseband signals equuivalent to complex values are a taken into account, raather than paassband signnals, and in thhis case, there wiill be a differrence of 3 dB B (see Figuree 1.79).

Fig gure 1.79. PS SDs

1.11. QAM Q modullation in an n AWGN ch hannel Our aim is to traansmit and reeceive data using u the 16 constellationns in the presence of Gaussiaan noise. Show the “plott” of the noiisy constellaation and estimatee the symboll error rate (S SER) for two o different siggnal/noise raatios. Creaate a 16-QA AM constellattion based on o the V.29 norm for teelephone line modems. - 5 5i -3 -3-3i -3i 3-3i 3 3+3i 3i -3+ +3i -1 -1i 1 1ii]; c = [-5 -5i M = lenngth(c); Generatte randomizeed signals. data = randi([0 r M-1],2000,1);

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Digital Communication Techniques

Modulate data using the MATLAB genqammod. function. It is necessary to use general QAM modulation because the constellation is not rectangular. modData = genqammod(data,c); Pass the signal through an AWGN channel with a signal/noise ratio of 20 dB. Show a scatter plot for the signal received with the reference constellation. h = scatterplot(rxSig); hold on scatterplot(c,[],[],'r*',h) grid Result: Scatter plot 6 4 2 0 -2 -4 -6 -6

-4

-2

0

2

4

6

In-Phase

Figure 1.80. Trace of scatter from a 16 QAM. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Demodulate the signal received using the genqamdemod function and determine the number of symbol errors S and the symbol error ratio.

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87

demodData = genqamdemod(rxSig,c); [numErrors,ser] = symerr(data,demodData) numErrors = 1 ser = 5.0000e-04 Repeat the transmission and demodulation process with an AWGN channel with an SNR of 10 dB. Determine the SER for the reduced signal/noise ratio. As predicted, running breaks down when the SNR diminishes. rxSig = awgn (modData,10,'measured'); demodData = genqamdemod(rxSig,c); [numErrors,ser] = symerr(data,demodData) numErrors = 462 ser = 0.2310 %% General QAM Modulation in an AWGN Channel, MathWorks, Inc. % Transmit and receive data using a nonrectangular 16-ary constellation in % the presence of Gaussian noise. Show the scatter plot of the noisy % constellation and estimate the symbol error rate (SER) for two different % signal-to-noise ratios. %% % Create a 16-QAM constellation based on the V.29 standard for % telephone-line modems. c = [-5 -5i 5 5i -3 -3-3i -3i 3-3i 3 3+3i 3i -3+3i -1 -1i 1 1i]; M = length(c); %% % Generate random symbols. data = randi([0 M-1],2000,1); %% % Modulate the data using the |genqammod| function. It is necessary to use % general QAM modulation because the custom constellation is not % rectangular. modData = genqammod(data,c); %% % Pass the signal through an AWGN channel having a 20 dB signal-to-noise % ratio (SNR). rxSig = awgn(modData,20,'measured'); %% % Display a scatter plot of the received signal along with the reference

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Digital Communication Techniques

% constellation, |c|. h = scatterplot(rxSig); hold on scatterplot(c,[],[],'r*',h) grid %% % Demodulate the received signal using the |genqamdemod| function and % determine the number of symbol errors and the symbol error ratio. demodData = genqamdemod(rxSig,c); [numErrors,ser] = symerr(data,demodData) %% % Repeat the transmission and demodulation process with an AWGN channel % having a 10 dB SNR. Determine the symbol error rate for the reduced % signal-to-noise ratio. As expected, the performance degrades when the SNR % is decreased. rxSig = awgn(modData,10,'measured'); demodData = genqamdemod(rxSig,c); [numErrors,ser] = symerr(data,demodData) Hence:

Figure 1.81. 16 constellations. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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89

1.11.1. QAM demodulation genqamdemod (general quadrature amplitude demodulation) %% General QAM Modulation and Demodulation %% % Create the points that describe a hexagonal constellation (MATLAB Inc.). inphase = [1/2 1 1 1/2 1/2 2 2 5/2]; quadr = [0 1 -1 2 -2 1 -1 0]; inphase = [inphase;-inphase]; inphase = inphase(:); quadr = [quadr;quadr]; quadr = quadr(:); const = inphase + 1i*quadr; %% % Plot the constellation. h = scatterplot(const); %% % Generate input data symbols. Modulate the symbols using this % constellation. x = [3 8 5 10 7]; y = genqammod(x,const); %% % Demodulate the modulated signal, |y|. z = genqamdemod(y,const); %% % Plot the modulated signal in same figure. hold on; scatterplot(y,1,0,'ro',h); legend('Constellation','Modulated signal'); %% % Determine the number of symbol errors between the demodulated data to the % original sequence. numErrs = symerr(x,z) y = genqammod(x,const); %% % Demodulate the modulated signal, |y|. z = genqamdemod(y,const); %% % Plot the modulated signal in same figure. hold on; scatterplot(y,1,0,'ro',h);

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legend(''Constellatioon','Modulateed signal'); %% % Deterrmine the nuumber of sym mbol errors between the demodulated d data to the % originnal sequencee. numErrrs = symerr(xx,z) Result:

Figure 1.82. Plot of a QAM Q constella ation. For a color version w k/gontrand/dig gital.zip of thiss figure, see www.iste.co.uk

Obseerve that theere is a goodd agreementt between thhe simulated and the theoretical data. or 1.11.2. Detecting phase erro

F Figure 1.83. Phase P detectio on (Matlab Incc)

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91

The signal can be written as: y(t) = x(t)ejφ + η(t), t = 1,2,…,N. We seek to express the value of de φ from the observation: y1(t) t = 1,2,…,N (i.e., to estimate the value B = ejφ). If we suppose η(t) assimilated to a Gaussian noise distribution, then an unbiased estimator may be: Best = 1/N*sum(y1(1 :N)) and φis = angle (Bis) The variance of Best is therefore: σ2/N. Phase offset:

Suppression of I/Q imbalance Systems (QPSK, …, OFDM) suffer from phase/quadrature-phase (IQ) imbalances in frontal analog processing, which can have a considerable impact on performances. Similarly, the local oscillator undergoes a carrier frequency shift. Since these modulations are very sensitive to carrier frequency shifts, this distortion should be considered when analyzing any schema for estimating/compensating IQ imbalance. Algorithms are developed for compensating these distortions in the digital domain.

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Digital Communication Techniques

%% Deleting the 'I/Q (Imbalance) of a QPSK signal % Mitigating the impacts of amplitude and phase imbalance % over a signal modulated by a QPSK using the system object: comm.iqimbalancecompensator (TM). %% % Creating a constellation diagram object. To indicate the non-value pairs % to ensure the constellation diagram shows only the last 100 % data symbols.constDiagram = comm.ConstellationDiagram(... 'SymbolsToDisplaySource','Property',... 'SymbolsToDisplay',100); %% % To create an I/Q imbalance compensater. iqImbComp = comm.IQImbalanceCompensator; %% % To generate random data symbols and apply %% modulation %QPSK data = randi([0 3],1e7,1); txSig = pskmod(data,4,pi/4); %% % To apply an imbalance in amplitude and phase to the signal transmitted. ampImb = 5; % dB phImb = 15; % deg gainI = 10.^(0.5*ampImb/20); gainQ = 10.^(-0.5*ampImb/20); imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180); imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180)); rxSig = imbI + imbQ; %% % To plot the constellation diagram of the signal received. Observe that this signal has undergone a shift in amplitude and phase. constDiagram(rxSig) %% % To apply the I/Q compensation algorithm and observe the constellation (b). % The constellation of the compensated signal (c) is almost identical to the reference constellation (a) compSig = iqImbComp(rxSig); constDiagram(compSig)

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93

(a)

(b)

(c)

Figure 1.84. Removing I/Q imbalance. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

1.12. Frequency-shift keying Frequency- and phase-shift key shifts are very similar. A frequency shift of +1 Hz “shifts” the phase to the rhythm of 360° per second, compared to the phase of a unshifted signal. The amplitude remains unchanged, and the frequency varies with the rhythm of the modulating frequency. FSK is an FM modulation using a primary antipolar signal (–1; +1).

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Digital Communicattion Techniquess

Fig gure 1.85. Pha ase shifting. For F a color verssion of thiss figure, see www.iste.co.uk w k/gontrand/dig gital.zip

1.12.1. Binary FS SK

(

)

The modulated signal is writtten as: m( t ) = A cos 2πf pt + Φ ( t )

1 dΦ ( t ) Δf = ak × g( t − kT ) with 2π dt 2 g(t) the rectangular waveform w off duration T and unitary amplitude. a

In thhe interval [kkT, (k+1)T[, we write:

T, (k+1)T[ we w can write after integrattion: As a result, in thee interval [kT Φ (t ) = π .Δf .ak .(t − kT ) + θ k

The modulated signal s can finally be wriitten in the innterval [kT, (k+1)T[ as: m(t ) = A cos ( 2π f p t + π .Δf .ak .(t − kT ) + θk )

Let us u examine performance p e, in the interval [0, T[ and a take as aan origin θ0 = 0: m(t ) = Acos ( 2π f pt + π .Δf .a0 .t. )

A t = T (by negativee value), we write: w m(T ) = Acos ( 2π f pT + π .Δf .a0 .T )

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95

Let us examine performance, in the interval [T, 2T[:

(

m( t ) = A cos 2πf p t + π .Δf .a1 .( t − T ) + θ 1

A

t

=

T,

we

write:

)

m (T ) = A cos ( 2π f pT + π .Δf .a1 .(T − T ) + θ1 )

= A cos(2π f pT + θ1 )

So that there can be phase continuity, it is necessary that: θ 1 = π .Δf .a0 .T A t = 2T (per negative value), we write:

(

)

m( 2T ) = A cos 2πf p ( 2T ) + π .Δf .a1 .( 2T − T ) + θ1 = A cos(2πf p ( 2T ) + π .Δf .a1 .T + θ1 )

Let us also examine performance in the interval [2T, 3T[:

(

m( t ) = A cos 2πf p t + π .Δf .a 2 .( t − 2T ) + θ 2

)

A t = 2T, we write:

(

)

m( 2T ) = A cos 2πf p ( 2T ) + π .Δf .a1 .( 2T − 2T ) + θ 2 = A cos( 2πf p ( 2T ) + θ 2 )

So that there can be phase continuity; it is necessary that:

θ 2 = π .Δf .a1 .T + θ1 Finally, the condition of continuity in the interval [kT, (k+1)T[ is written as:

θ k = π .Δf .a k − 1 .T + θ k − 1 1.13. Minimum-shift keying Minimum frequency keying produces a phase advance or a phase delay; frequency keying can be detected through sampling, using phase sampling at each symbol period. Phase shifts of π(2N + 1)/2 radian are easily detected with a I/Q demodulator. At even symbols, the I arm of the channel gives the data transmitted, while at odd symbols, it is the Q arm. This orthogonality between I and Q simplifies detection algorithms and consequently reduces the power in a mobile receiver. The minimum frequency keying that plots

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Digital Communication Techniques

the orthogonality of I and Q is the one that has a phase shift of π/2 radian per symbol as a consequence. FSK endowed with this deviation is called minimum-shift keying (MSK). The deviation should be precise so as to produce repeated phase shifts at 90°. FSK is used in a GSM cellular norm (global system for mobile communications). A phase shift of +90° represents one bit of information equal to “1”, whereas −90° represents “0”. The peakto-peak frequency offset of an FSK is equal to half the binary flow. The FSK and MSK produce constant envelope conveyor signals, which therefore have no amplitude variation. This is a desirable characteristic for improving transmitter power efficiency. Amplitude variations can exert nonlinearities in the amplifiers’ amplitude transfer function, producing spectral regrowth, a component of the power at the adjacent channel. Consequently, more effective but less linear amplifier scans can be used with constant envelope signals, reducing energy consumption. NOTE.– GMSK uses a Gaussian filter. The central lobe is kept, unlike the others (see Figure 1.87); – much used for 2G, or GSM, at around 270 kbits/s; – 99% of the power is found at a band of 240 kHz; – as for spectral efficiency, it is in the order of 1.3 bit/s.Hz; – also used for DECT (Digital European Cordless Telephone), with a flow of 1.13 Mbits/s and a BW of around 1.73 MHz; – spectral efficiency: around 2/3 bit/s.Hz.

Figure 1.86. Minimum-shift keying (MSK) modulation spectrum: continuous phase, modulation index: 0.5. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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97

Figure 1.87. Gaussian MS SK (GMSK); th he data are, fro om the outset,, processed ussing a Gaussia an filter. For a color version n of this figure,, see www.iste e.co.uk/gontra and/digital.zip

1.13.1. Bit error ratio r (BER)//Gaussian channel % Geneerate theoretiical BER datta for AWGN N channels % data for f several modulation m scchemas, conssidering a AW WGN channnel. % Creatte a vector off Eb/No valuues and defin ne the modulation order M M. clear all EbNo = (0:10)'; M = 4; %% K modulation using beraw wgn % Geneerate theoretiical BER datta for QPSK %==== ======== ======== ================ ======== ===== ===== ====== % functtion. berQ = berawgn(EbN No,'psk',M,'nnondiff'); %% % Geneerate equivallent data for DPSK and FSK. F berD = berawgn(EbN No,'dpsk',M); berF = berawgn(EbN b No,'fsk',M,'coherent');

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Digital Communication Techniques

%% % Plotting results. semilogy(EbNo,[berD berF berQ]) xlabel('Eb/No (dB)') ylabel('BER') legend('DPSK','FSK','QPSK') grid Result:

Figure 1.88. BER for different modulations. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

1.13.2. Typical analytical expressions used in “berawgn” The average bit error probability for each channel is: Pb = Pr(if e/1 sent) = Pr(if e/0 sent) Pb =

exp [ −

(

/ )

]d

Modulation

=

.

exp [ −



=

99

] dx

=

The demodulator output is none other than the multiplexed output of channels I, Q. The bit error rate for the output is the same as that for each channel. One symbol represents two bits from channels I, Q. A symbol error occurs if both are false. So the error probability per symbol will be: Ps = 1 – Pr (both bits are correct) = 1- ( 1 – Pb)2 = 2Pb – Pb2 =2



(coherent QPSK)

1.14. Amplitude-shift keying In this case, modulation only takes place on the in-phase carrier cos (ωpt + ϕp). There is no quadrature carrier. This modulation is sometimes called one dimensional. The modulated signal is then written as: m(t) = ∑_

.g(t − kT ).cos(ωpt + ϕp)

The waveform g(t) is rectangular of T duration and amplitude equal to 1 if t belongs to the interval [0, T[ and equal to 0 elsewhere. Remember that the Ik symbol takes its value in the (A1, A2, … AM) alphabet. In other words, this alphabet highlights the M = 2n possible amplitudes of the signal, the value n designating the groups of n bits or symbols to be transmitted. The carriers’ changes in amplitude will be produced at the rhythm R of the symbol transmission. 1.14.1. On–off keying One example of AM is (binary) OOK. Here, a single bit is transmitted per period T, and so n = 1 and M = 2. The symbol Ik takes its value in the

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Digital Communication Techniques

alphabet (0, I0). We therefore observe carrier extinctions on a chronogram when Ik = 0 (see Figure 1.90). We can note that this modulation is equivalent to an ASK with suppressed carrier (AM-P) by an often random binary unipolar signal (0; +1). The continuous component of the unipolar signal causes a line at frequency fp, for the PSD to which are added the lateral bands either side of the fp, corresponding to the binary, unipolar PSD signal. If the B2 filter is infinitely steep, then interference appears between moments (here: harmonic). We also have B2 = B1, B1 being the baseband. Then, we either reduce the primary band or we limit the smoothing, taking account of the central symmetry of the flanks in relation to fp± B1/2 (conditions directly related to Nyquist criteria).

Figure 1.89. On–off keying (OOK) amplitude modulation

Figure 1.90. OOK constellation

Modulation

101

On reception, demodulation is carried out by envelope detection. In the absence of noise, elevation to the square of the signal m(t) (i.e considering a signal proportional to power) induces a term at frequency 2fp, eliminated by filtering (using a lowpass circuit), and an information-carrying term, in baseband, proportional to  Ik 2.g(t − kT)2. The spectrum of the baseband signal is given by the Fourier transform of an impulse: ∝

( )=

(

) +

( )

The spectrum of the modulated signal is the same spectrum, shifted by ± fp, and includes a line at ± fp frequencies. 1.14.2. Modulation at “M states” In this case, we instead use symmetrical modulation. “ASK symmetrical modulations” There are always M = 2n amplitudes possible for the signal, but here the alphabet values are such that: Ai = (2i – M + 1).a0 with i = 1,2, … M. Following the values of n, we obtain: n = 1, M = 2;

-1.I0, 1. I0

n = 1, M = 4;

-3 I0, -1 I0, 1 I0, 3 I0

n = 1, M = 8;

-7 I0, -5 I0, -3 I0, -1 I0, 1 I0, 3 I0, 5 I0, 7 I0

The constellation of the modulation at M symmetrical states is given in Figure 1.91 for M taking values 2, 4 and 8.

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Digital Communication Techniques

Figure 1.91. Constellation of amplitude phase shift-keying at M states

The arrangement of the symbols in fact uses a Gray code: a single bit changes when we pass from one point to another. Symmetrical ASK 8 000 001 011 010 110 111 101 100 -7 I0, -5 I0, -3 I0, -1 I0, 1 I0, 3 I0, 5 I0, 7 I0 Symmetrical QPSK 00 01 11 10 -3 I0, -1 I0, 1 I0, 3 I0 Symmetrical BPSK 0 1 -1 I0, 1 I0 For example: chronogram of “symmetrical QPSK” (see Figure 1.92).

Figure 1.92. Symmetrical ASK

Modula ation

103

Figuure 1.93 shoows that tw wo bits, at each e period T, are trannsmitted simultanneously. It is therefore nott a question of o achieving g envelope deetection at reeception. s l QPSK The spectrum of symmetrical d does not shhow any linne and is The spectrum off the signal in baseband written as: ∝

( )=

−1 3

(

)

The spectrum off the modulaated signal is i the same spectrum shhifted by ± fp . Modula ation and de emodulation n Figuures 1.92 annd 1.93 shoow, respectiively, a sim mplified synoptic of coherennt (isochronous) modulatiion and demo odulation onn a single carr rrier.

F Figure 1.93. Modulation M on a single carrie er

Figure 1.94. Cohere ent demodulatiion on a single e carrier

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Digital Communication Techniques

On the side of the receiver, and assuming there is no noise, if we multiply the signal received m(t) =Ik .g(t − kT ).cos(ωpt + ϕp) by a sinusoidal wave coming from a local oscillator A.cos(ωpt + ϕp), we obtain: S1(t) = Ik.g(t − kT ).cos (ωpt + ϕp). A cos ((ωpt + ϕp) By developing this expression and by eliminating the term in cos(2ωpt) by filtering, we obtain: S1(t) = Α/2∗ Ik.g(t − kT).cos (ϕp- ϕl) If the receiver has a local oscillator synchronized in frequency and in phase over that of the transmitter, ϕp will be close to ϕl and, so cos(ϕp- ϕl) will be close to 1, and consequently S2(t) =Α/2∗Ik.g(t − kT). Thus, the signal S2(t) is, to a homothety, equal to the modulating train α(t) = S1(t) =Ik.g(t − kT) which is itself the signal carrying information. It remains for the receiver to recover the rhythm, of period T, of the symbols transmitted, to sample the signal S2(t) in the middle of each period, then to decide, via a comparator at (M-1) thresholds, of the value Ik received. “ASK M” performances We then express the error probability according to the N0/Eb ratio, in which Eb represents the energy sent per bit. Depending on this ratio, the error probability per symbol is given by the ratio:

Ps ( e ) =

 3log 2 M Eb  M −1 erfc  .  2 M  M − 1 N0 

1.15. Quadrature amplitude modulation Another member of the digital modulation family is QAM. It is used in applications including digital radio, microwaves, and DVB-C (Digital Video Broadcasting) end modems.

Modulation

105

Digital transmission: in cable and modems. In 16-QAM at 16 states, there are four I values and four Q values; in total, 16 states for the signal. 16 = 24: 4 bits can be sent per symbol: 2 bits for I, 2 bits for Q. The rate per symbol is therefore four times the rate per bit: thus, this modulation format produces a more spectrally efficient transmission than BPSK, QPSK, 8PSK QPSK (itself identical to 4QAM). Another form is 32QAM. In this, there are 16 values for I and l6 for Q. So 36 possible states (6 × 6 = 36). This is a lot (the closest power of 2 is 32). Thus, the symbol’s states in four corners, which require a great deal of power at transmission, are omitted; this avoids power peaks at transmission: 26 = 32, if there are 5 bits per symbol. In this case, the rate per symbol is a fifth of the binary flow. The current limit is around 512 QAM or indeed 1024. 256QAM: 16 values for I and 16 ones for Q. 16*16 = 256 possible states of the signal, this signal lies across eight bits. There is therefore good spectral efficiency, but the symbols are very close. Hence, there are errors resulting from noise and distortion. More power is therefore needed to transmit (by spreading the signal), and power efficiency is therefore lost compared to other modulations. Solution? Two quadrature carriers (see QAM). This is a modulation known as two-dimensional. 1.15.1. Limits on theoretical spectral efficiency Passband efficiency measures data processing via this or that modulation in a limited passband. Table 1.2 indicates the limits for the main types of modulation, of the efficiency of the theoretical passband. These are ideal values for modulators; since demodulators, filtering and transmission paths are never perfect.

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Digital Communication Techniques

Modulation format

Efficiency limits of the theoretical passband

MSK

1 bit/s/Hz

BPSK

1 bit/s/Hz

QPSL

2 bit/s/Hz

8PSK

3 bit/s/Hz

16 QAM

1 bit/s/Hz

32 QAM

5bit/s/Hz

64 QAM

6 bit/s/Hz

256 QAM

8 bit/s/Hz

Table 1.2. Some modulation formats and their spectral eficiency

1.15.2. I/Q imbalance function [z] = iq_imbalance(r,g,phi) % This function shows the I/Q imbalance in MATLAB. And the signal % phase error between the in=phase and quadrature components of the signal r in % baseband. The g parameter of the model represents the disparity between the Q/I % branches of the receiver, and “phi” represents the phase error of the local oscillator (in % degrees). Ri = real(r); Rq = imag(r); Zi = Ri; %I branch Zq = g*(-sin(phi/180*pi)*Ri + cos(phi/180*pi)*Rq);% Q branch crosstalk (diaphony) z = Zi + 1i*Zq; end function [Kest,Pest] = pilot_iq_imb_est(g,phi,dc_i,dc_q) % Length 64 – Long preambule as defined in EEE 802.11a preamble_freqDomain = [0,0,0,0,0,0,1,1,-1,-1,1,1,-1,1,... -1,1,1,1,1,1,1,-1,-1,1,1,-1,1,-1,1,1,1,1,... 0,1,-1,-1,1,1,-1,1,-1,1,-1,-1,-1,-1,-1,1,1,... -1,-1,1,-1,1,-1,1,1,1,1,0,0,0,0,0]; % representation in the frequency domain Preamble = ifft(preamble_freqDomain,64); % representation in the time domain

Modulation

107

% Define the model of the preambule known by the imbalance of C.C and Q.I. and estimate the r = receiver_impairments(preamble,g,phi,dc_i,dc_q); z = dc_compensation(r); % raises the C.C. imbalance before evaluating the Q.I imbalance. I = real(z); Q = imag(z); Kest = sqrt(sum((Q.*Q))./sum(I.*I)); % estimates the gain imbalance Pest = sum(I.*Q)./sum(I.*I); % estimates the disparity of the phase end function [r,n] = add_awgn_noise(s,SNR_dB,samplesPerSymbol) % Function that adds AWGN to the signal considered [r, n] = add_awgn_noise (s, SNR_dB) % adds the noise vector to AWGN at the signal ‘s’ % to produce the resulting signal vector ‘r’ of SNR. % r this sends the global signal (s, SNR_dB, samplesPerSymbol) % indicates the oversampling ratio used. % This also reflects the noise vector that is added to the signal ‘s’

function y = blind_iq_compensation(z) % Function to estimate and compensate the I/Q imbalance. During data transmission % % y = blind_iq_compensation (z) estimates and compensates this I/Q imbalance present at the level of the complex signal received at the processor, in baseband. . I = real(z); Q = imag(z); theta1 = (-1)*mean(sign(I).*Q); theta2 = mean(abs(I)); theta3 = mean(abs(Q)); c1 = theta1/theta2; c2 = sqrt((theta3^2-theta1^2)/theta2^2); yI = I; yQ = (c1*I+Q)/c2; y = (yI + 1i*yQ); end function [v] = dc_compensation(z) % Serves to estimate and remove impairments in QI branches.

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Digital Communication Techniques

% v = dc_compensation(z) cancels this DC alteration iDC = mean(real(z)); % estimated DC on the I branch 1i*qDC = mean(imag(z)); % estimated DC on the Q branch v = z-(iDC + 1i*qDC);% raises the estimated DC function [grayCoded] = dec2gray(decInput) % conversion decimal in Gray code % example: x = [0 1 2 3 4 5 6 7] %decimal % y = dec2gray(x) % returns y = [ 0 1 3 2 6 7 5 4] % Gray code [rows,cols] = size(decInput); grayCoded = zeros(rows,cols); for i = 1:rows for j = 1:cols grayCoded(i,j)=bitxor(bitshift(decInput(i,j),-1),decInput(i,j)); end end function [y] = dc_impairment(x,dc_i,dc_q) % Function for creating Compensator Coefficients in a complex model of % baseband [le = iq_imbalance] of y (x, dc_i, dc_q) represents the imbalance of % these coefficients between the in-phase and quadrature components of the % complex signal X. baseband. % The C.C polarizations linked to each I, each Q, are represented by parameters dc_i and dc_q. y = x + (dc_i +1i*dc_q); ASK and PSK are not very good choices for using the energy sent effectively when the number of points M is large. In ASK, the points of the constellation form a straight line, while for PSK, the points form a circle. Indeed, the probability of error is dependent on the minimal distance between the points of the constellation, and the best modulation is one that maximizes this distance for a given average power. A more suitable choice is therefore a modulation that distributes the points uniformly in the plot.

Modulation

109

It is known that the modulated signal m(t) can be written as: m(t) = I(t).cos(ω p t +ϕp) − Q(t).sin(ω pt +ϕ p ) and that the two signals I(t) and Q(t) have the expression: I(t) =  Ik (t )g(t − kT) and Q(t) = Qk (t)g(t − kT) The m(t) modulated signal is thus the sum of the two quadrature carriers, modulated in amplitude by signals I(t) and Q(t). 1.15.3. QAM-M constellations Symbols Ik and Qk have their respective values in the two M alphabets elements (A1, A2, … AM) and (B1, B2, … BM), which give rise to a modulation that possesses a number [of] E = M2 states. Each state is represented by a pairing (Ik, Qk ), otherwise known by a complex symbol: Zk = Ik + jQk. In the particularly common case where M can be written as M = 2n, the Ik s and the Qk s represent words of n bits. The complex symbol Zk = Ik + jQk can consequently represent a word of 2n bits. The benefit is that the signal m(t) is a combination of two quadrature carriers modulated in amplitude by independent symbols Ik and Qk. In addition, symbols Ik and Qk very often take their values in the same alphabet of M elements. For example, QAM-16 is created from symbols Ik and Qk taking their values in the alphabet {±d, ±3d} hence d is a given constant. A representation of the constellation of this modulation is given in Figure 1.95. QAM-16 has often been used for transmission on a public switched network telephone line (at 9600 bit/s) and for high capacity microwave transmissions (140 Mbits/s) developed in the 1980s. More generally, symbols Ik and Qk take their values in the alphabet {±d, ±3d, ±5d, …, ±(M-1)d} with M, e.g. QAM-4, QAM-16, QAM-64 and QAM-256.

110

Dig gital Communication Technique es

Figu ure 1.95. QAM M-16 and QAM M-64 constella ations

ation and de emodulation n Modula of two Wheen the signnal m(t) is obtained th hrough a combination c quadratuure carriers modulated in i amplitudee by indepenndent symbolls Ik and Qk, this simplifies thhe modulatorr and demod dulator. In fact, for the m modulator the incooming binary ry train {ik} is easily diivided into two t trains, {{Ik} and {Qk} (ssee Figure 1.996).

Figure 1.96. QAM-M modulator m

The reception off a QAM signnal uses a co oherent (isochhrone) demoodulation and theerefore requiires extractioon of a carrrier synchroonized in phhase and

Modulation

111

frequency with the carrier sent. The signal received is therefore demodulated in two branches: with the in-phase carrier and with the quadrature carrier. The demodulated signals are converted by two CANs; then, a logical decoding determines the symbols and regenerates the bitstream received. The synoptic schema-block of the QAM-M demodulator is very similar to the PSK demodulation. Spectral efficiency For a single modulation speed R = 1/T, the binary flow D = 1/Tb of the QAM-M is multiplied by n = log2M compared to that of QAM-2. In other words, for a given BW B, the spectral efficiency: η = Δ/Β is multiplied by n = log 2 M. M = 2n modulation; binary flow: D; spectral efficiency: η n

M = 2n

Modulation

Binary flow

Spectral efficiency η

1

2

QAM-4

D

η

2

4

QAM-2

2.D

2.η

4

16

QAM-16

4.D

4.η

6

64

QAM-64

6.D

6.η

8

256

QAM-256

8.D

8.η

Table 1.3. Gain obtained on the spectral efficiency and on the binary flow for different QAM-M modulations for a single modulation speed. We increase M, but with an increased complexity

“QAM”: a generalization of ASK and PSK By considering the signal m(t) during a period T, we have: m(t) = Ik.cos(ωpt +ϕp) − Qk.sin(ωpt +ϕp), m(t) = Re[ (Ik+j Qk).ej ( ωp t + φp)] with:

Zk = Ik + j Qk

112

Dig gital Communication Technique es

and ϼ = and

+

= arctang ( /

)±k

The signal m(t) is i then writteen as: m(t) = ϼ .cos (ωpt + ϕp + ϕk). odulation caan be consiidered a We can see veery well thaat QAM mo simultanneous modullation of the phase and th he amplitude. Thus: – PS SK phase modulation m caan be seen as a an ASK modulation but one where Ik is constant;; – sim milarly, ASK K AM can be b considered d as a QAM M modulationn, where the Qk are a null. Hencce, the namee “amplitude and phase shhift-keying” (APSK) sometim mes called QA AM; – CIR(4,4,4,4) C plitudes andd 4 phases,, whose modulationn at 4 amp constelllation is giveen in Figure 1.97, is a good g examplle of this (seee IUT – Internattional Union of Telecomm munications)).

Figure 1.9 97. APSK-16 constellation c

Compa arison of AS SK and PSK K ASK K and PSK can c be comppared as functions of M using curvees of the error prrobabilities per p symbol Ps(e). P For ex xample, for a probability of error

Modulation

113

per symbol Ps(e) of 10-5 and for a signal to noise ratio of Eb/N0 of 4 dB, ASK can only send 2 bits per symbol (M = 4), whereas PSK can send 3 (M = 8). This gives ASK a clear advantage for M, ranging from 2 to 16. For values of M higher than 16, degradation of PSK performances leads us to look for other modulations at the cost of increased modulator and demodulator complexity. From the point of view of simplicity of performance, it is ASK that has the advantage, as it is one-dimensional. The noise level and any other “distortion” gives the lowest error rate on bits since the BPSK system gives the greatest distance between the points of the signal. The probability of a BPSK’s bit error in an AWGN can be obtained by

Pe, BPSK = Q

2 Eb N0

where – Eb: energy per bit; – No: spectral of noise power; – Pe: PSD of noise. Q is calculated via the area under the Gaussian tail: Q ( x) = x  a dx =

1 2π

 e ∞ x

 t2 − 2 

  

dt

ax +C lna

% Demonstration of Eb/N0 Vs BER for baseband differentially encoded BPSK (MATLAB Inc.) clear all;clc; %––––Input Fields–––––––––––– N = 1000000; %Number of symbols to transmit EbN0dB = -4:2:20; % Eb/N0 range in dB for simulation %––––––––Transmitter––––––––––

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Digital Communication Techniques

m = rand(1,N) > 0.5; % random symbols from 0's and 1's b = filter(1,[1 -1],m,0); %IIR filter implementing the differential encoding b = mod(b,2); % XOR operation is equivalent to modulo-2 and binary negation s = 2*b-1; % BPSK antipodal Mapping SER = zeros(length(EbN0dB),1); % Place holder for SER values for each Eb/N0 EbN0lin = 10.^(EbN0dB/10); % Converting Eb/N0 from dB to linear scale for i = 1:1:length(EbN0dB), Esym = sum(abs(s).^2)/length(s); % calculate actual symbol energy from generated samples N0 = Esym/EbN0lin(i); % find the noise spectral density noiseSigma = sqrt(N /2); % standard deviation for AWGN Noise n = noiseSigma*(randn(1,N) + 1i*randn(1,N)); y = s + n;%received signal = signal + awgn noise %–––––––Receiver–––––––––– bCap = (y > = 0);% clipping detection (BPSK demod) mCap = filter([1 1],1,bCap,0); % FIR filter implementing the differential decoding mCap = mod(mCap,2); % binary messages, therefore modulo-2 SER(i) = sum((m~ = mCap))/N;%––– Symbol Error Rate Computation–– – end %–––Theoretical Symbol Error Rate–––––– theoreticalSER_DPSK = erfc(sqrt(EbN0lin)).*(1-0.5*erfc(sqrt(EbN0lin))); %Theoretical symbol error rate for DPSK theoreticalSER_BPSK = 0.5*erfc(sqrt(EbN0lin)); %Theoretical symbol error rate for BPSK %––––––Plotting––––––––––––– semilogy(EbN0dB,SER,'k*'); hold on; semilogy(EbN0dB,theoreticalSER_DPSK,'r-','LineWidth',1.0); semilogy(EbN0dB,theoreticalSER_BPSK,'b-','LineWidth',1.0); set(gca,'XLim',[-4 12]);set(gca,'YLim',[1E-6 1E0]);set(gca,'XTick',-4:2:12); title('Probability of Symbol Error for BPSK signals'); xlabel('E_b/N_0 (dB)');ylabel('Probability of Symbol Error - P_s'); legend('Simulated', 'Theoretical');grid on;

Modulation

115

Probability of Symbol Error for BPSK signals

100

Simulated Theoretical

Probability of Symbol Error - P s

10-1

10-2

10-3

10-4

10-5

10-6 -4

-2

0

2

4

6

8

10

12

E b /N0 (dB)

Figure 1.98. BPSK: probability of symbol error. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

The bit error probability for QPSK is similar to that for BPSK, but QPSK makes it possible to double the information transmitted, without increasing the BW transmitter. In addition, QPSK offers double the spectrum efficiency with the same energy efficiency. The bit error probability can be obtained using the following relationship;

,

=

1 2

2

where N0 is the noise spectral density. QPSK has the same probability of error as BPSK because of the 3 dB reduction in the error distance of QPSK, compensated by the 3 dB reduction in its BW. QAM involves two different signals at the same time (two-dimensional modulation). It can be seen as an AM or a PM. It is used mainly in digital

116

Digital Communication Techniques

telecommunications systems at very high data flows. Errors in information are reduced; the adjacent constellation points are well distributed. There are different forms of QAM but the most common are 16QAM, 64QAM, 128QAM and 256QAM. QAM M makes it possible to transmit more bits per symbol; this a priori makes it possible to transmit data in a much smaller passband. However, if the average energy of the constellation remains constant, the symbols should be very close to one another, which makes them more vulnerable to noise and other distortions, which leads to an error rate on the higher bits. This signal should be transmitted using more power so that the symbol spreads further, thus reducing this technology’s energy efficiency compared to other modulation techniques. Nevertheless, higher order QAM can transmit more data, which makes them more effective in terms of spectral transmission, but they are not as weak as lower order QAM. The general form of QAM M-ary signals can be expressed by: Si ( t ) =

2 E min 2 E min a1 cos ( 2π f C t ) + b1 sin ( 2π f C t ) , TS TS

0 ≤ t ≤ T , i = 1, 2,3,...M

where Emin is the energy of the signal with the lowest amplitude, and a1 and b1 are a pair of independent integers, chosen according to the location of the particular signal point. The average probability of error in an AWGN for a QAM M-ary can be represented by:

,

≅= 4 (1 −

1 √

)

2

Higher order QAM modulation schemas are therefore vulnerable to errors. Consequently, error correction encoding ensures that there is a greater chance of the signal surviving in AWGN and the Rayleigh channel along multiple paths and thus improves system performances. However, there is always a compromise in applications because a particular application may necessitate greater precision in data reception, whereas for another application, the desirable requirement may be the passband or the power available.

Modula ation

117

The choice of technology t u used is cruccial, as it grreatly influennces the characteeristics, perrformances and overaall physicall realizationn of a communnication systtem. Digital modulattion will coontinue to be pertinennt in the w world of communnication, voiice and high flow data, since s the com mmunication systems designer has the main m aim of o transmittin ng informattion in the shortest p at an affordabble cost and with the possiblee time, in the available passband, lowest error e rates poossible.

Figure 1.99. 1 Rayleigh h channel

1.16. Digital D communication ns transmittters Beloow is repressented a bloock schema (Figure 1.1100) from a digital communnications traansmitter. Itss ends are ussed for analog signals. T The first state, thhen, is to coonvert a conntinuous anaalog signal into discreeet digital binary train; t this is digitization. d

F Figure 1.100. Block schema a from a digita al communicattions transmittter

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Digital Communication Techniques

The following stage consists of adding coding, for example voice coding, for data compression. Some channel codings are then added. Channel coding encodes data so as to reduce as much as possible the effects of noise and interferences in the transmission path. It adds additional bits to the incoming data flow and removes redundancies. These additional bits are intended for error correction or sometimes for sending sequences of streams for identification or equalization. This can facilitate synchronization for different symbols. In symbol clock transitions, the carrier transmitted is the correct I/Q (or amplitude/phase) value to represent a given symbol (a specific point in the constellation). Then, the I/Q or amplitude/phase values of the carrier transmitted are changed to represent another symbol. The interval between these two moments is the period of the symbol clock. The symbol clock phase is correct when the symbol clock coincides with optimal instants to detect symbols. The next step in the transmitter: filtering. This is essential for good BW efficiency. Without filtering, signals would have transitions that were too quick between states and so a broader frequency spectrum needed to send information. A single filter creates a compact and spectrally efficient signal to do this, which can be placed on a carrier. The output of the channel coder is then introduced into the modulator. Since there are I and Q components in radio, half of the information can be sent in I, the other half on Q. This is why digital radios perform well with this type of digital signal. I and Q components are separated. The rest of the transmitter is similar to a typical RF transmitter or a microwave transmitter/receiver pairing. The signal’s frequency is converted into a higher RF – upconverted – intermediate frequency (IF). Undesirable signals, which would be produced by high frequency conversion, are then rejected though filtering.

1.16.1. A digital communications receiver The receiver is similar to the transmitter, but in reverse. The arriving signal is “down converted” to an “IF” and demodulated. The ability to demodulate the signal is hampered by factors including atmospheric noise; the signals and multiple paths compete with one another, indeed destroy one another and disappear.

Modula ation

119

F Figure 1.101. At A the receive er: demodulatio on

In geeneral, demoodulation invvolves the folllowing stagees: 1) ree-establishingg the carrier frequency (ccarrier lock);; 2) ree-establishingg the symboll clock; 3) deecompositionn of the signaal into I and Q componennts; 4) deetermining thhe I and Q vaalues for each symbol slicing; 5) deecoding and interleaving; 6) exxpansion of bit b flow; 7) diigital-analog conversion, if it takes pllace. , the signal starts off diggital and In more m and morre systems, nevertheless n remainss digital. It iss never analoog in the sense of a contiinuous analog signal, as for auudio. The main differencce between th he transmitteer and receivver is the questionn of re-establishing the carrier c and th he (symbol) clock. c The frrequency and the phase (or thhe synchronizzation) of thee symbol cloock should bee correct in the reeceiver to deemodulate thhe bits correctly and recoover the infoormation transmittted. A symbbol clock coould have the right frequuency but thhe wrong phase; if i the symbool clock is alligned with the transitions between symbols rather than t the sym mbols themsselves, the demodulation d n would be of poor quality. Sym mbol clocks arre usually fixxed in frequeency and thiss frequency iis known exactly by the transmitter and reeceiver. The difficulty lies in obtainiing them aligned in phase or by synchronnization. Theere is a varieety of techniqques and wo or three. If the signall’s amplitudee changes duuring the most syystems use tw modulattion, a receiiver can meeasure these variations. The transmiitter can send a specific s syncchronization signal or a predetermine p ed order of bbits such

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as 1010010101010100 to “train” thhe receiver’ss clock. In syystems with a pulsed carrier, the symboll clock can be aligned with the caarrier powerr. In the transmittter, RF carrrier and digiital data are well distribbuted, since they are produceed inside thee transmitter itself. This is not the case c for the rreceiver. The receiver can appproximate where w the carrrier is, but haas no inform mation on ming/synchronnization of symbol s clockks. One diffficulty in its phasse or the tim receiverr design is that of the algorithms for re-estaablishing thee coding channell. This can be facilitated by chan nnel coding carried outt on the transmittter. Com mpromises inn frequency, phase, syncchronization and modulaation are made too cancel inteerferences foor multi-userr communications system ms. It is necessary to measuure parameteers in RF digital commuunication systems to M inclu ude analyzinng the modullator and make thhe right comppromises. Measures demoduulator, characcterizing thee quality of the transmiitted signal, locating causes of o high BER R and studyinng new typess of modulatiion. Measurees on RF digital communicati c ions systems fall into mo ore or less fouur categoriess: power, frequency, synchronnization and precision p in modulation.

1.16.2. Measures of power Meaasures of poower includee the poweer of the caarrier and m measures associatted with am mplifier gain as well as loss of insertion in fillters and attenuattors. The siignals used in digital modulation are noise signals. Measurees of band power (pow wer integratted over a particular rrange of frequencies) or PSD D are often made. PSD measures noormalize pow wer in a B BW, usually y 1 Hz. certain

Figure e 1.102. Powe er of the adjacent channel (A Agilent)

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121

1.16.3. Power of the t adjacen nt channell The power of the adjacent channel is a measure of the interrferences n neighboringg channels. T This test created by a user thhat affects otther users in quantifiies the energgy of a digitaally modulatted RF signaal, which tacckles the predicteed transmisssion path inn an adjaceent channel. The resultt of the measureements is thhe ratio (oft ften in dB) of the pow wer measuredd in the adjacent channel at any power trransmitted. A similar measure is the ppower of an alternnative channnel that keepss the same raatio to two chhannels, far from the predicteed communiccation channel.

Figure e 1.103. Meassures of powerr and synchron nization

For pulsed sym mbols (such as TDMA),, power meeasures havee a time componnent and can also have a frequency component. Burst B profiless (power dependiing on time) have openinng and closin ng times, whiich can be m measured. Anotherr measure is that of averrage power, when the caarrier is on, aaveraged over maany “closed/oopened” cyclles.

1.16.4. Frequency y measures s Freqquency measures are often complex in digital syystems sincee factors other thhan pure toonalities cann be consideered. The occupied o BW W is an importaant measure. It ensures that the op perators remaain in the ppassband allocateed to them.

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Figure 1.104. Occupied bandwidth (Agilent). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

The occupied BW is a measure of how the frequency spectrum is covered by the considered signal. The units are in hertz, and the measure of the occupied BW generally involves a percentage or ratio of power. Typically, some of the power in a signal to be measured should be specified. A commonly used percentage is 99. A measure of power according to frequency (such as power in the band) is made to add power in order to achieve the percentage indicated. For example, one might indicate that “99%” of the power in this signal is contained in a BW of “30 kHz”. One could also indicate that the BW occupied by this signal is 30 kHz if the desired power ratio of 99% is known. The numbers of BWs typically occupied change drastically, depending on the symbol and filtering rate. The value is around 30 kHz for the NADC π/4 DQPSK signal and around 350 kHz for a GSM 0.3 GMSK signal. For digital video signals, the occupied BW is generally 6–8 MHz. Simple frequency measurement techniques are not often precise or sufficient enough to measure the central frequency. We calculate a “centroid” carrier, which is the frequency distribution center for the PSD for a modulated signal.

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1.16.5. Synchronization measures Synchronization measures are most often made in pulsed or in burst systems. The measures include intervals of repetition of pulses, in time, duty cycle, and the times between bit errors. Lighting or extinguishing times also involve power measures.

Modulation precision Measures of modulation precision lead us to measure how close constellation states or the signal trajectory relates to an (ideal) trajectory of the reference signal. Measures of modulation exactitude usually include precision demodulation of a signal and a comparison of this demodulated signal with the mathematically generated “reference” signal. The difference, or residue, between the two is the modulation error, and it can be expressed by a range of means including the amplitude of the error vector module (EVM), the phase error, the error on I and the error on Q. The reference signal is subtracted from the demodulated signal, starting from a residual error signal. Once the reference signal has been subtracted, it is easier to see small errors that have been drowned or obscured by the modulation itself. The error signal can be examined in many ways: in the time domain or (since it is a vector quantity) depending on its I/Q components. A frequency transformation can also be achieved and the spectral composition of this signal error can be observed alone. Let us return to the bases of the vector modulation: the bits are transferred on an RF carrier by changing the amplitude and phase of the carrier. At each transition of the symbol clock, the carrier occupies a unique position among several possible ones in the I/Q plane. Each position encodes a specific data symbol which is formed of one or more data bits. A constellation diagram indicates the effective positions (in other words, the size and phase relative to the carrier) for all the authorized 2n symbols. To demodulate the incoming data, the amplitude and phase of the signal received for each clock transition should be accurately determined. The drawing of the constellation diagram, including the ideal locations for the symbols, is decided generically by the modulation format chosen (BPSK, 16QAM, π/4 DQPSK, etc.). The trajectory taken by the signal from

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one digital location to another depends d on implementattion of the ddedicated system. At any a moment,, the phase or o amplitudee of the signnal can be m measured. These values v definee the real orr “measured d” phaser. At A the same time, an ideal “reference” phhaser can be calculated, given the knnowledge of the data t flow transmitted, symbol-cloock synchrronization, baseband filtering parametters, etc. The differences between th hese two phasers form tthe basis for EVM M measurem ments. The schema in Figure F 1.105 defines the EVM and several relateed limits. wn, the EVM M is the scallar distance between b the phaser’s tw wo limits, As show i.e. the importance of o the differeence vector. In other words, it is the residual noise annd the residdual distortioon that remaain after an ideal version of the signal has h been extracted.

Figure 1.10 05. Different ty ypes of errors

In standard s US S NADC-TD DMA (IS-54 4), the EVM M is defineed as a percentaage of the signal volttage at the symbols. In I the π/4 DQPSK modulattion format, these symbools all have the t same levvel of voltagee, which is not trrue of all foormats. IS-544 is currently the only standard s thatt defines EVM exxplicitly; so EVM can be b defined differently d thhan other moodulation formats. In a formatt such as 64Q QAM, the sy ymbols, repreesenting all llevels of M’s voltage, can be definned by an av verage level of voltage foor all the the EVM symbolss (a value cllose to the average a levell of the signaal) or by thee highest voltage of the four exterior sym mbols, althou ugh the errorr vector has a phase value, generally g random, linked to the symbo ol values.

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125

This is a function of the error itself (which may or may not be random) and the position of the data symbol on the constellation which, in practice, is random. A more interesting angle is measured between real and ideal phasers (1/Q phase error) that contains the information useful in signalreconstruction problems. Similarly, the I/Q amplitude error shows the difference in amplitude between the real and ideal signals. The EVM, as indicated in the standard, is the RMS value of the error values at the instant of the clock symbol’s transition. The trajectory errors between the symbols are avoided.

Trouble shooting error vector measurements Error vector amplitude measurements of the quantity concerned can, once correctly applied, provide a great deal of information on the quality of a digitally modulated signal. They can also indicate the exact causes of problems encountered by identifying precisely the type of degradation effective in a signal; indeed, they can even help identify their sources. EVM measurements are developed rapidly, having already been written in system standards such as NADC and PHS (phone standard), and they appear in several standards including those for digital visual transmission.

Amplitude versus phase error Different error mechanisms affect signals in different ways: the amplitude alone, the phase alone, or both simultaneously. The relative quantities of each type of error can quickly confirm or eliminate certain types of problems. The last diagnostic stage is to solve the EVM in its amplitude and phase error components, and to compare their relative sizes. When the average phase error (in degrees) is substantially larger than the average amplitude error (in percentage), an unwanted PM becomes the dominant error mode. This can be triggered by noise, inter-pairing problems in the frequency reference, phase-locked loops, or other frequency synthesizing stages. The residual AM is demonstrated by amplitude errors, which are substantially greater than angle errors.

IQ phase error over time The phase error is the instantaneous angle difference between the signal measured and the ideal reference signal. Once seen as a function of time (or of the symbol), this shows the modulation wave form of any PM signal

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(phase magnitude), m residual or interfering. Sinusoidal S waaves or otherr regular wave foorms indicatee an interferinng signal, a residual r PM//FM noise, etc.).

Figure e 1.106. Phas se error

An ideal i signal will w have a uniform u con nstellation, perfectly sym mmetrical with thee original. The T 1/Q imbaalance appeaars when thee constellatioon is not “squared”, i.e. whenn the height of the Q ax xis does not equal the brreadth of the I axxis. The quadrature q errror is seen at any tilt of o the consttellation. Quadratture error is caused wheen the phasee ratio betweeen vectors I and Q varies from fr 90°.

Figure 1.107 7. Phase noise versus time: appears rand dom (Agilent)

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The error vector as a function of time The EVM is therefore the difference between the incoming and the ideal reference generated internally. Seen as a function of the symbol or of time, the errors can be correlated to specific points on the incoming wave forms, such as peaks or passages to zero. The EVM is a scalar value (amplitude alone). Peak errors are produced with the signal’s peaks; they indicate compression or clipping phenomena. Error peaks correlated with minima suggest nonlinearities in the passage to zero.

Figure 1.108. EVM peaks (above) appear during the amplitude’s passage to zero (lower) (see phase noise)

One example of zero-crossing nonlinearities involves push-pull amplifiers (one of the components is a source (of current), whereas the other is a reservoir, or sink, for the current); the positive and negative halves of the signal are provided by additional transistors (e.g. Bipolar, NPN and PNP; MOS, N and P, mounted on a totem pole, in phase. This can be a real challenge, particularly in high power amplifiers) with precision to polarize and stabilize the amplifiers, one being dimmed while the other is on, without creating distortions. The critical moment is the passage to zero, an effect well-known in analog design (diode-mounted transistors are then added so that the currents are not cancelled on a voltage range around the commutation of additional transistors). The error spectrum (EVM in terms of frequency) The error spectrum is calculated using the waveform’s FFT and the EVM results in the frequency domain. In most digital systems, the non-uniform

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distribution noise, or the peaks of discreet signals, indicate the presence of coupled external interferences.

Figure 1.109. EVM peaks (above) appear during the amplitude’s passage to zero (below) (see phase noise – measures). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 1.110. RF spectrum (above) and error vector spectrum (below) (QPSK). For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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1.17. Applications 1.17.1. Domains There is a “plethora” of domains for applying the different digital transmission techniques that we have just demonstrated. Some are described below.

Telephone modems During the 1960s and 1970s, the transmission of data along telephone channels was at the origin of the development of many signal processing techniques in telecommunications. The transmission of a high flow on a telephone channel (on a frequency band of around 3,500 Hz) necessitated the manufacture of modulators with a large number of states: QAM-16, QAM32 and QAM-128. It was thought that flow would typically be limited to 9.6 kbit/s, since the S/B ratio of the links was prohibitive. In fact, coding and filtering techniques and the use of lattice constellations made it possible to cross a significant limitation on the quality and flow that could be attained.

Microwave radios When digitization began, microwave radios used simple modulations such as 4-PSK, but efficient use of the accessible radio spectrum pushed the development of microwave radios using modulations with a high number of states such as QAM-16 and QAM-64. It was QAM-16 modulation that made possible the transmission of a flow of 140 Mbit/s in the 6.4–7.1 GHz band for channels spaced at 40 MHz. Note that the transmitter should have a good linearity to transmit this type of modulation. The crucial problem in digital microwave radios is propagation by multiple paths, which greatly degrades quality and limits the possibility of high-capacity links. This phenomenon is accentuated when the number of modulation states increases. Unlike high-capacity microwave radios, there is low cost low BW radio (2 Mbit/s), in which spectral efficiency is not the first concern. The

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modulations used are generally MDF-PC at two or four states, which makes it possible to use a nonlinear amplification in the transmitter.

Satellite transmissions Satellite transmissions display significant attenuation in space and a more restricted power from their transmitter. This calls for power efficiency (immunity from noise) versus spectral efficiency of links. The modulations most often used are PSK-2, PSK-4 and PSK-8. With the latter, the amplifier allows the available power to be used efficiently. However, today, PSK-16 and QAM-16 modulations, linked to powerful coding, are in vogue. The standard in Europe for radio broadcasting digital television by satellite is based on a PSK-4.

Radiocommunications with cell phones We know that digital radiocommunications systems cover the entire world. Japanese and American cellular systems use a different modulation to the one used in European systems. The modulation used in the United States and in Japan is π/4-DQPSK, a PSK-4 whose axes turn by π/4 from one symbol to the next. Phase rotations of π are therefore forbidden in this modulation. Signal envelope passages through zero are thus avoided and this considerably reduces its temporal fluctuations. The modulation used in the European cellular system, called GSM (Groupe Spécial Mobile), is a constant-envelope modulation known by the name of GMSK (Gaussian minimum-shift keying). This is in fact a variant of MSK modulation, in which the impulses coming into the modulator are Gaussian. This temporal and spectral shaping smooths the signal’s phase trajectory and reduces its spectral occupation compared to initial MSK modulation; the Nyquist criteria is therefore respected. The data stream sent in a 200 kHz band is a multiplex of eight telephone channels. Taking into account the error correction code, the synchronization bits and identification of the channel, as well as other auxiliary data, the overall flow is around 270 kbit/s.

Modulation

Cellular systems

American

Japanese

Standard

IS-54/-56 P

DC

Frequency range

Rx:869-894 Tx:824-849

Rx:810-826 Tx:940-956

Rx:925-960 Tx:880-915

Cellular systems

American

Japanese

European

Standard

IS-54/-56 P

PDC

GSM

Frequency range

Rx:869-894 Tx:824-849

Rx:810-826 Tx:940-956

Rx:925-960 Tx:880-915

Number of channels

832

1,600

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Number of users per channel

3

3

8

131

European

Channel spacing

30 kHz

25 kHz

200 kHz

Modulation

π/4-DQPSK

π/4 DQPSK

GMSK

Binary flow

48.6 kbit/s

42 kbit/s

270 kbit/s

Table 1.4. Cellular systems

1.17.2. Digressions or precisions, around modulations QAM is a form of carrier modulation using modification of the amplitude of the carrier itself and a quadrature wave (a wave dephased by 90° with the carrier) according to the information transported by input signals.

Figure 1.111. Diagram of constellations for QAM at 16 states. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

In other words, this can be considered (using a complex notation) as a simple wave AM, expressed in complex, by a signal.

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This means that the amplitude and phase of the carrier are simultaneously modified according to the information to be transmitted. PM can also be considered as a particular instance of QAM, where only the phase varies. This remark can also be extended to FM, as this can be seen as a particular instance of PM. M = 64; refC = qammod(0:M-1,M);% - MATLAB Inc.constDiagram = comm.ConstellationDiagram(... 'SymbolsToDisplaySource','Property', ... 'SymbolsToDisplay',256, ... 'XLimits',[-10 10],'YLimits',[-10 10], ... 'ReferenceConstellation',refC); iqImbComp = comm.IQImbalanceCompensator('StepSizeSource','Input port', ... 'CoefficientOutputPort',true); nSym = 25000; data = randi([0 3],600,1); txSig = pskmod(data,4,pi/4,'gray'); iqImbComp = comm.IQImbalanceCompensator('AdaptInputPort',true, ... 'StepSize',0.001,'CoefficientOutputPort',true); ampImb = 5; % dB phImb = 15; % deg gainI = 10.^(0.5*ampImb/20); gainQ = 10.^(-0.5*ampImb/20); imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180); imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180)); rxSig = imbI + imbQ; [~,isCoef1] = iqImbComp(rxSig(1:200),true); [~,isCoef2] = iqImbComp(rxSig(201:400),false); [~,isCoef3] = iqImbComp(rxSig(401:600),true); isCoef = [isCoef1; isCoef2; isCoef3]; plot((1:600)',[real(isCoef) imag(isCoef)]) grid xlabel('Symbols') ylabel('Coefficient Value') legend('Real','Imaginary','location','best')

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Modulation

Figure 1.112. I/Q imbalance measure – compensation coefficients. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

The I/Q imbalance compensator system object compensates the imbalance between the in-phase and the quadrature components of a modulated signal. To compensate for the I/Q imbalance: – Define and configure the object IQImbalanceCompensator (Matlab Inc.). – Select “step” to compensate the I/Q imbalance according to the properties of comp.IQImbalanceCompensator. The “step’s” behavior is specific to each object in the toolbox. – The adaptive algorithm within the I/Q imbalance compensator is compatible with M-PSK, M-QAM and OFDM modulation schemas, where M > 2. Attenuate the impacts of the amplitude and phase imbalance on a signal modulated by QPSK using the system object: comm.IQImbalance Compensator.

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– Create an object “constellation” diagram. constDiagram = comm.ConstellationDiagram(... 'SymbolsToDisplaySource','Property', ... 'SymbolsToDisplay',100). – Create an I/Q imbalance compensator: iqImbComp = comm. IQImbalanceCompensator. – Produce random data symbols and apply QPSKdata = randi([0 3],1e7,1) modulation. txSig = pskmod(data,4,pi/4); % Apply amplitude and phase imbalance to the signal transmitted. ampImb = 5; % dB phImb = 15; % deg gainI = 10.^(0.5*ampImb/20); gainQ = 10.^(-0.5*ampImb/20); imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180); imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180)); rxSig = imbI + imbQ; % Plot of the constellation diagram of the signal received. Observe that the signal received really has undergone % shifts in amplitude and phase. constDiagram(rxSig) hMod = comm.PSKModulator(8); refC = constellation(hMod); hScope = comm.ConstellationDiagram(... 'SymbolsToDisplaySource','Property', ... 'SymbolsToDisplay',100, ... 'ReferenceConstellation',refC); hIQComp = comm.IQImbalanceCompensator('CoefficientSource','Input port'); data = randi([0 7],1000,1); txSig = step(hMod,data); ampImb = 5; % dB phImb = 15; % deg gainI = 10.^(0.5*ampImb/20); gainQ = 10.^(-0.5*ampImb/20); imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180); imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180)); rxSig = imbI + imbQ;

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step(hScope,rxSig); % Sequence: deletion of I/Q imbalance %Apply the I/Q compensation algorithm and display the constellation. %The constellation of compensated signals is almost aligned on the %reference constellation compCoef = iqimbal2coef(ampImb,phImb); compSig = step(hIQComp,rxSig,compCoef); step(hScope,compSig) % Object: comm.IQImbalanceCompensatorSystem ™ with external coefficients. % Create 8-PSK modulator system objects and a constellation diagram. % Use non-value pairs to ensure that the %constellation diagram displays only the last 100 data symbols and %to provide the reference constellation. hMod = comm.PSKModulator(8); refC = constellation(hMod); hScope = comm.ConstellationDiagram(... 'SymbolsToDisplaySource','Property', ... 'SymbolsToDisplay',100, ... 'ReferenceConstellation',refC); % Create an I/Q imbalance compensator object with an input port %for the algorithm’s coefficients. hIQComp = comm.IQImbalanceCompensator('CoefficientSource','Input port'); % Generate random data symbols (see random) and apply the modulation % 8-PSK. data = randi([0 7],1000,1); txSig = step(hMod,data); % Apply an amplitude and phase imbalance to the signal transmitted. ampImb = 5; % dB phImb = 15; % deg gainI = 10.^(0.5*ampImb/20); gainQ = 10.^(-0.5*ampImb/20); imbI = real(txSig)*gainI*exp(-0.5i*phImb*pi/180); imbQ = imag(txSig)*gainQ*exp(1i*(pi/2 + 0.5*phImb*pi/180)); rxSig = imbI + imbQ;

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%Plot of the constellation diagram of the signal received. Observe that the %signal received has shifted in amplitude and phase. step(hscope,rxsig);

Figure 1.113. Removal of I/Q imbalance

2 Some Developments in Modulation Techniques

2.1. Orthogonal frequency division multiplexing 2.1.1. Introduction Adaptation, coding, the information to be transmitted and the propagation channel are vital questions regarding telecommunications. Modulations are often used for several frequency-selective channels; blocks are modulated using Fourier transforms. This technique, called orthogonal frequency division multiplexing (OFDM), is a market leader; it converts a broad multipath channel into simple, single-path subchannels, which are easy to equalize. In addition, proper use of cyclical redundancies at transmission makes it possible to reduce the complexity of terminals via fast Fourier transforms (FFTs); there are also advantages (simplicity of equalization, use of FFT/Direct Cosine Transform (DCT) algorithms, which can be seen as a Fourier transform) and drawbacks (a lack of diversity). There is also a precursor system called Kineplex, for military radio links in a HF band: 1.8– 30 MHz. However, it is expensive to create orthogonal analog filters; this system has had scant success. At the start of the 1980s, digital modulators based on FFTs began to be introduced into multicarrier modulations. Hence it was easy to achieve speedy take-up of this technology. The algorithm for calculating the FFT was “invented” by Cooley and Tukey (among others), researchers at IBM at the start of the 1960s, and had a significant impact on the development of digital signal processing applications. Multicarrier systems, based on the FFT, are currently known by the name Discrete MultiTone (DMT) for filter networks.

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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2.1.2. Multicarrier modulations For multipath channels, traditional modulation techniques are very sensitive to intersymbol interferences (ISI). These interferences are stronger, and a symbol’s duration is weaker compared to the delay spread. In other words, a simple demodulation is preferred if the duration of useful symbols is high compared to the propagation channel delay. This advantage, actually for wireless communication, is a solution adapted for different types of highflow networks such as cellular networks or local radio loops and local wireless networks. The original idea of multicarrier modulation is to transform the equalization stage in the temporal field using a simpler equalization in the frequency domain. The benefit of multicarrier modulations is that they place information in a time-frequency window so that their duration is very high in relation to the delay spread. To describe the principle, we consider an electrical circuit for which the current response (the signal sent) is governed by a differential equation. It is not necessarily easy to interpret and solve the differential equation (even though it is a linear operator) to determine this current. Electricians know very well that it is much easier to estimate the circuit’s response at certain frequencies (than building Bode diagrams) and so determine the circuit’s transfer function (which corresponds to the transmission channel). Each frequency component of the input current is then filtered by the circuit’s response at this frequency. Once the transfer function is acquired, the current is determined by dividing the tension (in other words, the signal received) by the circuit’s transfer function (which, in this case, is none other than the impedance). Multicarrier modulations have been designed on the same principle. Indeed, the frequency representation of the equation gives us: Y(f) = H(f)S(f) + η(f) where – Y(f) represents the Fourier transform of the signal received r(t); – H(f) represents the Fourier transform of the channel c(t);

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139

– S(f) represents the Fourier transform of the signal transmitted S(f); – η(f) represents the Fourier transform of the noise n(t). The signal S(f) is transmitted on a number of sinusoids of different frequencies (called carrier frequencies). This operation is carried out with the help of a Fourier transform. At reception, the signal is demodulated with the help of an inverse Fourier transform: the signal obtained is then quite simply filtered by the channel’s transfer function; in other words, each component of the signal is multiplied by a coefficient corresponding to the channel’s frequency gain. It is then easy for the receiver to equalize the channel since it is enough to divide each signal received by the corresponding gain (in this case, we speak of scalar equalization). Consequently, at reception, intersymbol interference is suppressed and the symbols sent undergo only one attenuation. Each subchannel can then be considered as a single channel transmission, endowed with its own signal to noise ratio (attenuation function) and breadth Δf. It should be noted that the word “multi” is attached to multicarrier modulation, but with the transmission of a single data source. The channels include attenuations that depend on the frequency. The information signal is transmitted on each sinusoidal frequency carrier fi (i = 1, …, 4) and is attenuated by the gain |G(fi)|. Recall that OFDM, transmitting data by blocks, is a multicarrier modulation technique involving a fast Fourier transformation. As for digital implementation, for OFDM systems, the data flow from flow R (original flow) is multiplexed with the N/R flow into N parallel R/N flows. This is frequency multiplexing, the data being transmitted on N channels. To make this transfer, OFDM circuits transmit data by blocks by introducing redundancy to useful information, instead of transmitting it in series in single-carrier systems, permitting a simple (since it is scalar) inversion of the propagation channel (codes used for 3G, Wi-Fi, Bluetooth, GPS). The spread of the spectral power density at transmission on a broad range of frequencies lowers its average level (see Figure 2.1).

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Figure 2.1. Spectrum sp pread. For a co olor version off this figure, see www.iste e.co.uk/gontra and/digital.zip

Therrefore, severaal emitters can c coexist in n the same frrequency banndwidth. Wi-Fi and a GPS usee this techniique. For GP PS, for exam mple, the siggnal that gives thhe satellite localization l h a rate of has o flow of 50 bits/s. It iis mixed with a random r code of a 1.0233 Mbits/s flo ow rate. Thee spreading factor is SF = 200,400, which results in a lowering of the PSD by 43 dB. It rem minds us that thee GPS signaal is drowned in the bacckground nooise and is ttherefore unobserrvable directlly. For a Wi-Fi traansmission, the t signal iss mixed witth a pseudoo-random sequencce of 1 Mbitts/s rate, whiich gives a spectrum wiidth BW = 222 MHz. Dependding on the innitial signal rate (not sprread) and Wii-Fi standardd (b, g, n or ac), the spread signal then modulates a carrier in BPSK or Q QPSK, or several carriers withh a COFDM coding (see below). b mission strategies Transm Evenn though muulticarrier sysstems greatly y favor channnel equalizaation, the phenom mena of fadding (low traansfer functtion gain value v h(f)) are still circumsscribed. In fact, if the channel’s frequency gain g at a pparticular frequency is low, or indeed nulll, the inform mation carrieed by the sinnusoid at proaches to solving s this problem this freqquency is coompletely loost. Two app exist. el known att transmissio on Channe The channel’s transfer t funnction may be known by b the sendder: this r infoorms the happenss when the environmennt varies littlle and the receiver sender of o their know wledge of thee channel. Th his is the casse, for exampple, with multicarrrier transmiissions on ADSL A twisted d pairs. Sincce the channel varies

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little, the transmitter has access, via the receiver, to the channel’s transfer function during the transmission period. In this case, the optimal transmission strategy is known by the expression “water pouring”, thanks to Shannon’s work – an analogy of a reservoir of water with an irregular basin. Each of the available channels is part of a container with its own depth, governed by the reciprocal signal to noise ratio dependent on the channel’s frequency. To attribute the power, suppose that you are pouring water into this reservoir (the quantity depends on the maximum average transmission power desired). Once the water level is established, the greatest quantity of water is found in the deepest parts of the container. This involves allocating more power to the channel with the best signal-to-noise ratio (SNR). Note, however, that the distribution in each channel is not in proportion, but that it varies in a nonlinear fashion with the maximum average transmission power). At fixed power the power of the transmitter is limited to the standards and limits of minimal consumption when the channel’s amplitude is minimum (the same when it is maximum). In other words, the signal’s intensity profile should be adapted to this profile of the channel. Similarly, a builder may build an unstable house; on a fixed budget, he cannot change the terrain, nor reinforce the house foundations using additional foundation columns. He must therefore make the best use of the position using only the columns available, by anchoring them in solid places to ensure the building’s stability. Here, it is possible to adapt the signal to noise ratio for each subcarrier: – the power of each symbol sent (“power loading”); – the size of the constellation (“bit loading”). These techniques are known by the name of AOFDM (adaptive OFDM). Knowledge of the channel at the transmitter can be achieved in TDD (timedivision duplex) mode since, in this case, the channel from the transmitter to the receiver or from the receiver to the transmitter are the same. This technique is adapted to channels at slow variations with little mobility. Other transmission techniques for reaching the channel’s capacity The channel is known by the transmitter via a precoding and decision feedback equalization (using, for instance, MMSE-DFE – minimum meansquared-error decision-feedback equalizer). However, this occurs at the cost of complexity, which grows according to the flow and the channel’s length.

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CONCLUSION.– The concomitant use of a precoder, adapted to the channel profile, in transmission and a decision feedback equalizer, makes it possible to reach the channel’s capacity at the cost of greater complexity. Channel unknown by transmission When a channel is not known by the transmitter (i.e. as it happens in a wireless transmission), diversity and coding techniques are used to reduce the probability of error in messages transmitted. In this case, correction bits are added to the signal (the same as in cases where the channel is known by the transmitter). Coding To make transmissions more reliable, error correction bits are added, whose values depend on those of the bits of the signal the transmission accompanies. The coding’s operativity is characterized by the coefficient R, called code yield, which is introduced as redundancy (a yield of ½ is equal to sending twice the number of bits). However, the useful flow (information, not redundancy bits) can then be negligible in relation to the total flow. Research is still in the process of creating codes that are effective and easy to decode. In fact, in communication, the complexity of the terminals needs to be minimized; these should be compact and autonomous in energy. At reception, the recipient evaluates the specific redundancy bits and, as they know the rules that have created them and the “sequences” that appear most, they can mitigate divergences. However, an additional redundancy is needed during the transmission of redundant bits to the carrier of the channel affected by fading. As channel attenuation depends on frequency, redundant messages can pass without being distorted, at least at some frequencies (we call this scatter), via the following techniques: – Frequency interweaving: the redundant information is transmitted on different frequency carriers (indispensable) at two different instants. These different carriers should be separated by the channel’s coherence band (the coherence band is the band for which two frequency carriers are completely independent). During transmission on a limited band such as HiperLAN/2, the coherence band is in the order of 2.5 MHz for a band using 20 MHz (all systems, of course, have a limited band to ensure coexistence of other transmission systems).

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– Temporal interweaving: redundant information is, in this case, transmitted on the same carrier, but at two different moments. This supposes that the fading changes quickly (e.g. a moving receiver), so that the value of this carrier’s channel’s gain could have changed. Temporal diversity, however, needs a delay to correct and decode the information destined for the receiver. In fact, in a static environment, the latency (the time spent waiting for the channel’s gain value to have changed) can become prohibitive for real-time applications (voice, broadcasting video). Spatial interweaving Using several antennae at the transmitter and receiver can be a means of palliating the channels’ lack of diversity. In fact, it is possible to position several antennae on the terminal, as they are very small at high frequencies: in the future, for standards at 60 GHz, it will be possible to install several antennae on the transmitter. Redundant information then passes through several different channels and a proper recombination of the signal received permits the useful signal to be extracted. 2.1.3. General principles OFDM systems therefore break down the channel into N subchannels (or carriers) whose central frequencies are spaced at a multiple of the inverse of the symbol’s period 1/T. Unlike single carrier modulations, where the data are sent in series, the underlying principle is block transmission. Modulation of a block of symbols (an OFDM symbol) is carried out using an inverse Fourier transform. If s(k) = {sl(k), 1 ≤ l ≤N}, the vector transmitted (OFDM symbol) of dimension N. Each component l(k) is transmitted at instant (kN + l)T, where k ≥ 0. The positive integer l is the number of carriers on which the component sl(k) of the OFDM symbol is transmitted. The FFT modulator generates the digitized signal transmitted at instant iT at frequency N/T, xi = 1/N1/2. pi -kN exp(j2 li/N) Temporal functions {pi, i ≥ 0} are formatting functions with a baseband spectrum P(f). The sequence {xi} has a spectrum with period N/T and is formed of subchannels whose frequencies are multiples of 1/T (see Figure 2.2).

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Figure 2.2. SPECTRUM of a multicarrier modulation based on a Fourier transform. The different channels overlap while still remaining orthogonal

NOTE.– Because of the spectral overlap of subcarriers in the form of sinc function, low levels of disruption can suffice to “break” the orthogonality, hence the interference. For a rectangular filter, the vector of dimension N at instant (kN)T s(k) = {sl(k), 1 ≤ l ≤ N} can be obtained as a discrete Fourier transform of the vector x(k) = {xn(k), 1 ≤ n ≤ N }. Without distortion due to the channel, this schema might suffice. Nevertheless, the propagation channel c(t) (such that c(iT) = ci for any i} has a length L (c0 ≠ 0, …, cL-1 ≠ 0), while the last components of block N transmitted at instant (k-1)NT x(k-1) = {xn(k-1), 1 ≤ n ≤ N } (obtained after an inverse Fourier transform) interfere with the L first components of the block x(k) = {xn(k), 1 ≤ n ≤ N } following (L < N) resulting from the channel’s memory. Techniques for simplifying OFDM systems use simulated compensation and the cyclical prefix (Cyclic Tip); this means introducing redundancy and structuring it to transform the product into a classic convolution, a product of circular convolution. If the channel is formed of L-coefficients, then each block x(k) = {xn(k), 1 ≤ n ≤ N} is cyclically extended after the inverse Fourier transform of D coefficients such that D ≥ L − 1. A temporal vector of dimension N + D is then sent. Via a

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Fourier transform, the operation of cyclical convolution is then transformed into a scalar frequency product that is easily equalizable. 2.1.4. How to choose N? Choosing N depends on several factors: the type of channel (fast variation, length of the impulse response, etc.) and the complexity that you are willing to accept for the Fourier modulator. If you do not lose the useful rate, it makes sense to choose very far ahead the recharge factor D of N (N + D). In fact, for durable D, the resilience factor has been carried to 1 with the increase in the number of service providers. However, the complexity of the FFT modulator is created by the block’s size. In addition, the channel should not vary in the OFDM symbol so that the model closes the table. Finally, the distance between the carriers corresponds to a factor 1/NT, which decreases as N increases. There is therefore a gain in diversity when N increases. In practice, in some systems (see HIPERLAN) at reception, the first samples containing interferences from the previous block are left to one side. The Fourier transform of block N is therefore: {hnsn(k), 0 ≤ n ≤ N − 1}. Equalization is very simple; it is enough, therefore, to make scalar inversions instead of matrix inversions. Then, add a Gaussian noise and obtain the same frequency model by adding an independent Gaussian scalar frequency noise of the same variance on each carrier (the transformation of the Fourier transform of Gaussian probability density does not modify its statistic). 2.1.5. Practical aspects One of the great advantages of the OFDM transmission system is that it shares the complexity of l, equalization between transmitters and receivers, unlike the single carrier transfer. This requires only one simple and inexpensive receiver. The OFDM uses different variants: – Effective use of output frequencies compared to conventional multiplexing solutions. In fact, the overlap of the OFDM channel retains a complete orthogonality.

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– A digital equalization and a simple and optimal decoding via use of the guard interval (but a decrease in flow) and the Viterbi algorithm. What is more, the use of diverse coding systems such as convolutive coding and block (Reed–Solomon) codes are very effective. – Many carrier technologies are robust to noise, each carrier providing noise, independent of the other carriers. Unlike single-carrier modulations, where the noise can affect some symbols transmitted, the loss of one symbol caused by substantial noise will not affect the other symbols. – OFDM technologies permit great suppleness in bit/flow sharing in multiuser environments. In other words, depending on the channel’s instantaneous gain, each carrier can be coded independently of the others. It is possible to use a “water pouring” method when the channel is known at transmission. – Finally, we can note that channel estimation in the OFDM context is facilitated by sending learning sequences in the frequency domain. Identification of the channel’s coefficients does not require matrix inversion of a discrete inverse Fourier transform of the block of frequency symbols; OFDM can lead to a symbol (engendering directly generated constellations). This creates substantial constraints on the amplifiers and leads to substantial consumption. For example, if the frequency symbol vector gives [1, 1, 1, ... 1], a temporal signal [(N)1/2, 0, 0 ...] is obtained. The first component of the block of temporal symbols generated has a very high amplitude. To avoid clipping, which destroys the orthogonality of the carriers, reducing system performances, it is necessary to use a linear amplifier, raising the cost of the transmitter. Currently, transmission techniques are applied more to reducing the amplitude of the signal, also called peak to average power ratio (PAPR). In practice, most methods for reducing PAPR are based on modifying the signal sent with the help of a correction vector; this is added to the symbols, creating a new constellation with better properties. This causes an increase in complexity at the level of the transmitter. OFDM is generally very sensitive to problems with frequency offset and synchronization. In the first case, frequency offset causes interference between carriers, which can destroy the carriers’ orthogonality. In the second case, synchronization errors cause an offset on the symbols received. Single carrier compensation techniques are not well adapted to multicarrier techniques, and new approaches are being studied.

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In thhe new standaards, higher flows accenttuate these difficulties. It caan be noted that t the prevvious OFDM M model doess not apply w when the cyclic prefix p is smalller than the channel’s leength. A sym mbol sent on a carrier will be able to intterfere withh the symbo ols from adjjacent carrieers. One SL) is to redduce the lenggth of the solutionn (already used in the conntext of ADS channell, with the heelp of a pre-eequalizer. 2.1.6. COFDM C The main drawbaack of OFDM M techniquess is its lack of o diversity. OFD DM schemass have sacrifficed the div versity of sinngle carrierss for the benefit of simplifieed equalizattion. When a subcarrieer is attenuaated, the i definitivelly lost. In praactice, codedd OFDM informaation sent on this carrier is schemass, known byy the acronym m COFDM, make m it posssible to palliaate these drawbaccks. Among systems usinng COFDM, we can talk about: – diggital audio brroadcasting; replacing radio; – diggital video brroadcasting, adapted for digital televiision; – HIIPERLAN/2 and IEEE 8002.11a, dediccated to local wireless neetworks.

Figure 2.3 3. Schema of the t principle behind b a COFD DM system

The schema of the t principlee behind a COFDM C system is repressented in Figure 2.3. First, a brewing module is introduced at the startt of the transmisssion chain to distributee energy acrross all the bits. This pprecludes long seequences of 0 or 1, whhich can create lines in the spectruum. This

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operation is carried out by multiplying the signal by the output of a (pseudo-)random generator. Before strengthening protection for the bits against potential errors, a convolutive code is adopted. The role of this code is to link bits to one another, to recover the values in each case of erroneous transmission. In the context of standard HiperLAN/2, a convolutive coder of constraint length 7, each incident bit will generate two outgoing bits and will be linked to six preceding bits (the constraint length is the increased number of registers per unit). The coder’s performance is 0.5, as there are two output bits for one input bit. Output bit X is one “or exclusive” between bits 1, 2, 3, 4 and 7 while output bit Y is one “or exclusive” between bits 1, 3, 4, 6 and 7. To increase the flows and decrease the code’s redundancy, a punching module is used. This involves transmitting only some of the coder’s output bits. In the case of a ¾ yield, for 3 bits present at entry to the coder, the punching will only transmit 4 in place of the usual 6 bits. At the level of the decoder, the bits not transmitted are replaced by the value 0 and the errors due to the punching do not generally affect the system’s performances too much. The benefit of punching is that it can modify the yield of codes without implementing a new convolutive code. At the convolutive coder’s output, a frequency interweaver is used. The interweaver “disperses” the bits containing the same information over several carriers. In practice, the interweaver is a table that makes a position on a given carrier corresponds to each bit. If there are enough independent frequency carriers, we can then recover and reconstitute the information from samples that have not been attenuated. Hence, there is possibility of correcting a long sequence of erroneous consecutive bits. In this context, diversity is created by using joint interweaving and coding. At the interweaver’s output, the bits are modulated into symbols such as QPSK, QAM 16 or QAM 64. Depending on the size of the constellation sent, the flow will be changed.

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At the receiver side, the signal received, after FFT demodulation, is deinterlaced according to the sender’s correspondence table. Metrics, taking account of the constellation of symbols sent and the fading on each carrier, are then calculated. These are used by a Viterbi decoder (decoding a bitstream coded with the help of a convolution or a lattice code: see Figure 2.4). In order to correct errors, since the bits have not been attenuated or amplified, they have more weight in the decoding than the bits issuing from carriers with low gain. No equalization is carried out and decoding is a maximum likelihood decoding, because of the Viterbi algorithm. Performances are improved via the code’s constraint length and the size of the interweave. This can, however, lead to lattices at the highly complex Viterbi decoder (Figure 2.4), and consequent decoding delays. Finally, data packets are discarded at the decoder’s output.

Figure 2.4. Viterbi algorithm. The lattice makes it possible to visualize the decoding and grasp the temporal evolution (from left to right) of a state machine. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

2.1.7. Equalization and decoding Equalization makes it possible to compensate for the effect of the channel; the decoding makes it possible to reverse the transmitter’s coding algorithm. The coding operation serves to attenuate the effects of (Gaussian) noise resulting from system imperfections. This breaking down of the receiver into two modules is certainly not optimal but it is less complex than

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a joint equalization/decoding. As for COFDM, equalization and decoding are achieved concomitantly. 2.1.8. The multiuser context Allocation of resources between most users is another major challenge for network development, with a view to future use of OFDM. It has become clear that more and more known technologies, such as multiple frequency access, time attribution multiple access and code division multiple access are compatible with modulation at multiple carriers. They appear upstream of the transmission chain to distinguish users. Many of these operating techniques rely on the principle of re-using the spectrum: available radio stations are subdivided into a given geographical zone (the cell) and provide the same frequency subsystem at distant stations. One cell, whose size is defined by the pitch of fixed access points is, grosso modo, 2 km in diameter in urban areas, and more in rural areas. In very dense areas, mobile network capacity is increased by microcells of around 500 m. Access points use different methods to separate signals coming from various transmitters. The simplest and most used is frequency division multiple access: – One frequency is allocated to each user. The access point (or base station) knows which mobile uses which frequency and sorts OFDM signals, exactly the way in which we used to choose our preferred radio stations by turning the transistor button to the correct position. – Digitization of communications makes the same frequency attribution possible between several users. In time division multiple access, the mobile device sends or receives in time periods in the order of milliseconds. Fixed base stations know which device is transmitting in which time period and it reforms the message from blocks sent in this way. 2.1.9. Code division multiple access Code division multiple access is a competitive process. We need to remember that each mobile device has its own coding, which enables several users to send simultaneously on the same frequency band. The access point tests the data flows that enter this frequency band and set the correlations for

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each sequence of cells to find each bit transmitted to each. This method has already been normalized in third-generation mobile communications (UMTS). The cells communicate with one another in pairs (or binomes). Binomes wish to communicate with one another and do not care what the others are saying. Therefore, these conversations can take place by trying to define the different options for each conversation. Each couple wishes to speak in a large space; each in their turn. Each binome has the right to speak for 20 sec during which the other pairs are silent. They thus exchange speaking time in turns. Each pair speaks another language (another code) with a multiple access to the code distribution technique. Pairs can speak simultaneously within the room. The analogy here is that codes can be considered as languages. This also assumes that couples do not understand the languages spoken by other couples. Language is a filter through which a French man cannot understand a conversation with a German or Navajo couple. Of course, this technique has limits, since you can no longer adjust the binomes, since new conversations introduce new noise. Orthogonality also enables high spectral efficiency and total flow approaching the Nyquist flow, the passband being almost entirely used up. Orthogonal multiplexing produces an almost flat (white) frequency spectrum, which creates minimum interference with adjacent channels. Filtering separate from each subcarrier is not necessary for decoding, since a FFT is enough to separate the carriers from one another. The signal to be transmitted is generally repeated on different subcarriers. Thus, on a transmission channel with multiple paths where certain frequencies will be destroyed, due to the destructive combination of paths, the system will still be capable of recovering information lost on other carrier frequencies that will not have been destroyed. Each subcarrier is modulated independently using digital modulations: BPSK, QPSK, QAM16, QAM-64, etc. OFDM decoding requires very precise synchronization of the receiver’s frequency with the transmitter. Any deviation in frequency leads to the loss of the subcarriers’ orthogonality and consequently creates interference between them. This synchronization becomes difficult to carry out as soon as the receiver is in movement, in particular when the speed or direction varies or if many interference echoes are present.

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2.1.10. Schematic c ordinogra am

Figure 2.5. 2 Space–tim me coding

Beloow are repreesented typiccal spectrums: Figure 2.66(a), close tto a slot, and Figuure 2.6(b), thhat of the phyysical layer OFDM O 802.11a.

(a)

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(b) Figure 2.6. (a) A typical OFDM spectrum (measure). (b) Example: signal and parameters of the physical layer OFDM 802.11a. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

IEEE 802.11a/g and HIPERLAN/2 signals are of an impulse (or burst) type. The channel’s total bandwidth is 20 MHz with an occupied bandwidth of 16.6 MHz. A single OFDM symbol contains 52 subcarriers; 48 are data subcarriers and 4 are pilot subcarriers. In the center, “DC” or “Null”, the zero subcarrier is not used. All the data subcarriers use the same modulation format in a given burst. However, the modulation format can vary from one burst to another. The possible modulation formats of the data subcarrier are BPSK, QPSK, 16QAM and 64QAM. The pilot subcarriers are always modulated with the help of BPSK and a known amplitude and phase. Each OFDM subcarrier transports a single data symbol, or “constellation point”, as well as amplitude and phase information. This means that the amplitude and phase will vary for each subcarrier and OFDM symbol in the burst transmitted. To see how the OFDM operates, it is necessary to look at the receiver. This acts as a bank of demodulators, conveying each carrier continuously. The resulting signal is integrated over the symbol period to regenerate this carrier’s data. The same demodulator also demodulates the other carriers. Since the carriers’ spacing is equal to the reciprocal of the symbol period,

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they willl have a whole number of o cycles in the symbols’ period and the sum of their contributionns will be zerro; in other words, w there is i no interferrence.

Fiigure 2.7. Tra aditional view of o modulation carrying the reception r signa als

Figure 2.8 8. OFDM specctrum. For a co olor version off this figure, see www.iste e.co.uk/gontra and/digital.zip

One condition of OFDM traansmitting an nd receiving systems is tthat they h linearitty. Any nonllinearity will cause interrference betw ween the should have carriers because of intermodulaation distortiion. This wiill present uundesired signals that t would cause c interferrence and alter transmissiion orthogonnality. As for fo the equipm ment to be used, u the peak k to average level of multicarrier systemss such as OF FDM demandds that the final f RF ampplifier situateed at the transmittters’ outputt is capable of managin ng the peakss, while the average power is much weakker, which leeads to inefficiency.

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In some systems, the peaks are limited. Although this introduces distortion, and has a high level of data errors as a consequence, the system can rely on error correction to remove them. 2.1.11. Data in OFDM The data to be transmitted on an OFDM signal are therefore declined through the signal’s carriers, each to form part of the payload. This reduces the flow taken by each carrier. A lesser flow has the advantage that the interference linked to reflections is much less critical. This is achieved by adding a guard band time, or a guard interval, into the system (Figure 2.9). This ensures that the data are only collected/sampled when the signal is stable and no new delayed signal arrives, which would change the signal’s synchronization and phase. Direct signal

Reflection 1

Last reflection

Sampling window Guard interval

Figure 2.9. Guard intervals

The distribution of data across a large number of carriers in the OFDM signal has other advantages. Zeros caused by the effects of multiple paths or interference on a given frequency only affect a restricted number of carriers; the others receive correctly. By using error-coding techniques, which means adding other data to the signal transmitted, it makes it possible for corrupted data to be reformed in the receiver. This can be done because the error correction code is transmitted in a different part of the signal.

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2.1.12. OFDM: advantages and disadvantages Advantages OFDM is used in many wireless systems with high flows because of the many advantages it provides. – Immunity to selective fading: one of the main advantages of OFDM is that it is more resistant to selective frequency erasure than simple carrier systems because it divides the global channel into multiple narrowband signals that are allocated individually as flat fading subchannels. – Resilience to interferences: interference appearing on a channel can be in a limited band and as such will not affect all the subchannels. This means that not all the data will be lost. – Spectrum efficiency: when using recovery subcarriers in a closed space, one significant advantage of OFDM is that it permits effective use of the available spectrum. – Resilient to ISIs: another advantage of OFDM is that it is very resilient to intersymbol and interframe interference. This results from low flow on each of the subchannels. – Resilient to the effects of narrowband: using coding and interleafing adequate to the channel, it is possible to recover symbols lost due to the channel’s selective frequency and interference in narrowband. Not all data are lost. – Easier channel equalization: one of the questions with CDMA systems was the complexity of equalizing the channel, as the equalization had to be applied across the entire channel. One advantage of OFDM is that with the help of multiple subchannels, channel equalization becomes much simpler; the signal received is multiplied by a channel given by the reverse of its gain. Disadvantages While OFDM is currently in use, there remain some drawbacks that should be considered following its use. – High peak to average power ratio: an OFDM signal has noise, such as an amplitude variation over a relatively extended dynamic range, or an average peak power ratio. This affects the efficiency of the RF amplifiers,

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which should be linear and adapted to large amplitude variations; the amplifier cannot therefore function with a high yield. – Sensitive to the offset and the drift of the carrier frequency: simple carrier systems are less sensitive to this. – Variants: - OFDMI: there are several other OFDM variants. These follow the basic OFDM format, but have additional attributes or variations. - COFDM: this stands for coded orthogonal frequency division multiplexing and it is a form of OFDM where error correction coding is incorporated into the signal. - Flash OFDM: this is a variant of OFDM developed by Flareon: a “fast hopping” form of OFDM. Because of Flash-OFDM, you can remain connected to the Internet while you are moving (up to 280 km/h). It uses multiple tonalities and fast hopping to spread the signals over a given spectrum. - Orthogonal Frequency-Division Multiple Access (OFDMA): this deals with orthogonal multiple access frequency division. This provides opportunities for multiple accesses for applications such as cellular telecommunications using OFDM technologies. - VOFDM (Vector OFDM): this form of OFDM uses Multiple Input Multiple Output (MIMO) technology (see below). It was developed by Cisco Systems. MIMO uses several antennae to transmit and receive signals, so that the multipath effects are used to improve reception of the signal and the transmission speed supported by this range. - WOFDM (Wide-Band OFDM): OFDM has a wide band. The concept of this form of OFDM is that it uses a degree of spacing between channels that is large enough that no frequency error between the transmitter and receiver affects operating. It is particularly applicable in Wi-Fi systems. 2.1.13. Intermediate conclusion Over the course of the last five decades, a polyvalent modulation has been developed on the wireless telecommunications market. This is mainly due to the OFDM technique using a multi-modulator medium. The OFDM, underpinned by the FFTs, has simplified modulator complexity, solving the

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problem of equalization; but theoretical flow from the point of view of transmission is still a long way off because of the lack of low complexity coding/decoding algorithms. Many researchers and R&D engineers are already working on the next generations. In the presence of a multipath channel, reception of several echoes in phase opposition can give raise to fading (severe attenuation on the part of the frequency band). In the framework of an OFDM system, it is usually impossible to reconstruct the symbols transported by the subcarriers affected by these fading phenomena. This is explained by the fact that OFDM that is not pre-coded does not introduce redundancy (or frequency diversity). This drawback can be circumvented using COFDM, at the cost of a decrease in spectral efficiency. OFDM is a block transmission system; we usually introduce a guard interval between these blocks. This makes it possible to eliminate interference between successive blocks in the presence of multipath channels and to facilitate even more equalization on the condition that the guard interval has a duration higher than the last path’s arrival time. Two types of guard interval are currently in use: the cyclical prefix, which consists of recopying the last samples of the block at its terminals, and the loading or jam of zeros, which consists of inserting zeros at the start of the block. Both these techniques, however, lead to a drop in spectral efficiency. OFDM (or a similar technique) is used in: – digital terrestrial broadcasting (DVB-T, DVB-H); – digital terrestrial broadcasting (DAB); – digital terrestrial broadcasting (T-DMB); – digital broadcasting (DRM); – wired links, such as ADSL, VDSL, powerline communications, cable modems (standard Docsis);

(homeplug)

– wireless networks, based on standards 802.11a, 802.11g, 802.11n and 802.11ac (Wi-Fi), 802.16 (WiMAX) and HiperLAN; – new generation mobile networks (LTE → 5G) that use a multiple access technique based on OFDM called OFDMA.

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2.1.14. QPSK and OFDM with MATLAB system objects This example shows how to simulate a basic communication system in which the signal is a modulated QPSK, then undergoes orthogonal frequency division multiplexing. The signal is then passed through a channel with additive white Gaussian noise, before being demultiplexed and demodulated. To finish, the number of bit errors are calculated. The example presents the use of objects™ from the MATLAB® system. Set the simulation parameters as follows: M = 4; % Modulation alphabet k = log2(M); % Bits/symbol numSC = 128; % Number of OFDM subcarriers cpLen = 32; % OFDM cyclic prefix length maxBitErrors = 100; % Maximum number of bit errors maxNumBits = 1e7; % Maximum number of bits transmitted Construct System objects needed for the simulation: QPSK modulator, QPSK demodulator, OFDM modulator, OFDM demodulator, AWGN channel, and an error rate calculator. Use name-value pairs to set the object properties. Set the QPSK modulator and demodulator so that they accept binary inputs. qpskMod = comm.QPSKModulator('BitInput',true); qpskDemod = comm.QPSKDemodulator('BitOutput',true); Set the OFDM modulator and demodulator pair according to the simulation parameters. ofdmMod = comm.OFDMModulator('FFTLength',numSC,'CyclicPrefixLength',cp Len); ofdmDemod = comm.OFDMDemodulator('FFTLength',numSC,'CyclicPrefixLength', cpLen); Set the NoiseMethod property of the AWGN channel object to Variance and define the VarianceSource property so that the noise power can be set from an input port. channel = comm.AWGNChannel('NoiseMethod','Variance',...

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'VarianceSource','Input port'); Set the ResetInputPort property to true to enable the error rate calculator to be reset during the simulation. errorRate = comm.ErrorRate('ResetInputPort',true); Use the info function of the ofdmMod object to determine the input and output dimensions of the OFDM modulator ( Matlab Inc). ofdmDims = info(ofdmMod) ofdmDims = struct with fields: DataInputSize: [117 1] OutputSize: [160 1]

Determine the number of data subcarriers from the ofdmDims structure variable. numDC = ofdmDims.DataInputSize(1) numDC = 117

Determine the OFDM frame size (in bits) from the number of data subcarriers and the number of bits per symbol. frameSize = [k*numDC 1];

Set the SNR vector based on the desired Eb/No range, the number of bits per symbol, and the ratio of the number of data subcarriers to the total number of subcarriers. EbNoVec = (0:10)'; snrVec = EbNoVec + 10*log10(k) + 10*log10(numDC/numSC);

Initialize the BER and error statistics arrays. berVec = zeros(length(EbNoVec),3); errorStats = zeros(1,3);

Simulate the communication link over the range of Eb/No values. For each Eb/No value, the simulation runs until either maxBitErrors are recorded or the total number of transmitted bits exceeds maxNumBits. for m = 1:length(EbNoVec) snr = snrVec(m); while errorStats(2) < = maxBitErrors && errorStats(3) < = maxNumBits

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dataIn = randi([0,1],frameSize); % Generate random binary data qpskTx = qpskMod(dataIn); % Apply QPSK modulation txSig = ofdmMod(qpskTx); % Apply OFDM modulation powerDB = 10*log10(var(txSig)); % Calculate Tx signal power noiseVar = 10.^(0.1*(powerDB-snr)); % Calculate the noise variance rxSig = channel(txSig,noiseVar); % Pass the signal through a noisy channel qpskRx = ofdmDemod(rxSig); % Apply OFDM demodulation dataOut = qpskDemod(qpskRx); % Apply QPSK demodulation errorStats = errorRate(dataIn,dataOut,0); % Collect error statistics end berVec(m,:) = errorStats; errorStats = errorRate(dataIn,dataOut,1); calculator end

% Save BER data % Reset the error rate

Use the berawgn function to determine the theoretical BER for a QPSK system. berTheory = berawgn(EbNoVec,'psk',M,'nondiff'); Plot the theoretical and simulated data on the same graph to compare results. Figure semilogy(EbNoVec,berVec(:,1),'*') hold on semilogy(EbNoVec,berTheory) legend('Simulation','Theory','Location','Best') xlabel('Eb/No (dB)') ylabel('Bit Error Rate') grid on hold off

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Figure 2.10. BER versus Eb/No. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip Box 2.1. Bit error rate versus energy per bit. For a color version of this text, see www.iste.co.uk.uk/gontrand/digital.zip

2.1.15. FDM versus OFDM: difference between FDM and OFDM Note that ax-16d/16e, wlan-11g/11n and LTE technologies demand a higher flow and are mainly used for broadband internet service. In the case of FDM, the whole bandwidth is used by the user/subscriber, while in OFDM the bandwidth is divided into a number of narrowband channels and each is assigned to the user/subscriber. Consequently, the OFDM supports more subscribers/channels compared to FDM. OFDM in MATLAB takes account of the OFDM transmitter and receiver. The program covers the OFDM transmission and reception base chain (RF Wireless World: Kendo UI: java script library and Matlab: “open

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sources”1), transmission of binary data, data projection, IFFT and insertion of cyclic prefix (CP). In the time domain, these data pass through a noisy channel (see AWGN). The OFDM receiver is formed of a CP, FFT and “demapping” data and decodes these same data. OFDM transmitter part clc; clear all; close all; %.............................................................. % Initiation %.............................................................. no_of_data_bits = 64%Number of bits per channel extended to 128 M = 4 %Number of subcarrier channel n = 256;%Total number of bits to be transmitted at the transmitter block_size = 16; %Size of each OFDM block to add cyclic prefix cp_len = floor(0.1 * block_size); %Length of the cyclic prefix %............................................................ % Transmitter %......................................................... %......................................................... % Source generation and modulation %........................................................ % Generate random data source to be transmitted of length 64 data = randsrc(1, no_of_data_bits, 0:M-1); Figure(1),stem(data); grid on; xlabel('Data Points'); ylabel('Amplitude') title('Original Data ') % Perform QPSK modulation on the input source data qpsk_modulated_data = pskmod(data, M); Figure(2),stem(qpsk_modulated_data);title('QPSK Modulation') %............................................................ %............................................................. % Converting the series data stream into four parallel data stream to 1 https://www.rfwireless-world.com/source-code/MATLAB/OFDM-matlab-code.html.

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form % four sub carriers S2P = reshape(qpsk_modulated_data, no_of_data_bits/M,M) Sub_carrier1 = S2P(:,1) Sub_carrier2 = S2P(:,2) Sub_carrier3 = S2P(:,3) Sub_carrier4 = S2P(:,4) Figure(3), subplot(4,1,1),stem(Sub_carrier1),title('Subcarrier1'),grid on; subplot(4,1,2),stem(Sub_carrier2),title('Subcarrier2'),grid on; subplot(4,1,3),stem(Sub_carrier3),title('Subcarrier3'),grid on; subplot(4,1,4),stem(Sub_carrier4),title('Subcarrier4'),grid on; %.................................................................. %.................................................................. % IFFT OF FOUR SUB_CARRIERS %................................................................. %.............................................................. number_of_subcarriers = 4; cp_start = block_size-cp_len; ifft_Subcarrier1 = ifft(Sub_carrier1) ifft_Subcarrier2 = ifft(Sub_carrier2) ifft_Subcarrier3 = ifft(Sub_carrier3) ifft_Subcarrier4 = ifft(Sub_carrier4) Figure(4), subplot(4,1,1),plot(real(ifft_Subcarrier1),'r'), title('IFFT on all the sub-carriers') subplot(4,1,2),plot(real(ifft_Subcarrier2),'c') subplot(4,1,3),plot(real(ifft_Subcarrier3),'b') subplot(4,1,4),plot(real(ifft_Subcarrier4),'g') %........................................................... %........................................................... % ADD-CYCLIC PREFIX %.......................................................... %............................................................ for I = 1:number_of_subcarriers, ifft_Subcarrier(:,i) = ifft((S2P(:,i)),16)% 16 is the ifft point for j = 1:cp_len,

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cyclic_prefix(j,i) = ifft_Subcarrier(j + cp_start,i) end Append_prefix(:,i) = vertcat( cyclic_prefix(:,i), ifft_Subcarrier(:,i)) % Appends prefix to each subcarriers end A1 = Append_prefix(:,1); A2 = Append_prefix(:,2); A3 = Append_prefix(:,3); A4 = Append_prefix(:,4); Figure(5), subplot(4,1,1),plot(real(A1),'r'),title('Cyclic prefix added to all the sub-carriers') subplot(4,1,2),plot(real(A2),'c') subplot(4,1,3),plot(real(A3),'b') subplot(4,1,4),plot(real(A4),'g') Figure(11),plot((real(A1)),'r'),title('Orthogonality'),hold on,plot((real(A2)),'c'),hold on, plot((real(A3)),'b'),hold on,plot((real(A4)),'g'),hold on,grid on %Convert to serial stream for transmission [rows_Append_prefix cols_Append_prefix] = size(Append_prefix) len_ofdm_data = rows_Append_prefix*cols_Append_prefix % OFDM signal to be transmitted ofdm_signal = reshape(Append_prefix, 1, len_ofdm_data); Figure(6),plot(real(ofdm_signal)); xlabel('Time'); ylabel('Amplitude'); title('OFDM Signal');grid on; %............................................................... Passing time domain data through channel and AWGN %............................................................. channel = randn(1,2) + sqrt(-1)*randn(1,2); after_channel = filter(channel, 1, ofdm_signal); awgn_noise = awgn(zeros(1,length(after_channel)),0); recvd_signal = awgn_noise + after_channel; % With AWGN noise Figure(7),plot(real(recvd_signal)),xlabel('Time'); ylabel('Amplitude');

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title('OFDM Signal after passing through channel');grid on; %........................................................... OFDM receiver part %.......................................................... recvd_signal_paralleled = reshape(recvd_signal,rows_Append_prefix, cols_Append_prefix); %........................................................ %........................................................ % Remove cyclic Prefix %....................................................... %...................................................... recvd_signal_paralleled(1:cp_len,:)=[]; R1 = recvd_signal_paralleled(:,1); R2 = recvd_signal_paralleled(:,2); R3 = recvd_signal_paralleled(:,3); R4 = recvd_signal_paralleled(:,4); Figure(8),plot((imag(R1)),'r'),subplot(4,1,1),plot(real(R1),'r'), title('Cyclic prefix removed from the four subcarriers') subplot(4,1,2),plot(real(R2),'c') subplot(4,1,3),plot(real(R3),'b') subplot(4,1,4),plot(real(R4),'g') %................................................... %................................................... % FFT of recievied signal for i = 1:number_of_subcarriers, % FFT fft_data(:,i) = fft(recvd_signal_paralleled(:,i),16); end F1 = fft_data(:,1); F2 = fft_data(:,2); F3 = fft_data(:,3); F4 = fft_data(:,4); Figure(9), subplot(4,1,1),plot(real(F1),'r'),title('FFT of all the four sub-

Some Developments in Modulation Techniques

carriers') subplot(4,1,2),plot(real(F2),'c') subplot(4,1,3),plot(real(F3),'b') subplot(4,1,4),plot(real(F4),'g') %................................ %.............................. % Signal Reconstructed %.................................. %.................................. % Conversion to serial and demodulation a recvd_serial_data = reshape(fft_data, 1,(16*4)); qpsk_demodulated_data = pskdemod(recvd_serial_data,4); Figure(10) stem(data) hold on stem(qpsk_demodulated_data,'rx'); grid on;xlabel('Data Points');ylabel('Amplitude'); title('Received Signal with error')

Figure 2.11. Starting data. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Figure 2.12. QPSK modulations. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 2.13. Subcarriers. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Some Developments in Modulation Techniques

Figure 2.14. Inverse Fourier transforms of the subcarriers. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 2.15. Addition of prefixes. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Figure 2.16. Orthogonality. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure 2.17. The OFDM signal. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

2.2. A note on orthogonality Orthogonality in this context can be defined using an inner product such as ⟨ϕ1,ϕ2⟩ = ∫2π0ϕ1(x)ϕ2(x) dx. This is therefore null.

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This inner product measures scalar projections by finding the average of the two functions. Thus, we are looking at the integral of the product of two sines of different frequencies: ϕ1 = sin(x) and ϕ2 = sin(2x). Note that the frequency of ϕ1 is 1 and the frequency of ϕ2 is 2. The basic idea is as follows: if the frequencies of two sine curves are different, between 0 and 2π, the two sinusoidal curves are of opposite signs as much as they are of the same sign.

Figure 2.18. Two sinusoidal curves of different periods; with a null algebric sum. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Therefore, their product will be as positive as it is negative. In the integral, these positive contributions compensate for the negative contributions, leading to a zero average.

Figure 2.19. Null mean value: the positive surface equals the negative one. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Given that sin(mx)sin(nx) = (cos(m − n)x/2) − cos(m + n)x/2)), for m = 1 and n = 2, the integral is really null, since sin(0) = sin(2πmx) = 0. More generally, the functions, outside a certain space in R, form a vector space. They can be added, subtracted and scaled. Therefore, they can be applied to linear algebra. We can break down functions in [0,2π] on sinusoidal components by making their average with sine and cosine curves. An analogy is of lighting up the function and seeing how far its shadow is projected onto the sin(x) vector. The projection is: ⟨f(x),sin(x)⟩sin(x). As we have seen, the sines and cosines of different frequencies are orthogonal between them, since the average of the products is null. In fact, they form an orthonormal base of the vector space of functions in [0,2π] (basis of the Fourier analysis). We are dealing with a linear operator on the function vectors’ space on [0,2π]. By using integration by parts, we can see that it is a symmetrical linear operator (inducing a symmetrical matrix). The sines and cosines are eigenvectors. Vectors in R2 This is the pair: v = (v1,v2), where v1 and v2 are components of vector v. We will now try to define the orthogonality with the help of Pythagoras’ theorem: If u, v and w are vectors in R2, such that u = (u1, u2) and v = (v1, v2) and w = u + v. If u, v and w form a right triangle, with w being the hypotenuse, then it is true (Pythagoras’ theorem) that: ||w||2 = ||u + v||2 = ||u||2 + ||v||2 (u1 + v1)2 + (u2 + v2)2 = (u21 + u22) + (v21 + v22)

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(u21 + 2u1v1 + v21) + (u22 + 2u2v2 + v22) = (u21 + u22) + (v21 + v22) And so u1v1 + u2v2 = 0

(∗)

thus, for two vectors to be orthogonal, they should satisfy this condition, known as dot product or inner product. u⋅v = u1v1 + u2v2 Functions in L2([0,1]) We can now apply these same ideas to functions. So, to state that two functions are orthogonal means that their norms satisfy Pythagoras’ theorem. We define norm L2 as: ||f|| = (∫01|f(t)|2dt)1/2 So, Pythagoras’ theorem, for real functions f and g, is now given as: ||f + g||2 = ||f||2 + ||g||2 ∫01(f(t) + g(t))2dt = ∫01f(t)2dt + ∫01g(t)2dt ∫01 (f(t)2 + 2f(t)g(t) + g(t)2)dt = ∫01f(t)2dt + ∫01g(t)2dt We obtain: ∫01f(t)g(t)dt = 0 So, for two functions to be orthogonal in L2 ([0,1]), they should satisfy this condition. We will now do with functions in L2 ([0,1]) what we did with vectors in R2, and define the inner product of the real function f and g in L2 ([0,1]) as: (f,g) = ∫01f(t)g(t)dt) = 0 In conclusion, two vectors are orthogonal if their inner product is null, or in equivalent manner, when Pythagoras’ theorem is satisfied.

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2.3. Global System for Mobile Communications 2.3.1. Introduction This introduction of GSM (Global System for Mobile communications) tackles network architecture, network elements, system characteristics, applications and burst types2 – used to speak of sudden, very intense and very brief transmissions. The radio signal sent in a time-slot (TS). A “burst” is linked to the period of the carrier, which is modulated by a data flow: it represents the physical content during the TS, that is, GSM frame structure or frame hierarchy, logical channels, physical channels, physical layer or speech processing, GSM cell phone network input or the power setting procedure, protocol stacking, planning RF calls, up and down links. GSM comes from l.5G or second-generation technology with 5G, passing, of course, through various GSM standards: GSM900, EGSM900, GSM1800 and GSM1900; they differ mainly in band and bandwidth, and RF carrier frequency. The minimum size of an Ethernet frame is 64 octets. Why? By imposing this size, a station close to one of the terminals of the Ethernet bus would never to be able to inject its frame completely into the bus without being detected by another station at the other terminal of the bus it wishes to send. The minimum time to detect a collision is the signal propagation time, and the time taken to go from one terminal of the Ethernet bus to the other is the slot time. For the slot size, this is the volume of octets that can be transmitted during a slot time, i.e. 64 octets. At the time when 802.3 (10 Mb/s) was standardized, electronics made it possible to manage an Ethernet bus with a maximum length of 2,500 m (with maximum of 4 repeaters) (which is a slot time of 4.6 µs). Therefore, when higher speeds were introduced in the world of Ethernet, it was necessary to retain the maximum length of 2,500 m by increasing the slot time (and so the minimum size of an Ethernet frame so that collisions could be detected at the bus terminals), or keep the same slot time but decrease the cable’s length.

2 NOTE.– the Ethernet burst is strongly linked to the CSMA/CD (Carrier Sense Multiple Access with Collision Detection) protocol and its collision window.

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It was this last option that was kept when fast Ethernet was implemented, i.e. the slot time and the minimum Ethernet length of 64 octets were kept, but the maximum length of the Ethernet bus was kept at 100 m. With gigabit Ethernet, things became more complicated, because if one wanted to keep the same slot size, the maximum length of a gigabit cable could not exceed 10 m. Therefore, the maximum length of the cable was kept at 100 m (to make fast → giga migration easier) and the slot size was increased to 512 octets. But to make gigabit Ethernet compatible with Ethernet 10/100, it was necessary to keep the minimum Ethernet size of 64 octets. Consequently, when a giga interface needs to a send a frame of less than 512 octets, it completes (pads) the stream of 512 bits with special symbols called extension symbols and this insertion of special symbols by the Ethernet chipset is called a carrier extension process. But this is not very effective if the giga interface sends a large number of small frames. Burst timer gigabit Ethernet was therefore introduced, which is equal to the transmission time of 1,500 octets (a little less than a carrier detection cycle or the maximum size of 1,518 octets). The interface thus sends a “burst” of small packets delimited by the extension symbols and encapsulated in the Gig Ethernet frames. Many details would be needed to explain the various entities in this domain; this is not the purpose of this book, which is intended as a broad introduction. A large quantity of high-quality technical documentation exists. 2.3.2. Forming a GSM The GSM network is a mobile subset formed of base stations and networks and operation subsets. The following figure gives a synopsis of the complete GSM network. It is formed of cells, a transmission-reception area linked to a base station (base transceiver station [BTS]). Two frequencies of 890 and 915 MHz are used for uplinks of the mobile to the base and 935 and 960 MHz for downlinks. Each of these two bands is divided into 125 channels. For each user, the available passband is multiplexed in time and frequency.

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STN

Switched teelephone netwo ork (fixed)

MSC

Mobile swiitching center

BSC

Base statioon controller/mo obile switching center

BTS

Base transcceiver station

HLR

Home locaation register

VLR

Visitor locaation register

BSS

Base statioon subsystem

PSTN N!

Public swittched telephonee network

ISON N

Africa Technologyy dedicated to (east) (

SIM

Subscriber identity modulle

ME

Mobile equuipment Ta able 2.1. A GS SM

Figure 2.20. GSM netw work architectture. For a collor version w k/gontrand/dig gital.zip of thiss figure, see www.iste.co.uk

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Figure 2.21. Composition of a GSM. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Mobile station: the mobile station is a GSM device including the PSD (design signal processing) block, a microprocessor with a dedicated function, and the SIM (subscriber identity module) card. Base station subsystem: the subset of BSC stations (including the low BTS transmitter–receiver station) and the base. To provide the GSM service, the region/town on the ground is divided into cells. Cell size is usually 100 m–35 km. All these BTS are connected with a BSC. This BSC is responsible for RF tasks and transfer in BSS, i.e. between two BTS. Network subsystem (NSS): this subsystem provides the interface between the cellular system and the telephone network with a switched circuit, i.e. public switched telephone network (PSTN). It operates switching and the functions linked to operation and maintenance. The NSS operates call processing functions such as call installation, switching, changes and transfers between BSCs. The NSS takes care of security and functions linked to authentication. There are various network elements in this subset as mentioned in the GSM network architecture above. They are explained below. These are fundamentally data base elements. HLR: this stores and provides linked information for each permanent subscriber. VLR: this stores information linked to subscribing visitors on devices, subscriber network, location, etc.

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Authentication center (AUC): authentication center is used to authenticate activities in the system. It holds the encryption key (key A5) and authentication key (key A3) in HLR and VLR. It provides security while maintaining the type of equipment available in GSM interfaces. 2.4. MIMO 2.4.1. Introduction MIMO: this is a multiplexing technique for mobile and wireless networks, enabling data transfers at longer range and with a higher flow than with antennae using the single-input single-output (SISO) technique, although older Wi-Fi networks or standard GSM networks use a single antenna at the transmitter and receiver, and MIMO uses several. MIMO technology, which offers enormous advantages at relatively low cost for wireless networks, has become the core of standard 802.11n. MIMO has proven itself so innovative that it has also been adapted for use in other wireless networks, such as 4G mobile networks. Wireless networks built under former standards 802.11a/b/g, to operate effectively, require minimization of multitrack power, whereas multiantennae MIMO wireless networks benefit from this. Multitrack is a phenomenon that occurs during a radio transmission when the signal sent is reflected by objects situated between the transmitter and receiver. The signals reflected are connected to the receiver’s antenna and interfere with one another. EXAMPLE.–

11 Mbps, range: ~30 m, frequency: 2.4 GHz and 802.11a/802.11g (instantaneous speed: 54 Mbps, range: tens of meters: 5 GHz or 2.4 GHz). 2.4.2. Principles

MIMO technologies therefore use antenna networks at transmission and/or reception to increase the SNR and/or the transmission flow. This also makes it possible to decrease the transmission level of radio signals and so to

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reduce surrounding electromagnetic pollutiion, but alsoo to prolongg battery my, in this case, in a telephone. Several anttennae are uused for autonom transmisssion and recception (2×2, 4×2, 2×4, 4×4, 4 etc.). – A new n dimensiion: space: - spatial multipplexing to im mprove flow (Bell Laboratories Layeer SpaceBLAST]); Time [B - spatial diverssity to make the link viab ble: block sppace-time codding and trellis coding c (spacce-time blockk code [STB BC] and spaace-time trelllis code [STTC]). – Wee are interestted in MIMO O 2×2 schem mas using spattial diversityy: - im mproving linnk robustnesss; - “simple” “ scheema for impllementation.

Figu ure 2.22. Tran nsmission bea am/beam formation

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h11

+

h21

h12 h22

TX

+

RX

Figure 2.23. Transmissions between transmitters and receivers

Multiple entry and exit system

H12

H21

H11

H H =  11  H 21

H12  H 22 

H22 Cdiversity = log2det[I +(PT/2σ2 )·H H †]= W hen the λi are the eigenvalues λi of HHa

P P     = log2 1 + T 2 λ1  + log2 1 + T 2 λ2   2σ   2σ  Transmitter

λ1 λ2

Reciever

m=min(nr , nt ) parallel channels (Equal power allocated to each pipe) Figure 2.24. Multiple inputs and outputs

We can consider three main categories: – MIMO spatial diversity: with MIMO, a single message is transmitted simultaneously on different antennae at transmission. The signals received on each of the antennae are then re-sent in-phase and summed coherently. A simplified version only uses the signal from a single antenna, the one that

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receives the best signal at a given instant (polarized antennae). This makes it possible to increase the transmission’s SNR (thanks to diversity gain). For this technique to be effective, MIMO subchannels must be independent of one another. – MIMO spatial multiplexing: each message is divided into submessages. These submessages are transmitted simultaneously on each of the transmission antennae. The signals received on the reception antennae are reassembled to reform the original message. As for MIMO diversity, the propagation subchannels should be decorrelated. MIMO multiplexing makes it possible to increase transmission flows (thanks to multiplexing gain). MIMO diversity and multiplexing techniques can be applied jointly, e.g. for a 5 × 5 MIMO system (i.e. five transmission and five reception antennae), a 2 × 2 MIMO subsystem can be configured to perform multiplexing and create a 3 × 3 MIMO to achieve MIMO diversity. – MIMO – beamforming (spatial filtering, lane forming): we use the MIMO antenna network to orient and manage the radiowave beam (amplitude and phase of the beam). We can then create constructive/ destructive lobes and optimize a transmission between transmitter and target. Beamforming (spatial filtering) techniques make it possible both to extend radio coverage (of a base station or an access point, for example) and to reduce interferences between users, and ambient electromagnetic pollution (by targeting the receiver as much as possible). We distinguish two MIMO variants depending on the number of users receiving data simultaneously on the same carriers: – The SU-MIMO (Single User), certainly the most used, makes it possible to send data by different antennae to a single user, at a given instant; it makes it necessary to have several antennae in each receiver. This mode makes it possible to reach a unitary flow peak higher than MU-MIMO. – MU-MIMO (Multi User) makes it possible to share the radio flow and transmit data flows to two (or more) users, with four antennae in transmission and two antennae in each receiver. It uses the “spatial multiplexing” mode and makes it possible to increase the spectral efficiency of the radio cell (the global flow) without imposing a high number of antennae for each terminal. When the transmitter knows the channel, the maximum possible capacity is given by the water-filling algorithm.

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2.4.3. Uses MIMO technology is mainly used in standards: Wi-Fi (IEEE 802.11n and 802.11ac), WiMax (IEEE 802.16e), and mobile telephony3 HSPA+, LTE and LTE Advanced standards. Freebox v5 HD, ADSL, and FAI Free use the MIMO technique to transmit the HD video (replaced by a CPL link in January 2008) between its two cases, as well as for the computer link. The Orange Livebox also uses Wi-Fi in MIMO 4 × 4 mode to connect the TV box. Having made an excursion into the domaine of digital modulation, we will make some succint disgressions in afferent signal processing, as this mathematical aspect is indipensable.

3 Source: https://fr.wikipedia.org/wiki/Téléphonie_mobile.

3 Signal Proc cessing g: Samp pling

3.1. Z-ttransforms s The z-transform m is the brooadest conceept for convverting discrrete-time m is the brroadest conccept for coonverting series. The Laplacce transform continuoous-time proocesses. For example, thee Laplace traanform enabbles us to transforrm a differenntial equationn, as well as its problemss of initial vaalues and correspoonding limitss, into a spacce in which the equationn can be solvved using ordinaryy algebra. Sppace commuutation for traansforming calculation c pproblems into alggebraic operrations on trransforms iss called opeerational calcculation. Laplacee and z-transfforms are thee most signifficant methodds for this. 3.1.1. Transforms T s

Figure 3.1. Laplace and z-transforms z

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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The z-transforms, f*(t), is the Lapalace transforms of a sampled f’(t) function. ∗(

)=

∗(

)=

( ).

) , f(t) = 0 for t < 0

= 0∞ ( − (

). ( −

)

Let us take the Laplace transform: X (or F) ( )=

∗(

)=

∗(

)=

). ( −

(

(

(

)

). ( −

). ( −

)

)

.

( −

)

Change in variable is given as: Z = esTs X*(s) = becomes X(Z), such that: ( ) = [ ( )] =

(

).

3.1.2. Inverse z-transform The inverse z-transform can be derived using the Cauchy integal theorem. Start with the z-transform.

Multiply both sides by zK-1 and integrate with a contour integral for which the contour integration endorses the origin and is found entirely in X(z)’s convergence region.

Signal Pro ocessing: Samp pling

The above equatiion is the invverse z-transfform.

Figure 3.2. Z-transforms s’ properties

Z Figure 3.3. Z-transforms; convolutions

185

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Figu ure 3.4. Links between z-tra ansforms, Fou urier z-transforrms and Lapla ace ztransforrms. For a colo or version of th his figure, see e www.iste.co.uk/gontrand/d digital.zip

Figure 3.5. Con ntinuous/discre ete; temporal//frequency. Fo or a color version of thiss figure, see www.iste.co.uk w k/gontrand/dig gital.zip

Signal Pro ocessing: Samp pling

187

Signnals: – Thhis is a physiccal representtation of the information:: - current, c volattage, luminouus flux, noisee, etc. – Infformation caan be linked to t various meedia: - sound, imagee, video, text,, etc. – Siggnals that aree achieved phhysically: - bounded b enerrgy, - bounded b ampplitude, - bounded b specctral frequenccy. – Siggnals can be classed in diifferent famiilies: - deterministic d signals (periiodic or not), - evolution e of f(t) f predictedd by a mathem matical model, - raandom signaals (stationaryy, order 1 or order 2, or not), n - sttatistical obsservations. 3.2. Ba asics of sig gnal proces ssing In the t followiing, we will w present, synopticallly, the baasics of signal processing. p W note thaat there are many bookks that disccuss this We domain.

Figure 3.6. The T signal processing chain n

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NOISE.–– Any random m disruptionn phenomeno on is called “nnoise”. The information can notably be disrupted d: – during acquisittion; – aloong the transm mission channnel; – at reception. r

Figure 3.7. Chain noise

In geeneral, naturaal signals aree represented d by continuoous-time funnctions.

Figure 3.8. Analog/contin nuous signal

Conttinuous-timee functions can c be descrribed using only o one parrt of the signal, since s all real signals are bounded b in frequency. f In prractice, samppling consistts of allowin ng the signall to pass oveer a very brief insstant at regullar times periods.

Signal Pro ocessing: Samp pling

189

Figure 3.9. Discretize ed signal

Figu ure 3.10. Sam mpling

The convolutionn function of o a linear time-invarian t nt system (L LTIS) is defined according too its impulse response h(tt).

Figure 3.1 11. Notion of convolution c

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Figu ure 3.12. Notio on of “Dirac” signal s convolu utions

Figure 3.13. 3 Multipliccation of a sign nal using a Dirrac comb

3.3. Re eal discrete ezation pro ocessing 3.3.1. Real R discre etization co omb

Figure e 3.14. Real sa ampling

Signal Pro ocessing: Samp pling

191

We approach thhe Dirac coomb when t comes close to 0, whhile still maintainning productt Aτ = 1. The average valuue of the signnal g(t) is Ao o = Aτ/Ts. R samplled signal 3.3.2. Real

Figure 3.15. Sample ed signal

3.3.3. Blocked, B sa ampled sig gnal Therre is a sustainned samplingg, as shown in i Figure 3.116.

Figure 3.16 6. Blocked, sam mpled signal

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3.3.4. Model M of re eal sampled d signals

F Figure 3.17. Relationship R between convo olution and imp pulse responsse

We should consider the imppulse response of the forrmatted circuuit. This amountss to saying thhat the signaal can be mod deled by a coonvolution prroduct: y = {S(t) ⋅ g(t)} y(t) g * h(t) 3.3.5. Uniform U qu uantifying

Fig gure 3.18. Sig gnal digitization n and quantifyying

3.3.6. Signal S quan ntification step: roun nding If {nn - (/2)} < S(t) < {n  + (/2)}, then Sq(t) = n. NOTE.–– All values inn the range are a representted by the sam me quantizedd level.

Signal Processing: Sampling

193

3.3.7. Signal quantification step: troncature If n < S(t) < (n + 1) , then Sq(t) = n. 3.3.8. Quantification solution This is the quantification step. It is linked to the binary coding that follows quantification. EXAMPLE.– If S(t) is a variant voltage of 0 to 1 V and the digital signal is represented by a binary word of 8 bits, this can take 28 possible values. → 256 quantification levels. Hence a solution of:  = 1 / (256 - 1) = 3.92 mV General case:  = V / (2n - 1) 3.3.9. Additive white Gaussian noise (AWGN): a simple but effective model We will indicate a simple model showing the way in which noise affects the reception of a signal sent in a channel and processed by the receiver. In this model, the noise is: Additive: take a given value y{k] at the kth instance of sampling. The receiver interprets it as the sum of two components: the first is the silent component y0[k], i.e. the sampling value, which would have been received at the kth instance of sampling in the absence of noise, because of the input wave form that has passed through the channel in the presence of only the distortion, and the second is the w[k] component, independent of noise of the input wave form. We can thus write: y[k] = y0[k] + w[k]. In the absence of distortion, and what we will assume here, y0[k] will be one or the other: V0 or V1. 3.3.10. Quantification error and quantification noise The value Sq(t), in the case of quantification by rounding, can be distanced from the real value of a maximum quantity (/2).

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The minimum quuantification error: ε = /2.  ncy noise Fs F and an am mplitude The quantificatioon introducees a frequen noise: /2.  3.3.11. In practice e, sample and a hold an nd CAN

Figure 3.19. “Sample and d hold”/ADC (a analog-to-digittal converter)

nd synchronizzes the converter. The signal g(t) coommands thee sampler an

Figure 3.20. 3 Fourier transform t

Signal Pro ocessing: Samp pling

195

3.3.12. Spectra of periodic signals s S = S(t + nT S(t) T) The signal can be broken down into o a discrete sum of siinusoidal functionns.

Figure 3.21. Fourier co oefficients

Figure 3.22 2. Spectrums: module/ampliitude and argu ument/phase

3.3.13. Non-perio odic signal spectrums s The signal can be broken dow wn by consid dering a periiodic functioon whose period T tends towaard infinity. Condditions: – S(tt) is boundedd in time;

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– inttegral betweeen -∞ and +∞ ∞ is finite; – finnite number of o maximum ms and discon ntinuities. Limiit of cn:

S(t) is therefore broken downn into Fourieer integrals. The spectrall density is a continuous funcction of ν.

F Figure 3.23. Power P spectral density (PSD D)

Figure 3.24. Results/c conclusion

Signal Pro ocessing: Samp pling

197

3.3.14. PSD versu us delay ation Transla

Conssequence: deelay theorem m: the modulee spectral dennsity is not m modified when thhe signal is delayed: ρ = ρτ(ν) ρ(ν) Dirac fu unction

3.3.15. FT of a prroduct: the Planchere el theorem i the convoluution producct of the FT and a reciprocaally, A prroduct’s FT is

FT of o a sampled signal. The signal S*(t) is first consiidered as a direct d productt of S(t) by tthe comb functionn: S*(t) = S(tt) . PgnTs (t).

Finaally, we have the transform m of the sam mpled signal:

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3.3.16. Periodic signal s before samplin ng

Figure 3.25. Sp pectrum of con ntinuous signa als

ed signals 3.3.17. Spectrum of sample

Figu ure 3.26. Spectrum of samp pled period sig gnals

Signal Pro ocessing: Samp pling

199

Figure 3.2 27. Some specctrums: period dic signal beforre sampling

Figure 3.28. Spectru um of an aperiiodic sampled window

3.3.18. Condition ns for samp pling freque ency Specctrum boundeed in frequenncy → νmax Twoo cases should be consideered: νmax > Fs / 2 νmax < Fs / 2 – Inn the first innstance (Fs > 2 νmax), the t represenntative motiff ρ(ν) is distinguuished withouut “distoritioon” in ρ*(ν) by b repeating all the Fs. – Inn the secondd case (Fs < 2 νmax), the t representtative motiff ρ(ν) is distortedd in ρ*(ν): reecovery in Fe/2. F

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SHANNON’S THEOREM.– To be able to reform a signal from its samples, the sampling frequency must be at least equal to 2 vmax (2 points, at least “taken on the smallest period Tmin”, so 4 points for 2Tmin, 6 points for 3Tmin, 8 points for 4Tmin...). 3.4. Coding techniques (summary) Entropy and capacity are given as: n

n

i =1

i =1

H ( X ) = E ( h( X )) =  p i . log(1 p i ) = − p i . log( p i )

R = H max ( X ) − H ( X ) n

m

H ( X , Y ) = −   p ( x i , y j ). log( p ( x i , y j )) i =1 j =1 n

m

H ( X / Y ) = −   p ( x i , y j ). log( p ( x i / y j )) i =1 j =1

n

m

I ( X ; Y ) =  p( xi , y j ). log( i =1 j =1

p( xi , y j ) p( xi ). p( y j )

)

I ( X ; Y ) = H ( X ) + H (Y ) − H ( X , Y ) I ( X ; Y ) = H ( X ) − H ( X / Y ) = H (Y ) − H (Y / X )

C = Max( I ( X ;Y ))

ηc =

I ( X ;Y ) C

NOTE.– A Hamming code is a linear block code in which the columns of the control matrix H are binary representations of words from 1 to 2n - 1 (n ∈ N*).

Signal Processing: Sampling

Source coding N

l =  p ( s i ).l i i =1

η=

H (S ) l . log D

Channel coding Code group (Hamming)

v = i.G  z1  z = H .v ′ T =  :   z m  t

G.H = 0

Cyclical coding k bits of information. n = size of code word degree of g(x): m = n-k v( x) = i(x) × g( x) or

v( x) = c( x) + x m .i( x)

 x m .i ( x)   c( x) = Reste   g ( x) 

Table 3.1. Hamming code

201

4 A Little on Ass sociated d Hardw ware

4.1. Vo oltage-conttrolled osciillator We present, p heraafter, a synopptic of a volltage-controlled oscillator, kernel of the so-called s phaase-locked looop, with its associate noiise phase.

Figure 4.1. Voltage-contro V olled oscillatorr

For color versions of thee figures in this chapter, see ww ww.iste.co.uk/ggontrand/digitall.zip.

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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Figure 4.2. Power: harmonic rejection

Figure 4.3. Pushing/pulling of a VCO

A Little on Associated Hardw ware

Figurre 4.4. Phase e noise

Figurre 4.5. Phase noise caused by parasite siignals. L stands forr local oscillato or We note that LO

205

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Figu ure 4.6. Blockk schema of a phase-locked d loop

Figure 4.7. Diagram circuit of a VCO

A Little on Associated Hardw ware

Figure 4.8. Adaptation n at output

Figure 4.9. Adap ptation and lay yout of an ante enna

207

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Fiigure 4.10. Im mportant param meters of a VC CO

Figure 4.11. Sketch of o layout

A Little on Associated Hardware

209

4.2. Impulse sensitivity function In this section, we use a linear model varying in time to analyze the sensitivity of a VCO (voltage-controlled oscillator) to substrate disturbances. The method is based on calculating the impulse sensitivity function (ISF), which represents the excess phase after the application of a disturbance impulse to an oscillator circuit. By varying the appearance of the disruption impulse event during an oscillator period, we can reach the ISF function: Γ(τ). Note that we do not consider the amplitude offset as it disappears with time (at low injection); however, the phase shift is preserved. The disruption impulse can be a current impulse in a capacitative node (injected charge), or a voltage pulse on an inductive node. The ISF function Γ(τ ) has no dimension and has the same period as that of the oscillator (or we can consider that Γ (ωcτ ) has a period of 2π. For greater clarity, we will consider the pseudo-ISF function Γ ϕ (τ ) in the following part of this section. Γϕ (τ ) has the radian dimension (A.s) or radian (Vs) depending on the type of disruption (current or voltage). Finally, we can write the excess phase for an impulse response: hϕ (t ,τ ) = Γ ϕ (ω cτ ) ⋅ u (t − τ )

[4.1]

where u ( t ) is a step function. In other words, the function Γ ϕ (τ ) is a direct representation of the excess phase (offset), normalized by the injected charge (for the impulse current). Because of its periodicity, Γ ϕ (τ ) can be developed in a Fourier series as follows: Γ ϕ (ω cτ ) =

c0 ∞ +  c n ⋅ cos (nω cτ + θ n ) 2 n =1

[4.2]

Considering the harmonic disruption (current or voltage) defined by its size and angular frequency ω m (frequency: f m ):

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p (t ) = A ⋅ cos (ω m t ) ⋅ u (t − t 0 )

[4.3]

Because of the linear time-varying system, offset is obtained according to the expression: t

t

−∞

t0

ϕ (t ) =  Γϕ (ω cτ ) ⋅ p(τ ) ⋅ dτ =  Γϕ (ω cτ ) ⋅ A ⋅ cos(ω mτ ) ⋅ dτ

[4.4a]

This offset can then be written as follows:

ϕ (t ) = A ⋅

∞ c sin [(nω ± ω )t + θ ] c0 sin (ω m t ) c m n + ϕ 0 (t 0 ) + A⋅  n 2ω m 2(nω c ± ω m ) 1

[4.4b]

In [4.4], only the term calculated with ω m = nω c + Δω (with Δω restart:with(plots):with(LinearAlgebra): > assume(n,posint):assume(N,posint):assume(Ym, positive): assume(T,positive): A3.3.2. Analytical expression of Fourier series General term of the series of the Fourier series > u:=n->a(n)*cos(2*n*Pi/T*t)+b(n)*sin(2*n*Pi/T*t);

 2nπ t   2nπ t  u : n → a (n) cos   + b(n) sin    T   T  Expression of a truncated Fourier series at N terms > sdf:=N->a0+Sum(u(n),n=1..N);

Appendix 3

N

sdf := N → a 0 +  u ( n) n =1

A3.3.3. Expression of a Fourier series > fourier_series:=sdf('infinity'); ∞  2n  π t  fourier _ serie := a0 +  ( a(n  ) cos    T  n  =1  2n ~ π t   + b(n ~) sin    T ~ 

A3.3.4. General term of the series of the Fourier series > us:=n->s(n)*cos(2*n*Pi/T*t-alpha(n));

 2nπ t  us := n → s (n) cos  − α ( n)   T  A3.3.5. Expression of a truncated Fourier series at N terms > sdfs:=N->s0+Sum(us(n),n=1..N); N

sdfs := N → s0 +  us (n) n =1

A3.3.6. Expression of a Fourier series > fourier_series:=sdfs('infinity'); ∞  2n  π t  fourier _ serie := s0 +  ( s (n  ) cos  − + α (n ~)  T   n  =1

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A3.3.7. Expression of coefficients of the function’s Fourier series f(t) > coef_a0:= a0=int(f(t),t=-T/2..T/2)/T; > coef_an:=an=2/T* int(f(t)*cos(2*n*Pi/T*t),t=T/2..T/2); > coef_bn:=bn=2/T* int(f(t)*sin(2*n*Pi/T*t),t=T/2..T/2); 1

coef _ a0 := a0 =



2

T~

f (t )dt

1 − T~ 2

T~

 2T ~   2n ~ π t    2  − 1T ~ f (t ) cos   dt  2  T ~    coef _ an := an =  T~ 1

 2T ~   2n ~ π t    2  − 1T ~ f (t ) sin   dt  2  T ~    coef _ bn := bn =  T~ 1

Choice of function f(t) to be processed (triangular, rectangular...) Below are some examples: you can choose one by removing the # sign at the beginning of the line that interests you. WARNING.– Leave only one active line (i.e., not starting with #). IMPORTANT.– The chosen function must be T-periodic. A rectified sine or cosine has a period divided by 2. The program is not written for this type of function. As given below, you have to define the functions between −T / 2 and T / 2.

Appendix 3

271

> Sig:=f(t)=piecewise(-T/2< t and t < 0, 4*Ym*(t+T/4)/T,0 < t and t < T/2, 4*Ym*(T/4t)/T):titre:=" Triangular signal ": > #Sig:=f(t)=piecewise(-T/2< t and t < 0,-Ym, 0 < t and t < T/2,Ym):titre:=" Symetric rectangular signal ": > #Sig:=f(t)=piecewise(-T/2< t and t < 0,0, 0 < t and t < T/2,Ym):titre:=" rectangular signal ": > #Sig:=f(t)=piecewise(-T/2< t and t < 0, t*2*Ym/T,0 < t and t < T/2, t*2*Ym/T):titre:=" Positive triangular signal ": > #Sig:= f(t) = piecewise(0 #Sig:= f(t) = piecewise(-T/2< t and t < T/2,exp(2*Pi*abs(t)/T)):titre:="Ugly signal ": > #Sig:= f(t) = piecewise(-T/2< t and t < T/2,abs(sin(Pi/T*t))):titre:=" Sine rectified":

Here, one selects arbitrarily the value 1 for the period of the function as well as for its amplitude. >plot(subs(Ym=1,T=1,rhs(Sig)), t=-1..1,title=titre,discont=true);

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The following line extends the function f(t) over several periods. > Signal:=unapply(Sig,t)(t-floor((2*t+T)/2/T)): > plot(subs(Ym=1,T=1,rhs(Signal)),t=2..2,title=titre,discont=true);

Appendix 3

273

Analytical calculation of the Fourier coefficients of the function f(t) chosen: let an and bn, or sn and phin. > a0:= eval(rhs(subs(Sig,coef_a0))); > a:=n -> eval(rhs(subs(Sig,coef_an))):a(n); > b:=n -> eval(rhs(subs(Sig,coef_bn))):b(n);

a0 := 0 −

4Ym ~ ( (−1)n ~ − 1) n ~2 π 2 0

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Numerical values of the an: > seq([evalf(subs(Ym=1,n=k,a(n)))],k=1..15);

Numerical values of the bn: > seq([evalf(subs(Ym=1,n=k,b(n)))],k=1..15);

A3.4. Plot of ab and bn according to n The an is plotted in blue, and the bn is plotted in red. COMMENT ON FIGURES A3.3 AND A3.4.– The writing of this graph is not obvious. For the purposes of the analytic calculation of a(n) and b(n), we must declare n as integer, but this obviously poses a problem for Maple in making a classical plot. What is done here is to draw a point by point. It is necessary to generate a sequence of points of coordinates (k, a (k)) with k a non-integer variable but to which one will take only whole values. >pointplot({seq([k,evalf(subs(Ym=1,n=k,a(n)))],k=1. .15)},color=blue,title="an: function of n",labels=['n','an'], symbolsize=30); > pointplot({seq([k,evalf(subs(Ym=1,n=k,b(n)))],k=1.. 15)},color=red,title="bn: function de n",labels=['n','bn'], symbolsize=30);

Appendix 3

Figure A3.3. an: function of n. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Figure A3.4. bn: function of n. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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> s0:=a0; > s:=n->(a(n)^2+b(n)^2)^(1/2):s(n); >#alpha:= n>`if`(s(n)=0,0,`if`(a(n)=0,Pi/2*sign(b(n)),arctan(b(n)/a (n)))); # this line creates an error in Maple 16

> #alpha:=n->`if`(a(n)=0,Pi/2,arctan(b(n)/a(n))); >alpha:= n->``if`(s(n)=0,0,`if`(a(n)=0,Pi/2*b(n)/abs(b(n)), arctan(b(n)/(a(n)+1e-6))));

s0 := 0

16

Ym ~ 2 ( ( −1) n ~ − 1)

2

n ~4 π 4

 α := n → if  s(n) = 0,0, if  

 1 π b( n)    a(n) = 0,  ,   2 ( ) b n  

  b( n) arctan   a(n) + 0.000001      >

seq([evalf(subs(Ym=1,n=k,s(n)))],k=1..15);

Appendix 3

277

Numerical values of sn: >seq([evalf(subs(Ym=1,n=k,alpha(n)))],k=1..15);

A3.5. Plot of sn and alphan according to n The sn is plotted in blue, the alphan in red. >pointplot({seq([k,evalf(subs(Ym=1,n=k,s(n)))],k=1. .15)}, font=[COURIER,14],color=blue,labels=['n','sn'],labe lfont= ['COURIER',14],title="sn, function de n",titlefont=['COURIER',14], symbolsize=30); > pointplot({seq([k,evalf(subs(Ym=1,n=k,alpha(n)))],k =1..15)}, font=[COURIER,14],color=blue,labels=['n','alphan'], labelfont=['COURIER',14],title="alphan : function of n",titlefont=['COURIER',14], symbolsize=30);

Figure A3.5. sn: function of n. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

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Figure A3.6. alphan: function of n. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

A3.6. Graphical representation of the signal reconstitution from the Fourier series Change here the number of harmonics for the Fourier series: > Nmax:=30: > partial_SF:=sdf(N);

partial _ SF :=

N~



n ~ =1





 2n ~ π t  4Ym ~ ( (−1) n ~ − 1) cos    T~  − n ~2π 2 > p1:=animate(plot,[subs(Ym=1,T=1,sdf(N)),t=2..2,color=blue,legend="Signal synthesized"],N=1..Nmax,frames=Nmax): > p2:=plot(subs(Ym=1,T=1,rhs(Signal)),t=2..2,Reconstituted signal=1.2..1.2,color=red,legend="Original signal"): > display(p2,p1,axes=BOXED);

Appendix 3

Figure A3.7. Reconsituted signal

>N1:=plot(subs(Ym=1,T=1,N=1,sdf(N)),t=1..1,color=red,legend="Signal synthetized n=1"): >N2:=plot(subs(Ym=1,T=1,N=2,sdf(N)),t=1..1,color=magenta,legend="Signal synthetized n=1..2"): >N3:=plot(subs(Ym=1,T=1,N=3,sdf(N)),t=1..1,color=green,legend="Signal synthetized n=1..3"): >N5:=plot(subs(Ym=1,T=1,N=5,sdf(N)),t=1..1,color=blue,legend="Signal synthetized n=1..5"): >N7:=plot(subs(Ym=1,T=1,N=7,sdf(N)),t=1..1,color=orange,legend="Signal synthetized n=1..7"):

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>N9:=plot(subs(Ym=1,T=1,N=9,sdf(N)),t=1..1,color=cyan,legend="Signal synthetized n=1..9"): >N11:=plot(subs(Ym=1,T=1,N=11,sdf(N)),t=1..1,color=gold,legend="Signal synthetized n=1..11"): > N13:=plot(subs(Ym=1,T=1,N=13,sdf(N)),t=1..1,color=violet,legend="Signal synthetized n=1..13"): > Nref:=plot(subs(Ym=1,T=1,rhs(Signal)),t=1..1,color=black,legend="Signal original"): > display(Nref,N1,N2,N3,N5,N7,N9,N11,N13,axes=BOXED);

Figure A3.8. Signal synthetized with different numbers of harmonics. For a color version of this figure, see www.iste.co.uk/gontrand/digital.zip

Appendix 3

281

> N:='N': Scroll through the animation above to visualize the construction of the series, harmonic by harmonic. A3.7. Manual definition of Fourier coefficients (amplitude and phase) > restart: > data := {s0=0, s1=0.9, phi1=0, s3=0.1, phi3=0, s5=0.07, phi5=0, s7=0.03, phi7=0, s9=0, phi9=0, omega=2*3.1415927*400};

data := {ω = 2513.274160, φ1 = 0, φ 3 = 0, φ 5 = 0, φ 7 = 0, φ 9 = 0, s0 = 0, s1 = 0.9, s3 = 0.1, s5 = 0.07, s7 = 0.03, s9 = 0} A3.7.1. Calculation of f(t) over three periods >f:=s0+s1*cos(omega*t-phi1)+s3*cos(omega*3*tphi3)+s5*cos(omega*5*t-phi5)+s7*cos(omega*7*tphi7)+s9*cos(omega*9*t-phi9);

f := s0 + s1 cos(ω t − φ 1) + s3 cos(3 ω t − φ 3) + s5 cos(5 ω t − φ 5) + s7 cos(7 ω t − φ7 ) + s9 cos(9 ω t − φ 9) > plot(subs(data,f), t = -0..0.0075, axes=BOXED, labels= ['t', 'signal_synthetized']);

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Figure A3.9. A Synthesiz zed signal

The Faast Fourier Tra ansform The FFT F uses the formalism fo of thhe DFT (Disccrete Fourier Transform) T coomplex FFT method by J.W W. Cooley and d J.W. Tukey (1965) ( Deccompositions by alternatinng the signal by N = 2 m points in thhe time domainn in N signalls of 1 point.. At each tim me, the sequennce of even ssamples forms a new signal and a the sequeence of odd saamples forms the other new w signal. There are log2N suuccessive decompositions in i all (16→44, 512→7, 40096→12 etc…).. Theere is anotherr method of decomposition d n: this amounnts to reorderring the samplees according to t an index whose w bits are inverted com mpared to the starting index. To find the TF spectrum off each 1-point signal, this stage is trivvial: the spectruum of a 1-point signal is eqqual to itself. An equally trivial demonsstration: numbeers in normal order o and num mbers in reversse order.

Append dix 3

283

To combine N frequency spectrums at 1 point each in i such a waay as to “undo”” the decomposition in thee time domain n at the first stage. It is thherefore necessaary in the frrequency dom main to makee the correspoonding movee to the followiing one in thee time domain:

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Dig gital Communication Technique es

Lett us take one of o the stages of o this recombination:

Thiis recombinatiion in the tim me domain can n be made in the t following way by dilutingg the two signnals to combinne with 0, offsset and additioon:

B A3.1. The Box e fast Fourier transform t (FFT)

The time-frequeency corresppondence of this operatiing sequencce is the followinng:

Append dix 3

The block diagraam of the lastt stage or syn nthesis stagee is therefore:

1

1

1!

1 2!



285

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Figure A3.10. FFT flowchart

Here is the program in Maple that makes the Fourier transform using this method. A3.8. FFT with Matlab restart; with(inttrans); # Definition of the parameters used for the functions f:=4; N:=512; a:=0;

Appendix 3

287

b:=2*Pi; inter:=b-a; Te:=evalf(inter/N); # Definition of the Dirac Di:=array(1..N); for i to N do Di[i]:= exp(-(a+(i-1)*Te)*(a+(i-1)*Te)); end do: # Definition of the cosine C:=array(1..N); for i to N do C[i]:= cos(f*(a+(i-1)*Te)); end do: # Definition of the sine S:=array(1..N); for i to N do S[i]:= sin(f*(a+i*Te)); end do: with(plots): with(DiscreteTransforms); # Definition of the function to be transformed: we can choose between sine (S), cosine (C) or Dirac (Di) Z:=Vector(N,i->evalf(S[i]),datatype=complex[8]); Z1:=FourierTransform(Z); Z2:=InverseFourierTransform(Z1);

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# We take the amplitude of the transform Z11:=array(1..N); for i from 1 to N do Z11[i]:=Re(sqrt(Z[i]*conjugate(Z[i]))): end do: Z22:=array(1..N); for i from 1 to N do Z22[i]:=Re(Z2[i]): end do: # Plot of the original and transformed functions pointplot({seq([i,Re(Z[i])],i=1..N)},labels=["abscisse", "ordonnée"],labeldirections=[HORIZONTAL,VERTICAL]); legend=["Fonction de départ"]; pointplot({seq([i,Re(Z1[i])],i=1..N)},labels=["abscisse", "ordonnée"],labeldirections=[HORIZONTAL,VERTICAL]); legend=["Transformée de Fourier de la fonction de départ"]; PHASE:=array(1..N): for i from 1 to N do PHASE[i]:=argument(Z1[i]): end do: PHAS:=array(1..N): for i from 1 to N do PHAS[i]:=arctan(Im(Z1[i])/Re(Z1[i])): end do: # Plot of the phases pointplot({seq([i,(PHASE[i])],i=1..N)}); pointplot({seq([i,(PHAS[i])],i=1..N)});

Appendix 3

Figure A3.11. M = 8. FFT by decimation in time

Zeros padding:

yn = xn1{o.. N −1} [n] ^

Xk =

1 ^ k  Y  N  NTe 

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Figure A3.12. Some Fourier transforms

References

Agilent Technologies. (2000). Agilent AN 1298: Digital Modulation in Communication Systems – An Introduction. Agilent Technologies, Santa Clara, CA. Akansu, A.N., Duhamel, P., Lin, X., de Courville, M. (1998). Orthogonal transmultiplexers in communications: A review. IEEE Trans. Signal Process., 4, 979–995. Debbah, M. (2002). OFDM. Report, Supélec and Gif sur Yvette, Australia. Degauque, M. (1998). Transmission numérique sur porteuse : ASK, FSK et PSK. Report, CNAM, Bordeaux. Hajimiri, A., Lee, T.H. (1998). A general theory of phase noise in electrical oscillators. IEEE J. Solid-State Circuits, 33(2), 179–194. Girard, P.R., Clarysse, P., Pujol, R., Gouttte, R., Delachartre, P. Hypequaternions: A new tool for physics. Adv. Appl. Clifford Algebras, 28(3), 1–14. Kadionik, P. (2000). Bases de transmission numériques : les modulations numériques. Report, ENSEIRB, Bordeaux. Shannon, C.E. (1949). Communication in the presence of noise. Proc. IRE, 37(1), 10–21. Sharma, D.-K., Mishra, A., Saxena, R. (2010). Analog and digital modulation techniques: An overview. Int. J. Comput. Sci. Commun. Technol., 3(1), 551–561. Viswanathan, M. (2019). Digital Modulations Using MATLAB: Build Simulation Models from Scratch. Independently published. Xiong, F. (2000). Digital Modulation Techniques. Artech House, Boston, MA.

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

Index

A ADSL (asymmetric digital subscriber line), 140, 147, 158, 182 APSK (amplitude and phase-shift keying), 112 ASK (amplitude-shift keying), 14, 17, 100–102, 104, 108, 111–113, 255, 256, 258, 261 automatic gain control, 60 AWGN (added white Gaussian noise), 54, 60, 81–88, 97, 107, 113, 114, 116, 159, 163, 165, 193 B baseband, 1, 3, 15, 17, 20, 36, 49, 84, 85, 100, 101, 103, 106–108, 113, 124, 143, 258 BiCMOS (bipolar complementary metal–oxide–semiconductor), 213 BPSK (binary phase-shift keying), 22, 27, 31–33, 36, 37, 40, 42, 53, 102, 105, 106, 113–115, 123, 140, 151, 153, 232, 262

envelope, 30, 31, 96, 101, 103, 130, 236, 238, 240, 245 EVM (error vector module), 63, 81, 123–125, 127, 128 eye diagram, 21– 23 F, I FFT (fast Fourier transform), 54, 57, 60, 127, 137, 139, 143, 145, 149, 151, 157, 163, 166, 217 282, 284, 286, 289 imbalance, 91–93, 106–108, 126, 133–136 ISF (impulse sensitivity function), 209–211, 215 M, O MATLAB, 45, 50, 53–55, 60, 61, 63, 66, 67, 74, 79, 81, 84, 86, 89, 90, 106, 113, 132, 133, 159, 160, 162, 163, 232, 286 oscillator, 53, 91, 104, 106, 203, 209, 210, 213–215, 217, 219, 221, 223, 225, 235, 238, 241, 249, 250

D, E design, 4, 54, 120, 127, 138, 177, 219, 231, 232

Digital Communication Techniques, First Edition. Christian Gontrand. © ISTE Ltd 2020. Published by ISTE Ltd and John Wiley & Sons, Inc.

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P, R

S, T, V

phase noise, 63, 64, 66, 71, 72, 74–78, 126, 127, 128, 205, 210, 216, 228, 230 shift, 14, 22, 28, 31, 38, 40, 53, 59, 94, 95, 102, 112, 209, 214, 237 PLL (phase-locked loop), 54, 61, 206, 219–222, 226–229, 238, 241, 251 PSD (power spectral density), 16, 39, 40, 45, 65, 84, 100, 113, 120, 122, 140, 177, 197, 217 raised cosine, 39, 41, 55, 57, 58, 60, 61, 63, 66, 67, 72, 259 RF (radiofrequency), 2, 5, 16, 24, 31, 50, 118, 120, 121, 123, 128, 154, 156, 162, 174, 177, 219, 235

Shannon, 9, 141, 200 trellis, 179 VCO (voltage-controlled oscillator), 204, 209, 219, 221–225, 227, 235, 238, 241

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