Communication
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`
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`meleiaee wsd
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`1
`
`LGE 1021
`
`

`

`
`
`FourthEdition
`
`Digital
`
`Communications
`
`=—
`
`2
`
`

`

`A classic in the field, Digital Communications makes
`
`an excellent grTi| | ii
`
`STATE UNIVERSITY LIBRA
`
`
`
`
`A000054535237
`comprehensive reference book for the professional. This newbien maintait
`earlier editions with its comprehensive coverage, accuracy, and excellent explanations.
`It
`continues to provide the flexible organization that makes this book a goodfit for a one- or two-
`semester course. Features of the new edition of Digital Communicationsinclude:
`
`I
`
`Complete and thoroughintroduction to the analysis and design of digital communication
`systems that makes this book a must-havereference.
`Flexible organization that makesit useful in a one- or two-semester course.
`Excellent end-of-chapter problemsthat challenge the student.
`An accompanying website that provides presentation material and solutions for
`instructors.
`
`This new edition of John Proakis’ best-selling Digital Communications is up to date with new
`coverageof currenttrendsin the field. New topics have been addedthat include:
`
`Serial and Parallel Concatenated Codes
`Punctured Convolutional Codes
`Turbo TCM
`Turbo Equalization
`Spatial Multiplexing
`Digital Cellular CDMA System Based on DS Spread Spectrum
`Reduced Complexity ML Detectors
`
`,
`
`McGraw-Hill Higher Education =
`
`A Division of The McGraw-Hill Companies
`
`
`
`ISBN O-O?-232111-3
`
`0072°321111
`
`www-mhhe-com
`
`
`
`3
`
`

`

`
`
`Digital Communications
`
`Fourth Edition
`
`aTTA—
`
`JOHN G. PROAKIS
`
`Department ofElectrical and Computer Engineering
`Northeastern University
`
`{i
`
`Boston
`
`Burr Ridge, IL Dubuque, IA Madison, WI
`Bangkok Bogota Caracas Lisbon
`
`NewYork San Francisco
`
`St. Louis
`
`London Madrid Mexico City Milan
`New Delhi
`Seoul
`Singapore
`Sydney Taipei Toronto
`
`4
`
`

`

`——
`i
`
`
`
`McGraw-Hill Higher Education $7
`
`A Division of The McGraw-Hill Companies
`
`DIGITAL COMMUNICATIONS
`Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc., 122] Avenue of
`the Americas, New York, NY, 10020. Copyright ©2001, 1995, 1989, 1983, by The McGraw-
`Hill Companies, Inc. All rights reserved. No part ofthis publication may be reproduced or
`distributed in any form or by any means, or stored in a database or retrieval system, without the
`prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any
`network or other electronic storage or transmission, or broadcast for distance learning.
`Some ancillaries, including electronic and print components, may not be available to customers
`outside the United States.
`
`This book is printed on acid-free paper.
`
`4567890 DOC/DOC 09876543
`
`ISBN 0-07-232111-3
`
`Publisher: Thomas Casson
`Sponsoring editor: Catherine Fields Shultz
`Developmental editor: Emily J. Gray
`Marketing manager: John Wannemacher
`Project manager: Craig S. Leonard
`Production supervisor: Rose Hepburn
`Senior designer: Kiera Cunningham
`New media: Christopher Styles
`Compositor: Interactive Composition Corporation
`Typeface: 10.5/12 Times Roman
`Printer: Quebecor World Fairfield, Inc.
`
`Library of Congress Cataloging-in-Publication Data
`Proakis, John G.
`Digital communications / John G. Proakis.4th ed.
`p. cm.
`ISBN 0-07-232111-3
`1. Digital communications. I. Title.
`TK5103.7.P76 2000
`621.382-de21
`
`00-025305
`
`www.mhhe.com
`
`5
`
`

`

`[BRIEF CONTENTS §&
`
`80
`
`148
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`231
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`333
`
`376
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`416
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`548
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`598
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`660
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`709
`
`726
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`800
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`896
`
`939
`
`943
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`949
`961
`
`963
`
`Introduction
`Probability and Stochastic Processes
`Source Coding
`4 Characterization of Communication Signals and Systems
`5 OptimumReceivers for the Additive White Gaussian
`Noise Channel
`6 Carrier and Symbol Synchronization
`7 Channel Capacity and Coding
`8 Block and Convolutional Channel Codes
`9
`Signal Design for Band-Limited Channels
`10 Communication Through Band-Limited Linear Filter Channels
`11 Adaptive Equalization
`12 Multichannel and Multicarrier Systems
`13 Spread Spectrum Signals for Digital Communications
`14 Digital Communications through Fading Multipath Channels
`15 Multiuser Communications
`
`1 2 3
`
`Appendix A The Levinson-Durbin Algorithm
`Appendix B- Error Probability for Multichannel Binary Signals
`Appendix C_ Error Probabilities for Adaptive Reception of
`M-PhaseSignals
`Appendix D Square-Root Factorization
`
`References and Bibliography
`
`6
`
`

`

`
`
`[CONTENTS
`
`Preface
`
`1
`
`|
`
`|
`
`Introduction
`
`1.1 Elements of a Digital Communication System
`1.2 Communication Channels and Their Characteristics
`1.3 Mathematical Models for Communication Channels
`1.4. A Historical Perspective in the Developmentof Digital Communications
`1.5 Overview of the Book
`1.6 Bibliographical Notes and References
`
`2 Probability and Stochastic Processes
`2.1. Probability
`2.1.1 Random Variables, Probability Distributions, and Probability
`Densities | 2.1.2 Functions of Random Variables | 2.1.3 Statistical
`Averages of Random Variables | 2.1.4 Some Useful Probability
`Distributions | 2.1.5 Upper Bounds onthe Tail Probability | 2.1.6
`Sums of Random Variables and the Central Limit Theorem
`2.2 Stochastic Processes
`2.2.1 Statistical Averages | 2.2.2 Power Density Spectrum/ i273
`Response of a Linear Time-Invariant System to a RandomInput
`Signal | 2.2.4 Sampling Theoremfor Band-Limited Stochastic
`Processes | 2.2.5 Discrete-Time Stochastic Signals and Systems P2236
`Cyclostationary Processes
`2.3. Bibliographical Notes and References
`Problems
`
`3 Source Coding
`3.1. Mathematical Models for Information Sources
`3.2 A Logarithmic Measure of Information
`3.2.1 Average Mutual Information and Entropy | 3.2.2 Information
`Measures for Continuous Random Variables
`3.3. Coding for Discrete Sources
`3.3.1 Coding for Discrete Memoryless Sources | 3.3.2 Discrete
`Stationary Sources | 3.3.3 The Lempel-Ziv Algorithm
`
`xix
`
`I
`
`l
`3
`10
`13
`15
`16
`
`17
`17
`
`61
`
`75
`75
`
`80
`80
`82
`
`90
`
`xl
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`7
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`

`

`xii
`
`Contents
`
`103
`Coding for Analog Sources—OptimumQuantization
`3.4
`3.4.1 Rate-Distortion Function | 3.4.2 Scalar Quantization|3.4.3
`Vector Quantization
`Coding Techniques for Analog Sources
`oro)
`3.5.1 Temporal Waveform Coding|3.5.2 Spectral Waveform
`Coding | 3.5.3 Model-Based Source Coding
`Bibliographical Notes and References
`Problems
`
`140
`141
`
`3.6
`
`Characterization of Communication Signals and Systems
`148
`Representation of Band-Pass Signals and Systems
`4.1
`4.1.1 Representation of Band-Pass Signals|4.1.2 Representation of
`Linear Band-Pass Systems|4.1.3 Response ofa Band-Pass Systemto
`a Band-Pass Signal | 4.1.4 Representation of Band-Pass Stationary
`Stochastic Processes
`Signal Space Representations
`4.2
`4.2.1 Vector Space Concepts|4.2.2 Signal Space Concepts|4.2.3
`Orthogonal Expansions ofSignals
`168
`Representation of Digitally Modulated Signals
`4.3
`4.3.1 Memoryless Modulation Methods|4.3.2 Linear Modulation with
`Memory | 4.3.3 Non-linear Modulation Methods with Memory—
`CPFSK and CPM
`201
`Spectral Characteristics of Digitally Modulated Signals
`4.4
`4.4.1 Power Spectra of Linearly Modulated Signals|4.4.2 Power
`Spectra of CPFSK and CPMSignals|4.4.3 Power Spectra of
`Modulated Signals with Memory
`Bibliographical Notes and References
`Problems
`
`148
`
`221
`222
`
`4.5
`
`Optimum Receivers for the Additive White Gaussian Noise
`Channel
`Optimum Receiver for Signals Corrupted by Additive White Gaussian
`Noise
`5.1.1 Correlation Demodulator | 5.1.2 Matched-Filter Demodulator
`5.1.3 The Optimum Detector | 5.1.4 The Maximum-Likelihood
`Sequence Detector | 5.1.5 A Symbol-by-Symbol MAP Detector for
`Signals with Memory
`Performance of the Optimum Receiver for Memoryless Modulation
`5.2.1 Probability of Error for Binary Modulation | 5.2.2 Probability of
`Error for M-ary Orthogonal Signals | 5.2.3 Probability of Error for
`M-ary Biorthogonal Signals|5.2.4 Probability of Error for Simplex
`Signals | 5.2.5 Probability of Error for M-ary Binary-Coded
`
`Signals | 5.2.6 Probability of Error for M-ary PAM|3.2.7
`Probability of Error for M-ary PSK | 5.2.8 Differential PSK (DPSK)
`
`5.1
`
`5.2
`
`bt ha pei
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`ht tat —
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`8
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`

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`
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`XH
`
`Contents
`
`283
`
`and Its Performance | 5.2.9 Probability of Error for QAM | 5.2.10
`Comparison of DigitalModulation Methods
`5.3 OptimumReceiver for CPM Signals
`5.3.1 Optimum Demodulation and Detection of CPM | 5.3.2
`Performance of CPMSignals | 5.3.3 Symbol-by-Symbol Detection of
`CPMSignals | 5.3.4 Suboptinum Demodulation and Detection of
`CPM Signals
`5.4 Optimum Receiver for Signals with Random Phase in AWGN Channel 300
`5.4.] Optimum Receiver for Binary Signals | 3.4.2 Optimum Receiver
`for M-ary Orthogonal Signals | 5.4.3 Probability of Error for Envelope
`Detection of M-ary Orthogonal Signals | 5.4.4 Probability of Error for
`Envelope Detection of Correlated Binary Signals
`5.5 Performance Analysis for Wireline and Radio Communication Systems
`5.5.1 Regenerative Repeaters | 5.5.2 Link Budget Analysis in Radio
`Communication Systems
`5.6 Bibliographical Notes and References
`Problems
`
`313
`
`g
`
`6 Carrier and Symbol Synchronziation
`6.1
`Signal Parameter Estimation
`6.1.1 The Likelihood Function | 6.1.2 Carrier Recovery and Symbol
`Synchronization in Signal Demodulation
`6.2. Carrier Phase Estimation
`6.2.1 Maximum-Likelihood Carrier Phase Estimation | 6.2.2 The
`Phase-Locked Loop | 6.2.3 Effect of Additive Noise on the Phase
`Estimate | 6.2.4 Decision-Directed Loops|6.2.5 Non-Decision-
`Directed Loops
`6.3 Symbol Timing Estimation
`6.3.1 Maximum-Likelihood Timing Estimation | 6.3.2 Non-Decision-
`Directed Timing Estimation
`Joint Estimation of Carrier Phase and Symbol Timing
`6.4
`6.5 Performance Characteristics of ML Estimators
`6.6 Bibliographical Notes and References
`Problems
`
`366
`368
`371
`372
`
`333
`S55
`
`338
`
`359
`
`7 Channel Capacity and Coding
`7.1 Channel Models and Channel Capacity
`7.1.1 Channel Medels | 7.1.2 Channel Capacity | 7.1.3 Achieving
`Channel Capacity with Orthogonal Signals | 7.1.4 Channel Reliability
`Functions
`392
`7.2 RandomSelection of Codes
`7.2.1 Random Coding Based on M-ary Binary-Coded Signals|7.2.2
`Random Coding Based on M-ary Multiamplitude Signals | 7.2.3
`Comparison of Rg with the Capacity of the AWGN Channel
`7.3 Communication System Design Based on the Cutoff Rate
`
`376
`376
`
`402
`
`9
`
`

`

`XIV
`
`7.4 Bibliographical Notes and References
`Problems
`
`8 Block and Convolutional Channel Codes
`
`Contents
`
`408
`409
`
`416
`
`8.1 Linear Block Codes
`8.1.1 The Generator Matrix and the Parity Check Matrix | 8.1.2 Seme
`Specific Linear Block Codes | 8.1.3 Cyclic Codes | 8.1.4 Optimum
`Soft-Decision Decoding of Linear Block Codes
`| 8.1.5 Hard-Decision
`Decoding of Linear Block Codes | 8.1.6 Comparison of Performance
`Between Hard-Decision and Soft-Decision Decoding | 8.1.7 Bounds on
`Minimum Distance of Linear Block Codes | 8.1.8 Nonbinary Block
`Codes and Concatenated Block Codes | 8.1.9 Interleaving of Coded
`Data for Channels with Burst Errors | 8.1.10 Serial and Parallel
`Concatenated Block Codes
`8.2 Convolutional Codes
`8.2.1 The Transfer Function of a Convolutional Code | 8.2.2 Optimum
`Decoding of Convolutional Codes—The Viterbi Algorithm|8.2.3
`Probability of Error for Soft-Decision Decoding | 8.2.4 Probability of
`Error for Hard-Decision Decoding|8.2.5 Distance Properties of
`Binary Convolutional Codes|8.2.6 Punctured Convolutional Codes
`8.2.7 Other Decoding Algorithms for Convolutional Codes | 8.2.8
`Practical Considerations in the Application of Convolutional Codes|
`8.2.9 Nonbinary Dual-k Codes and Concatenated Codes | 8.2.10
`Parallel and Serial Concatenated Convolutional Codes
`8.3. Coded Modulation for Bandwidth-Constrained Channels—Trellis-Coded
`Modulation
`8.4 Bibliographical Notes and References
`Problems
`
`416
`
`471
`
`522
`539
`541
`
`9 Signal Design for Band-Limited Channels
`9,1. Characterization of Band-Limited Channels
`9.2 Signal Design for Band-Limited Channels
`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 IST | 9.2.4 Signal Design for Channels with
`Distortion
`9.3 Probability of Error in Detection of PAM
`9.3.1 Probability of Error for Detection of PAM with Zero IST | 9.3.2
`Probability of Error for Detection of Partial-Response Signals
`9.4 Modulation Codes for Spectrum Shaping
`9.5 Bibliographical Notes and References
`Problems
`
`548
`548
`554
`
`574
`
`578
`588
`588
`
`
`
`10
`
`10
`
`

`

`Contents
`
`AV
`
`10 Communication Through Band-Limited Linear Filter Channels
`Optimum Receiver for Channels with ISI and AWGN
`10.1
`10.1.1 Optimum Maximum-Likelihood Receiver | 10.1.2 A Discrete-
`Time Model for a Channel with ISI | 10.1.3 The Viterbi Algorithm
`for the Discrete-Time White Noise Filter Model
`| 10.1.4 Performance
`of MLSE for Channels with [ST
`Linear Equalization
`10.2.1 Peak Distortion Criterion | 10.2.2 Mean-Square-Error (MSE)
`Criterion | 10.2.3 Performance Characteristics of the MSE
`Equalizer | 10.2.4 Fractionally Spaced Equalizers | 10.2.5 Baseband
`and Passband Linear Equalizers
`Decision-Feedback Equalization
`10.3.1 Coefficient Optimization | 10.3.2 Performance Characteristics
`of DFE | 10.3.3 Predictive Decision-Feedback Equalizer | 10.3.4
`Equalization at the Transmitter—Tomlinson—Harashima Precoding
`Reduced Complexity ML Detectors
`Iterative Equalization and Decoding—Turbo Equalization
`Bibliographical Notes and References
`Problems
`
`10.2
`
`10.3
`
`10.4
`
`10.5
`10.6
`
`11
`
`me
`
`Adaptive Equalization
`11.1
`Adaptive Linear Equalizer
`11.1.1 The Zero-Forcing Algorithm | 11.1.2 The LMS Algorithm |
`11.1.3 Convergence Properties of the LMS Algorithm | 11.1.4 Excess
`MSE Due to Noisy Gradient Estimates | 11.1.5 Accelerating the
`Initial Convergence Rate in the LMS Algorithm | 11.1.6 Adaptive
`Fractionally Spaced Equalizer—The Tap Leakage Algorithm | 11.1.7
`An Adaptive Channel Estimator forML Sequence Detection
`Adaptive Decision-Feedback Equalizer
`Adaptive Equalization of Trellis-Coded Signals
`Recursive Least-Squares Algorithms for Adaptive Equalization
`11.4.1 Recursive Least-Squares (Kalman) Algorithm | 11.4.2 Linear
`Prediction and the Lattice Filter
`Self-Recovering (Blind) Equalization
`11.5.1 Blind Equalization Based on the Maximum-Likelihood
`Criterion | 11.5.2 Stochastic Gradient Algorithms | 11.5.3 Blind
`Equalization Algorithms Based on Second- and Higher-Order Signal
`Statistics
`Bibliographical Notes and References
`Problems
`
`11.3
`11.4
`
`12
`
`Multichannel and Multicarrier Systems
`12.1 Multichannel Digital Communications in AWGN Channels
`12.1.1 Binary Signals | 12.1.2 M-ary Orthogonal Signals
`
`11
`
`598
`
`599
`
`616
`
`638
`
`647
`649
`651
`652
`
`660
`
`660
`
`677
`678
`682
`
`704
`705
`
`709
`
`709
`
`11
`
`

`

`XVI
`
`Contents
`
`12.2. Multicarrier Communications
`| 12.2.2 An
`12.2.1 Capacity of a Nonideal Linear Filter Channel
`FFT-Based Multicarrier System|12.2.3 Minimizing Peak-to-Average
`Ratio in the Multicarrier Systems
`12.3. Bibliographical Notes and References
`Problems
`
`723
`724
`
`715
`
`726
`13. Spread Spectrum Signals for Digital Communications
`728
`13.1 Model of Spread Spectrum Digital Communication System
`729
`13.2 Direct Sequence Spread SpectrumSignals
`13.2.1 Error Rate Performance of the Decoder|13.2.2 Some
`Applications of DS Spread Spectrum Signals|13.2.3 Effect of Pulsed
`Interference on DS Spread Spectrum Systems|13.2.4 Excision of
`NarrowbandInterference in DS Spread Spectrum Systems | 13.2.5
`Generation of PN Sequences
`Frequency-HoppedSpread SpectrumSignals
`13.3.1 Performance of FH Spread Spectrum Signals in an AWGN
`Channel|13.3.2 Performance of FH Spread Spectrum Signals in
`Partial-Band Interference|13.3.3 A CDMA System Based on FH
`Spread Spectrum Signals
`13.4 Other Types of Spread SpectrumSignals
`13.5 Synchronization of Spread Spectrum Systems
`13.6 Bibliographical Notes and References
`Problems
`
`nal
`
`784
`786
`792
`794
`
`13.3
`
`800
`801
`
`14 Digital Communications through Fading Multipath Channels
`14.1 Characterization of Fading Multipath Channels
`14.1.1 Channel Correlation Functions and Power Spectra | 14.1.2
`Statistical Models for Fading Channels
`14.2. The Effect of Signal Characteristics on the Choice of a Channel Model 814
`14.3.
`Frequency-Nonselective, Slowly Fading Channel
`$16
`14.4 Diversity Techniques for Fading Multipath Channels
`821
`14.4.1 Binary Signals|14.4.2 Multiphase Signals | 14.4.3 M-ary
`Orthogonal Signals
`14.5 Digital Signaling over a Frequency-Selective, Slowly Fading Channel
`14.5.1 A Tapped-Delay-Line Channel Model
`| 14.5.2 The RAKE
`Demodulator|14.5.3 Performance of RAKE Demodulator | 14.5.4
`Receiver Structures for Channels with Intersymbel Interference
`14.6 Coded Waveforms for Fading Channels
`!4.6.1 Probability of Error for Soft-Decision Decoding of Linear
`Binary Block Codes|14.6.2 Probability of Error for Hard-Decision
`Decoding of Linear Binary Block Cades|14.6.3 Upper Bounds on
`the Performance of Convolutional Codes for a Rayleigh Fading
`Channel
`| 14.6.4 Use of Constant-Weight Codes and Concatenated
`Codes for a Fading Channel|14.6.5 System Design Based on the
`
`840
`
`852
`
`12
`
`12
`
`

`

`
`
`XVII
`
`Contents
`
`Cutoff Rate | 14.6.6 Performance of Coded Phase-Coherent
`Communication Systems—Bit-Interleaved Coded Modulation | 14.6.7
`Trellis-Coded Modulation
`14.7. Multiple-Antenna Systems
`14.8 Bibliographical Notes and References
`Problems
`
`878
`885
`887
`
`896
`15 Multiuser Communications
`896
`15.1
`Introduction to Multiple Access Techniques
`899
`15.2. Capacity of Multiple Access Methods
`905
`15.3. Code-Division Multiple Access
`15.3.1 CDMASignal and Channel Models|15.3.2 The Optimum
`Receiver|15.3.3 Suboptimum Detectors|15.3.4 Successive
`Interference Cancellation | 15.3.5 Performance Characteristics of
`Detectors
`922
`15.4 Random Access Methods
`15.4.1 ALOHA Systems and Protocols|15.4.2 Carrier Sense
`Systems and Protocols
`15.5 Bibliographical Notes and References
`Problems
`
`931
`933
`
`Appendix A The Levinson—Durbin Algorithm
`
`Appendix B_ Error Probability for Multichannel Binary Signals
`
`Appendix C_ Error Probabilities for Adaptive Reception of /-Phase
`Signals
`C.1. Mathematical Model for M/-Phase Signaling Communication
`System
`C2. Characteristic Function and Probability Density Function of
`the Phase 4
`C.3 Error Probabilities for Slowly Rayleigh Fading Channels
`C.4. Error Probabilities for Time-Invariant and Ricean Fading
`Channels
`
`Appendix D Square-Root Factorization
`
`References and Bibliography
`
`Index
`
`939
`
`943
`
`949
`
`949
`
`952
`953
`
`956
`
`961
`
`963
`
`993
`
`
`
`13
`
`13
`
`

`

`bo
`
`Digital Communications
`
`
`Information
`
`
`Source
`Channel
`Digital
`source and
`encoder
`encoder
`modulator
`input transducer
`
`
`
`Channel
`
`
`
`
`
`
`Source
`Channel
`Digital
`Output
`transducer
`decoder
`decoder
`demodulator
`
`
`
`
`
`Output
`signal
`
`FIGURE 1.1-1
`Basic elements of a digital communication system.
`
`through the channel. Thus. the added redundancyserves to increase the relia-
`bility of the received data and improvesthe fidelity of the received signal. In
`effect. redundancyin the information sequence aids the receiver in decoding the
`desired information sequence. For example, a (trivial) form of encoding of the
`binary information sequenceis simply to repeat each binarydigit 77 times, where
`m is some positive integer. More sophisticated (nontrivial) encoding involves
`taking & information bits at a time and mapping each k-bit sequence into a
`unique 77-bit sequence, called a code word. The amount of redundancyintroduced
`by encoding the data in this manneris measured bythe ratio ”/k. The reciprocal
`of this ratio. namely k/n, is called the rate of the code or, simply. the coderate.
`The binary sequence at the output of the channel encoderis passed to the
`digital modulator. which serves as the interface to the communication channel.
`Since nearlyall 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 sequenceis to
`be transmitted one bit at a time at some uniform rate R bits per second (bits/s).
`The digital modulator may simply mapthe binary digit 0 into a waveform so(1)
`and the binarydigit 1 into a waveform s,(rf). 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 = 2’ distinct waveforms s,(r), i = 0,1,..., M — 1, one waveform for
`each of the 2” possible b-bit sequences. We call this M-ary modulation (M > 2).
`Note that a newh-bit sequence enters the modulator every b/R seconds. Hence,
`whenthe channel bit rate R is fixed, the amount of time available to transmit one
`of the M waveforms corresponding to a b-bit sequenceis 4 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
`
`14
`
`14
`
`

`

`Chapter Four: Characterization of Communication Signals and Systems
`
`169
`
`Whenthe mapping fromthe digital sequence {a,,} to waveformsis performed
`under the constraint that a waveform transmitted in any time interval depends
`on one or more previously transmitted waveforms, the modulatoris said to have
`memory. On the other hand, when the mapping from the sequence {a,} to the
`waveforms{s,,(¢)} is performed without anyconstraint on previouslytransmitted
`waveforms, the modulator is called memoryless.
`In addition to classifying the modulator as either memoryless or having
`memory, we mayclassifyit as either Jinear or non-linear. Linearity of a modula-
`tion method requires that the principle of superposition applies in the mapping
`of the digital sequence into successive waveforms. In non-linear modulation, the
`superposition principle does not applyto signals transmitted in successive time
`intervals. We shall begin by describing memoryless modulation methods.
`
`4.3.1 Memoryless Modulation Methods
`
`Asindicated above, the modulator in a digital communication system maps a
`sequence ofbinary digits into a set of corresponding signal waveforms. These
`waveforms may differ in either amplitude or in phase or in frequency. or some
`combination of two or more signal parameters. We consider each of these signal
`types separately, beginning with digital pulse amplitude modulation (PAM). In
`all cases. we assume that the sequence of binarydigits at the inputto the mod-
`ulator occurs at a rate of R bits/s.
`.
`
`Pulse-amplitude-modulated (PAM) signals. In digital PAM, the signal wave-
`forms maybe represented as
`
`S(t) = Re[4,2(te*"]
`= A,,,g(t) cos 27f.t,
`
`i=. Bie,
`
`Oe ee
`
`(4.3-1)
`’
`
`where {4,,, 1 < 1m < M} denote the set ofWM possible amplitudes corresponding
`to M = 2* possible k-bit blocks of symbols. The signal amplitudes 4,, take the
`discrete values (levels)
`
`A, =Qm-1-M)d,
`
`ialaeM
`
`(4.3—2)
`
`where 2dis the distance between adjacent signal amplitudes. The waveform g(f)
`is a real-valued signal pulse whose shape influences the spectrum of the trans-
`mitted signal, as we shall observe later. The symbol rate for the PAMsignal is
`R/k. This is the rate at which changes occurin the amplitude of the carrier to
`reflect
`the transmission of new information. The time interval 7, = 1/R is
`called the bit interval and the time interval T= k/R=kT), is called the symbol
`interval.
`
`15
`
`15
`
`

`

`170
`
`Digital Communications
`
`The M PAM signals have energies
`
`mf
`
`0
`
`Sin(0) dt
`Cm = |
`tA,
`g(t) dt
`2
`:
`=
`J0
`Sea 5A mee
`denotes the energy in the pulse g(f). Clearly, these signals are one-
`where €,
`dimensional (V = 1), and, hence, are represented by the general form
`
`2
`
`ih
`
`ne
`(4.3—3)
`ano
`
`Sm(t) = SnfO
`
`where/() is defined as the unit-energy signal waveform given as
`[2.
`f(t) = ./=g(t) cos 2zf,t
`Ve,
`
`(4.34)
`
`‘
`(4.3-5)
`
`and
`
`Sie aloe
`
`m=1,2,...,JVW
`
`(4.3-6)
`
`The corresponding signal space diagrams for M =2, M=4, and M =8 are
`shown in Figure 4.3—1. Digital PAM is also called amplitude-shift keving (ASK).
`The mapping or assignment of & information bits to the M@ = 2* possible
`signal amplitudes may be done in a numberof ways. The preferred assignment is
`one in which the adjacent signal amplitudes differ by one binary digit as illu-
`strated in Figure 4.3—1. This mapping is called Gray encoding. It is important in
`the demodulation of the signal because the most likely errors caused by noise
`involve the erroneous selection of an adjacent amplitude to the transmitted signal
`amplitude. In such a case, only a single bit error occurs in the 4-bit sequence.
`Wenote that the Euclidean distance between anypair of signal points is
`
`é
`/
`dy) = Vv (Sip, vr a)
`
`2
`
`= SEe|Am = A,|
`
`= d,/2E,|m — n|
`
`(4.3-7)
`
`Hence, the distance between a pair of adjacent signal points, 1.e., the minimum
`Euclidean distance, is
`
`(e)
`(4.3-8)
`Ce in = dV/2€,
`The carrier-modulated PAM signal represented by Equation 4.3-1 is a
`double-sideband (DSB) signal and requires twice the channel bandwidth ofthe
`equivalent low-pass signal for transmission. Alternatively, we may use single-
`sideband (SSB) PAM, whichhas the re resentati
`Jer
`sideband):
`
`16
`
`16
`
`

`

`
`
`Chapter Four: Characterization of Communication Signals and Systems
`
`171
`
`4
`
`(a)M=2
`
`FIGURE 4.3-1
`Signal space diagramfor digital
`PAM signals.
`
`00
`
`01
`
`1
`
`10
`
`
`
`(c)M=8
`
`s(t) = Re{A4,,[g() +ja(nler™"},
`m=1,2,...,M
`(4.3-9)
`where g(¢) is the Hilbert transformofg(r). Thus,thebandwidthoftheSSBsignal
`is half that of the DSB signal-
`igita
`signal is also appropriate for transmission over a channel
`that does not require carrier modulation. In this case, the signal waveform may
`
`be simply represented
`as
`
`Sm(t) = Ang(t),
`
`Fy Me eee IVE
`
`(4.3—10)
`
`This is nowcalled a baseband signal. For example a four-amplitude level base-
`band PAM signal is illustrated in Figure 4.3-2a. The carrier-modulated version
`of the signal is shown in Figure 4.3—2b.
`In the special case of M =2signals, the binary PAM waveforms have the
`special property that
`
`Hence, these two signals have the same energyand across-correlation coefficient
`of —1. Such signals are called antipodal.
`
`s(e— —s>(t)
`
`In digital phase modulation, the Msignal wave-
`
`~>Phase-modulated signals.
`forms are represented as
`S(t) = Relg(1e?tr— Mee),
`= e(t)oos|2a +a(in = |
`a 2n
`,
`2
`:
`aa,
`;
`= g(t) cos = (m — 1)cos2zf,t — g(t) sin =, (m — 1)sin2zf.t
`
`m=1,2,...,M,
`
`O<1<T
`
`where g(/) is the signal pulse shape and 6,, = 2m(m — 1)/M,m = 1,2,..., M, are
`the M possible phases of the carrier that convey the transmitted information.
`Digital phase modulation is usually called_phase-shift keying (PSK).
`
`(4.3-11)
`
`17
`
`17
`
`

`

`172
`
`Digital Communications
`
`Signal
`
`amplitude
`
`
`Data:
`
`11
`
`10
`
`00
`
`01
`
`11
`
`00
`
`(a) Baseband PAM signal
`
`(b) Bandpass PAM signal
`
`FIGURE 4.3-2
`Baseband and band-pass PAM signals.
`
`Wenote that these signal waveforms have equal energy, 1.e.,
`
`(4.3-12)
`
`r 4a | S;,(t)dt
`= 3, g(t)dt =35€,
`
`ef
`
`2l
`
`Furthermore, the signal waveforms may be represented as a linear combination
`of two orthonormal signal waveforms, /\(‘) and f3(¢), 1.e.,
`
`Sp(0) aa Smif (t) =r Sm2f2(t)
`
`(4.3-13)
`
`where
`
`
`
`AW =|Fate2mfet (4.3—-14)
`
`2
`
`2
`
`AY) = - ze) sin 2af..t
`
`(4.3-15)
`
`g
`
`18
`
`

`

`Chapter Four: Characterization of Communication Signals and Systems
`
`173
`
`and the two-dimensional vectors §,, = [5,,1 52] are given by
`
`(ge
`
`Sn = cos(m— 1)
`
`Codec?)
`
`\[=2sin2(m SEs
`
`Lf ioRe M
`
`(4.3-16)
`
`Signal space diagrams for M = 2, 4, and 8 are shown in Figure 4.3-3. We note
`that M = 2 corresponds to one-dimensional signals, which are identical to binary
`PAM signals.
`As is the case of PAM. the mapping or assignment of k information bits to
`the M = 2* possible phases may be done in a numberof ways. The preferred
`assignment is Gray encoding, so that the mostlikely errors caused by noisewill
`result in a singlebit error in the k-bit symbol.
`The Euclidean distance between signal points is
`(e)
`isa = Sin iD S,||
`
`= {e-{ - Coss(im = nh
`
`Qn
`d
`
`1/2
`
`(4.3-17)
`
`The minimum Euclidean distance correspondsto the case in which |m —n| = 1,
`i.e., adjacent signal phases. In this case,
`SEISfpeA—————
`:
`20
`
`ea fe( = cos)
`
`(4.3-18)
`
`FIGURE 4.3-3
`Signal space diagrams for PSK signals.
`
`0
`
`1
`
`M=2
`
`(BPSK)
`
`ll
`
`00
`
`M=8
`(Octal PSK)
`
`10
`
`M=4
`
`(QPSK)
`
`19
`
`19
`
`

`

`174
`
`Digital Communications
`
`A variant of 4-phase PSK (QPSK), called 7/4-QPSK is obtained byintro-
`ducing an additional 2/4 phase shift in the carrier phaseiin each symbolinterval.
`This phase shift facilitates(symbol synchronization.Jaa
`Quadrature amplitude modulation. The bandwidth efficiency of PAM/SSB
`can also be obtained by simultaneously impressing two separate k-bit symbols
`from the information sequence {a,,} on two quadrature carriers cos 27f,t and
`sin 27f.t. The resulting modulation technique is called quadrature PAM or
`QAM. and the corresponding signal waveforms may be expressed as
`
` m=1,2,....M,
`Smt) = Re[(Ame tiAm)g(Der"],
`= Ame8(t) cos 27fot — Ams8(t) sin 27771
`
`O<1<T
`
`(4.3-19)
`
`where 4,,. and A,,,, are the information-bearing signal amplitudes of the quad-
`rature carriers and g(f) is the signal pulse.
`Alternatively, the QAM signal waveforms maybe expressed as
`
`nl)
`
`l| Re[Vneae]
`Il
`Vng(t) cos(27f,.t + 6,,)
`
`(4.3-20)
`
`where’ Vin. = V Aine + Ams and Gy = tan7!(A,,,/Amc). From this expression,it is
`apparentthat the QAMsignal waveforms maybe viewed as combined amplitude
`and phase modulation.
`,
`Infact. we mayselect any combination of M,-level PAM and M-phase PSK
`to construct an M = M,M, combined PAM-PSK signalconstellation. If M, =
`2” and M, = 2”,
`the combined PAM-PSK signal constellation results in the
`simultaneous transmission of m+n =log M,M)> binarydigits occurring at a
`symbol rate R/(m-+n). Examples of signal space diagrams for combined
`PAM-PSK are shown in Figure 4.34, for M = 8 and M = 16.
`As in the case of PSK signals, the QAM signal waveforms mayberepre-
`sented as a linear combination of two orthonormal signal waveforms, f{(f) and
`A(t). ie.
`
`Sm(t) = Smif (D) + Smofo(d)
`
`(4.321)
`
`where
`
`2A= jecos27f..t
`;
`f(t) = — Ee(t) sin 2zf..t
`
`5
`
`(4,.3—22)
`
`20
`
`20
`
`

`

`
`
`Chapter Four: Characterization of Communication Signals and Systems
`
`175
`
`FIGURE 4.34
`Examples of combined PAM—
`PSK signal space diagrams.
`
`M 8
`
`and
`
`M=16
`
`Sn = [Sint Sy]
`
`=AneEx AmoEel
`is the energyof the signal pulse g(¢).
`where €,
`The Euclidean distance between anypair ofsignal vectorsis
`
`(4.3-23)
`
`A) = |S. — Spl
`
`(4.3—24) = V SEgl(Ame ¥ Ane)” ate (Ams = Ans)|
`In the special case where the signal amplitudes takethe set of discrete va
`{2m —1—M)d. m=1,2,...,M}. the signal space diagram is rectangular, as
`showr-in-Figere_4.3—-5. In this case, the Euclidean distance between adjacent
`points, i.e., the minimum distance, is
`d=Oe,
`min
`
`(4.3-25)
`
`which is the same result as for PAM.
`
`FIGURE 4.3-5
`Several signal space diagrams for
`
`rectangular QAM.
`
`21
`
`21
`
`

`

`176
`
`Digital Communications
`
`It is apparent from the discussion above that the
`Multidimensional signals.
`digital modulation of the carrier amplitude and phase allows us to construct
`signal waveforms that correspond to two-dimensional vectors and signal space
`diagrams. If we wish to construct signal waveforms corresponding to higher-
`dimensional vectors, we mayuse either the time domain orthe frequency domain
`or both in orderto increase the number of dimensions.
`Suppose we have N-dimensional signal vectors. For any V. we may subdi-
`vide a time interval of length T, = NT into Nsubintervals of length T = 7,/N.
`In each subinterval of length 7, we may use binary PAM (a one-dimensional
`signal) to transmit an element of the N-dimensional signal vector. Thus. the NV
`time slots are used to transmit the N-dimensional signal vector. If N is even, a
`
`time slot of length 7 maybe used to simultaneously transmittwo components of
`the N-dimensional vector by modulating the amplitude of quadrature carriers
`independently by the corresponding components. In this manner, the N-dimen-
`sional signal vector is transmitted in 4NT seconds (4Ntimeslots).
`Alternatively, a frequency band of width NAfmay be subdivided into N
`frequencyslots each of width Af. An N-dimensional signal vector can be trans-
`mitted over the channel by simultaneously modulating the amplitude of Ncar-
`riers. one in each of the N frequency slots. Care must be taken to provide
`sufficient frequency separation Af between successive carriers so that there is
`no cross-talk interference among the signals on the N carriers. If quadrature
`carriers are used in each frequencyslot. the N-dimensional vector (even NV)
`may be transmitted in } NV frequencyslots, thus reducing the channel bandwidth
`utilization bya factorof 2.
`More generally, we mayuse both the time and frequency domainsjointly to
`transmit an N-dimensional signal vector. For example, Figure 4.3-6 illustrates a
`subdivision of the time and frequencyaxes into 12 slots. Thus, an V = 12-dimen-
`sional signal vector may be transmitted by PAM or an N= 24-dimensional
`signal vector may be transmitted by use of two quadrature carriers (QAM) in
`each slot.
`
`Orthogonal multidimensional signals. As a speci