SIGNALS AND SYSTEM BOOK

Signals and Systems Using MATLAB Luis F Chaparro Department of Electrical and Computer Engineering University of Pittsburgh AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Elsevier, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK c 2011 Elsevier Inc All rights reserved Copyright No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than R as may be noted herein) MATLAB is a trademark of The MathWorks, Inc and is used with permission The MathWorks R does not warrant the accuracy of the text or exercises in this book This books use or discussion of MATLAB software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical R approach or particular use of the MATLAB software Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein Library of Congress Cataloging-in-Publication Data Chaparro, Luis F R Signals and systems using MATLAB / Luis F Chaparro p cm ISBN 978-0-12-374716-7 Signal processing–Digital techniques System analysis MATLAB I Title TK5102.9.C472 2010 621.382’2–dc22 2010023436 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our Web site at www.elsevierdirect.com Printed in the United States of America 10 11 12 13 To my family, with much love Contents PREFACE xi ACKNOWLEDGMENTS xvi Part Introduction CHAPTER From the Ground Up! 0.1 Signals and Systems and Digital Technologies 0.2 Examples of Signal Processing Applications 0.3 0.2.1 Compact-Disc Player 0.2.2 Software-Defined Radio and Cognitive Radio 0.2.3 Computer-Controlled Systems Analog or Discrete? 0.4 0.3.1 Continuous-Time and Discrete-Time Representations 0.3.2 Derivatives and Finite Differences 0.3.3 Integrals and Summations 0.3.4 Differential and Difference Equations Complex or Real? 10 12 13 16 20 0.5 0.4.1 Complex Numbers and Vectors 0.4.2 Functions of a Complex Variable 0.4.3 Phasors and Sinusoidal Steady State 0.4.4 Phasor Connection Soft Introduction to MATLAB 20 23 24 26 29 0.5.1 Numerical Computations 0.5.2 Symbolic Computations Problems 30 43 53 Part CHAPTER iv Theory and Application of Continuous-Time Signals and Systems 63 Continuous-Time Signals 65 1.1 Introduction 65 1.2 Classification of Time-Dependent Signals 66 Contents 1.3 Continuous-Time Signals 67 1.4 1.3.1 Basic Signal Operations—Time Shifting and Reversal 1.3.2 Even and Odd Signals 1.3.3 Periodic and Aperiodic Signals 1.3.4 Finite-Energy and Finite Power Signals Representation Using Basic Signals 71 75 77 79 85 1.4.1 1.4.2 1.4.3 1.4.4 1.5 Complex Exponentials Unit-Step, Unit-Impulse, and Ramp Signals Special Signals—the Sampling Signal and the Sinc Basic Signal Operations—Time Scaling, Frequency Shifting, and Windowing 1.4.5 Generic Representation of Signals What Have We Accomplished? Where Do We Go from Here? 85 88 100 102 105 106 Problems 108 CHAPTER Continuous-Time Systems 117 2.1 Introduction 117 2.2 System Concept 118 2.3 2.2.1 System Classification 118 LTI Continuous-Time Systems 119 2.4 2.3.1 Linearity 2.3.2 Time Invariance 2.3.3 Representation of Systems by Differential Equations 2.3.4 Application of Superposition and Time Invariance 2.3.5 Convolution Integral 2.3.6 Causality 2.3.7 Graphical Computation of Convolution Integral 2.3.8 Interconnection of Systems—Block Diagrams 2.3.9 Bounded-Input Bounded-Output Stability What Have We Accomplished? Where Do We Go from Here? 120 125 130 135 136 143 145 147 153 156 Problems 157 CHAPTER The Laplace Transform 165 3.1 Introduction 165 3.2 The Two-Sided Laplace Transform 166 3.3 3.2.1 Eigenfunctions of LTI Systems 167 3.2.2 Poles and Zeros and Region of Convergence 172 The One-Sided Laplace Transform 176 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 Linearity Differentiation Integration Time Shifting Convolution Integral 185 188 193 194 196 v vi Contents 3.4 Inverse Laplace Transform 197 3.5 3.4.1 Inverse of One-Sided Laplace Transforms 3.4.2 Inverse of Functions Containing e−ρs Terms 3.4.3 Inverse of Two-Sided Laplace Transforms Analysis of LTI Systems 3.6 3.5.1 LTI Systems Represented by Ordinary Differential Equations 214 3.5.2 Computation of the Convolution Integral 221 What Have We Accomplished? Where Do We Go from Here? 226 197 209 212 214 Problems 226 CHAPTER Frequency Analysis: The Fourier Series 237 4.1 Introduction 237 4.2 Eigenfunctions Revisited 238 4.3 Complex Exponential Fourier Series 245 4.4 Line Spectra 248 4.5 4.4.1 Parseval’s Theorem—Power Distribution over Frequency 248 4.4.2 Symmetry of Line Spectra 250 Trigonometric Fourier Series 251 4.6 Fourier Coefficients from Laplace 255 4.7 Convergence of the Fourier Series 265 4.8 Time and Frequency Shifting 270 4.9 Response of LTI Systems to Periodic Signals 273 4.9.1 Sinusoidal Steady State 274 4.9.2 Filtering of Periodic Signals 276 4.10 Other Properties of the Fourier Series 279 4.10.1 Reflection and Even and Odd Periodic Signals 4.10.2 Linearity of Fourier Series—Addition of Periodic Signals 4.10.3 Multiplication of Periodic Signals 4.10.4 Derivatives and Integrals of Periodic Signals 4.11 What Have We Accomplished? Where Do We Go from Here? 279 282 284 285 289 Problems 290 CHAPTER Frequency Analysis: The Fourier Transform 299 5.1 Introduction 299 5.2 From the Fourier Series to the Fourier Transform 300 5.3 Existence of the Fourier Transform 302 5.4 Fourier Transforms from the Laplace Transform 302 5.5 Linearity, Inverse Proportionality, and Duality 304 5.5.1 5.5.2 5.5.3 Linearity 304 Inverse Proportionality of Time and Frequency 305 Duality 310 Contents 5.6 Spectral Representation 313 5.7 5.6.1 Signal Modulation 5.6.2 Fourier Transform of Periodic Signals 5.6.3 Parseval’s Energy Conservation 5.6.4 Symmetry of Spectral Representations Convolution and Filtering 313 317 320 322 327 5.8 5.7.1 Basics of Filtering 5.7.2 Ideal Filters 5.7.3 Frequency Response from Poles and Zeros 5.7.4 Spectrum Analyzer Additional Properties 329 332 337 341 344 5.9 5.8.1 Time Shifting 344 5.8.2 Differentiation and Integration 346 What Have We Accomplished? What Is Next? 350 Problems 350 CHAPTER Application to Control and Communications 359 6.1 Introduction 359 6.2 System Connections and Block Diagrams 360 6.3 Application to Classic Control 363 6.4 6.3.1 Stability and Stabilization 369 6.3.2 Transient Analysis of First- and Second-Order Control Systems 371 Application to Communications 377 6.5 6.4.1 AM with Suppressed Carrier 6.4.2 Commercial AM 6.4.3 AM Single Sideband 6.4.4 Quadrature AM and Frequency-Division Multiplexing 6.4.5 Angle Modulation Analog Filtering 379 380 382 383 385 390 6.6 6.5.1 Filtering Basics 6.5.2 Butterworth Low-Pass Filter Design 6.5.3 Chebyshev Low-Pass Filter Design 6.5.4 Frequency Transformations 6.5.5 Filter Design with MATLAB What Have We Accomplished? What Is Next? 390 393 396 402 405 409 Problems 409 Part CHAPTER Theory and Application of Discrete-Time Signals and Systems 417 Sampling Theory 419 7.1 Introduction 419 vii viii Contents 7.2 Uniform Sampling 420 7.3 7.2.1 Pulse Amplitude Modulation 7.2.2 Ideal Impulse Sampling 7.2.3 Reconstruction of the Original Continuous-Time Signal 7.2.4 Signal Reconstruction from Sinc Interpolation 7.2.5 Sampling Simulation with MATLAB The Nyquist-Shannon Sampling Theorem 7.4 7.3.1 Sampling of Modulated Signals 438 Practical Aspects of Sampling 439 7.5 7.4.1 Sample-and-Hold Sampling 7.4.2 Quantization and Coding 7.4.3 Sampling, Quantizing, and Coding with MATLAB What Have We Accomplished? Where Do We Go from Here? 420 421 428 432 433 437 439 441 444 446 Problems 447 CHAPTER Discrete-Time Signals and Systems 451 8.1 Introduction 451 8.2 Discrete-Time Signals 452 8.3 8.2.1 Periodic and Aperiodic Signals 8.2.2 Finite-Energy and Finite-Power Discrete-Time Signals 8.2.3 Even and Odd Signals 8.2.4 Basic Discrete-Time Signals Discrete-Time Systems 454 458 461 465 478 8.3.1 8.3.2 481 8.4 Recursive and Nonrecursive Discrete-Time Systems Discrete-Time Systems Represented by Difference Equations 8.3.3 The Convolution Sum 8.3.4 Linear and Nonlinear Filtering with MATLAB 8.3.5 Causality and Stability of Discrete-Time Systems What Have We Accomplished? Where Do We Go from Here? 486 487 494 497 502 Problems 502 CHAPTER The Z-Transform 511 9.1 Introduction 511 9.2 Laplace Transform of Sampled Signals 512 9.3 Two-Sided Z-Transform 515 9.4 9.3.1 Region of Convergence 516 One-Sided Z-Transform 521 9.4.1 9.4.2 9.4.3 Computing the Z-Transform with Symbolic MATLAB 522 Signal Behavior and Poles 522 Convolution Sum and Transfer Function 526 Contents 9.5 9.4.4 Interconnection of Discrete-Time Systems 537 9.4.5 Initial and Final Value Properties 539 One-Sided Z-Transform Inverse 542 9.6 9.5.1 Long-Division Method 9.5.2 Partial Fraction Expansion 9.5.3 Inverse Z-Transform with MATLAB 9.5.4 Solution of Difference Equations 9.5.5 Inverse of Two-Sided Z-Transforms What Have We Accomplished? Where Do We Go from Here? 542 544 547 550 561 564 Problems 564 CHAPTER 10 Fourier Analysis of Discrete-Time Signals and Systems 571 10.1 Introduction 571 10.2 Discrete-Time Fourier Transform 572 10.2.1 Sampling, Z-Transform, Eigenfunctions, and the DTFT 10.2.2 Duality in Time and Frequency 10.2.3 Computation of the DTFT Using MATLAB 10.2.4 Time and Frequency Supports 10.2.5 Parseval’s Energy Result 10.2.6 Time and Frequency Shifts 10.2.7 Symmetry 10.2.8 Convolution Sum 10.3 Fourier Series of Discrete-Time Periodic Signals 573 575 577 580 585 587 589 595 596 10.3.1 Complex Exponential Discrete Fourier Series 10.3.2 Connection with the Z-Transform 10.3.3 DTFT of Periodic Signals 10.3.4 Response of LTI Systems to Periodic Signals 10.3.5 Circular Shifting and Periodic Convolution 10.4 Discrete Fourier Transform 599 601 602 604 607 614 10.4.1 DFT of Periodic Discrete-Time Signals 10.4.2 DFT of Aperiodic Discrete-Time Signals 10.4.3 Computation of the DFT via the FFT 10.4.4 Linear and Circular Convolution Sums 10.5 What Have We Accomplished? Where Do We Go from Here? 614 616 617 622 628 Problems 629 CHAPTER 11 Introduction to the Design of Discrete Filters 639 11.1 Introduction 639 11.2 Frequency-Selective Discrete Filters 641 11.2.1 Linear Phase 641 11.2.2 IIR and FIR Discrete Filters 643 ix C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an 10.2 Discrete-Time Fourier Transform Downsampler x [n] M ↓ x [Mn] Decimator x [n] H (z) x [n] M ↓ x [M n] LPF Upsampler FIGURE 10.3 Top: Downsampler and decimator Bottom: upsampler and interpolator x [n] L ↑ Interpolator x [n/L] x [n] L ↑ x˜ [n] x [n/L] H (z) LPF %%%%%%%%%%%%%%%%%%%%%%%%% % Example 10.6 -down sampling and decimation %%%%%%%%%%%%%%%%%%%%%%%%% x = [ones(1,10) zeros(1,100)];Nx = length(x); n1 = 0:19; % first signal % Nx = 200;n = 0:Nx - 1; x = cos(pi ∗ n/4); % second signal y = x(1:2:Nx - 1); % downsampling with M = X = fft(x);Y = fft(y); % ffts of original and downsampled signals L = length(X);w = 0:2 ∗ pi/L:2 ∗ pi - ∗ pi/L;w1 = (w - pi)/pi; % frequency range z = decimate(x,2,‘fir’); % decimation with M = Z = fft(z); % fft of decimated signal %%%%%%%%%%%%%%%%%%%%%%%%% % interpolation %%%%%%%%%%%%%%%%%%%%%%%%% s = interp(y,2); As shown in Figure 10.4, the rectangular pulse is not band limited to π/2 since it has frequency components beyond π/2, while the sinusoid is band limited The DTFT of the downsampled rectangular pulse (a narrower pulse) is not an expanded version of the DTFT of the pulse, while the DTFT of the downsampled sinusoid is an expanded version The MATLAB function decimate uses an FIR low-pass filter to smooth out x[n] to a frequency of π/2 before downsampling In the case of the sinusoid, which satisfies the downsampling condition, the downsampling and the decimation provide the same results, but not for the rectangular pulse The original discrete-time signal can be recovered by interpolation This procedure is composed of upsampling followed by low-pass filtering The MATLAB function interp is used to that effect If we use the downsampled signal as input to this function, we obtain slightly better results for the sinusoid than for the pulse when comparing the interpolated signal to the original signal The results are shown in Figure 10.5 The error s[n] − x[n] is shown also The signal s[n] is the interpolation of the downsampled signal y[n] n 10.2.5 Parseval’s Energy Result Just like in the case of continuous-time signals, the energy or power of a discrete-time signal x[n] can be equally computed in either the time or the frequency domain Stt.010.Mssv.BKD002ac.email.ninhd.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj.dtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn 585 C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems 10 |X (e jω )| x [n] 0.5 0 10 y [n] |Y (e jω )| 10 0.5 −0.5 0.5 −0.5 ω/π 0.5 −0.5 0.5 −0.5 0.5 −0.5 ω/π 0.5 10 |Z (e jω )| z [n] −1 15 0.5 −0.5 10 0.5 −1 15 10 n −1 15 (a) 100 |X (e jω )| x [n] −1 10 100 |Y (e jω )| y [n] 50 −1 15 −1 10 50 −1 15 100 |Z (e jω )| z [n] 586 −1 10 n 50 −1 15 (b) FIGURE 10.4 Downsampling of (a) non-band-limited and (b) band-limited discrete-time signals The signals x[n] correspond to the original signals, while y[n] and z[n] are their downsampled and decimated signals, respectively The corresponding magnitude spectra are shown Notice the difference between the downsampled and the decimated signals, they are identical when the signals are band-limited, slightly different otherwise Stt.010.Mssv.BKD002ac.email.ninhd.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj.dtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Interpolated Original 10 n 12 14 16 s [n], x [n] 0.8 0.6 0.4 0.2 −0.2 −0.4 10 n 12 14 16 Interpolated Original −0.5 −1 0 0.5 18 s [n]−x [n] s [n]−x [n] s [n], x [n] 10.2 Discrete-Time Fourier Transform n 10 12 14 16 0.1 0.05 −0.05 −0.1 −0.15 18 (a) n 10 12 14 16 (b) FIGURE 10.5 Interpolation of (a) non-band-limited and (b) bandlimited discrete-time signals The interpolated signal is compared to the original signal, and the interpolation error is shown The errors signals show that the original signal can be recovered almost exactly when the signal satisfies the bandlimiting condition, not otherwise If the DTFT of a finite-energy signal x[n] is X(e jω ), we have that the energy of the signal is given by ∞ X Ex = |x[n]| = 2π n=−∞ Zπ |X(e jω )|2 dω (10.18) −π The magnitude square |X(e jω )|2 has the units of energy per radian, and so it is called an energy density When |X(e jω )|2 is plotted against frequency ω, the plot is called the energy spectrum of the signal, or how the energy of the signal is distributed over frequencies 10.2.6 Time and Frequency Shifts Shifting in time does not change the frequency content of a signal Thus, the magnitude of the signal DTFT is not affected, only the phase is Indeed, if x[n] has a DTFT X(e jω ), then the DTFT of x[n − N] for some integer N is X F(x[n − N]) = x[n − N]e−jωn n = X x[m]e−jω(m+N) = e−jωN X(e jω ) m If x[n] has a DTFT X(e jω ) = |X(e jω )|e jθ (ω) where θ (ω) is the phase, the shifted signal x1 [n] = x[n − N] has a DTFT of X1 (e jω ) = X(e jω )e−jωN = |X(e jω )|e−j(ωN−θ (ω)) Stt.010.Mssv.BKD002ac.email.ninhd.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj.dtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn 587 C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an 588 CH A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems In a dual way, when we multiply a signal by a complex exponential e jω0 n for some frequency ω0 , the spectrum of the signal is shifted in frequency So if x[n] has a DTFT X(e jω ), the modulated signal x[n]e jω0 n has as DTFT X(e j(ω−ω0 ) ) Indeed, the DTFT of x1 [n] = x[n]e jω0 n is X X X1 (e jω ) = x1 [n]e−jωn = x[n]e−j(ω−ω0 )n = X(e j(ω−ω0 ) ) n n The following pairs illustrate the duality in time and frequency shifts: if the DTFT of x[n] is X(e jω ) then x[n − N] ⇔ X(e jω )e−jωN x[n]e jω0 n ⇔ X(e j(ω−ω0 ) ) (10.19) Remark The signal x[n]e jω0 n was called modulated because x[n] modulates the complex exponential or discrete-time sinusoids It can be written as x[n] cos(ω0 n) + jx[n] sin(ω0 n) n Example 10.7 The DTFT of x[n] = cos(ω0 n), −∞ < n < ∞, cannot be found from the Z-transform or from the sum defining the DTFT as x[n] is not a finite-energy signal Use the frequency-shift and the timeshift properties to find the DTFTs of x[n] = cos(ω0 n) and y[n] = sin(ω0 n) Solution Using Euler’s identity we have that x[n] = cos(ω0 n) = e jω0 n + e−jω0 n and so the DTFT of x[n] is given by X(e jω ) = F[0.5e jω0 n ] + F[0.5e−jω0 n ] = F[0.5]ω−ω0 + F[0.5]ω+ω0 = π[δ(ω − ω0 ) + δ(ω + ω0 )] Since y[n] = sin(ω0 n) = cos(ω0 n − π/2) = cos(ω0 (n − π/(2ω0 )) = x[n − π/(2ω0 )] we have that according to the time-shift property its DTFT is given by Y(e jω ) = X(e jω )e−jωπ/(2ω0 ) h i = π δ(ω − ω0 )e−jωπ/(2ω0 ) + δ(ω + ω0 )e−jωπ/(2ω0 ) Stt.010.Mssv.BKD002ac.email.ninhd.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj.dtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an 10.2 Discrete-Time Fourier Transform h i = π δ(ω − ω0 )e−jπ/2 + δ(ω + ω0 )e jπ/2 = −jπ[δ(ω − ω0 ) − δ(ω + ω0 )] Thus, the frequency content of the cosine and the sine is concentrated at the frequency ω0 Although the sinusoids are infinite-energy signals they have finite power and their spectra can be measured with a spectrum analyzer, which displays how the power is distributed over the frequencies n 10.2.7 Symmetry When plotting or displaying the spectrum of a real-valued discrete-time signal it is important to know that it is only necessary to show the magnitude and the phase spectra for frequencies [0 π], since the magnitude and the phase of X(e jω ) are even and odd functions of ω, respectively This can be shown by considering a real-valued discrete-time signal x[n], with inverse DTFT given by x[n] = 2π Zπ X(e jω )e jωn dω −π and its complex conjugate is x [n] = 2π ∗ Zπ jω X (e )e ∗ −jωn dω = 2π −π Zπ 0 X∗ (e−jω )e jω n dω0 −π Since x[n] = x∗ [n], as x[n] is real, comparing the above integrals we have that X(e jω ) = X∗ (e−jω ) |X(e jω )|e jθ(ω) = |X(e−jω )|e−jθ(−ω) Re[X(e jω )] + jIm[X(e jω )] = Re[X(e−jω )] − jIm[X(e−jω )] or that the magnitude is an even function of ω—that is, |X(e jω )| = |X(e−jω )| (10.20) and that the phase is an odd function of ω, or θ (ω) = −θ (−ω) (10.21) Likewise, the real and the imaginary parts of X(e jω ) are also even and odd functions of ω: Re[X(e jω )] = Re[X(e−jω )] Im[X(e jω )] = −Im[X(e−jω )] Stt.010.Mssv.BKD002ac.email.ninhd.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj.dtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn (10.22) 589 C.vT.Bg.Jy.Lj.Tai lieu Luan vT.Bg.Jy.Lj van Luan an.vT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.LjvT.Bg.Jy.Lj Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an 590 CH A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems n Example 10.8 For the signal x[n] = α n u[n], < α < 1, find the magnitude and the phase of its DTFT X(e jω ) Solution The DTFT of x[n] is X(e jω ) = 1 jω = z=e −1 − αz − αe−jω since the Z-transform has a region of convergence |z| > α that includes the unit circle Its magnitude is |X(e jω )| = p (1 − α cos(ω))2 + α sin2 (ω) which is an even function of ω given that cos(ω) = cos(−ω) and sin2 (−ω) = (− sin(ω))2 = sin2 (ω) The phase is given by   α sin(ω) θ (ω) = − tan−1 − α cos(ω) which is an odd function of ω As functions of ω, the numerator is odd and the denominator is even, so that the argument of the inverse tangent is odd, which is in turn odd n n Example 10.9 For a discrete-time signal x[n] = cos(ω0 n + φ) −π ≤φ