Fundamentals of Signals and Systems Using the Web and MATLAB Edward W. Kamen Bonnie S Heck Third Edition

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1 Fundamentals of Signals and Systems Using the Web and MATLAB Edward W. Kamen Bonnie S Heck Third Edition

2 Pearson Education Limited Edinburgh Gate Harlow Essex CM JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 14 All rights reserved. o part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6 1 Kirby Street, London EC1 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners. ISB 1: ISB 13: British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

3 Section 4.3 DFT of Truncated Signals Hence, the truncated signal given by (4.49) can be expressed in the form - 1 x [n] = x[n]pcn - d (4.5) where x[n] is the original discrete-time signal whose values are known only for n =, 1, Á, - 1. ow let P1V denote the DTFT of the rectangular pulse pcn d. Setting q = 1-1/ in the result in Example 4.11 gives P1V = sin[v/] - 1V/ e-j1 sin1v/ Then, by the DTFT property involving multiplication of signals (see Table 4.), taking the DTFT of both sides of (4.5) results in the DTFT X 1V of the truncated signal x [n] given by X 1V = X1V * P1V = 1 p X1V - lp1l dl (4.51) p L-p where X1V is the DTFT of x[n]. Thus, the -point DFT X k of the truncated signal x [n] [defined by (4.49)] is given by X k = [X1V * P1V] V = pk/, k =, 1, Á, - 1 (4.5) By (4.5) it is seen that the distortion in X k from the desired values X1pk/ can be characterized in terms of the effect of convolving P1V with the spectrum X1V of the signal. If x[n] is not suitably small for n 6 and n Ú, in general, the sidelobes that exist in the amplitude spectrum ƒ P1V ƒ will result in sidelobes in the amplitude spectrum ƒ X1V * P1V ƒ. This effect is shown in the following example: Example 4.1 -Point DFT Consider the discrete-time signal x[n] = 1.9 n u[n], n Ú, which is plotted in Figure From the results in Example 4., the DTFT is 1 X1V = 1 -.9e -jv 1 x[n] n FIGURE 4.13 Signal in Example

4 Chapter 4 Fourier Analysis of Discrete-Time Signals The amplitude spectrum ƒ X1V ƒ is plotted in Figure ote that, since the signal varies rather slowly, most of the spectral content over the frequency range from to p is concentrated near the zero point V =. For = 1 the amplitude of the -point DFT of the signal is shown in Figure This plot was obtained by the commands = 1; n = :-1; x =.9.^n; Xk = dft(x); k = n; stem(k,abs(xk), filled ) Comparing Figures 4.14 and 4.15, we see that the amplitude of the 1-point DFT is a close approximation to the amplitude spectrum ƒ X1V ƒ. This turns out to be the case since x[n] is small for n Ú 1 and is zero for n 6. ow consider the truncated signal x[n] shown in Figure The amplitude of the 1-point DFT of the truncated signal is plotted in Figure This plot was generated by the following commands: = 1; n = :-1; x =.9.^n; x(1:1) = zeros(1,1); Xk = dft(x); k = n; stem(k,abs(xk), filled ) 1 X 5.5π 1π 1.5π π FIGURE 4.14 Amplitude spectrum of signal in Example Xk k FIGURE 4.15 Amplitude of 1-point DFT. 191

5 Section 4.3 DFT of Truncated Signals 1 x[n] n FIGURE 4.16 Truncated signal in Example Xk k FIGURE 4.17 Amplitude of 1-point DFT of truncated signal. Comparing Figures 4.17 and 4.15 reveals that the spectral content of the truncated signal has higher frequency components than those of the signal displayed in Figure 4.13.The reason for this is that the truncation at n = 11 causes an abrupt change in the signal magnitude, which introduces high-frequency components in the signal spectrum (as displayed by the DFT). The next example shows that the sidelobes in the amplitude spectrum ƒ X1V * P1V ƒ can produce a phenomenon whereby spectral components can leak into various frequency locations as a result of the truncation process. Example 4.13 DFT of Truncated Sinusoid Suppose that the signal x[n] is the infinite-duration sinusoid x[n] = 1cos V n, - q 6 n 6 q. From Table 4.1, we see that the DTFT of x[n] is the impulse train q a p[d1v + V - pi + d1v - V - pi] i =-q 19

6 Chapter 4 Fourier Analysis of Discrete-Time Signals A plot of the DTFT of cos V n for -p V p is shown in Figure From the figure it is seen that, over the frequency range -p V p, all the spectral content of the signal cos V n is concentrated at V = V and V = -V. ow consider the truncated sinusoid x [n] =1cos V n pcn d, where V and pcn p d is the shifted rectangular pulse defined in Example 4.1, where is an odd integer with Ú 3. Then by definition of the truncated signal is given by pcn d, cos V n, n =, 1, Á, - 1 x [n] = e, all other n As given in Example 4.1, the DTFT P1V of the pulse pcn d is P1V = sin[v/] - 1V/ e-j1 sin1v/ Then, by the DTFT property involving multiplication of signals, the DTFT X 1V of the truncated signal x [n] is given by Using the shifting property of the impulse (see Section 1.1) yields p X 1V = 1 P1V - lp[d1l + V + d1l - V ] dl p L-p X 1V = 1 [P1V + V + P1V - V ] ow the relationship (4.48) holds for the truncated signal x and thus the -point DFT X [n], k of x [n] is given by X k = X a pk b = 1 pk cpa + V b + Pa pk - V bd, k =, 1, Á, - 1 (π) (π) (repeats) π W W π W FIGURE 4.18 DTFT of x[n] = cos V n with -p V p. 193

7 Section 4.3 DFT of Truncated Signals where Pa pk sinca ; V bapk ; V bd b = sinca pk expc -ja - 1 ba pk ; V ; V bd, b n d k =, 1,, Á, - 1 Suppose that V = 1pr/ for some integer r where r - 1. This is equivalent to assuming that cos V n goes through r complete periods as n is varied from n = to n = - 1. Then Pa pk sinca ; V b = sinca k =, 1, Á, - 1 ; pr bapk bd pk ; pr b n d expc -jq pk ; pr d, = sin1pk ; pr p1k ; r expc -jq d, k =, 1, Á, - 1 pk ; pr sina b and thus Pa pk - V b = e, k = r, k =, 1, Á, r - 1, r + 1, Á, - 1 Pa pk + V b = e, k = - r, k =, 1, Á, - r - 1, - r + 1, Á, - 1 Finally, the DFT X k is given by X k = e, k = r, k = - r, all other k for k

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