Signals and Systems
SIGNALS AND SYSTEMS I. RAVI KUMAR Head Department of Electronics and Communication Engineering Sree Visvesvaraya Institute of Technology and Science Mahabubnagar, Andhra Pradesh New Delhi-110001 2009
SIGNALS AND SYSTEMS I. Ravi Kumar 2009 by PHI Learning Private Limited, New 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-978-81-203-3851-7 The export rights of this book are vested solely with the publisher. Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Baba Barkha Nath Printers, Bahadurgarh, Haryana-124507.
To my beloved students and my gurus
Contents Preface Acknowledgements xiii xv 1. SIGNALS ANALYSIS 1 60 1.1 Introduction 1 1.2 Classification of Signals 1 1.3 Continuous-Time and Discrete-Time Signals 2 1.3.1 Representation of Discrete-Time Signals 2 1.4 Analog and Digital Signals 3 1.5 Real and Complex Signals 3 1.6 Deterministic and Random Signals 4 1.7 Even and Odd Signals 4 1.8 Periodic and Non-periodic Signals 5 1.9 Energy and Power Signals 7 1.10 Elementary Signals 8 1.10.1 Exponential Signals 8 1.10.2 Periodicity of Discrete-Time Complex Exponential Signal 9 1.10.3 Sinusoidal Signals 10 1.10.4 Relation between Sinusoidal and Complex Exponential Signals 11 1.10.5 Exponentially Damped Sinusoidal Signals 11 1.10.6 Unit Step Function 12 1.10.7 Unit Ramp Function 13 1.10.8 Unit Impulse Function (Dirac Delta Function) 14 1.10.9 Signum Function 15 Important Formulae 16 Solved Problems 20 Objective Type Questions 55 Review Questions 59 vii
viii Contents 2. VECTOR SPACE CONCEPTS 61 81 2.1 Vector Space 61 2.2 Axioms of Vector Space 62 2.3 Concept of Linear Independence 62 2.4 Basis and Dimension 63 2.5 Orthogonal Vector Space 63 2.6 Inner Product 64 2.6.1 Axioms of Inner Product 64 2.7 Norm 64 2.7.1 Properties of Norm 65 Important Formulae 65 Solved Problems 67 Objective Type Questions 78 Review Questions 80 3. SIGNAL SPACE CONCEPTS 82 110 3.1 Analogy between Vectors and Signals 82 3.2 Orthogonal Signal Space 83 3.3 Signal Approximation Using Orthogonal Functions 83 3.4 Mean Square Error (MSE) 85 3.5 Closed or Complete Set of Orthogonal Functions 86 3.6 Orthogonality in Complex Functions 87 3.7 Gram Schmidt Procedure 88 Important Formulae 90 Solved Problems 92 Objective Type Questions 108 Review Questions 109 4. FOURIER SERIES 111 166 4.1 Introduction 111 4.2 Fourier Series Dirichlet s Conditions 111 4.2.1 Fourier Series 111 4.2.2 Dirichlet s Conditons 112 4.3 Trigonometric (Sinusoidal) Fourier Series 114 4.4 Complex Exponential Fourier Series 115 4.5 Relation between Trigonometric and Complex Fourier Series 115 4.6 Concept of Negative Frequency 116 4.7 Representation of Fourier Series of Continuous-Time Periodic Signals 116 4.8 Complex Fourier Spectrum 117 4.9 Representation of Arbitrary Function 118 4.10 Properties of Fourier Series 118 4.11 Gibb s Phenomenon 120
Contents ix Important Formulae 120 Solved Problems 122 Objective Type Questions 160 Review Questions 164 5. FOURIER TRANSFORMS 167 242 5.1 Introduction 167 5.2 Deriving Fourier Transform from Fourier Series 167 5.3 Properties of Fourier Transforms 170 5.4 Fourier Transforms of Standard Signals 176 5.4.1 Single-sided Exponential Function: e at u(t). 176 5.4.2 Double-sided Exponential Function: e a t 177 5.4.3 Ê t ˆ ÏA, t < t /2 Gate Function: p Á = Ì ËT Ó0, t < t /2 179 5.5 Unit Impulse Function 180 5.5.1 Properties of Impulse Functions 180 5.5.2 Shifting Property/Sampling Property 181 5.6 Signum Function 181 5.7 Fourier Transforms Involving Impulse Function and Signum Function 182 5.7.1 Fourier Transform of Impulse Function: d (t) 182 5.7.2 Fourier Transform of Constant Function 183 5.7.3 Fourier Transform of Signum Function 184 5.7.4 Fourier Transform of Unit Step Function 185 5.8 Fourier Transform of Periodic Functions 186 5.9 Introduction to Hilbert Transform 187 5.9.1 Properties of Hilbert Transform 188 5.9.2 Applications of Hilbert Transform 189 Important Formulae 190 Solved Problems 192 Objective Type Questions 229 Review Questions 236 6. SIGNAL TRANSMISSION THROUGH LINEAR SYSTEMS 243 290 6.1 Introduction 243 6.2 Systems 243 6.3 Classification of Systems 244 6.3.1 Linear and Non-linear Systems 244 6.3.2 Time-Invariant and Time-Varying Systems 244 6.3.3 Causal and Non-casual Systems 244 6.3.4 Static and Dynamic Systems 245 6.3.5 Stable and Unstable Systems 245 6.4 BIBO Stability Criterion 245 6.5 Linear Time-Invariant and Linear Time-Variant (LTI and LTV) Systems 246 6.5.1 Properties 246 6.6 Transfer Function of LTI System 248
x Contents 6.7 Unit Impulse Response of LTI System 249 6.8 Distortionless Transmission 250 6.9 Signal Bandwidth and System Bandwidth 251 6.10 Causality and Physical Realization 252 6.11 Paley Wiener Criterion 252 6.12 Filter Characteristics of Linear Systems 254 6.13 Ideal Filter Characteristics 254 6.14 Bandwidth and Rise Time 255 Important Formulae 258 Solved Problems 259 Objective Type Questions 286 Review Questions 288 7. CONVOLUTION AND CORRELATION OF SIGNALS 291 339 7.1 Introduction 291 7.2 Concept of Convolution 291 7.3 Convolution Theorems 292 7.3.1 Time Convolution Theorem 292 7.3.2 Frequency Convolution Theorem 293 7.4 Graphical Convolution 294 7.5 Energy Density Spectrum 294 7.6 Power Density Spectrum 296 7.7 Comparison of ESD and PSD 299 7.8 Cross-correlation 300 7.9 Cross-correlation of Energy and Power Signals 301 7.9.1 Cross-correlation of Energy Signals 301 7.9.2 Cross-correlation of Periodic or Power Signals 302 7.10 Autocorrelation 303 7.10.1 Autocorrelation for Energy Signals 303 7.10.2 Autocorrelation for Periodic Signals 304 7.11 Relation between Autocorrelation and Spectral Densities 305 7.12 Relation between Convolution and Correlation 307 7.13 Detection of Periodic Signals in Presence of Noise by Correlation 307 7.13.1 Detection by Autocorrelation 307 7.13.2 Detection by Cross-correlation 308 Important Formulae 308 Solved Problems 310 Objective Type Questions 334 Review Questions 337 8. SAMPLING THEORY 340 368 8.1 Introduction 340 8.2 Sampling Theorem 340 8.3 Nyquist Rate and Nyquist Interval 343 8.4 Reconstruction of Signal 344
Contents xi 8.5 Effects of Under Sampling Aliasing 345 8.6 Sampling of Band-Pass Signals 346 8.7 Sampling Techniques 347 8.8 Ideal or Instantaneous Sampling 348 8.9 Flat-Top Sampling 348 8.10 Natural Sampling 350 8.11 Comparison of Various Sampling Methods 351 Important Formulae 351 Solved Problems 352 Objective Type Questions 364 Review Questions 367 9. LAPLACE TRANSFORMS 369 465 9.1 Introduction 369 9.2 The Laplace Transform 369 9.2.1 Definition 369 9.2.2 Existence of Laplace Transform 370 9.2.3 Advantages of Laplace Transforms 370 9.2.4 Limitations of Laplace Transforms 370 9.2.5 Applications of Laplace Transforms 370 9.3 Relation between Laplace Transform and Fourier Transform 370 9.4 Concept of Region of Convergence (ROC) for Laplace Transforms 371 9.4.1 Region of Convergence (ROC) 371 9.4.2 Properties of ROC 371 9.5 Inverse Laplace Transform 372 9.5.1 Methods of Finding Inverse Laplace Transform 372 9.6 Properties of Laplace Transforms 373 9.6.1 Linearity (or Superposition) Property 373 9.6.2 Shifting in s-domain (First Translation Theorem) 374 9.6.3 Time Shifting Property (Second Translation Theorem) 374 9.6.4 Time-Scaling or Scale Change Property 375 9.6.5 Time Reversal Property 375 9.6.6 Differentiation in Time-domain 375 9.6.7 Differentiation in s-domain 376 9.6.8 Integration in Time-domain 377 9.6.9 Convolution Property 377 9.6.10 Conjugate Property 378 9.6.11 Initial Value Theorem 378 9.6.12 Final Value Theorem 378 9.7 Laplace Transform of Periodic Functions 379 9.8 Laplace Transform of Certain Signals Using Waveform Synthesis 380 9.9 Laplace Transform Solution to Differential Equations 381 Important Formulae 383 Solved Problems 385 Objective Type Questions 459 Review Questions 462
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