Table Of Content

Signals and Systems Signals and Systems THIRD EDITION A. ANAND KUMAR Principal College of Engineering K.L. University Green Fields, Vaddeswaram Guntur District Andhra Pradesh PHI Learning (iivctic Limited Delhi-110092 2013 SIGNALS AND SYSTEMS, Third Edition A. Anand Kumar © 2018 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-4840-0 The export rights of this book are vested solely with the publisher. Fourth Printing (Third Edition) i : oe September 2013 Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Mohan Makhijani at Rekha Printers Private Limited, New Delhi-110020. Contents Preface ... Symbols, Notations and Abbreviations 1. SIGNALS... Wel, TPO MUCHO tiseccccscsscseernrcesnerecie 1.2 Representation of Discrete-time Signal: 1.2.1 Graphical Representation 1.2.2 Functional Representation ........0.....0... 1.2.3. Tabular Representation .. 1.2.4 Sequence Representation 1.3 Elementary Signals ....c.ccccccccc 1.3.1 Unit Step Function 1.3.2 Unit Ramp Function ...... 1.3.3. Unit Parabolic Function 1.3.4 Unit Impulse Function 1.3.5 Sinusoidal Signal ..... 1.3.6 Real Exponential Signal .... , 1.3.7 Complex Exponential Signal...... 1.3.8 Rectangular Pulse Function 1.3.9 Triangular Pulse Function . 1. 1 3.10 Signum Function......... 3.11 Sine Function ... 1.3.12 Gaussian Function . 1.4 Basic Operations on Signal: 1.4.1 Time Shifting .. 1.4.2 Time Reversal . viii | Contents 1.4.3. Amplitude Scaling ......... iT 1.4.4 Time Scalin, 1.4.5 Signal Addition 1.4.6 Signal Multiplication. s 1.5 Classification of Signals .. 1.5.1 Deterministic and Random Signals 1.5.2 Periodic and Non-periodic Signals .. 1.5.3 Energy and Power Signals .. 1.5.4 Causal and Non-causal Signals 1.5.5 Even and Odd Signals .........000..00.++ MatLab Programs Short Questions with Answe: Review Questions . Fill in the Blank. Objective Type Questions Problems .. 2. SYSTEMS «+ 113-170 2.1 Introduction 113 2.2 Classification of Systems - 113 2.2.1 Lumped Parameter and Distributed Parameter Systems 115 2.2.2 Static and Dynamic Systems -....::cceccee 115 2.2.3. Causal and Non-causal Systems sx: 116 2.2.4 Linear and Non-linear System: . 118 2.2.5 Time-invariant and Time-varying Systema 125 2.2.6 Stable and Unstable Systems .. 7 . 138 2.2.7 Invertible and Non-invertible Systems . fs 156 2.2.8 FIR and IIR Systems............ MatLab Programs Short Questions with Answers Review Question Fill in the Blanks . Objective Type Questions Problems 3. SIGNAL ANALYSIS ..... +. 171-202 3.1 Introduction .........+ 171 3.2 Analogy between Vectors and Signals 171 3.3 Graphical Evaluation of a Component of one Function in the Other ss... 180 Orthogonal Vector Space ......... vont 181 Orthogonal Signal Space 0... . 182 3.5.1 Approximation of a Function by a 2 Set ‘of Matwaly Orthogonal Functions 183 3.5.2 Evaluation of Mean Square Error ............. mavossemcees UBD. 3.5.3. Representation of a Function by a Closed ora Complete ‘Set St Mutually Orthogonal Functions .......0..cscccccsc cece ieceeseeeesseeeesees 185 Contents| ix 3.6 Orthogonality in Complex Functions ......::000 e 187 MatLab Programs ... 193 Short Questions with Answers 197 Review Question: 200 Fill in the Blank 200 Objective Type Questions 201 Problems 202 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALG............ 203-295 4.1 Introduction ... 4.2 Representation of Fourier Series 4.3. Existence of Fourier Series ........ 4.4 Trigonometric Form of Fourier Series ....... : : seve 4.4.1 Evaluation of Fourier Coefficients of the Trigonometric Fourier Series ..... 205 4.5 Cosine Representation (Alternate Form of the Trigonometric Representation) . 4.6 Wave Symmetry .. . 46:1 ven or Mirror Swinney 4.6.2 Odd or Rotation Symmetry 4.6.3. Half Wave Symmetry 4.6.4 Quarter Wave Symmetry 4.7 Exponential Fourier Series ......... 4.7.1 Determination of the Coefficients SE “BRpoASALAl Fourier ‘S888 4.7.2 Trigonometric Fourier Series from Exponential Fourier Series 4.7.3. Exponential Fourier Series from Trigonometric Fourier Series.. 4.7.4 Cosine Fourier Series from Exponential Fourier Series ... 4.8 Fourier Spectrum .. 4.9 Power Representation Using the Fourier Series . 4.10 Gibbs Phenomenon. 4.11 Properties of Continuous: Time "Fourier Seaies 4.11.1 Linearity Property .......0..0.0c:cc cee 4.11.2 Time Shifting Property .. 4.11.3 Time Reversal Property. 4.11.4 Time Scaling Property....... 4.11.5 Time Differentiation Propert 4.11.6 Time Integration Property , 4.11.7 Convolution Theorem or Property... : 4.11.8 Modulation or Multiplication Property ..... 4.11.9 Conjugation and Conjugate Symmetry Property . 4.11.10 Parseval’s Relation or Theorem or Property ........ MatLab Programs... Short Questions with Answe: Review Questions Fill in the Blank: Objective Type Questions Problems ..... x | Contents 5. FOURIER TRANSFORMS 5.1 Introduction ......... . 5.2 Fourier Transform Representation of Non-periodic Functions... + 296-409 we 296 warren 2D: $:2:1 Derivation of the’ Fontier Transform 6f a Nonpérodic Signal front the Fourier Series of a Periodic Signal ....... Magnitude and Phase Representation of Fourier Transform Existence of Fourier Transforms... Fourier Transforms of Standard Signals 5.5.1 Impulse Function 6(f) .. Single-sided Real Exponential Function ¢ Double-sided Real Exponential Function ¢ Complex Exponential Function ¢/"... Constant Amplitude (1) . Signum Function sgn (1) . Unit Step Function u(0) ..... Rectangular Pulse (Gate pulse) IT (it) or rect (it) Triangular Pulse A(t/t) .. Cosine Wave cos Wot. 5.5.11 Sine Wave sin @ot .. 5.6 Properties of Continuous Time Fourier Transform 5.6.1 Linearity Property ... 5.6.2 Time Shifting Bropeity 5.6.3 Frequency Shifting Property (Multipli 5.6.4 Time Reversal Property . 5.6.5 Time Scaling Property... . 5.6.6 Differentiation in Time Domain Property............. . s 5.6.7 Differentiation in Frequency Domain Property . 316 5.6.8 Time Integration Property .........- 5.6.9 Convolution Property or Theorem. 5.6.10 Multiplication Property or Theorem... 5.6.11 Duality (Symmetry) Property... 5.6.12 Modulation Property ... 5.6.13 Conjugation Property .. 5.6.14 Autocorrelation Property .. ss 536.15 Parieval’s’Rélation of Theotém or Property ee) uk U() alt] tion by an Exponential). 313 5.6.16 Area under the Curve 323 5.6.17 Fourier Transform of Complex and Real Functions... . 323 5.7 Fourier Transform of a Periodic Signal . . 325 5.8 System Analysis with Fourier Transform . 371 5.9 Introduction to Hilbert Transform.. . 380 MatLab Programs 3 Short Questions with Answers Review Question. Fill in the Blank. Objective Type Question Problems Contents | xi 6. Pa SIGNAL TRANSMISSION THROUGH LINEAR SYSTEMS 6.1 Introduction .. 6.2 Systems 6.3 Properties of Linear Time Invariant Systems . 6.3.1 The Commutative Property 6.3.2 The Distributive Property . 6.3.3 The Associative Property ... “ 6.3.4 Systems with and without Memory .. 6.3.5 Causality ...... 6.3.6 Stability .......... 6.3.7 Invertibility ... 6.3.8 The Unit Step Response... 6.4 Transfer Function of an LTI System 6.5 Filter Characteristics of Linear Systems....... 6.6 Distortionless Transmission through a System 6.7 Signal Bandwidth .... 6.8 System Bandwidth. 6.9 Ideal Filter Characteristics .. — 421 6.10 Causality and Paley-Wiener Criterion for Phy 1 Realization . z . 424 6.11 Relationship between Bandwidth and Rise Time ......0...0.:0.cccccceceseeesteeeeesereees 424 447 451 451 -» 410-456 we 410 . 410 413 413 . 413 414 .414 . 414 415 415. 415 416 417 418 420 . 421 Short Questions with Answers Review Questions . Fill in the Blanks . Objective Type Questions 453 Problems... 1. 454 CONVOLUTION AND CORRELATION OF SIGNALS 457-540 457 7.1 Introduction ............ o - 457 7.2 Concept of Convolution .. . 7.3 Properties of Convolution... .. 459 7.4 Convolution Theorems ......:.... .. 463 - 463 . 464 . 466 483 7.4.1 Time Convolution Theorem. 7.4.2 Frequency Convolution Theorem .... 7.5 Graphical Procedure to Perform Convolution. 7.6 Signal Comparison: Correlation of Function: 7.6.1 Cross Correlation . . 485 7.6.2 Autocorrelation ... . 488 7.7 Energy Density Spectrum 491 7.8 Power Density Spectrum ......... .. 493 7.9 Relation between Autocorrelation Function and Energy/Power Spectra Density Function .. eee 497 7.9.1 Relation between ESD and Autocorrelation Function R()... .. 497 7.9.2 Relation between Autocorrelation Function R(t) and Power Spectral Density (PSD) .. 7.10 Relation between Convolution and Correlation : . 498 498 xii | Contents 7.11 Detection of Periodic Signals in the Presence of Noise by Correlation .................. 516 7A1.1 Detection by Autocorrelation .... 7.11.2 Detection by Cross Correlation 7.12 Extraction of a Signal from Noise by Filtering MatLab Programs Short Questions with Review Questions . Fill in the Blank: Objective Type Questions Problems ...... SAMPLING...... 8.1 Introduction 8.2 Sampling. 8.3 Sampling Theorem 8.4 Nyquist Rate of Sampling . 8.5 Effects of under Sampling—Ali 8.6 Anti-Aliasing Filter... 8.7 Sampling Techniques .... . 8.7.1 Ideal or Impulse Sampling .. 8.7.2 Natural Sampling ..... 8.7.3. Flat Top Sampling .. 8.8 Data Reconstruction ..... 8.8.1 Ideal Reconstruction Filter 8.8.2 Zero Order Hold ... c 8.8.3 Transfer Function of a Zero Order Hold. 8.9 Sampling of Band Pass Signals . MatLab Programs Short Questions Review Questions Fill in the Blank. Objective Type Questions Problems .. ith Answers LAPLACE TRANSFORNMS .... 9.1 Introduction ..........cccee 9.2 Region of Convergence . 9.3 Existence of Laplace Transform 9.4 Advantages and Limitations of Laplace Transform . 9.5 Relation between Fourier Transform and Laplace Transform. 9.6 One-sided (Unilateral) Laplace Transform of Some Commonly Used Signal 9.6.1 Impulse Function [x() = 5()] 9.6.2 Step Function [x(f) = u(t)] .. 9.6.3 Ramp Function [x(‘) = tu(t)] 9.6.4 Parabolic Function [x()) = 7 u()). Contents | _ xiii o7 98 9.9 9.10 911 9.12 9.13 MatLab Programs Short Questions with Answers Review Questions Fill in the Blank: Objective Type Questions Problems ..... 9.6.5 Real Exponential Function [x(t) = e“ u()] .. 598 9.6.6 Complex Exponential Function [x() = e/° uo). . 599 9.6.7 Sine and Cosine Functions [x(t) = cos wt u(t), sin rt ot ud] 599 9.6.8 Hyperbolic sine and cosine Functions [x(f) = sinh @f u(f), cosh wt nf (DI . 600 9.6.9 Damped sine and cosine Functions [x(/) = e“ sin of u(d)] .. 9.6.10 Damped Hyperbolic sine and cosine Functions [x(1) = e~" Properties and Theorems of Laplace Transform ... 9.7.1 Linearity Property . 9.7.2 Time Shift Property .... 9.7.3. Time Scaling Property... 9.7.4 Time Reversal Property. 9.7.5 Transform of Derivatives Property .. 9.7.6 Transform of Integrals Property ....... 9.7.7 Differentiation in s-domain Property .. sinh of uy] . 601 9.7.8 Frequency Shift Property . . 622 9.7.9 Time Convolution Property 622 9.7.10 Multiplication or Modulation or Convolution in s . 623 9.7.11 Conjugation and Conjugate Symmetry Property ..... 9.7.12 Parseval’s Relation or Theorem or Property .. 9.7.13 Initial Value Theorem ... 9.7.14 Final Value Theorem........... a 9.7.15 Time Periodicity Property (Laplac é Transform of Periodie Functions) Inversion of Unilateral Laplace Transform ..... 9.8.1 Distinct Poles 9.8.2 Multiple Poles . 9.8.3 Complex Roots Inversion of Bilateral Laplace Transform . ROCs for Various Classes of Signals .... 9.10.1 Right-sided Signals . 9.10.2 Left-sided Signals .. 9.10.3 Two-sided Signals . 9.10.4 Finite Duration Signals 9.10.5 Properties of ROC .. Solution of Differential Equations Using Laplace “Transform . Waveform Synthesis Circuit Analysis Using Laplace Transforms 9.13.1 Step and Impulse Responses of Series R-L Circuit. 9.13.2. Step and Impulse Responses of Series R-C Circuit 9.13.3 Step Response of Series R-L-C Circuit 0.0.0.0...