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**Unformatted text preview: **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 , n 3 , otherwise (a) Show that the analytical expression for DTFT of x [ n ] is X d ( ) = ( e- j ( 3 2- 2 ) . sin(2 ) cos( / 2) [0 , 2 ] , 6 = 4 = Hint: You can consider X d ( ) as the sum of a finite geometric series. Plot its magnitude and phase for 0 < 2 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 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 ] , 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);...

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