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# assn9 - Assignment 9 Due 1(a Compute the discrete-time...

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Unformatted text preview: Assignment 9 Due April 16, 2009 1.(a) Compute the discrete-time Fourier Transform (DTFT) of the signal = (b) Use DTFT properties to compute the DTFT of the signal y[n] = (n + 2. An LTI system is described the following difference equation y[n] = + %y[n — 1]. (a) Determine the frequency response H(ej‘“’) for the system. (b) Determine the impulse response h[n] for the system. 3.(a) Let 3:0?) be a signal with Nyquist rate wg. Determine the Nyquist rate for the signal 2:05) cos(2wgt). (b) If the signal \$(t) with Nyquist rate 1500s is passed through a ﬁlter with H (jw) = O for iw| > 10001r, can the signal be recovered? Justify your answer. 4. Consider the continuous-time signal 360:) = cos(407rt). We want to sample 233,05) to obtain the continuous and sampled signal 3:3,(t) = 22;“, 33C(kT)6(t — M“), where T is the sampling period. (3.) Plot XCUw), the Fourier transform of area). (b) Plot Xp(jw), the Fourier transform of 22190:), when T = 1/50. Does aliasing occur? Justify your answer. (c) Plot Xp(jw) when T = 1 / 30. Does aliasing occur? Justify your answer. Hint: Xporu) = 1% 221-00 Xcow ~ 52%))- 5.(a) Compute the continuous-time Fourier Transform of the signal shown in the ﬁgure below. — ‘2 (b) Use the Fourier Transform properties to compute the Fourier transform of the signal y(t). we; Assignment 9 -- MATLAB A. Pulse -> sinc -> pulse by FFT and IFFT o The script computes the FFT of a pulse followed by reconstruction of the pulse ﬁom the frequency-domain representation x : ones(50,1); % Original time-domain Signal x_fft1 : fft(x); % Take N = 50 point FFT (same no. of points as original signal) X_fft2 = fft(X,128); % Take N : 128 point FFT x_synth1 I ifft(x_fftl); x_synth2 = ifft(x_fft2); figure subplot(3,1,1) stem(0:1:49,x),tit1e('Actua1 time~domain signal') subp10t(3,1,2) stem(rea1(x_synth1)),tit1e('Synthesized signall'),axis([0 50 0 1]) subplot(3,1,3) stem(rea1(xﬂsynth2)),tit1e('Synthesized signa12'),axis([0 128 0 1]) 0 Run the script. Label the X and y axes appropriately. Deliverables: Plots of the synthesized signal for both the cases B. Effect of lowpass ﬁltering with different bandwidths on signal synthesis 6 The script shows the effect of bandwidth of a lowpass ﬁlter during the synthesis of O the time-domain signal. 0\0 Original time—domain signal Take N = 256 points FFT Lowpass filtered BW = 64 points Lowpass filtered BW = 128 points y_bw3 = yifft(1:256); Lowpass filtered BW = 256 points y_synth = real(ifft(y_bw1,512)); % Synthesized signall y_synth2 = real(ifft(y_bw2,512)); % Synthesized signalZ y_synth3 real(ifft(y_bw3,512)); % Synthesized signal3 figure subplot(4,l,l) stem(0:1:49,y),title(‘Actual time—domain signal') ) ) y = ones(50,1); y_fft = fftlyr512); y_bwl = y_fft(1:64); y_bw2 = y_fft(1:128); o\° o\° o\° o\° El subplot(4,1,2 plot(y_synthl ,title('Synthesized signall'),axis([0 511 0 ll); subplot(4,1,3) plot(y_synth2),title('Synthesized signal2'),axis([0 511 0 1]); subplot(4,l,4) plot(y_synth3),title('Synthesized signalB'),axis([O 511 0 ll); Run the script. Label the x and y axes appropriately. What is the effect of increasing the bandwidth of the low-pass filter on the reconstructed signal? Which of them gives a closer match to the original signal? Deliverables: Plots of the reconstructed signals for the three different bandwidths. Inferences drawn ...
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