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Unformatted text preview: Assignment 9 Due April 16, 2009 1.(a) Compute the discretetime 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 continuoustime 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% 22100 Xcow ~ 52%)) 5.(a) Compute the continuoustime 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 frequencydomain representation x : ones(50,1); % Original timedomain 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 timedomain 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 lowpass 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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 Spring '10
 Antonia
 Digital Signal Processing, Lowpass

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