PRACTICAL SIGNAL PROCESSING AND ITS APPLICATIONS With Solved Homework Problems

PRACTICAL SIGNAL PROCESSING AND ITS APPLICATIONS With Solved Homework Problems

ADVANCED SERIES IN ELECTRICAL AND COMPUTER ENGINEERING Editors: W.-K. Chen (University of Illinois, Chicago, USA) Y.-F. Huang (University of Notre Dame, USA) The purpose of this series is to publish work of high quality by authors who are experts in their respective areas of electrical and computer engineering. Each volume contains the state-of-the-art coverage of a particular area, with emphasis throughout on practical applications. Sufficient introductory materials will ensure that a graduate and a professional engineer with some basic knowledge can benefit from it. Published: Vol. 20: Computational Methods with Applications in Bioinformatics Analysis edited by Jeffrey J. P. Tsai and Ka-Lok Ng Vol. 18: Broadband Matching: Theory and Implementations (Third Edition) by Wai-Kai Chen Vol. 17: Practical Signal Processing and Its Applications: With Solved Homework Problems by Sharad R Laxpati and Vladimir Goncharoff Vol. 16: Design Techniques for Integrated CMOS Class-D Audio Amplifiers by Adrian I. Colli-Menchi, Miguel A. Rojas-Gonzalez and Edgar Sanchez-Sinencio Vol. 15: Active Network Analysis: Feedback Amplifier Theory (Second Edition) by Wai-Kai Chen (University of Illinois, Chicago, USA) Vol. 14: Linear Parameter-Varying System Identification: New Developments and Trends by Paulo Lopes dos Santos, Teresa Paula Azevedo Perdicoúlis, Carlo Novara, Jose A. Ramos and Daniel E. Rivera Vol. 13: Semiconductor Manufacturing Technology by C. S. Yoo Vol. 12: Protocol Conformance Testing Using Unique Input/Output Sequences by X. Sun, C. Feng, Y. Shen and F. Lombardi Vol. 11: Systems and Control: An Introduction to Linear, Sampled and Nonlinear Systems by T. Dougherty Vol. 10: Introduction to High Power Pulse Technology by S. T. Pai and Q. Zhang For the complete list of titles in this series, please visit http://www.worldscientific.com/series/asece

Advanced Series in Electrical and Computer Engineering Vol. 17 PRACTICAL SIGNAL PROCESSING AND ITS APPLICATIONS With Solved Homework Problems Sharad R Laxpati Vladimir Goncharoff University of Illinois at Chicago, USA World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Laxpati, S. R., author. Goncharoff, Vladimir, author. Title: Practical signal processing and its applications : with solved homework problems / by Sharad R. Laxpati (University of Illinois at Chicago, USA), Vladimir Goncharoff (University of Illinois at Chicago, USA). Description: [Hackensack] New Jersey : World Scientific, [2017] Series: Advanced series in electrical and computer engineering ; volume 17 Identifiers: LCCN 2017036466 ISBN 9789813224025 (hc : alk. paper) Subjects: LCSH: Signal processing--textbooks. Classification: LCC TK5102.9.L39 2017 DDC 621.382/2--dc23 LC record available at https://lccn.loc.gov/2017036466 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit http://www.worldscientific.com/worldscibooks/10.1142/10551#t=suppl Desk Editor: Suraj Kumar Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore

Dedication We dedicate this work to our spouses, Maureen Laxpati and Marta Goncharoff, in sincere appreciation of their love and support.

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Preface The purpose of this book is two-fold: to emphasize the similarities in the mathematics of continuous and discrete signal processing, and as the title suggests to include practical applications of theory presented in each chapter. It is an enlargement of the notes we have developed over four decades while teaching the course Discrete and Continuous Signals & Systems at the University of Illinois at Chicago (UIC). The textbook is intended primarily for sophomore and junior-level students in electrical and computer engineering, but will also be useful to engineering professionals for its background theory and practical applications. Students in related majors at UIC may take this course, generally during their junior year, as a technical elective. Prerequisites are courses on differential equations and electrical circuits, but most students in other majors acquire sufficient background in introductory mathematics and physics courses. There is a plethora of texts on signal processing; some of them cover mostly analog signals, some mostly digital signals, and others include both digital and analog signals within each chapter or in separate chapters. We have found that we can give students a better understanding in less time by presenting analog and digital signal processing concepts in parallel (students like this approach). The mathematics of digital signal processing is not much different from the mathematics of analog signal processing: both require an understanding of signal transforms, the frequency domain, complex number algebra, and other useful operations. Thus, we wrote most chapters in this textbook to emphasize parallelism between analog and digital signal processing theories: there is a vii

viii Preface topic-by-topic, equation-by-equation match between digital/analog chapter pairs {2, 3}, {4, 5} and {9, 10}, and a somewhat looser correspondence between chapter pairs {7, 8} and {11, 12}. We hope that because of this textbook organization, even when reading only the analog or only the digital chapters of the textbook, readers will be able to quickly locate and understand the corresponding parallel-running descriptions in the other chapters. However, this textbook is designed to teach students all the material in Chapters 1-10 during a one-semester course. Sampling theory (Ch. 6) is presented at an early stage to explain the close relationship between continuous- and discrete-time domains. The Fourier series is introduced as the special case of Fourier transform operating on a periodic waveform, and the DFT is introduced as the special case of discrete-time Fourier transform operating on a periodic sequence; this is a more satisfactory approach in our opinion. Chapters {11, 12} provide useful applications of Z and Laplace transform analysis; as time permits, the instructor may include these when covering Chapters {9, 10}. To maintain an uninterrupted flow of concepts, we avoid laborious derivations without sacrificing mathematical rigor. Readers who desire mathematical details will find them in the footnotes and in cited reference texts. For those who wish to immediately apply what they have learned, plenty of MATLAB examples are given throughout. And, of course, students will appreciate the Appendix with its 100 pages of fullyworked-out homework problems. This textbook provides a fresh and different approach to a first course in signal processing at the undergraduate level. We believe its parallel continuous-time/discrete-time approach will help students understand and apply signal processing concepts in their further studies.

Preface ix Overview of material covered: Chapter 1: Overview of the goals, topics and tools of signal processing. Chapters 2, 3: Time domain signals and their building blocks, manipulation of signals with various time-domain operations, using these tools to create new signals. Chapters 4, 5: Fourier transform to the frequency domain and back to time domain, operations in one domain and their effect in the other, justification for using the frequency domain. Chapter 6: Relationship between discrete-time and continuoustime signals in both time and frequency domains; sampling and reconstruction of signals. Chapters 7, 8: Time and frequency analysis of linear systems, ideal and practical filtering. Chapters 9, 10: Generalization of the Fourier transform to the Z/Laplace transform, and justification for doing that. Chapters 11, 12: Useful applications of Z/S domain signal and system analysis. Appendix: Solved sample problems for material in each chapter. The flowchart in Fig. 1.4, p. 13, shows this textbook s organization of material that makes it possible to follow either discrete- or continuoustime signal processing, or follow each chapter in numerical sequence. We recommend the following schedule for teaching a 15-week semesterlong university ECE course on introductory signal processing:

x Preface Chapter # lectures 1 1 Introductory lecture Continuous-time 3 4 Continuous-time 5 6 Discrete-time 2 2 Discrete-time 4 4 both 6 4 Expand if necessary Continuous-time 8 5 Discrete-time 7 5 Continuous-time 10 (& 12) 5 Examples from Ch.12 as needed Discrete-time 9 (& 11) 5 Examples from Ch.11 as needed 41 lectures total We are indebted to our UIC faculty colleagues for their comments about and use of the manuscript in the classroom, and to the publisher s textbook reviewers. We also thank our many students who, over the years, have made teaching such a rewarding profession for us, with special thanks to those students who have offered their honest comments for improving this textbook s previous editions. Sharad R. Laxpati and Vladimir Goncharoff

Contents Dedication... v Preface... vii List of Tables... xxiii List of Figures... xxv Chapter 1: Introduction to Signal Processing... 1 1.1 Analog and Digital Signal Processing... 1 1.2 Signals and their Usefulness... 2 1.2.1 Radio communications... 2 1.2.2 Data storage... 2 1.2.3 Naturally-occurring signals... 2 1.2.4 Other signals... 3 1.3 Applications of Signal Processing... 3 1.4 Signal Processing: Practical Implementation... 4 1.5 Basic Signal Characteristics... 5 1.6 Complex Numbers... 7 1.6.1 Complex number math refresher... 7 1.6.2 Complex number operations in MATLAB... 8 1.6.3 Practical applications of complex numbers... 8 1.7 Textbook Organization... 12 1.8 Chapter Summary and Comments... 14 1.9 Homework Problems... 15 Chapter 2: Discrete-Time Signals and Operations... 17 2.1 Theory... 17 2.1.1 Introduction... 17 2.1.2 Basic discrete-time signals... 18 2.1.2.1 Impulse function... 18 2.1.2.2 Periodic impulse train... 19 2.1.2.3 Sinusoid... 20 xi

xii Contents 2.1.2.4 Complex exponential... 21 2.1.2.5 Unit step function... 21 2.1.2.6 Signum function... 21 2.1.2.7 Ramp function... 22 2.1.2.8 Rectangular pulse... 22 2.1.2.9 Triangular pulse... 23 2.1.2.10 Exponential decay... 23 2.1.2.11 Sinc function... 24 2.1.3 Signal properties... 25 2.1.3.1 Energy and power sequences... 25 2.1.3.2 Summable sequences... 26 2.1.3.3 Periodic sequences... 26 2.1.3.4 Sum of periodic sequences... 27 2.1.3.5 Even and odd sequences... 27 2.1.3.6 Right-sided and left-sided sequences... 28 2.1.3.7 Causal, anticausal sequences... 28 2.1.3.8 Finite-length and infinite-length sequences... 29 2.1.4 Signal operations... 29 2.1.4.1 Time shift... 29 2.1.4.2 Time reversal... 31 2.1.4.3 Time scaling... 31 2.1.4.4 Cumulative sum and backward difference... 32 2.1.4.5 Conjugate, magnitude and phase... 32 2.1.4.6 Equivalent signal expressions... 33 2.1.5 Discrete convolution... 34 2.1.5.1 Convolution with an impulse... 34 2.1.5.2 Convolution of two pulses... 35 2.1.6 Discrete-time cross-correlation... 36 2.2 Practical Applications... 37 2.2.1 Discrete convolution to calculate the coefficient values of a polynomial product... 37 2.2.2 Synthesizing a periodic signal using convolution... 38 2.2.3 Normalized cross-correlation... 39 2.2.4 Waveform smoothing by convolving with a pulse... 41 2.2.5 Discrete convolution to find the Binomial distribution... 42 2.3 Useful MATLAB Code... 43 2.3.1 Plotting a sequence... 43 2.3.2 Calculating power of a periodic sequence... 44 2.3.3 Discrete convolution... 44 2.3.4 Moving-average smoothing of a finite-length sequence... 44 2.3.5 Calculating energy of a finite-length sequence... 46

Contents xiii 2.3.6 Calculating the short-time energy of a finite-length sequence... 46 2.3.7 Cumulative sum and backward difference operations... 47 2.3.8 Calculating cross-correlation via convolution... 48 2.4 Chapter Summary and Comments... 49 2.5 Homework Problems... 50 Chapter 3: Continuous-Time Signals and Operations... 55 3.1 Theory... 55 3.1.1 Introduction... 55 3.1.2 Basic continuous-time signals... 56 3.1.2.1 Impulse function... 56 3.1.2.2 Periodic impulse train... 57 3.1.2.3 Sinusoid... 58 3.1.2.4 Complex exponential... 59 3.1.2.5 Unit step function... 59 3.1.2.6 Signum function... 59 3.1.2.7 Ramp function... 60 3.1.2.8 Rectangular pulse... 60 3.1.2.9 Triangular pulse... 61 3.1.2.10 Exponential decay... 61 3.1.2.11 Sinc function... 62 3.1.3 Signal properties... 62 3.1.3.1 Energy and power signals... 62 3.1.3.2 Integrable signals... 63 3.1.3.3 Periodic signals... 64 3.1.3.4 Sum of periodic signals... 64 3.1.3.5 Even and odd signals... 65 3.1.3.6 Right-sided and left-sided signals... 66 3.1.3.7 Causal, anticausal signals... 66 3.1.3.8 Finite-length and infinite-length signals... 66 3.1.4 Continuous-time signal operations... 67 3.1.4.1 Time delay... 67 3.1.4.2 Time reversal... 68 3.1.4.3 Time scaling... 68 3.1.4.4 Cumulative integral and time differential... 69 3.1.4.5 Conjugate, magnitude and phase... 70 3.1.4.6 Equivalent signal expressions... 70

xiv Contents 3.1.5 Convolution... 71 3.1.5.1 Convolution with an impulse... 71 3.1.5.2 Convolution of two pulses... 72 3.1.6 Cross-correlation... 72 3.2 Practical Applications... 74 3.2.1 Synthesizing a periodic signal using convolution... 74 3.2.2 Waveform smoothing by convolving with a pulse... 74 3.2.3 Practical analog cross-correlation... 75 3.2.4 Normalized cross-correlation as a measure of similarity... 76 3.2.5 Application of convolution to probability theory... 77 3.3 Useful MATLAB Code... 77 3.3.1 Plotting basic signals... 78 3.3.2 Estimating continuous-time convolution... 79 3.3.3 Estimating energy and power of a signal... 81 3.3.4 Detecting pulses using normalized correlation... 82 3.3.5 Plotting estimated probability density functions... 84 3.4 Chapter Summary and Comments... 86 3.5 Homework Problems... 87 Chapter 4: Frequency Analysis of Discrete-Time Signals... 93 4.1 Theory... 93 4.1.1 Discrete-Time Fourier Transform (DTFT)... 93 4.1.2 Fourier transforms of basic signals... 95 4.1.2.1 Exponentially decaying signal... 95 4.1.2.2 Constant value... 97 4.1.2.3 Impulse function... 98 4.1.2.4 Delayed impulse function... 98 4.1.2.5 Signum function... 98 4.1.2.6 Unit step function... 100 4.1.2.7 Complex exponential function... 100 4.1.2.8 Sinusoid... 101 4.1.2.9 Rectangular pulse function... 102 4.1.3 Fourier transform properties... 105 4.1.3.1 Linearity... 105 4.1.3.2 Time shifting... 107 4.1.3.3 Time/frequency duality... 108 4.1.3.4 Convolution... 108 4.1.3.5 Modulation... 110 4.1.3.6 Frequency shift... 112 4.1.3.7 Time scaling... 112 4.1.3.8 Parseval s Theorem... 114

Contents xv 4.1.4 Graphical representation of the Fourier transform... 118 4.1.4.1 Rectangular coordinates... 119 4.1.4.2 Polar coordinates... 120 4.1.4.3 Graphing the amplitude of F (e jω )... 121 4.1.4.4 Logarithmic scales and Bode plots... 122 4.1.5 Fourier transform of periodic sequences... 123 4.1.5.1 Comb function... 123 4.1.5.2 Periodic signals as convolution with a comb function... 124 4.1.5.3 Discrete Fourier Transform (DFT)... 126 4.1.5.4 Time-frequency duality of the DFT... 129 4.1.5.5 Fast Fourier Transform (FFT)... 130 4.1.5.6 Parseval s Theorem... 131 4.1.6 Summary of Fourier transformations for discrete-time signals... 133 4.2 Practical Applications... 134 4.2.1 Spectral analysis using the FFT... 134 4.2.1.1 Frequency resolution... 134 4.2.1.2 Periodic sequence... 135 4.2.1.3 Finite-length sequence... 137 4.2.2 Convolution using the FFT... 142 4.2.3 Autocorrelation using the FFT... 143 4.2.4 Discrete Cosine Transform (DCT)... 145 4.3 Useful MATLAB Code... 146 4.3.1 Plotting the spectrum of a discrete-time signal... 146 4.4 Chapter Summary and Comments... 152 4.5 Homework Problems... 154 Chapter 5: Frequency Analysis of Continuous-Time Signals... 157 5.1 Theory... 157 5.1.1 Fourier Transform... 157 5.1.2 Fourier transforms of basic signals... 159 5.1.2.1 Exponentially decaying signal... 159 5.1.2.2 Constant value... 160 5.1.2.3 Impulse function... 161 5.1.2.4 Delayed impulse function... 161 5.1.2.5 Signum function... 162 5.1.2.6 Unit step function... 163 5.1.2.7 Complex exponential function... 163 5.1.2.8 Sinusoid... 164 5.1.2.9 Rectangular pulse function... 165

xvi Contents 5.1.3 Fourier transform properties... 167 5.1.3.1 Linearity... 168 5.1.3.2 Time shifting... 169 5.1.3.3 Time/frequency duality... 170 5.1.3.4 Convolution... 171 5.1.3.5 Modulation... 172 5.1.3.6 Frequency shift... 174 5.1.3.7 Time scaling... 174 5.1.3.8 Parseval s Theorem... 176 5.1.4 Graphical representation of the Fourier transform... 179 5.1.4.1 Rectangular coordinates... 180 5.1.4.2 Polar coordinates... 181 5.1.4.3 Graphing the amplitude of F (ω )... 182 5.1.4.4 Logarithmic scales and Bode plots... 183 5.1.5 Fourier transform of periodic signals... 184 5.1.5.1 Comb function... 184 5.1.5.2 Periodic signals as convolution with a comb function... 185 5.1.5.3 Exponential Fourier Series... 187 5.1.5.4 Trigonometric Fourier Series... 190 5.1.5.5 Compact Trigonometric Fourier Series... 193 5.1.5.6 Parseval s Theorem... 193 5.1.6 Summary of Fourier transformations for continuous-time signals... 195 5.2 Practical Applications... 196 5.2.1 Frequency scale of a piano keyboard... 196 5.2.2 Frequency-domain loudspeaker measurement... 198 5.2.3 Effects of various time-domain operations on frequency magnitude and phase... 199 5.2.4 Communication by frequency shifting... 201 5.2.5 Spectral analysis using time windowing... 204 5.2.6 Representing an analog signal with frequency-domain samples... 207 5.3 Useful MATLAB Code... 208 5.4 Chapter Summary and Comments... 215 5.5 Homework Problems... 216 Chapter 6: Sampling Theory and Practice... 219 6.1 Theory... 219 6.1.1 Sampling a continuous-time signal... 219 6.1.2 Relation between CTFT and DTFT based on sampling... 221

Contents xvii 6.1.3 Recovering a continuous-time signal from its samples... 224 6.1.3.1 Filtering basics... 224 6.1.3.2 Frequency domain perspective... 224 6.1.3.3 Time domain perspective... 230 6.1.4 Oversampling to simplify reconstruction filtering... 234 6.1.5 Eliminating aliasing distortion... 235 6.1.5.1 Anti-alias post-filtering... 236 6.1.5.2 Anti-alias pre-filtering... 237 6.1.6 Sampling bandpass signals... 238 6.1.7 Approximate reconstruction of a continuous-time signal from its samples... 239 6.1.7.1 Zero-order hold method... 239 6.1.7.2 First-order hold method... 242 6.1.8 Digital-to-analog conversion... 245 6.1.9 Analog-to-digital conversion... 245 6.1.10 Amplitude quantization... 246 6.1.10.1 Definition... 246 6.1.10.2 Why quantize?... 248 6.1.10.3 Signal to quantization noise power ratio (SNR Q ).. 249 6.1.10.4 Non-uniform quantization... 253 6.2 Practical Applications... 255 6.2.1 Practical digital-to-analog conversion... 255 6.2.2 Practical analog-to-digital conversion... 256 6.2.2.1 Successive approximation ADC... 256 6.2.2.2 Logarithmic successive approximation ADC... 258 6.2.2.3 Flash ADC... 259 6.2.2.4 Delta-Sigma (ΔΣ) ADC... 260 6.2.3 Useful MATLAB Code... 261 6.2.3.1 Amplitude quantization... 261 6.3 Chapter Summary and Comments... 263 6.4 Homework Problems... 265 Chapter 7: Frequency Analysis of Discrete-Time Systems... 269 7.1 Theory... 269 7.1.1 Introduction... 269 7.1.2 Linear shift-invariant discrete-time system... 270 7.1.2.1 Impulse response... 270 7.1.2.2 Input/output relations... 273 7.1.3 Digital filtering concepts... 277 7.1.3.1 Ideal lowpass filter... 278 7.1.3.2 Ideal highpass filter... 279

xviii Contents 7.1.3.3 Ideal bandpass filter... 279 7.1.3.4 Ideal band-elimination filter... 280 7.1.4 Discrete-time filter networks... 282 7.1.4.1 Digital filter building blocks... 282 7.1.4.2 Linear difference equations... 283 7.1.4.3 Basic feedback network... 285 7.1.4.4 Generalized feedback network... 288 7.1.4.5 Generalized feed-forward network... 290 7.1.4.6 Combined feedback and feed-forward network... 293 7.2 Practical Applications... 295 7.2.1 First-order digital filters... 295 7.2.1.1 Lowpass filter... 297 7.2.1.2 Highpass filter... 301 7.2.2 Second-order digital filters... 303 7.2.2.1 Bandpass filter... 304 7.2.2.2 Notch filter... 306 7.2.2.3 Allpass filter... 307 7.2.3 Specialized digital filters... 308 7.2.3.1 Comb filter... 308 7.2.3.2 Linear-phase filter... 310 7.2.4 Interpolation and Decimation... 313 7.2.4.1 Interpolation by factor a... 313 7.2.4.2 Decimation by factor b... 315 7.2.5 Nyquist frequency response plot... 317 7.3 Useful MATLAB Code... 319 7.3.1 Plotting frequency response of filter described by a difference equation... 319 7.3.2 FIR filter design by windowing the ideal filter s impulse response... 321 7.3.3 FIR filter design by frequency sampling... 328 7.4 Chapter Summary and Comments... 330 7.5 Homework Problems... 332 Chapter 8: Frequency Analysis of Continuous-Time Systems... 335 8.1 Theory... 335 8.1.1 Introduction... 335 8.1.2 Linear Time-Invariant Continuous System... 336 8.1.2.1 Input/output relation... 338 8.1.2.2 Response to e jω 0t... 339

Contents xix 8.1.3 Ideal filters... 340 8.1.3.1 Ideal lowpass filter... 341 8.1.3.2 Ideal highpass filter... 341 8.1.3.3 Ideal bandpass filter... 342 8.1.3.4 Ideal band-elimination filter... 343 8.2 Practical Applications... 344 8.2.1 RLC circuit impedance analysis... 344 8.2.2 First order passive filter circuits... 346 8.2.2.1 Highpass filter... 346 8.2.3 Second order passive filter circuits... 350 8.2.3.1 Bandpass filter... 350 8.2.3.2 Band-elimination filter... 353 8.2.4 Active filter circuits... 353 8.2.4.1 Basic feedback network... 353 8.2.4.2 Operational amplifier... 355 8.2.4.3 Noninverting topology... 357 8.2.4.4 Inverting topology... 358 8.2.4.5 First-order active filter... 359 8.2.4.6 Second-order active filter... 361 8.3 Useful MATLAB Code... 362 8.3.1 Sallen-Key circuit frequency response plot... 362 8.3.2 Calculating and plotting impedance of a one-port network... 362 8.4 Chapter Summary and Comments... 365 8.5 Homework Problems... 367 Chapter 9: Z-Domain Signal Processing... 369 9.1 Theory... 369 9.1.1 Introduction... 369 9.1.2 The Z transform... 370 9.1.3 Region of convergence... 372 9.1.4 Z transforms of basic signals... 374 9.1.4.1 Exponentially decaying signal... 374 9.1.4.2 Impulse sequence... 375 9.1.4.3 Delayed impulse sequence... 375 9.1.4.4 Unit step sequence... 375 9.1.4.5 Causal complex exponential sequence... 376 9.1.4.6 Causal sinusoidal sequence... 376 9.1.4.7 Discrete ramp sequence... 377 9.1.5 Table of Z transforms... 378

xx Contents 9.1.6 Z transform properties... 381 9.1.6.1 Linearity... 381 9.1.6.2 Time shifting... 383 9.1.6.3 Convolution... 384 9.1.6.4 Time multiplication... 384 9.1.6.5 Conjugation... 385 9.1.6.6 Multiplication by n in the time domain... 385 9.1.6.7 Multiplication by a n in the time domain... 386 9.1.6.8 Backward difference... 386 9.1.6.9 Cumulative sum... 386 9.1.7 Table of Z transform properties... 387 9.1.8 Z transform of linear difference equations... 388 9.1.9 Inverse Z transform of rational functions... 389 9.1.9.1 Inverse Z transform yielding finite-length sequences... 389 9.1.9.2 Long division method... 390 9.1.9.3 Partial fraction expansion method... 391 9.2 Chapter Summary and Comments... 396 9.3 Homework Problems... 398 Chapter 10: S-Domain Signal Processing... 399 10.1 Theory... 399 10.1.1 Introduction... 399 10.1.2 Laplace transform... 400 10.1.3 Region of convergence... 402 10.1.4 Laplace transforms of basic signals... 404 10.1.4.1 Exponentially decaying signal... 404 10.1.4.2 Impulse function... 405 10.1.4.3 Delayed impulse function... 405 10.1.4.4 Unit step function... 406 10.1.4.5 Complex exponential function... 406 10.1.4.6 Sinusoid... 407 10.1.4.7 Ramp function... 408 10.1.5 Table of Laplace transforms... 408 10.1.6 Laplace transform properties... 410 10.1.6.1 Linearity... 411 10.1.6.2 Time shifting... 413 10.1.6.3 Frequency shifting duality... 414 10.1.6.4 Time scaling... 414 10.1.6.5 Convolution... 415 10.1.6.6 Time multiplication... 415

Contents xxi 10.1.6.7 Time differentiation... 417 10.1.6.8 Time integration... 418 10.1.7 Table of Laplace transform properties... 418 10.1.8 Inverse Laplace transform of rational functions... 420 10.1.8.1 Partial fraction expansion method... 420 10.2 Chapter Summary and Comments... 427 10.3 Homework Problems... 429 Chapter 11: Applications of Z-Domain Signal Processing... 431 11.1 Introduction... 431 11.2 Applications of Pole-Zero Analysis... 431 11.2.1 Poles and zeros of realizable systems... 432 11.2.2 Frequency response from H (z)... 432 11.2.3 Frequency response from pole/zero locations... 433 11.2.3.1 Magnitude response... 434 11.2.3.2 Phase response... 438 11.2.4 Effect on H (e jω ) of reciprocating a pole... 439 11.2.5 System stability... 441 11.2.5.1 Causal systems... 441 11.2.5.2 Anticausal systems... 441 11.2.5.3 Stabilizing an unstable causal system... 442 11.2.6 Pole-zero plots of basic digital filters... 443 11.2.6.1 Lowpass filter... 444 11.2.6.2 Highpass filter... 445 11.2.6.3 Bandpass digital filter... 446 11.2.6.4 Notch filter... 447 11.2.6.5 Comb filter... 448 11.2.6.6 Allpass filter (real pole and zero)... 449 11.2.6.7 Allpass filter (complex conjugate poles and zeros)... 450 11.2.7 Minimum-phase system... 451 11.2.8 Digital filter design based on analog prototypes... 451 11.2.8.1 Impulse-invariant transformation... 452 11.2.8.2 Bilinear transformation... 455 11.3 Chapter Summary and Comments... 458 11.4 Homework Problems... 460 Chapter 12: Applications of S-Domain Signal Processing... 463 12.1 Introduction... 463

xxii Contents 12.2 Linear System Analysis in the S-Domain... 464 12.2.1 Linear time-invariant continuous system... 464 12.2.2 Frequency response from H (s)... 466 12.3 Applications of Pole-Zero Analysis... 468 12.3.1 Poles and zeros of realizable systems... 469 12.3.2 Frequency response from pole/zero locations... 469 12.3.2.1 Magnitude response... 469 12.3.2.2 Phase response... 474 12.3.3 Effect on H (ω) of mirroring a pole about the jω axis... 475 12.3.4 System stability... 476 12.3.4.1 Causal systems... 477 12.3.4.2 Anticausal systems... 477 12.3.4.3 Stabilizing an unstable causal system... 477 12.3.5 Pole-zero plots of basic analog filters... 479 12.3.5.1 Lowpass filter... 480 12.3.5.2 Highpass filter... 481 12.3.5.3 Bandpass filter... 482 12.3.5.4 Notch (band-elimination) filter... 483 12.3.6 Minimum-phase system... 484 12.4 Circuit Analysis in the S-Domain... 484 12.4.1 Transient Circuit Analysis... 491 12.4.2 Passive ladder analysis using T matrices... 494 12.5 Solution of Linear Differential Equations... 498 12.6 Relation Between Transfer Function, Differential Equation, and State Equation... 501 12.6.1 Differential equation from H (s)... 501 12.6.2 State equations from H (s)... 502 12.7 Chapter Summary and Comments... 504 12.8 Homework Problems... 506 Appendix: Solved Homework Problems... 509 Bibliography... 605 Index... 607

List of Tables Table 4.1. Table of discrete-time Fourier transform pairs... 103 Table 4.2. Table of discrete-time Fourier transform properties... 115 Table 4.3. Summary of Fourier transformations for discrete-time signals.... 133 Table 5.1. Table of continuous-time Fourier transform pairs... 165 Table 5.2. Table of continuous-time Fourier transform properties... 177 Table 5.3. Summary of Fourier transformations for continuous-time signals.... 195 Table 8.1. Voltage-current characteristics of R,L,C components in time and frequency domains... 345 Table 9.1. Regions of convergence for the Z transforms of various types of sequences.... 373 Table 9.2. Table of Z transform pairs. (region of convergence is for a causal time signal)... 379 Table 9.3. Table of Z transform properties... 387 xxiii

xxiv List of Tables Table 10.1. Table of Laplace transform pairs. (region of convergence is for a causal time signal)... 409 Table 10.2. Table of Laplace transform properties... 418 Table 12.1. V-I characteristic of R, L, and C in time and s-domains... 486

List of Figures Fig. 1.1. Fig. 1.2 Shown is a 50-millisecond span of a continuous-time speech signal. Its nearly-periodic nature is the result of vocal cord vibrations during vowel sounds... 5 A discrete-time signal obtained by sampling a sine wave... 6 Fig. 1.3. Phasor diagram graphical solution for 2 cos 100t + 45 + 3 sin 100t 90 2.1 cos 100t + 138.... 12 Fig. 1.4. Fig. 2.1. Textbook chapter organization, showing the parallelism between discrete-time and continuous-time domains.... 13 Example of a sequence x(n) as a function of its index variable n... 17 Fig. 2.2. Impulse sequence (n)... 18 Fig. 2.3. Delayed impulse sequence (n 4)... 18 Fig. 2.4. Impulse train 4(n)... 19 Fig. 2.5. Impulse train 3(n)... 19 Fig. 2.6. Impulse train 2(n)... 20 Fig. 2.7. A sinusoidal sequence (A = 1, = 1, = /3)... 20 Fig. 2.8. Unit step function sequence u(n)... 21 Fig. 2.9. Signum function sequence sgn(n)... 22 xxv

xxvi List of Figures Fig. 2.10. Ramp function sequence r(n)... 22 Fig. 2.11. Rectangular pulse sequence rect 4 (n)... 23 Fig. 2.12. Triangular pulse sequence 5 (n)... 23 Fig. 2.13. Exponentially decaying sequence u(n) (0.8) n... 24 Fig. 2.14. Discrete-time sequence sinc (n)... 24 Fig. 2.15. Fig. 2.16. Pulse rect 4 (n + 1), a time-shifted version of sequence rect 4 (n)... 30 Pulse rect 4 n 2, a time-shifted version of sequence rect 4 (n)... 30 Fig. 2.17. Delayed impulse sequence δ(n 4)... 30 Fig. 2.18. Delayed exponentially decaying sequence... 30 Fig. 2.19. Sequence u ( n 2)... 31 Fig. 2.20. Rectangular pulse sequence x(n)... 40 Fig. 2.21. Fig. 2.22. Fig. 2.23. Signal y (n), composed of noise plus rectangular pulses at various delays and amplitudes... 40 Normalized cross-correlation C xy (n) between x(n) and y(n). Notice that rectangular pulses in y(n) (Fig. 2.20) were detected as peaks of the triangular pulses... 40 Short-time normalized cross-correlation STC xy (n) between x(n) and y (n). Rectangular pulses in y (n) were detected as locations where STC xy (n) 1... 41 Fig. 2.24. MATLAB plot of a Binomial(50, 0.5) distribution... 43

List of Figures xxvii Fig. 2.25. An example of using MATLAB s stem function to plot a sequence... 44 Fig. 2.26. A noisy sinusoidal sequence before smoothing... 45 Fig. 2.27. A noisy sinusoidal sequence after smoothing... 46 Fig. 2.28. Sequence x(n)... 47 Fig. 2.29. Calculated short-time energy of the sequence x(n) in Fig. 2.27... 47 Fig. 2.30. Autocorrelation of random noise... 49 Fig. 3.1. Impulse function δ(t )... 56 Fig. 3.2. Shifted impulse function δ(t + π)... 56 Fig. 3.3. Fig. 3.4. Multiplying δ(t t 0 ) by signal x(t ) gives the same product as does multiplying δ(t t 0 ) by the constant c = x (t 0 )... 57 Impulse train δ 4.2 (t ). (When not specified, assume each impulse area = 1.)... 58 Fig. 3.5. A sinusoidal signal (A = 1, ω = 1, θ = π /3)... 58 Fig. 3.6. Unit step function u(t )... 59 Fig. 3.7. Signum function sgn(t )... 60 Fig. 3.8. Ramp function r (t )... 60 Fig. 3.9. Rectangular pulse function rect(t )... 61 Fig. 3.10. Triangular pulse function Δ(t )... 61

xxviii List of Figures Fig. 3.11. Exponentially decaying signal u (t )e 0.223t... 61 Fig. 3.12. Function sinc(t )... 62 Fig. 3.13. Signal rect t 1, which is rect(t ) after 1-sec delay... 67 Fig. 3.14. Signal rect(t + 1/2), which is rect(t ) after 1/2-sec advance... 67 Fig. 3.15. Delayed impulse function δ(t 4)... 67 Fig. 3.16. Delayed exponentially decaying signal... 68 Fig. 3.17. Signal u (( t ) 2)... 68 Fig. 3.18. Triangular pulse function Δ(t ), before (dotted line) and after (solid line) smoothing via convolution with pulse 5rect(5t)... 75 Fig. 3.19. Practical analog cross-correlation technique... 76 Fig. 3.20. MATLAB plot of y(t ) = sin(2πt )... 78 Fig. 3.21. MATLAB plot of yt ( ) u t 1.5 rect( t/ 2) Δ t ut 1.5... 79 Fig. 3.22. MATLAB plot of 2rect(t) convolved with Δ t 1... 81 Fig. 3.23. Original signal y(t ) that is composed of three triangular pulses.... 83 Fig. 3.24. Triangular pulse x(t ) used for waveform matching... 83 Fig. 3.25. Signal z(t ) = y(t ) + noise added... 84 Fig. 3.26. Normalized cross-correlation result C xz (t ), showing locations and polarities of triangular pulses that were detected in the noise waveform z(t )... 84

List of Figures xxix Fig. 3.27. Estimated PDF of r.v. X... 86 Fig. 3.28. Estimated PDF of r.v. Y... 86 Fig. 3.29. Estimated PDF of random variable Z = X + Y, demonstrating the fact that fz a fx a * fy a... 86 Fig. 4.1. Sequence x(n) to be transformed to the frequency domain in Example 4.1... 106 Fig. 4.2. From Example 4.1: F{x(n)} = X(e jω ) = 2 + 4 7 cos(kω)+ 14 k 1 k 8 cos(kω )... 107 Fig. 4.3. The spectrum X(e jω ) = F{x(n)} in Example 4.2... 110 10 Fig. 4.4. Plot of F {rect 10 (n)} = 1 + 2 k 1 cos(kω )... 111 Fig. 4.5. Plot of F rect 10 (n) cos (π 6)n =1+ 10 k 1 cos k ω π/6 + cos k(ω + π/6)... 112 Fig. 4.6. A graph of 7{sinc(3.5ω) * δ π (ω)}e j100ω vs. ω/π... 119 Fig. 4.7. A 3-D graph of complex-valued F(e jω )... 119 Fig. 4.8. Re{F(e jω )} vs. ω, corresponding to Fig. 4.7... 120 Fig. 4.9. Im{F(e jω )} vs. ω, corresponding to Fig. 4.7... 120 Fig. 4.10. A graph of (1 + 2 2 k 1 cos(kω) )e jω vs. ω... 121 Fig. 4.11. A graph of { 1 + 2 k cos(kω) e jω } vs. ω... 121 Fig. 4.12. A graph of 1 + 2 2 k 1 cos(kω) vs. ω... 122 Fig. 4.13. A graph of 1 + 2 2 k 1 cos(kω) vs. ω... 122 Fig. 4.14. Impulse train δ 4 (n)... 124

xxx List of Figures Fig. 4.15. Impulse train (π /2)δ π /2 (ω) = F {δ 4 (n)}... 124 Fig. 4.16. Rectangular pulse rect 5 (n )... 125 Fig. 4.17. Impulse train δ 20 (n )... 125 Fig. 4.18. Periodic signal f p (n ) = rect 5 (n ) * δ 20 (n )... 125 Fig. 4.19. Discrete Fourier Transform spectrum for periodic signal f p (n ) = rect (n ) * δ 20 (n ) = (1 20) 19 F p (k) j(k 2π/ 20)n k 0 e... 129 Fig. 4.20. A plot of periodic discrete-time sequence x(n ) = cos(2πn/10)... 136 Fig. 4.21. Fig. 4.22. Fig. 4.23. Fig. 4.24. Fig. 4.25. A plot of the spectrum of x(n ) =cos(2πn/10), calculated using the Fast Fourier Transform (FFT)... 137 Samples of X(e jω ) = F {x(n)} found using the FFT method, when x(n ) = {0.0975, 0.2785, 0.5469, 0.9575, 0.9649} for 0 n 4 and x(n ) = 0 elsewhere... 139 Samples of X(e jω ) found using the FFT method for the same x(n ) as in Fig. 4.22, this time zero-padding with 100 zeros prior to taking the FFT... 140 Samples of X(e jω ) = F {x(n)} found using the FFT method, when x(n) = {1, 1, 1, 1, 1} for 0 n 4 and x(n ) = 0 elsewhere... 141 Samples of X (e jω ) found using the FFT method for the same x(n) as in Fig. 4.24, this time zero-padding with 100 zeros prior to taking the FFT... 141 Fig. 4.26. Plot of phase spectrum H(e jω ) from Example 4.8... 147 Fig. 4.27. Plot of magnitude spectrum H(e jω ) from Example 4.8... 147

List of Figures xxxi Fig. 4.28. Plot of magnitude spectrum H(e jω ) from Example 4.9... 148 Fig. 4.29. Plot of phase spectrum H(e jω ) from Example 4.9... 148 Fig. 4.30. Plot of magnitude spectrum H(e jω ) from Example 4.10... 149 Fig. 4.31. Plot of phase spectrum H(e jω ) from Example 4.10... 150 Fig. 4.32. Fig. 4.33. Fig. 5.1. Plot of magnitude spectrum H(e jω ) from Example 4.11... 151 Plot of magnitude spectrum H(e jω ) 2 in db, from Example 4.12... 152 Sequence x(t ) to be transformed to the frequency domain in Example 5.1... 168 Fig. 5.2. From Example 5.1: F {x(t )} =X(ω) = 2sinc(2ω )+ 3sinc(ω )... 169 Fig. 5.3. Fig. 5.4. Fig. 5.5. The spectrum X(ω) = 2 sinc(ω) rect(ω 2π ) in Example 5.3... 172 Plot of F {rect(t 3π)} = 3π sinc(ω3π 2), from Example 5.4... 173 Plot of F {rect(t 3π)cos(4πt)} = (3π 2)sinc (ω 4π)3π 2 +(3π 2)sinc((ω + 4π)3π 2), from Example 5.4... 174 Fig. 5.6. A graph of sinc(ω/2)e j20ω vs. ω/π... 180 Fig. 5.7. A 3-D graph of complex-valued F(ω)... 180 Fig. 5.8. Re{F(ω)} vs. ω, corresponding to Fig. 5.7... 181 Fig. 5.9. Im{F(ω)}vs. ω, corresponding to Fig. 5.7... 181

xxxii List of Figures Fig. 5.10. A graph of sinc(ω/2)e jω/5 vs. ω... 182 Fig. 5.11. A graph of {sinc(ω/2)e jω/5 } vs. ω... 182 Fig. 5.12. A graph of sinc(ω/2) vs. ω... 183 Fig. 5.13. A graph of sinc(ω/2) vs. ω... 183 Fig. 5.14. Impulse train δ 4 (t )... 185 Fig. 5.15. Impulse train (π/2)δ π /2 (ω ) = F {δ 4 (t )}... 185 Fig. 5.16. Rectangular pulse rect(t )... 186 Fig. 5.17. Impulse train δ 2 (t )... 186 Fig. 5.18. Periodic signal f p (t ) = rect(t ) * δ 2 (t )... 186 Fig. 5.19. Exponential Fourier Series spectrum for periodic signal f p (t ) = rect(t ) * δ 2 (t ) = D n e jn(π)t... 190 n Fig. 5.20. Periodic signal f p (t ) =rect(t ) * δ (t ), in Example 5.7... 192 Fig. 5.21. Plot of Trigonometric Fourier Series coefficients a 0 a 40 for periodic signal f p (t ) = rect(t ) * δ 4 (t ) in Example 5.7... 192 Fig. 5.22. Frequencies of piano keys over the middle octave, with A-440 tuning. Note that each key frequency value increases in frequency by factor 1.0595... 197 Fig. 5.23. Sample baseband spectrum M(ω)... 202 Fig. 5.24. Spectrum of m(t ) cos(ω 0 t ), which is 1 M ω ω 2 0 + 1 M(ω + ω 2 0 )... 203 Fig. 5.25. Sample baseband spectrum M(ω)... 203

List of Figures xxxiii Fig. 5.26. Magnitude spectrum of X (ω) = F {cos(ω 0 t) rect(t T )} (sinusoid multiplied by a rectangular time window) shown near ω = ω 0, from Example 5.8... 205 Fig. 5.27. Magnitude spectrum of X (ω) = F {cos(ω 0 t) Δ(t T )} (sinusoid multiplied by a triangular time window) shown near ω = ω 0, from Example 5.8... 206 Fig. 5.28. Plot of H(ω) = jω/(5 + jω) vs. ω, in Example 5.9.. 209 Fig. 5.29. Plot of H(ω) = ( jω/(5 + jω)) vs. ω, in Example 5.9... 209 Fig. 5.30. Plot of H(ω) = 10/(10 + jω) on a log scale vs. ω, from Example 5.10... 210 Fig. 5.31. Fig. 5.32. Fig. 5.33. Fig. 5.34. Fig. 5.35. Plot of H(ω) 2 = 10/(10 + jω) 2 in db vs. ω, from Example 5.11... 211 Plot of H(ω) = 10/(10 + jω) vs. ω using a log-log scale, from Example 5.12... 212 Plot of H(ω) 2 = 10/(10 + jω) 2 in db, vs. ω on a log scale, from Example 5.13... 213 Plot of H (f ) = 10/(10 + j 2πf ) in db, vs. f on a log scale, from Example 5.14... 214 Plot of H (f ) = (10/(10 + j 2πf )) in degrees vs. f on a log scale, in Example 5.15.... 214 Fig. 6.1. Sampling-based Fourier transform relationships.... 223 Fig. 6.2. Signal spectrum before and after sampling.... 225 Fig. 6.3. Ideal lowpass filtering to recover F(ω ) from F s (ω).... 226

xxxiv Fig. 6.4. List of Figures Non-ideal lowpass filtering to recover F(ω ) from F s (ω).... 227 Fig. 6.5. Ideal lowpass filtering of F s (ω) to recover F(ω ), when ω s = 2ω max.... 228 Fig. 6.6. Fig. 6.7. Fig. 6.8. Fig. 6.9. Fig. 6.10. Fig. 6.11. Fig. 6.12. The case where ω s < 2ω max, producing aliasing distortion.... 228 Sine waves at different frequencies can give identical samples if at least one of them is undersampled.... 229 Identical spectra result when cos(3t) and cos(5t) are sampled at ω s = 8 rad/sec (T s = 2π/8 sec), which demonstrates aliasing... 230 Reconstructing x(t ) as the sum of weighted sinc functions.... 232 Ideal LPF impulse response (sinc), and its magnitude spectrum (rect).... 234 Non-ideal LPF impulse response (truncated, delayed sinc), and its magnitude spectrum (rect pulse with overshoot).... 234 The spectral consequences of reconstructing a Nyquistrate-sampled signal using a non-ideal lowpass filter.... 235 Fig. 6.13. Post-filtering to remove aliasing distortion.... 236 Fig. 6.14. Pre-filtering to prevent aliasing distortion.... 237 Fig. 6.15. An example of undersampling a bandpass signal without destructive overlap-adding of adjacent spectral copies.... 238

List of Figures xxxv Fig. 6.16. Reconstructing x(t) as the sum of weighted rect functions: (a) x s (t) * rect(t /T s ); (b) x s (t) * rect((t T s /2)/T s ).... 240 Fig. 6.17. Sample-and-hold circuit to obtain x s (t) * rect((t T s /2)/T s ) from x(t ).... 241 Fig. 6.18. Fig. 6.19. Fig. 6.20. Fig. 6.21. Fig. 6.22. Fig. 6.23. Fig. 6.24. (a) Original spectrum, (b) spectral distortion due to sample/hold process, and (c) the resulting product of these: T s sinc(ωt s /2) X s (ω)..... 242 Reconstructing x(t ) as the sum of weighted triangular pulse functions (first-order hold filter).... 243 (a) Original spectrum, (b) spectral distortion due to 1 st -order hold process, and (c) the resulting product of these: T s sinc 2 (ωt s /2) X s (ω)... 244 Circuit diagram symbol for a digital-to-analog converter (DAC)... 245 Circuit diagram symbol for the analog-to-digital converter (ADC)... 246 Input-output description of a uniform quantizer that rounds input value x to the nearest integer value.... 247 Probability density functions of input and output signals to a uniform quantizer, as described above..... 249 Fig. 6.25. Noise-additive model for a quantizer.... 250 Fig. 6.26. Input-output description of a non-uniform quantizer.... 254 Fig. 6.27. Simulating a non-uniform quantization characteristic.... 255 Fig. 6.28. A simple digital-to-analog converter.... 256

xxxvi Fig. 6.29. List of Figures A digital-to-analog converter using an R-2R ladder network.... 256 Fig. 6.30. Successive approximation analog-to-digital converter.... 257 Fig. 6.31. Logarithmic successive approximation analog-to-digital converter... 258 Fig. 6.32. Flash analog-to-digital converter.... 259 Fig. 6.33. Delta-Sigma analog-to-digital converter.... 260 Fig. 6.34. Non-uniform quantization of a sinusoid (from Example 6-1).... 262 Fig. 6.35. Gray-level image before and after 2-level quantization... 263 Fig. 7.1. Fig. 7.2. Fig. 7.3. Fig. 7.4. Fig. 7.5. Measuring the impulse response of a discrete-time system.... 270 The relationship between input and output signals of a linear, shift invariant system is completely described by the system s impulse response.... 271 The effect of passing eigenfunction e jω 0n through a linear, shift-invariant discrete-time system (H (e jω 0) is a complex constant).... 272 The input/output relationships of an LSI system, shown in both time and frequency domains.... 274 Frequency responses of the four ideal digital filter types (the period centered at ω = 0 is highlighted).... 278 Fig. 7.6. Fig. 7.7. Time-domain representation of the sample delay function.... 282 Frequency-domain representation of the sample delay function.... 283

List of Figures xxxvii Fig. 7.8. Fig. 7.9. LSI system model in the form of a M th -order causal linear difference equation.... 284 Magnitude-squared vs. frequency plot of H (e jω ) = 1/(1 0.5e jω ).... 285 Fig. 7.10. Fig. 7.11. Fig. 7.12. Time domain description of a basic discrete-time feedback network.... 286 Frequency domain description of the basic feedback network in Fig. 7.10, having transfer function Y (e jω ) X (e jω ) = H 1 (e jω ) 1 H (e jω )H 2 (e jω ).... 287 Network used to implement the transfer function H (e jω )=1/(1 0.5e jω ).... 288 Fig. 7.13. General M th -order discrete-time feedback network.... 289 Fig. 7.14. Fig. 7.15. General M th order discrete-time feed-forward network.... 291 General M th order discrete-time network as a cascade of feedback and feed-forward networks.... 293 Fig. 7.16. Simplified general M th order discrete-time network.... 294 Fig. 7.17. General 1 st order discrete-time network.... 296 Fig. 7.18. First-order lowpass digital filter network, having transfer function H LPF (e jω )=Y(e jω ) X (e jω ) = (C 1 + C 1 e jω ) (1 α e jω )... 298 Fig. 7.19. First-order lowpass digital filter network, simplified.... 298 Fig. 7.20. Magnitude and phase of H (e jω ) in Example 7.3 (first-order digital lowpass filter).... 299

xxxviii Fig. 7.21. Fig. 7.22. List of Figures Magnitude-squared and phase of H LPF (e jω ) in Example 7.3 (first-order digital lowpass filter).... 300 First-order highpass digital filter network, having transfer function j j j j j H ( e ) Y( e )/ X( e ) C C e /1 e... 301 HPF 2 2 Fig. 7.23. First-order highpass digital filter network, simplified.... 302 Fig. 7.24(a). Magnitude-squared value of H (e jω ) in Example 7.4 (first-order digital highpass filter).... 302 Fig. 7.24(b). Phase of H (e jω ) in Example 7.4 (first-order digital highpass filter).... 303 Fig. 7.25. General 2 nd order discrete-time network.... 304 Fig. 7.26. Discrete-time network to implement a simple bandpass filter having peak gain at frequency ω 0 (0 < ω 0 < π).... 305 Fig. 7.27(a). 10 log 10 H(e jω ) 2 (db) in Example 7.5 (2 nd order digital bandpass filter).... 305 Fig. 7.27(b). Arg{H (e jω )} (radians) in Example 7.5 (2 nd order digital bandpass filter)... 306 Fig. 7.28. H N (e jω ) 2 of the notch filter in Example 7.6.... 307 Fig. 7.29. H N (e jω ) of the notch filter in Example 7.6.... 307 Fig. 7.30. Principal value of phase of H AP (e jω ) in Eq. (7.70); the magnitude response H AP (e jω ) = 1 (simple allpass filter).... 308 Fig. 7.31. Magnitude response of comb filter having impulse response h CF (n) = 1 δ (n 5). The frequency range is π ω π.... 309

List of Figures xxxix Fig. 7.32. Magnitude response of comb filter having impulse response h CF (n) = 1 + δ (n 4). The frequency range is π ω π.... 309 Fig. 7.33. Spectrum F(e jω ).... 313 Fig. 7.34. Spectrum G(e jω )=F(e j 2ω ) (a = 2)... 313 Fig. 7.35. Spectrum of Y (e jω )= LPF{G(e jω )} (dotted lines indicate the spectral copies eliminated by the lowpass filter when a = 2).... 314 Fig. 7.36. Original sequence f (n ) vs. time index value n.... 314 Fig. 7.37. Fig. 7.38. Sequence g(m ) (f (n ) after up-sampling by factor a = 3) vs. time index value m... 314 Sequence y(m ) (f (n ) after up-sampling by factor a = 3 and lowpass filtering with bandwidth ω s 2a = π 3) vs. time index value m... 315 Fig. 7.39. Interpolator that operates on f (n ) to produce y (m ).... 315 Fig. 7.40. Graphical depiction of Eq. (7.79) when up-sampling factor b = 3... 316 Fig. 7.41. Aliased result after decimation described in Fig. 7.39 (b = 3)... 316 Fig. 7.42. Decimator that operates on f (n ) to produce y(m )... 317 Fig. 7.43. Resulting spectrum after the decimation described in Fig. 7.41, this time with proper anti-aliasing lowpass filtering prior to down-sampling by factor (b = 3)... 317 Fig. 7.44. Magnitude-squared plot of H (e jω )=1 (2e jω 1) vs. ω, in db, from Example 7.7.... 318 Fig. 7.45. Nyquist plot of H (e jω )=1 (2e jω 1) vs. ω, from Example 7.7... 319

xl Fig. 7.46. Fig. 7.47. Fig. 7.48. Fig. 7.49. Fig. 7.50. Fig. 7.51. Fig. 7.52. Fig. 7.53. List of Figures Magnitude spectrum corresponding to difference equation in Example 7.7... 320 Phase spectrum corresponding to difference equation in Example 7.7... 321 A plot of the half-raised (Hanning) window for M = 41... 322 A plot of an ideal lowpass filter s impulse response h ILP (n) vs. n, over 30 n 30, when cutoff frequency ω = π 4... 322 A plot of h ILP (n)w (n) vs. n, over 30 n 30, where w(n) is a 41-pt. Hanning window. Figure 7.49 shows h ILP (n) without windowing... 323 Frequency response H LP (e jω )=F{h ILP (n)w (n )} of FIR filter having cutoff frequency ω 0 = π 4, which was obtained by multiplying an ideal LPF s impulse response with a 41-pt. Hanning window. The ideal LPF s magnitude response is the dotted line... 323 A plot of h ILP (n)w (n ) vs. n (cutoff frequency ω = π /2), over 100 n 100, where w (n) is a 201-pt. Hanning window... 324 H LP (e jω ) vs. ω in db, where w (n ) is a 201-pt. Hanning window, corresponding to the impulse response function shown in Fig. 7.52... 324 Fig. 7.54. H LP (e jω ) vs. ω in db (cutoff frequency ω = π /2), where w (n ) is a 1001-pt. Hanning window... 325

List of Figures xli Fig. 7.55. A plot of h ILP (n)w (n ) vs. n (cutoff frequency ω = 3π /4), over 20 n 20, where w (n ) is a 41-pt. Hamming window... 326 Fig. 7.56. H HP (e jω ) vs. ω in db, where w (n ) is a 41-pt. Hamming window, corresponding to the impulse response function shown in Fig. 7.55... 326 Fig. 7.57. A plot of h IBP (n)w (n ) vs. n (passband: π 4 < ω < π 2), over 100 n 100, where w (n ) is a 201-pt. Hamming window... 327 Fig. 7.58. Fig. 7.59. Fig. 7.60. Fig. 7.61. H BP (e jω ) vs. ω in db, where w (n ) is a 201-pt. Hamming window, corresponding to the impulse response function shown in Fig. 7.57... 327 A plot of h(n )w(n ) vs. n, over 25 n 25, where w (n ) is a 51-pt. Hamming window. The filter was designed by sampling a desired H (e jω )... 329 H (e jω ) vs. ω, where w(n ) is a 51-pt. Hamming window, corresponding to the impulse response function shown in Fig. 7.59... 329 H (e jω ) vs. ω, where w (n ) is a 1001-pt. Hamming window, obtained by sampling a desired H (e jω ) (dotted line in Fig. 7.60) in frequency... 330 Fig. 8.1. A linear and time-invariant continuous-time system.... 336 Fig. 8.2. Fig. 8.3. The input/output relationships of an LTIC system, shown in both time and frequency domains.... 338 Frequency magnitude responses of the four ideal analog filter types.... 340

xlii Fig. 8.4. List of Figures (a) Basic RC highpass filter network; (b) network transformed to impedance values.... 346 Fig. 8.5. H HP (ω) = jω/( jω + ω H ) vs. ω, for ω H = 10 rad/sec.... 347 Fig. 8.6. H HP (ω) = ( jω/( jω +ω H )) vs. ω, for ω H = 10 rad/sec.... 348 Fig. 8.7. (a) Basic LR lowpass filter network; (b) network transformed to impedance values.... 349 Fig. 8.8. H LP (ω) = ω L /( jω + ω L ) vs. ω, for ω L = 10.... 349 Fig. 8.9. Fig. 8.10. Fig. 8.11. H LP (ω) = (ω L /( jω + ω L )) in degrees, vs. ω, for ω L = 10 rad/sec.... 350 (a) Basic RLC bandpass filter network; (b) network transformed to impedance values.... 351 H BP (ω) = jωrc/((1 ω 2 LC ) + jωrc ) vs. ω, for peak frequency ω = 10 rad/sec. (R = 10 Ω, L = 1 H, C = 4/375 F).... 352 Fig. 8.12. H BP (ω) = jωrc/((1 ω 2 LC ) + jωrc ) vs. ω, for peak frequency ω 0 = 10 rad/sec. (R = 10 Ω, L = 1 H, C = 4/375 F).... 352 Fig. 8.13. Time domain description of a basic continuous-time feedback network.... 354 Fig. 8.14. Frequency domain description of the basic feedback network in Fig. 8.13, having transfer function Y (ω) X (ω) = H (ω) (1 + H (ω)h 2 (ω))... 355