Digital Communications 2

Humans have always used communication systems: first with smoke Ruyet Le Didier Pischella Mylène signals, and then with the telegraph and telephone. These systems and their technological advances have changed our lifestyle profoundly. Digital Nowadays, smartphones enable us to make calls, watch videos and use social networks, and with this comes the emergence of the connected man and the wider applications of smart objects. All current and future communication systems rely on a digital communication chain that Communications 2 consists of a source and a destination separated by a transmission channel, which may be a portion of a cable, an optical fiber, a wireless mobile or satellite channel. Whichever channel, the processing blocks implemented in the communication chain have the same basis. Digital Modulations This second volume of Digital Communications concerns the blocks located after channel coding in the communication chain. The authors present baseband and sine waveform transmissions and the different steps required at the receiver to perform detection, namely 2 Communications Digital synchronization and channel estimation. Two variants of these blocks used in current and future systems, multi-carrier modulations and coded modulations, are also detailed. Mylène Pischella and Didier Le Ruyet

Mylène Pischella is Associate Professor in Telecommunications at CNAM in Paris, France. Her research interests include resource allocation in multi-cellular networks, and cooperative and relay-assisted networks. Didier Le Ruyet is Professor of Signal Processing for Communications at CNAM in Paris, France. His research interests include multi-user cooperative communications, detection algorithms for wireless communication, multicarrier modulation and cognitive radio.

Z(7ib8e8-CBIEGI( www.iste.co.uk

Digital Communications 2

Series Editor Pierre-Noël Favennec

Digital Communications 2

Digital Modulations

Mylène Pischella Didier Le Ruyet

First published 2015 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 John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com

© ISTE Ltd 2015 The rights of Mylène Pischella and Didier Le Ruyet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2015946706

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-846-8 Contents

Preface ...... xi

List of Acronyms ...... xiii

Notations ...... xvii

Introduction ...... xix

Chapter 1. Background ...... 1 1.1. Introduction ...... 1 1.2. Common operations and functions ...... 1 1.2.1. Convolution ...... 1 1.2.2. Scalar product ...... 2 1.2.3. Dirac function, Dirac impulse and Kronecker’s symbol ... 2 1.2.4. Step function ...... 3 1.2.5. Rectangular function ...... 3 1.3. Common transforms ...... 3 1.3.1. Fourier transform ...... 3 1.3.2. The z transform ...... 6 1.4. Probability background ...... 6 1.4.1. Discrete random variables ...... 7 1.4.2. Continuous random variables ...... 9 1.4.3. Jensen’s inequality ...... 9 1.4.4. Random signals ...... 10 1.5. Background on digital signal processing ...... 13 vi Digital Communications 2

1.5.1. Sampling ...... 13 1.5.2. Discrete, linear and time-invariant systems ...... 14 1.5.3. Finite impulse response filters ...... 17 1.5.4. Infinite impulse response filters ...... 17

Chapter 2. Baseband Transmissions ...... 19 2.1. Introduction ...... 19 2.2. Line codes ...... 20 2.2.1. Non-return to zero (NRZ) code ...... 20 2.2.2. Unipolar return-to-zero (RZ) code ...... 23 2.2.3. Bipolar return-to-zero (RZ) code ...... 25 2.2.4. Manchester code ...... 25 2.2.5. Alternate mark inversion code ...... 26 2.2.6. Miller code ...... 28 2.2.7. Non-return to zero inverted (NRZI) ...... 31 2.2.8. Multi level transmit 3 (MLT-3) code ...... 32 2.2.9. RLL(d,k) codes ...... 33 2.2.10. M-ary NRZ code ...... 35 2.3. Additive white Gaussian noise channel ...... 36 2.4. Optimum reception on the additive white Gaussian noise channel ...... 38 2.4.1. Introduction ...... 38 2.4.2. Modulator’s block diagram ...... 39 2.4.3. Optimum receiver for the additive white Gaussian noise channel ...... 44 2.4.4. Evaluation of the bit error rate for the binary NRZ signal on the additive white Gaussian noise channel ...... 52 2.5. Nyquist criterion ...... 60 2.5.1. Introduction ...... 60 2.5.2. Transmission channel ...... 61 2.5.3. Eye diagram ...... 62 2.5.4. Nyquist criterion ...... 63 2.5.5. Transmit and receive ﬁlters with matched ﬁlter ...... 66 2.6. Conclusion ...... 68 Contents vii

2.7. Exercises ...... 69 2.7.1. Exercise 1: power spectrum density of several line codes ...... 69 2.7.2. Exercise 2: Manchester code ...... 70 2.7.3. Exercise 3: study of a magnetic recording system ...... 70 2.7.4. Exercise 4: line code and erasure ...... 72 2.7.5. Exercise 5: 4 levels NRZ modulation ...... 73 2.7.6. Exercise 6: Gaussian transmit ﬁlter ...... 74 2.7.7. Exercise 7: Nyquist criterion ...... 75 2.7.8. Exercise 8: raised cosine ﬁlter ...... 76

Chapter 3. Digital Modulations on Sine Waveforms ...... 77 3.1. Introduction ...... 77 3.2. Passband transmission and equivalent baseband chain ...... 78 3.2.1. Narrowband signal ...... 78 3.2.2. Filtering of a narrowband signal in a passband channel ... 82 3.2.3. Complex order of a second-order stationary random process ...... 84 3.2.4. Synchronous detection ...... 90 3.3. Linear digital modulations on sine waveforms ...... 92 3.3.1. Main characteristics of linear digital modulations ...... 92 3.3.2. Parameters of an M-symbols modulation ...... 96 3.3.3. Amplitude shift keying ...... 98 3.3.4. Phase shift keying ...... 106 3.3.5. Quadrature amplitude modulations ...... 113 Eb 3.3.6. Link between N0 and signal-to-noise ratio depending on the power values ...... 119 3.3.7. Power spectrum density of regular modulations ...... 120 3.3.8. Conclusion ...... 121 3.4. Frequency shift keying ...... 122 3.4.1. Deﬁnitions ...... 122 3.4.2. Discontinuous-phase FSK ...... 124 3.4.3. Continuous-phase FSK ...... 126 3.4.4. Demodulation ...... 126 3.4.5. GMSK modulation ...... 130 3.4.6. Performances ...... 132 3.5. Conclusion ...... 135 viii Digital Communications 2

3.6. Exercises ...... 135 3.6.1. Exercise 1: constellations of 8-QAM ...... 135 3.6.2. Exercise 2: irregular ASK modulation ...... 136 3.6.3. Exercise 3: comparison of two PSK ...... 137 3.6.4. Exercise 4: comparison of QAM and PSK modulations ... 137 3.6.5. Exercise 5: comparison of 8-PSK and 8-QAM modulations ...... 138 3.6.6. Exercise 6: comparison of 2-FSK and 2-ASK modulations ...... 139 3.6.7. Exercise 7: comparison of 16-QAM and 16-FSK ...... 140

Chapter 4. Synchronization and Equalization ...... 141 4.1. Introduction ...... 141 4.2. Synchronization ...... 142 4.2.1. Frequency shift correction ...... 144 4.2.2. Time synchronization ...... 150 4.2.3. Channel estimate with training sequence ...... 153 4.2.4. Cramer–Rao’s bound ...... 154 4.3. Equalization ...... 157 4.3.1. Channel generating distortions ...... 158 4.3.2. Discrete representation of a channel with inter-symbol interference and preprocessing ...... 159 4.3.3. Linear equalization ...... 162 4.3.4. Decision-feedback equalization ...... 177 4.3.5. Maximum likelihood sequence estimator ...... 180 4.4. Conclusion ...... 186 4.5. Exercises ...... 187 4.5.1. Exercise 1: estimation of a constant signal from noisy observations ...... 187 4.5.2. Exercise 2: frequency shift correction ...... 188 4.5.3. Exercise 3: zero-forcing equalization ...... 188 4.5.4. Exercise 4: MMSE equalization ...... 189 4.5.5. Exercise 5: MMSE-DFE equalization ...... 190 4.5.6. Exercise 6: MLSE equalization with one shift register ... 190 4.5.7. Exercise 7: MLSE equalization with two shift registers ... 190 Contents ix

Chapter 5. Multi-carrier Modulations ...... 193 5.1. Introduction ...... 193 5.2. General principles of multi-carrier modulation ...... 196 5.2.1. Parallel transmission on subcarriers ...... 196 5.2.2. Non-overlapping multi-carrier modulations: FMT ...... 197 5.2.3. Overlapping multi-carrier modulations ...... 198 5.2.4. Chapter’s structure ...... 199 5.3. OFDM ...... 199 5.3.1. Transmission and reception in OFDM ...... 201 5.3.2. Cyclic preﬁx principle ...... 202 5.3.3. Optimum power allocation in OFDM ...... 209 5.3.4. PAPR ...... 215 5.3.5. Sensitivity to asynchronicity ...... 218 5.3.6. OFDM synchronization techniques ...... 219 5.4. FBMC/OQAM ...... 225 5.4.1. Principles of continuous-time FBMC/OQAM ...... 225 5.4.2. Discrete-time notations for FBMC/OQAM ...... 231 5.4.3. Prototype ﬁlter ...... 233 5.5. Conclusion ...... 236 5.6. Exercises ...... 236 5.6.1. Exercise 1 ...... 236 5.6.2. Exercise 2 ...... 237

Chapter 6. Coded Modulations ...... 239 6.1. Lattices ...... 240 6.1.1. Deﬁnitions ...... 240 6.1.2. Group properties of a lattice ...... 245 6.1.3. Lattice classiﬁcation ...... 248 6.1.4. Lattice performances on the additive white Gaussian noise channel ...... 251 6.2. Block-coded modulations ...... 255 6.2.1. Main algebraic constructions of lattices ...... 256 6.2.2. Construction of block-coded modulations ...... 259 6.3. Trellis-coded modulations ...... 270 6.3.1. Construction of trellis-coded modulations ...... 271 6.3.2. Decoding of trellis-coded modulations ...... 275 6.4. Conclusion ...... 276 x Digital Communications 2

Appendices ...... 277

Appendix A ...... 279

Appendix B ...... 285

Bibliography ...... 291

Index ...... 297

Summary of Volume 1 ...... 299 Preface

Humans have always used communication systems: in the past, native Americans used clouds of smoke, then Chappe invented his telegraph and Edison the telephone, which has deeply changed our lifestyle. Nowadays, smartphones enable us to make calls, watch videos and communicate on social networks. The future will see the emergence of the connected man and wider applications of smart objects. All current and future communication systems rely on a digital communication chain that consists of a source and a destination separated by a transmission channel, which may be a portion of a cable, an optical fiber, a wireless mobile or satellite channel. Whichever the channel, the processing blocks implemented in the communication chain have the same basis. This book aims at detailing them, across two volumes: – the first volume deals with source coding and channel coding. After a presentation of the fundamental results of information theory, the different lossless and lossly source coding techniques are studied. Then, error- correcting-codes (block codes, convolutional codes and concatenated codes) are theoretically detailed and their applications provided; – the second volume concerns the blocks located after channel coding in the communication chain. It first presents baseband and sine waveform transmissions. Then, the different steps required at the receiver to perform detection, namely synchronization and channel estimation, are studied. Two variants of these blocks which are used in current and future systems, multicarrier modulations and coded modulations, are finally detailed.

This book arises from the long experience of its authors in both the business and academic sectors. The authors are in charge of several diploma xii Digital Communications 2 and higher-education teaching modules at Conservatoire national des arts et métiers (CNAM) concerning digital communication, information theory and wireless mobile communications.

The different notions in this book are presented with an educational objective. The authors have tried to make the fundamental notions of digital communications as understandable and didactic as possible. Nevertheless, some more advanced techniques that are currently strong research topics but are not yet implemented are also developed.

Digital Communications may interest students in the ﬁelds of electronics, telecommunications, signal processing, etc., as well as engineering and corporate executives working in the same domains and wishing to update or complete their knowledge on the subject.

The authors would like to thank their colleagues from CNAM, and especially from the EASY department.

Mylène Pischella would like to thank her daughter Charlotte and husband Benjamin for their presence, affection and support.

Didier Le Ruyet would like to thank his parents and his wife Christine for their support, patience and encouragements during the writing of this book.

Mylène PISCHELLA Didier LE RUYET Paris, France August 2015 List of Acronyms

ADSL: assymetric digital subscriber line

ASK: amplitude shift keying

BER: bit error rate

CD: compact disc

CFM: constellation’s ﬁgure of merit

CMT: cosine modulated multitone

DFE: decision feedback equalizer

DWMT: discrete wavelet multitone

EFM: eight-to-fourteen modulation

EGF: extended Gaussian function

FBMC: ﬁlter bank multicarrier

FDDI: ﬁber distributed data interface

FFT: fast Fourier transform

FIR: ﬁnite impulse response xiv Digital Communications 2

FMT: ﬁltered multi-tone

FSK: frequency shift keying

GMSK: Gaussian minimum shift keying

GSM: global system for mobile communications

HDMI: high-deﬁnition multimedia interface

IFFT: inverse fast Fourier transform

IIR: inﬁnite impulse response

IOTA: isotropic orthogonal transform algorithm

MAP: maximum a posteriori

MIMO: multiple input, multiple output

ML: maximum likelihood

MLSE: maximum likelihood sequence estimator

MLT-3: multilevel transmit 3

MMSE: minimum mean square error

MSK: minimum shift keying

NRZ: non-return to zero

NRZI: non-return to zero inverted

OFDM: orthogonal frequency division multiple access

OQAM: offset quadrature amplitude modulation

PAPR: peak to average power ratio

PSK: phase shift keying

QAM: quadrature amplitude modulation List of Acronyms xv

QPSK: quadrature phase shift keying

RFID: radio frequency identiﬁcation

RLL: run length limited

RZ: return to zero

SCH: synchronization channel

SMT: staggered multitone

SNR: signal-to-noise ratio

VDSL: very-high-bit-rate digital subscriber line

ZF: zero forcing

ZF-DFE: zero-forcing decision feedback equalizer

Notations

B: bandwidth

Bc: coherence bandwidth

b(t): baseband additive white Gaussian noise

ck = ak + jbk: baseband complex symbol, with ak its real part and bk its imaginary part

dmin: minimum distance

E[x]: expected value of a random variable x

Eb: average energy per bit

Es: average energy per symbol

fc: carrier frequency of narrowband passband signal

gc(t): impulse response of the transmit baseband channel

g(t): transmit ﬁlter

gr(t): matched ﬁlter at the receiver

γxx(f): power spectrum density of a random process x

h(t): impulse response of the narrowband passband channel xviii Digital Communications 2

H(z): linear digital equalization ﬁlter

N: noise power,or number of samples in an OFDM or FBMC symbol

NCP: number of samples in OFDM cyclic preﬁx

n(t): noise ﬁltered by the receive ﬁlter

N0: one-sided noise power spectrum density

M: alphabet size of the modulation’s constellation

P : signal power

Pe: error probability per symbol

p: probability of a random variable

p(t)=g(t) ∗ gc(t) ∗ gr(t): complete equivalent channel

Ψ: phase of the passband signal

Rss(t): autocorrelation function of random process s

r(t): output signal of narrowband passband channel

s(t): input signal of narrowband passband channel

T : symbol period

Tb: transmission duration of a bit

Tm: maximum delay of the channel

TN : transmission duration of an OFDM or FBMC symbol

u(t): step function

x(t): transmitted signal on the baseband channel

y: received signal at channel’s output after matched ﬁlter and sampling

y(t): received baseband signal after matched ﬁlter Introduction

The fundamental objective of a communication system is to reproduce at a destination point, either exactly or approximately, a message selected at another point. This was theorized by Claude Shannon in 1948 [SHA 48].

The communication chain is composed of a source (also called transmitter) and a destination (also called receiver). They are separated by a transmission channel, which may, for instance, be a wired cable if we consider assymetric digital subscriber line (ADSL) transmission, optical ﬁber, a wireless mobile channel between a base station and a mobile terminal or between a satellite and its receiver, a hard drive and so forth. The latter example indicates that when we refer to point, we may consider either location or time. The main issue faced by communication systems is that the channel is subject to additional noise, and may also introduce some distortions on the transmitted signal. Consequently, advanced techniques must be implemented in order to decrease the impact of noise and distortions on the performances as much as possible, so that the receiver signal may be as similar as possible to the transmitted signal.

The performance of a transmission system is evaluated by either computing or measuring the error probability per received information bit at the receiver, also called the bit error rate. The other major characteristics of a communication system are its complexity, its bandwidth, its consumed and transmitted power and the useful data rate that it can transmit. The bandwidth of many communication systems is limited; it is thus highly important to maximize the spectral efﬁciency, which is deﬁned as the ratio between the binary data rate and the bandwidth. Nevertheless, this should be done without increasing the bit error rate. xx Digital Communications 2

This book consists of two volumes. It aims at detailing all steps of the communication chain, represented in Figure I.1. Source and channel coding are deeply studied in the first volume: Digital Communications 1: Source and Channel Coding. Even though both volumes can be read independently, we will sometimes refer to some of the notions developed in the first volume. The present volume focuses on the following blocks: modulation, synchronization and equalization. Once source and channel coding have been performed, data have first been compressed, and then encoded by adding some well-chosen redundancy that protects the transmitted bits from the channel’s disruptions. In this second volume, we are first located at the modulator’s input. The modulator takes the binary data at the output of the channel encoder in order to prepare them for transmission on the channel.

Figure I.1. Block diagram of the communication chain

The coded bits are then associated with symbols that will be transmitted during a given symbol period. They are shaped by a transmit filter before being sent on the channel. At the stage, transmission techniques vary, depending on whether baseband transmission (when the signal is carrier by the null frequency) or sine waveform transmission takes place. In the latter case, the frequency carrier is far larger than the bandwidth. In the former case, symbols are carried by the specific value of a line code; whereas in the latter case, they are carried by either the signal’s amplitude, phase or carrier frequency value. Baseband transmissions are detailed in Chapter 2. Digital modulations on sine waveforms are studied in Chapter 3. Furthermore, Introduction xxi advanced digital modulation techniques, called coded modulations, are presented in Chapter 6. They associate modulations and channel encoding in order to increase the spectral efficiency. Chapter 2 introduces some fundamental notions such as optimum detection on the additive white Gaussian noise channel, and Chapter 3 explains the connection between baseband transmissions and sine waveform transmissions. Modulations’ performances are determined in terms of bit error rate depending on the signal-to-noise ratio and on the bandwidth.

The demodulator aims at extracting samples while maximizing a given criterion (signal-to-noise ratio, bit error rate, etc.). The detector’s objective is to determine the most likely transmitted symbol. If channel decoding takes hard inputs, the detector directly provides the estimated binary sequence. If channel decoding takes soft inputs, then the detector provides soft information on bits, which generally are logarithms of likelihood ratio.

In order to have an efﬁcient detection, the input samples should be as close as possible to the transmitted ones. Yet, the channel may generate distortions on the transmitted signal, as well as delays which may result in a loss of the symbol period, or frequency shifts. Demodulation and detection must consequently be preceded by a synchronization at the receiver with the transmitted signal in order to correct these shifts. Thus, if the channel has introduced distortions that lead to intersymbol interference, an equalization step must be added in order to estimate the transmitter symbols. These two steps are presented in Chapter 4. We can notice that synchronization may be blind, which means that it is performed without any channel knowledge, in which case its performances will be far lower than if synchronization is based on a training sequence. Similarly, equalization requires channel estimation, which must be implemented prior to these two steps.

In order not to use equalization at the receiver, most recent communication systems split the original wideband channel into several subchannels, where each subchannel only produces a constant signal attenuation but no distortion. This step modiﬁes the modulation block and is thus located before transmission on the channel. Multi-carrier modulations are detailed in Chapter 5.

Finally, we can notice that some useful mathematics and digital signal processing background are provided in Chapter1. xxii Digital Communications 2

The summary of the contributions of this book shows that we have tried to be as exhaustive as possible, while still studying a large spectrum of the digital communications domain. Some topics have not been considered due to lack of space. For instance, the wireless mobile channel’s characteristics or the multiple-access techniques on this channel have not been mentioned, as well as multiple antenna multiple input, multiple output (MIMO) techniques. Moreover, we have only focused on the most modern techniques, only considering older techniques with an educational objective, when the latter allow us to better understand more complex modern techniques.

Most of the chapters detail fundamental notions of digital communications and are thus necessary to comprehend the whole communication chain. They provide several degrees of understanding by giving an introduction to these techniques, while still giving some more advanced details. In addition, they are illustrated by examples of implementations in current communications systems. Some exercises are proposed at the end of each chapter, so that the readers may take over the presented notions.

We can nevertheless notice that Chapter 6 proposes an opening to advanced modulation and coding techniques. It may be read as a complement in order to strengthen knowledge. 1

Background

1.1. Introduction

This chapter provides some background necessary for the remainder of this book. The common operations and functions are presented ﬁrst. The common transforms required for calculations are detailed in section 1.3. Then, some background on discrete and continuous probabilities is provided in section 1.4. Finally, some elements of digital signal processing are recalled in section 1.5.

1.2. Common operations and functions

1.2.1. Convolution

The convolution between two signals s(t) and h(t) is deﬁned as:

∞ r(t)=(s ∗ h)(t)= s(u)h(t − u)du [1.1] −∞

We will write it as (s ∗ h)(t)=s(t) ∗ h(t) with a notational abuse.

The convolution is linear and invarious to time. Consequently, for any continuous signals s1(t) and s2(t), any complex values α1,α2 and any time delay t1,t2, we can write:

(α1s1(t − t1)+α2s2(t − t2)) ∗ h(t)

= α1(s1 ∗ h)(t − t1)+α2(s2 ∗ h)(t − t2) [1.2]

Digital Communications 2: Digital Modulations, First Edition. Mylène Pischella and Didier Le Ruyet. © ISTE Ltd 2015. Published by ISTE Ltd and John Wiley & Sons, Inc. 2 Digital Communications 2

1.2.2. Scalar product

The scalar product between two continuous signals s(t) and r(t) is deﬁned as: ∞ s, r = s(t)r∗(t)dt [1.3] −∞

For any continuous signals s1(t), s2(t), r1(t) and r2(t) and any complex values α1,α2, the following linearity properties hold:

α1s1 + α2s2,r = α1 s1,r + α2 s2,r ∗ ∗ s, α1r1 + α2r2 = α1 s, r1 + α2 s, r2 [1.4]

T For vectors of size m × 1, s =[s0,s1, ..., sm−1] and T r =[r0,r1, ..., rm−1] , the scalar product is similarly deﬁned:

m−1 ∗ H s, r = skrk = r s [1.5] k=0

1.2.3. Dirac function, Dirac impulse and Kronecker’s symbol

The continuous Dirac function, denoted as t → δ(t), is deﬁned in the following way: for any ﬁnite-energy signal s(t) and any real value τ,

∞ s(t)δ(t − τ)dt = s(t − τ) [1.6] −∞

From [1.6] and [1.1], the convolution of any function s(t) by a Dirac function delayed by τ is equal to the function s(t) delayed by τ:

s(t) ∗ δ(t − τ)=s(t − τ) [1.7]

The Dirac impulse is deﬁned for discrete signals as follows: 1 if n =0 δn = [1.8] 0 if n =0 Background 3

The Kronecker’s symbol is a function of two variables which is equal to 1 if both variables are equal, and to 0 elsewhere. 1 if n = n0 δn,n0 = [1.9] 0 if n = n0

As a result, the Dirac impulse is equal to the Kronecker symbol when n0 =0.

1.2.4. Step function

The step function is a continuous function which is equal to 0 when the input value is negative, and equal to 1 when the input value is positive: 0 if t ≤ 0 u(t)= [1.10] 1 if t>0

1.2.5. Rectangular function

The rectangular function is a continuous function which is equal to 0 outside of a time interval T that starts at t =0: 0 if t/∈ [0,T[ ΠT (t)= [1.11] 1 if t ∈ [0,T[

1.3. Common transforms

1.3.1. Fourier transform

1.3.1.1. Fourier transform of a continuous signal The Fourier transform is a particularly important tool of the ﬁeld of digital communications. It allows us to study a signal no longer in the time domain, but in the frequency domain. The spectral properties of a signal are more relevant to characterize it than its time properties. For instance, due to its spectral properties, a signal can be determined as baseband or passband, its bandwidth can be characterized and so on. 4 Digital Communications 2

The Fourier transform of a continuous signal s(t) is:

∞ TF[s](f)= s(t)e−j2πftdt [1.12] −∞ It is often denoted by S(f)=TF[s](f).

The inverse Fourier transform allows us to recover the time signal s(t) from its Fourier transform S(f):

∞ TF−1[S](t)= S(f)ej2πftdf [1.13] −∞

The Fourier transform is linear. In digital communications, the following property is of particular interest: the Fourier transform of a convolutional product is equal to the product of both Fourier transforms. The opposite property also stands:

TF[s ∗ h](f)=TF[s] × TF[h] ⇔ TF[s × h](f)=TF[s] ∗ TF[h] [1.14]

We list here the Fourier transform of some functions that are useful in digital communications and signal processing: – the Fourier transform of the Dirac function is a constant: s(t)=δ(t) ⇔ S(f)=1; – the Fourier transform of the rectangular function between −T/2 and T/2 − ⇔ sin(πfT) is a cardinal sine function: s(t)=ΠT (t T/2) S(f)=T πfT ; – the Fourier transform of a cosine is a sum of two Dirac functions: s(t)= 1 cos(2πf0t) ⇔ S(f)= 2 (δ(f − f0)+δ(f + f0)); – the Fourier transform of a sine is a difference between two Dirac 1 functions: s(t) = sin(2πf0t) ⇔ S(f)= 2 (δ(f − f0) − δ(f + f0)).

1.3.1.2. Discrete Fourier transform

Le x be a discrete signal composed of N samples, x =[x0,x1, ..., xN−1], with sampling frequency Fe and sampling period Te =1/Fe. xn = x(nTe) is the sample corresponding to time nT . Background 5

Its Fourier transform is deﬁned as:

N−1 1 −j2πfnTe TF[x](f)=X(f)=√ xne [1.15] N n=0

The discrete Fourier transform is generally determined for some discrete frequency values: N−1 kFe 1 − 2 nk X = √ x e j π N [1.16] N n N n=0

The N samples of the Fourier transform are denoted as kFe X =[X0,X1, ..., XN−1] with Xk = X N . The inverse discrete Fourier transform is similarly deﬁned: N−1 −1 1 kFe 2 nk TF [X](t)=x(nT )=√ X ej π N [1.17] e N N k=0

Energy is maintained between the time sample’s vector x and the frequency sample vector X. This is proven by Parseval’s equality:

N−1 N−1 1 2 1 ∗ √ |x(nTe)| = √ x(nTe)x (nTe) N n=0 N n=0 N−1 N−1 1 ∗ kFe − 2 nk = √ x(nT ) X e j π N e N N n=0 k=0 N−1 N−1 ∗ kFe 1 − 2 nk = X √ x(nT )e j π N N e k=0 N n=0 −1 N kF kF = X∗ e X e N N k=0 N−1 2 kFe = X [1.18] N k=0 6 Digital Communications 2

The discrete Fourier transform and its inverse can be implemented with low complexity by using the fast Fourier transform (FFT). This recursive algorithm was established by Cooley and Tuckey in 1965.ForN samples, the FFT requires N log2(N) operations, whereas a direct application of equation [1.17] would require N 2 operations.

1.3.2. The z transform

The z transform is used in the ﬁelds of signal processing and digital communications to model ﬁltering operations, and especially time delays.

It is applied on discrete sampled signals x =[x0,x1, ..., xN−1], and is deﬁned as follows:

N−1 −n TZ[x](z)=X(z)= xnz [1.19] n=0

We can see that the z transform is equal, up to a proportionality factor, to the discrete Fourier transform [1.19] when z = ej2πfTe .

1.4. Probability background

Probability theory is a mathematical domain that describes and models random processes. In this section, we present a summary of this theory. We recommend for further reading [PAP 02] and [DUR 10].

Let X be an experiment or an observation that can be repeated under similar circumstances several times. At each repetition, the result of this observation is an event denoted by x , which can take several possible outcomes. The set of these values is denoted by AX .

The result X = x of this observation is not known before it takes place. X is consequently called a random variable. It is modeled by the frequency of appearance of all its outcomes.

Two classes of random variables can be distinguished: – discrete random variables, when the set of outcomes is discrete;