Digital communications john g proakis, masoud salehi 5th edition

C O N T E N T S1.1 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 Histori

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

www.elsolucionario.org

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

www.elsolucionario.org

B R I E F C O N T E N T S

Chapter 9 Digital Communication Through Band-Limited

Chapter 12 Spread Spectrum Signals for Digital Communications 762

Chapter 13 Fading Channels I: Characterization and Signaling 830

Appendices

Appendix B Error Probability for Multichannel Binary Signals 1090

Appendix C Error Probabilities for Adaptive Reception of M-Phase

C O N T E N T S

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

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

2.5 Limit Theorems for Sums of Random Variables 63

2.6–1 Complex Random Vectors

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

2.8–1 Sampling Theorem for Band-Limited Random Processes / 2.8–2 The Karhunen-Lo`eve Expansion

2.10 Bibliographical Notes and References 82

3.1 Representation of Digitally Modulated Signals 95

3.2–1 Pulse Amplitude Modulation (PAM) / 3.2–2 Phase Modulation / 3.2–3 Quadrature Amplitude

Modulation / 3.2–4 Multidimensional Signaling

3.3–1 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

4.1–1 Optimal Detection for a General Vector Channel

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 Optimal Detection and Error Probability for Band-Limited

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 Optimal Detection and Error Probability for Power-Limited

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

4.5 Optimal Detection in Presence of Uncertainty:

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

4.9–1 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

5.1–1 The Likelihood Function / 5.1–2 Carrier Recovery and Symbol Synchronization in Signal Demodulation

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

5.3–1 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

6.1 Mathematical Models for Information Sources 331

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

6.4–1 Entropy and Mutual Information for Continuous Random Variables / 6.4–2 The Rate Distortion Function

6.5–1 Channel Models / 6.5–2 Channel Capacity

6.6 Achieving Channel Capacity with Orthogonal Signals 367

6.8–1 Bhattacharyya and Chernov Bounds / 6.8–2 Random Coding

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

7.3–1 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

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

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

7.8–1 Shortening and Lengthening / 7.8–2 Puncturing and Extending / 7.8–3 Expurgation and Augmentation

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

7.13–1 Product Codes / 7.13–2 Concatenated Codes

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

8.2–1 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

8.4–1 Rate-Compatible Punctured Convolutional Codes

8.5 Other Decoding Algorithms for Convolutional Codes 525

8.6 Practical Considerations in the Application of

8.7 Nonbinary Dual-k Codes and Concatenated Codes 537

8.8 Maximum a Posteriori Decoding of Convolutional

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

8.10–1 Tanner Graphs / 8.10–2 Factor Graphs / 8.10–3 The Sum-Product Algorithm / 8.10–4 MAP Decoding Using the Sum-Product Algorithm

8.11 Low Density Parity Check Codes 568

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

9.4–1 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

9.5–1 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

9.7 Iterative Equalization and Decoding—Turbo

10.1–1 The Zero-Forcing Algorithm / 10.1–2 The LMS Algorithm / 10.1–3 Convergence Properties of the LMS

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

10.3 Adaptive Equalization of Trellis-Coded Signals 706

10.4 Recursive Least-Squares Algorithms for Adaptive

10.4–1 Recursive Least-Squares (Kalman) Algorithm / 10.4–2 Linear Prediction and the Lattice Filter

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

11.1 Multichannel Digital Communications in AWGN

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

Spectrum Systems / 12.2–4 Excision of Narrowband Interference in DS Spread Spectrum Systems / 12.2–5 Generation of PN Sequences

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

Chapter 13 Fading Channels I: Characterization

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

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

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

13.6–1 Performance Degradation of an OFDM System due to Doppler Spreading / 13.6–2 Suppression of ICI in OFDM Systems

14.1–1 Capacity of Finite-State Channels

14.2–1 The Ergodic Capacity of the Rayleigh Fading Channel / 14.2–2 The Outage Capacity of Rayleigh Fading Channels

14.4 Performance of Coded Systems In Fading Channels 919

14.4–1 Coding for Fully Interleaved Channel Model

14.5–1 TCM Systems for Fading Channels / 14.5–2 Multiple Trellis-Coded Modulation (MTCM)

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

14.8–1 Channel Cutoff Rate for Fully Interleaved Fading Channels with CSI at Receiver

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

15.2–1 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

15.4–1 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

15.5 Bibliographical Notes and References 1021

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

16.5–1 ALOHA Systems and Protocols / 16.5–2 Carrier Sense Systems and Protocols

A.1 Eigenvalues and Eigenvectors of a Matrix 1086

Appendix B Error Probability for Multichannel Binary Signals 1090

Appendix C Error Probabilities for Adaptive Reception

C.3 Error Probabilities for Slowly Fading Rayleigh Channels 1100

C.4 Error Probabilities for Time-Invariant and Ricean Fading

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 newchapter 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 forstudents in electrical engineering It is also designed to serve as a text for self-studyand 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 andprior 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 randomvariables, 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 linkbudget 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 directed methods are described

non-decision-Chapter 6 provides an introduction to topics in information theory, including

lossless source coding, lossy data compression, channel capacity for different channelmodels, 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 softdecision and hard decision decoding methods are described, and their performance inAWGN 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, 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

trel-channels, the BCJR algorithm for iterative decoding of turbo codes, and the sum-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 analyempha-sis 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 designeither 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

communication through fading channels, multiple-antenna systems, and multiuser munications.

com-The authors and McGraw-Hill would like to thank the following reviewers for theirsuggestions on selected chapters of the fifth edition manuscript:

Paul Salama, Indiana University/Purdue University, Indianapolis; Dimitrios inakos, University of Toronto, and Ender Ayanoglu, University of California, Irvine.

Hatz-Finally, the first author wishes to thank Gloria Doukakis for her assistance in typingparts 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

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 tion sequence, is passed to the channel encoder The purpose of the channel encoder

informa-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

FIGURE 1.1–1

Basic elements of a digital communication system

the fidelity of the received signal In effect, redundancy in the information sequenceaids 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) ing involves taking k information bits at a time and mapping each k-bit sequence into

encod-a unique n-bit sequence, cencod-alled encod-a code word The encod-amount of redundencod-ancy 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 transmittingelectrical signals (waveforms), the primary purpose of the digital modulator is to mapthe 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 s i (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 theatmosphere (free space) On the other hand, telephone channels usually employ a variety

of physical media, including wire lines, optical fiber cables, and wireless (microwaveradio) Whatever the physical medium used for transmission of the information, theessential feature is that the transmitted signal is corrupted in a random manner by a

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 withwhich 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 theoutput 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 transmissionover the channel are signal attenuation, amplitude and phase distortion, and multipathdistortion.

The effects of noise may be minimized by increasing the power in the transmittedsignal However, equipment and other practical constraints limit the power level inthe transmitted signal Another basic limitation is the available channel bandwidth

A bandwidth constraint is usually due to the physical limitations of the medium andthe electronic components used to implement the transmitter and the receiver Thesetwo limitations constrain the amount of data that can be transmitted reliably over anycommunication 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 arebasically 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 usablebandwidth of several megahertz (MHz) Figure 1.2–1 illustrates the frequency range ofguided electromagnetic channels, which include waveguides and optical fibers.Signals transmitted through such channels are distorted in both amplitude andphase and further corrupted by additive noise Twisted-pair wireline channels are alsoprone to crosstalk interference from physically adjacent channels Because wirelinechannels carry a large percentage of our daily communications around the country andthe world, much research has been performed on the characterization of their trans-mission properties and on methods for mitigating the amplitude and phase distortionencountered in signal transmission In Chapter 9, we describe methods for designingoptimum transmitted signals and their demodulation; in Chapter 10, we consider thedesign of channel equalizers that compensate for amplitude and phase distortion onthese channels

Fiber-Optic Channels

Optical fibers offer the communication system designer a channel bandwidth that isseveral orders of magnitude larger than coaxial cable channels During the past twodecades, optical fiber cables have been developed that have a relatively low signal atten-uation, and highly reliable photonic devices have been developed for signal generationand signal detection These technological advances have resulted in a rapid deploy-ment of optical fiber channels, both in domestic telecommunication systems as well asfor transcontinental communication With the large bandwidth available on fiber-opticchannels, it is possible for telephone companies to offer subscribers a wide array oftelecommunication services, including voice, data, facsimile, and video

The transmitter or modulator in a fiber-optic communication system is a lightsource, either a light-emitting diode (LED) or a laser Information is transmitted byvarying (modulating) the intensity of the light source with the message signal The lightpropagates through the fiber as a light wave and is amplified periodically (in the case of

FIGURE 1.2–1

Frequency range for guided wirechannel

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

10 of the wavelength Consequently, a radio station transmitting in the

amplitude-modulated (AM) frequency band, say at f c = 1 MHz [corresponding to a wavelength

ofλ = c/f c= 300 meters (m)], requires an antenna of at least 30 m Other importantcharacteristics and attributes of antennas for wireless transmission are described inChapter 4

Figure 1.2–2 illustrates the various frequency bands of the electromagnetic trum The mode of propagation of electromagnetic waves in the atmosphere and in

spec-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 ofpropagation 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 nals being reflected (bent or refracted) from the ionosphere, which consists of several

sig-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 ferent delays It generally results in intersymbol interference in a digital communicationsystem Moreover, the signal components arriving via different propagation paths may

dif-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 isthe dominant propagation mode Additive noise in the HF range is a combination ofatmospheric noise and thermal noise

Sky-wave ionospheric propagation ceases to exist at frequencies above imately 30 MHz, which is the end of the HF band However, it is possible to haveionospheric scatter propagation at frequencies in the range 30–60 MHz, resulting fromsignal scattering from the lower ionosphere It is also possible to communicate overdistances of several hundred miles by use of tropospheric scattering at frequencies inthe range 40–300 MHz Troposcatter results from signal scattering due to particles

approx-in the atmosphere at altitudes of 10 miles or less Generally, ionospheric scatter andtropospheric 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 littleloss and make satellite and extraterrestrial communications possible Hence, at fre-quencies in the very high frequency (VHF) band and higher, the dominant mode ofelectromagnetic propagation is LOS propagation For terrestrial communication sys-tems, this means that the transmitter and receiver antennas must be in direct LOS withrelatively little or no obstruction For this reason, television stations transmitting in theVHF and ultra high frequency (UHF) bands mount their antennas on high towers toachieve 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 andvideo transmission at frequencies above 1 gigahertz (GHz) have antennas mounted ontall towers or on the top of tall buildings

The dominant noise limiting the performance of a communication system in VHFand UHF ranges is thermal noise generated in the receiver front end and cosmic noisepicked 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) inlight rain to about 0.3 dB/km in heavy rain At 100 GHz, the attenuation ranges fromabout 0.1 dB/km in light rain to about 6 dB/km in heavy rain Hence, in this frequencyrange, heavy rain introduces extremely high propagation losses that can result in serviceoutages (total breakdown in the communication system)

At frequencies above the extremely high frequency (EHF) band, we have the frared and visible light regions of the electromagnetic spectrum, which can be used

in-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 atextremely low frequencies However, the transmission of signals at such low frequencies

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 tosignal 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 jacent tracks is generally present in the readback signal of a storage system, just as is

ad-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 109bitsper square inch has been demonstrated in magnetic disk storage systems The speed atwhich data can be written on a disk or tape and the speed at which it can be read backare 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 digitalmagnetic or optical storage system In the readback process, the signal is demodulatedand 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 physicalchannels, we find it convenient to construct mathematical models that reflect the mostimportant characteristics of the transmission medium Then, the mathematical model forthe channel is used in the design of the channel encoder and modulator at the transmitterand the demodulator and channel decoder at the receiver Below, we provide a briefdescription of the channel models that are frequently used to characterize many of thephysical channels that we encounter in practice

The Additive Noise Channel

The simplest mathematical model for a communication channel is the additive noise

channel, illustrated in Figure 1.3–1 In this model, the transmitted signal s(t) is corrupted

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 communicationsystem or from interference encountered in transmission (as in the case of radio signaltransmission)

If the noise is introduced primarily by electronic components and amplifiers at thereceiver, 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 ourcommunication system analysis and design Channel attenuation is easily incorporatedinto the model When the signal undergoes attenuation in transmission through the

FIGURE 1.3–1

The additive noise channel

whereα 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)

=

 ∞

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

A good model for multipath signal propagation through physical channels, such asthe ionosphere (at frequencies below 30 MHz) and mobile cellular radio channels, is aspecial case of (1.3–3) in which the time-variant impulse response has the form

where the {a k (t) } represents the possibly time-variant attenuation factors for the L

multipath propagation paths and{τ k} are the corresponding time delays If (1.3–4) issubstituted into (1.3–3), the received signal has the form

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 SamuelMorse 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 anddashes (code words) In this code, more frequently occurring letters are represented byshort 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 digitalcommunication system (telegraphy), the beginnings of what we now regard as moderndigital communications stem from the work of Nyquist (1924), who investigated theproblem of determining the maximum signaling rate that can be used over a telegraphchannel of a given bandwidth without intersymbol interference He formulated a model

of a telegraph system in which a transmitted signal has the general form

n

where g(t) represents a basic pulse shape and {a n} 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

pulse caused no intersymbol interference at the sampling time k /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



sin[2πW(t − n/2W)]

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 A δ 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 A δ

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 formation were precursors to the work of Shannon (1948a,b), who established the

in-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 concon-straint, 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

where P is the average transmitted power and N0 is the power spectral density of theadditive 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 establishedbasic 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 nication systems based on a geometrical approach Kotelnikov’s approach was laterexpanded by Wozencraft and Jacobs (1965)

commu-Following Shannon’s publications came the classic work of Hamming (1950) onerror-detecting and error-correcting codes to combat the detrimental effects of channelnoise Hamming’s work stimulated many researchers in the years that followed, and avariety of new and powerful codes were discovered, many of which are used today inthe implementation of modern communication systems

The increase in demand for data transmission during the last four decades, coupledwith 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 performanceachieved have served as benchmarks for any given communication system design Thetheoretical 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 todesign and develop more efficient digital communication systems

There have been many new advances in the area of digital communications ing the early work of Shannon, Kotelnikov, and Hamming Some of the most notableadvances are the following:

follow-• The development of new block codes by Muller (1954), Reed (1954), Reed andSolomon (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 Hocquenghem (BCH) codes, e.g., the Berlekamp–Massey algorithm (see Chien,1964; Berlekamp, 1968)

Bose–Chaudhuri-• The development of convolutional codes and decoding algorithms by Wozencraftand 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-productdecoding algorithm by Gallager (1963)

• The development of turbo codes and iterative decoding by Berrou et al (1993)

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 tion signals, their demodulation, their error rate performance in additive, white Gaussian

modula-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, andbasic information theoretic limits on channel capacity, source information rates, and

channel coding rates

The design of efficient modulators and demodulators for linear filter channels withdistortion 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

Chapter 12 presents an introduction to direct sequence and frequency hopped spreadspectrum signals and systems and an evaluation of their performance under worst-case

interference conditions

The design of signals and coding techniques for digital communication throughfading 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 ing the performance of wireless communication systems through signal diversity and

improv-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 communicationsystems 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 IEEEPress in a book edited by Sloane and Wyner (1993) and previously in Russia in abook 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).

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 withthe 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

TABLE 2.0–1

Table of Fourier Transform Properties

Linearity αx1(t) + βx2(t) αX1( f ) + β X2( f )

Conjugacy x∗(t) X∗(− f ) Time-scaling (a= 0) x(at) |a|1 Xf

a



Time-shift x(t − t0 ) e − j2π f t0X ( f )

Modulation e j 2 π f0t x(t) X ( f − f0 ) Convolution x(t)  y(t) X ( f )Y ( f )

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 allcases, the spectral characteristics of the information sequence do not directly match thespectral characteristics of the communication channel, and hence the information signalcannot be directly transmitted over the channel In many cases the information signal

is a low frequency (baseband) signal, and the available spectrum of the communicationchannel is at higher frequencies Therefore, at the transmitter the information signal istranslated to a higher frequency signal that matches the properties of the communicationchannel This is the modulation process in which the baseband information signal isturned 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

TABLE 2.0–2

Table of Fourier Transform Pairs

Time Domain Frequency Domain

δ(t − t0 ) e − j2π f t0

e j 2 π f0t δ( f − f0 ) cos(2π f0t) 12δ( f − f0 ) + 1

2δ( f + f0 ) sin(2π f0t) 1

2 j δ( f − f0 ) − 1

2 j δ( f + f0 )

(t) sinc( f ) sinc(t) ( f )

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

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 )= 0when| 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 lowpasssignals are low frequency signals, which means that in the time domain, they are slowlyvarying 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

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 Withthis definition we can state that in general and for all signals, real or complex, thebandwidth is defined as one-half of the frequency support

In practice, the spectral characteristics of the message signal and the communication

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

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

around some frequency± f0which is far from zero More formally, we define a bandpass

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 f0is 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 f0is 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∗

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