Schaum's Outline of Signals and Systems

SCHaums outlines Signals and Systems HWEI P HSU PhD 571 fullysolved problems with stepbystep solutions 20 problemsolving videos online 20 New MATLAB videos Concise explanations of all course concepts Explanations with abundant illustrative examples Signals and Systems 00HsuSignalsFMSchaums design 310819 429 PM Page i 00HsuSignalsFMSchaums design 310819 429 PM Page ii This page intentionally left blank Signals and Systems Fourth Edition Hwei P Hsu PhD Schaums Outline Series New York Chicago San Francisco Athens London Madrid Mexico City Milan New Delhi Singapore Sydney Toronto 00HsuSignalsFMSchaums design 310819 429 PM Page iii HWEI P HSU received his BS from National Taiwan University and his MS and PhD from Case Institute of Technology He has published several books including Schaums Outline of Analog and Digital Communications and Schaums Outline of Probability Random Variables and Random Processes Copyright 2020 2014 2011 1995 by McGrawHill Education All rights reserved Except as permitted under the United States Copyright Act of 1976 no part of this publication may be reproduced or distributed in any form or by any means or stored in a database or retrieval system without the prior written permission of the publisher ISBN 9781260454253 MHID 1260454258 The material in this eBook also appears in the print version of this title ISBN 9781260454246 MHID 126045424X eBook conversion by codeMantra Version 10 All trademarks are trademarks of their respective owners Rather than put a trademark symbol after every occurrence of a trade marked name we use names in an editorial fashion only and to the benefit of the trademark owner with no intention of infringe ment of the trademark Where such designations appear in this book they have been printed with initial caps McGrawHill Education eBooks are available at special quantity discounts to use as premiums and sales promotions or for use in corporate training programs To contact a representative please visit the Contact Us page at wwwmhprofessionalcom McGrawHill Education the McGrawHill Education logo Schaums and related trade dress are trademarks or registered trade marks of McGrawHill Education andor its affiliates in the United States and other countries and may not be used without written permission All other trademarks are the property of their respective owners McGrawHill Education is not associated with any product or vendor mentioned in this book TERMS OF USE This is a copyrighted work and McGrawHill Education and its licensors reserve all rights in and to the work Use of this work is subject to these terms Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work you may not decompile disassemble reverse engineer reproduce modify create derivative works based upon transmit distribute disseminate sell publish or sublicense the work or any part of it without McGrawHill Educations prior consent You may use the work for your own noncommercial and personal use any other use of the work is strictly prohibited Your right to use the work may be terminated if you fail to comply with these terms THE WORK IS PROVIDED AS IS McGRAWHILL EDUCATION AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE AND EXPRESSLY DISCLAIM ANY WARRANTY EXPRESS OR IMPLIED INCLUD ING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE McGrawHill Education and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free Neither McGrawHill Education nor its licensors shall be liable to you or anyone else for any inaccuracy error or omission regardless of cause in the work or for any damages resulting therefrom McGrawHill Education has no responsibility for the content of any information accessed through the work Under no circumstances shall McGrawHill Education andor its licensors be liable for any indirect incidental special punitive consequential or similar damages that result from the use of or inability to use the work even if any of them has been advised of the possibility of such damages This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract tort or otherwise v Preface to The Second Edition The purpose of this book like its previous edition is to provide the concepts and theory of signals and systems needed in almost all electrical engineering fields and in many other engineering and science disciplines as well In the previous edition the book focused strictly on deterministic signals and systems This new edition expands the contents of the first edition by adding two chapters dealing with random signals and the response of linear systems to random inputs The background material on probability needed for these two chapters is included in Appendix B I wish to express my appreciation to Ms Kimberly Eaton and Mr Charles Wall of the McGrawHill Schaum Series for inviting me to revise the book HWEI P HSU Shannondell at Valley Forge Audubon Pennsylvania 00HsuSignalsFMSchaums design 310819 429 PM Page v vi Preface to The First Edition The concepts and theory of signals and systems are needed in almost all electrical engineering fields and in many other engineering and scientific disciplines as well They form the foundation for further studies in areas such as communication signal processing and control systems This book is intended to be used as a supplement to all textbooks on signals and systems or for selfstudy It may also be used as a textbook in its own right Each topic is introduced in a chapter with numerous solved problems The solved problems constitute an integral part of the text Chapter 1 introduces the mathematical description and representation of both continuoustime and discrete time signals and systems Chapter 2 develops the fundamental inputoutput relationship for linear timeinvariant LTI systems and explains the unit impulse response of the system and convolution operation Chapters 3 and 4 explore the transform techniques for the analysis of LTI systems The Laplace transform and its application to con tinuoustime LTI systems are considered in Chapter 3 Chapter 4 deals with the ztransform and its application to discretetime LTI systems The Fourier analysis of signals and systems is treated in Chapters 5 and 6 Chapter 5 considers the Fourier analysis of continuoustime signals and systems while Chapter 6 deals with discretetime signals and systems The final chapter Chapter 7 presents the state space or state variable concept and analysis for both discretetime and continuoustime systems In addition background material on matrix analysis needed for Chapter 7 is included in Appendix A I am grateful to Professor Gordon Silverman of Manhattan College for his assistance comments and careful review of the manuscript I also wish to thank the staff of the McGrawHill Schaum Series especially John Aliano for his helpful comments and suggestions and Maureen Walker for her great care in preparing this book Last I am indebted to my wife Daisy whose understanding and constant support were necessary factors in the completion of this work HWEI P HSU Montville New Jersey 00HsuSignalsFMSchaums design 310819 429 PM Page vi vii To the Student To understand the material in this text the reader is assumed to have a basic knowledge of calculus along with some knowledge of differential equations and the first circuit course in electrical engineering This text covers both continuoustime and discretetime signals and systems If the course you are taking cov ers only continuoustime signals and systems you may study parts of Chapters 1 and 2 covering the continuous time case Chapters 3 and 5 and the second part of Chapter 7 If the course you are taking covers only discretetime signals and systems you may study parts of Chapters 1 and 2 covering the discretetime case Chapters 4 and 6 and the first part of Chapter 7 To really master a subject a continuous interplay between skills and knowledge must take place By study ing and reviewing many solved problems and seeing how each problem is approached and how it is solved you can learn the skills of solving problems easily and increase your store of necessary knowledge Then to test and reinforce your learned skills it is imperative that you work out the supplementary problems hints and answers are provided I would like to emphasize that there is no short cut to learning except by doing 00HsuSignalsFMSchaums design 310819 429 PM Page vii viii Contents CHAPTER 1 Signals and Systems 1 11 Introduction 1 12 Signals and Classification of Signals 1 13 Basic ContinuousTime Signals 6 14 Basic DiscreteTime Signals 11 15 Systems and Classification of Systems 14 Solved Problems 17 CHAPTER 2 Linear TimeInvariant Systems 51 21 Introduction 51 22 Response of a ContinuousTime LTI System and the Convolution Integral 51 23 Properties of ContinuousTime LTI Systems 53 24 Eigenfunctions of ContinuousTime LTI Systems 54 25 Systems Described by Differential Equations 54 26 Response of a DiscreteTime LTI System and Convolution Sum 56 27 Properties of DiscreteTime LTI Systems 57 28 Eigenfunctions of DiscreteTime LTI Systems 58 29 Systems Described by Difference Equations 59 Solved Problems 60 CHAPTER 3 Laplace Transform and ContinuousTime LTI Systems 101 31 Introduction 101 32 The Laplace Transform 101 33 Laplace Transforms of Some Common Signals 105 34 Properties of the Laplace Transform 106 35 The Inverse Laplace Transform 109 36 The System Function 110 37 The Unilateral Laplace Transform 113 Solved Problems 116 CHAPTER 4 The zTransform and DiscreteTime LTI Systems 148 41 Introduction 148 42 The zTransform 148 43 zTransforms of Some Common Sequences 152 44 Properties of the zTransform 153 00HsuSignalsFMSchaums design 310819 429 PM Page viii 45 The Inverse zTransform 156 46 The System Function of DiscreteTime LTI Systems 158 47 The Unilateral zTransform 160 Solved Problems 160 CHAPTER 5 Fourier Analysis of ContinuousTime Signals and Systems 193 51 Introduction 193 52 Fourier Series Representation of Periodic Signals 193 53 The Fourier Transform 196 54 Properties of the ContinuousTime Fourier Transform 200 55 The Frequency Response of ContinuousTime LTI Systems 203 56 Filtering 206 57 Bandwidth 209 Solved Problems 210 CHAPTER 6 Fourier Analysis of DiscreteTime Signals and Systems 261 61 Introduction 261 62 Discrete Fourier Series 261 63 The Fourier Transform 263 64 Properties of the Fourier Transform 267 65 The Frequency Response of DiscreteTime LTI Systems 271 66 System Response to Sampled ContinuousTime Sinusoids 273 67 Simulation 274 68 The Discrete Fourier Transform 275 Solved Problems 278 CHAPTER 7 State Space Analysis 329 71 Introduction 329 72 The Concept of State 329 73 State Space Representation of DiscreteTime LTI Systems 330 74 State Space Representation of ContinuousTime LTI Systems 332 75 Solutions of State Equations for DiscreteTime LTI Systems 334 76 Solutions of State Equations for ContinuousTime LTI Systems 337 Solved Problems 340 CHAPTER 8 Random Signals 392 81 Introduction 392 82 Random Processes 392 83 Statistics of Random Processes 394 84 Gaussian Random Process 400 Solved Problems 401 CHAPTER 9 Power Spectral Density and Random Signals in Linear System 417 91 Introduction 417 92 Correlations and Power Spectral Densities 417 93 White Noise 419 94 Response of Linear System to Random Input 421 Solved Problems 424 Contents ix 00HsuSignalsFMSchaums design 310819 429 PM Page ix APPENDIX A Review of Matrix Theory 443 A1 Matrix Notation and Operations 443 A2 Transpose and Inverse 446 A3 Linear Independence and Rank 447 A4 Determinants 448 A5 Eigenvalues and Eigenvectors 450 A6 Diagonalization and Similarity Transformation 451 A7 Functions of a Matrix 452 A8 Differentiation and Integration of Matrices 458 APPENDIX B Review of Probability 459 B1 Probability 459 B2 Random Variables 464 B3 TwoDimensional Random Variables 468 B4 Functions of Random Variables 470 B5 Statistical Averages 473 APPENDIX C Properties of Linear TimeInvariant Systems and Various Transforms 478 C1 ContinuousTime LTI Systems 478 C2 The Laplace Transform 478 C3 The Fourier Transform 480 C4 DiscreteTime LTI Systems 481 C5 The zTransform 482 C6 The DiscreteTime Fourier Transform 483 C7 The Discrete Fourier Transform 485 C8 Fourier Series 485 C9 Discrete Fourier Series 486 APPENDIX D Review of Complex Numbers 487 D1 Representation of Complex Numbers 487 D2 Addition Multiplication and Division 488 D3 The Complex Conjugate 488 D4 Powers and Roots of Complex Numbers 488 APPENDIX E Useful Mathematical Formulas 489 E1 Summation Formulas 489 E2 Eulers Formulas 489 E3 Trigonometric Identities 489 E4 Power Series Expansions 490 E5 Exponential and Logarithmic Functions 490 E6 Some Definite Integrals 490 Schaums Signals and Systems Videos 491 Schaums Signals and Systems MATLAB Videos 492 MATLAB Prints for Online Videos 493 INDEX 509 The laptop icon next to an exercise indicates that the exercise is also available as a video with stepbystep instructions These videos are available on the Schaumscom website by following the instructions on the inside front cover Contents x 00HsuSignalsFMSchaums design 210919 1157 AM Page x The concept and theory of signals and systems are needed in almost all electrical engineering fields and in many other engineering and scientific disciplines as well In this chapter we introduce the mathematical description and representation of signals and systems and their classifications We also define several important basic signals essential to our studies A signal is a function representing a physical quantity or variable and typically it contains information about the behavior or nature of the phenomenon For instance in an RC circuit the signal may represent the voltage across the capacitor or the current flowing in the resistor Mathematically a signal is represented as a function of an independent variable t Usually t represents time Thus a signal is denoted by xt where xₜt and xₙn are real signals and j 1 Note that in Eq 11 t represents either a continuous or a discrete variable Where xₜt 12 xt xt even part of xt xₙn 12 xn xn even part of xn xₒt 12 xt xt odd part of xt xₒn 12 xn xn odd part of xn Periodic discretetime signals are defined analogously A sequence discretetime signal xn is periodic with period N if there is a positive integer N for which xn N xn all n The unit step function ut also known as the Heaviside unit function is defined as ut1 t0 0 t0 which is shown in Fig 14a Note that it is discontinuous at t 0 and that the value at t 0 is undefined Similarly the shifted unit step function ut t0 is defined as utt01 tt0 0 tt0 which is shown in Fig 14b The unit impulse function δt also known as the Dirac delta function plays a central role in system analysis Traditionally δt is often defined as the limit of a suitably chosen conventional function having unity area over an infinitesimal time interval as shown in Fig 15 and possesses the following properties δt0 t0 t0 tεtε δτdτ1 If gt is a generalized function its nth generalized derivative gnt is defined by the following relation tφtgntdt1n tφntgtdt where φt is a testing function which can be differentiated an arbitrary number of times and vanishes outside some fixed interval and φnt is the nth derivative of φt Thus by Eqs 128 and 120 the derivative of δt can be defined as tφtδtdtφ0 Note that if s σ a real number then Eq 135 reduces to a real exponential signal xt eσt As illustrated in Fig 18 if σ 0 then xt is a growing exponential and if σ 0 then xt is a decaying exponential A continuoustime sinusoidal signal can be expressed as xt A cosω0t θ where A is the amplitude real ω0 is the radian frequency in radians per second and θ is the phase angle in radians The sinusoidal signal xt is shown in Fig 19 and it is periodic with fundamental period T0 2πω0 The reciprocal of the fundamental period T0 is called the fundamental frequency f0 f0 1T0 Hertz Hz The unit step sequence un is defined as un 1 n 0 0 n 0 which is shown in Fig 110a Note that the value of un at n 0 is defined unlike the continuoustime step function ut at t 0 and equals unity Similarly the shifted unit step sequence un k is defined as un k 1 n k 0 n k which is shown in Fig 110b Unlike the continuoustime unit impulse function δt δn is defined without mathematical complication or difficulty From definitions 145 and 146 it is readily seen that xnδn x0δn 147 xnδn k xkδn k 148 which are the discretetime counterparts of Eqs 125 and 126 respectively From definitions 143 to 146 δn and un are related by δn un un 1 149 un k δk k0 δn k 150 which are the discretetime counterparts of Eqs 130 and 131 respectively Using definition 146 any sequence xn can be expressed as xn k xkδn k which corresponds to Eq 127 in the continuoustime signal case The complex exponential sequence is of the form xn ejΩ0n 152 Again using Eulers formula xn can be expressed as xn ejΩ0n cosΩ0n j sinΩ0n 153 Thus xn is a complex sequence whose real part is cosΩ0n and imaginary part is sinΩ0n Periodicity of ejΩ0n In order for ejΩ0n to be periodic with period N 0 Ω0 must satisfy the following condition Prob 111 Ω0 2π m N positive integer 154 Thus the sequence ejΩ0n is not periodic for any value of Ω0 It is periodic only if Ω02π is a rational number Note that this property is quite different from the property that the continuoustime signal ejΩt is periodic for any value of Ωt Thus if Ω0 satisfies the periodicity condition in Eq 154 Ω0 0 and N and m have no factors in common then the fundamental period of the sequence xn in Eq 152 is N0 given by N0 m 2π Ω0 155 Another very important distinction between the discretetime and continuoustime complex exponentials is that the signals ejΩn are all distinct for distinct values of Ω0 but that this is not the case for the signals ejΩn A sinusoidal sequence can be expressed as xn A cosΩ0n θ 158 If n is dimensionless then both Ω0 and θ have units of radians Two examples of sinusoidal sequences are shown in Fig 113 As before the sinusoidal sequence in Eq 158 can be expressed as A cosΩ0n θ A ReejΩ0n θ 159 As we observed in the case of the complex exponential sequence in Eq 152 the same observations Eqs 154 and 156 also hold for sinusoidal sequences For instance the sequence in Fig 113a is periodic with fundamental period 12 but the sequence in Fig 113b is not periodic where T is the operator representing some welldefined rule by which x is transformed into y Relationship 160 is depicted as shown in Fig 114a Multiple input andor output signals are possible as shown in Fig 114b We will restrict our attention for the most part in this text to the singleinput singleoutput case B Deterministic and Stochastic Systems If the input and output signals x and y are deterministic signals then the system is called a deterministic system If the input and output signals x and y are random signals then the system is called a stochastic system C ContinuousTime and DiscreteTime Systems If the input and output signals x and y are continuoustime signals then the system is called a continuoustime system Fig 115a If the input and output signals are discretetime signals or sequences then the system is called a discretetime system Fig 115b Note that in a continuoustime system the input xt and output yt are often expressed by a differential equation see Prob 132 and in a discretetime system the input xn and output yn are often expressed by a difference equation see Prob 137 D Systems with Memory and without Memory A system is said to be memoryless if the output at any time depends on only the input at that same time Otherwise the system is said to have memory An example of a memoryless system is a resistor R with the input xt taken as the current and the voltage taken as the output yt The inputoutput relationship Ohms law of a resistor is yt R xt 161 A second example of a system with memory is a capacitor C with the current as the input xt and the voltage as the output yt then yt 1C t xτ dτ 162 A second example of a system with memory is a discretetime system whose input and output sequences are related by yn kn xk 163 E Causal and Noncausal Systems A system is called causal if its output at the present time depends on only the present andor past values of the input Thus in a causal system it is not possible to obtain an output before an input is applied to the system A system is called noncausal or anticipative if its output at the present time depends on future values of the input Example of noncausal systems are yt xt 1 164 yn xn 165 Note that all memoryless systems are causal but not vice versa F Linear Systems and Nonlinear Systems If the operator T in Eq 160 satisfies the following two conditions then T is called a linear operator and the system represented by a linear operator T is called a linear system 1 Additivity Given that Tx₁ y₁ and Tx₂ y₂ then Tx₁ x₂ y₁ y₂ 166 for any signals x₁ and x₂ 2 Homogeneity or Scaling Tαx αy for any signals x and any scalar α Any system that does not satisfy Eq 166 andor Eq 167 is classified as a nonlinear system Eqs 166 and 167 can be combined into a single condition as Tα₁x₁ α₂x₂ α₁y₁ α₂y₂ 168 where α₁ and α₂ are arbitrary scalars Eq 168 is known as the superposition property Examples of linear systems are the resistor Eq 161 and the capacitor Eq 162 Examples of nonlinear systems are y x² 169 y cos x 170 Note that a consequence of the homogeneity or scaling property Eq 167 of linear systems is that a zero input yields a zero output This follows readily by setting α 0 in Eq 167 This is another important property of linear systems G TimeInvariant and TimeVarying Systems A system is called timeinvariant if a time shift delay or advance in the input signal causes the same time shift in the output signal Thus for a continuoustime system the system is timeinvariant if Txt τ yt τ 171 for any real value of τ For a discretetime system the system is timeinvariant or shiftinvariant if Txn k yn k 172 for any integer k A system which does not satisfy Eq 171 continuoustime system or Eq 172 discretetime system is called a timevarying system To check a system for timeinvariance we can compare the shifted output with the output produced by the shifted input Probs 133 to 139 H Linear TimeInvariant Systems If the system is linear and also timeinvariant then it is called a linear timeinvariant LTI system I Stable Systems A system is boundedinputboundedoutput BIBO stable if for any bounded input x defined by x k₁ 173 the corresponding output y is also bounded defined by y k₂ 174 where k₁ and k₂ are finite real constants An unstable system is one in which not all bounded inputs lead to bounded output For example consider the system where output yn is given by yn n 1un and input xn un is the unit step sequence In this case the input un 1 but the output yn increases without bound as n increases J Feedback Systems A special class of systems of great importance consists of systems having feedback In a feedback system the output signal is fed back and added to the input to the system as shown in Fig 116 A discretetime signal xn is shown in Fig 119 Sketch and label each of the following signals a xn2 b x2n c xn d xn 2 Given the continuoustime signal specified by xt1t 1t1 0 otherwise determine the resultant discretetime sequence obtained by uniform sampling of xt with a sampling interval of a 025 s b 05 s and c 10 s Using the discretetime signals x1n and x2n shown in Fig 122 represent each of the following signals by a graph and by a sequence of numbers a y1nx1nx2n b y2n2x1n c y3nx1nx2n 18 Show that a If xt and xn are even then aa xt dt 2 0a xt dt 175a nkk xn x0 2 n1k xn 175b b If xt and xn are odd then x0 0 and x0 0 176 aa xt dt 0 and nkk xn 0 177 a We can write aa xt dt 0a xt dt 0a xt dt Letting τ λ in the first integral on the righthand side we get 0a xt dt 0a xλdλ 0a xλ dλ Since xt is even that is xλ xλ we have 0a xλ dλ 0a xt dt Hence aa xt dt 0a xt dt 0a xt dt 2 0a xt dt Similarly nkk xn nk1 xn x0 n1k xn Letting n m in the first term on the righthand side we get mkk xm m1k xm Since xn is even that is xm xm we have mkk xm m1k xm nkk xn Hence nkk xn nk1 xn x0 n1k xn b Since xt and xn are odd that is xt xt and xn xn we have x0 x0 and x0 x0 16 Find the even and odd components of xt ejt Let xet and xot be the even and odd components of ejt respectively ejt xet xot From Eqs 15 and 16 and using Eulers formula we obtain xet 12 ejt ejt cost xot 12 ejt ejt j sint 17 Show that the product of two even signals or of two odd signals is an even signal and that the product of an even and an odd signal is an odd signal Let xt x1tx2t If x1t and x2t are both even then xt x1tx2t x1tx2t xt and xt is even If x1t is even and x2t is odd then xt x1tx2t x1tx2t xt and xt is odd Note that in the above proof variable t represents either a continuous or a discrete variable Hence x0 x0 x0 x0 0 x1 x0 x0 x0 0 Similarly a xt dt a xt dt a xλ dλ a xt dt a xλ dλ a xt dt a xt dt a xλ dλ 0 and m k xn 1 n k xn x0 k n 1 xn m k xm x0 k n 1 xn m k xm x0 k n 1 xn m k xn x0 k n 1 xn x0 0 in view of Eq 176 19 Show that the complex exponential signal xt e jω 0 t is periodic and that its fundamental period is 2πω 0 By Eq 17 xt will be periodic if e jω 0 t T e jω 0 t Since e jω 0 t T e jω 0 te jω 0 T we must have e jω 0 T 1 178 If ω 0 0 then xt 1 which is periodic for any value of T If ω 0 0 Eq 178 holds if ω 0 T m2π or T m 2π ω 0 m positive integer Thus the fundamental period T 0 the smallest positive T of xt is given by 2πω 0 We note that cosω 0 t T θ cosω 0 t θ ω 0 T cosω 0 t θ if ω 0 T m2π or T m 2π ω 0 m positive integer Thus the fundamental period T 0 of xt is given by 2πω 0 111 Show that the complex exponential sequence xn e jΩ 0 n is periodic only if Ω 0 2π is a rational number By Eq 19 xn will be periodic if e jΩ 0 n N e jΩ 0 n or e jΩ 0 N 1 179 Equation 179 holds only if Ω 0 N m2π m positive integer or Ω 0 2π m N rational numbers Thus xn is periodic only if Ω 0 2π is a rational number 112 Let xt e jω 0 t with radian frequency ω 0 and fundamental period T 0 2πω 0 Consider the discretetime sequence xn obtained by uniform sampling of xt with sampling interval T s that is xn xnT s e jω 0 nT s Find the condition on the value of T s so that xn is periodic If xn is periodic with fundamental period N 0 then e jω 0 n N 0 T s e jω 0 nT s e jω 0 N 0 T s Thus we must have e jω 0 N 0 T s 1 ω 0 N 0 T s 2π or T s N 0 m2π m positive integer or T s T 0 m N 0 181 Thus xn is periodic if the ratio T s T 0 of the sampling interval and the fundamental period of xt is a rational number Note that the above condition is also true for sinusoidal signals xt cosω 0 t θ 113 Consider the sinusoidal signal xt cos 15t a Find the value of sampling interval T s such that xn xnT s is a periodic sequence b Find the fundamental period of xn xnT s if T s 01 seconds a The fundamental period of xt is T 0 2πω 0 2π15 By Eq 181 xn xnT s is periodic if T s T 0 T s 2π15 m N 0 where m and N 0 are positive integers Thus the required value of T s is given by T s m N 0 T 0 m 2π 15 183 b Substituting T s 01T π10 in Eq 182 we have T s T 0 π10 2π15 15 20 3 4 Thus xn xnT s is periodic By Eq 182 N 0 m T 0 T s 4 3 The smallest positive integer N 0 is obtained with m 3 Thus the fundamental period of xn x01n is N 0 4 114 Let x 1 t and x 2 t be periodic signals with fundamental periods T 1 and T 2 respectively Under what conditions is the sum xt x 1 t x 2 t periodic and what is the fundamental period of xt if it is periodic Since x 1 t and x 2 t are periodic with fundamental periods T 1 and T 2 respectively we have x 1 t x 1 t T 1 x 1 t mT 1 m positive integer x 2 t x 2 t T 2 x 2 t kT 2 k positive integer Thus xt x 1 t mT 1 x 2 t kT 2 In order for xt to be periodic with period T one needs xt T x 1 t T x 2 t T x 1 t mT 1 x 2 t kT 2 Thus we must have mT 1 kT 2 T 184 or T 1 T 2 k m rational number 185 In other words the sum of two periodic signals is periodic only if the ratio of their respective periods can be expressed as a rational number Then the fundamental period is the least common multiple of T₁ and T₂ and is given by Eq 184 if the integers m and k are relative prime If the ratio T₁T₂ is an irrational number then the signals x₁n and x₂n do not have a common period and xt cannot be periodic Let x₁n and x₂n be periodic sequences with fundamental periods N₁ and N₂ respectively Under what conditions is the sum xn x₁n x₂n periodic and what is the fundamental period of xn if it is periodic Since x₁n and x₂n are periodic with fundamental periods N₁ and N₂ respectively we have x₁n N₁ x₁n and x₂n N₂ x₂n Thus xn x₁n mN₁ x₂n kN₂ In order for xn to be periodic with period N one needs xn N x₁n N x₂n N x₁n mN₁ x₂n kN₂ Thus we must have mN₁ kN₂ N Since we can always find integers m and k to satisfy Eq 186 it follows that the sum of two periodic sequences is also periodic and its fundamental period is the least common multiple of N₁ and N₂ E int0infty xt2 e2at dt frac12a infty E limT o infty intT2T2 xt2 dt limT o infty intT2T2 xt2 dt limT o infty fracT3 infty P limN o infty frac12N 1 sumn NN xn2 limN o infty frac12N 1 sumn NN xn2 limN o infty frac12N 1 N 1 frac12 infty The unit step function ut can be defined as a generalized function by the following relation ϕtutdt0ϕtdt 198 where ϕt is a testing function which is integrable over 0 t Using this definition show that ut1 t0 0 t0 Rewrite Eq 198 as ϕtutdtϕtdt0ϕtdt0ϕtdt we obtain 0ϕtutdt0ϕt1utdt This can be true only if ϕtutdt0 and 0ϕt1utdt0 Now using Eq 120 for ϕ0 we obtain ϕtδtdt1aϕ01aϕtdtϕ01adt for any ϕt Then by the equivalence property 199 we obtain δat1aδt δt ut fracdutdt a intinftyinfty varphit deltat dt varphi0 where varphi0 fracd varphitdtbiggt0 a intinftyinfty varphit deltat dt intinftyinfty varphit deltat dt varphi0 Thus the inputoutput relationship of the RC circuit is described by a firstorder linear differential equation with constant coefficients a Find the inputoutput relationship The system is timeinvariant 137 Find the inputoutput relation of the feedback system shown in Fig 137 From Fig 137 the input to the unit delay element is xn yn Thus the output yn of the unit delay element is Eq 1111 yn xn 1 yn 1 Rearranging we obtain yn yn 1 xn 1 Thus the inputoutput relation of the system is described by a firstorder difference equation with constant coefficients 138 A system has the inputoutput relation given by yn Txn n xn Determine whether the system is a memoryless b causal c linear d timeinvariant or e stable a Since the output value at n depends on only the input value at n the system is memoryless b Since the output does not depend on the future input values the system is causal c Let xn α₁x₁n α₂x₂n Then yn Txn nα₁x₁n α₂x₂n α₁n x₁n α₂n x₂n Thus the superposition property 168 is satisfied and the system is linear d Let y₁n be the response to x₁n xn n₀ Then y₁n Txn n xn n₀ yn n₀ n n₀xn n₀ y₁n Hence the system is not timeinvariant e Let xn un Then yn n un Thus the bounded unit step sequence produces an output sequence that grows without bound Fig 138 and the system is BIBO stable 139 A system has the inputoutput relation given by yn Txn xk₀n where k₀ is a positive integer Is the system timeinvariant Let y₁n be the response to xn xn n₀ Then y₁n Txn xk₀n n₀ But yn n₀ xk₀n n₀ y₁n Hence the system is not timeinvariant unless k₀ 1 Note that the system described by Eq 1114 is called a compressor It creates the output sequence by selecting every k₀th sample of the input sequence Thus it is obvious that this system is timevarying 140 Consider the system whose inputoutput relation is given by the linear equation y ax b If b 0 then the system is not linear because x 0 implies y b 0 If b 0 then the system is linear 141 The system represented by T in Fig 139 is known to be timeinvariant When the inputs to the system are x₁n x₂n and x₃n the outputs of the system are y₁n y₂n and y₃n as shown Determine whether the system is linear From Fig 139 it is seen that x₁n x₁n x₃n 2 Thus if T is linear then Tx₁n Tx₁n Tx₃n 2 y₁n y₂n 2 which is shown in Fig 140 From Figs 139 and 140 we see that y₁n y₁n y₂n 2 Hence the system is not linear yt xt 1 yn y0zn yn λzn yn Txn n xn a yt 2xt c xn un un 1 Linear TimeInvariant Systems 21 Introduction Two most important attributes of systems are linearity and timeinvariance In this chapter we develop the fundamental inputoutput relationship for systems having these attributes It will be shown that the inputoutput relationship for LTI systems is described in terms of a convolution operation The importance of the convolution operation in LTI systems stems from the fact that knowledge of the response of an LTI system to the unit impulse input allows us to find its output to any input signals Specifying the inputoutput relationships for LTI systems by differential and difference equations will also be discussed C Convolution Integral Equation 25 defines the convolution of two continuoustime signals xt and ht denoted by yt xt ht xτht τ dτ 26 Equation 26 is commonly called the convolution integral Thus we have the fundamental result that the output of any continuoustime LTI system is the convolution of the input xt with the impulse response ht of the system Fig 21 Continuoustime LTI system In many applications the step response st is also a useful characterization of the system The step response st can be easily determined by Eq 210 that is st ht ut t hτut τ dτ ht dτ 212 Thus the step response st can be obtained by integrating the impulse response ht Differentiating Eq 212 with respect to t we get ht st dst dt 213 Thus the impulse response ht can be determined by differentiating the step response st C Stability The BIBO boundedinputboundedoutput stability of an LTI system Sec 15H is readily ascertained from its impulse response It can be shown Prob 213 that a continuoustime LTI system is BIBO stable if its impulse response is absolutely integrable that is hτ dτ 221 24 Eigenfunctions of ContinuousTime LTI Systems In Chap 1 Prob 144 we saw that the eigenfunctions of continuoustime LTI systems represented by T are the complex exponentials eστ with s a complex variable That is Teστ λeστ 222 where λ is the eigenvalue of T associated with eστ Setting xt eστ in Eq 210 we have yt Teστ hτeσττ dτ hτeσt dτ eστ 223 Hseστ λeστ 224 where λ Hs hτeστ dτ 224 25 Systems Described by Differential Equations A Linear ConstantCoefficient Differential Equations A general Nthorder linear constantcoefficient differential equation is given by N k0 ak dkyt dk M k0 bk dkit dk 225 where coefficients ak and bk are real constants The order N refers to the highest derivative of yt in Eq 225 Such differential equations play a central role in describing the inputoutput relationships of a wide variety of electrical mechanical chemical and biological systems For instance in the RC circuit considered in Prob 132 the input xt vt and the output yt vt are related by a firstorder constantcoefficient differential equation Eq 1105 dyt dt 1 RC yt 1 RC xt 226 D Properties of the Convolution Sum The following properties of the convolution sum are analogous to the convolution integral properties shown in Sec 23 1 Commutative xn hn hn xn 236 2 Associative xn hn h2n xn hn h2n 237 3 Distributive xn h1n h2n xn h1n xn h2n 238 E Convolution Sum Operation Again applying the commutative property 236 of the convolution sum to Eq 235 we obtain yn hn xn k hkxn k 239 which may at times be easier to evaluate than Eq 235 Similar to the continuoustime case the convolution sum Eq 235 operation involves the following four steps 1 The impulse response hk is timereversed that is reflected about the origin to obtain hk and then shifted by n to form hn k hk n which is a function of k with parameter n 2 Two sequences xk and hn k are multiplied together for all values of k with n fixed at some value 3 The product xkhn k is summed overall k to produce a single output sample yn 4 Steps 1 to 3 are repeated as n varies over to to produce the entire output yn F Step Response The step response sn of a discretetime LTI system with the impulse response hn is readily obtained from Eq 239 as sn hn un k hkun k n hk 240 From Eq 240 we have hn sn sn 1 241 Equations 240 and 241 are the discretetime counterparts of Eqs 212 and 213 respectively C Impulse Response Unlike the continuoustime case the impulse response hn of a discretetime LTI system described by Eq 253 or equivalently by Eq 254 can be determined easily as hn 1a0 k0M bk δnk k1N ak hnk For the system described by Eq 255 the impulse response hn is given by hn 1a0 k0M bk δnk bna0 0 n M 0 otherwise Note that the impulse response for this system has finite terms that is it is nonzero for only a finite time duration Because of this property the system specified by Eq 255 is known as a finite impulse response FIR system On the other hand a system whose impulse response is nonzero for an infinite time duration is said to be an infinite impulse response IIR system Examples of finding impulse responses are given in Probs 244 and 245 In Chap 4 we will find the impulse response by using transform techniques SOLVED PROBLEMS Responses of a ContinuousTime LTI System and Convolution 21 Verify Eqs 27 and 28 that is a xt ht ht xt b xt h1t h2t xt h1t h2t a By definition 26 we have xt ht xτhτ t dτ By changing the variable t τ λ we have xt ht xλhλ dλ hλxt dλ ht xt b Let xt ft and ht ht ft Then f1t xτhτ t dτ and xt h1t h2t f1t h2t f1t xτh1τ σ dσ xτh2τ t dτ xt ht ht f2t Thus xt ht ht xt ht ht xt f2t Thus we can write the output yt as yt frac1alpha1 ealpha tut 264 25 Compute the output yt for a continuoustime LTI system whose impulse response ht and the input xt are given by ht ealpha tut xt ealpha ut T alpha 0 Since ut auut au begincases 1 0 t T t 0 0 extotherwise endcases CHAPTER 2 Linear TimeInvariant Systems CHAPTER 2 Linear TimeInvariant Systems CHAPTER 2 Linear TimeInvariant Systems CHAPTER 2 Linear TimeInvariant Systems 69 b Since both x₁τ and x₂τ are periodic with the same period T₀ x₁τx₂τ τ is also periodic with period T₀ Then using property 188 Prob 117 we obtain fτ ₀ᶦT₀ x₁τ x₂τ τ dτ ₀ᶦT₀ x₁τ x₂τ τ dτ for an arbitrary a c We evaluate the periodic convolution graphically Signals xτ xτ and xτxτ are sketched in Fig 213a from which we obtain fτ A²τ 0 τ T₀2 A²τ T₀ T₀2 τ T₀ and fτ T₀ fτ which is plotted in Fig 213b a xt A T₀ T₀ T₀2 0 T₀2 τ 0 τ T₀2 xτ 0 τ T₀2 T₀ 0 T₀ T₀2 0 T₀2 τ xtxτ A² 0 τ T₀2 b xt A T₀ T₀ T₀2 0 T₀2 τ T₀ τ T₀ T₀ 0 T₀ T₀2 0 T₀2 τ xτ A² T₀2 τ T₀ fτ A²T₀2 2T₀ T₀ T₀ 0 T₀2 2T₀ t Fig 213 491 Schaums Signals and Systems Videos 1 Problem 146 Signals and Systems express a signal in terms of unit step functions 2 Problem 156 Signals and Systems determine if a signal is linear timeinvariant andor causal 3 Problem 161 Signals and Systems determine if a system is invertible 4 Problem 246 Linear TimeInvariant Systems find the convolution of a pair of signals 5 Problem 258 Linear TimeInvariant Systems find the different equation for a 2nd order circuit 6 Problem 264 Linear TimeInvariant Systems find the output of a discretetime system 7 Problem 343 Laplace Transform and ContinuousTime LTI Systems find the Laplace transform of a signal 8 Problem 349 Laplace Transform and ContinuousTime LTI Systems find the inverse Laplace transform of a signal 9 Problem 355 Laplace Transform and ContinuousTime LTI Systems find the transfer function of a system described by a block diagram 10 Problem 448 The zTransform and DiscreteTime LTI Systems find the ztransform of a discretetime system 11 Problems 453454 The zTransform and DiscreteTime LTI Systems find the inverse zTransform of a discretetime signal 12 Problem 456 The zTransform and DiscreteTime LTI Systems use zTransforms to find the transfer function and difference equation for a system 13 Problem 561 Fourier Analysis of ContinuousTime Signals and Systems find the trigonometric and complex exponential Fourier series of a continuoustime signal 14 Problems 567569 Fourier Analysis of ContinuousTime Signals and Systems find the Fourier transform of a continuoustime signal 15 Problem 575 Fourier Analysis of ContinuousTime Signals and Systems find the frequency response and type of a filter 16 Problem 662 Fourier Analysis of DiscreteTime Signals and Systems find the discrete Fourier series for a periodic sequence 17 Problem 671 Fourier Analysis of DiscreteTime Signals and Systems find the frequency and impulse response of a causal discretetime LTI system 18 Problem 765 State Space Analysis find the state space representation of a system 19 Problem 768 State Space Analysis find the state space representation of a system and determine whether it is asymptotically andor BIBO stable 20 Problem 773 State Space Analysis use the state space method to solve a linear differential equation 10HsuSignalsApp 83119 531 PM Page 491 492 Schaums Signals and Systems MATLAB Videos 1 Problems 1112 plot continuoustime and discretetime signal transformations 2 Problem 15 plot even and odd components of continuoustime and discretetime signals 3 Problem 116 determine if given are periodic and the fundamental period 4 Problem 120 determine whether a signal is an energy or power signal 5 Problem 25 compute yt by using convolution for a continuoustime LTI system 6 Problem 224 find the impulse response and step response of a given system 7 Problem 230 compute yn by using convolution for a discretetime LTI system 8 Problem 343 find the Laplace transform of given signals 9 Problem 349 find the inverse Laplace transform of given signals 10 Problem 353 find the output yt of a given CT LTI system given ht and the input xt 11 Problem 441 find the zTransform of a given signal 12 Problem 453 find the inverse zTransform of a given signal 13 Problem 563 find the Fourier series representation of a given signal 14 Problem 569 find the inverse Fourier transform of a given signal 15 Problem 571 find the Fourier transform of a given signal 16 Problem 662 find the discrete Fourier series of a given signal 17 Problem 665 find the Fourier transform of a given sequence 18 Problem 667 find the inverse Fourier transform of a discretetime system 19 Problem 762 find the system function and whether a discretetime LTI system is controllable of observable given a state space representation 20 Problem 774 find the state space representation of a system and whether it is controllable or observable given a state space representation 10HsuSignalsApp 83119 531 PM Page 492 MATLAB Prints for Online Videos 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 493 MATLAB Prints for Online Videos 494 494 Problem 11 Problem 12 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 494 MATLAB Prints for Online Videos 495 Problem 15b Even and Odd Signals Problem 15a Original Signals 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 495 MATLAB Prints for Online Videos 496 496 Problem 116c 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 496 MATLAB Prints for Online Videos 497 Problem 116d 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 497 MATLAB Prints for Online Videos 498 Problem 116g 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 820 PM Page 498 MATLAB Prints for Online Videos 499 Problem 116h 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 499 MATLAB Prints for Online Videos 500 Problem 25 Continuoustime Convolution Problem 224a 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 500 MATLAB Prints for Online Videos 501 Problem 224b Problem 230 Discretetime Convolution 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 501 MATLAB Prints for Online Videos 502 Problem 232 Problem 510a 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 502 MATLAB Prints for Online Videos 503 Problem 510b Problem 569a 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 503 Problem 571 Problem 569b MATLAB Prints for Online Videos 504 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 504 MATLAB Prints for Online Videos 505 Problem 662a Problem 662b 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 505 MATLAB Prints for Online Videos 506 Problem 665a Problem 662c 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 506 MATLAB Prints for Online Videos 507 Problem 665b Problem 667a 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 507 MATLAB Prints for Online Videos 508 Problem 667b 11HsuMATLAB Prints for Online VideosNEWSchaums design 160919 821 PM Page 508 509 Absolute bandwidth 209 Accumulation 155 Additivity 16 Adjoint or adjugate matrix 449 Advance unit 154 Aliasing 253 Algebra of events 459 Allpass filter 301 Amplitude distortion 206 Amplitude spectrum 195 Analog signals 2 Analytic signal 257 Anticausal sequence 58 Anticausal signals 53 Aperiodic sequences see Nonperiodic sequences Aperiodic signals see Nonperiodic signals Asymptotically stable systems 337 340 Autocorrelation 396 417 timeaveraged 399 Autocovariance 396 Auxiliary conditions difference equations 59 differential equations 55 Average power 5 normalized 5 Bandlimited signal 209 252 Bandlimited white noise 420 Bandpass signal 209 Bandwidth absolute 209 energy containment 251 equivalent 249 filter or system 209 signal 209 3dB or half power 209 Bayes rule 463 Bayestheorem 464 Bernoulli distribution 466 Binomial distribution 466 Bilateral or twosided Laplace transform 101 Bilateral or twosided ztransform 149 Bilinear transformation 307 Bode plots 240 Boundedinputboundedoutput BIBO stability 17 54 58 71 90 111 131 180 337 340 Canonical simulation the first form 347 353 the second form 348 354 Canonical State representation the first form 347 353 the second form 348 354 CauchySchwarz inequlity 476 Causal sequence 58 Causal signal 53 Causal system 16 Causality 44 53 55 58 89 111 158 CayleyHamilton theorem 335 339 360 454 Chain 394 Characteristic equation 335 450 Characteristic function see Eigenfunction Characteristic polynomial 450 Characteristic values see Eigenvalues Chebyshev inequlity 476 Circular convolution 277 Circular shift 276 Cofactor 448 Complement 459 Complex frequency 199 Complex numbers 487 Complex random process 394 Complex signals 2 Compressor 43 Conditional probability 462 Connection between the Fourier transform continuoustime and the Laplace transform 198 the Fourier transform discretetime and the ztransform 266 Constituent matrix 336 340 456 Continuoustime LTI systems 51 causality 53 111 described by differential equations 54 206 eigenfunctions 54 frequency response 203 impulse response 51 properties 53 response 51 stability 54 state space representation 332 step response 52 system or transfer function 110 Continuoustime signals 1 Continuoustime systems 15 Controllability matrix 368 380 Controllable systems 368 380 INDEX 12HsuSignalIndex 91619 358 PM Page 509 Convolution circular 277 continuoustime 52 discretetime 56 in frequency 201 integral 52 periodic 67 86 properties 52 57 sum 56 Convolution property discrete Fourier transform DFT 277 Fourier transform continuoustime 201 235 Fourier transform discretetime 269 291 Laplace transform 108 ztransform 155 168 Convolution theorem frequency 201 233 time 201 231 Correlation coefficient 475 Correlations 424 Counting process 403 Covariance 475 Covariance stationary 410 Covariance matrix 400 Crosscorrelation 396 417 Crosscovariance 397 Crosspower spectral density 419 Decimationinfrequency 320 Decimationintime 317 Degenerative circuits 143 Delay unit 41 154 Determinants 448 Laplace expansion 448 Deterministic signals 3 DFS see Discrete Fourier series DFT see Discrete Fourier transform DFT matrix 315 Diagonal matrix 443 Diagonalization matrix 451 Difference equations 59 recursive 59 Differential equations 54 homogeneous solution 54 particular solution 54 Digital signals 2 Digital simulation of analog signals 274 Dirac delta function δfunction see Unit impulse function Dirichlet conditions for Fourier series 195 for Fourier transforms 198 Discrete Fourier series DFS 261 278 properties 262 Discrete Fourier transform DFT definition 275 inverse 275 Npoint 276 properties 276 Discrete frequency or line spectra 195 Discretetime LTI systems causality 58 158 described by difference equations 59 eigenfunctions 58 finite impulse response FIR 60 impulse response 56 infinite impulse response IIR 60 properties 57 response 56 stability 58 159 state space representation 330 step response 57 system function 158 Discretetime signals 1 Discretetime systems 15 Distortionless transmission 205 Distribution Bernoulli 466 binomial 466 exponential 467 normal or Gaussian 468 Poisson 466 uniform 467 Distribution function 465 cumulative cdf 465 Duality property discrete Fourier series 262 discrete Fourier transform 277 Fourier transform continuoustime 200 223 Fourier transform discretetime 268 Durationlimited signal 258 Eigenfunctions or characteristic function 46 of continuoustime LTI systems 54 of discretetime LTI systems 58 Eigenvalues or characteristic values 46 96 335 450 Eigenvectors 335 450 Energy containment bandwidth 251 Energy content 5 normalized 5 Energydensity spectrum 202 Energy signal 5 Energy theorem 202 Ensemble 392 average 395 Equivalence property 34 Equivalent bandwidth 249 Even signal 3 Events 459 algebra of 459 certain 459 elementary 459 equally likely 462 independent 463 null 459 Equally likely events 462 Ergodicity 399 Ergodic in the autocorrelation 399 in the mean 399 Expectation or mean 473 Exponential distribution 467 Exponential sequences complex 12 real 13 Index 510 12HsuSignalIndex 91619 358 PM Page 510 Exponential signals complex 8 real 9 Fast Fourier transform FFT decimationinfrequency algorithm 320 decimationintime algorithm 317 Feedback systems 17 FFT see Fast Fourier transform Filter bandwidth 209 ideal band pass 207 ideal band stop 207 ideal frequencyselective 206 ideal lowpass 207 ideal highpass 207 narrowband 209 nonideal frequencyselective 208 Filtering 206 Finalvalue theorem unilateral Laplace transform 135 unilateral ztransform 187 Finiteduration signal 104 Finite impulse response FIR 60 Finite sequence 152 FIR see Finite impulse response First difference sequence 269 Fourier series coefficients 194 complex exponential 194 convergence 195 discrete DFS 261 278 harmonic form 195 trigonometric 194 Fourier spectra 198 265 Fourier transform continuoustime 198 convergence 198 definition 198 inverse 198 properties 200 tables 202 203 Fourier transform discretetime 265 convergence 266 definition 265 inverse 265 properties 267 tables 270 271 Frequency angular 193 fundamental 193 complex 198 convolution theorem 201 fundamental 9 10 193 radian 9 Frequency response continuoustime LTI systems 203 237 discretetime LTI systems 271 294 Frequency selective filter 206 Frequency shifting 200 219 267 276 Gain 205 Gaussian pulse 236 Gaussian or normal random process 400 Generalized derivatives 8 Generalized functions 7 Harmonic component 195 Hilbert transform 245 Homogeneity 16 Identity matrix 335 444 IIR see Infinite impulse response Impulseinvariant method 306 Impulse response continuoustime LTI systems 51 discretetime LTI systems 56 Impulse train periodic 216 Independent events 463 Independent increments 403 Index set 392 Infinite impulse response IIR 60 Initial condition 55 Initial rest 55 Initial state 381 Initialvalue theorem unilateral Laplace transform 135 unilateral ztransform 186 Initially relaxed condition see Initial rest Interconnection of systems 72 112 Intersection 459 Inverse transform see Fourier Laplace etc Invertible system 48 Jacovian 473 Joint cumulative distribution function cdf 468 distribution function 468 probability density function pdf 469 probability mass function pmf 468 Jointly widesense stationary WSS 398 Laplace transform bilateral twosided 101 definition 101 inverse 109 properties 106 120 region of convergence ROC 102 tables 105 109 unilateral onesided 101 113 134 Leftsided signal 104 Line spectra 195 Linear system 16 response to random input 421 423 Linear timeinvariant LTI system 16 continuoustime 51 discretetime 56 Linearity 16 55 106 Magnitude response 204 272 Magnitude spectrum 195 198 Marginal distribution function 468 pdf 469 pmf 469 Markov inequlity 476 Index 511 12HsuSignalIndex 91619 358 PM Page 511 Matrix or matrices characteristic equation 450 characteristic polynomial 450 conformable 445 constituent 336 340 456 controllability 368 380 covariance 400 diagonal 443 diagonalization 451 differentiation 458 eigenvalues 450 eigenvectors 450 function of 452 idempotent 456 identity or unit 335 444 integration 458 inverse 446 449 minimal polynomials 361 455 nilpotent 361 nonsingular 331 450 observability 369 381 power 452 rank 448 similar 331 452 singular 450 skewsymmetric 446 spectral decomposition 335 340 456 spectrum 456 statetransition 335 symmetric 446 system 331 transpose 446 Mean 395 Modulation theorem 228 Moment 474 Mutually exclusive or disjoint events 459 Narrowband random process 420 Ndimensional state equations 331 Nilpotent 361 Noncausal system 16 Nonideal frequencyselective filter 208 Nonlinear system 16 Nonperiodic or aperiodic sequence 5 signals 4 Nonrecursive equation 59 Nonsingular matrix 331 450 Normal or Gaussian distribution 468 Normalized average power 5 Normalized energy content 5 Npoint DFT 276 Sequence 276 Null event 459 Nyquist sampling interval 254 Nyquist sampling rate 254 Observability matrix 369 381 Observable system 369 381 Odd signal 3 Orthogonal random variables 475 sequences 278 signals 210 Parsevals identity see Parsevals theorem Parsevals relation 202 discrete Fourier series DFS 284 discrete Fourier transform DFT 277 Fourier series 221 Fourier transform continuoustime 202 233 234 Fourier transform discretetime 270 periodic sequences 284 periodic signals 221 Parsevals theorem discrete Fourier series DFS 263 284 discrete Fourier transform DFT 277 Fourier series 196 Fourier transform continuoustime 202 234 Fourier transform discretetime 270 Partial fraction expansion 110 158 Pass band 206 Period 4 fundamental 4 Periodic convolution continuoustime 67 discretetime 86 Periodic impulse train 216 Periodic sequences 261 Periodic signals 4 Phase distortion 206 Phase response 204 272 Phase shifter 245 Phase spectrum 195 198 Poisson distribution 466 random process 407 Poles 103 Power 5 average 5 Power series expansion 157 Power signals 5 Power spectral density or power spectrum 417 418 424 cross 419 Probability 459 axiomatic definition 460 conditional 462 density function pdf 467 mass function pmf 465 measure 460 total 463 Random sequence 394 Random signals 3 392 Random binary signal 401 429 Random experiment 459 Random or stochastic processes 392 atocorrelation 396 autocovariance 396 continuousparameter 394 crosscorrelation 396 crosscovariance 397 description 394 discreteparameter 394 independent 397 orthogonal 397 Index 512 12HsuSignalIndex 91619 358 PM Page 512 parameter set 394 probabilistic expressions 394 realization 392 state space 394 statistics of 394 strictsense stationary SSS 397 uncorrelated 397 widesense stationary WSS 397 Random variable rv 464 Bernoulli 466 binomial 466 continuous 467 exponential 467 normal or Gaussian 468 Poisson 466 twodimensional 468 uniform 467 Real signals 2 Recursive equation 59 Region of convergence ROC Laplace transform 102 ztransform 149 Relationship between the DFT and the DFS 276 the DFT and the discretetime Fourier transform 276 Response frequency 203 237 271 294 impulse 51 56 magnitude 204 272 phase 204 272 step 52 57 system 273 to random input 421 431 zeroinput 55 zerostate 55 Rightsided signal 104 Rise time 250 Sampled signal ideal 252 Sample space 392 459 Sample function 392 Samples 2 Sampling 1 interval 2 Nyquist 254 rate or frequency 252 274 Nyquist 254 Sampling theorem in the frequency domain 258 uniform 254 Sequence 1 complex exponential 12 exponential 13 finite 152 first difference 269 left sided 152 nonperiodic or aperiodic 5 Npoint 276 orthogonal 278 periodic 5 rightsided 152 sinusoidal 14 twosided 152 Siftinvariant 16 Simple random walk 404 Shifting in the sdomain 106 Signal bandwidth 209 Signals analog 2 analytical 257 anticausal 53 bandlimited 209 252 254 bandpass 209 causal 53 complex 2 complex exponential 8 continuoustime 1 deterministic 3 digital 2 discretetime 1 durationlimited 258 energy 5 even 3 finiteduration 104 Gaussian pulse 236 highpass 209 ideal sampled 252 leftsided 104 lowpass 209 nonperiodic or aperiodic 4 odd 3 periodic 4 power 5 random 3 392 random binary 401 429 real 2 real exponential rightsided 104 sinusoidal 9 telegraph 427 428 timelimited 104 twosided 104 Signum function 254 Similar matrices 331 452 Similarity transformation 331 451 Simulation 274 304 by bilinear transformation 307 canonical 347 348 impulseinvariance method 306 Singular matrix 450 Sinusoidal sequences 14 Sinusoidal signals 9 Spectral coefficients 262 Spectral decomposition 335 340 456 Spectrum or spectra 195 amplitude 195 discrete frequency 195 energydensity 202 Fourier 198 265 line 195 magnitude 195 198 phase 195 198 splane 102 Index 513 12HsuSignalIndex 91619 358 PM Page 513 Stability asymptotical 337 340 boundedinputboundedoutput BIBO 17 54 58 71 90 111 131 180 337 340 Stable systems 17 Standard deviation 475 State 329 394 State equations continuoustime 333 337 discretetime 331 334 State space 329 State space representation continuoustime LTI systems 332 discretetime LTI systems 331 334 canonical the first form 347 the second form 348 Statetransition matrix 335 State variables 329 State vectors 330 Stationarity 397 Stationary strictsense SSS 397 widesense WSS 397 Statistical or ensemble average 395 473 Step response 52 57 Stop band 206 Superposition property 16 Systems causal and noncausal 16 continuoustime and discretetime 15 continuoustime LTI 51 controllable 368 described by difference equations 59 91 described by differential equations 54 75 discretetime LTI 56 feedback 17 interconnection of 112 invertible 48 linear and nonlinear 16 linear timeinvariant LTI 17 51 memoryless 15 multipleinput multipleoutput 331 observable 369 stable 17 timeinvariant and timevarying 16 with and without memory 15 System function continuoustime LTI systems 110 129 338 discretetime LTI systems 158 176 337 System representation 14 System response 273 Telegraph signal 427 428 Testing function 7 3dB bandwidth 209 Time averages 399 Timeaveraged autocorrelation 399 mean 399 Time convolution theorem 201 231 Time delay 205 Timeinvariance 55 Timeinvariant systems 16 Time reversal 118 155 200 268 277 Time scaling 107 200 268 Time shifting 106 154 200 267 276 Timevarying systems 16 Total probability 463 Transfer function 111 Transform circuits 114 Transforms see Fourier Laplace etc Twodimensional rv 468 Twosided signal 104 Uniform distribution 467 Uniform sampling theorem 254 Unilateral Laplace transform 101 134 Unilateral ztransform 149 184 Union 459 Unitadvance operator 154 Unit circle 150 Unitdelay operator 154 Unitdelay element 41 Unit impulse function 6 Unit impulse sequence 11 Unit ramp function 40 Unit sample response 56 See also Impulse response Unit sample sequence see Unit impulse sequence Unit step function 6 33 Unit step sequence 11 Variance 474 Vector mean 400 Venn diagram 459 White noise 419 bandlimited 420 zplane 150 ztransform bilateral or twosided 148 definition 148 inverse 156 properties 153 166 region of convergence ROC 149 tables 153 156 unilateral or onesided 149 184 Zeroinput response 55 Zero padding 276 Zerostate response 55 Zeros 103 Index 514 12HsuSignalIndex 91619 358 PM Page 514 To Access Your Online andor Mobile Prep Course ON YOUR SMARTPHONE ON YOUR DESKTOPLAPTOP Just go to the app store search for Schaums download the app and start studying Type in Schaumscom click through to your desired title and start studying 1 Go to wwwschaumscom 2 Click on the banner to launch the mobile web app 3 Find your desired discipline and click through to your title 4 Once you have opened your title the videos or audio content will open on the main Dashboard page Book Online Mobile YOU ARE ALL SET STEPS TO ACCESS EASY OUTLINE DIAGNOSTIC TEST 1 Go to Schaumscom 2 Search for your books title and then click on your books cover 3 Scroll down and click on the Downloads Resources tab 4 Click on the bullet labeled Online Diagnostic Test Hsu 126045424XSchaums Outline of Signals and Systems 4eFINALindd 2 83019 953 AM