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A FOUNDATION IN DIGITAL COMMUNICAT ION 2ND EDITION BY AMOS LAPIDOTH
A FOUNDATION IN DIGITAL COMMUN ICAT ION 2ND EDITION BY AMOS LAPIDOTH
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, A F O U N DAT I O N I N D I G I TA L C O M M U N I C AT I O NSecond Edition
Written in the intuitive yet rigorous style that readers of A Foundation in Digital
Communication have come to expect, the second edition adds new chapters on the
Radar Problem (with Lyapunov’s theorem) and Intersymbol Interference channels.
An added bonus is the treatment of the baseband representation of passband noise.
But much more has changed. Most notably, the derivation of the optimal receiver
for the additive white Gaussian noise channel has been completely rewritten. It is
now simpler, more geometric, and more intuitive, and yet the end result is stronger
and more general: it easily generalizes to multiple-antenna channels and to the
Radar Problem.
The definition of the power spectral density of nonstationary stochastic pro-
cesses such as QAM signals remains, but new are the connections with the average
autocovariance function and its Fourier Transform. Also unchanged is the empha-
sis on the geometry of the space of energy-limited signals and on the isometry
properties of the Fourier Transform, the baseband representation of passband
signals, and complex sampling.
With the additional topics and over 150 new problems, the book can now be
used for a one- or two-semester graduate course on digital communications or for
a course on stochastic processes and detection theory.
A M O S L A P I D O T H is Professor of Information Theory at ETH Zurich, the Swiss
Federal Institute of Technology, and a Fellow of the IEEE. He received his Ph.D.
in Electrical Engineering from Stanford University, and has held the positions of
Assistant and Associate Professor at the Massachusetts Institute of Technology.
,
,A F O U N DAT I O N I N D I G I TA L
COM M UNI CAT ION
Second Edition
A M O S L A P I D OT H
ETH Zurich
, University Printing House, Cambridge CB2 8BS, United Kingdom
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www.cambridge.org
Information on this title: www.cambridge.org/9781107177321
DOI: 10.1017/9781316822708
c Amos Lapidoth 2017
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2017
Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall
A catalogue record for this publication is available from the British Library.
ISBN 978-1-107-17732-1 Hardback
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accurate or appropriate.
,To my family
,
,Contents
Preface to the Second Edition xvi
Preface to the First Edition xviii
Acknowledgments for the Second Edition xxvi
Acknowledgments for the First Edition xxvii
1 Some Essential Notation 1
2 Signals, Integrals, and Sets of Measure Zero 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Integrating Complex-Valued Signals . . . . . . . . . . . . . . . . . . . 5
2.4 An Inequality for Integrals . . . . . . . . . . . . . . . . . . . . . . . . 6
2.5 Sets of Lebesgue Measure Zero . . . . . . . . . . . . . . . . . . . . . 7
2.6 Swapping Integration, Summation, and Expectation . . . . . . . . . . 10
2.7 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 The Inner Product 14
3.1 The Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 When Is the Inner Product Defined? . . . . . . . . . . . . . . . . . . 17
3.3 The Cauchy-Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . 18
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 The Cauchy-Schwarz Inequality for Random Variables . . . . . . . . . 23
3.6 Mathematical Comments . . . . . . . . . . . . . . . . . . . . . . . . 23
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 The Space L2 of Energy-Limited Signals 27
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 L2 as a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Subspace, Dimension, and Basis . . . . . . . . . . . . . . . . . . . . 29
4.4 u2 as the “length” of the Signal u(·) . . . . . . . . . . . . . . . . 31
4.5 Orthogonality and Inner Products . . . . . . . . . . . . . . . . . . . . 33
4.6 Orthonormal Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.7 The Space L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
vii
,viii Contents
4.8 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Convolutions and Filters 54
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Time Shifts and Reflections . . . . . . . . . . . . . . . . . . . . . . . 54
5.3 The Convolution Expression . . . . . . . . . . . . . . . . . . . . . . . 55
5.4 Thinking About the Convolution . . . . . . . . . . . . . . . . . . . . 55
5.5 When Is the Convolution Defined? . . . . . . . . . . . . . . . . . . . 56
5.6 Basic Properties of the Convolution . . . . . . . . . . . . . . . . . . . 58
5.7 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.8 The Matched Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.9 The Ideal Unit-Gain Lowpass Filter . . . . . . . . . . . . . . . . . . . 61
5.10 The Ideal Unit-Gain Bandpass Filter . . . . . . . . . . . . . . . . . . 62
5.11 Young’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.12 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 The Frequency Response of Filters and Bandlimited Signals 65
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Review of the Fourier Transform . . . . . . . . . . . . . . . . . . . . 65
6.3 The Frequency Response of a Filter . . . . . . . . . . . . . . . . . . . 77
6.4 Bandlimited Signals and Lowpass Filtering . . . . . . . . . . . . . . . 80
6.5 Bandlimited Signals Through Stable Filters . . . . . . . . . . . . . . . 90
6.6 The Bandwidth of a Product of Two Signals . . . . . . . . . . . . . . 91
6.7 Bernstein’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.8 Time-Limited and Bandlimited Signals . . . . . . . . . . . . . . . . . 94
6.9 A Theorem by Paley and Wiener . . . . . . . . . . . . . . . . . . . . 97
6.10 Picket Fences and Poisson Summation . . . . . . . . . . . . . . . . . 97
6.11 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Passband Signals and Their Representation 104
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.2 Baseband and Passband Signals . . . . . . . . . . . . . . . . . . . . . 104
7.3 Bandwidth around a Carrier Frequency . . . . . . . . . . . . . . . . . 107
7.4 Real Passband Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.5 The Analytic Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.6 Baseband Representation of Real Passband Signals . . . . . . . . . . 119
7.7 Energy-Limited Passband Signals . . . . . . . . . . . . . . . . . . . . 133
7.8 Shifting to Passband and Convolving . . . . . . . . . . . . . . . . . . 141
7.9 Mathematical Comments . . . . . . . . . . . . . . . . . . . . . . . . 142
7.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8 Complete Orthonormal Systems and the Sampling Theorem 147
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Complete Orthonormal System . . . . . . . . . . . . . . . . . . . . . 147
8.3 The Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
, Contents ix
8.4 The Sampling Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.5 The Samples of the Convolution . . . . . . . . . . . . . . . . . . . . 156
8.6 Closed Subspaces of L2 . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.7 An Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.8 Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . . . . 161
8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
9 Sampling Real Passband Signals 168
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.2 Complex Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
9.3 Reconstructing xPB from its Complex Samples . . . . . . . . . . . . . 170
9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10 Mapping Bits to Waveforms 176
10.1 What Is Modulation? . . . . . . . . . . . . . . . . . . . . . . . . . . 176
10.2 Modulating One Bit . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.3 From Bits to Real Numbers . . . . . . . . . . . . . . . . . . . . . . . 178
10.4 Block-Mode Mapping of Bits to Real Numbers . . . . . . . . . . . . . 179
10.5 From Real Numbers to Waveforms with Linear Modulation . . . . . . 181
10.6 Recovering the Signal Coefficients with a Matched Filter . . . . . . . 182
10.7 Pulse Amplitude Modulation . . . . . . . . . . . . . . . . . . . . . . 184
10.8 Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
10.9 Uncoded Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10.10 Bandwidth Considerations . . . . . . . . . . . . . . . . . . . . . . . . 188
10.11 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 189
10.12 Some Implementation Considerations . . . . . . . . . . . . . . . . . . 191
10.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
11 Nyquist’s Criterion 195
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
11.2 The Self-Similarity Function of Energy-Limited Signals . . . . . . . . 196
11.3 Nyquist’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
11.4 The Self-Similarity Function of Integrable Signals . . . . . . . . . . . 208
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12 Stochastic Processes: Definition 213
12.1 Introduction and Continuous-Time Heuristics . . . . . . . . . . . . . 213
12.2 A Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
12.3 Describing Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 216
12.4 Additional Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
13 Stationary Discrete-Time Stochastic Processes 219
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13.2 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
13.3 Wide-Sense Stationary Stochastic Processes . . . . . . . . . . . . . . 220
13.4 Stationarity and Wide-Sense Stationarity . . . . . . . . . . . . . . . . 221
13.5 The Autocovariance Function . . . . . . . . . . . . . . . . . . . . . . 222
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