digital-communication-proakis-salehi-5th-edition.pdf

Proakis-27466 pro57166˙fm September 26, 2007 12:35
Digital Communications
Fifth Edition
John G. Proakis
Professor Emeritus, Northeastern University
Department of Electrical and Computer Engineering,
University of California, San Diego
Masoud Salehi
Department of Electrical and Computer Engineering,
Northeastern University

DIGITAL COMMUNICATIONS, FIFTH EDITION
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Library of Congress Cataloging-in-Publication Data
Proakis, John G.
Digital communications / John G. Proakis, Masoud Salehi.—5th ed.
p. cm.
Includes index.
ISBN 978–0–07–295716–7—ISBN 0–07–295716–6 (hbk. : alk. paper) 1. Digital communications.
I. Salehi, Masoud. II. Title.
TK5103.7.P76 2008
621.382—dc22
2007036509
www.mhhe.com

D E D I C A T I O N
To
Felia, George, and Elena
John G. Proakis
To
Fariba, Omid, Sina, and My Parents
Masoud Salehi
iii

B R I E F C O N T E N T S
Preface xvi
Chapter 1 Introduction 1
Chapter 2 Deterministic and Random Signal Analysis 17
Chapter 3 Digital Modulation Schemes 95
Chapter 4 Optimum Receivers for AWGN Channels 160
Chapter 5 Carrier and Symbol Synchronization 290
Chapter 6 An Introduction to Information Theory 330
Chapter 7 Linear Block Codes 400
Chapter 8 Trellis and Graph Based Codes 491
Chapter 9 Digital Communication Through Band-Limited
Channels 597
Chapter 10 Adaptive Equalization 689
Chapter 11 Multichannel and Multicarrier Systems 737
Chapter 12 Spread Spectrum Signals for Digital Communications 762
Chapter 13 Fading Channels I: Characterization and Signaling 830
Chapter 14 Fading Channels II: Capacity and Coding 899
Chapter 15 Multiple-Antenna Systems 966
Chapter 16 Multiuser Communications 1028
Appendices
Appendix A Matrices 1085
Appendix B Error Probability for Multichannel Binary Signals 1090
Appendix C Error Probabilities for Adaptive Reception of M-Phase
Signals 1096
Appendix D Square Root Factorization 1107
References and Bibliography 1109
Index 1142
v

C O N T E N T S
Preface xvi
Chapter 1 Introduction 1
1.1 Elements of a Digital Communication System 1
1.2 Communication Channels and Their Characteristics 3
1.3 Mathematical Models for Communication Channels 10
1.4 A Historical Perspective in the Development of
Digital Communications 12
1.5 Overview of the Book 15
1.6 Bibliographical Notes and References 15
Chapter 2 Deterministic and Random Signal Analysis 17
2.1 Bandpass and Lowpass Signal Representation 18
2.1–1 Bandpass and Lowpass Signals / 2.1–2 Lowpass
Equivalent of Bandpass Signals / 2.1–3 Energy
Considerations / 2.1–4 Lowpass Equivalent of a
Bandpass System
2.2 Signal Space Representation of Waveforms 28
2.2–1 Vector Space Concepts / 2.2–2 Signal Space
Concepts / 2.2–3 Orthogonal Expansions of Signals /
2.2–4 Gram-Schmidt Procedure
2.3 Some Useful Random Variables 40
2.4 Bounds on Tail Probabilities 56
2.5 Limit Theorems for Sums of Random Variables 63
2.6 Complex Random Variables 63
2.6–1 Complex Random Vectors
2.7 Random Processes 66
2.7–1 Wide-Sense Stationary Random Processes / 2.7–2
Cyclostationary Random Processes / 2.7–3 Proper and
Circular Random Processes / 2.7–4 Markov Chains
2.8 Series Expansion of Random Processes 74
2.8–1 Sampling Theorem for Band-Limited Random
Processes / 2.8–2 The Karhunen-Loève Expansion
2.9 Bandpass and Lowpass Random Processes 78
vi

Contents vii
Problems 82
Chapter 3 Digital Modulation Schemes 95
3.1 Representation of Digitally Modulated Signals 95
3.2 Memoryless Modulation Methods 97
3.2–1 Pulse Amplitude Modulation (PAM) / 3.2–2 Phase
Modulation / 3.2–3 Quadrature Amplitude
Modulation / 3.2–4 Multidimensional Signaling
3.3 Signaling Schemes with Memory 114
3.3–1 Continuous-Phase Frequency-Shift Keying
(CPFSK) / 3.3–2 Continuous-Phase Modulation (CPM)
3.4 Power Spectrum of Digitally Modulated Signals 131
3.4–1 Power Spectral Density of a Digitally Modulated Signal
with Memory / 3.4–2 Power Spectral Density of Linearly
Modulated Signals / 3.4–3 Power Spectral Density of
Digitally Modulated Signals with Finite Memory / 3.4–4
Power Spectral Density of Modulation Schemes with a Markov
Structure / 3.4–5 Power Spectral Densities of CPFSK and
CPM Signals
Problems 148
Chapter 4 Optimum Receivers for AWGN Channels 160
4.1 Waveform and Vector Channel Models 160
4.1–1 Optimal Detection for a General Vector Channel
4.2 Waveform and Vector AWGN Channels 167
4.2–1 Optimal Detection for the Vector AWGN
Channel / 4.2–2 Implementation of the Optimal Receiver for
AWGN Channels / 4.2–3 A Union Bound on the Probability of
Error of Maximum Likelihood Detection
4.3 OptimalDetectionandErrorProbabilityforBand-Limited
Signaling 188
4.3–1 Optimal Detection and Error Probability for ASK or
PAM Signaling / 4.3–2 Optimal Detection and Error
Probability for PSK Signaling / 4.3–3 Optimal Detection and
Error Probability for QAM Signaling / 4.3–4 Demodulation
and Detection
4.4 OptimalDetectionandErrorProbabilityforPower-Limited
Signaling 203
4.4–1 Optimal Detection and Error Probability for Orthogonal
Signaling / 4.4–2 Optimal Detection and Error Probability
for Biorthogonal Signaling / 4.4–3 Optimal Detection and
Error Probability for Simplex Signaling

viii Contents
4.5 Optimal Detection in Presence of Uncertainty:
Noncoherent Detection 210
4.5–1 Noncoherent Detection of Carrier Modulated
Signals / 4.5–2 Optimal Noncoherent Detection of FSK
Modulated Signals / 4.5–3 Error Probability of Orthogonal
Signaling with Noncoherent Detection / 4.5–4 Probability of
Error for Envelope Detection of Correlated Binary
Signals / 4.5–5 Differential PSK (DPSK)
4.6 A Comparison of Digital Signaling Methods 226
4.6–1 Bandwidth and Dimensionality
4.7 Lattices and Constellations Based on Lattices 230
4.7–1 An Introduction to Lattices / 4.7–2 Signal
Constellations from Lattices
4.8 Detection of Signaling Schemes with Memory 242
4.8–1 The Maximum Likelihood Sequence Detector
4.9 Optimum Receiver for CPM Signals 246
4.9–1 Optimum Demodulation and Detection of CPM /
4.9–2 Performance of CPM Signals / 4.9–3 Suboptimum
Demodulation and Detection of CPM Signals
4.10 Performance Analysis for Wireline and Radio
Communication Systems 259
4.10–1 Regenerative Repeaters / 4.10–2 Link Budget
Analysis in Radio Communication Systems
Problems 266
Chapter 5 Carrier and Symbol Synchronization 290
5.1 Signal Parameter Estimation 290
5.1–1 The Likelihood Function / 5.1–2 Carrier Recovery and
Symbol Synchronization in Signal Demodulation
5.2 Carrier Phase Estimation 295
5.2–1 Maximum-Likelihood Carrier Phase Estimation /
5.2–2 The Phase-Locked Loop / 5.2–3 Effect of Additive
Noise on the Phase Estimate / 5.2–4 Decision-Directed
Loops / 5.2–5 Non-Decision-Directed Loops
5.3 Symbol Timing Estimation 315
5.3–1 Maximum-Likelihood Timing Estimation /
5.3–2 Non-Decision-Directed Timing Estimation
5.4 Joint Estimation of Carrier Phase and Symbol Timing 321
5.5 Performance Characteristics of ML Estimators 323
Problems 327
Chapter 6 An Introduction to Information Theory 330
6.1 Mathematical Models for Information Sources 331

Contents ix
6.2 A Logarithmic Measure of Information 332
6.3 Lossless Coding of Information Sources 335
6.3–1 The Lossless Source Coding Theorem / 6.3–2 Lossless
Coding Algorithms
6.4 Lossy Data Compression 348
6.4–1 Entropy and Mutual Information for Continuous
Random Variables / 6.4–2 The Rate Distortion Function
6.5 Channel Models and Channel Capacity 354
6.5–1 Channel Models / 6.5–2 Channel Capacity
6.6 Achieving Channel Capacity with Orthogonal Signals 367
6.7 The Channel Reliability Function 369
6.8 The Channel Cutoff Rate 371
6.8–1 Bhattacharyya and Chernov Bounds / 6.8–2 Random
Coding
Problems 381
Chapter 7 Linear Block Codes 400
7.1 Basic Definitions 401
7.1–1 The Structure of Finite Fields / 7.1–2 Vector Spaces
7.2 General Properties of Linear Block Codes 411
7.2–1 Generator and Parity Check Matrices / 7.2–2 Weight
and Distance for Linear Block Codes / 7.2–3 The Weight
Distribution Polynomial / 7.2–4 Error Probability of Linear
Block Codes
7.3 Some Specific Linear Block Codes 420
7.3–1 Repetition Codes / 7.3–2 Hamming Codes /
7.3–3 Maximum-Length Codes / 7.3–4 Reed-Muller
Codes / 7.3–5 Hadamard Codes / 7.3–6 Golay Codes
7.4 Optimum Soft Decision Decoding of Linear
Block Codes 424
7.5 Hard Decision Decoding of Linear Block Codes 428
7.5–1 Error Detection and Error Correction Capability of
Block Codes / 7.5–2 Block and Bit Error Probability for Hard
Decision Decoding
7.6 Comparison of Performance between Hard Decision and
Soft Decision Decoding 436
7.7 Bounds on Minimum Distance of Linear Block Codes 440
7.7–1 Singleton Bound / 7.7–2 Hamming Bound /
7.7–3 Plotkin Bound / 7.7–4 Elias Bound / 7.7–5
McEliece-Rodemich-Rumsey-Welch (MRRW) Bound /
7.7–6 Varshamov-Gilbert Bound
7.8 Modified Linear Block Codes 445
7.8–1 Shortening and Lengthening / 7.8–2 Puncturing and
Extending / 7.8–3 Expurgation and Augmentation

x Contents
7.9 Cyclic Codes 447
7.9–1 Cyclic Codes — Definition and Basic Properties /
7.9–2 Systematic Cyclic Codes / 7.9–3 Encoders for Cyclic
Codes / 7.9–4 Decoding Cyclic Codes / 7.9–5 Examples of
Cyclic Codes
7.10 Bose-Chaudhuri-Hocquenghem (BCH) Codes 463
7.10–1 The Structure of BCH Codes / 7.10–2 Decoding
BCH Codes
7.11 Reed-Solomon Codes 471
7.12 Coding for Channels with Burst Errors 475
7.13 Combining Codes 477
7.13–1 Product Codes / 7.13–2 Concatenated Codes
Problems 482
Chapter 8 Trellis and Graph Based Codes 491
8.1 The Structure of Convolutional Codes 491
8.1–1 Tree, Trellis, and State Diagrams / 8.1–2 The Transfer
Function of a Convolutional Code / 8.1–3 Systematic,
Nonrecursive, and Recursive Convolutional Codes /
8.1–4 The Inverse of a Convolutional Encoder and
Catastrophic Codes
8.2 Decoding of Convolutional Codes 510
8.2–1 Maximum-Likelihood Decoding of Convolutional
Codes — The Viterbi Algorithm / 8.2–2 Probability of
Error for Maximum-Likelihood Decoding of Convolutional
Codes
8.3 Distance Properties of Binary Convolutional Codes 516
8.4 Punctured Convolutional Codes 516
8.4–1 Rate-Compatible Punctured Convolutional Codes
8.5 Other Decoding Algorithms for Convolutional Codes 525
8.6 Practical Considerations in the Application of
Convolutional Codes 532
8.7 Nonbinary Dual-k Codes and Concatenated Codes 537
8.8 Maximum a Posteriori Decoding of Convolutional
Codes — The BCJR Algorithm 541
8.9 Turbo Codes and Iterative Decoding 548
8.9–1 Performance Bounds for Turbo Codes / 8.9–2 Iterative
Decoding for Turbo Codes / 8.9–3 EXIT Chart Study of
Iterative Decoding
8.10 Factor Graphs and the Sum-Product Algorithm 558
8.10–1 Tanner Graphs / 8.10–2 Factor Graphs / 8.10–3 The
Sum-Product Algorithm / 8.10–4 MAP Decoding Using the
Sum-Product Algorithm

Contents xi
8.11 Low Density Parity Check Codes 568
8.11–1 Decoding LDPC Codes
8.12 Coding for Bandwidth-Constrained Channels — Trellis
Coded Modulation 571
8.12–1 Lattices and Trellis Coded Modulation /
8.12–2 Turbo-Coded Bandwidth Efficient Modulation
Problems 590
Chapter 9 Digital Communication Through Band-Limited
Channels 597
9.1 Characterization of Band-Limited Channels 598
9.2 Signal Design for Band-Limited Channels 602
9.2–1 Design of Band-Limited Signals for No Intersymbol
Interference—The Nyquist Criterion / 9.2–2 Design of
Band-Limited Signals with Controlled ISI—Partial-Response
Signals / 9.2–3 Data Detection for Controlled ISI /
9.2–4 Signal Design for Channels with Distortion
9.3 Optimum Receiver for Channels with ISI and AWGN 623
9.3–1 Optimum Maximum-Likelihood Receiver /
9.3–2 A Discrete-Time Model for a Channel with ISI /
9.3–3 Maximum-Likelihood Sequence Estimation (MLSE) for
the Discrete-Time White Noise Filter Model /
9.3–4 Performance of MLSE for Channels with ISI
9.4 Linear Equalization 640
9.4–1 Peak Distortion Criterion / 9.4–2 Mean-Square-Error
(MSE) Criterion / 9.4–3 Performance Characteristics of the
MSE Equalizer / 9.4–4 Fractionally Spaced
Equalizers / 9.4–5 Baseband and Passband Linear Equalizers
9.5 Decision-Feedback Equalization 661
9.5–1 Coefficient Optimization / 9.5–2 Performance
Characteristics of DFE / 9.5–3 Predictive Decision-Feedback
Equalizer / 9.5–4 Equalization at the
Transmitter—Tomlinson–Harashima Precoding
9.6 Reduced Complexity ML Detectors 669
9.7 Iterative Equalization and Decoding—Turbo
Equalization 671
Problems 674
Chapter 10 Adaptive Equalization 689
10.1 Adaptive Linear Equalizer 689
10.1–1 The Zero-Forcing Algorithm / 10.1–2 The LMS
Algorithm / 10.1–3 Convergence Properties of the LMS

xii Contents
Algorithm / 10.1–4 Excess MSE due to Noisy Gradient
Estimates / 10.1–5 Accelerating the Initial Convergence Rate
in the LMS Algorithm / 10.1–6 Adaptive Fractionally Spaced
Equalizer—The Tap Leakage Algorithm / 10.1–7 An Adaptive
Channel Estimator for ML Sequence Detection
10.2 Adaptive Decision-Feedback Equalizer 705
10.3 Adaptive Equalization of Trellis-Coded Signals 706
10.4 Recursive Least-Squares Algorithms for Adaptive
Equalization 710
10.4–1 Recursive Least-Squares (Kalman)
Algorithm / 10.4–2 Linear Prediction and the Lattice Filter
10.5 Self-Recovering (Blind) Equalization 721
10.5–1 Blind Equalization Based on the Maximum-Likelihood
Criterion / 10.5–2 Stochastic Gradient Algorithms /
10.5–3 Blind Equalization Algorithms Based on Second- and
Higher-Order Signal Statistics
Problems 732
Chapter 11 Multichannel and Multicarrier Systems 737
11.1 Multichannel Digital Communications in AWGN
Channels 737
11.1–1 Binary Signals / 11.1–2 M-ary Orthogonal Signals
11.2 Multicarrier Communications 743
11.2–1 Single-Carrier Versus Multicarrier
Modulation / 11.2–2 Capacity of a Nonideal Linear Filter
Channel / 11.2–3 Orthogonal Frequency Division
Multiplexing (OFDM) / 11.2–4 Modulation and
Demodulation in an OFDM System / 11.2–5 An FFT
Algorithm Implementation of an OFDM System / 11.2–6
Spectral Characteristics of Multicarrier Signals / 11.2–7 Bit
and Power Allocation in Multicarrier Modulation / 11.2–8
Peak-to-Average Ratio in Multicarrier Modulation / 11.2–9
Channel Coding Considerations in Multicarrier Modulation
Problems 760
Chapter 12 Spread Spectrum Signals for Digital
Communications 762
12.1 Model of Spread Spectrum Digital Communication
System 763
12.2 Direct Sequence Spread Spectrum Signals 765
12.2–1 Error Rate Performance of the Decoder /
12.2–2 Some Applications of DS Spread Spectrum
Signals / 12.2–3 Effect of Pulsed Interference on DS Spread

Contents xiii
Spectrum Systems / 12.2–4 Excision of Narrowband
Interference in DS Spread Spectrum Systems /
12.2–5 Generation of PN Sequences
12.3 Frequency-Hopped Spread Spectrum Signals 802
12.3–1 Performance of FH Spread Spectrum Signals in an
AWGN Channel / 12.3–2 Performance of FH Spread
Spectrum Signals in Partial-Band Interference / 12.3–3 A
CDMA System Based on FH Spread Spectrum Signals
12.4 Other Types of Spread Spectrum Signals 814
12.5 Synchronization of Spread Spectrum Systems 815
Problems 823
Chapter 13 Fading Channels I: Characterization
and Signaling 830
13.1 Characterization of Fading Multipath Channels 831
13.1–1 Channel Correlation Functions and Power
Spectra / 13.1–2 Statistical Models for Fading Channels
13.2 The Effect of Signal Characteristics on the Choice of a
Channel Model 844
13.3 Frequency-Nonselective, Slowly Fading Channel 846
13.4 Diversity Techniques for Fading Multipath Channels 850
13.4–1 Binary Signals / 13.4–2 Multiphase Signals / 13.4–3
M-ary Orthogonal Signals
13.5 Signaling over a Frequency-Selective, Slowly Fading
Channel: The RAKE Demodulator 869
13.5–1 A Tapped-Delay-Line Channel Model / 13.5–2 The
RAKE Demodulator / 13.5–3 Performance of RAKE
Demodulator / 13.5–4 Receiver Structures for Channels with
Intersymbol Interference
13.6 Multicarrier Modulation (OFDM) 884
13.6–1 Performance Degradation of an OFDM System due to
Doppler Spreading / 13.6–2 Suppression of ICI in OFDM
Systems
Problems 891
Chapter 14 Fading Channels II: Capacity and Coding 899
14.1 Capacity of Fading Channels 900
14.1–1 Capacity of Finite-State Channels
14.2 Ergodic and Outage Capacity 905
14.2–1 The Ergodic Capacity of the Rayleigh Fading
Channel / 14.2–2 The Outage Capacity of Rayleigh Fading
Channels
14.3 Coding for Fading Channels 918

xiv Contents
14.4 Performance of Coded Systems In Fading Channels 919
14.4–1 Coding for Fully Interleaved Channel Model
14.5 Trellis-Coded Modulation for Fading Channels 929
14.5–1 TCM Systems for Fading Channels / 14.5–2 Multiple
Trellis-Coded Modulation (MTCM)
14.6 Bit-Interleaved Coded Modulation 936
14.7 Coding in the Frequency Domain 942
14.7–1 Probability of Error for Soft Decision Decoding of
Linear Binary Block Codes / 14.7–2 Probability of Error for
Hard-Decision Decoding of Linear Block Codes / 14.7–3
Upper Bounds on the Performance of Convolutional Codes for
a Rayleigh Fading Channel / 14.7–4 Use of Constant-Weight
Codes and Concatenated Codes for a Fading Channel
14.8 The Channel Cutoff Rate for Fading Channels 957
14.8–1 Channel Cutoff Rate for Fully Interleaved Fading
Channels with CSI at Receiver
Problems 961
Chapter 15 Multiple-Antenna Systems 966
15.1 Channel Models for Multiple-Antenna Systems 966
15.1–1 Signal Transmission Through a Slow Fading
Frequency-Nonselective MIMO Channel / 15.1–2 Detection
of Data Symbols in a MIMO System / 15.1–3 Signal
Transmission Through a Slow Fading Frequency-Selective
MIMO Channel
15.2 Capacity of MIMO Channels 981
15.2–1 Mathematical Preliminaries / 15.2–2 Capacity of a
Frequency-Nonselective Deterministic MIMO
Channel / 15.2–3 Capacity of a Frequency-Nonselective
Ergodic Random MIMO Channel / 15.2–4 Outage
Capacity / 15.2–5 Capacity of MIMO Channel When the
Channel Is Known at the Transmitter
15.3 Spread Spectrum Signals and Multicode Transmission 992
15.3–1 Orthogonal Spreading Sequences / 15.3–2
Multiplexing Gain Versus Diversity Gain / 15.3–3 Multicode
MIMO Systems
15.4 Coding for MIMO Channels 1001
15.4–1 Performance of Temporally Coded SISO Systems in
Rayleigh Fading Channels / 15.4–2 Bit-Interleaved Temporal
Coding for MIMO Channels / 15.4–3 Space-Time Block
Codes for MIMO Channels / 15.4–4 Pairwise Error
Probability for a Space-Time Code / 15.4–5 Space-Time
Trellis Codes for MIMO Channels / 15.4–6 Concatenated
Space-Time Codes and Turbo Codes

Contents xv
Problems 1021
Chapter 16 Multiuser Communications 1028
16.1 Introduction to Multiple Access Techniques 1028
16.2 Capacity of Multiple Access Methods 1031
16.3 Multiuser Detection in CDMA Systems 1036
16.3–1 CDMA Signal and Channel Models / 16.3–2 The
Optimum Multiuser Receiver / 16.3–3 Suboptimum
Detectors / 16.3–4 Successive Interference
Cancellation / 16.3–5 Other Types of Multiuser
Detectors / 16.3–6 Performance Characteristics of Detectors
16.4 Multiuser MIMO Systems for Broadcast Channels 1053
16.4–1 Linear Precoding of the Transmitted Signals / 16.4–2
Nonlinear Precoding of the Transmitted Signals—The QR
Decomposition / 16.4–3 Nonlinear Vector
Precoding / 16.4–4 Lattice Reduction Technique for
Precoding
16.5 Random Access Methods 1068
16.5–1 ALOHA Systems and Protocols / 16.5–2 Carrier
Sense Systems and Protocols
Problems 1078
Appendix A Matrices 1085
A.1 Eigenvalues and Eigenvectors of a Matrix 1086
A.2 Singular-Value Decomposition 1087
A.3 Matrix Norm and Condition Number 1088
A.4 The Moore–Penrose Pseudoinverse 1088
Appendix B Error Probability for Multichannel Binary Signals 1090
Appendix C Error Probabilities for Adaptive Reception
of M-Phase Signals 1096
C.1 Mathematical Model for an M-Phase Signaling Communi-
cation System 1096
C.2 Characteristic Function and Probability Density Function of
the Phase θ 1098
C.3 Error Probabilities for Slowly Fading Rayleigh Channels 1100
C.4 Error Probabilities for Time-Invariant and Ricean Fading
Channels 1104
Appendix D Square Root Factorization 1107
References and Bibliography 1109
Index 1142

P R E F A C E
It is a pleasure to welcome Professor Masoud Salehi as a coauthor to the fifth edition
of Digital Communications. This new edition has undergone a major revision and
reorganization of topics, especially in the area of channel coding and decoding. A new
chapter on multiple-antenna systems has been added as well.
The book is designed to serve as a text for a first-year graduate-level course for
students in electrical engineering. It is also designed to serve as a text for self-study
and as a reference book for the practicing engineer involved in the design and analysis
of digital communications systems. As to background, we presume that the reader has
a thorough understanding of basic calculus and elementary linear systems theory and
prior knowledge of probability and stochastic processes.
Chapter 1 is an introduction to the subject, including a historical perspective and
a description of channel characteristics and channel models.
Chapter 2 contains a review of deterministic and random signal analysis, including
bandpass and lowpass signal representations, bounds on the tail probabilities of random
variables, limit theorems for sums of random variables, and random processes.
Chapter 3 treats digital modulation techniques and the power spectrum of digitally
modulated signals.
Chapter 4 is focused on optimum receivers for additive white Gaussian noise
(AWGN) channels and their error rate performance. Also included in this chapter is
an introduction to lattices and signal constellations based on lattices, as well as link
budget analyses for wireline and radio communication systems.
Chapter 5 is devoted to carrier phase estimation and time synchronization methods
based on the maximum-likelihood criterion. Both decision-directed and non-decision-
directed methods are described.
Chapter 6 provides an introduction to topics in information theory, including
lossless source coding, lossy data compression, channel capacity for different channel
models, and the channel reliability function.
Chapter 7 treats linear block codes and their properties. Included is a treatment
of cyclic codes, BCH codes, Reed-Solomon codes, and concatenated codes. Both soft
decision and hard decision decoding methods are described, and their performance in
AWGN channels is evaluated.
Chapter 8 provides a treatment of trellis codes and graph-based codes, includ-
ing convolutional codes, turbo codes, low density parity check (LDPC) codes, trel-
lis codes for band-limited channels, and codes based on lattices. Decoding algo-
rithms are also treated, including the Viterbi algorithm and its performance on AWGN
xvi

Preface xvii
channels,theBCJRalgorithmforiterativedecodingofturbocodes,andthesum-product
algorithm.
Chapter 9 is focused on digital communication through band-limited channels.
Topics treated in this chapter include the characterization and signal design for band-
limited channels, the optimum receiver for channels with intersymbol interference and
AWGN, and suboptimum equalization methods, namely, linear equalization, decision-
feedback equalization, and turbo equalization.
Chapter 10 treats adaptive channel equalization. The LMS and recursive least-
squares algorithms are described together with their performance characteristics. This
chapter also includes a treatment of blind equalization algorithms.
Chapter 11 provides a treatment of multichannel and multicarrier modulation.
Topics treated include the error rate performance of multichannel binary signal and
M-ary orthogonal signals in AWGN channels; the capacity of a nonideal linear filter
channel with AWGN; OFDM modulation and demodulation; bit and power alloca-
tion in an OFDM system; and methods to reduce the peak-to-average power ratio in
OFDM.
Chapter 12 is focused on spread spectrum signals and systems, with emphasis
on direct sequence and frequency-hopped spread spectrum systems and their perfor-
mance. The benefits of coding in the design of spread spectrum signals is emphasized
throughout this chapter.
Chapter 13 treats communication through fading channels, including the charac-
terization of fading channels and the key important parameters of multipath spread and
Doppler spread. Several channel fading statistical models are introduced, with empha-
sis placed on Rayleigh fading, Ricean fading, and Nakagami fading. An analysis of the
performance degradation caused by Doppler spread in an OFDM system is presented,
and a method for reducing this performance degradation is described.
Chapter 14 is focused on capacity and code design for fading channels. After intro-
ducing ergodic and outage capacities, coding for fading channels is studied. Bandwidth-
efficient coding and bit-interleaved coded modulation are treated, and the performance
of coded systems in Rayleigh and Ricean fading is derived.
Chapter 15 provides a treatment of multiple-antenna systems, generally called
multiple-input, multiple-output (MIMO) systems, which are designed to yield spatial
signal diversity and spatial multiplexing. Topics treated in this chapter include detection
algorithms for MIMO channels, the capacity of MIMO channels with AWGN without
and with signal fading, and space-time coding.
Chapter 16 treats multiuser communications, including the topics of the capacity
of multiple-access methods, multiuser detection methods for the uplink in CDMA
systems, interference mitigation in multiuser broadcast channels, and random access
methods such as ALOHA and carrier-sense multiple access (CSMA).
With 16 chapters and a variety of topics, the instructor has the flexibility to design
either a one- or two-semester course. Chapters 3, 4, and 5 provide a basic treatment of
digital modulation/demodulation and detection methods. Channel coding and decoding
treated in Chapters 7, 8, and 9 can be included along with modulation/demodulation
in a one-semester course. Alternatively, Chapters 9 through 12 can be covered in place
of channel coding and decoding. A second semester course can cover the topics of

xviii Preface
communicationthroughfadingchannels,multiple-antennasystems,andmultiusercom-
munications.
The authors and McGraw-Hill would like to thank the following reviewers for their
suggestions on selected chapters of the fifth edition manuscript:
Paul Salama, Indiana University/Purdue University, Indianapolis; Dimitrios Hatz-
inakos, University of Toronto, and Ender Ayanoglu, University of California, Irvine.
Finally, the first author wishes to thank Gloria Doukakis for her assistance in typing
parts of the manuscript. We also thank Patrick Amihood for preparing several graphs
in Chapters 15 and 16 and Apostolos Rizos and Kostas Stamatiou for preparing parts
of the Solutions Manual.

Proakis-27466 book September 25, 2007 11:6
1
Introduction
In this book, we present the basic principles that underlie the analysis and design
of digital communication systems. The subject of digital communications involves the
transmission of information in digital form from a source that generates the information
to one or more destinations. Of particular importance in the analysis and design of
communication systems are the characteristics of the physical channels through which
the information is transmitted. The characteristics of the channel generally affect the
design of the basic building blocks of the communication system. Below, we describe
the elements of a communication system and their functions.
1.1
ELEMENTS OF A DIGITAL COMMUNICATION SYSTEM
Figure 1.1–1 illustrates the functional diagram and the basic elements of a digital
communication system. The source output may be either an analog signal, such as an
audio or video signal, or a digital signal, such as the output of a computer, that is discrete
in time and has a finite number of output characters. In a digital communication system,
the messages produced by the source are converted into a sequence of binary digits.
Ideally, we should like to represent the source output (message) by as few binary digits
as possible. In other words, we seek an efficient representation of the source output
that results in little or no redundancy. The process of efficiently converting the output
of either an analog or digital source into a sequence of binary digits is called source
encoding or data compression.
The sequence of binary digits from the source encoder, which we call the informa-
tion sequence, is passed to the channel encoder. The purpose of the channel encoder
is to introduce, in a controlled manner, some redundancy in the binary information
sequence that can be used at the receiver to overcome the effects of noise and inter-
ference encountered in the transmission of the signal through the channel. Thus, the
added redundancy serves to increase the reliability of the received data and improves
1

2 Digital Communications
FIGURE 1.1–1
Basic elements of a digital communication system.
the fidelity of the received signal. In effect, redundancy in the information sequence
aids the receiver in decoding the desired information sequence. For example, a (trivial)
form of encoding of the binary information sequence is simply to repeat each binary
digit m times, where m is some positive integer. More sophisticated (nontrivial) encod-
ing involves taking k information bits at a time and mapping each k-bit sequence into
a unique n-bit sequence, called a code word. The amount of redundancy introduced by
encoding the data in this manner is measured by the ratio n/k. The reciprocal of this
ratio, namely k/n, is called the rate of the code or, simply, the code rate.
The binary sequence at the output of the channel encoder is passed to the digital
modulator, which serves as the interface to the communication channel. Since nearly
all the communication channels encountered in practice are capable of transmitting
electrical signals (waveforms), the primary purpose of the digital modulator is to map
the binary information sequence into signal waveforms. To elaborate on this point, let
us suppose that the coded information sequence is to be transmitted one bit at a time at
some uniform rate R bits per second (bits/s). The digital modulator may simply map the
binary digit 0 into a waveform s0(t) and the binary digit 1 into a waveform s1(t). In this
manner, each bit from the channel encoder is transmitted separately. We call this binary
modulation. Alternatively, the modulator may transmit b coded information bits at a
time by using M = 2b
distinct waveforms si (t), i = 0, 1, . . . , M − 1, one waveform
for each of the 2b
possible b-bit sequences. We call this M-ary modulation (M > 2).
Note that a new b-bit sequence enters the modulator every b/R seconds. Hence, when
the channel bit rate R is fixed, the amount of time available to transmit one of the M
waveforms corresponding to a b-bit sequence is b times the time period in a system
that uses binary modulation.
The communication channel is the physical medium that is used to send the signal
from the transmitter to the receiver. In wireless transmission, the channel may be the
atmosphere (free space). On the other hand, telephone channels usually employ a variety
of physical media, including wire lines, optical fiber cables, and wireless (microwave
radio). Whatever the physical medium used for transmission of the information, the
essential feature is that the transmitted signal is corrupted in a random manner by a

Chapter One: Introduction 3
variety of possible mechanisms, such as additive thermal noise generated by electronic
devices; man-made noise, e.g., automobile ignition noise; and atmospheric noise, e.g.,
electrical lightning discharges during thunderstorms.
At the receiving end of a digital communication system, the digital demodulator
processes the channel-corrupted transmitted waveform and reduces the waveforms to
a sequence of numbers that represent estimates of the transmitted data symbols (binary
or M-ary). This sequence of numbers is passed to the channel decoder, which attempts
to reconstruct the original information sequence from knowledge of the code used by
the channel encoder and the redundancy contained in the received data.
A measure of how well the demodulator and decoder perform is the frequency with
which errors occur in the decoded sequence. More precisely, the average probability
of a bit-error at the output of the decoder is a measure of the performance of the
demodulator–decoder combination. In general, the probability of error is a function of
the code characteristics, the types of waveforms used to transmit the information over
the channel, the transmitter power, the characteristics of the channel (i.e., the amount
of noise, the nature of the interference), and the method of demodulation and decoding.
These items and their effect on performance will be discussed in detail in subsequent
chapters.
As a final step, when an analog output is desired, the source decoder accepts the
output sequence from the channel decoder and, from knowledge of the source encoding
method used, attempts to reconstruct the original signal from the source. Because of
channel decoding errors and possible distortion introduced by the source encoder,
and perhaps, the source decoder, the signal at the output of the source decoder is an
approximation to the original source output. The difference or some function of the
difference between the original signal and the reconstructed signal is a measure of the
distortion introduced by the digital communication system.
1.2
COMMUNICATION CHANNELS AND THEIR CHARACTERISTICS
As indicated in the preceding discussion, the communication channel provides the con-
nection between the transmitter and the receiver. The physical channel may be a pair of
wires that carry the electrical signal, or an optical fiber that carries the information on a
modulated light beam, or an underwater ocean channel in which the information is trans-
mitted acoustically, or free space over which the information-bearing signal is radiated
by use of an antenna. Other media that can be characterized as communication channels
are data storage media, such as magnetic tape, magnetic disks, and optical disks.
One common problem in signal transmission through any channel is additive noise.
In general, additive noise is generated internally by components such as resistors and
solid-state devices used to implement the communication system. This is sometimes
called thermal noise. Other sources of noise and interference may arise externally to
the system, such as interference from other users of the channel. When such noise
and interference occupy the same frequency band as the desired signal, their effect
can be minimized by the proper design of the transmitted signal and its demodulator at

the receiver. Other types of signal degradations that may be encountered in transmission
over the channel are signal attenuation, amplitude and phase distortion, and multipath
distortion.
The effects of noise may be minimized by increasing the power in the transmitted
signal. However, equipment and other practical constraints limit the power level in
the transmitted signal. Another basic limitation is the available channel bandwidth.
A bandwidth constraint is usually due to the physical limitations of the medium and
the electronic components used to implement the transmitter and the receiver. These
two limitations constrain the amount of data that can be transmitted reliably over any
communication channel as we shall observe in later chapters. Below, we describe some
of the important characteristics of several communication channels.
Wireline Channels
The telephone network makes extensive use of wire lines for voice signal transmission,
as well as data and video transmission. Twisted-pair wire lines and coaxial cable are
basically guided electromagnetic channels that provide relatively modest bandwidths.
Telephone wire generally used to connect a customer to a central office has a bandwidth
of several hundred kilohertz (kHz). On the other hand, coaxial cable has a usable
bandwidth of several megahertz (MHz). Figure 1.2–1 illustrates the frequency range of
guided electromagnetic channels, which include waveguides and optical fibers.
Signals transmitted through such channels are distorted in both amplitude and
phase and further corrupted by additive noise. Twisted-pair wireline channels are also
prone to crosstalk interference from physically adjacent channels. Because wireline
channels carry a large percentage of our daily communications around the country and
the world, much research has been performed on the characterization of their trans-
mission properties and on methods for mitigating the amplitude and phase distortion
encountered in signal transmission. In Chapter 9, we describe methods for designing
optimum transmitted signals and their demodulation; in Chapter 10, we consider the
design of channel equalizers that compensate for amplitude and phase distortion on
these channels.
Fiber-Optic Channels
Optical fibers offer the communication system designer a channel bandwidth that is
several orders of magnitude larger than coaxial cable channels. During the past two
decades, optical fiber cables have been developed that have a relatively low signal atten-
uation, and highly reliable photonic devices have been developed for signal generation
and signal detection. These technological advances have resulted in a rapid deploy-
ment of optical fiber channels, both in domestic telecommunication systems as well as
for transcontinental communication. With the large bandwidth available on fiber-optic
channels, it is possible for telephone companies to offer subscribers a wide array of
telecommunication services, including voice, data, facsimile, and video.
The transmitter or modulator in a fiber-optic communication system is a light
source, either a light-emitting diode (LED) or a laser. Information is transmitted by
varying (modulating) the intensity of the light source with the message signal. The light
propagates through the fiber as a light wave and is amplified periodically (in the case of

FIGURE 1.2–1
Frequency range for guided wire
channel.
digital transmission, it is detected and regenerated by repeaters) along the transmission
path to compensate for signal attenuation. At the receiver, the light intensity is detected
by a photodiode, whose output is an electrical signal that varies in direct proportion
to the power of the light impinging on the photodiode. Sources of noise in fiber-optic
channels are photodiodes and electronic amplifiers.
Wireless Electromagnetic Channels
In wireless communication systems, electromagnetic energy is coupled to the prop-
agation medium by an antenna which serves as the radiator. The physical size and
the configuration of the antenna depend primarily on the frequency of operation. To
obtain efficient radiation of electromagnetic energy, the antenna must be longer than

1
10
of the wavelength. Consequently, a radio station transmitting in the amplitude-
modulated (AM) frequency band, say at fc = 1 MHz [corresponding to a wavelength
of λ = c/fc = 300 meters (m)], requires an antenna of at least 30 m. Other important
characteristics and attributes of antennas for wireless transmission are described in
Chapter 4.
Figure 1.2–2 illustrates the various frequency bands of the electromagnetic spec-
trum. The mode of propagation of electromagnetic waves in the atmosphere and in
FIGURE 1.2–2
Frequency range for wireless electromagnetic channels. [Adapted from Carlson (1975), 2nd
edition, c
McGraw-Hill Book Company Co. Reprinted with permission of the publisher.]

FIGURE 1.2–3
Illustration of ground-wave propagation.
free space may be subdivided into three categories, namely, ground-wave propagation,
sky-wave propagation, and line-of-sight (LOS) propagation. In the very low frequency
(VLF) and audio frequency bands, where the wavelengths exceed 10 km, the earth
and the ionosphere act as a waveguide for electromagnetic wave propagation. In these
frequency ranges, communication signals practically propagate around the globe. For
this reason, these frequency bands are primarily used to provide navigational aids from
shore to ships around the world. The channel bandwidths available in these frequency
bands are relatively small (usually 1–10 percent of the center frequency), and hence the
information that is transmitted through these channels is of relatively slow speed and
generally confined to digital transmission. A dominant type of noise at these frequen-
cies is generated from thunderstorm activity around the globe, especially in tropical
regions. Interference results from the many users of these frequency bands.
Ground-wave propagation, as illustrated in Figure 1.2–3, is the dominant mode of
propagation for frequencies in the medium frequency (MF) band (0.3–3 MHz). This is
the frequency band used for AM broadcasting and maritime radio broadcasting. In AM
broadcasting, the range with ground-wave propagation of even the more powerful radio
stations is limited to about 150 km. Atmospheric noise, man-made noise, and thermal
noise from electronic components at the receiver are dominant disturbances for signal
transmission in the MF band.
Sky-wave propagation, as illustrated in Figure 1.2–4, results from transmitted sig-
nals being reflected (bent or refracted) from the ionosphere, which consists of several
layers of charged particles ranging in altitude from 50 to 400 km above the surface of
the earth. During the daytime hours, the heating of the lower atmosphere by the sun
causes the formation of the lower layers at altitudes below 120 km. These lower layers,
especially the D-layer, serve to absorb frequencies below 2 MHz, thus severely limiting
sky-wave propagation of AM radio broadcast. However, during the nighttime hours, the
electron density in the lower layers of the ionosphere drops sharply and the frequency
absorption that occurs during the daytime is significantly reduced. As a consequence,
powerful AM radio broadcast stations can propagate over large distances via sky wave
over the F-layer of the ionosphere, which ranges from 140 to 400 km above the surface
of the earth.
FIGURE 1.2–4
Illustration of sky-wave propagation.

A frequently occurring problem with electromagnetic wave propagation via sky
wave in the high frequency (HF) range is signal multipath. Signal multipath occurs
when the transmitted signal arrives at the receiver via multiple propagation paths at dif-
ferent delays. It generally results in intersymbol interference in a digital communication
system. Moreover, the signal components arriving via different propagation paths may
add destructively, resulting in a phenomenon called signal fading, which most people
have experienced when listening to a distant radio station at night when sky wave is
the dominant propagation mode. Additive noise in the HF range is a combination of
atmospheric noise and thermal noise.
Sky-wave ionospheric propagation ceases to exist at frequencies above approx-
imately 30 MHz, which is the end of the HF band. However, it is possible to have
ionospheric scatter propagation at frequencies in the range 30–60 MHz, resulting from
signal scattering from the lower ionosphere. It is also possible to communicate over
distances of several hundred miles by use of tropospheric scattering at frequencies in
the range 40–300 MHz. Troposcatter results from signal scattering due to particles
in the atmosphere at altitudes of 10 miles or less. Generally, ionospheric scatter and
tropospheric scatter involve large signal propagation losses and require a large amount
of transmitter power and relatively large antennas.
Frequencies above 30 MHz propagate through the ionosphere with relatively little
loss and make satellite and extraterrestrial communications possible. Hence, at fre-
quencies in the very high frequency (VHF) band and higher, the dominant mode of
electromagnetic propagation is LOS propagation. For terrestrial communication sys-
tems, this means that the transmitter and receiver antennas must be in direct LOS with
relatively little or no obstruction. For this reason, television stations transmitting in the
VHF and ultra high frequency (UHF) bands mount their antennas on high towers to
achieve a broad coverage area.
In general, the coverage area for LOS propagation is limited by the curvature of
the earth. If the transmitting antenna is mounted at a height h m above the surface of
the earth, the distance to the radio horizon, assuming no physical obstructions such
as mountains, is approximately d =
√
15h km. For example, a television antenna
mounted on a tower of 300 m in height provides a coverage of approximately 67 km.
As another example, microwave radio relay systems used extensively for telephone and
video transmission at frequencies above 1 gigahertz (GHz) have antennas mounted on
tall towers or on the top of tall buildings.
The dominant noise limiting the performance of a communication system in VHF
and UHF ranges is thermal noise generated in the receiver front end and cosmic noise
picked up by the antenna. At frequencies in the super high frequency (SHF) band above
10 GHz, atmospheric conditions play a major role in signal propagation. For example,
at 10 GHz, the attenuation ranges from about 0.003 decibel per kilometer (dB/km) in
light rain to about 0.3 dB/km in heavy rain. At 100 GHz, the attenuation ranges from
about 0.1 dB/km in light rain to about 6 dB/km in heavy rain. Hence, in this frequency
range, heavy rain introduces extremely high propagation losses that can result in service
outages (total breakdown in the communication system).
At frequencies above the extremely high frequency (EHF) band, we have the in-
frared and visible light regions of the electromagnetic spectrum, which can be used
to provide LOS optical communication in free space. To date, these frequency bands

have been used in experimental communication systems, such as satellite-to-satellite
links.
Underwater Acoustic Channels
Over the past few decades, ocean exploration activity has been steadily increasing.
Coupled with this increase is the need to transmit data, collected by sensors placed
under water, to the surface of the ocean. From there, it is possible to relay the data via
a satellite to a data collection center.
Electromagnetic waves do not propagate over long distances under water except at
extremelylowfrequencies.However,thetransmissionofsignalsatsuchlowfrequencies
is prohibitively expensive because of the large and powerful transmitters required. The
attenuation of electromagnetic waves in water can be expressed in terms of the skin
depth, which is the distance a signal is attenuated by 1/e. For seawater, the skin depth
δ = 250/
√
f , where f is expressed in Hz and δ is in m. For example, at 10 kHz, the
skin depth is 2.5 m. In contrast, acoustic signals propagate over distances of tens and
even hundreds of kilometers.
An underwater acoustic channel is characterized as a multipath channel due to
signal reflections from the surface and the bottom of the sea. Because of wave mo-
tion, the signal multipath components undergo time-varying propagation delays that
result in signal fading. In addition, there is frequency-dependent attenuation, which is
approximately proportional to the square of the signal frequency. The sound velocity
is nominally about 1500 m/s, but the actual value will vary either above or below the
nominal value depending on the depth at which the signal propagates.
Ambient ocean acoustic noise is caused by shrimp, fish, and various mammals.
Near harbors, there is also man-made acoustic noise in addition to the ambient noise.
In spite of this hostile environment, it is possible to design and implement efficient and
highly reliable underwater acoustic communication systems for transmitting digital
signals over large distances.
Storage Channels
Information storage and retrieval systems constitute a very significant part of data-
handling activities on a daily basis. Magnetic tape, including digital audiotape and
videotape, magnetic disks used for storing large amounts of computer data, optical
disks used for computer data storage, and compact disks are examples of data storage
systems that can be characterized as communication channels. The process of storing
data on a magnetic tape or a magnetic or optical disk is equivalent to transmitting
a signal over a telephone or a radio channel. The readback process and the signal
processing involved in storage systems to recover the stored information are equivalent
to the functions performed by a receiver in a telephone or radio communication system
to recover the transmitted information.
Additive noise generated by the electronic components and interference from ad-
jacent tracks is generally present in the readback signal of a storage system, just as is
the case in a telephone or a radio communication system.
The amount of data that can be stored is generally limited by the size of the disk
or tape and the density (number of bits stored per square inch) that can be achieved by

the write/read electronic systems and heads. For example, a packing density of 109
bits
per square inch has been demonstrated in magnetic disk storage systems. The speed at
which data can be written on a disk or tape and the speed at which it can be read back
are also limited by the associated mechanical and electrical subsystems that constitute
an information storage system.
Channel coding and modulation are essential components of a well-designed digital
magnetic or optical storage system. In the readback process, the signal is demodulated
and the added redundancy introduced by the channel encoder is used to correct errors
in the readback signal.
1.3
MATHEMATICAL MODELS FOR COMMUNICATION CHANNELS
In the design of communication systems for transmitting information through physical
channels, we find it convenient to construct mathematical models that reflect the most
importantcharacteristicsofthetransmissionmedium.Then,themathematicalmodelfor
the channel is used in the design of the channel encoder and modulator at the transmitter
and the demodulator and channel decoder at the receiver. Below, we provide a brief
description of the channel models that are frequently used to characterize many of the
physical channels that we encounter in practice.
The Additive Noise Channel
The simplest mathematical model for a communication channel is the additive noise
channel,illustratedinFigure1.3–1.Inthismodel,thetransmittedsignal s(t)iscorrupted
by an additive random noise process n(t). Physically, the additive noise process may
arise from electronic components and amplifiers at the receiver of the communication
system or from interference encountered in transmission (as in the case of radio signal
transmission).
If the noise is introduced primarily by electronic components and amplifiers at the
receiver, it may be characterized as thermal noise. This type of noise is characterized
statistically as a Gaussian noise process. Hence, the resulting mathematical model
for the channel is usually called the additive Gaussian noise channel. Because this
channel model applies to a broad class of physical communication channels and because
of its mathematical tractability, this is the predominant channel model used in our
communication system analysis and design. Channel attenuation is easily incorporated
into the model. When the signal undergoes attenuation in transmission through the
FIGURE 1.3–1
The additive noise channel.

FIGURE 1.3–2
The linear filter channel with
additive noise.
channel, the received signal is
r(t) = αs(t) + n(t) (1.3–1)
where α is the attenuation factor.
The Linear Filter Channel
In some physical channels, such as wireline telephone channels, filters are used to en-
sure that the transmitted signals do not exceed specified bandwidth limitations and thus
do not interfere with one another. Such channels are generally characterized mathemat-
ically as linear filter channels with additive noise, as illustrated in Figure 1.3–2. Hence,
if the channel input is the signal s(t), the channel output is the signal
r(t) = s(t) c(t) + n(t)
=
∞
−∞
c(τ)s(t − τ) dτ + n(t)
(1.3–2)
where c(t) is the impulse response of the linear filter and denotes convolution.
The Linear Time-Variant Filter Channel
Physical channels such as underwater acoustic channels and ionospheric radio chan-
nels that result in time-variant multipath propagation of the transmitted signal may be
characterized mathematically as time-variant linear filters. Such linear filters are charac-
terized by a time-variant channel impulse response c(τ; t), where c(τ; t) is the response
of the channel at time t due to an impulse applied at time t − τ. Thus, τ represents the
“age” (elapsed-time) variable. The linear time-variant filter channel with additive noise
is illustrated in Figure 1.3–3. For an input signal s(t), the channel output signal is
r(t) = s(t) c(τ; t) + n(t)
=
∞
−∞
c(τ; t)s(t − τ) dτ + n(t)
(1.3–3)
FIGURE 1.3–3
Linear time-variant filter channel with
additive noise.

A good model for multipath signal propagation through physical channels, such as
the ionosphere (at frequencies below 30 MHz) and mobile cellular radio channels, is a
special case of (1.3–3) in which the time-variant impulse response has the form
c(τ; t) =
L

k=1
ak(t)δ(τ − τk) (1.3–4)
where the {ak(t)} represents the possibly time-variant attenuation factors for the L
multipath propagation paths and {τk} are the corresponding time delays. If (1.3–4) is
substituted into (1.3–3), the received signal has the form
r(t) =
L

k=1
ak(t)s(t − τk) + n(t) (1.3–5)
Hence, the received signal consists of L multipath components, where the kth compo-
nent is attenuated by ak(t) and delayed by τk.
The three mathematical models described above adequately characterize the great
majority of the physical channels encountered in practice. These three channel models
are used in this text for the analysis and design of communication systems.
1.4
A HISTORICAL PERSPECTIVE IN THE DEVELOPMENT
OF DIGITAL COMMUNICATIONS
It is remarkable that the earliest form of electrical communication, namely telegraphy,
was a digital communication system. The electric telegraph was developed by Samuel
Morse and was demonstrated in 1837. Morse devised the variable-length binary code
in which letters of the English alphabet are represented by a sequence of dots and
dashes (code words). In this code, more frequently occurring letters are represented by
short code words, while letters occurring less frequently are represented by longer code
words. Thus, the Morse code was the precursor of the variable-length source coding
methods described in Chapter 6.
Nearly 40 years later, in 1875, Emile Baudot devised a code for telegraphy in which
every letter was encoded into fixed-length binary code words of length 5. In the Baudot
code, binary code elements are of equal length and designated as mark and space.
Although Morse is responsible for the development of the first electrical digital
communication system (telegraphy), the beginnings of what we now regard as modern
digital communications stem from the work of Nyquist (1924), who investigated the
problem of determining the maximum signaling rate that can be used over a telegraph
channel of a given bandwidth without intersymbol interference. He formulated a model
of a telegraph system in which a transmitted signal has the general form
s(t) =

n
ang(t − nT ) (1.4–1)

where g(t) represents a basic pulse shape and {an} is the binary data sequence of {±1}
transmitted at a rate of 1/T bits/s. Nyquist set out to determine the optimum pulse shape
that was band-limited to W Hz and maximized the bit rate under the constraint that the
pulsecausednointersymbolinterferenceatthesamplingtimek/T, k = 0, ±1, ±2, . . . .
His studies led him to conclude that the maximum pulse rate is 2W pulses/s. This rate
is now called the Nyquist rate. Moreover, this pulse rate can be achieved by using
the pulses g(t) = (sin 2πWt)/2πWt. This pulse shape allows recovery of the data
without intersymbol interference at the sampling instants. Nyquist’s result is equivalent
to a version of the sampling theorem for band-limited signals, which was later stated
precisely by Shannon (1948b). The sampling theorem states that a signal of bandwidth
W can be reconstructed from samples taken at the Nyquist rate of 2W samples/s using
the interpolation formula
s(t) =

n
s

n
2W

sin[2πW(t − n/2W)]
2πW(t − n/2W)
(1.4–2)
In light of Nyquist’s work, Hartley (1928) considered the issue of the amount
of data that can be transmitted reliably over a band-limited channel when multiple
amplitude levels are used. Because of the presence of noise and other interference,
Hartley postulated that the receiver can reliably estimate the received signal amplitude
to some accuracy, say Aδ. This investigation led Hartley to conclude that there is a
maximum data rate that can be communicated reliably over a band-limited channel
when the maximum signal amplitude is limited to Amax (fixed power constraint) and
the amplitude resolution is Aδ.
Another significant advance in the development of communications was the work
of Kolmogorov (1939) and Wiener (1942), who considered the problem of estimating a
desired signal waveform s(t) in the presence of additive noise n(t), based on observation
of the received signal r(t) = s(t) + n(t). This problem arises in signal demodulation.
Kolmogorov and Wiener determined the linear filter whose output is the best mean-
square approximation to the desired signal s(t). The resulting filter is called the optimum
linear (Kolmogorov–Wiener) filter.
Hartley’s and Nyquist’s results on the maximum transmission rate of digital in-
formation were precursors to the work of Shannon (1948a,b), who established the
mathematical foundations for information transmission and derived the fundamental
limits for digital communication systems. In his pioneering work, Shannon formulated
the basic problem of reliable transmission of information in statistical terms, using
probabilistic models for information sources and communication channels. Based on
such a statistical formulation, he adopted a logarithmic measure for the information
content of a source. He also demonstrated that the effect of a transmitter power con-
straint, a bandwidth constraint, and additive noise can be associated with the channel
and incorporated into a single parameter, called the channel capacity. For example,
in the case of an additive white (spectrally flat) Gaussian noise interference, an ideal
band-limited channel of bandwidth W has a capacity C given by
C = W log2

1 +
P
W N0

bits/s (1.4–3)

where P is the average transmitted power and N0 is the power spectral density of the
additive noise. The significance of the channel capacity is as follows: If the information
rate R from the source is less than C(R C), then it is theoretically possible to achieve
reliable (error-free) transmission through the channel by appropriate coding. On the
other hand, if R C, reliable transmission is not possible regardless of the amount of
signal processing performed at the transmitter and receiver. Thus, Shannon established
basic limits on communication of information and gave birth to a new field that is now
called information theory.
Another important contribution to the field of digital communication is the work
of Kotelnikov (1947), who provided a coherent analysis of the various digital commu-
nication systems based on a geometrical approach. Kotelnikov’s approach was later
expanded by Wozencraft and Jacobs (1965).
Following Shannon’s publications came the classic work of Hamming (1950) on
error-detecting and error-correcting codes to combat the detrimental effects of channel
noise. Hamming’s work stimulated many researchers in the years that followed, and a
variety of new and powerful codes were discovered, many of which are used today in
the implementation of modern communication systems.
The increase in demand for data transmission during the last four decades, coupled
with the development of more sophisticated integrated circuits, has led to the develop-
ment of very efficient and more reliable digital communication systems. In the course
of these developments, Shannon’s original results and the generalization of his results
on maximum transmission limits over a channel and on bounds on the performance
achieved have served as benchmarks for any given communication system design. The
theoretical limits derived by Shannon and other researchers that contributed to the de-
velopment of information theory serve as an ultimate goal in the continuing efforts to
design and develop more efficient digital communication systems.
There have been many new advances in the area of digital communications follow-
ing the early work of Shannon, Kotelnikov, and Hamming. Some of the most notable
advances are the following:
• The development of new block codes by Muller (1954), Reed (1954), Reed and
Solomon (1960), Bose and Ray-Chaudhuri (1960a,b), and Goppa (1970, 1971).
• The development of concatenated codes by Forney (1966a).
• The development of computationally efficient decoding of Bose–Chaudhuri-
Hocquenghem (BCH) codes, e.g., the Berlekamp–Massey algorithm (see Chien,
1964; Berlekamp, 1968).
• The development of convolutional codes and decoding algorithms by Wozencraft
and Reiffen (1961), Fano (1963), Zigangirov (1966), Jelinek (1969), Forney (1970b,
1972, 1974), and Viterbi (1967, 1971).
• The development of trellis-coded modulation by Ungerboeck (1982), Forney et al.
(1984), Wei (1987), and others.
• The development of efficient source encodings algorithms for data compression, such
as those devised by Ziv and Lempel (1977, 1978), and Linde et al. (1980).
• The development of low-density parity check (LDPC) codes and the sum-product
decoding algorithm by Gallager (1963).
• The development of turbo codes and iterative decoding by Berrou et al. (1993).

1.5
OVERVIEW OF THE BOOK
Chapter 2 presents a review of deterministic and random signal analysis. Our primary
objectives in this chapter are to review basic notions in the theory of probability and
random variables and to establish some necessary notation.
Chapters 3 through 5 treat the geometric representation of various digital modula-
tion signals, their demodulation, their error rate performance in additive, white Gaussian
noise (AWGN) channels, and methods for synchronizing the receiver to the received
signal waveforms.
Chapters 6 to 8 treat the topics of source coding, channel coding and decoding, and
basic information theoretic limits on channel capacity, source information rates, and
channel coding rates.
The design of efficient modulators and demodulators for linear filter channels with
distortion is treated in Chapters 9 and 10. Channel equalization methods are described
for mitigating the effects of channel distortion.
Chapter 11 is focused on multichannel and multicarrier communication systems,
their efficient implementation, and their performance in AWGN channels.
Chapter12presentsanintroductiontodirectsequenceandfrequencyhoppedspread
spectrum signals and systems and an evaluation of their performance under worst-case
interference conditions.
The design of signals and coding techniques for digital communication through
fading multipath channels is the focus of Chapters 13 and 14. This material is especially
relevant to the design and development of wireless communication systems.
Chapter 15 treats the use of multiple transmit and receive antennas for improv-
ing the performance of wireless communication systems through signal diversity and
increasing the data rate via spatial multiplexing. The capacity of multiple antenna
systems is evaluated and space-time codes are described for use in multiple antenna
communication systems.
Chapter 16 of this book presents an introduction to multiuser communication
systems and multiple access methods. We consider detection algorithms for uplink
transmission in which multiple users transmit data to a common receiver (a base
station) and evaluate their performance. We also present algorithms for suppressing
multiple access interference in a broadcast communication system in which a transmit-
ter employing multiple antennas transmits different data sequences simultaneously to
different users.
1.6
BIBLIOGRAPHICAL NOTES AND REFERENCES
There are several historical treatments regarding the development of radio and telecom-
munications during the past century. These may be found in the books by McMahon
(1984), Millman (1984), and Ryder and Fink (1984). We have already cited the classi-
cal works of Nyquist (1924), Hartley (1928), Kotelnikov (1947), Shannon (1948), and

Hamming (1950), as well as some of the more important advances that have occurred
in the field since 1950. The collected papers by Shannon have been published by IEEE
Press in a book edited by Sloane and Wyner (1993) and previously in Russia in a
book edited by Dobrushin and Lupanov (1963). Other collected works published by
the IEEE Press that might be of interest to the reader are Key Papers in the Development
of Coding Theory, edited by Berlekamp (1974), and Key Papers in the Development of
Information Theory, edited by Slepian (1974).

2
Deterministic and Random Signal Analysis
In this chapter we present the background material needed in the study of the following
chapters. The analysis of deterministic and random signals and the study of different
methods for their representation are the main topics of this chapter. In addition, we
also introduce and study the main properties of some random variables frequently
encountered in analysis of communication systems. We continue with a review of
random processes, properties of lowpass and bandpass random processes, and series
expansion of random processes.
Throughout this chapter, and the book, we assume that the reader is familiar with
the properties of the Fourier transform as summarized in Table 2.0–1 and the important
Fourier transform pairs given in Table 2.0–2.
In these tables we have used the following signal definitions.
(t) =
⎧
⎪
⎨
⎪
⎩
1 |t| 1
2
1
2
t = ±1
2
0 otherwise
sinc(t) =

sin(πt)
πt
t = 0
1 t = 0
and
sgn(t) =
⎧
⎪
⎨
⎪
⎩
1 t 0
−1 t 0
0 t = 0
(t) = (t) (t) =
⎧
⎪
⎨
⎪
⎩
t + 1 −1 ≤ t 0
−t + 1 0 ≤ t 1
0 otherwise
The unit step signal u−1(t) is defined as
u−1(t) =
⎧
⎪
⎨
⎪
⎩
1 t 0
1
2
t = 0
0 t 0
We also assume that the reader is familiar with elements of probability, random
variables, and random processes as covered in standard texts such as Papoulis and Pillai
(2002), Leon-Garcia (1994), and Stark and Woods (2002).
17

TABLE 2.0–1
Table of Fourier Transform Properties
Property Signal Fourier Transform
Linearity αx1(t) + βx2(t) αX1( f ) + βX2( f )
Duality X(t) x(− f )
Conjugacy x∗
(t) X∗
(− f )
Time-scaling (a = 0) x(at) 1
|a|
X
f
a

Time-shift x(t − t0) e− j2π f t0
X( f )
Modulation ej2π f0t
x(t) X( f − f0)
Convolution x(t) y(t) X( f )Y( f )
Multiplication x(t)y(t) X( f ) Y( f )
Differentiation dn
dtn x(t) ( j2π f )n
X( f )
Differentiation in frequency tn
x(t)
j
2π
n dn
d f n X( f )
Integration
t
−∞
x(τ) dτ X( f )
j2π f
+ 1
2
X(0)δ( f )
Parseval’s theorem
∞
−∞
x(t)y∗
(t) dt =
∞
−∞
X( f )Y∗
( f ) d f
Rayleigh’s theorem
∞
−∞
|x(t)|2
dt =
∞
−∞
|X( f )|2
d f
2.1
BANDPASS AND LOWPASS SIGNAL REPRESENTATION
As was discussed in Chap. 1, the process of communication consists of transmission
of the output of an information source over a communication channel. In almost all
cases, the spectral characteristics of the information sequence do not directly match the
spectral characteristics of the communication channel, and hence the information signal
cannot be directly transmitted over the channel. In many cases the information signal
is a low frequency (baseband) signal, and the available spectrum of the communication
channel is at higher frequencies. Therefore, at the transmitter the information signal is
translated to a higher frequency signal that matches the properties of the communication
channel. This is the modulation process in which the baseband information signal is
turned into a bandpass modulated signal. In this section we study the main properties
of baseband and bandpass signals.
2.1–1 Bandpass and Lowpass Signals
In this section we will show that any real, narrowband, and high frequency signal—
called a bandpass signal—can be represented in terms of a complex low frequency

Chapter Two: Deterministic and Random Signal Analysis 19
TABLE 2.0–2
Table of Fourier Transform Pairs
Time Domain Frequency Domain
δ(t) 1
1 δ( f )
δ(t − t0) e− j2π f t0
ej2π f0t
δ( f − f0)
cos(2π f0t) 1
2
δ( f − f0) + 1
2
δ( f + f0)
sin(2π f0t) 1
2 j
δ( f − f0) − 1
2 j
δ( f + f0)
(t) sinc( f )
sinc(t) ( f )
(t) sinc2
( f )
sinc2
(t) ( f )
e−αt
u−1(t), α 0 1
α+ j2π f
te−αt
u−1(t), α 0 1
(α+ j2π f )2
e−α|t|
(α 0) 2α
α2+(2π f )2
e−πt2
e−π f 2
sgn(t) 1
jπ f
u−1(t) 1
2
δ( f ) + 1
j2π f
1
2
δ(t) + j 1
2πt
u−1( f )
δ
(t) j2π f
δ(n)
(t) ( j2π f )n
1
t
− jπ sgn( f )
∞
n=−∞
δ(t − nT0) 1
T0
∞
n=−∞
δ f −
n
T0
signal, called the lowpass equivalent of the original bandpass signal. This result makes
it possible to work with the lowpass equivalents of bandpass signals instead of directly
working with them, thus greatly simplifying the handling of bandpass signals. That is
so because applying signal processing algorithms to lowpass signals is much easier due
to lower required sampling rates which in turn result in lower rates of the sampled data.
The Fourier transform of a signal provides information about the frequency content,
or spectrum, of the signal. The Fourier transform of a real signal x(t) has Hermitian
symmetry, i.e., X(− f ) = X∗
( f ), from which we conclude that |X(− f )| = |X( f )| and
X∗
( f ) = − X( f ). In other words, for real x(t), the magnitude of X( f ) is even and

W
X( f )
f
FIGURE 2.1–1
The spectrum of a real-valued lowpass
(baseband) signal.
its phase is odd. Because of this symmetry, all information about the signal is in the
positive (or negative) frequencies, and in particular x(t) can be perfectly reconstructed
by specifying X( f ) for f ≥ 0. Based on this observation, for a real signal x(t), we
define the bandwidth as the smallest range of positive frequencies such that X( f ) = 0
when | f | is outside this range. It is clear that the bandwidth of a real signal is one-half
of its frequency support set.
A lowpass, or baseband, signal is a signal whose spectrum is located around the
zero frequency. For instance, speech, music, and video signals are all lowpass signals,
although they have different spectral characteristics and bandwidths. Usually lowpass
signals are low frequency signals, which means that in the time domain, they are slowly
varying signals with no jumps or sudden variations. The bandwidth of a real lowpass
signal is the minimum positive W such that X( f ) = 0 outside [−W, +W]. For these
signals the frequency support, i.e., the range of frequencies for which X( f ) = 0, is
[−W, +W]. An example of the spectrum of a real-valued lowpass signal is shown in
Fig. 2.1–1. The solid line shows the magnitude spectrum |X( f )|, and the dashed line
indicates the phase spectrum X( f ).
We also define the positive spectrum and the negative spectrum of a signal x(t) as
X+( f ) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
X( f ) f 0
1
2
X(0) f = 0
0 f 0
X−( f ) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
X( f ) f 0
1
2
X(0) f = 0
0 f 0
(2.1–1)
It is clear that X+( f ) = X( f )u−1( f ), X−( f ) = X( f )u−1(− f ) and X( f ) = X+( f ) +
X−( f ). For a real signal x(t), since X( f ) is Hermitian, we have X−( f ) = X∗
+(− f ).
For a complex signal x(t), the spectrum X( f ) is not symmetric; hence, the signal
cannot be reconstructed from the information in the positive frequencies only. For
complex signals, we define the bandwidth as one-half of the entire range of frequencies
over which the spectrum is nonzero, i.e., one-half of the frequency support of the signal.
This definition is for consistency with the definition of bandwidth for real signals. With
this definition we can state that in general and for all signals, real or complex, the
bandwidth is defined as one-half of the frequency support.
Inpractice,thespectralcharacteristicsofthemessagesignalandthecommunication
channel do not always match, and it is required that the message signal be modulated
by one of the many different modulation methods to match its spectral characteristics to

X( f )
X( f ) X( f )
f
f0
f0
FIGURE 2.1–2
The spectrum of a real-valued bandpass signal.
the spectral characteristics of the channel. In this process, the spectrum of the lowpass
message signal is translated to higher frequencies. The resulting modulated signal is a
bandpass signal.
A bandpass signal is a real signal whose frequency content, or spectrum, is located
aroundsomefrequency± f0 whichisfarfromzero.Moreformally,wedefineabandpass
signal to be a real signal x(t) for which there exists positive f0 and W such that the
positive spectrum of X( f ), i.e., X+( f ), is nonzero only in the interval [ f0 − W/2, f0 +
W/2], where W/2 f0 (in practice, usually W f0). The frequency f0 is called the
central frequency. Obviously, the bandwidth of x(t) is at most equal to W. Bandpass
signals are usually high frequency signals which are characterized by rapid variations
in the time domain.
An example of the spectrum of a bandpass signal is shown in Figure 2.1–2. Note
that since the signal x(t) is real, its magnitude spectrum (solid line) is even, and its phase
spectrum (dashed line) is odd. Also, note that the central frequency f0 is not necessarily
the midband frequency of the bandpass signal. Due to the symmetry of the spectrum,
X+( f ) has all the information that is necessary to reconstruct X( f ). In fact we can write
X( f ) = X+( f ) + X−( f ) = X+( f ) + X∗
+(− f ) (2.1–2)
which means that knowledge of X+( f ) is sufficient to reconstruct X( f ).
2.1–2 Lowpass Equivalent of Bandpass Signals
We start by defining the analytic signal, or the pre-envelope, corresponding to x(t) as
the signal x+(t) whose Fourier transform is X+( f ). This signal contains only positive
frequency components, and its spectrum is not Hermitian. Therefore, in general, x+(t)
is a complex signal. We have
x+(t) = F −1
[X+( f )]
= F −1
[X( f )u−1( f )]
= x(t)
1
2
δ(t) + j
1
2πt

=
1
2
x(t) +
j
2

x(t)
(2.1–3)

Xl( f ) 2X( f f0 )
f
FIGURE 2.1–3
The spectrum of the lowpass equivalent of the
signal shown in Figure 2.1–2.
where
x(t) = 1
πt
x(t) is the Hilbert transform of x(t). The Hilbert transform of x(t) is
obtained by introducing a phase shift of −π
2
at positive frequency components of x(t)
and π
2
at negative frequencies. In the frequency domain we have
F

x(t)

= − jsgn( f )X( f ) (2.1–4)
Some of the properties of the Hilbert transform will be covered in the problems at the
end of this chapter.
Now we define xl(t), the lowpass equivalent, or the complex envelope, of x(t), as
the signal whose spectrum is given by 2X+( f + f0), i.e.,
Xl( f ) = 2X+( f + f0) = 2X( f + f0)u−1( f + f0) (2.1–5)
Obviously the spectrum of xl(t) is located around the zero frequency, and therefore it is
in general a complex lowpass signal. This signal is called the lowpass equivalent or the
complex envelope of x(t). The spectrum of the lowpass equivalent of the signal shown
in Figure 2.1–2 is shown in Figure 2.1–3.
Applying the modulation theorem of the Fourier transform, we obtain
xl(t) = F −1
[Xl( f )]
= 2x+(t)e− j2π f0t
= (x(t) + j
x(t))e− j2π f0t
(2.1–6)
= (x(t) cos 2π f0t +
x(t) sin 2π f0t)
+ j(
x(t) cos 2π f0t − x(t) sin 2π f0t) (2.1–7)
From Equation 2.1–6 we can write
x(t) = Re

xl(t)ej2π f0t

(2.1–8)
This relation expresses any bandpass signals in terms of its lowpass equivalent. Using
Equations 2.1–2 and 2.1–5, we can write
X( f ) = 1
2

Xl( f − f0) + X∗
l (− f − f0)

(2.1–9)
Equations 2.1–8, 2.1–9, 2.1–5, and 2.1–7 express x(t) and xl(t) in terms of each other
in the time and frequency domains.
The real and imaginary parts of xl(t) are called the in-phase component and the
quadrature component of x(t), respectively, and are denoted by xi (t) and xq(t). Both
xi (t) and xq(t) are real-valued lowpass signals, and we have
xl(t) = xi (t) + jxq(t) (2.1–10)

Comparing Equations 2.1–10 and 2.1–7, we conclude that
xi (t) = x(t) cos 2π f0t +
x(t) sin 2π f0t
xq(t) =
x(t) cos 2π f0t − x(t) sin 2π f0t
(2.1–11)
Solving Equation 2.1–11 for x(t) and
x(t) gives
x(t) = xi (t) cos 2π f0t − xq(t) sin 2π f0t

x(t) = xq(t) cos 2π f0t + xi (t) sin 2π f0t
(2.1–12)
Equation 2.1–12 shows that any bandpass signal x(t) can be expressed in terms of
two lowpass signals, namely, its in-phase and quadrature components.
Equation 2.1–10 expresses xl(t) in terms of its real and complex parts. We can
write a similar relation in polar coordinates expressing x(t) in terms of its magnitude
and phase. If we define the envelope and phase of x(t), denoted by rx (t) and θx (t),
respectively, by
rx (t) =

x2
i (t) + x2
q (t) (2.1–13)
θx (t) = arctan
xq(t)
xi (t)
(2.1–14)
we have
xl(t) = rx (t)ejθx (t)
(2.1–15)
Substituting this result into Equation 2.1–8 gives
x(t) = Re

rx (t)ej(2π f0t+θx (t))

(2.1–16)
resulting in
x(t) = rx (t) cos (2π f0t + θx (t)) (2.1–17)
A bandpass signal and its envelope are shown in Figure 2.1–4.
FIGURE 2.1–4
A bandpass signal. The dashed curve denotes the envelope.

It is important to note that xl(t)—and consequently xi (t), xq(t), rx (t), and θx (t)—
depends on the choice of the central frequency f0. For a given bandpass signal x(t),
different values of f0—as long as X+( f ) is nonzero only in the interval [ f0 −W/2, f0 +
W/2], where W/2 f0—yield different lowpass signals xl(t). Therefore, it makes
more sense to define the lowpass equivalent of a bandpass signal with respect to a
specific f0. Since in most cases the choice of f0 is clear, we usually do not make this
distinction.
Equations 2.1–12 and 2.1–17 provide two methods for representing a bandpass
signal x(t) in terms of two lowpass signals, one in terms of the in-phase and quadrature
components and one in terms of the envelope and the phase. The two relations given in
Equations 2.1–8 and 2.1–12 that express the bandpass signal in terms of the lowpass
component(s) define the modulation process, i.e., the process of going from lowpass to
bandpass. The system that implements this process is called a modulator. The structure
of a general modulator implementing Equations 2.1–8 and 2.1–12 is shown in Fig-
ure 2.1–5(a) and (b). In this figure double lines and double blocks indicate complex
values and operations.
Similarly, Equations 2.1–7 and 2.1–11 represent how xl(t), or xi (t) and xq(t), can
be obtained from the bandpass signal x(t). This process, i.e., extracting the lowpass
signal from the bandpass signal, is called the demodulation process and is shown in
Figure 2.1–6(a) and (b). In these block diagrams the block denoted by H represents
a Hilbert transform, i.e., an LTI system with impulse response h(t) = 1
πt
and transfer
function H( f ) = − jsgn( f ).
xl(t) x(t)

xl(t) x(t)
xi(t)
xq(t)
cos 2␲f0t
sin 2␲f0t

xl(t)
2x(t)
x(t)
Re(·)
f0
Modulator
(a) (b)
(c)
ej2␲f0
t
FIGURE 2.1–5
A complex (a) and real (b) modulator. A general representation for a modulator is
shown in (c).

x(t) xl(t)
f0
Demodulator
(a) (b)
(c)
xi(t)
xq(t)
x(t)
cos 2␲f0t
sin 2␲f0t

sin 2␲f0t
cos 2␲f0t

Ᏼ

xl(t)
x(t)
ej2␲f0
t
2x(t)

Ᏼ
x̂(t)
j
x̂(t)
FIGURE 2.1–6
A complex (a) and real (b) demodulator. A general representation for a demodulator is
shown in (c).
2.1–3 Energy Considerations
In this section we study the relation between energy contents of the signals introduced
in the preceding pages. The energy of a signal x(t) is defined as
Ex =
∞
−∞
|x(t)|2
dt (2.1–18)
and by Rayleigh’s relation from Table 2.0–1 we can write
Ex =
∞
−∞
|x(t)|2
dt =
∞
−∞
|X( f )|2
dt (2.1–19)
Since there is no overlap between X+( f ) and X−( f ), we have X+( f )X−( f ) = 0,
and hence
Ex =
∞
−∞
|X+( f ) + X−( f )|2
d f
=
∞
−∞
|X+( f )|2
d f +
∞
−∞
|X−( f )|2
d f
= 2
∞
−∞
|X+( f )|2
d f
= 2Ex+
(2.1–20)

On the other hand,
Ex = 2
∞
−∞
|X+( f )|2
d f
= 2
∞
−∞

Xl( f )
2

2
d f
=
1
2
Exl
(2.1–21)
This shows that the energy in the lowpass equivalent signal is twice the energy in the
bandpass signal.
We define the inner product of two signals x(t) and y(t) as
x(t), y(t) =
∞
−∞
x(t)y∗
(t) dt =
∞
−∞
X( f )Y∗
( f ) d f (2.1–22)
where we have used Parseval’s relation from Table 2.0–1. Obviously
Ex = x(t), x(t) (2.1–23)
In Problem 2.2 we prove that if x(t) and y(t) are two bandpass signals with lowpass
equivalents xl(t) and yl(t) with respect to the same f0, then
x(t), y(t) =
1
2
Re [ xl(t), yl(t) ] (2.1–24)
The complex quantity ρx,y, called the cross-correlation coefficient of x(t) and y(t), is
defined as
ρx,y =
x(t), y(t)

Ex Ey
(2.1–25)
and represents the normalized inner product between two signals. From Exl
= 2Ex and
Equation 2.1–24 we can conclude that if x(t) and y(t) are bandpass signals with the
same f0, then
ρx,y = Re (ρxl ,yl
) (2.1–26)
Two signals are orthogonal if their inner product (and subsequently, their ρ) is
zero. Note that if ρxl ,yl
= 0, then using Equation 2.1–26, we have ρx,y = 0; but the
converse is not necessarily true. In other words, orthogonality in the baseband implies
orthogonality in the pass band, but not vice versa.
EXAMPLE 2.1–1. Assume that m(t) is a real baseband signal with bandwidth W, and
define two signals x(t) = m(t) cos 2π f0t and y(t) = m(t) sin 2π f0t, where f0 W.
Comparing these relations with Equation 2.1–12, we conclude that
xi (t) = m(t) xq(t) = 0
yi (t) = 0 yq(t) = −m(t)

or, equivalently,
xl(t) = m(t)
yl(t) = − jm(t)
Note that here
ρxl ,yl
= j
∞
−∞
m2
(t) = jEm
Therefore,
ρx,y = Re (ρxl ,yl
) = Re ( jEm) = 0
This means that x(t) and y(t) are orthogonal, but their lowpass equivalents are not
orthogonal.
2.1–4 Lowpass Equivalent of a Bandpass System
A bandpass system is a system whose transfer function is located around a frequency
f0 (and its mirror image − f0). More formally, we define a bandpass system as a system
whose impulse response h(t) is a bandpass signal. Since h(t) is bandpass, it has a
lowpass equivalent denoted by hl(t) where
h(t) = Re

hl(t)ej2π f0t

(2.1–27)
If a bandpass signal x(t) passes through a bandpass system with impulse response
h(t), then obviously the output will be a bandpass signal y(t). The relation between the
spectra of the input and the output is given by
Y( f ) = X( f )H( f ) (2.1–28)
Using Equation 2.1–5, we have
Yl( f ) = 2Y( f + f0)u−1( f + f0)
= 2X( f + f0)H( f + f0)u−1( f + f0)
=
1
2
[2X( f + f0)u−1( f + f0)] [2H( f + f0)u−1( f + f0)]
=
1
2
Xl( f )Hl( f ) (2.1–29)
where we have used the fact that for f − f0, which is the range of frequencies of
interest, u2
−1( f + f0) = u−1( f + f0) = 1. In the time domain we have
yl(t) =
1
2
xl(t) hl(t) (2.1–30)
Equations 2.1–29 and 2.1–30 show that when a bandpass signal passes through a
bandpass system, the input-output relation between the lowpass equivalents is very
similar to the relation between the bandpass signals, the only difference being that for
the lowpass equivalents a factor of 1
2
is introduced.

2.2
SIGNAL SPACE REPRESENTATION OF WAVEFORMS
Signal space (or vector) representation of signals is a very effective and useful tool in
the analysis of digitally modulated signals. We cover this important approach in this
section and show that any set of signals is equivalent to a set of vectors. We show that
signals have the same basic properties of vectors. We study methods of determining an
equivalent set of vectors for a set of signals and introduce the notion of signal space
representation, or signal constellation, of a set of waveforms.
2.2–1 Vector Space Concepts
A vector v in an n-dimensional space is characterized by its n components v1 v2 · · · vn.
Let v denote a column vector, i.e., v = [v1 v2 · · · vn]t
, where At
denotes the transpose
of matrix A. The inner product of two n-dimensional vectors v1 = [v11 v12 · · · v1n]t
and v2 = [v21 v22 · · · v2n]t
is defined as
v1, v2 = v1 · v2 =
n
i=1
v1i v∗
2i = vH
2 v1 (2.2–1)
where AH
denotes the Hermitian transpose of the matrix A, i.e., the result of first
transposing the matrix and then conjugating its elements. From the definition of the
inner product of two vectors it follows that
v1, v2 = v2, v1
∗
(2.2–2)
and therefore,
v1, v2 + v2, v1 = 2 Re [ v1, v2 ] (2.2–3)
A vector may also be represented as a linear combination of orthogonal unit vectors
or an orthonormal basis ei , 1 ≤ i ≤ n, i.e.,
v =
n
i=1
vi ei (2.2–4)
where, by definition, a unit vector has length unity and vi is the projection of the vector
v onto the unit vector ei , i.e., vi = v, ei . Two vectors v1 and v2 are orthogonal if
v1, v2 = 0. More generally, a set of m vectors vk, 1 ≤ k ≤ m, are orthogonal if
vi , v j = 0 for all 1 ≤ i, j ≤ m, and i = j. The norm of a vector v is denoted by v
and is defined as
v = ( v, v )1/2
=

n
i=1
|vi |2 (2.2–5)
which in the n-dimensional space is simply the length of the vector. A set of m vec-
tors is said to be orthonormal if the vectors are orthogonal and each vector has a

unit norm. A set of m vectors is said to be linearly independent if no one vector can be
represented as a linear combination of the remaining vectors. Any two n-dimensional
vectors v1 and v2 satisfy the triangle inequality
v1 + v2 ≤ v1 + v2 (2.2–6)
with equality if v1 and v2 are in the same direction, i.e., v1 = av2 where a is a positive
real scalar. The Cauchy–Schwarz inequality states that
| v1, v2 | ≤ v1 · v2 (2.2–7)
with equality if v1 = av2 for some complex scalar a. The norm square of the sum of
two vectors may be expressed as
v1 + v2
2
= v1
2
+ v2
2
+ 2 Re [ v1, v2 ] (2.2–8)
If v1 and v2 are orthogonal, then v1, v2 = 0 and, hence,
v1 + v2
2
= v1
2
+ v2
2
(2.2–9)
This is the Pythagorean relation for two orthogonal n-dimensional vectors. From matrix
algebra, we recall that a linear transformation in an n-dimensional vector space is a
matrix transformation of the form v
= Av, where the matrix A transforms the vector
v into some vector v
. In the special case where v
= λv, i.e.,
Av = λv
where λ is some scalar, the vector v is called an eigenvector of the transformation and
λ is the corresponding eigenvalue.
Finally, let us review the Gram–Schmidt procedure for constructing a set of or-
thonormal vectors from a set of n-dimensional vectors vi , 1 ≤ i ≤ m. We begin by
arbitrarily selecting a vector from the set, say, v1. By normalizing its length, we obtain
the first vector, say,
u1 =
v1
v1
(2.2–10)
Next, we may select v2 and, first, subtract the projection of v2 onto u1. Thus, we obtain
u
2 = v2 − ( v2, u1 )u1 (2.2–11)
Then we normalize the vector u
2 to unit length. This yields
u2 =
u
2
u
2
(2.2–12)
The procedure continues by selecting v3 and subtracting the projections of v3 into u1
and u2. Thus, we have
u
3 = v3 − ( v3, u1 )u1 − ( v3, u2 )u2 (2.2–13)
Then the orthonormal vector u3 is
u3 =
u
3
u
3
(2.2–14)

By continuing this procedure, we construct a set of N orthonormal vectors, where
N ≤ min(m, n).
2.2–2 Signal Space Concepts
As in the case of vectors, we may develop a parallel treatment for a set of signals. The
inner product of two generally complex-valued signals x1(t) and x2(t) is denoted by
x1(t), x2(t) and defined as
x1(t), x2(t) =
∞
−∞
x1(t)x∗
2 (t) dt (2.2–15)
similar to Equation 2.1–22. The signals are orthogonal if their inner product is zero.
The norm of a signal is defined as
x(t) =
∞
−∞
|x(t)|2
dt
1/2
=

Ex (2.2–16)
where Ex is the energy in x(t). A set of m signals is orthonormal if they are orthogonal
and their norms are all unity. A set of m signals is linearly independent if no signal can
be represented as a linear combination of the remaining signals. The triangle inequality
for two signals is simply
x1(t) + x2(t) ≤ x1(t) + x2(t) (2.2–17)
and the Cauchy–Schwarz inequality is
| x1(t), x2(t) | ≤ x1(t) · x2(t) =

Ex1
Ex2
(2.2–18)
or, equivalently,

∞
−∞
x1(t)x∗
2 (t) dt

≤

∞
−∞
|x1(t)|2
dt

1/2

∞
−∞
|x2(t)|2
dt

1/2
(2.2–19)
with equality when x2(t) = ax1(t), where a is any complex number.
2.2–3 Orthogonal Expansions of Signals
In this section, we develop a vector representation for signal waveforms, and thus we
demonstrate an equivalence between a signal waveform and its vector representation.
Suppose that s(t) is a deterministic signal with finite energy
Es =
∞
−∞
|s(t)|2
dt (2.2–20)
Furthermore, suppose that there exists a set of functions {φn(t), n = 1, 2, . . . , K} that
are orthonormal in the sense that
φn(t), φm(t) =
∞
−∞
φn(t)φ∗
m(t) dt =

1 m = n
0 m = n
(2.2–21)

We may approximate the signal s(t) by a weighted linear combination of these func-
tions, i.e.,

s(t) =
K
k=1
skφk(t) (2.2–22)
where {sk, 1 ≤ k ≤ K} are the coefficients in the approximation of s(t). The approx-
imation error incurred is
e(t) = s(t) −
s(t)
Let us select the coefficients {sk} so as to minimize the energy Ee of the approximation
error. Thus,
Ee =
∞
−∞
|s(t) −
s(t)|2
dt (2.2–23)
=
∞
−∞

s(t) −
K
k=1
skφk(t)

2
dt (2.2–24)
The optimum coefficients in the series expansion of s(t) may be found by differentiating
Equation 2.2–23 with respect to each of the coefficients {sk} and setting the first deriva-
tives to zero. Alternatively, we may use a well-known result from estimation theory
based on the mean square error criterion, which, simply stated, is that the minimum
of Ee with respect to the {sk} is obtained when the error is orthogonal to each of the
functions in the series expansion. Thus,
∞
−∞

s(t) −
K
k=1
skφk(t)

φ∗
n (t) dt = 0, n = 1, 2, . . . , K (2.2–25)
Since the functions {φn(t)} are orthonormal, Equation 2.2–25 reduces to
sn = s(t), φn(t) =
∞
−∞
s(t)φ∗
n (t) dt, n = 1, 2, . . . , K (2.2–26)
Thus, the coefficients are obtained by projecting the signal s(t) onto each of the
functions {φn(t)}. Consequently,
s(t) is the projection of s(t) onto the K-dimensional
signal space spanned by the functions {φn(t)}, and therefore it is orthogonal to the error
signal e(t) = s(t) −
s(t), i.e., e(t),
s(t) = 0. The minimum mean-square approxima-
tion error is
Emin =
∞
−∞
e(t)s∗
(t) dt (2.2–27)
=
∞
−∞
|s(t)|2
dt −
∞
−∞
K
k=1
skφk(t)s∗
(t) dt (2.2–28)
= Es −
K
k=1
|sk|2
(2.2–29)

which is nonnegative, by definition. When the minimum mean square approximation
error Emin = 0,
Es =
K
k=1
|sk|2
=
∞
−∞
|s(t)|2
dt (2.2–30)
Under the condition that Emin = 0, we may express s(t) as
s(t) =
K
k=1
skφk(t) (2.2–31)
where it is understood that equality of s(t) to its series expansion holds in the sense that
the approximation error has zero energy.
When every finite energy signal can be represented by a series expansion of the
form in Equation 2.2–31 for which Emin = 0, the set of orthonormal functions {φn(t)}
is said to be complete.
EXAMPLE 2.2–1. TRIGONOMETRIC FOURIER SERIES: Consider a finite energy real sig-
nal s(t) that is zero everywhere except in the range 0 ≤ t ≤ T and has a finite number
of discontinuities in this interval. Its periodic extension can be represented in a Fourier
series as
s(t) =
∞
k=0
ak cos
2πkt
T
+ bk sin
2πkt
T

(2.2–32)
where the coefficients {ak, bk} that minimize the mean square error are given by
a0 =
1
T
T
0
s(t) dt
ak =
2
T
T
0
s(t) cos
2πkt
T
dt, k = 1, 2, 3, . . .
bk =
2
T
T
0
s(t) sin
2πkt
T
dt, k = 1, 2, 3, . . .
(2.2–33)
The set of functions {1/
√
T ,
√
2/T cos 2πkt/T ,
√
2/T sin 2πkt/T } is a complete
set for the expansion of periodic signals on the interval [0, T ], and, hence, the series
expansion results in zero mean square error.
EXAMPLE 2.2–2. EXPONENTIAL FOURIER SERIES: Consider a general finite energy sig-
nal s(t) (real or complex) that is zero everywhere except in the range 0 ≤ t ≤ T and
has a finite number of discontinuities in this interval. Its periodic extension can be
represented in an exponential Fourier series as
s(t) =
∞
n=−∞
xn ej2π n
T t
(2.2–34)
where the coefficients {xn} that minimize the mean square error are given by
xn =
1
T
∞
−∞
x(t)e− j2π n
T t
dt (2.2–35)

The set of functions {
√
1/T ej2π n
T t
} is a complete set for expansion of periodic signals
on the interval [0, T ], and, hence, the series expansion results in zero mean square
error.
2.2–4 Gram-Schmidt Procedure
Now suppose that we have a set of finite energy signal waveforms {sm(t), m = 1, 2, . . . ,
M} and we wish to construct a set of orthonormal waveforms. The Gram-Schmidt
orthogonalization procedure allows us to construct such a set. This procedure is similar
to the one described in Section 2.2–1 for vectors. We begin with the first waveform
s1(t), which is assumed to have energy E1. The first orthonormal waveform is simply
constructed as
φ1(t) =
s1(t)
√
E1
(2.2–36)
Thus, φ1(t) is simply s1(t) normalized to unit energy. The second waveform is con-
structed from s2(t) by first computing the projection of s2(t) onto φ1(t), which is
c21 = s2(t), φ1(t) =
∞
−∞
s2(t)φ∗
1 (t) dt (2.2–37)
Then c21φ1(t) is subtracted from s2(t) to yield
γ2(t) = s2(t) − c21φ1(t) (2.2–38)
This waveform is orthogonal to φ1(t), but it does not have unit energy. If E2 denotes
the energy of γ2(t), i.e.,
E2 =
∞
−∞
γ 2
2(t) dt
the normalized waveform that is orthogonal to φ1(t) is
φ2(t) =
γ2(t)
√
E2
(2.2–39)
In general, the orthogonalization of the kth function leads to
φk(t) =
γk(t)
√
Ek
(2.2–40)
where
γk(t) = sk(t) −
k−1
i=1
cki φi (t) (2.2–41)
cki = sk(t), φi (t) =
∞
−∞
sk(t)φ∗
i (t) dt, i = 1, 2, . . . , k − 1 (2.2–42)
Ek =
∞
−∞
γ 2
k(t) dt (2.2–43)

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