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Principal Communication Systems Engineering with inserted solution manual[John G. Proakis, Masoud Salehi]
Communication Systems Engineering with inserted solution manual[John G. Proakis, Masoud Salehi]John G. Proakis, Masoud Salehi
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Proakis-50210 proa-fm August 9, 2001 14:2 COMMUNICATION SYSTEMS ENGINEERING John G. Proakis Masoud Salehi 2nd Ed. Upper Saddle River, New Jersey 07458 i Proakis-50210 proa-fm August 9, 2001 14:2 To Felia, George, and Elena. —John G. Proakis To Fariba, Omid, Sina, and my parents. —Masoud Salehi Library of Congress Cataloging-in-Publication Data CIP data available on file. Vice President and Editorial Director, ECS: Marcia J. Horton Publisher: Tom Robbins Editorial Assistant: Jody McDonnell Vice President and Director of Production and Manufacturing, ESM: David W. Riccardi Executive Managing Editor: Vince O'Brien Managing Editor: David A. George Production Editor: Chanda Wakefield Director of Creative Services: Paul Belfanti Creative Director: Carole Anson Art Director: Jayne Conte Art Editor: Greg Dulles Manufacturing Manager: Trudy Pisciotti Manufacturing Buyer: Lynda Castillo Marketing Manager: Holly Stark Marketing Assistant: Karen Moon c 2002 by Prentice-Hall, Inc. Upper Saddle River, New Jersey All rights reserved. No part of this book may be reproduced, in any form or by any means without permission in writing from the publisher. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN 0-13-061793-8 Pearson Education Ltd., London Pearson Education Australia Pty. Ltd., Sydney Pearson Education Singapore, Pte. Ltd. Pearson Education North Asia Ltd., Hong Kong Pearson Education Canada, Inc., Toronto Pearson Educacíon de Mexico, S.A. de C.V. Pearson Education—Japan, Tokyo Pearson Education Malaysia, Pte. Ltd. ii Proakis-50210 proa-fm August 3, 2001 15:53 Contents 1 PREFACE xi INTRODUCTION 1 1.1 1.2 Historical Review Elements of an Electrical Communication System 4 1.2.1 1.2.2 2 1 Digital Communication System, 7 Early Work in Digital Communications, 10 1.3 Communication Channels and Their Characteristics 12 1.4 Mathematical Models for Communication Channels 19 1.5 Organization of the Book 1.6 Further Reading 22 23 FREQUENCY DOMAIN ANALYSIS OF SIGNALS AND SYSTEMS 2.1 Fourier Series 2.1.1 2.2 24 Fourier Series for Real Signals: the Trigonometric Fourier Series, 29 Fourier Transforms 2.2.1 24 31 Fourier Transform of Real, Even, and Odd Signals, 35 iii Proakis-50210 proa-fm August 3, 2001 15:53 iv Contents 2.2.2 2.2.3 2.3 Basic Properties of the Fourier Transform, 36 Fourier Transform for Periodic Signals, 39 Power and Energy 2.3.1 2.3.2 2.4 Sampling of Bandlimited Signals 2.5 Bandpass Signals 2.6 Further Reading Problems 3 45 49 57 57 ANALOG SIGNAL TRANSMISSION AND RECEPTION 3.1 3.2 3.3 Amplitude Modulation (AM) 71 Double-Sideband Suppressed Carrier AM, 71 Conventional Amplitude Modulation, 78 Single-Sideband AM, 81 Vestigial-Sideband AM, 85 Implementation of AM Modulators and Demodulators, 88 Signal Multiplexing, 94 Angle Modulation 96 3.3.1 3.3.2 3.3.3 3.4 Representation of FM and PM Signals, 97 Spectral Characteristics of Angle-Modulated Signals, 101 Implementation of Angle Modulators and Demodulators, 107 Radio and Television Broadcasting 3.4.1 3.4.2 3.4.3 115 AM Radio Broadcasting, 115 FM Radio Broadcasting, 116 Television Broadcasting, 120 3.5 Mobile Radio Systems 3.6 Further Reading Problems 128 131 131 RANDOM PROCESSES 4.1 4.2 70 Introduction to Modulation 70 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.2.6 4 40 Energy-Type Signals, 41 Power-Type Signals, 42 Probability and Random Variables Random Processes: Basic Concepts 4.2.1 4.2.2 4.2.3 4.2.4 144 144 159 Description of Random Processes, 162 Statistical Averages, 164 Stationary Processes, 166 Random Processes and Linear Systems, 174 Proakis-50210 proa-fm August 3, 2001 15:53 Contents 4.3 v Random Processes in the Frequency Domain 4.3.1 4.3.2 4.4 Gaussian and White Processes 4.4.1 4.4.2 4.5 Bandlimited Processes and Sampling 4.6 Bandpass Processes 4.7 Further Reading 192 194 201 202 EFFECT OF NOISE ON ANALOG COMMUNICATION SYSTEMS 5.1 Effect of Noise on Linear-Modulation Systems 5.1.1 5.1.2 5.1.3 5.1.4 5.2 5.3 217 234 Threshold Effect in Angle Modulation, 244 Pre-emphasis and De-emphasis Filtering, 248 5.4 Comparison of Analog-Modulation Systems 5.5 Effects of Transmission Losses and Noise in Analog Communication Systems 252 5.5.1 5.5.2 5.5.3 5.5.4 5.6 251 Characterization of Thermal Noise Sources, 253 Effective Noise Temperature and Noise Figure, 254 Transmission Losses, 257 Repeaters for Signal Transmission, 258 Further Reading Problems 261 261 INFORMATION SOURCES AND SOURCE CODING 6.1 Modeling of Information Sources 6.1.1 6.1.2 225 The Phase-Locked Loop (PLL), 226 Effect of Additive Noise on Phase Estimation, 229 Effect of Noise on Angle Modulation 5.3.1 5.3.2 217 Effect of Noise on a Baseband System, 218 Effect of Noise on DSB-SC AM, 218 Effect of Noise on SSB AM, 220 Effect of Noise on Conventional AM, 221 Carrier-Phase Estimation with a Phase-Locked Loop (PLL) 5.2.1 5.2.2 6 186 Gaussian Processes, 186 White Processes, 188 Problems 5 177 Power Spectrum of Stochastic Processes, 177 Transmission over LTI Systems, 183 268 Measure of Information, 269 Joint and Conditional Entropy, 271 267 Proakis-50210 proa-fm August 3, 2001 15:53 vi Contents 6.2 6.3 Source-Coding Theorem Source-Coding Algorithms 276 6.3.1 6.3.2 6.4 6.8 Scalar Quantization, 291 Vector Quantization, 300 Waveform Coding 6.6.1 6.6.2 6.6.3 6.7 302 Pulse-Code Modulation (PCM), 302 Differential Pulse-Code Modulation (DPCM), 307 Delta Modulation (M), 310 Analysis-Synthesis Techniques 312 Digital Audio Transmission and Digital Audio Recording 6.8.1 6.8.2 316 Digital Audio in Telephone Transmission Systems, 317 Digital Audio Recording, 319 6.9 The JPEG Image-Coding Standard 6.10 Further Reading Problems 7 282 Mutual Information, 283 Differential Entropy, 284 Rate-Distortion Function, 285 Quantization 290 6.5.1 6.5.2 6.6 The Huffman Source-Coding Algorithm, 276 The Lempel-Ziv Source-Coding Algorithm, 280 Rate-Distortion Theory 6.4.1 6.4.2 6.4.3 6.5 273 323 327 327 DIGITAL TRANSMISSION THROUGH THE ADDITIVE WHITE GAUSSIAN NOISE CHANNEL 7.1 Geometric Representation of Signal Waveforms 7.2 Pulse Amplitude Modulation 345 7.3 Two-dimensional Signal Waveforms 341 350 7.3.1 Baseband Signals, 350 7.3.2 Two-dimensional Bandpass Signals—Carrier-Phase Modulation, 354 7.3.3 Two-dimensional Bandpass Signals—Quadrature Amplitude Modulation, 357 7.4 Multidimensional Signal Waveforms 7.4.1 7.4.2 7.4.3 7.4.4 360 Orthogonal Signal Waveforms, 360 Biorthogonal Signal Waveforms, 365 Simplex Signal Waveforms, 366 Binary-Coded Signal Waveforms, 367 340 Proakis-50210 proa-fm August 3, 2001 15:53 Contents 7.5 vii Optimum Receiver for Digitally Modulated Signals in Additive White Gaussian Noise 370 7.5.1 7.5.2 7.5.3 7.5.4 Correlation-Type Demodulator, 370 Matched-Filter-Type Demodulator, 375 The Optimum Detector, 381 Demodulation and Detection of Carrier-Amplitude Modulated Signals, 386 7.5.5 Demodulation and Detection of Carrier-Phase Modulated Signals, 388 7.5.6 Demodulation and Detection of Quadrature Amplitude Modulated Signals, 396 7.5.7 Demodulation and Detection of Frequency-Modulated Signals, 398 7.6 Probability of Error for Signal Detection in Additive White Gaussian Noise 405 7.6.1 7.6.2 7.6.3 7.6.4 7.6.5 7.6.6 7.6.7 7.6.8 7.6.9 7.6.10 7.7 Performance Analysis for Wireline and Radio Communication Channels 436 7.7.1 7.7.2 7.8 Regenerative Repeaters, 437 Link Budget Analysis for Radio Channels, 438 Symbol Synchronization 442 7.8.1 7.8.2 7.8.3 7.8.4 7.8.5 7.9 Probability of Error for Binary Modulation, 405 Probability of Error for M-ary PAM, 408 Probability of Error for Phase-Coherent PSK Modulation, 413 Probability of Error for DPSK, 417 Probability of Error for QAM, 418 Probability of Error for M-ary Orthogonal Signals, 423 Probability of Error for M-ary Biorthogonal Signals, 428 Probability of Error for M-ary Simplex Signals, 429 Probability of Error for Noncoherent Detection of FSK, 430 Comparison of Modulation Methods, 432 Early–Late Gate Synchronizers, 443 Minimum Mean-Square-Error Method, 445 Maximum-Likelihood Methods, 448 Spectral Line Methods, 449 Symbol Synchronization for Carrier-Modulated Signals, 451 Further Reading Problems 8 452 453 DIGITAL TRANSMISSION THROUGH BANDLIMITED AWGN CHANNELS 8.1 Digital Transmission through Bandlimited Channels 8.1.1 474 474 Digital PAM Transmission through Bandlimited Baseband Channels, 478 8.1.2 Digital Transmission through Bandlimited Bandpass Channels, 480 Proakis-50210 proa-fm August 3, 2001 15:53 viii Contents 8.2 The Power Spectrum of Digitally Modulated Signals 482 8.2.1 8.2.2 8.3 Signal Design for Bandlimited Channels 8.3.1 8.3.2 8.4 490 Design of Bandlimited Signals for Zero ISI—The Nyquist Criterion, 492 Design of Bandlimited Signals with Controlled ISI—Partial Response Signals, 497 Probability of Error in Detection of Digital PAM 8.4.1 8.4.2 8.4.3 8.5 The Power Spectrum of the Baseband Signal, 483 The Power Spectrum of a Carrier-Modulated Signal, 488 499 Probability of Error for Detection of Digital PAM with Zero ISI, 500 Symbol-by-Symbol Detection of Data with Controlled ISI, 501 Probability of Error for Detection of Partial Response Signals, 504 Digitally Modulated Signals with Memory 507 8.5.1 8.5.2 8.5.3 Modulation Codes and Modulation Signals with Memory, 508 The Maximum-Likelihood Sequence Detector, 521 Maximum-Likelihood Sequence Detection of Partial Response Signals, 525 8.5.4 The Power Spectrum of Digital Signals with Memory, 530 8.6 System Design in the Presence of Channel Distortion 8.6.1 8.6.2 8.7 Multicarrier Modulation and OFDM 556 8.7.1 8.8 An OFDM System Implemented via the FFT Algorithm, 557 Further Reading Problems 9 534 Design of Transmitting and Receiving Filters for a Known Channel, 535 Channel Equalization, 538 560 561 CHANNEL CAPACITY AND CODING 9.1 9.2 Modeling of Communication Channels Channel Capacity 9.2.1 9.3 Transmission of Analog Sources by PCM, 590 A Tight Bound on Error Probability of Orthogonal Signals, 592 The Promise of Coding, 595 Linear Block Codes 9.5.1 9.5.2 9.6 579 Gaussian Channel Capacity, 583 Coding for Reliable Communication 591 9.4.1 9.4.2 9.5 576 Bounds on Communication 586 9.3.1 9.4 576 Cyclic Codes 9.6.1 601 Decoding and Performance of Linear Block Codes, 606 Burst-Error-Correcting-Codes, 614 615 The Structure of Cyclic Codes, 615 Proakis-50210 proa-fm August 3, 2001 15:53 Contents 9.7 ix Convolutional Codes 623 9.7.1 9.7.2 Basic Properties of Convolutional Codes, 624 Optimum Decoding of Convolutional Codes—The Viterbi Algorithm, 629 9.7.3 Other Decoding Algorithms for Convolutional Codes, 634 9.7.4 Bounds on Error Probability of Convolutional Codes, 634 9.8 Complex Codes Based on Combination of Simple Codes 9.8.1 9.8.2 9.8.3 9.8.4 9.8.5 9.9 Coding for Bandwidth-Constrained Channels 9.9.1 9.9.2 9.10 646 Combined Coding and Modulation, 647 Trellis-Coded Modulation, 649 Practical Applications of Coding 9.10.1 9.10.2 9.10.3 9.11 638 Product Codes, 639 Concatenated Codes, 640 Turbo Codes, 640 The BCJR Algorithm, 642 Performance of Turbo Codes, 644 655 Coding for Deep-Space Communications, 656 Coding for Telephone-Line Modems, 657 Coding for Compact Discs, 658 Further Reading Problems 661 661 10 WIRELESS COMMUNICATIONS 10.1 674 Digital Transmission on Fading Multipath Channels 674 10.1.1 10.1.2 10.1.3 Channel Models for Time-Variant Multipath Channels, 676 Signal Design for Fading Multipath Channels, 684 Performance of Binary Modulation in Frequency Nonselective Rayleigh Fading Channels, 686 10.1.4 Performance Improvement Through Signal Diversity, 689 10.1.5 Modulation and Demodulation on Frequency Selective Channels— The RAKE Demodulator, 694 10.1.6 Multiple Antenna Systems and Space-Time Codes, 697 10.2 Continuous Carrier-Phase Modulation 702 10.2.1 10.2.2 10.2.3 10.2.4 10.2.5 10.3 Continuous-Phase FSK (CPFSK), 702 Continuous-Phase Modulation (CPM), 711 Spectral Characteristics of CPFSK and CPM Signals, 715 Demodulation and Detection of CPM Signals, 720 Performance of CPM in AWGN and Rayleigh Fading Channels, 726 Spread-Spectrum Communication Systems 10.3.1 10.3.2 10.3.3 729 Model of a Spread-Spectrum Digital Communication System, 730 Direct-Sequence Spread-Spectrum Systems, 731 Some Applications of DS Spread-Spectrum Signals, 742 Proakis-50210 proa-fm August 3, 2001 15:53 x Contents 10.3.4 10.3.5 10.3.6 10.3.7 10.4 Digital Cellular Communication Systems 10.4.1 10.4.2 10.5 Effect of Pulsed Interference and Fading, 746 Generation of PN Sequences, 748 Frequency-Hopped Spread Spectrum, 752 Synchronization of Spread-Spectrum Systems, 758 766 The GSM System, 766 CDMA System Based on IS-95, 768 Further Reading Problems 774 775 APPENDIX A: THE PROBABILITY OF ERROR FOR MULTICHANNEL RECEPTION OF BINARY SIGNALS 782 REFERENCES 785 INDEX 794 Proakis-50210 proa-fm August 3, 2001 15:53 Preface The objective of this book is to provide an introduction to the basic principles in the analysis and design of communication systems. It is primarily intended for use as a text for a first course in communications, either at a senior level or at a first-year graduate level. BROAD TOPICAL COVERAGE Although we have placed a very strong emphasis on digital communications, we have provided a review of important mathematical foundational topics and a solid introduction to analog communications. The major topics covered are: • • • A review of frequency domain analysis of signals and systems, and the characterization of random processes (Chapters 2 and 4) An introduction to analog signal transmission and reception (Chapters 3 and 5) An introduction to digital communications (Chapters 6–10) EMPHASIS ON DIGITAL COMMUNICATIONS Our motivation for emphasizing digital communications is due to the technological developments that have occurred during the past five decades. Today, digital communication systems are in common use and generally carry the bulk of our daily information transmission through a variety of communications media, such as wireline telephone channels, microwave radio, fiber optic channels, and satellite channels. We are currently witnessing an explosive growth in the development of personal communication systems xi Proakis-50210 proa-fm August 3, 2001 xii 15:53 Preface and ultrahigh speed communication networks, which are based on digital transmission of the information, whether it is voice, still images, or video. We anticipate that, in the near future, we will witness a replacement of the current analog AM and FM radio and television broadcast by digital transmission systems. The development of sophisticated, high-speed digital communication systems has been accelerated by concurrent developments in inexpensive high speed integrated circuits (IC) and programmable digital signal processing chips. The developments in Microelectronic IC fabrication have made possible the implementation of high-speed, high precision A/D converters, of powerful error-correcting coders/decoders, and of complex digital modulation techniques. All of these technological developments point to a continuation in the trend toward increased use of digital communications as a means for transmitting information. OVERVIEW OF THE TEXT It is assumed that students using this book have a basic understanding of linear system theory, both continuous and discrete, including a working knowledge of Fourier series and Fourier transform techniques. Chapter 2 provides a review of basic material on signals and systems and establishes the necessary notation used in subsequent chapters. It is also assumed that students have had a first course in probability. Such courses are currently required in many undergraduate electrical engineering and computer engineering programs. Chapter 4 provides a review of probability and random processes to the extent that is necessary for a first course in communications. Chapter 3 treats modulation and demodulation of analog signals. This treatment includes amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM). Radio and television broadcasting and mobile radio cellular systems are discussed as examples of analog communication systems. Chapter 5 continues the treatment of analog communication systems by analyzing the effect of additive noise in the demodulation of AM, FM, and PM signals. The phase-locked loop, which is used for estimating the phase of a sinusoidal carrier in both analog and digital communication systems is also described in Chapter 5. The chapter concludes with a treatment of the effect of transmission losses and the characterization of noise sources in communication systems. A logical beginning in the introduction of digital communication systems analysis and design is the characterization of information sources and source encoding. Chapter 6 is devoted to this topic. In this chapter we introduce the reader to the modeling of information sources, both discrete and continuous (analog), and the basic mathematical concepts of entropy and mutual information. Our discussion of source encoding for discrete sources includes the Huffman coding algorithm and the Lempel-Ziv algorithm. For the case of analog sources, we treat both scalar and vector quantization and describe the common waveform-coding techniques, namely, PCM, DPCM, and DM. We also describe the LPC-based source modeling method. As practical examples of the sourcecoding methods described in this chapter we cite the digital speech transmission systems Proakis-50210 proa-fm August 3, 2001 15:53 Preface xiii in the telephone plant, the digital audio recording systems as embodied in the compact disc (CD) player and the JPEG image-coding standard. Digital modulation and demodulation techniques are described in Chapter 7. Binary and nonbinary modulation methods are described based on a geometric representation of signals, and their error-rate performance is evaluated and compared. This chapter also describes symbol synchronization methods for digital communication systems. Chapter 8 treats digital transmission through bandlimited AWGN channels. In this chapter we derive the power-spectral density of linearly modulated baseband signals and consider the problem of signal design for a bandlimited channel. We show that the effect of channel distortion is to introduce intersymbol interference (ISI), which can be eliminated or minimized by proper signal design. The use of linear and nonlinear adaptive equalizers for reducing the effect of ISI is also described. Chapter 9 treats the topic of channel coding and decoding. The capacity of a communication channel is first defined, and the capacity of the Gaussian channel is determined. Linear block codes and convolutional codes are introduced and appropriate decoding algorithms are described. The benefits of coding for bandwidth constrained channels are also described. The final section of this chapter presents three practical applications of coding. The last chapter of this book treats topics in wireless communications. First, we consider the characterization of fading multipath channels and describe the effects of such channels on wireless digital communication systems. The design of signals that are effective in mitigating this type of channel distortion is also considered. Second, we describe the class of continuous-phase modulated signals, which are especially suitable for digital communication in wireless channels. Finally, we treat the class of spreadspectrum signals, which are suitable for multi-user wireless communication systems. EXAMPLES AND HOMEWORK PROBLEMS We have included a large number of carefully chosen examples and homework problems. The text contains over 180 worked-out examples and over 480 problems. Examples and problems range from simple exercises to more challenging and thoughtprovoking problems. A Solutions Manual is available free to all adopting faculty, which is provided in both typeset form and as a diskette formatted in LATEX. Solutions are not available for sale to students. This will enable instructors to print out solutions in any configuration easily. COURSE OPTIONS This book can serve as a text in either a one- or two-semester course in communication system. An important consideration in the design of the course is whether or not the students have had a prior course in probability and random processes. Another important consideration is whether or not analog modulation and demodulation techniques are to be covered. Here, we outline three scenarios. Others are certainly possible. Proakis-50210 proa-fm August 3, 2001 15:53 xiv Preface 1. A one-term course in analog and digital communication: Selected review sections from Chapters 2 and 4, all of chapters 3, 5, 7, and 8, and selections from chapters 6, 9, and 10. 2. A one-term course in digital communication: Selected review sections from Chapters 2 and 4, and Chapters 6–10. 3. A two-term course sequence on analog and digital communications: (a) Chapters 2–6 for the first course. (b) Chapters 7–10 for the second course. We wish to thank Gloria Doukakis for her assistance in the preparation of the manuscript. John Proakis Adjunct Professor, University of California at San Diego and Professor Emeritus, Masoud Salehi Northeastern University Proakis-50210 book August 3, 2001 13:2 1 Introduction Every day, in our work and in our leisure time, we come in contact with and use a variety of modern communication systems and communication media, the most common being the telephone, radio, television, and the Internet. Through these media we are able to communicate (nearly) instantaneously with people on different continents, transact our daily business, and receive information about various developments and events of note that occur all around the world. Electronic mail and facsimile transmission have made it possible to rapidly communicate written messages across great distances. Can you imagine a world without telephones, radio, and TV? Yet, when you think about it, most of these modern-day communication systems were invented and developed during the past century. Here, we present a brief historical review of major developments within the last two hundred years that have had a major role in the development of modern communication systems. 1.1 HISTORICAL REVIEW Telegraphy and Telephony. One of the earliest inventions of major significance to communications was the invention of the electric battery by Alessandro Volta in 1799. This invention made it possible for Samuel Morse to develop the electric telegraph, which he demonstrated in 1837. The first telegraph line linked Washington with Baltimore and became operational in May 1844. Morse devised the variable-length binary code given in Table 1.1, in which letters of the English alphabet were represented by a sequence of dots and dashes (code words). In this code, more frequently occurring letters are represented by short code words, while letters occurring less frequently are represented by longer code words. 1 Proakis-50210 book August 3, 2001 13:2 2 Introduction TABLE 1.1 A B C D E F G H I J K L M Chapter 1 MORSE CODE ·— —··· —·—· —·· · ··—· ——· ···· ·· ·——— —·— ·—·· —— N O P Q R S T U V W X Y Z —· ——— ·——· ——·— ·—· ··· — ··— ···— ·—— —··— —·—— ——·· (a) Letters Period (·) ·—·—·— Comma (,) ——··—— Interrogation (?) ··——·· Quotation Mark (") · — · · — · Colon (:) ———··· Semicolon (;) —·—·—· Parenthesis ( ) —·——·— ·———— ··——— ···—— ····— ··· · · —···· ——··· ———·· ————· ————— 1 2 3 4 5 6 7 8 9 0 (b) Numbers Wait sign (AS) Double dash (break) Error sign Fraction bar (/) End of message (AR) End of transmission (SK) ·—··· —···— ··· · ···· —··—· ·—·—· ···—·— (c) Punctuation and Special Characters The Morse code was the precursor to the variable-length source-coding methods that are described in Chapter 6. It is remarkable that the earliest form of electrical communications that was developed by Morse, namely telegraphy, was a binary digital communication system in which the letters of the English alphabet were efficiently encoded into corresponding variable-length code words having binary elements. Nearly forty years later, in 1875, Émile Baudot developed a code for telegraphy in which each letter was encoded into fixed-length binary code words of length 5. In the Baudot code the binary code elements have equal length and are designated as mark and space. An important milestone in telegraphy was the installation of the first transatlantic cable in 1858 that linked the United States and Europe. This cable failed after about four weeks of operation. A second cable was laid a few years later and became operational in July 1866. Telephony came into being with the invention of the telephone in the 1870s. Alexander Graham Bell patented his invention of the telephone in 1876, and in 1877 established the Bell Telephone Company. Early versions of telephone communication systems were relatively simple and provided service over several hundred miles. Significant advances in the quality and range of service during the first two decades of the twentieth century resulted from the invention of the carbon microphone and the induction coil. Proakis-50210 book August 3, 2001 Section 1.1 13:2 Historical Review 3 The invention of the triode amplifier by Lee De Forest in 1906 made it possible to introduce signal amplification in telephone communication systems and, thus, to allow for telephone signal transmission over great distances. For example, transcontinental telephone transmission became operational in 1915. Two world wars and the Great Depression during the 1930s must have been a deterrent to the establishment of transatlantic telephone service. It was not until 1953, when the first transatlantic cable was laid, that telephone service became available between the United States and Europe. Automatic switching was another important advance in the development of telephony. The first automatic switch, developed by Strowger in 1897, was an electromechanical step-by-step switch. This type of switch was used for several decades. With the invention of the transistor, electronic (digital) switching became economically feasible. After several years of development at the Bell Telephone Laboratories, a digital switch was placed in service in Illinois in June 1960. During the past thirty years there have been numerous significant advances in telephone communications. Fiber optic cables are rapidly replacing copper wire in the telephone plant and electronic switches have replaced the old electromechanical systems. Wireless Communications. The development of wireless communications stems from the works of Oersted, Faraday, Gauss, Maxwell, and Hertz. In 1820, Oersted demonstrated that an electric current produces a magnetic field. On August 29, 1831, Michael Faraday showed that an induced current is produced by moving a magnet in the vicinity of a conductor. Thus, he demonstrated that a changing magnetic field produces an electric field. With this early work as background, James C. Maxwell in 1864 predicted the existence of electromagnetic radiation and formulated the basic theory that has been in use for over a century. Maxwell's theory was verified experimentally by Hertz in 1887. In 1894, a sensitive device that could detect radio signals, called the coherer, was used by its inventor Oliver Lodge to demonstrate wireless communication over a distance of 150 yards at Oxford, England. Guglielmo Marconi is credited with the development of wireless telegraphy. Marconi demonstrated the transmission of radio signals at a distance of approximately 2 kilometers in 1895. Two years later, in 1897, he patented a radio telegraph system and established the Wireless Telegraph and Signal Company. On December 12, 1901, Marconi received a radio signal at Signal Hill in Newfoundland, which was transmitted from Cornwall, England, a distance of about 1700 miles. The invention of the vacuum tube was especially instrumental in the development of radio communication systems. The vacuum diode was invented by Fleming in 1904 and the vacuum triode amplifier was invented by De Forest in 1906, as previously indicated. The invention of the triode made radio broadcast possible in the early part of the twentieth century. Amplitude modulation (AM) broadcast was initiated in 1920 when radio station KDKA, Pittsburgh, went on the air. From that date, AM radio broadcasting grew rapidly across the country and around the world. The superheterodyne AM radio receiver, as we know it today, was invented by Edwin Armstrong during World War I. Another significant development in radio communications was the invention Proakis-50210 book August 3, 2001 4 13:2 Introduction Chapter 1 of Frequency modulation (FM), also by Armstrong. In 1933, Armstrong built and demonstrated the first FM communication system. However, the use of FM was slow to develop compared with AM broadcast. It was not until the end of World War II that FM broadcast gained in popularity and developed commercially. The first television system was built in the United States by V. K. Zworykin and demonstrated in 1929. Commercial television broadcasting began in London in 1936 by the British Broadcasting Corporation (BBC). Five years later the Federal Communications Commission (FCC) authorized television broadcasting in the United States. The Past Fifty Years. The growth in communications services over the past fifty years has been phenomenal. The invention of the transistor in 1947 by Walter Brattain, John Bardeen, and William Shockley; the integrated circuit in 1958 by Jack Kilby and Robert Noyce; and the laser by Townes and Schawlow in 1958, have made possible the development of small-size, low-power, low-weight, and high-speed electronic circuits which are used in the construction of satellite communication systems, wideband microwave radio systems, and lightwave communication systems using fiber optic cables. A satellite named Telstar I was launched in 1962 and used to relay TV signals between Europe and the United States. Commercial satellite communication services began in 1965 with the launching of the Early Bird satellite. Currently, most of the wireline communication systems are being replaced by fiber optic cables which provide extremely high bandwidth and make possible the transmission of a wide variety of information sources, including voice, data, and video. Cellular radio has been developed to provide telephone service to people in automobiles, buses, and trains. High-speed communication networks link computers and a variety of peripheral devices literally around the world. Today we are witnessing a significant growth in the introduction and use of personal communications services, including voice, data, and video transmission. Satellite and fiber optic networks provide high-speed communication services around the world. Indeed, this is the dawn of the modern telecommunications era. There are several historical treatments in the development of radio and telecommunications covering the past century. We cite the books by McMahon, entitled The Making of a Profession—A Century of Electrical Engineering in America (IEEE Press, 1984); Ryder and Fink, entitled Engineers and Electronics (IEEE Press, 1984); and S. Millman, Ed., entitled A History of Engineering and Science in the Bell System— Communications Sciences (1925–1980) (AT & T Bell Laboratories, 1984). 1.2 ELEMENTS OF AN ELECTRICAL COMMUNICATION SYSTEM Electrical communication systems are designed to send messages or information from a source that generates the messages to one or more destinations. In general, a communication system can be represented by the functional block diagram shown in Figure 1.1. The information generated by the source may be of the form of voice (speech source), a picture (image source), or plain text in some particular language, such as English, Japanese, German, French, etc. An essential feature of any source that generates infor- Proakis-50210 book August 3, 2001 Section 1.2 13:2 Elements of an Electrical Communication System Information source and input transducer 5 Transmitter Channel Output signal Output transducer Receiver Figure 1.1 Functional block diagram of a communication system. mation is that its output is described in probabilistic terms; i.e., the output of a source is not deterministic. Otherwise, there would be no need to transmit the message. A transducer is usually required to convert the output of a source into an electrical signal that is suitable for transmission. For example, a microphone serves as the transducer that converts an acoustic speech signal into an electrical signal, and a video camera converts an image into an electrical signal. At the destination, a similar transducer is required to convert the electrical signals that are received into a form that is suitable for the user; e.g., acoustic signals, images, etc. The heart of the communication system consists of three basic parts, namely, the transmitter, the channel, and the receiver. The functions performed by these three elements are described next. The Transmitter. The transmitter converts the electrical signal into a form that is suitable for transmission through the physical channel or transmission medium. For example, in radio and TV broadcast, the Federal Communications Commission (FCC) specifies the frequency range for each transmitting station. Hence, the transmitter must translate the information signal to be transmitted into the appropriate frequency range that matches the frequency allocation assigned to the transmitter. Thus, signals transmitted by multiple radio stations do not interfere with one another. Similar functions are performed in telephone communication systems where the electrical speech signals from many users are transmitted over the same wire. In general, the transmitter performs the matching of the message signal to the channel by a process called modulation. Usually, modulation involves the use of the information signal to systematically vary either the amplitude, frequency, or phase of a sinusoidal carrier. For example, in AM radio broadcast, the information signal that is transmitted is contained in the amplitude variations of the sinusoidal carrier, which is the center frequency in the frequency band allocated to the radio transmitting station. This is an example of amplitude modulation. In FM radio broadcast, the information signal that is transmitted is contained in the frequency variations of the sinusoidal carrier. This is an example of frequency modulation. Phase modulation (PM) is yet a third method for impressing the information signal on a sinusoidal carrier. Proakis-50210 book August 3, 2001 6 13:2 Introduction Chapter 1 In general, carrier modulation such as AM, FM, and PM is performed at the transmitter, as indicated above, to convert the information signal to a form that matches the characteristics of the channel. Thus, through the process of modulation, the information signal is translated in frequency to match the allocation of the channel. The choice of the type of modulation is based on several factors, such as the amount of bandwidth allocated, the types of noise and interference that the signal encounters in transmission over the channel, and the electronic devices that are available for signal amplification prior to transmission. In any case, the modulation process makes it possible to accommodate the transmission of multiple messages from many users over the same physical channel. In addition to modulation, other functions that are usually performed at the transmitter are filtering of the information-bearing signal, amplification of the modulated signal, and in the case of wireless transmission, radiation of the signal by means of a transmitting antenna. The Channel. The communications channel is the physical medium that is used to send the signal from the transmitter to the receiver. In wireless transmission, the channel is usually the atmosphere (free space). On the other hand, telephone channels usually employ a variety of physical media, including wirelines, optical fiber cables, and wireless (microwave radio). Whatever the physical medium for signal transmission, the essential feature is that the transmitted signal is corrupted in a random manner by a variety of possible mechanisms. The most common form of signal degradation comes in the form of additive noise, which is generated at the front end of the receiver, where signal amplification is performed. This noise is often called thermal noise. In wireless transmission, additional additive disturbances are man-made noise, and atmospheric noise picked up by a receiving antenna. Automobile ignition noise is an example of man-made noise, and electrical lightning discharges from thunderstorms is an example of atmospheric noise. Interference from other users of the channel is another form of additive noise that often arises in both wireless and wireline communication systems. In some radio communication channels, such as the ionospheric channel that is used for long range, short-wave radio transmission, another form of signal degradation is multipath propagation. Such signal distortion is characterized as a nonadditive signal disturbance which manifests itself as time variations in the signal amplitude, usually called fading. This phenomenon is described in more detail in Section 1.3. Both additive and nonadditive signal distortions are usually characterized as random phenomena and described in statistical terms. The effect of these signal distortions must be taken into account on the design of the communication system. In the design of a communication system, the system designer works with mathematical models that statistically characterize the signal distortion encountered on physical channels. Often, the statistical description that is used in a mathematical model is a result of actual empirical measurements obtained from experiments involving signal transmission over such channels. In such cases, there is a physical justification for the mathematical model used in the design of communication systems. On the other hand, in some communication system designs, the statistical characteristics of the channel Proakis-50210 book August 3, 2001 Section 1.2 13:2 Elements of an Electrical Communication System 7 may vary significantly with time. In such cases, the system designer may design a communication system that is robust to the variety of signal distortions. This can be accomplished by having the system adapt some of its parameters to the channel distortion encountered. The Receiver. The function of the receiver is to recover the message signal contained in the received signal. If the message signal is transmitted by carrier modulation, the receiver performs carrier demodulation in order to extract the message from the sinusoidal carrier. Since the signal demodulation is performed in the presence of additive noise and possibly other signal distortion, the demodulated message signal is generally degraded to some extent by the presence of these distortions in the received signal. As we shall see, the fidelity of the received message signal is a function of the type of modulation, the strength of the additive noise, the type and strength of any other additive interference, and the type of any nonadditive interference. Besides performing the primary function of signal demodulation, the receiver also performs a number of peripheral functions, including signal filtering and noise suppression. 1.2.1 Digital Communication System Up to this point we have described an electrical communication system in rather broad terms based on the implicit assumption that the message signal is a continuous timevarying waveform. We refer to such continuous-time signal waveforms as analog signals and to the corresponding information sources that produce such signals as analog sources. Analog signals can be transmitted directly via carrier modulation over the communication channel and demodulated accordingly at the receiver. We call such a communication system an analog communication system. Alternatively, an analog source output may be converted into a digital form and the message can be transmitted via digital modulation and demodulated as a digital signal at the receiver. There are some potential advantages to transmitting an analog signal by means of digital modulation. The most important reason is that signal fidelity is better controlled through digital transmission than analog transmission. In particular, digital transmission allows us to regenerate the digital signal in long-distance transmission, thus eliminating effects of noise at each regeneration point. In contrast, the noise added in analog transmission is amplified along with the signal when amplifiers are used periodically to boost the signal level in long-distance transmission. Another reason for choosing digital transmission over analog is that the analog message signal may be highly redundant. With digital processing, redundancy may be removed prior to modulation, thus conserving channel bandwidth. Yet a third reason may be that digital communication systems are often cheaper to implement. In some applications, the information to be transmitted is inherently digital; e.g., in the form of English text, computer data, etc. In such cases, the information source that generates the data is called a discrete (digital) source. In a digital communication system, the functional operations performed at the transmitter and receiver must be expanded to include message signal discretization at Proakis-50210 book August 3, 2001 13:2 8 Introduction Information source and input transducer Source encoder Channel encoder Chapter 1 Digital modulator Channel Output signal Output transducer Source decoder Channel decoder Digital demodulator Figure 1.2 Basic elements of a digital communication system. the transmitter and message signal synthesis or interpolation at the receiver. Additional functions include redundancy removal, and channel coding and decoding. Figure 1.2 illustrates the functional diagram and the basic elements of a digital communication system. The source output may be either an analog signal, such as audio or video signal, or a digital signal, such as the output of a computer which is discrete in time and has a finite number of output characters. In a digital communication system, the messages produced by the source are usually converted into a sequence of binary digits. Ideally, we would 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 a digital source into a sequence of binary digits is called source encoding or data compression. We shall describe source-encoding methods in Chapter 6. The sequence of binary digits from the source encoder, which we call the information sequence is passed to the channel encoder. The purpose of the channel encoder is to introduce, in a controlled manner, some redundancy in the binary information sequence which can be used at the receiver to overcome the effects of noise and interference 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 the fidelity of the received signal. In effect, redundancy in the information sequence aids the receiver in decoding the desired information sequence. For example, a (trivial) form of encoding of the binary information sequence is simply to repeat each binary digit m times, where m is some positive integer. More sophisticated (nontrivial) encoding involves taking k information bits at a time and mapping each k-bit sequence into a unique n-bit sequence, called a code word. The amount of redundancy introduced by encoding the data in this manner is measured by the ratio n/k. The reciprocal of this ratio, namely, k/n, is called the rate of the code or, simply, the code rate. The binary sequence at the output of the channel encoder is passed to the digital modulator, which serves as the interface to the communications channel. Since nearly all of the communication channels encountered in practice are capable of transmitting electrical signals (waveforms), the primary purpose of the digital modulator is to map Proakis-50210 book August 3, 2001 Section 1.2 13:2 Elements of an Electrical Communication System 9 the binary information sequence into signal waveforms. To elaborate on the point, let us suppose that the coded information sequence is to be transmitted one bit at a time at some uniform rate R bits/s. The digital modulator may simply map the binary digit 0 into a waveform s0 (t) and the binary digit 1 into a waveform s1 (t). In this manner, each bit from the channel encoder is transmitted separately. We call this binary modulation. Alternatively, the modulator may transmit b coded information bits at a time by using M = 2b distinct waveforms si (t), i = 0, 1, . . . , M − 1, one waveform for each of the 2b possible b-bit sequences. We call this M-ary modulation (M > 2). Note that a new b-bit sequence enters the modulator every b/R seconds. Hence, when the channel bit rate R is fixed, the amount of time available to transmit one of the M waveforms corresponding to a b-bit sequence is b times the time period in a system that uses binary modulation. At the receiving end of a digital communications system, the digital demodulator processes the channel-corrupted transmitted waveform and reduces each waveform to a single number that represents an estimate of the transmitted data symbol (binary or Mary). For example, when binary modulation is used, the demodulator may process the received waveform and decide on whether the transmitted bit is a 0 or a 1. In such a case, we say the demodulator has made a binary decision. As one alternative, the demodulator may make a ternary decision; that is, it decides that the transmitted bit is either a 0 or 1 or it makes no decision at all, depending on the apparent quality of the received signal. When no decision is made on a particular bit, we say that the demodulator has inserted an erasure in the demodulated data. Using the redundancy in the transmitted data, the decoder attempts to fill in the positions where erasures occurred. Viewing the decision process performed by the demodulator as a form of quantization, we observe that binary and ternary decisions are special cases of a demodulator that quantizes to Q levels, where Q ≥ 2. In general, if the digital communications system employs M-ary modulation, where m = 0, 1, . . . , M −1 represent the M possible transmitted symbols, each corresponding to b = log2 M bits, the demodulator may make a Q-ary decision, where Q ≥ M. In the extreme case where no quantization is performed, Q = ∞. When there is no redundancy in the transmitted information, the demodulator must decide which of the M waveforms was transmitted in any given time interval. Consequently Q = M, and since there is no redundancy in the transmitted information, no discrete channel decoder is used following the demodulator. On the other hand, when there is redundancy introduced by a discrete channel encoder at the transmitter, the Qary output from the demodulator occurring every b/R seconds is fed to the 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 encoder perform is the frequency with which errors occur in the decoded sequence. More precisely, the average probability of a bit-error at the output of the decoder is a measure of the performance of the demodulator-decoder combination. In general, the probability of error is a function of the code characteristics, the types of waveforms used to transmit the information over the channel, the transmitter power, the characteristics of the channel; i.e., the amount of noise, the nature of the interference, etc., and the method of demodulation and decoding. These items and their effect on performance will be discussed in detail in Chapters 7–9. Proakis-50210 book August 3, 2001 13:2 10 Introduction Chapter 1 As a final step, when an analog output is desired, the source decoder accepts the output sequence from the channel decoder and, from knowledge of the sourceencoding method used, attempts to reconstruct the original signal from the source. Due to 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 communications system. 1.2.2 Early Work in Digital Communications Although Morse is responsible for the development of the first electrical digital communication system (telegraphy), the beginnings of what we now regard as modern digital communications stem from the work of Nyquist (1924), who investigated the problem of determining the maximum signaling rate that can be used over a telegraph channel of a given bandwidth without intersymbol interference. He formulated a model of a telegraph system in which a transmitted signal has the general form s(t) = an g(t − nT ) n where g(t) represents a basic pulse shape and {an } is the binary data sequence of {±1} transmitted at a rate of 1/T bits/sec. Nyquist set out to determine the optimum pulse shape that was bandlimited to W Hz and maximized the bit rate 1/T under the constraint that the pulse caused no intersymbol interference at the sampling times k/T, k = 0, ±1, ±2, . . . . His studies led him to conclude that the maximum pulse rate 1/T is 2W pulses/sec. This rate is now called the Nyquist rate. Moreover, this pulse rate can be achieved by using the pulses g(t) = (sin 2π W t)/2π W t. This pulse shape allows the recovery of the data without intersymbol interference at the sampling instants. Nyquist's result is equivalent to a version of the sampling theorem for bandlimited signals, which was later stated precisely by Shannon (1948). The sampling theorem states that a signal s(t) of bandwidth W can be reconstructed from samples taken at the Nyquist rate of 2W samples/sec using the interpolation formula n sin 2π W (t − n/2W ) s s(t) = 2W 2π W (t − n/2W ) n In light of Nyquist's work, Hartley (1928) considered the issue of the amount of data that can be transmitted reliably over a bandlimited channel when multiple amplitude levels are used. Due to the presence of noise and other interference, Hartley postulated that the receiver can reliably estimate the received signal amplitude to some accuracy, say Aδ . This investigation led Hartley to conclude that there is a maximum data rate that can be communicated reliably over a bandlimited channel, when the maximum signal amplitude is limited to Amax (fixed power constraint) and the amplitude resolution is Aδ . Proakis-50210 book August 3, 2001 Section 1.2 13:2 Elements of an Electrical Communication System 11 Another significant advance in the development of communications was the work of 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. 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 (Wiener) filter. Hartley's and Nyquist's results on the maximum transmission rate of digital information were precursors to the work of Shannon (1948a,b) who established the mathematical foundations for information theory 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 constraint, a bandwidth constraint, and additive noise can be associated with the channel and incorporated into a single parameter, called the channel capacity. For example, in the case of an additive white (spectrally flat) Gaussian noise interference, an ideal bandlimited channel of bandwidth W has a capacity C given by P bits/s C = W log2 1 + W N0 where P is the average transmitted power and N0 is the power-spectral density of the additive noise. The significance of the channel capacity is as follows: If the information rate R from the source is less than C (R < C), then it is theoretically possible to achieve reliable transmission through the channel by appropriate coding. On the other hand if R > C, reliable transmission is not possible regardless of the amount of signal processing performed at the transmitter and receiver. Thus, Shannon established basic limits on communication of information and gave birth to a new field that is now called information theory. Initially, the fundamental work of Shannon had a relatively small impact on the design and development of new digital communications systems. In part, this was due to the small demand for digital information transmission during the decade of the 1950s. Another reason was the relatively large complexity and, hence, the high cost of digital hardware required to achieve the high efficiency and the high reliability predicted by Shannon's theory. Another important contribution to the field of digital communications is the work of Kotelnikov (1947) which provided a coherent analysis of the various digital communication systems based on a geometrical approach. Kotelnikov's approach was later expanded by Wozencraft and Jacobs (1965). The increase in the demand for data transmission during the last three decades, coupled with the development of more sophisticated integrated circuits, has led to the development of very efficient and more reliable digital communications systems. In the course of these developments, Shannon's original results and the generalization Proakis-50210 book August 3, 2001 12 13:2 Introduction Chapter 1 of his results on maximum transmission limits over a channel and on bounds on the performance achieved, have served as benchmarks relative to which any given communications system design is compared. The theoretical limits derived by Shannon and other researchers that contributed to the development of information theory serve as an ultimate goal in the continuing efforts to design and develop more efficient digital communications systems. Following Shannon's publications came the classic work of Hamming (1950) on error detecting and error-correcting codes to combat the detrimental effects of channel noise. Hamming's work stimulated many researchers in the years that followed and a variety of new and powerful codes were discovered, many of which are used today in the implementation of modern communication systems. 1.3 COMMUNICATION CHANNELS AND THEIR CHARACTERISTICS As indicated in our preceding discussion, the communication channel provides the connection 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 transmitted 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, its effect can be minimized by proper design of the transmitted signal and its demodulator at the receiver. Other types of signal degradations that may be encountered in transmission over the channel are signal attenuation, amplitude and phase distortion, and multipath distortion. The effects of noise may be minimized by increasing the power in the transmitted signal. However, equipment and other practical constraints limit the power level in the transmitted signal. Another basic limitation is the available channel bandwidth. A bandwidth constraint is usually due to the physical limitations of the medium and the electronic components used to implement the transmitter and the receiver. These two limitations result in constraining the amount of data that can be transmitted reliably over any communications channel. Shannon's basic results relate the channel capacity to the available transmitted power and channel bandwidth. Next, 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- book August 3, 2001 Section 1.3 13:2 Communication Channels and Their Characteristics 13 lines and coaxial cable are basically guided electromagnetic channels which provide relatively modest bandwidths. Telephone wire generally used to connect a customer to a central office has a bandwidth of several hundred kilohertz (KHz). On the other hand, coaxial cable has a usable bandwidth of several megahertz (MHz). Figure 1.3 illustrates the frequency range of guided electromagnetic channels which includes waveguides and optical fibers. Signals transmitted through such channels are distorted in both amplitude and phase and further corrupted by additive noise. Twisted-pair wireline channels are also prone to crosstalk interference from physically adjacent channels. Because wireline channels carry a large percentage of our daily communications around the country and Ultraviolet 1015 Hz Visible light 106 m Infrared 1014 Hz 100 mm 100 GHz 1 cm Waveguide 10 GHz 10 cm 1m 100 MHz 10 m Coaxial cable channels Frequency 1 GHz Wavelength Proakis-50210 10 MHz 100 m 1 MHz 1 km 100 kHz 10 km Wireline channels 10 kHz 100 km 1 kHz Figure 1.3 Frequency range for guided wireline channels. Proakis-50210 book August 3, 2001 14 13:2 Introduction Chapter 1 the world, much research has been performed on the characterization of their transmission properties and on methods for mitigating the amplitude and phase distortion encountered in signal transmission. In Chapter 8, we describe methods for designing optimum transmitted signals and their demodulation, including the design of channel equalizers that compensate for amplitude and phase distortion. Fiber Optic Channels. Optical fibers offer the communications system designer a channel bandwidth that is several orders of magnitude larger than coaxial cable channels. During the past decade optical fiber cables have been developed which have a relatively low signal attenuation and highly reliable photonic devices have been developed for signal generation and signal detection. These technological advances have resulted in a rapid deployment of optical fiber channels both in domestic telecommunication systems as well as for transatlantic and transpacific communications. With the large bandwidth available on fiber optic channels it is possible for the telephone companies to offer subscribers a wide array of telecommunication services, including voice, data, facsimile, and video. The transmitter or modulator in a fiber optic communication system is a light source, either a light-emitting diode (LED) or a laser. Information is transmitted by varying (modulating) the intensity of the light source with the message signal. The light propagates through the fiber as a light wave and is amplified periodically (in the case of 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. It is envisioned that optical fiber channels will replace nearly all wireline channels in the telephone network in the next few years. Wireless Electromagnetic Channels. In radio communication systems, electromagnetic energy is coupled to the propagation medium by an antenna which serves as the radiator. The physical size and the configuration of the antenna depend primarily on the frequency of operation. To obtain efficient radiation of electromagnetic energy, the antenna must be longer than 1/10 of the wavelength. Consequently, a radio station transmitting in the AM frequency band, say at 1 MHz (corresponding to a wavelength of λ = c/ f c = 300 m) requires an antenna of at least 30 meters. Figure 1.4 illustrates the various frequency bands of the electromagnetic spectrum. The mode of propagation of electromagnetic waves in the atmosphere and in free space may be subdivided into three categories, namely, ground-wave propagation, sky-wave propagation, and line-of-sight (LOS) propagation. In the VLF and ELF 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 from 1–10% of the center frequency), and hence, the information that is transmitted through August 3, 2001 Section 1.3 13:2 Communication Channels and Their Characteristics Frequency band Use Ultraviolet Visible light 106 m 1015 Hz Experimental Infrared Millimeter waves 1 cm Super High Frequency (SHF) 10 cm Ultra High Frequency (UHF) 1m Very High Frequency (VHF) 10 m High Frequency (HF) 100 m 1014 Hz Experimental Navigation Satellite to satellite Microwave relay Earth—satellite Radar 100 GHz Microwave radio 10 GHz 1 GHz UHF TV Mobile, aeronautical VHF TV and FM Broadcast Mobile radio Business Amateur radio International radio Citizen's band 100 MHz Shortwave radio 10 MHz Medium Frequency (MF) AM broadcast 1 MHz Low Frequency (LF) Aeronautical Navigation Radio teletype 100 kHz 1 km 10 km 15 Frequency book Wavelength Proakis-50210 Very Low Frequency (VLF) 10 kHz Audio band 1 kHz Longwave radio 100 km Figure 1.4 Frequency range for wireless electromagnetic channels. (Adapted from Carlson, c 1975 McGraw-Hill. Reprinted with permission of the publisher.) Sec. Ed.; these channels is relatively slow speed and, generally, confined to digital transmission. A dominant type of noise at these frequencies 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, illustrated in Figure 1.5, is the dominant mode of propagation for frequencies in the MF band (0.3–3 MHz). This is the frequency band Proakis-50210 book August 3, 2001 13:2 16 Introduction Earth Chapter 1 Figure 1.5 Illustration of ground-wave propagation. Ionosphere Earth Figure 1.6 Illustration of sky-wave propagation. used for AM broadcasting and maritime radio broadcasting. In AM broadcast, the range with ground-wave propagation of even the more powerful radio stations is limited to about 100 miles. Atmospheric noise, man-made noise, and thermal noise from electronic components at the receiver are dominant disturbances for signal transmission of MF. Sky-wave propagation, as illustrated in Figure 1.6, results from transmitted signals being reflected (bent or refracted) from the ionosphere, which consists of several layers of charged particles ranging in altitude from 30–250 miles 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 75 miles. 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 night-time hours the electron density in the lower layers of the ionosphere drops sharply and the frequency absorption that occurs during the day time 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 90–250 miles above the surface of the earth. A frequently occurring problem with electromagnetic wave propagation via skywave in the HF frequency range is signal multipath. Signal multipath occurs when the transmitted signal arrives at the receiver via multiple propagation paths at different delays. Signal multipath generally results in intersymbol interference in a digital communication system. Moreover, the signal components arriving via different propagation paths may add destructively, resulting in a phenomenon called signal fading, which most people have experienced when listening to a distant radio station at night, when sky-wave is the dominant propagation mode. Additive noise at HF is a combination of atmospheric noise and thermal voice. Sky-wave ionospheric propagation ceases to exist at frequencies above approximately 30 MHz, which is the end of the HF band. However, it is possible to have Proakis-50210 book August 3, 2001 Section 1.3 13:2 Communication Channels and Their Characteristics 17 ionospheric scatter propagation at frequencies in the range of 30–60 MHz, resulting from signal scattering from the lower ionosphere. It is also possible to communicate over distances of several hundred miles by use of tropospheric scattering at frequencies in the range of 40–300 MHz. Troposcatter results from signal scattering due to particles in the atmosphere at altitudes of 10 miles or less. Generally, ionospheric scatter and tropospheric scatter involve large signal propagation losses and require a large amount of transmitter power and relatively large antennas. Frequencies above 30 MHz propagate through the ionosphere with relatively little loss and make satellite and extraterrestrial communications possible. Hence, at frequencies in the VHF band and higher, the dominant mode of electromagnetic propagation is line-of-sight (LOS) propagation. For terrestrial communication systems, this means that the transmitter and receiver antennas must be in direct LOS with relatively little or no obstruction. For this reason television stations transmitting in the VHF and UHF frequency bands mount their antennas on high towers in order to achieve a broad coverage area. In general, the coverage area for LOS propagation is limited by the curvature of the earth. If the transmitting antenna is mounted at a height h feet above the surface of the earth, the distance to the radio√horizon, assuming no physical obstructions such a mountains, is approximately d = 2h miles. For example, a TV antenna mounted on a tower of 1000 ft in height provides a coverage of approximately 50 miles. As another example, microwave radio relay systems used extensively for telephone and video transmission at frequencies above 1 GHz have antennas mounted on tall towers or on the top of tall buildings. The dominant noise limiting the performance of communication systems in the VHF and UHF frequency ranges is thermal noise generated in the receiver front end and cosmic noise picked up by the antenna. At frequencies in the SHF band above 10 GHz, atmospheric conditions play a major role in signal propagation. Figure 1.7 illustrates the signal attenuation in dB/mile due to precipitation for frequencies in the range of 10–100 GHz. We observe that heavy rain introduces extremely high propagation losses that can result in service outages (total breakdown in the communication system). At frequencies above the EHF band, we have the infrared and visible light regions of the electromagnetic spectrum which can be used to provide LOS optical communication in free space. To date, these frequency bands have been used in experimental communication systems, such as satellite-to-satellite links. Underwater Acoustic Channels. Over the past few decades, ocean exploration activity has been steadily increasing. Coupled with this increase in ocean exploration is the need to transmit data, collected by sensors placed underwater, 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 underwater, except at extremely 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 book August 3, 2001 15:18 18 Introduction Chapter 1 10.0 Heavy rain (1 in. / h) Medium rain (0.1 in. / h) 1.0 Attenuation, dB/mi Proakis-50210 Light rain (0.05 in. / h) 0.1 Fog (0.01 in. / h) 0.01 0.001 10 30 Frequency, GHz 100 Figure 1.7 Signal attenuation due to precipitation. (From Ziemer and Tranter c Houghton Mifflin Reprinted with permission of the publisher.) (1990); depth, which √ is the distance a signal is attenuated by 1/e. For sea water, the skin depth δ = 250/ f , where f is expressed in Hz and δ is in meters. 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. A shallow water acoustic channel is characterized as a multipath channel due to signal reflections from the surface and the bottom of the sea. Due to wave motion, the signal multipath components undergo time-varying propagation delays which result in signal fading. In addition, there is frequency-dependent attenuation, which is approximately proportional to the square of the signal frequency. 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. Proakis-50210 book August 3, 2001 Section 1.4 13:2 Mathematical Models for Communication Channels 19 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 our data-handling activities on a daily basis. Magnetic tape, including digital audio tape and video tape, magnetic disks used for storing large amounts of computer data, and optical disks used for computer data storage, music (compact disks), and video 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 is 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 adjacent tracks is generally present in the readback signal of a storage system, just as is the case in a telephone or a radio communication system. The amount of data that can be stored is generally limited by the size of the disk or tape and the density (number of bits stored per square inch) that can be achieved by the write/read electronic systems and heads. For example, a packing density of 109 bits/sq. in. has been recently demonstrated in an experimental magnetic disk storage system. (Current commercial magnetic storage products achieve a much lower density.) The speed at which data can be written on a disk or tape and the speed at which it can be read back is also limited by the associated mechanical and electrical subsystems that constitute an information storage system. Channel coding and modulation are essential components of a well-designed digital magnetic or optical storage system. In the readback process, the signal is demodulated and the added redundancy introduced by the channel encoder is used to correct errors in the readback signal. 1.4 MATHEMATICAL MODELS FOR COMMUNICATION CHANNELS In the design of communication systems for transmitting information through physical channels, we find it convenient to construct mathematical models that reflect the most important characteristics of the transmission medium. Then, the mathematical model for the channel is used in the design of the channel encoder and modulator at the transmitter and the demodulator and channel decoder at the receiver. Next, we provide a brief description of the channel models that are frequently used to characterize many of the physical channels that we encounter in practice. The Additive Noise Channel. The simplest mathematical model for a communication channel is the additive noise channel, illustrated in Figure 1.8. In this model the transmitted signal s(t) is corrupted by an additive random noise process Proakis-50210 book August 3, 2001 13:2 20 Introduction Chapter 1 Channel s(t) r(t) s(t) n(t) n(t) Figure 1.8 The additive noise channel. n(t). Physically, the additive noise process may arise from electronic components and amplifiers at the receiver of the communication system, or from interference encountered in transmission, as in the case of radio signal transmission. If the noise is introduced primarily by electronic components and amplifiers at the receiver, it may be characterized as thermal noise. This type of noise is characterized statistically as a Gaussian noise process. Hence, the resulting mathematical model for the channel is usually called the additive Gaussian noise channel. Because this channel model applies to a broad class of physical communication channels and because of its mathematical tractability, this is the predominant channel model used in our communication system analysis and design. Channel attenuation is easily incorporated into the model. When the signal undergoes attenuation in transmission through the channel, the received signal is r (t) = as(t) + n(t) (1.4.1) where a represents the attenuation factor. The Linear Filter Channel. In some physical channels such as wireline telephone channels, filters are used to ensure that the transmitted signals do not exceed specified bandwidth limitations and, thus, do not interfere with one another. Such channels are generally characterized mathematically as linear filter channels with additive noise, as illustrated in Figure 1.9. Hence, if the channel input is the signal s(t), the s (t) Linear filter h (t) r (t) s (t) h(t) n(t) n (t) Channel Figure 1.9 The linear filter channel with additive noise. Proakis-50210 book August 3, 2001 Section 1.4 s(t) 13:2 Mathematical Models for Communication Channels Linear time-invariant filter h( ; t) 21 r (t) n (t) Figure 1.10 Linear time-variant filter channel with additive noise. Channel channel output is the signal r (t) = s(t) h(t) + n(t) +∞ = h(τ )s(t − τ ) dτ + n(t) −∞ (1.4.2) where h(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 channels which result in time-variant multipath propagation of the transmitted signal may be characterized mathematically as time-variant linear filters. Such linear filters are characterized by time-variant channel impulse response h(τ ; t) where h(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 Figure 1.10. For an input signal s(t), the channel output signal is r (t) = s(t) h(τ ; t) + n(t) +∞ h(τ ; t)s(t − τ ) dτ + n(t) = −∞ (1.4.3) A good model for multipath signal propagation through physical channels, such as the ionosphere (at frequencies below 30 MHz) and mobile cellular radio channels, is a special case of Equation (1.4.3) is which the time-variant impulse response has the form h(τ ; t) = L ak (t)δ(τ − τk ) (1.4.4) k=1 where the {ak (t)} represent the possibly time-variant attenuation factors for the L multipath propagation paths. If Equation (1.4.4) is substituted into Equation (1.4.3), the received signal has the form r (t) = L ak (t)s(t − τk ) + n(t) (1.4.5) k=1 Hence, the received signal consists of L multipath components, where each component is attenuated by {ak } and delayed by {τk }. Proakis-50210 book August 3, 2001 15:18 22 Introduction Chapter 1 The three mathematical models described above adequately characterize a large majority of physical channels encountered in practice. These three channel models are used in this text for the analysis and design of communication systems. 1.5 ORGANIZATION OF THE BOOK Before we embark on the analysis and design of communication systems, we provide a brief review of basic frequency-domain characteristics of signals and linear systems in Chapter 2. Emphasis is placed on the Fourier series and the Fourier transform representation of signals and the use of transforms in linear systems analysis. The process of sampling a bandlimited analog signal is also considered. In Chapter 3, we treat the modulation and demodulation of analog signals. This chapter provides detailed description of amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM). As examples of analog signal transmission and reception, we consider radio and television broadcasting, and mobile radio cellular communication systems. In Chapter 4, we present a review of the basic definitions and concepts in probability and random processes. These topics are particularly important in our study of electrical communications, because information sources produce random signals at their output and communication channels generally corrupt the transmitted signals in a random manner, through the addition of noise and other channel distortion. Special emphasis is placed on Gaussian random processes, which provide mathematically tractable models for additive noise disturbances. Both time domain and frequency domain representations of random signals are presented. Chapters 5 provides a treatment of the effects of additive noise in the demodulation of amplitude modulated (AM) and angle modulated (FM, PM) analog signals and a comparison of these analog signal modulations in terms of their signal-to-noise ratio performance. Also discussed in this chapter is the problem of estimating the carrier phase using a phase-locked loop (PLL). Finally, we describe the characterization of thermal noise and the effect of transmission losses in analog communication systems. The remainder of the book is focused on digital communication systems. Chapter 6 is devoted to the modeling and characterization of information sources and source coding. In this chapter, we introduce a measure for the information content of a discrete source and describe two algorithms, the Huffman algorithm and the LempelZiv algorithm, for efficient encoding of the source output. The major part of the chapter is devoted to the problem of encoding the outputs of analog sources. Several waveformencoding methods are described, including pulse-code modulation (PCM), differential PCM, and delta modulation (DM). We also describe a model-based approach to analog source encoding, based on linear prediction. As practical examples of the theory presented in this chapter, we consider digital audio recording systems, digital audio transmission in telephone systems, and the JPEG image-coding standard. Chapter 7 treats modulation methods for digital signal transmission through an additive white Gaussian noise channel. Various types of binary and nonbinary Proakis-50210 book August 3, 2001 Section 1.6 13:2 Further Reading 23 modulation methods are described based on a geometric representation of signals and their performance is evaluated in terms of the probability of error. A link budget analysis for radio communication systems is also described. The final topic of this chapter is focused on signal synchronization methods for digital communication systems. In Chapter 8, we consider the problem of digital communication through bandlimited, additive white Gaussian noise channels. In particular, we describe the design of bandlimited signal waveforms for such channels and the spectral characteristics of digitally modulated signals. Digitally modulated signals with memory, their modulation, and their spectral characteristics are also described. The effect of channel distortion on the transmitted signals is characterized in terms of intersymbol interference (ISI), and the design of adaptive equalizers for suppressing ISI is described. Channel coding and decoding is the topic of Chapter 9. In this chapter, we describe the concept of channel capacity, and derive the capacity of an additive white Gaussian noise channel. Linear block codes and convolutional codes are considered for enhancing the performance of a digital communication system in the presence of additive noise. Decoding algorithms for both block codes and convolutional codes are also described. The final topic of this chapter provides a treatment of trellis-coded modulation, which is widely used in the implementation of high speed modems. The last chapter of the book, Chapter 10, focuses on topics dealing with wireless communications. We begin by characterizing channel fading and multipath effects in wireless communication systems and describe the design of signals that are effective in mitigating these channel distortions. Then, we describe the class of continuous-phase modulated signals, which are especially suitable for digital communications in wireless channels due to their constant amplitude and narrow bandwidth characteristics. Finally, we treat the class of spread-spectrum signals, which are suitable for multi-user wireless communication systems. As examples of practical wireless communication systems, we briefly describe two digital cellular communication systems, the pan European GSM system and the North-American IS-95 CDMA system. In an introductory book of this level, we have not attempted to provide a large number of references to the technical literature. However, we have included in each chapter several supplementary references for further reading, including textbooks and basic or tutorial treatments of important topics. References are cited by giving the author's name, with the year of publication in parentheses; e.g., Nyquist (1924). 1.6 FURTHER READING We have already cited several historical treatments of radio and telecommunications during the past century. These include the book by McMahon (1984), Ryder and Fink (1984), and Millman (1984). The classical works of Nyquist (1924), Hartley (1928), Kotelnikov (1947), Shannon (1948), and Hamming (1950) are particularly important because they lay the groundwork of modern communication systems engineering. Proakis-50210 book August 3, 2001 13:6 2 Frequency Domain Analysis of Signals and Systems In this chapter, we review the basics of signals and linear systems in the frequency domain. The motivation for studying these fundamental concepts stems from the basic role they play in modeling various types of communication systems. In particular signals are used to transmit the information over a communication channel. Such signals are usually called information-bearing signals. In the transmission of an information-bearing signal over a communication channel, the shape of the signal is changed, or distorted, by the channel. In other words, the output of the communication channel, which is called the received signal, is not an exact replica of the channel input due to the channel distortion. The communication channel is an example of a system; i.e., an entity that produces an output signal when excited by an input signal. A large number of communication channels can be modeled closely by a subclass of systems called linear systems. Linear systems are a large subclass of systems which arise naturally in many practical applications and are rather easy to analyze. We have devoted this entire chapter to the study of the basics of signals and linear systems in the frequency domain. 2.1 FOURIER SERIES The analysis of signals and linear systems in the frequency domain is based on representation of signals in terms of the frequency variable and this is done through employing Fourier series and Fourier transforms. Fourier series is applied to periodic signals whereas the Fourier transform can be applied to periodic and nonperiodic signals. 24 Proakis-50210 book August 3, 2001 Section 2.1 13:6 Fourier Series 25 Theorem 2.1.1. [Fourier Series] Let the signal x(t) be a periodic signal with period T0 . If the following conditions (known as the Dirichlet conditions) are satisfied 1. x(t) is absolutely integrable over its period; i.e., T0 |x(t)| dt < ∞, 0 2. The number of maxima and minima of x(t) in each period is finite, 3. The number of discontinuities of x(t) in each period is finite, then x(t) can be expanded in terms of the complex exponential signals {e +∞ x± (t) = xn e j2π Tn t j2π Tn t +∞ 0 } n=−∞ as (2.1.1) 0 n=−∞ where 1 xn = T0 α+T0 α for some arbitrary α and x(t) x± (t) = x(t + ) + x(t − ) 2 x(t)e − j2π Tn t 0 dt (2.1.2) if x(t) is continuous at t if x(t) is discontinuous at t (2.1.3) Some observations concerning this theorem are in order. • • • • • The coefficients xn are called the Fourier series coefficients of the signal x(t). These are, in general, complex numbers. The parameter α in the limits of the integral is arbitrary. It can be chosen to simplify computation of the integral. Usually α = 0 or α = −T0 /2 are good choices. For all practical purposes, x± (t) equals x(t). From now on, we will use x(t) instead of x± (t) but will keep in mind that the Fourier series expansion, at points of discontinuity of x(t), gives the midpoint between right and left limits of the signal. The Dirichlet conditions are only sufficient conditions for the existence of the Fourier series expansion. For some signals that do not satisfy these conditions, we can still find the Fourier series expansion. The quantity f 0 = T10 is called the fundamental frequency of the signal x(t). We observe that the frequencies of the complex exponential signals are multiples of this fundamental frequency. The nth multiple of the fundamental frequency (for positive n's) is called the nth harmonic. Proakis-50210 book August 3, 2001 13:6 26 Frequency Domain Analysis of Signals and Systems • • Chapter 2 Conceptually this is a very important result. It states that the periodic signal x(t) can be described by the period T0 (or fundamental frequency f 0 ) and the sequence of complex numbers {xn }; i.e., to describe x(t) it is sufficient to specify a countable set of (in general complex) numbers. This is a considerable reduction in complexity for describing x(t), since to define x(t) for all t, we have to specify its values on an uncountable set of points. The Fourier series expansion can be expressed in terms of the angular frequency ω0 = 2π f 0 , by xn = ω0 2π α+ ω2π 0 α x(t)e− jnω0 t dt (2.1.4) and +∞ x(t) = xn e jnω0 t (2.1.5) n=−∞ • In general, xn = |xn |e j xn . |xn | gives the magnitude of the nth harmonic and xn gives its phase. Figure 2.1 shows a graph of the magnitude and phase of various harmonics in x(t). This type of graph is called the discrete spectrum of the periodic signal x(t). xn 4 3 2 1 0 1 2 1 2 3 4 n 4 n xn 4 2 3 1 3 0 Figure 2.1 The discrete spectrum of x(t). Proakis-50210 book August 3, 2001 Section 2.1 13:6 Fourier Series 27 x (t) 1 T0 2 2 T0 t Figure 2.2 Periodic signal x(t). Example 2.1.1 Let x(t) denote the periodic signal depicted in Figure 2.2 and described analytically by +∞ x(t) = n=−∞ t − nT0 τ (2.1.6) where 1, |t| < 12 (t) = 12 , |t| = 12 0, otherwise is a rectangular pulse. Determine the Fourier series expansion for this signal. Solution We first observe that the period of the signal is T0 and xn = 1 T0 1 = T0 + T0 2 x(t)e − t − jn 2π T 0 dt T0 2 + τ2 − τ2 1e t − jn 2π T 0 dt = 1 T0 − jn π τ + jn π τ e T0 − e T0 T0 − jn2π = 1 sin πn τ = sinc T0 nπ τ T0 nτ T0 (2.1.7) where we have used the definition of the sinc function sinc(t) = sin(π t)/(π t). Therefore, x(t) = +∞ τ nτ jn 2π t sinc e T0 T0 T0 n=−∞ A graph of the Fourier series coefficients for is shown in Figure 2.3. (2.1.8) Proakis-50210 book August 3, 2001 13:6 28 Frequency Domain Analysis of Signals and Systems Chapter 2 T0 x sinc T0 T0 42 0 2 4 x Figure 2.3 The discrete spectrum of the rectangular pulse train. Example 2.1.2 Determine the Fourier series expansion for the signal x(t) shown in Figure 2.4 and described by +∞ x(t) = (−1)n (t − n) n=−∞ Solution Here T0 = 2 and it is convenient to choose α = − 12 . Then, 1 xn = 2 = 1 2 =− 3 2 x(t)e− jnπt dt − 12 1 2 − 12 e− jnπt dt − 1 2 3 2 e− jnπt dt 1 2 1 1 nπ nπ 3π π e− j 2 − e j 2 − e− jn 2 − e− jn 2 j2πn − j2π n nπ nπ 1 1 − jnπ − sin sin e nπ 2 nπ 2 1 nπ = (1 − cos(nπ )) sin nπ 2 = 2 nπ 2 = − nπ 0 n = 4k + 1 n = 4k + 3 n even (2.1.9) Proakis-50210 book August 3, 2001 Section 2.1 13:6 Fourier Series 29 1 2 5 2 1 3 2 3 2 1 2 1 2 1 5 2 2 1 t Figure 2.4 Signal x(t). From these values of xn we have the following Fourier series expansion for x(t) 2 jπ t 2 j3π t 2 j5π t + e− j3π t ) + + e− j5πt ) − · · · (e + e− jπt ) − (e (e π 3π 5π 4 4 4 = cos(π t) − cos(3π t) + cos(5π t) − · · · π 3π 5π x(t) = = ∞ 4 (−1)k cos(2k + 1)πt π 2k + 1 (2.1.10) k=0 2.1.1 Fourier Series for Real Signals: the Trigonometric Fourier Series If the signal x(t) is a real signal satisfying the conditions of the Fourier series theorem, then there exist alternative ways to expand the signal. For real x(t), we have 1 α+T0 j2π n t x(t)e T0 dt x−n = T0 α α+T0 ∗ 1 − j2π Tn t 0 dt x(t)e = T0 α ∗ = xn (2.1.11) This means that for a real, periodic x(t), the positive and negative coefficients are conjugates. Hence, |xn | has even symmetry and xn has odd symmetry with respect to the n = 0 axis. An example of the discrete spectrum for a real signal is shown in Figure 2.5. From x−n = xn∗ it follows that if we denote xn = an −2jbn , then x−n = an +2jbn and therefore, for n ≥ 1, xn e j2π Tn t 0 + x−n e − j2π Tn t 0 an − jbn j2π Tn t an + jbn − j2π Tn t 0 + 0 e e 2 2 n n = an cos 2π t + bn sin 2π t T0 T0 = (2.1.12) (2.1.13) Proakis-50210 book August 3, 2001 13:6 30 Frequency Domain Analysis of Signals and Systems Chapter 2 xn 3 2 1 0 1 0 1 2 3 n 3 n xn 3 1 2 2 Figure 2.5 Discrete spectrum of a real-valued signal. Since x0 is real and given as x0 = a0 , 2 we conclude that ∞ n n a0 + an cos 2π t + bn sin 2π t x(t) = 2 T0 T0 n=1 (2.1.14) This relation, which only holds for a real periodic signal x(t), is called the trigonometric Fourier series expansion. To obtain an and bn , we have 1 an − jbn = xn = 2 T0 α+T0 α x(t)e − j2π Tn t 0 dt, and, therefore, an − jbn 1 = 2 T0 α α+T0 n x(t) cos 2π t T0 j dt − T0 α α+T0 n x(t) sin 2π t T0 dt From this we obtain n x(t) cos 2π t dt T0 α α+T0 2 n x(t) sin 2π t dt bn = T0 α T0 2 an = T0 α+T0 (2.1.15) (2.1.16) Proakis-50210 book August 3, 2001 Section 2.2 13:6 Fourier Transforms 31 There exists still a third way to represent the Fourier series expansion of a real signal. Noting that n j2π n t − j2π Tn t 0 = 2|x | cos 2π t + xn (2.1.17) xn e T0 + x−n e n T0 we have ∞ n |xn | cos 2π t + xn x(t) = x0 + 2 T0 n=1 (2.1.18) In summary, for a real periodic signal x(t), we have three alternative ways to represent the Fourier series expansion x(t) = +∞ xn e j2π Tn t 0 (2.1.19) n=−∞ ∞ n n a0 an cos 2π t + bn sin 2π t + = 2 T0 T0 n=1 ∞ n |xn | cos 2π t + xn = x0 + 2 T0 n=1 where the corresponding coefficients are obtained from bn 1 α+T0 an − j2π Tn t 0 dt = −j x(t)e xn = T0 α 2 2 α+T0 2 n x(t) cos 2π t dt an = T0 α T0 α+T0 2 n x(t) sin 2π t dt bn = T0 α T0 1 an2 + bn2 |xn | = 2 bn xn = −arctan an (2.1.20) (2.1.21) (2.1.22) (2.1.23) (2.1.24) (2.1.25) (2.1.26) 2.2 FOURIER TRANSFORMS The Fourier transform is the extension of the Fourier series to the general class of signals (periodic and nonperiodic). Here, as in Fourier series, the signals are expressed in terms of complex exponentials of various frequencies, but these frequencies are not discrete. Hence, in this case, the signal has a continuous spectrum as opposed to a discrete spectrum. We can formally state the Fourier transform theorem as follows. Proakis-50210 book August 3, 2001 32 13:6 Frequency Domain Analysis of Signals and Systems Chapter 2 Theorem 2.2.1. [Fourier Transform] If the signal x(t) satisfies certain conditions known as the Dirichlet conditions, namely, 1. x(t) is absolutely integrable on the real line; i.e., +∞ |x(t)| dt < ∞, −∞ 2. The number of maxima and minima of x(t) in any finite interval on the real line is finite, 3. The number of discontinuities of x(t) in any finite interval on the real line is finite, then, the Fourier transform (or Fourier integral) of x(t), defined by +∞ X( f ) = x(t)e− j2π f t dt (2.2.1) exists and the original signal can be obtained from its Fourier transform by +∞ X ( f )e j2π f t d f x± (t) = (2.2.2) −∞ −∞ where x± (t) is defined as in Theorem 2.1.1. We make the following observations concerning the Fourier transform. • • X ( f ) is in general a complex function. Its magnitude |X ( f )| and phase X ( f ) represent the amplitude and phase of various frequency components in x(t). The function X ( f ) is sometimes referred to as the spectrum† of the signal x(t). To denote that X ( f ) is the Fourier transform of x(t), the following notation is frequently employed X ( f ) = [x(t)] To denote that x(t) is the inverse Fourier transform of X ( f ), the following notation is used x(t) = −1 [X ( f )] Sometimes the following notation is used as a shorthand for both relations x(t) ⇔ X ( f ) • If the variable in the Fourier transform is chosen to be ω rather than f , then we have +∞ x(t)e− jωt dt X (ω) = −∞ † later. Sometimes X ( f ) is referred to as voltage spectrum, as opposed to power spectrum to be defined Proakis-50210 book August 3, 2001 Section 2.2 13:6 Fourier Transforms 33 and x(t) = • 1 2π +∞ −∞ X (ω)e jωt dω The Fourier transform and the inverse Fourier transform relations can be written as +∞ +∞ − j2π f τ x(τ )e dτ e j2π f t d f x(t) = −∞ +∞ = −∞ +∞ −∞ −∞ e j2π f (t−τ ) d f x(τ ) dτ (2.2.3) On the other hand, x(t) = +∞ −∞ δ(t − τ )x(τ ) dτ (2.2.4) where δ(t) is the unit impulse, which is defined as a signal that is zero everywhere except at t = 0 and has unit area. Comparing Equation (2.2.3) with Equation (2.2.4), we obtain +∞ δ(t − τ ) = e j2π f (t−τ ) d f (2.2.5) −∞ or, in general, δ(t) = +∞ −∞ e j2π f t d f (2.2.6) Hence, the spectrum of δ(t) is equal to unity over all frequencies. Example 2.2.1 Determine the Fourier transform of the signal Solution We have [ (t)] = (t)e− j2π f t dt −∞ = +∞ + 12 (t). e− j2π f t dt − 12 = 1 [e− jπ f − e jπ f ] − j2π f = sin π f πf = sinc( f ) (2.2.7) Proakis-50210 book August 3, 2001 13:6 34 Frequency Domain Analysis of Signals and Systems (t) Chapter 2 1 1/2 1 2 t 1 2 Sinc ( f ) 1 4 2 5 3 2 1 1 4 3 5 f Figure 2.6 transform. (t) and its Fourier Therefore, [ (t)] = sinc( f ) Figure 2.6 illustrates the Fourier transform relationship for this signal. Example 2.2.2 Find the Fourier transform of an impulse signal x(t) = δ(t). Solution The Fourier transform can be obtained by [δ(t)] = +∞ −∞ δ(t)e− j2π f t dt =1 (2.2.8) where we have used the sifting property of δ(t).† This shows that all frequencies are present in the spectrum of δ(t) with unity magnitude and zero phase. The graphs of x(t) and its Fourier transform are given in Figure 2.7. Similarly from the relation +∞ −∞ δ( f )e j2π f t dt = 1 we conclude that [1] = δ( f ) † The sifting property of δ(t) states that ∞ −∞ x(t)δ(t − t0 ) dt = x(t0 ). Proakis-50210 book August 3, 2001 Section 2.2 13:6 Fourier Transforms 35 (t) t 1 f Figure 2.7 Impulse signal and its spectrum. 2.2.1 Fourier Transform of Real, Even, and Odd Signals The Fourier transform relation can be written in general as +∞ [x(t)] = x(t)e− j2π f t dt −∞ +∞ = −∞ For real x(t), both integrals x(t) cos(2π f t) dt − j −∞ x(t) sin(2π f t) dt +∞ −∞ and +∞ x(t) cos(2π f t) dt +∞ −∞ x(t) sin(2π f t) dt are real and therefore denote the real and imaginary parts of X ( f ), respectively. Since cosine is an even function and sine is an odd function, we see that for real x(t) the real part of X ( f ) is an even function of f and the imaginary part is an odd function of f . Therefore, in general, for real x(t), the transform X ( f ) is a Hermitian function X (− f ) = X ∗ ( f ) This is equivalent to the following relations: Re[X (− f )] = Re[X ( f )] Im[X (− f )] = −Im[X ( f )] |X (− f )| = |X ( f )| X (− f ) = − X ( f ) Typical plots of |X ( f )| and X ( f ) for a real x(t) are given in Figure 2.8. Proakis-50210 book August 3, 2001 36 13:6 Frequency Domain Analysis of Signals and Systems Chapter 2 X( f ) f X( f ) f Figure 2.8 Magnitude and phase of the spectrum of a real signal. If in addition to being real, x(t) is an even signal, then the integral +∞ x(t) sin(2π f t) dt −∞ vanishes because the integrand is the product of an even and odd signal and, therefore, odd. Hence, the Fourier transform X ( f ) will be real and even. Similarly if x(t) is real and odd, the real part of its Fourier transform vanishes and X ( f ) will be imaginary and odd. 2.2.2 Basic Properties of the Fourier Transform In this section, we summarize the basic properties of the Fourier transform. Linearity Property. The Fourier transform operation is linear. That is, if x1 (t) and x2 (t) are signals possessing Fourier transforms X 1 ( f ) and X 2 ( f ), respectively, the Fourier transform of αx1 (t) + βx2 (t) is α X 1 ( f ) + β X 2 ( f ), where α and β are two arbitrary (real or complex) scalars. Duality Property. If X ( f ) = [x(t)] then x( f ) = [X (−t)] Proakis-50210 book August 3, 2001 Section 2.2 13:6 Fourier Transforms 37 and x(− f ) = [X (t)] Time Shift Property. A shift of t0 in the time origin causes a phase shift of −2π f t0 in the frequency domain. In other words, [x(t − t0 )] = e− j2π f t0 [x(t)] Scaling Property. For any real a = 0, we have 1 f [x(at)] = X |a| a Convolution Property. forms, then (2.2.9) If the signals x(t) and y(t) both possess Fourier trans- [x(t) y(t)] = [x(t)] · [y(t)] = X ( f ) · Y ( f ) (2.2.10) The Fourier transform of x(t)e j2π f0 t is X ( f − f 0 ), and Modulation Property. the Fourier transform of x(t) cos(2π f 0 t) is 1 X( 2 f − f 0 ) + 12 X ( f + f 0 ) Parseval's Property. If the Fourier transforms of the signals x(t) and y(t) are denoted by X ( f ) and Y ( f ), respectively, then ∞ ∞ ∗ x(t)y (t) dt = X ( f )Y ∗ ( f ) d f (2.2.11) −∞ −∞ Rayleigh's Property. If X ( f ) is the Fourier transform of x(t), then ∞ ∞ 2 |x(t)| dt = |X ( f )|2 d f −∞ −∞ Autocorrelation Property. The (time) autocorrelation function of the signal x(t) is denoted by Rx (τ ) and is defined by ∞ x(t)x ∗ (t − τ ) dt (2.2.12) Rx (τ ) = −∞ The autocorrelation property states that [Rx (τ )] = |X ( f )|2 (2.2.13) Differentiation Property. The Fourier transform of the derivative of a signal can be obtained from the relation d (2.2.14) x(t) = j2π f X ( f ) dt Proakis-50210 book August 3, 2001 38 13:6 Frequency Domain Analysis of Signals and Systems Chapter 2 Integration Property. The Fourier transform of the integral of a signal can be determined from the relation X( f ) 1 x(τ ) dτ = + X (0)δ( f ) j2π f 2 −∞ t (2.2.15) ∞ Moments Property. If [x(t)] = X ( f ), then −∞ t n x(t) dt, the nth moment of x(t), can be obtained from the relation ∞ j n dn t n x(t) dt = X ( f ) (2.2.16) 2π d f n −∞ f =0 Table 2.1 gives a collection of