SIGNALS AND SYSTEMS I. RAVI KUMAR

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2 Signals and Systems

3 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

4 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 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 and Printed by Baba Barkha Nath Printers, Bahadurgarh, Haryana

5 To my beloved students and my gurus

6 Contents Preface Acknowledgements xiii xv 1. SIGNALS ANALYSIS Introduction Classification of Signals Continuous-Time and Discrete-Time Signals Representation of Discrete-Time Signals Analog and Digital Signals Real and Complex Signals Deterministic and Random Signals Even and Odd Signals Periodic and Non-periodic Signals Energy and Power Signals Elementary Signals Exponential Signals Periodicity of Discrete-Time Complex Exponential Signal Sinusoidal Signals Relation between Sinusoidal and Complex Exponential Signals Exponentially Damped Sinusoidal Signals Unit Step Function Unit Ramp Function Unit Impulse Function (Dirac Delta Function) Signum Function 15 Important Formulae 16 Solved Problems 20 Objective Type Questions 55 Review Questions 59 vii

7 viii Contents 2. VECTOR SPACE CONCEPTS Vector Space Axioms of Vector Space Concept of Linear Independence Basis and Dimension Orthogonal Vector Space Inner Product Axioms of Inner Product Norm Properties of Norm 65 Important Formulae 65 Solved Problems 67 Objective Type Questions 78 Review Questions SIGNAL SPACE CONCEPTS Analogy between Vectors and Signals Orthogonal Signal Space Signal Approximation Using Orthogonal Functions Mean Square Error (MSE) Closed or Complete Set of Orthogonal Functions Orthogonality in Complex Functions Gram Schmidt Procedure 88 Important Formulae 90 Solved Problems 92 Objective Type Questions 108 Review Questions FOURIER SERIES Introduction Fourier Series Dirichlet s Conditions Fourier Series Dirichlet s Conditons Trigonometric (Sinusoidal) Fourier Series Complex Exponential Fourier Series Relation between Trigonometric and Complex Fourier Series Concept of Negative Frequency Representation of Fourier Series of Continuous-Time Periodic Signals Complex Fourier Spectrum Representation of Arbitrary Function Properties of Fourier Series Gibb s Phenomenon 120

8 Contents ix Important Formulae 120 Solved Problems 122 Objective Type Questions 160 Review Questions FOURIER TRANSFORMS Introduction Deriving Fourier Transform from Fourier Series Properties of Fourier Transforms Fourier Transforms of Standard Signals Single-sided Exponential Function: e at u(t) Double-sided Exponential Function: e a t Ê t ˆ ÏA, t < t /2 Gate Function: p Á = Ì ËT Ó0, t < t / Unit Impulse Function Properties of Impulse Functions Shifting Property/Sampling Property Signum Function Fourier Transforms Involving Impulse Function and Signum Function Fourier Transform of Impulse Function: d (t) Fourier Transform of Constant Function Fourier Transform of Signum Function Fourier Transform of Unit Step Function Fourier Transform of Periodic Functions Introduction to Hilbert Transform Properties of Hilbert Transform Applications of Hilbert Transform 189 Important Formulae 190 Solved Problems 192 Objective Type Questions 229 Review Questions SIGNAL TRANSMISSION THROUGH LINEAR SYSTEMS Introduction Systems Classification of Systems Linear and Non-linear Systems Time-Invariant and Time-Varying Systems Causal and Non-casual Systems Static and Dynamic Systems Stable and Unstable Systems BIBO Stability Criterion Linear Time-Invariant and Linear Time-Variant (LTI and LTV) Systems Properties Transfer Function of LTI System 248

9 x Contents 6.7 Unit Impulse Response of LTI System Distortionless Transmission Signal Bandwidth and System Bandwidth Causality and Physical Realization Paley Wiener Criterion Filter Characteristics of Linear Systems Ideal Filter Characteristics Bandwidth and Rise Time 255 Important Formulae 258 Solved Problems 259 Objective Type Questions 286 Review Questions CONVOLUTION AND CORRELATION OF SIGNALS Introduction Concept of Convolution Convolution Theorems Time Convolution Theorem Frequency Convolution Theorem Graphical Convolution Energy Density Spectrum Power Density Spectrum Comparison of ESD and PSD Cross-correlation Cross-correlation of Energy and Power Signals Cross-correlation of Energy Signals Cross-correlation of Periodic or Power Signals Autocorrelation Autocorrelation for Energy Signals Autocorrelation for Periodic Signals Relation between Autocorrelation and Spectral Densities Relation between Convolution and Correlation Detection of Periodic Signals in Presence of Noise by Correlation Detection by Autocorrelation Detection by Cross-correlation 308 Important Formulae 308 Solved Problems 310 Objective Type Questions 334 Review Questions SAMPLING THEORY Introduction Sampling Theorem Nyquist Rate and Nyquist Interval Reconstruction of Signal 344

10 Contents xi 8.5 Effects of Under Sampling Aliasing Sampling of Band-Pass Signals Sampling Techniques Ideal or Instantaneous Sampling Flat-Top Sampling Natural Sampling Comparison of Various Sampling Methods 351 Important Formulae 351 Solved Problems 352 Objective Type Questions 364 Review Questions LAPLACE TRANSFORMS Introduction The Laplace Transform Definition Existence of Laplace Transform Advantages of Laplace Transforms Limitations of Laplace Transforms Applications of Laplace Transforms Relation between Laplace Transform and Fourier Transform Concept of Region of Convergence (ROC) for Laplace Transforms Region of Convergence (ROC) Properties of ROC Inverse Laplace Transform Methods of Finding Inverse Laplace Transform Properties of Laplace Transforms Linearity (or Superposition) Property Shifting in s-domain (First Translation Theorem) Time Shifting Property (Second Translation Theorem) Time-Scaling or Scale Change Property Time Reversal Property Differentiation in Time-domain Differentiation in s-domain Integration in Time-domain Convolution Property Conjugate Property Initial Value Theorem Final Value Theorem Laplace Transform of Periodic Functions Laplace Transform of Certain Signals Using Waveform Synthesis Laplace Transform Solution to Differential Equations 381 Important Formulae 383 Solved Problems 385 Objective Type Questions 459 Review Questions 462

11 Signals And Systems 25% OFF Publisher : PHI Learning ISBN : Author : KUMAR, I. RAVI Type the URL : 4 Get this ebook

Theory and Problems of Signals and Systems

SCHAUM'S OUTLINES OF Theory and Problems of Signals and Systems HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University