Simulation of Communication Systems

Applications of Communications Theory Series Editor: R. W. Lucky, AT&T Bell Laboratories

Recent volumes in the series:

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COMPUTER NETWORK ARCHITECTURES AND PROTOCOLS Second Edition • Edited by Carl A. Sunshine

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DATA TRANSPORTATION AND PROTECTION John E. Hershey and R. K. Rao Yarlagadda

DEEP SPACE TELECOMMUNICATIONS SYSTEMS ENGINEERING 'Edited by Joseph H. Yuen

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MODELING AND ANALYSIS OF COMPUTER COMMUNICATIONS NETWORKS Jeremiah F. Hayes

MODERN TELECOMMUNICATIONS E. Bryan Carne

OPTICAL CHANNELS: Fibers, Clouds, Water, and the Atmosphere Sherman Karp, Robert M. Gagliardi, Steven E. Moran, and Larry B. Stotts

PRACTICAL COMPUTER DATA COMMUNICATIONS William J. Barksdale

SIMULATION OF COMMUNICATION .SYSTEMS Michel C. Jeruchim, Philip Balaban, and K. Sam Shanmugan

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Simulation of Communication Systems

Michel c. Jemchim GE Aerospace Philadelphia, Pennsylvania

Philip Balaban AT & T BeII Laboratories Holmdel, New Jersey

K. Sam Shanmugan University of Kansas Lawrence, Kansas and Comdisco Systems Foster City, California

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Llbrary of Congress Cataloglng-In-Publlcatlon Data

JeruchlN, Mlchel C. Slmulatlon of communlcatlon syste.s I Mlchel C. JeruchlN, Phl11p

Balaban, K. SaI Shanmugan. p. CN. -- CAppllcatlons of conmunlcatlons theory)

Includes blbllographlcal references and Index. ISBN 978-1-4613-6451-1 ISBN 978-1-4615-3298-9 (eBook) DOI 10.1007/978-1-4615-3298-9 1. Telecomnunlcatlon systems--CoMputer slnul.tlon. 1. Balaban,

Phl11p. II. Shanllugan, K. Su, 1943- III. Tltle. IV. Serles. TK6102.5.J47 1992 621.382·01'13--dc20 91-48369

ISBN 978-1-4613-6451-1

© 1992 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1992

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Radha, Kannon, and Ravi -KSS

Preface

Simulation may be defined as the discipline whose objective is to imitate one or more aspects of reality in a way that is as close to that reality as possible; indeed, an apt synonym that is gaining some currency is artificial reality. Under this definition, simulation is a very old discipline. Probably the first applications of simulation were to scale models of various types of dynamical structures or mechanical devices. Man has always looked for ways to "try things out" before building the real thing; this is the motivation behind any form of simulation. Thus, simulation of communication systems is concerned with imitating some aspects of the behavior of communication systems. It is implicit in our use of simulation that the medium (so to speak) for carrying it out is the digital computer. Computer-based modeling and simulation of communication systems has only developed in the last 20 years or so, since the advent of modern digital computers. A variety of modeling and simulation techniques have been developed and described in widely scattered journals, but until now there has not been a single volume devoted to the subject. We have tried to provide a unified framework that describes both the disciplines involved and the methods of modeling and simulating communication systems and subsystems.

In the electronic era, the first type of computer simulation, in today's use of the term, took shape in the form of analog computers. Analog computers were flexible arrangements of physical components that could be interconnected to "look" like some system of interest, subject of course, to the fundamental limitation that any system to be simulated had to be representable by some combination of the available building blocks. The nature of these building blocks was generally more suitable for representing other types of systems, e.g., control systems, than communication systems. Digital computers represent a fundamentally different approach to problem solving. Basically, the digital computer allows us to manipulate abstractions. If we represent each element of the physical system by a mathematical

vii

vIII Prerace

model (the abstraction) and "connect" these models, the digital computer can serve as the "laboratory" for that system. We refer to such models as software models. We can change parameters of the model at will, or parts of the models themselves, and observe the consequences "on paper" without having to build hardware until we are reasonably satisfied that a design under consideration will do the job.

The simulation of communication systems is concerned with imitating some aspects of the behavior of communications systems without building hardware, although this distinction will no doubt become increasingly blurred as it becomes more practical to fuse hardware and software models. However, because this type of hybrid simulation is in its infancy we shall not further consider it in this book; we shall deal solely with software modeling. Generally speaking, the simulation of communication systems is computationally intensive. As mentioned, it was only about 20 years ago that digital computers attained the necessary processing speed to make such simulations practical. Since that time, the application of simulation to communication systems has become increasingly common, to the point where it is safe to say that today simulation is the major tool for analysis and design of communication systems. Prior to the widespread use of simulation the preliminary analysis and design of communication systems used to be accomplished by traditional methods of analysis. Such methods attempted to develop formulas relating the various design parameters to the performance metrics of interest. It usually happened, for most practical systems, that varying degrees 9f approximation or simplification had to be made in order to render the analysis tractable. Simulation, on the other hand, is not generally hindered by the factors that make analysis difficult; this accounts in large part for the usefulness of simulation. This should not be construed, however, as implying that analysis can be replaced with simulation. Indeed, analysis and simulation should be viewed as mutually complementary. We shall see in the text that some of the most efficient techniques for performance evaluation combine methods of both analysis and simulation.

The connotation of the term "communication system" is generally much broader than our intended use. What we mean by this term is any system intended for the transmission of information. This information may be analog or digital in nature, although our emphasis will be on the latter. The manifestation of the transmission at the receiver is generally in the form of a continuous waveform which is processed to recover the original information. We are considering systems that can be represented by a block diagram, the individual blocks of which include all the functions or generic operations that might be normally encountered. We shall spend some time describing such generic blocks and the considerations related to their simulation.

Preface Ix

The simulation of communication systems is a multidisciplinary activity that combines elements from a number of diverse areas of specialization, some of which may be regarded .as belonging to traditional communications disciplines and some as perhaps. belonging more firmly on the simulation side. Among the traditional areas of communications we can list, of course, communication systems themselves; their theory of operation; and the mathematical representation of systems and their stimuli, namely, signals and noise, both in continuous time and in discrete time. Although these subjects are conventional in a sense, they must be treated with an eye toward their application in the simulation context and this requires a perspective not often imparted in the standard treatments. Subjects that are more shaded toward simulation, per se, are the generation of pseudorandom sequences intended to emulate specific types of random processes; the application of statistical theory to the interpretation and reliability of Monte Carlo esti­mates of various performance measures; the development of alternative (other than Monte Carlo) estimators; and, of course, an entire set of notions that we gather under the heading of simulation methodology, which, among other things, treats the relationship between accuracy and computational burden, and seeks to place the role of simulation within the larger context of communication systems engineering.

We have organized the text at a fairly detailed level, down to four and sometimes five levels of subheadings, so the reader can get a good idea of the actual topics covered by perusing the Table of Contents. As can be seen, our emphasis is on the principles and techniques of simulation, rather than on the actual construction of simulation software. The latter is a software development task that is not within our intended scope, although we shall spend a little time outlining the desirable features and organization of a simulation environment.

As is implied in the foregoing, our view is that the practice of simulation is an integrated application of many disciplines. Our experience indicates to us that difficulties or misapplications of simulation more often than not stem from a lack of understanding of some aspect in the chain of topics that constitute the art and science of simulation. This view is reflected in the structure of this book, which covers both fundamental theory slanted toward the simulation application, as well asmore purely simulation-specific techniques. Given the potentially tremendous breadth of topics that could fall within the scope of the book, it was necessary to limit ourselves so as to produce a volume of reasonable size. The specific coverage reflects our personal tendencies, but we do provide a large set of references for the· reader wishing to pursue topics that we treat lightly.

We have designed the book to appeal to multiple audiences. It can serve as a handy reference to the practitioner or as a source of self-study. It can also serve as a text for a one- or two-semester graduate course. One

x Preface

of us has used the manuscript in conjunction with a one-semester graduate seminar by selecting particular topics; a thorough coverage would probably require two semesters. As an aid to the instructor, we have provided a set of simulation-related problems and computer projects of varying difficulty. Some of the latter may be group efforts toward a semester project. Many of the problems or projects have been deliberately phrased in a way which may lead to somewhat different interpretations. There is not necessarily a single right answer. This we have done to mirror the ambiguities and uncertainties that one meets in the application of simulation to real-world problems. In this way, we hope to give the student something of the flavor that he will encounter in practice.

The inclusion of such a wide range of topics in the book was greatly facilitated by the assistance we received from many of our colleagues with expertise in specific areas, who gave unselfishly their time, advice, and, in some cases, actual manuscript. It is our pleasure to acknowledge their contributions.

To our colleague and good friend, Dr. Robert J. Wolfe, we would like to extend our gratitude for countless discussions and suggestions regarding fine points of mathematics and statistics which are reflected throughout, but in Chapter 5 in particular, which he reviewed thoroughly; Section 5.4 contains information originated by him. To another colleague and friend, Dr. John H. Moore, we would also like to express our appreciation for providing the information on, and the figure of, the rain model (Section 4.9.12) and helpful discussions on the waveguide model (Section 4.9.2.1); the material in Chapter 6 on systems engineering methodology has evolved during the last dozen years in no small part from numerous brainstorming sessions with John. To our colleague, Professor J. K. Townsend, we are deeply indebted for drafting and redrafting Case Study 2 (Section 7.2). We would also like to mention that the material on optical fiber modeling (Section 4.9.2.2) was largely extracted from his doctoral dissertation. To another mutual colleague, Dr. William Turin, we are most appreciative for providing the material in Section 4.9.4 on discrete channel modeling as well as the associated problems. We are much indebted to Dr. Aly Elrefaie who provided the original draft for material on optical amplifier modeling in Section 2.11.8.7 and for the material on laser modeling in Section 4.2.2. We would also like to express our appreciation to Dr. Adel A. M. Saleh for helpful discussions on optical devices. Finally, we would like to acknowl­edge the substantial analytical work of Professor Vasant Prabhu in develop­ing the moment method for Case Study 1 (Section 7.1); we would also like to thank him as well as Dr. Hector Corrales for their general contributions to this case study.

The book as a whole has benefited greatly from the thorough and critical reading of the first draft by Professor Ezio Biglieri, Dr. Robert

Preface xl

Harris, and Professor William Tranter. The present version incorporates many of their suggestions.

The authors would also like to commend Joyce Ciallella for her skilled typing of the manuscript and Vera Sehon and Megan Gannon for their drafting of the figures.

We would like, finally, to thank our silent partners-our families-for their forbearance during the several years of writing; to them we dedicate this work.

Michel C. Jeruchim Philip Balaban K. Sam Shanmugan

Contents

Chapter 1. Introduction 1.1. Methods of Performance Evaluation. . . . . . . . . • . . . . . . . . . . . . . . . . . . 1 1.2. Simulation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3. The Application of Simulation to the Design of Communication

Systems..................................................... 4 1.4. Historical Perspective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5. Outline of the Book .......................................... 8

References .................................................. 11

Chapter 2. Representation of Signals and Systems In Simulation

2.1. Introduction................................................. 13 2.1.1. Signals ............ ;.................................. 14

2.1.1.1. Continuous Signals ............................ 14 2.1.1.2. Discrete-Time Signals. . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2. Systems.............................................. 18 2.1.2.1. Properties of Systems........................... 18 2.1.2.2. Block Diagram Representation of Systems. . . . . . . . 20

2.2. Linear Time-Invariant Systems .............. : . . . . . . . . . . . . . . . . . . 21 2.2.1. Continuous Linear Time-Invariant Systems. . . . . . . . . . . . . . . . 21

2.2.1.1. The Impulse Response ......................... 21 2.2.1.2. The Convolution Integral. . . . . . . . . .. . . . . . . . . . . . . 22 2.2.1.3. Properties of the Convolution. . . . . . . . . . . . . . . . . . . 22

2.2.2. Discrete Linear Time-Invariant Systems. . . . . . . . . . . . . . . . . . . 22 2.2.2.1. The Impulse Response ......................... 22 2.2.2.2. Convolution Sum (Discrete Convolution) . . . . . . . . . 22 2.2.2.3. Properties of the Discrete Convolution ........... 24

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xlv Contents

2.3. Frequency Domain Representation .......... . . . . . . . . . . . . . . . . . . . 24 2.3.1. The Fourier Transform. . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . 24 2.3.2. Frequency Domain Representation of Periodic Continuous

Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . 25 2.3.2.1. The Fourier Series .................. , . . . . . . . . . . 25 2.3.2.2. Parseval's Theorem for Periodic Signals .......... 26

2.3.3. The Fourier Transform .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.3.1. Convergence.................................. 28 2.3.3.2. Properties of the Fourier Transform. . . . . . . . . . . . . . 29

2.3.4. The Frequency Response ............................... 30 2.3.4.1. Interconnection of Systems in the Frequency

Domain ........ ~ .................. , . . . . . . . . . . 31 2.3.4.2. Parseval's Theorem for Continuous Signals. . . . . . . . 32

2.3.5. Gibbs' Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.6. Relationship between the Fourier Transform and the Fourier

Series ..................................... , . . . . . . . . . . 33 2.3.6.1. Fourier Series Coefficients ...................... 34

2.3.7. The Fourier Transform of a Periodic Signal. . . .. . . . . . . . . . . 34 2.3.7.1. Periodic Convolution. . . . . . . . . . . . . . . .. . . . . . . . . . . 35 2.3.7.2. The Poisson Sumation Formula. . . . . . .. . . . . . . . . . . 36

2.4. Low-Pass Equivalent Signals and Systems ....................... 36 2.4.1. The Hilbert Transform ................................. 37 2.4.2. Properties of the Hilbert Transform .................. . . . . 39 2.4.3. Low-Pass Equivalent Modulated Signals. . . . . . . . . . . . . . . . . . 40 2.4.4. Hilbert Transform in System Analysis. . . . . . . . .. . . . . . . . . . . 41 2.4.5. Practical Considerations in Modeling of Low-Pass

Equivalents for Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.5. Sampling and Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 45

2.5.1. Impulse Sampling ..................................... 46 2.5.2. Sampling Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.5.2.1. Interpolation...................... . . . . . . . . . . . . 49 2.5.2.2. Aliasing: The Effect of Undersampling ........ . . . 49

2.6. Characterization of L TI Systems Using the Laplace Transform . . . . . 50 2.6.1. The Laplace Transform. . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . 51

2.6.1.1. Convergence and Stability. . . . . . . . . . ... . . . . . . . . . 51 2.6.2. Inverse Laplace Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.3. Properties of the Laplace Transform ..................... 52 2.6.4. Transfer or System Function ............................ 53 2.6.5. Interconnections of LTI Systems (Block Diagrams) ........ 54 2.6.6. Systems Characterized by Linear Constant-Coefficient

Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.6.6.1. Properties of the Transfer Function for Linear

Constant-Coefficient Differential Equations ....... 57 2.6.6.2. Realizations of Rational Transfer Functions Using

Biquadratic Expansion ......................... 58 2.6.7. Frequency Response ................................... 60 2.6.8. Low-Pass Equivalents of Bandpass Filters Represented by

Contents xv

Rational Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.6.9. Continuous Classical Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.6.9.1. Frequency Transformation. . . . . . . . . . .. . . . . . . . . . . 69 2.6.9.2. Low-Pass Equivalent Classical Filters ............ 71

2.7. Representation of Continuous Systems by Discrete Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.7.1. The z-Transform....................................... 73

2.7.1.1. Convergence and Stability. . . . . . . .. . . . . . . . . . . . . . 74 2.7.1.2. Table of Simple z-Transforms. . . . . . . . . . . . . . . . . . . 74 2.7.1.3. Properties of the z-Transform. . . . . . . .. . . . . . . . . . . 74

2.7.2. Systems Characterized by Linear Constant-Coefficient Difference Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.7.2.1. Structures of Recursive Discrete Filters

Implemented in Simulation Models . . . . . . . . . . . . . . 77 2.7.2.2. The Cascade Interconnections of Biquadratic

Canonic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.7.2.3. The Parallel Realization. . . . . . . . . . . . . . . . . . . . . . . . 80

2.7.3. Transformations between Continuous Time and Discrete Time Systems Represented by Rational Functions. . . . . . . . . . 80 2.7.3.1. Impulse Invariant Transformation ..... " . . . . . . . . . 81 2.7.3.2. The Bilinear Transformation. . . . . .. . . . . . . . . . . . . . 84 2.7.3.3. Effect of Mapping on Low-Pass Equivalent Filters

Represented by Rational Functions .............. 88 2.7.4. Finite Impulse Response (FIR) Discrete Systems .......... 88

2.7.4.1. Modeling of FIR Filters. .. . . . . . . .. . . . . . . . . . . . . . 89 2.7.4.2. Windowing..................... . . . . . . . . . . . . . . 89 2.7.4.3. Realization of FIR Filters. . . . . . . . . . . . . . . . . . . . . . . 90 2.7.4.4. Discussion on FIR Filter Modeling. . . . . . . . . . . . . . 90 2.7.4.5. Note on FIR Filter Design. . . . . . . . . . . . . . . . . . . . . . 91

2.8. Fourier Analysis for Discrete-Time Systems. . . . . . . . . . . . . . . . . . . . . . 91 2.8.1. Introduction.......................................... 91 2.8.2. The Discrete Fourier Transform ......................... 92 2.8.3. The Fast Fourier Transform (FFT) .......... , . . . . . . . . . . . . 94 2.8.4. Properties of the Discrete Fourier Transform . . . . . . . . . . . . . . 95

2.8.4.1. Periodic or Circular Properties .................. 95 2.8.4.2. The Periodic Time-Shift Property. . . . . . . . . . . . . . . . . 96 2.8.4.3. The Periodic or Circular Convolution ............ 96 2.8.4.4. The Discrete Periodic Convolution Theorem ...... 97 2.8.4.5. The Discrete Frequency Response ............... 98 2.8.4.6. Relationship between the Bandwidth and the

Duration of the Impulse Response . . . . . . . . . . . . . . . 98 2.8.4.7. Relationship between the DFT and the z-Transform 99 2.8.4.8. Increasing the Frequency Resolution of the DFT .. 99

2.8.5. Discrete Signal Processing (FIR Filtering) ................ 100 2.8.6. Frequency Domain FIR Filtering for Nonperiodic Signals .. 101

2.8.6.1. Difference between Periodic and Linear Convolution .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 101

xvi Contents

2.8.6.2. Linear Convolution for a Signal of Arbitrary Duration via the FFT .......................... 103

2.8.6.3. The Overlap-and-Add (OA) Method.......... ... 103 2.8.6.4. The Overlap-and-Save (OS) Method ............. 105 2.8.6.5. Efficiency of the Linear Convolution via the FFT . . 105

2.8.7. Implications of Frequency Domain FIR Filtering. . . . . . . . . . 107 2.8.7.1. Block Processing Using the OA and OS Methods. . 108 2.8.7.2. Gibbs' Phenomenon Distortion. . . . . ... . . . . . . . . . . 108

2.9. The Process of Mapping Continuous Signals and Systems into Discrete Signals and Systems for Simulation. . . . . . . . . . . . . . . . . . . . . 108 2.9.1. Preparation of Signals and Systems for Discrete Simulation 109 2.9.2. Mapping of Continuous Filters into Discrete Filters. ... . . . . 110

2.9.2.1. Finite-Impulse-Response (FIR) Filters.. ... . . . . . . . 110 2.9.2.2. Infinite-Impulse-Response (IIR) Filters. . . . . . . . . . . 114

2.9.3. Effects of Finite Word Length in Simulation of Digital Filters ......... '. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.9.3.1. Roundoff Noise in Simulations of IIR Filters. . . . . . 122 2.9.3.2. Roundoff Noise in Simulations of FIR Filters ..... 122 2.9.3.3. Effects of Quantization in Computation of the Fast-

Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.9.4. A Guide to the Selection of the Proper Method for Filter

Simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.10. Linear Time-Variant (LTV) Systems ............................ 125

2.10.1. The Impulse Response ................................. 125 2.10.1.1. Computation of the Superposition for LTV Systems 126

2.10.2. Computation of the Impulse Response for a Linear Differential Equation with Time-Variant Coefficients. . . . . . . 127

2.10.3. Properties of Linear Time-Variant Systems................ 129 2.10.3.1. Frequency-Domain Representation of Time-Variant

Systems ...................................... 130 2.10.3.2. Bandwidth Relations in Time-Variant Systems. . . . . 131 2.10.3.3. Sampling Rate............................ ..... 131

2.10.4. Models for LTV Systems. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 131 2.10.4.1. Separable Models. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 132 2.10.4.2. Discrete (Sampling) Models. . . . . . . . .. .. . . . . . . . . . 132

2.10.5. Interconnections of Time-Variant Linear Systems. . . . . . . . . . 134 2.10.5.1. The Algebra of LTV Systems. . . . . . . ... . . . . . . . . . . 135 2.10.5.2. The Feedback System.......................... 139

2.10.6. Interconnections of LTV Systems in the Frequency Domain 140 2.11. Nonlinear Systems........................................... 141

2.11.1. Introduction.......................................... 141 2.11.2. Simulation of Nonlinear Systems. . . . . . . . . . . . . .. . . . . . .. . . 141 2.11.3. Estimating the Sampling Rate for Nonlinear Systems. . . . . . . 142 2.11.4. Modeling Considerations for Nonlinear Systems. . . . . . . . . . . 143 2.11.5. Block Models for Memoryless Nonlinearities . .. . . . . . . . . . . . 144

2.11.5.1. Memoryless Baseband Nonlinearities. . . . . . . . . . . . . 144 2.11.5.2. Memoryless Bandpass Nonlinearities............. 145

Contents xvII

2.11.5.3. Low-Pass Equivalent of a Bandpass Nonlinearity. . 149 2.11.5.4. The Limiter Family. .. . . . . . . . . . . . . .. . . . . . . . . . . . 150 2.11.5.5. Setting the Operating Point of a Memoryless

Nonlinearity. . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . 152 2.11.6. Block Models for Nonlinearities with Memory. . . . . . . . . . . . . 153 2.11.7. Analytical Approach to Block Models. . . . . . .. . . . . . . . . . . . . 156

2.11.7.1. Modeling a Memoryless Baseband Nonlinearity 157 2.11.7.2. Modeling a Memoryless Bandpass Nonlinearity... 157 2.11.7.3. Baseband Nonlinearity with Memory-Volterra

Series Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . 161 2.11.7.4. Bandpass Nonlinearities with Memory-Volterra

Series Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2.11.8. Nonlinear Differential Equations ........................ 163

2.11.8.1. Introduction .................................. 163 2.11.8.2. Outline of Numerical Methods ...... ; . . . . . . . . . . . 164 2.11.8.3. Truncation Error of Integration Formulas. . . . . . . . . 166 2.11.8.4. Stability of Integration Formulas. . ... . . . . . . . . . . . . 168 2.11.8.5. The Use of Implicit and Explicit

Integration Formulas in Simulation.. . . . . . . . . . . . . 169 2.11.8.6. Accuracy and Stability Control . . . . . . . . . . . . . . . . . . 172 2.11.8.7. Application of Numerical Methods ....... ; . . . . . . . 173

2.12. Summary.................................................... 177 2.13. Appendix................................................... 179 2.14. Problems and Projects. . . . . . . . . .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 182

References .................................................. 185

Chapter 3. Simulation of Random Variables and Random Processes

3.1. Introduction................................................. 189 3.2. Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.2.1. Basic Concepts, Definitions, and Notations. . . . . . . . . . . . . . . 192 3.2.1.1. Averages ...................................... 193

3.2.2. Multidimensional Random Variables (Random Vectors) .... 194 3.2.3. Complex Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

3.3. Univariate Models............................................ 198 3.3.1. Univariate Models-Discrete. . . . . . . . . . . . . .. . . . . • . . . . . . . . 198

3.3.1.1. Uniform...................................... 199 3.3.1.2. Binomial..................................... 199 3.3.1.3. Negative Binomial. . . .. . . . . . . .. . .. . . . . . . . . . . . . . 200 3.3.1.4. Poisson......................... . . . . . . . . . . . . . . 200

3.3.2. Univariate Models-Continuous ......................... 201 3.3.2.1. Uniform....................... .. . . . . . . . . .. . . . 201 3.3.2.2. Gaussian (Normal) ............................ 201

xviii Contents

3.3.2.3. Exponential........................ . . . . . . . . . . . 202 3.3.2.4. Gamma...................................... 203 3.3.2.5. Rayleigh .............. ,............ . . . . . . . . . . . 203 3.3.2.6. Chi-Square.................................... 204 3.3.2.7. Student's t • • . • . • • • . . • • • • • . • . • • . • . .. . . . • • • . . . • • 205 3.3.2.8. F-Distribution................................. 205 3.3.2.9. Generalized Exponential. . . . . . . . . . . . . . . . . . . . . . . . 205

3.4. Multivariate Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 3.4.1. MultinomiaL.......................................... 206 3.4.2. Multivariate Gaussian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

3.4.2.1. Properties of the Multivariate Gaussian Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

3.4.2.2. Moments of Multivariate Gaussian pdf. . . . . . . . . . . 209 3.5. Transformations (Functions) of Random Variables ............... 210

3.5.1. Scalar-Valued Function of One Random Variable. . . . . . . . . . 212 3.5.1.1. Discrete Case ...................... , . . . . . . . . . . 212 3.5.1.2. Continuous Case .............................. 212

3.5.2. Functions of Several Random Variables .................. 214 3.5.2.1. Special Case-Linear Transformation ............ 215 3.5.2.2. Sum of Random Variables ...................... 216 3.5.2.3. Order Statistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

3.5.3. Nonlinear Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 3.5.3.1. Moment-Based Techniques. . . . . . . . . . . . . . . . . . . . . . 218 3.5.3.2. Monte Carlo Simulation Techniques ............. 219

3.6. Bounds and Approximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.6.1. Chebyshev's Inequality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.6.2. Chernoff Bound ....................................... 220 3.6.3. Union Bound....... .................................. 221 3.6.4. Central Limit Theorem ................................. 222 3.6.5. Approximate Computation of Expected Values . . . . . . . . . . . . 223

3.6.5.1. Series Expansion Technique ........ " . . . . . . . . . . . 224 3.6.5.2. Moments of Finite Sums of Random Variables .... 225 3.6.5.3. Quadrature Approximations. . . . . . . . . . . . . . . . . . . . . 226

3.7. Random Processes ........................................... 229 3.7.1. Basic Definitions and Notations. . . . . . . . . . . . . . . . . . . . . . . . . 229 3.7.2. Methods of Description ................................ 232

3.7.2.1. Joint Distribution. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 232 3.7.2.2. Analytical Description using Random Variables.... 232 3.7.2.3. Average Values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 3.7.2.4. Two or More Random Processes................. 234

3.7.3. Stationarity, Time Averaging and Ergodicity............... 235 3.7.3.1. Time Averages ................................ 236 3.7.3.2. Ergodicity.................................... 237

3.7.4. Correlation and Power Spectral Density Function of Stationary Random Processes ........................... 239 3.7-4.1. Autocorrelation Function and its Properties. . . . . . . 239

Contents xix

3.7.4.2. Cross-Correlation Function and its Properties. . . . . 240 3.7.4.3. Power Spectral Density. . . . . . . . . . . . ... . . . . . . . . . . 240 3.7.4.4. Low-Pass and Bandpass Processes. . . . . . . . . . . . . . . 242 3.7.4.5. Power and Bandwidth Calculations. . . . . . . . . . . . . . 242

3.7.5. Cross-Power Spectral Density Function and its Properties. . . 243 3.7.6. Power Spectral Density Functions of Random Sequences. . . 244

3.8. Random Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 3.8.1. Random Sequences.................................... 245

3.8.1.1. Independent Sequences. . . . . .. . . . . ... . . . . . . . . . . . 245 3.8.1.2. Markov Sequences. . . . . . . . . . . . . . . ... . . . . . . . . . . . 245 3.8.1.3. Autoregressive and Moving Average (ARMA)

Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 3.8.2. M -ary Digital Waveforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

3.8.2.1. Random Binary Waveform. . . . . . . . . . . . . . . . . . . . . . 249 3.8.3. Poisson Process .......................•................ 250 3.8.4. Shot (Impulse) Noise .................................. 251 3.8.5. Gaussian Process ...................................... 253

3.8.5.1. Definition of a Gaussian Process ...... , , , , . . . . . . . 253 3.8.5.2. Models of White and Band-Limited White Noise .. 255 3.8.5.3. Quadrature Representation of Bandpass (Gaussian)

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 3.9. Transformation of Random Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . 259

3.9.1. Response of Linear Time-Invariant Causal (LTIVC) System 260 3.9.1.1. Stationarity..................... . . . . . . . . . . . . . . 260 3.9.1.2. Probability Distribution. . . . . . . . . . . . . . . . . . . . . . . . . 260 3.9.1.3. Mean, Autocorrelation, and Power Spectral Density

Functions .................................... 260 3.9.2. Filtering................................ . . . . . . . . . . . . . . 261 3.9.3. Integration ................... "......................... 263 3.9.4. Response of Nonlinear and Time-Varying Systems. . . . . . . . . 265

3.9.4.1. Nonlinear Systems. . . . . . . . . . . . . . . .. . . . . . . . . . . . . 265 3.9.4.2. Time-Varying Systems. . . . . . . . . . . .. . . . . . . . . . . . . . 265

3.10. Sampling and Quantizing...................................... 266 3.10.1. Sampling............................................. 266

3.10.1.1. Sampling of Low-Pass Random Processes. . . . . . . . . 266 3.10.1.2. Aliasing Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 3.10.1.3. Sampling Rate for Simulations .................. 268 3.10.1.4. Sampling of Bandpass Random Process .......... 270

3.10.2. Quantization.......................................... 270 3.10.2.1. Uniform Quantizing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 3.10.2.2. Nonuniform Quantizer ......................... 272

3.11. Computer Generation of Random Numbers and Sequences. . . . . . . . 273 3.11.1. Generation of Uniform Random Numbers................ 273 3.11.2. Methods of Generating Random Numbers from an Arbitrary

pdf.................................................. 275 3.11.2.1. Transform Method (Analytical)................ .. 275

xx Contents

3.11.2.2. Transform Method (Empirical) . . . . . . . . . . . . . . . . . . 278 3.11.2.3. Transform Method for Discrete Random Variables 278 3.11.2.4. Acceptance/Rejection Method of Generating

Random Numbers............................. 279 3.11.3. Generating Gaussian Random Variables. . . . . .. . . . . . . . . . . . 281 3.11.4. Generating Independent Random Sequences. ... . . . . . . . . . • 282

3.11.4.1. White Gaussian Noise. . . . . . . . . . . . . .. . . . . . . . . . . . 282 3.11.4.2. Random Binary Sequence and Random Binary

Waveform .... '.' . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.11.4.3. Pseudorandom Binary Sequences ..•............. 284 3.11.4.4. M-ary PN Sequences.......................... 287

3.11.S. Generating Correlated Random Sequences. . . . . . . . . . . . . . . . 290 3.12. Testing of Random Number Generators......................... 292

3.12.1. Stationarity and Uncorrelatedness. . . . . . . . . . . . . . . . . . . . . . . . 293 3.12.2. Goodness-of-Fit Tests. . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . 29S

3.13. Summary..................................................... 297 3.14. Problems and Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

References .................................................. 300

Chapter 4. Modeling of Communication Systems

4.1. Introduction................................................. 303 4.2. Radiofrequency and Optical Sources. . . . . . . . . . . . . . . . . . . . . . . . .. . . 30S

4.2.1. Radiofrequency Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30S 4.2.2. Optical Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30S

4.3. Information Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 4.3.1. Analog Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 309 4.3.2. Digital Signals ........................................ 311

4.4. Source Encoders/Decoders. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 313 4.4.1. Quantization.......................................... 314 4.4.2. Differential Quantization ............................... 316 4.4.3. Encoding the Output of Discrete Information Sources . . . . . . 317

4.S. Baseband Modulation: Formatting; Line Coding. . . . . . . . . . . . . . . . . 319 4.S.1. Logical-to-Logical Mapping I: Binary Differential Encoding 320 4.S.2. Logical-to-Logical Mapping II: Correlative Coding ........ 320 4.5.3. Logical-to-Real Mapping I: Non-Return-to-Zero (NRZ)

Binary Signaling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 4.5.4. Logical-to-Real Mapping II: NRZ M-ary Signaling (PAM) 322 4.5.S. Logical-to-Real Mapping III: Return-to-Zero (RZ) Binary

Signaling .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 4.S.6. Logical-to-Real Mapping IV: Biphase Signaling or

Manchester Code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 4.S.7. Logical-to-Real Mapping V: Miller Code or Delay

Modulation . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 4.S.8. Logical-to-Real Mapping VI: Partial Response Signaling. . . . 323

Contents xxi

4.6. RF and Optical Modulation ................................... 326 4.6.1. Analog Modulation.................................... 326 4.6.2. Digital Quadrature Modulation. . . . . . . . . . . . . . . . . . . . . . . . • . 328 4.6.3. Continuous Phase Modulation (CPM): CPFSK; MSK...... 329

4.6.3.1. Continuous Phase Modulation................... 329 4.6.3.2. Continuous-Phase Frequency-Shift-Keying: CPFSK 331 4.6.3.3. Minimum-Shirt-Keying: MSK . . . . . . . . . . . . . . . . . . . 333

4.6.4. Some Implementation Notes............................ 333 4.7. Demodulation............................................... 336

4.7.1. Coherent Demodulation................................ 336 4.7.2. Noncoherent Demodulation............................. 339

4.7.2.1. Amplitude Demodulation. . . . . . . . . . . . . . . . . . . . . . . 340 4.7.2.2. Discriminator Detection of PM/FM Signals....... 340 4.7.2.3. PLL Demodulation of PM/FM Signals........... 341

4.8. Filtering....................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 4.8.1. Filters for Spectral Shaping. . . . . . .. ... .. . . ... . . .. . . . . . .. 344 4.8.2. Filters for Pulse Shaping. . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 4.8.3. Linear Minimum MSE Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 4.8.4. Filters for Minimizing Noise and Distortion. . . . . . . . . . . . . . . 348 4.8.5. Matched Filters. . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . 350 4.8.6. Adaptive Filtering (Equalization) ........................ 352 4.8.7. Filters Specified by Simple Functions in the Frequency

Domain 359 4.8.8. Tabular Filters for Masks and Measurements. . . . . . . . . . . . . . 361

4.9. Communication Channels and Models. . . .. . . . . . . • . . . . . . . . . . . . . . 362 4.9.1. The Almost Free-Space Channel......................... 363

4.9.1.1. Clear-Air Atmospheric (Tropospheric) Channel. . . . 364 4.9.1.2. The Rainy-Atmospheric Channel ................ 365 4.9.1.3. The Ionospheric Phase Channel ........... ,. . . .. 366

4.9.2. Conducting and Guided Wave Media .................... 368 4.9.2.1. Rectangular Waveguide Medium. . . . . . . . . . . . . . . . . 369 4.9.2.2. The Fiber Optic Channel .... ~ . . . .. . . . . . . . . . . . . . 370

4.9.3. Multipath Channels. . .. . .. . . . . .. . . . .. . .. .. . . . . . . . . . . . . . 374 4.9.3.1. Discrete Multipath............................. 374 4.9.3.2. Diffuse Multipath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 4.9.3.3. Combined Discrete and Diffuse Multipath ....... . 378 4.9.3.4. Specific Multipath Models: Radio-Relay Link;

Mobile Radio Link ............................ 379 4.9.3.5. Simulation of Multipath Channels ............... 386

4.9.4. Discrete Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 4.9.4.1. Memoryless Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . 387 4.9.4.2. Channels with Memory......................... 390

4.10. Multiplexing/Multiple Access. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 4.10.1. Basic Principles ....................................... 396 4.10.2. Issues in the Simulation of Multiple Access Methods... . . . . 399

4.11. Noise and Interference........................................ 401 4.11.1. Thermal (Gaussian) Noise.............................. 402

xxII Contents

4.11.2. Impulsive Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 4.11.3. Interference........................................... 405

4.12. Error Control Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 4.12.1. Block Codes: General Principles. . . . . . . . . . . . .. . . . . . . . . . . . 409

4.12.1.1. Block Encoders. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 410 4.12.2. Convolutional Codes: General Principles. . . . . . . . . . . . . . . . . 412

4.12.2.1. Convolutional Encoders. . . . . . . . . . . . . . . . . . . . . . . . 412 4.12.3. Block Decoders ....................................... 415 4.12.4. Soft Decision Decoding. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 416 4.12.5. Convolutional Decoders. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 416 4.12.6. Interleaving, Nonbinary Codes and Concatenation. . . . . . . . . 419 4.12.7. Simulation of Coded Communication Links. . . .. . . . . . . . . . . 423

4.13. Synchronization.............................................. 424 4.13.1. Carrier R~covery-BPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 4.13.2. The Phase-Locked Loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 4.13.3. Timing Recovery Scheme-BPSK........................ 435 4.13.4. Carrier Recovery-QPSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 4.13.5. Timing Recovery-QPSK ............................... 442

4.14. Spread Spectrum Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 4.15. Coded Modulation........................................... 449 4.16. Summary..................................................... 454 4.17. Problems and Projects. . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

References .................................................. 458

Chapter 5. Estimation of Performan~e Measures from Simulation 5.1. Preliminaries ..................... ;.......................... 463

5.1.1. Random Process Model: Stationarity and Ergodicity ....... 463 5.1.2. Basic Notation and Definitions. . . . . . . . . . . . . . ... . . . . . . . . . 465 5.1.3. Quality of an Estimator: Bias, Variance, Confidence Interval

and Time-Reliability Product. . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 5.1.3.1. Bias of an Estimator. . . . . . . . . . . . . . . .. . . . . . . . . . . 467 5.1.3.2. Variance of an Estimator. . . . . . . . . . . . . . . . . . . . . . . 467 5.1.3.3. Confidence Interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 5.1.3.4. Time-Reliability Product ....................... 470

5.2. Estimating the Average Level of a Waveform .................... 470 5.2.1. Form of the Estimator. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 470 5.2.2. Expected (Mean) Value of the Estimator ................. 470 5.2.3. Variance of the Estimator. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 471 5.2.4. Mixture (Signal Plus Noise) Processes. . . . . . . . . . . . . . . . . . . . 474 5.2.5. Confidence Interval Conditioned on the Signal.. . . . . . . . . . . 474

5.3. Estimating the Average Power (Mean-Square Value) of a Waveform 475 5.3.1. Form of the Estimator for Average Power. . . . . .. . . . . . . . . . . 476

Contents xxIII

5.3.2. Expected Value of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . 477 5.3.3. Variance of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

5.4. Estimating the Signal-to-Noise Ratio (SNR) .......... . . . . . . . . . . . 479 5.4.1. Introduction.......................................... 479 5.4.2. Form of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 5.4.3. Statistical Properties of the Estimator. . . . . . . . . . . . . . . . . . . . . 482 5.4.4. Implementing the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

5.5. Estimating the Probability Density or Distribution Function of the Amplitude of a Waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 5.5.1. The Empirical Distribution. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 487 5.5.2. The Empirical Probability Density Function-Histogram. . . . 488

5.5.2.1. Form of the Estimator. . . . . . . . . . . . .. . . . . . . . . . . . . 488 5.5.2.2. Expectation of the Estimator. . . . . . .. . . . . . . . . . . . . 489 5.5.2.3. Variance of the Estimator. . . . . . . . . . . . . . . . . . . . . . . 491

5.6. Estimating the Error Probability (Bit-Error-Rate) of a Digital System...................................................... 492 5.6.1. The Monte Carlo Method. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 496

5.6.1.1. Confidence Interval: Binomial Distribution. . . . . . . . 498 5.6.1.2. Confidence Interval: Poisson Approximation. . . . . . 498 5.6.1.3. Confidence Interval: Normal Approximation ..... '. 500 5.6.1.4. Mean and Variance of Monte Carlo Estimator. . . . . 501 5.6.1.5. Effect of Dependent Errors ..................... 502

5.6.2. Importance Sampling .................................. 503 5.6.2.1. Form of the Estimator. . . . . . . . . . . . .. . . . . . . . . . . . . 504 5.6.2.2. Choosing a Biased Density. . . . . . . . . . . . . . . . . . . . . . 505 5.6.2.3. Implementation of the Estimator. . . . . . . . . . . . . . . . . 507 5.6.2.4. Bias of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . 509 5.6.2.5. Variance (Time-Reliability Product) of the

Estimator 510 5.6.2.6. Some Considerations on Implementing and Using

Importance Sampling .......................... 513 5.6.3. Extreme Value Theory. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 515 5.6.4. Tail Extrapolation ..................................... 516

5.6.4.1. Form of the Estimator.......................... 518 5.6.4.2. Asymptotic Bias of the Estimator . . . . . . . . . . . . . . . . 520 5.6.4.3. Variance of the Estimator. . . . . . . . . .. . . . . . . . . . . . . 521 5.6.4.4. Summary of the Simulation Procedure for

Implementing Tail Extrapolation ................ 521 5.6.5. Quasianalytical (Semianalytic) Estimation. . . . . . • . . . . . . . . . . 523

5.6.5.1. Form of the Estimator and Computational Procedure for Binary or Quaternary Systems with a Generalized Exponential Distribution ............ 525

5.6.5.2. Reliability of the Estimator ..................... 527 5.6.5.3. Some Considerations on Implementing QA ....... 527

5.6.6. Summary and Comparison of BER Estimation Techniques . . 528 5.7. Estimating the Power Spectral Density (PSD) of a Process......... 531

5.7.1. Form of the Estimator.................................. 531

xxiv Contents

5.7.1.1. The Correlogram, or Indirect Method. . . . . . . . . . . . 531 5.7.1.2. The Periodogram or Direct Method. .. . . . . . . . . . . . 533

5.7.2. Modified Form of the Estimator: Windowing and Averaging 534 5.7.3. Expected Value of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . 538 5.7.4. Variance of the Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 5.7.5. Some Considerations on Implementing PSD Estimators:

Summary of the Simulation Procedure. . . . . . . . . . . . . . . . . . . . 540 5.7.5.1. Welch Periodogram Procedure (Direct Method) 540 5.7.5.2. Windowed Correlogram Procedure (Indirect M~~d) ~1

5.8. Visual Indicators of Performance and Related Bounds ............ 542 5.8.1. Eye Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 5.8.2. Scatter Diagrams ...................................... 544

5.9. Summary........................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 5.10. Problems and Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 546

References .................................................. 547

Chapter 6. Simulation and Modeling Methodology 6.1. Simulation Environment ...................................... 552

6.1.1. Features of the Software Environment. . . . . . . . . . . . . . . . . . . . 553 6.1.2. Components of the Software Environment. . . . . . . . . . . . . . . . 555 6.1.3. Hardware Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 6.1.4. Miscellaneous......................................... 557

6.2. Modeling Considerations. . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 6.2.1. Basic Concepts of Modeling ............................ 561 6.2.2. Cascaded Linear Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 6.2.3. Hardwired Synchronization: Phase and Timing Bias. . . . . . . . 565 6.2.4. Distribution of Phase and Timing Jitter Processes:

Replacement by a Single Approximately Equivalent Process 567 6.2.5. Effect of Synchronization Errors by Statistical Averaging. ... 571 6.2.6. Estimating Initial Carrier and Symbol Synchronization ..... 573 6.2.7. Block Estimator Structures. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 575 6.2.8. Simulation of Feedback Loops: Application to Phase-Locked

Carrier Tracking Loop and Phase-Locked Demodulator .... 578 6.2.8.1. Modeling Considerations .. ;. . . . . . . . . . . . . .. . . . . . 578 6.2.8.2. Stand-Alone PLL Model. . . . . . . . . . . . . . . . . . . . . . . . 579 6.2.8.3. Assembled PLL Model ......................... 585 ·6.2.8.4. The Phased-Locked Loop as a Phase Tracker. . . . . . 592 6.2.8.5.· The Phase-Locked Loop as an FM Demodulator. . . 593 6.2.8.6. Effect of Delay on the Performance of the

Assembled PLL Model ...... . . . . . . . . . . . . . . . . . . . 596 6.2.9. Multirate Sampling .................................... 597 6.2.10. Simulating a Hypothetical System. . . . . . . . . . . .. . . . . . . . . . . . 600

Contents xxv

6.3. Performance Evaluation Techniques ............................ 603 6.3.1. Basic QA Technique (QA-l) ............................ 603

6.3.1.1. Basic QA Technique for QAM Signals (QA-1.1) ... 604 6.3.2. Mixed QA Technique (QA-2) ........................... 609

6.3.2.1. MQA Variation 1 (QA-2.l}...................... 609 6.3.2.2. MQA Variation 2 (QA-2.2) ......... " . . . . . . . . . . . 611

6.3.3. QA Technique for Coded Systems with Hard Decision Decoding (QA-3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613

6.3.3.1. Independent-Error Channel (Interleaving): Method QA-3.1 ....................................... 614

6.3.3.2. Dependent-Error Channel: Method QA-3.2 ....... 616 6.3.4. QA Technique for Convolutionally Coded Systems with

Soft-Decision Decoding (QA-4) ......................... 618 6.3.5. QA Technique for Speeding up Equalizer Convergence

(QA-5) ............................................... 619 6.3.6. Simulation-Based Moment Evaluation for Analytic

Performance Evaluation (QA-6) ......................... 620 6.4. Error Sources in Simulation ................................... 622

6.4.1. Errors in System Modeling. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 622 6.4.2. Errors in Device Modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 6.4.3. Errors in Random Process Modeling ..................... 625 6.4.4. Processing Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

6.5. Validation................................................... 629 6.5.1. Validating Models of Devices. . . . . . . . . . . . ... . . . . . . . . . . . . 630 6.5.2. Validating Random Process Models. . . . . . . . . . . . . . . . . . . . . . 632 6.5.3. Validating the System Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 6.5.4. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

6.6. The Role of Simulation in Communication System Engineering .... 636 6.7. Summary.................................................... 640 6.8. Appendix: The "Equivalent Phase Noise" Process. . . . . . . . . . . . . . . . 641 6.9. Problems and Projects ........................................ 646

References .................................................. 648

Chapter 7. Three Case Studies 7.1. Case Study I: 64-QAM Equalized Digital Radio Link in a Fading

Environment ................................................ 652 7.1.1. Methods of Performance Evaluation Using Simulation...... 653

7.1.1.1. Selected Channel Snapshot Methodology. . . . . . . . . 654 7.1.1.2. Stochastic Channel Sequence Methodology ....... 654

7.1.2. The Channel Model.................................... 654 7.1.3. The Equalizer Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

7.1.3.1. The Stochastic Gradient Algorithm. . . . . . . . . . . . . . . 655 7.1.3.2. Covariance Matrix Inversion. . . . . . . . . . . . . . . . . . . . 656

xxvi Contents

7.1.4. The System Model. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 657 7.1.4.1. Modulator.................................... 657 7.1.4.2. Filters............................ . . . . . . . . . . . . 658 7.1.4.3. Demodulator.................................. 659 7.1.4.4. Equalizer..................................... 659 7.1.4.5. Detector...................................... 659 7.1.4.6. Synchronization............................... 660

7.1.5. The Selected Channel Snapshot Simulation ............... 660 7.1.5.1. Simulation Procedure .......................... 661 7.1.5.2. Calibration Procedure. . . . . . . . . . . . . . . . . . . . . . . . . . 661 7.1.5.3. Estimation of Error Probability. . . . . . . . . . . . . . . . . . 662 7.1.5.4. Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

7.1.6. The Stochastic Channel Sequence Simulation. . . .. . . . . . . . . . 663 7.1.6.1. Evaluation of Error Probability. . . . . . .. . . . . . . . . . . 665 7.1.6.2. Simulation Procedure .......................... 669 7.1.6.3. Evaluation of the Performance of Digital Radio

Using the Stochastic Channel Simulation ......... 670 7.1.6.4. Simulation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 670

7.1.7. Conclusions........................................... 670 7.2. Case Study II: Lightwave Communications Link................. 672

7.2.1. Block Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673 7.2.2. Photodetector Modeling. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 676 7.2.3. Tradeoff Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682

7.2.3.1. System Assumptions ........................... 683 7.2.3.2. Model Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 7.2.3.3. Nonideal Rise- and Fall-Time Sensitivity ......... 684 7.2.3.4. WDM and Optical Filtering ................. , . . . 686 7.2.3.5. Effects of LED Center Wavelength Drift. .. . . . . . . . 687 7.2.3.6. Photodetector Comparison. . . . . . . . . . . . . . . . . . . . . . 688 7.2.3.7. Timing Jitter Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

7.3. Case Study III: A Satellite System Example. . . . . . . . .. . . . . . . . . . . . 690 7.3.1. Transponder Simulation Block Diagram. . . . . . . . . . . . . . . . . . 691 7.3.2. System Simulation Block Diagram ....................... 694 7.3.3. Tradeoff Studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 696 References 701

Appendixes A. A Collection of Useful Results for the Error Probability of Digital

Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

B. Gaussian Tail Probabilities Q(x) and an Approximation Q(x) ....... 715

C. Coefficients of the Hermite Polynomials ............... " . . . . . . . . . . 717

D. Some Abscissas and Weights for Gaussian Quadrature Integration. . . . 719

E. Chi-Square Probabilities ........................................ 721

Index............................................................ 723