Cooper probabilistic methods of signal and system analysis, 3rd ed

www.elsolucionario.net Probabilistic Methods of Signal and System Analysis www.elsolucionario.net THIRD EDITION George R Cooper Clare D McGillem www.elsolucionario.net Probabilistic Methods of Signal and System Analysis www.elsolucionario.net Third Edition www.elsolucionario.net THE OXFORD SERIES IN ELECTRICAL AND COMPUTER ENGINEERING SERIES EDITORS Adel S Sedra, Series Editor, Electrical Engineering Michael R Lightner, Series Editor, Computer Engineering SERIES TITLES Allen and Holberg, CMOS Analog Circuit Design Bobi,:ow, Elementary Linear Circuit Analysis, 2nd Ed Bobrow, Fundamentals of Electrical Engineering, 2nd Ed Campbell, The Science and Engineering of Microelectronic Fabrication Chen, Analog & Digital Control System Design Chen, Linear System Theory and Design, 3rd Ed Comer, Digital Logic and State Machine Design, 3rd Ed Cooper and McGillem, Probabilistic Methods of Signal and System Analysis, 3rd Ed Franco, Electric Circuits Fundr;imentals Fortney, Principles Of Electronics: Analog & Digital Granzow, Digital Transmission Lines Guru and Hiziroglu, Electric Machinery & Transformers, 2nd Ed Boole and Boole, A Modern Short Course In Engineering Electromagnetics Jones, Introduction to Optical Fiber Communication Systems Krein, Elements of Power Electronics Kuo, Digital Control Systems, 3rd Ed Lathi, Modern Digital and Analog Communications Systems, 3rd Ed McGillem and Cooper, Continuous and Discrete· Signal and System Analysis, 3rd Ed Miner, Lines and Electromagnetic Fields for Engineers Roberts and Sedra, SPICE, 2nd Ed Roulston, An Introduction to the Physics of Semiconductor Devices Sadiku, Elements of Electromagnetics, 2nd Ed Santina, Stubberud, and Hostetter, Digital Control System Design, 2nd Ed Schwarz, Electromagnetics for Engineers Schwarz and Oldham, Electrical Engineering: An Introduction, 2nd Ed Sedra and Smith, Microelectronic Circuits, 4th Ed Stefani, Savant, Shahian, and Hostetter, Design of Feedback Control Systems, 3rd Ed Van Valkenburg, Anµlog Filter Design Warner and Grung,i Semiconductor Device Electronics Wolovich, Automatic Control Systems Yariv, Optical Electronics in Modem Communications, 5th Ed www.elsolucionario.net Chen, System and Signal Analysis, 2nd Ed www.elsolucionario.net CONTENTS ��� Preface , xi 1 Introduction to Probabil ity 1-3 Engineering Applications of Probability Random Experiments and Events Definitions of Probability 1-4 The Relative-Frequency Approach 1-6 The Axiomatic Approach 1-8 Independence 1-10 Bernoulli Trials 1-5 1-7 1-9 1-11 13 Elementary Set Theory Conditional Probability www.elsolucionario.net 1-1 1-2 19 22 27 Combined Experiments 29 31 Applications of Bernoulli Trials 35 38 References 50 Problems Random Variables sz 2-1 Concept of a Random Variable 2-2 Distribution Functions 52 54 57 2-3 Density Functions 2-4 Mean Values and Moments 2-5 The Gaussian Random Variable 2-6 Density Functions Related to Gaussian 2-8 Conditional Probability Distribution andJ)ensity Functions 2-9 Examples and Applications 2- 63 67 Other Probability Density Functions I 02 77 87 97 www.elsolucionario.net CONT E NTS vi Problems 109 119 References S everal Random Variab l es 120 120 3-1 Two Random Variables 3-2 Conditional Probability-Revisited 3-3 Statistical Independence 124 130 132 3-4 Correlation between Random Variables 3-5 Density Function of the Sum of Two Random Variables Probability Density Function of a Function of Two Random Variables 3-7 142 The Characteristic Function Problems 148 152 158 References 159 El em ents of Statistics 4-1 'Introduction 159 160 4-2 Sampling Theory-The Sample Mean 4-3 Sampling Theory-The Sample Variance 166 4-4 Sampling Distributions and Confidence Intervals 4-5 Hypothesis Testing 169 173 4-6 Curve Fitt_ing and Linear Regression 4-7 Correlation between Two Sets of Data Problems 177 182 184 References 188 Random Processes 189 189 5-1 Introduction 5-2 Continqous and Discrete Random Processes 5-3 Deterministic and Nondetermipistic Random Processes 5-4 Stationary and Nonstationary Random Processes 5-5 Ergodic and Nonergodic Random Processes 5-6 Measurement of Process Parameters 5-7 191 205 References 208 195 197 199 Smoothing Data with a Moving Window Average Problems www.elsolucionario.net 3-6 136 203 194 www.elsolucionario.net C O NTE NTS Correlation Functions 6-2 Introduction 209 209 Properties of Autocorrelation Functions 6.;.5 Examples of Autocorrelation Functions 6-7 Properties of Crosscorrelation Functions 6-4 6-6 6-8 6-9 216 220 Measurement of Autocorrelation Functions Crosscorrelation Functions 227 230 232 Examples and Applications of Crosscorrelation Functions Correlation Matrices for Sampled Functions Problems References 2s1 Introduction 257 7-2 Relation of Spectral Density to the Fourier Transform 7-4 Spectral Density and the Complex Frequency Plane 7-3 7-5 7-6 Properties of Spectral Density 259 263 Mean-Square Values from Spectral Density 271 274 Relation of Spectral Density to the Autocorrelation Function 7-7 White Noise 7-9 Autocorrelation Function Estimate of Spectral Density 7-11 Examples and Applications of Spectral Density 7-8 7-10 287 Cross-Spectral Density 189 Periodogram Estimate of Spectral Density Problems References 309 315 322 8-1 Introduction 8-3 Mean and Mean-Square Value of System Output 8-4 8-5 8-6 8-7 8-8 292 301 Respo, n se of Linear Systems to Random Inputs 8-2 234 240 245 256 Spectral Density 7-1 213 Example: Autocorrelation Function of a Binary Process 6-3 323 Analysis in the Time Domain 324 330 335 Examples of Time-Domain System Analysis 339 Analysis in the Frequency Domain 345 Spectral Density at the System Output 346 Autocorrelation Function of System Output Crosscorrelation between Input and Output 326 323 281 www.elsolucionario.net 6-1 vii www.elsolucionario.net C O N T E NTS viii Cross-Spectral Densities between Input and Output 8-10 Examples of Frequency-Domain Analysis 352 8-11 Nurp.erical Computation of System Output 359 Problems 368 380 References , Opti m um Li near System s 381 9-1 Introduction 9-2 Criteria of Optimality 9-3 Restrictions on the Optimum System 9-4 Optimization by Parameter Adjustment 381 · 382 384 385 9-5 Systems That Maximize Signal-to-Noise Ratio 9-6 Systems That Minimize Mean-Square Error Problems 350 395 www.elsolucionario.net 8-9 402 412 References 418 Appendices A Mathematical Tab l es A-1 419 Trigonometric Identities A-2 Indefinite Integrals A-3 Definite Integrals 419 420 421 A-4 Fourier Transform Operations A-5 Fourier Transforms A-6 One-Sided Laplace Transforms 422 423 423 B Freq uently Encount e red Probab i l ity Distri b utions B-1 Discrete Probability Functions B-2 Continuous Distributions C B i n o m i al Coeffi cients 425 427 43 D Nor mal Probabil ity Distri b ution Function E The Q-Function 434 -; Student's t Distri bution Function G Com p uter Com p utati ons 438 436 432 425 www.elsolucionario.net C O N T E NTS Ix H Table of Co rrelation Function-Spectral Density Pai rs 466 Conto u r Integration 475 www.elsolucionario.net Index 467 The goals of the Third Edition are essentially the same as those of the earlier editions, viz., to provide an introduction to the applications of probability theory to the solution of problems arising in the analysis of signals and systems that is appropriate for engineering students at the junior or senior level However, it may also serve graduate students and engineers as a concise review of material that they previously encountered in widely scattered sources This edition differs from the first and second in several respects In this edition use of the computer is introduced both in text examples and in selected problems The computer examples are carried out using MATLAB and the problems are such that they can be handled with the Student Edition of MATLAB as well as with other computer mathematics applications In addition to the introduction of computer usage in solving problems involving statistics and random processe� other changes have also been made In particular, a number of new sections have been added, virtually all of the exercises have been modified or changed, a number of the problems have been modified, and a number of new problems have been added Since this is an engineering text, the treatment is heuristic rather than rigorous, and the student will find many examples of the application of these concepts to engineering problems However, it is not completely devoid of the mathematical subtleties, and considerable attention has been devoted to pointing out some of the difficulties that make a more advanced study of the subject essential if one is to master it The authors believe that the educational process is best served by repeated exposure to difficult subject matter; this text is intended to be the first exposure to probability and random processes and, we hope, not the last The book is not comprehensive, but deals selectively with those topics that the authors have found most useful in the solution of engineering problems A brief discussion of some of the significant features of this book will help set the stage for a discussion of the various ways it can be used Elementary concepts of discrete probability are introduced in Chapter 1: first from the intuitive standpoint of the relative frequency approach and then from the more rigorous standpoint of axiomatic probability Simple examples illustrate all these concepts and are more meaningful to engineers than are the traditional examples of selecting red and white balls from urns The concept of a random variable is introduced in Chapter along with the ideas of probability distribution and density functions, mean values, and conditional probability A significant feature of this chapter is an extensive discussion of MATLAB is the registered trademark of The MathWorks, Inc., Natick, MA xi www.elsolucionario.net PREFACE www.elsolucionario.net www.elsolucionario.net P R E FA C E many different probability density functions and the physical situations in which they may occur Chapter extends the random variable concept to situations involving two or more random variables and introduces the concepts of statistical independence and correlation In Chapter 4, sampling theory, as applied to statistical estimation, is considered in some detail and a thorough discussion of sample mean and sample varianoe is given The distribution of the sample is described and the use of confidence intervals in making statistical decisions is both considered and illustrated by many examples of hypothesis testing The problem of fitting smooth curves to experimental data is analyzed, and the use of linear regression is illustrated by practical examples The problem of determining the correlation between data sets is examiried A general discussion of random processes and their classification is given in Chapter The emphasis here is on selecting probability models that are useful in solving engineering problems Accordingly, a great deal of attention is devoted to the physical significance of the various process classifications, with no attempt at mathematical rigor A unique feature of this chapter, which is continued in subsequent chapters, is an introduction to the practical problem of estimating the mean of a random process from an observed sample function The technique of smoothing data with a moving window is discussed Properties and applications of autocorrelation and crosscorrelation functions are discussed in Chapter Many examples are presented in an attempt to develop some insight into the nature of correlation functions The important problem of estimating autocorrelation functions is discussed in some detail and illustrated with several computer examples Chapter turns to a frequency-domain representation of random processes by introducing the concept of spectral density Unlike most texts, which simply define spectral density as the Fourier transform of the correlation function, a more fundamental approach is adopted here iri order to bring out the physical significance of the concept This chapter is the most difficult one in the book, but the authors believe the material should be presented in this way Methods of estimating the spectral density from the autocorrelation function and from the periodogram are developed and illustrated with appropriate computer-based examples The use of window functions to improve estimates is illustrated as well as the use of the computer to carry out integration of the spectral density using both the real and complex frequency representations Chapter utilizes the concepts of correlation functions and spectral density to analyze the response of linear systems to random inputs In a sense, this chapter is a culmination of all that preceded it, and is particularly significant to engineers who must use these concepts It contains many examples that are relevant to engineering probiems and emphasizes the need for mathematical models that are both realistic and manageable The comJ.lmtation of system output through simulation is examined and illustrated with computer examples, Chapter extends the concepts of systems analysis to consider systems that are optimum in some sense Both the Classical matched filter for known signals and the Wiener filter for random signals are considered from an elementary standpoint Computer examples of optimization are considered and illustrated with an example of an adaptive filter Several Appendices are included to provide useful mathematical and statistical tables and data Appendix G contains a detailed discussion, with examples, of the application of computers to the analysis of signals and systems and can serve as an introduction to some of the ways MATLAB can be used to solve such problems · www.elsolucionario.net xii www.elsolucionario.net APPENDIX H A � Rx(T) T - -T T +1 ,, l i • w, - A\ -0-0� + , Sx(w) 2a £ - ;Tl - 'f!, a + w2 I T l :o;: T e-a:rl - T sin (wT/2) (wT/2 )2 , l rl > T a a2 + (w - w o ) COS W 0T + a2 + 2'11"8 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