PRACTICAL SIGNAL PROCESSING AND ITS APPLICATIONS With Solved Homework Problems

Transcription

1 PRACTICAL SIGNAL PROCESSING AND ITS APPLICATIONS With Solved Homework Problems

2 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

3 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

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ 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 ISBN (hc : alk. paper) Subjects: LCSH: Signal processing--textbooks. Classification: LCC TK L DDC /2--dc23 LC record available at 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 Desk Editor: Suraj Kumar Typeset by Stallion Press enquiries@stallionpress.com Printed in Singapore

5 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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7 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

8 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.

9 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:

10 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

11 Contents Dedication... v Preface... vii List of Tables... xxiii List of Figures... xxv Chapter 1: Introduction to Signal Processing Analog and Digital Signal Processing Signals and their Usefulness Radio communications Data storage Naturally-occurring signals Other signals Applications of Signal Processing Signal Processing: Practical Implementation Basic Signal Characteristics Complex Numbers Complex number math refresher Complex number operations in MATLAB Practical applications of complex numbers Textbook Organization Chapter Summary and Comments Homework Problems Chapter 2: Discrete-Time Signals and Operations Theory Introduction Basic discrete-time signals Impulse function Periodic impulse train Sinusoid xi

12 xii Contents Complex exponential Unit step function Signum function Ramp function Rectangular pulse Triangular pulse Exponential decay Sinc function Signal properties Energy and power sequences Summable sequences Periodic sequences Sum of periodic sequences Even and odd sequences Right-sided and left-sided sequences Causal, anticausal sequences Finite-length and infinite-length sequences Signal operations Time shift Time reversal Time scaling Cumulative sum and backward difference Conjugate, magnitude and phase Equivalent signal expressions Discrete convolution Convolution with an impulse Convolution of two pulses Discrete-time cross-correlation Practical Applications Discrete convolution to calculate the coefficient values of a polynomial product Synthesizing a periodic signal using convolution Normalized cross-correlation Waveform smoothing by convolving with a pulse Discrete convolution to find the Binomial distribution Useful MATLAB Code Plotting a sequence Calculating power of a periodic sequence Discrete convolution Moving-average smoothing of a finite-length sequence Calculating energy of a finite-length sequence... 46

13 Contents xiii Calculating the short-time energy of a finite-length sequence Cumulative sum and backward difference operations Calculating cross-correlation via convolution Chapter Summary and Comments Homework Problems Chapter 3: Continuous-Time Signals and Operations Theory Introduction Basic continuous-time signals Impulse function Periodic impulse train Sinusoid Complex exponential Unit step function Signum function Ramp function Rectangular pulse Triangular pulse Exponential decay Sinc function Signal properties Energy and power signals Integrable signals Periodic signals Sum of periodic signals Even and odd signals Right-sided and left-sided signals Causal, anticausal signals Finite-length and infinite-length signals Continuous-time signal operations Time delay Time reversal Time scaling Cumulative integral and time differential Conjugate, magnitude and phase Equivalent signal expressions... 70

14 xiv Contents Convolution Convolution with an impulse Convolution of two pulses Cross-correlation Practical Applications Synthesizing a periodic signal using convolution Waveform smoothing by convolving with a pulse Practical analog cross-correlation Normalized cross-correlation as a measure of similarity Application of convolution to probability theory Useful MATLAB Code Plotting basic signals Estimating continuous-time convolution Estimating energy and power of a signal Detecting pulses using normalized correlation Plotting estimated probability density functions Chapter Summary and Comments Homework Problems Chapter 4: Frequency Analysis of Discrete-Time Signals Theory Discrete-Time Fourier Transform (DTFT) Fourier transforms of basic signals Exponentially decaying signal Constant value Impulse function Delayed impulse function Signum function Unit step function Complex exponential function Sinusoid Rectangular pulse function Fourier transform properties Linearity Time shifting Time/frequency duality Convolution Modulation Frequency shift Time scaling Parseval s Theorem

15 Contents xv Graphical representation of the Fourier transform Rectangular coordinates Polar coordinates Graphing the amplitude of F (e jω ) Logarithmic scales and Bode plots Fourier transform of periodic sequences Comb function Periodic signals as convolution with a comb function Discrete Fourier Transform (DFT) Time-frequency duality of the DFT Fast Fourier Transform (FFT) Parseval s Theorem Summary of Fourier transformations for discrete-time signals Practical Applications Spectral analysis using the FFT Frequency resolution Periodic sequence Finite-length sequence Convolution using the FFT Autocorrelation using the FFT Discrete Cosine Transform (DCT) Useful MATLAB Code Plotting the spectrum of a discrete-time signal Chapter Summary and Comments Homework Problems Chapter 5: Frequency Analysis of Continuous-Time Signals Theory Fourier Transform Fourier transforms of basic signals Exponentially decaying signal Constant value Impulse function Delayed impulse function Signum function Unit step function Complex exponential function Sinusoid Rectangular pulse function

16 xvi Contents Fourier transform properties Linearity Time shifting Time/frequency duality Convolution Modulation Frequency shift Time scaling Parseval s Theorem Graphical representation of the Fourier transform Rectangular coordinates Polar coordinates Graphing the amplitude of F (ω ) Logarithmic scales and Bode plots Fourier transform of periodic signals Comb function Periodic signals as convolution with a comb function Exponential Fourier Series Trigonometric Fourier Series Compact Trigonometric Fourier Series Parseval s Theorem Summary of Fourier transformations for continuous-time signals Practical Applications Frequency scale of a piano keyboard Frequency-domain loudspeaker measurement Effects of various time-domain operations on frequency magnitude and phase Communication by frequency shifting Spectral analysis using time windowing Representing an analog signal with frequency-domain samples Useful MATLAB Code Chapter Summary and Comments Homework Problems Chapter 6: Sampling Theory and Practice Theory Sampling a continuous-time signal Relation between CTFT and DTFT based on sampling

17 Contents xvii Recovering a continuous-time signal from its samples Filtering basics Frequency domain perspective Time domain perspective Oversampling to simplify reconstruction filtering Eliminating aliasing distortion Anti-alias post-filtering Anti-alias pre-filtering Sampling bandpass signals Approximate reconstruction of a continuous-time signal from its samples Zero-order hold method First-order hold method Digital-to-analog conversion Analog-to-digital conversion Amplitude quantization Definition Why quantize? Signal to quantization noise power ratio (SNR Q ) Non-uniform quantization Practical Applications Practical digital-to-analog conversion Practical analog-to-digital conversion Successive approximation ADC Logarithmic successive approximation ADC Flash ADC Delta-Sigma (ΔΣ) ADC Useful MATLAB Code Amplitude quantization Chapter Summary and Comments Homework Problems Chapter 7: Frequency Analysis of Discrete-Time Systems Theory Introduction Linear shift-invariant discrete-time system Impulse response Input/output relations Digital filtering concepts Ideal lowpass filter Ideal highpass filter

18 xviii Contents Ideal bandpass filter Ideal band-elimination filter Discrete-time filter networks Digital filter building blocks Linear difference equations Basic feedback network Generalized feedback network Generalized feed-forward network Combined feedback and feed-forward network Practical Applications First-order digital filters Lowpass filter Highpass filter Second-order digital filters Bandpass filter Notch filter Allpass filter Specialized digital filters Comb filter Linear-phase filter Interpolation and Decimation Interpolation by factor a Decimation by factor b Nyquist frequency response plot Useful MATLAB Code Plotting frequency response of filter described by a difference equation FIR filter design by windowing the ideal filter s impulse response FIR filter design by frequency sampling Chapter Summary and Comments Homework Problems Chapter 8: Frequency Analysis of Continuous-Time Systems Theory Introduction Linear Time-Invariant Continuous System Input/output relation Response to e jω 0t

19 Contents xix Ideal filters Ideal lowpass filter Ideal highpass filter Ideal bandpass filter Ideal band-elimination filter Practical Applications RLC circuit impedance analysis First order passive filter circuits Highpass filter Second order passive filter circuits Bandpass filter Band-elimination filter Active filter circuits Basic feedback network Operational amplifier Noninverting topology Inverting topology First-order active filter Second-order active filter Useful MATLAB Code Sallen-Key circuit frequency response plot Calculating and plotting impedance of a one-port network Chapter Summary and Comments Homework Problems Chapter 9: Z-Domain Signal Processing Theory Introduction The Z transform Region of convergence Z transforms of basic signals Exponentially decaying signal Impulse sequence Delayed impulse sequence Unit step sequence Causal complex exponential sequence Causal sinusoidal sequence Discrete ramp sequence Table of Z transforms

20 xx Contents Z transform properties Linearity Time shifting Convolution Time multiplication Conjugation Multiplication by n in the time domain Multiplication by a n in the time domain Backward difference Cumulative sum Table of Z transform properties Z transform of linear difference equations Inverse Z transform of rational functions Inverse Z transform yielding finite-length sequences Long division method Partial fraction expansion method Chapter Summary and Comments Homework Problems Chapter 10: S-Domain Signal Processing Theory Introduction Laplace transform Region of convergence Laplace transforms of basic signals Exponentially decaying signal Impulse function Delayed impulse function Unit step function Complex exponential function Sinusoid Ramp function Table of Laplace transforms Laplace transform properties Linearity Time shifting Frequency shifting duality Time scaling Convolution Time multiplication

21 Contents xxi Time differentiation Time integration Table of Laplace transform properties Inverse Laplace transform of rational functions Partial fraction expansion method Chapter Summary and Comments Homework Problems Chapter 11: Applications of Z-Domain Signal Processing Introduction Applications of Pole-Zero Analysis Poles and zeros of realizable systems Frequency response from H (z) Frequency response from pole/zero locations Magnitude response Phase response Effect on H (e jω ) of reciprocating a pole System stability Causal systems Anticausal systems Stabilizing an unstable causal system Pole-zero plots of basic digital filters Lowpass filter Highpass filter Bandpass digital filter Notch filter Comb filter Allpass filter (real pole and zero) Allpass filter (complex conjugate poles and zeros) Minimum-phase system Digital filter design based on analog prototypes Impulse-invariant transformation Bilinear transformation Chapter Summary and Comments Homework Problems Chapter 12: Applications of S-Domain Signal Processing Introduction

22 xxii Contents 12.2 Linear System Analysis in the S-Domain Linear time-invariant continuous system Frequency response from H (s) Applications of Pole-Zero Analysis Poles and zeros of realizable systems Frequency response from pole/zero locations Magnitude response Phase response Effect on H (ω) of mirroring a pole about the jω axis System stability Causal systems Anticausal systems Stabilizing an unstable causal system Pole-zero plots of basic analog filters Lowpass filter Highpass filter Bandpass filter Notch (band-elimination) filter Minimum-phase system Circuit Analysis in the S-Domain Transient Circuit Analysis Passive ladder analysis using T matrices Solution of Linear Differential Equations Relation Between Transfer Function, Differential Equation, and State Equation Differential equation from H (s) State equations from H (s) Chapter Summary and Comments Homework Problems Appendix: Solved Homework Problems Bibliography Index

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

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

25 List of Figures Fig 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 Phasor diagram graphical solution for 2 cos 100t sin 100t cos 100t Fig Fig Textbook chapter organization, showing the parallelism between discrete-time and continuous-time domains Example of a sequence x(n) as a function of its index variable n Fig Impulse sequence (n) Fig Delayed impulse sequence (n 4) Fig Impulse train 4(n) Fig Impulse train 3(n) Fig Impulse train 2(n) Fig A sinusoidal sequence (A = 1, = 1, = /3) Fig Unit step function sequence u(n) Fig Signum function sequence sgn(n) xxv

26 xxvi List of Figures Fig Ramp function sequence r(n) Fig Rectangular pulse sequence rect 4 (n) Fig Triangular pulse sequence 5 (n) Fig Exponentially decaying sequence u(n) (0.8) n Fig Discrete-time sequence sinc (n) Fig Fig Pulse rect 4 (n + 1), a time-shifted version of sequence rect 4 (n) Pulse rect 4 n 2, a time-shifted version of sequence rect 4 (n) Fig Delayed impulse sequence δ(n 4) Fig Delayed exponentially decaying sequence Fig Sequence u ( n 2) Fig Rectangular pulse sequence x(n) Fig Fig Fig Signal y (n), composed of noise plus rectangular pulses at various delays and amplitudes 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 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) Fig MATLAB plot of a Binomial(50, 0.5) distribution... 43

27 List of Figures xxvii Fig An example of using MATLAB s stem function to plot a sequence Fig A noisy sinusoidal sequence before smoothing Fig A noisy sinusoidal sequence after smoothing Fig Sequence x(n) Fig Calculated short-time energy of the sequence x(n) in Fig Fig Autocorrelation of random noise Fig Impulse function δ(t ) Fig Shifted impulse function δ(t + π) Fig Fig 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 ) Impulse train δ 4.2 (t ). (When not specified, assume each impulse area = 1.) Fig A sinusoidal signal (A = 1, ω = 1, θ = π /3) Fig Unit step function u(t ) Fig Signum function sgn(t ) Fig Ramp function r (t ) Fig Rectangular pulse function rect(t ) Fig Triangular pulse function Δ(t )... 61

28 xxviii List of Figures Fig Exponentially decaying signal u (t )e 0.223t Fig Function sinc(t ) Fig Signal rect t 1, which is rect(t ) after 1-sec delay Fig Signal rect(t + 1/2), which is rect(t ) after 1/2-sec advance Fig Delayed impulse function δ(t 4) Fig Delayed exponentially decaying signal Fig Signal u (( t ) 2) Fig Triangular pulse function Δ(t ), before (dotted line) and after (solid line) smoothing via convolution with pulse 5rect(5t) Fig Practical analog cross-correlation technique Fig MATLAB plot of y(t ) = sin(2πt ) Fig MATLAB plot of yt ( ) u t 1.5 rect( t/ 2) Δ t ut Fig MATLAB plot of 2rect(t) convolved with Δ t Fig Original signal y(t ) that is composed of three triangular pulses Fig Triangular pulse x(t ) used for waveform matching Fig Signal z(t ) = y(t ) + noise added Fig Normalized cross-correlation result C xz (t ), showing locations and polarities of triangular pulses that were detected in the noise waveform z(t )... 84

29 List of Figures xxix Fig Estimated PDF of r.v. X Fig Estimated PDF of r.v. Y Fig Estimated PDF of random variable Z = X + Y, demonstrating the fact that fz a fx a * fy a Fig Sequence x(n) to be transformed to the frequency domain in Example Fig From Example 4.1: F{x(n)} = X(e jω ) = cos(kω)+ 14 k 1 k 8 cos(kω ) Fig The spectrum X(e jω ) = F{x(n)} in Example Fig Plot of F {rect 10 (n)} = k 1 cos(kω ) Fig Plot of F rect 10 (n) cos (π 6)n =1+ 10 k 1 cos k ω π/6 + cos k(ω + π/6) Fig A graph of 7{sinc(3.5ω) * δ π (ω)}e j100ω vs. ω/π Fig A 3-D graph of complex-valued F(e jω ) Fig Re{F(e jω )} vs. ω, corresponding to Fig Fig Im{F(e jω )} vs. ω, corresponding to Fig Fig A graph of ( k 1 cos(kω) )e jω vs. ω Fig A graph of { k cos(kω) e jω } vs. ω Fig A graph of k 1 cos(kω) vs. ω Fig A graph of k 1 cos(kω) vs. ω Fig Impulse train δ 4 (n)

30 xxx List of Figures Fig Impulse train (π /2)δ π /2 (ω) = F {δ 4 (n)} Fig Rectangular pulse rect 5 (n ) Fig Impulse train δ 20 (n ) Fig Periodic signal f p (n ) = rect 5 (n ) * δ 20 (n ) Fig 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 Fig A plot of periodic discrete-time sequence x(n ) = cos(2πn/10) Fig Fig Fig Fig Fig A plot of the spectrum of x(n ) =cos(2πn/10), calculated using the Fast Fourier Transform (FFT) Samples of X(e jω ) = F {x(n)} found using the FFT method, when x(n ) = {0.0975, , , , } for 0 n 4 and x(n ) = 0 elsewhere 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 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 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 Fig Plot of phase spectrum H(e jω ) from Example Fig Plot of magnitude spectrum H(e jω ) from Example

31 List of Figures xxxi Fig Plot of magnitude spectrum H(e jω ) from Example Fig Plot of phase spectrum H(e jω ) from Example Fig Plot of magnitude spectrum H(e jω ) from Example Fig Plot of phase spectrum H(e jω ) from Example Fig Fig Fig Plot of magnitude spectrum H(e jω ) from Example Plot of magnitude spectrum H(e jω ) 2 in db, from Example Sequence x(t ) to be transformed to the frequency domain in Example Fig From Example 5.1: F {x(t )} =X(ω) = 2sinc(2ω )+ 3sinc(ω ) Fig Fig Fig The spectrum X(ω) = 2 sinc(ω) rect(ω 2π ) in Example Plot of F {rect(t 3π)} = 3π sinc(ω3π 2), from Example Plot of F {rect(t 3π)cos(4πt)} = (3π 2)sinc (ω 4π)3π 2 +(3π 2)sinc((ω + 4π)3π 2), from Example Fig A graph of sinc(ω/2)e j20ω vs. ω/π Fig A 3-D graph of complex-valued F(ω) Fig Re{F(ω)} vs. ω, corresponding to Fig Fig Im{F(ω)}vs. ω, corresponding to Fig

32 xxxii List of Figures Fig A graph of sinc(ω/2)e jω/5 vs. ω Fig A graph of {sinc(ω/2)e jω/5 } vs. ω Fig A graph of sinc(ω/2) vs. ω Fig A graph of sinc(ω/2) vs. ω Fig Impulse train δ 4 (t ) Fig Impulse train (π/2)δ π /2 (ω ) = F {δ 4 (t )} Fig Rectangular pulse rect(t ) Fig Impulse train δ 2 (t ) Fig Periodic signal f p (t ) = rect(t ) * δ 2 (t ) Fig Exponential Fourier Series spectrum for periodic signal f p (t ) = rect(t ) * δ 2 (t ) = D n e jn(π)t n Fig Periodic signal f p (t ) =rect(t ) * δ (t ), in Example Fig Plot of Trigonometric Fourier Series coefficients a 0 a 40 for periodic signal f p (t ) = rect(t ) * δ 4 (t ) in Example Fig Frequencies of piano keys over the middle octave, with A-440 tuning. Note that each key frequency value increases in frequency by factor Fig Sample baseband spectrum M(ω) Fig Spectrum of m(t ) cos(ω 0 t ), which is 1 M ω ω M(ω + ω 2 0 ) Fig Sample baseband spectrum M(ω)

33 List of Figures xxxiii Fig 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 Fig 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 Fig Plot of H(ω) = jω/(5 + jω) vs. ω, in Example Fig Plot of H(ω) = ( jω/(5 + jω)) vs. ω, in Example Fig Plot of H(ω) = 10/(10 + jω) on a log scale vs. ω, from Example Fig Fig Fig Fig Fig Plot of H(ω) 2 = 10/(10 + jω) 2 in db vs. ω, from Example Plot of H(ω) = 10/(10 + jω) vs. ω using a log-log scale, from Example Plot of H(ω) 2 = 10/(10 + jω) 2 in db, vs. ω on a log scale, from Example Plot of H (f ) = 10/(10 + j 2πf ) in db, vs. f on a log scale, from Example Plot of H (f ) = (10/(10 + j 2πf )) in degrees vs. f on a log scale, in Example Fig Sampling-based Fourier transform relationships Fig Signal spectrum before and after sampling Fig Ideal lowpass filtering to recover F(ω ) from F s (ω)

34 xxxiv Fig List of Figures Non-ideal lowpass filtering to recover F(ω ) from F s (ω) Fig Ideal lowpass filtering of F s (ω) to recover F(ω ), when ω s = 2ω max Fig Fig Fig Fig Fig Fig Fig The case where ω s < 2ω max, producing aliasing distortion Sine waves at different frequencies can give identical samples if at least one of them is undersampled 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 Reconstructing x(t ) as the sum of weighted sinc functions Ideal LPF impulse response (sinc), and its magnitude spectrum (rect) Non-ideal LPF impulse response (truncated, delayed sinc), and its magnitude spectrum (rect pulse with overshoot) The spectral consequences of reconstructing a Nyquistrate-sampled signal using a non-ideal lowpass filter Fig Post-filtering to remove aliasing distortion Fig Pre-filtering to prevent aliasing distortion Fig An example of undersampling a bandpass signal without destructive overlap-adding of adjacent spectral copies

35 List of Figures xxxv Fig 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 ) Fig Sample-and-hold circuit to obtain x s (t) * rect((t T s /2)/T s ) from x(t ) Fig Fig Fig Fig Fig Fig Fig (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) Reconstructing x(t ) as the sum of weighted triangular pulse functions (first-order hold filter) (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) Circuit diagram symbol for a digital-to-analog converter (DAC) Circuit diagram symbol for the analog-to-digital converter (ADC) Input-output description of a uniform quantizer that rounds input value x to the nearest integer value Probability density functions of input and output signals to a uniform quantizer, as described above Fig Noise-additive model for a quantizer Fig Input-output description of a non-uniform quantizer Fig Simulating a non-uniform quantization characteristic Fig A simple digital-to-analog converter

36 xxxvi Fig List of Figures A digital-to-analog converter using an R-2R ladder network Fig Successive approximation analog-to-digital converter Fig Logarithmic successive approximation analog-to-digital converter Fig Flash analog-to-digital converter Fig Delta-Sigma analog-to-digital converter Fig Non-uniform quantization of a sinusoid (from Example 6-1) Fig Gray-level image before and after 2-level quantization Fig Fig Fig Fig Fig Measuring the impulse response of a discrete-time system The relationship between input and output signals of a linear, shift invariant system is completely described by the system s impulse response The effect of passing eigenfunction e jω 0n through a linear, shift-invariant discrete-time system (H (e jω 0) is a complex constant) The input/output relationships of an LSI system, shown in both time and frequency domains Frequency responses of the four ideal digital filter types (the period centered at ω = 0 is highlighted) Fig Fig Time-domain representation of the sample delay function Frequency-domain representation of the sample delay function

37 List of Figures xxxvii Fig Fig LSI system model in the form of a M th -order causal linear difference equation Magnitude-squared vs. frequency plot of H (e jω ) = 1/(1 0.5e jω ) Fig Fig Fig Time domain description of a basic discrete-time feedback network 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 ) Network used to implement the transfer function H (e jω )=1/(1 0.5e jω ) Fig General M th -order discrete-time feedback network Fig Fig General M th order discrete-time feed-forward network General M th order discrete-time network as a cascade of feedback and feed-forward networks Fig Simplified general M th order discrete-time network Fig General 1 st order discrete-time network Fig 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 ) Fig First-order lowpass digital filter network, simplified Fig Magnitude and phase of H (e jω ) in Example 7.3 (first-order digital lowpass filter)

38 xxxviii Fig Fig List of Figures Magnitude-squared and phase of H LPF (e jω ) in Example 7.3 (first-order digital lowpass filter) 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 HPF 2 2 Fig First-order highpass digital filter network, simplified Fig. 7.24(a). Magnitude-squared value of H (e jω ) in Example 7.4 (first-order digital highpass filter) Fig. 7.24(b). Phase of H (e jω ) in Example 7.4 (first-order digital highpass filter) Fig General 2 nd order discrete-time network Fig Discrete-time network to implement a simple bandpass filter having peak gain at frequency ω 0 (0 < ω 0 < π) Fig. 7.27(a). 10 log 10 H(e jω ) 2 (db) in Example 7.5 (2 nd order digital bandpass filter) Fig. 7.27(b). Arg{H (e jω )} (radians) in Example 7.5 (2 nd order digital bandpass filter) Fig H N (e jω ) 2 of the notch filter in Example Fig H N (e jω ) of the notch filter in Example Fig Principal value of phase of H AP (e jω ) in Eq. (7.70); the magnitude response H AP (e jω ) = 1 (simple allpass filter) Fig Magnitude response of comb filter having impulse response h CF (n) = 1 δ (n 5). The frequency range is π ω π

39 List of Figures xxxix Fig Magnitude response of comb filter having impulse response h CF (n) = 1 + δ (n 4). The frequency range is π ω π Fig Spectrum F(e jω ) Fig Spectrum G(e jω )=F(e j 2ω ) (a = 2) Fig Spectrum of Y (e jω )= LPF{G(e jω )} (dotted lines indicate the spectral copies eliminated by the lowpass filter when a = 2) Fig Original sequence f (n ) vs. time index value n Fig Fig Sequence g(m ) (f (n ) after up-sampling by factor a = 3) vs. time index value m 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 Fig Interpolator that operates on f (n ) to produce y (m ) Fig Graphical depiction of Eq. (7.79) when up-sampling factor b = Fig Aliased result after decimation described in Fig (b = 3) Fig Decimator that operates on f (n ) to produce y(m ) Fig 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) Fig Magnitude-squared plot of H (e jω )=1 (2e jω 1) vs. ω, in db, from Example Fig Nyquist plot of H (e jω )=1 (2e jω 1) vs. ω, from Example

40 xl Fig Fig Fig Fig Fig Fig Fig Fig List of Figures Magnitude spectrum corresponding to difference equation in Example Phase spectrum corresponding to difference equation in Example A plot of the half-raised (Hanning) window for M = A plot of an ideal lowpass filter s impulse response h ILP (n) vs. n, over 30 n 30, when cutoff frequency ω = π 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 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 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 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 Fig H LP (e jω ) vs. ω in db (cutoff frequency ω = π /2), where w (n ) is a 1001-pt. Hanning window

41 List of Figures xli Fig 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 Fig 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 Fig 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 Fig Fig Fig Fig 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 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 ) H (e jω ) vs. ω, where w(n ) is a 51-pt. Hamming window, corresponding to the impulse response function shown in Fig 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 Fig A linear and time-invariant continuous-time system Fig Fig The input/output relationships of an LTIC system, shown in both time and frequency domains Frequency magnitude responses of the four ideal analog filter types

42 xlii Fig List of Figures (a) Basic RC highpass filter network; (b) network transformed to impedance values Fig H HP (ω) = jω/( jω + ω H ) vs. ω, for ω H = 10 rad/sec Fig H HP (ω) = ( jω/( jω +ω H )) vs. ω, for ω H = 10 rad/sec Fig (a) Basic LR lowpass filter network; (b) network transformed to impedance values Fig H LP (ω) = ω L /( jω + ω L ) vs. ω, for ω L = Fig Fig Fig H LP (ω) = (ω L /( jω + ω L )) in degrees, vs. ω, for ω L = 10 rad/sec (a) Basic RLC bandpass filter network; (b) network transformed to impedance values 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) Fig 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) Fig Time domain description of a basic continuous-time feedback network Fig 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))