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# HW3_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 3 Wednesday, September 10, 2008 Prof. Bresler, Prof. Jones Due by September 17, 2008 Problem 1 (10 points) Let X [ k ] denotes the N-point DFT of the discrete-time signal x [ n ] (0 n N - 1) (a) Show that X [0] = 0 if x [ n ] = - x [ N - 1 - n ]. (b) Given N is an even number, show that X [ N/ 2] = 0 if x [ n ] = x [ N - 1 - n ]. Problem 2 (25 points) Let x [ n ] be a discrete-time sequence: x [ n ] = ( - 1) n , 0 n 3 0 , otherwise (a) Show that the analytical expression for DTFT of x [ n ] is X d ( ω ) = ( e - j ( 3 ω 2 - π 2 ) . sin(2 ω ) cos( ω/ 2) ω [0 , 2 π ] , ω 6 = π 4 ω = π Hint: You can consider X d ( ω ) as the sum of a finite geometric series. Plot its magnitude and phase for 0 ω < 2 π using the following matlab function and com- mands: w = [0:128-1]/128*2*pi; Xdtft = zeros(1,128); Xdtft(65) = 4; Xdtft([1:64,66:end]) = ... exp(-i*(3/2)*w([1:64,66:end])+i*pi/2).*sin(2*w([1:64,66:end])) ... ./cos(w([1:64,66:end])/2); mag_Xdtft = abs(Xdtft); ang_Xdtft = angle(Xdtft); figure; subplot(2, 1, 1); plot(w/(2*pi), mag_Xdtft); grid on; 1

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xlabel(’\omega / (2\pi)’); ylabel(’|X_d(\omega)|’); title(’DTFT of x[n]’); subplot(2, 1, 2); plot(w/(2*pi), ang_Xdtft); grid on; xlabel(’\omega/ (2\pi)’); ylabel(’phase of X_d(\omega) in radians’); (b) Compute the 4-point DFT of x [ n ] , 0 n < 4 and stem plot its magnitude and phase. You can use the following commands: x = [1, -1, 1, -1]; M = 4; Xdft_4 = fft(x, M); m = 0:(M-1); mag_Xdft_4 = abs(Xdft_4); ang_Xdft_4 = angle(Xdft_4); figure; subplot(2, 1, 1); stem(m, mag_Xdft_4); grid on; title(’magnitude of 4-point DFT of x’); xlim([-0.2 4]); xlabel(’m’); ylabel(’|X[m]|’);
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HW3_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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