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EE 453 Homework #11: Due April 23, 2010 April 15, 2010 1. Consider the nite-length discrete time sequence x[n] = [1 0 0 0 1 0 0 0]. (a) Determine an...

Could i please get help on 1, 2, and 5 on the attached file.

EE 453 Homework #11: Due April 23, 2010 April 15, 2010 1. Consider the finite-length discrete time sequence x[n] = [1 0 0 0 1 0 0 0]. (a) Determine an expression for the DTFT of this signal. Also, determine an ex- pression for the magnitude of the DTFT and obtain a sketch for the frequency range [0, 2 π ]. (b) Use your results in (a) to determine (by hand) a general expression (as a function of k) for the N-point DFT of the discrete-time signal for the following N i. N=8 ii. N=32 iii. N=128 For each N-point DFT, calculate X(4). What effect does increasing N have on the DFT? (c) For the same Ns as in (b), use the MATLAB command fft to calculate the N-point DFTs of the discrete-time sequence. Use the stem command to plot the DFT magnitude for each N above. Verify that the MATLAB results agree with your hand-calculated results in (b) by examining the vector entry for k=4. Recall that MATLAB starts indexing at 1, not 0, so k=8 corresponds to the 5th vector location in MATLAB. 2. An analog signal x(t) (unknown length) is sampled at Fs = 80 kHz and its frequency spectrum is estimated via a 1000 point DFT. (a) What is the frequency spacing/resolution (in Hz) between spectrum samples? (b) What are the frequency-domain limitations on x(t) to avoid frequency-domain aliasing when calculating the DFT? 1
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(c) What are the time-domain limitations on x(t) to avoid rippling or smearing in the DFT? (d) Discuss the tradeoffs in determining what to use for Fs, N, and L when estimating the Fourier Transform using the DFT. 3. The ability to resolve (separately identify) closely spaced sinusoids using the DFT is a common problem in digital signal processing applications. As we discussed in class, if L is too small (too much windowing of the data), the main lobes of the sinusoids start combining and the resulting DFT plot shows only a single sinusoid. Consider the analog signal x ( t ) = 4 cos (2 π 1500 t ) + 5 cos (2 π 1750 t ) (1) sampled using a sampling frequency of Fs = 20,000 Hz. (a) Use the fft command in MATLAB to generate DFT plots (magnitude only) for the following cases. Due to the large N that were using, use plot, not stem when you generate the frequency plot. For large N, it looks nicer drawing a smooth figure. Plot the DFT magnitude data vs. continuous-time frequency F. You can use the fftplot MATLAB code in the notes. Make sure to use fftshift so that you get frequencies in the range [-Fs/2 Fs/2). i. L = 20, N=1000, Rectangular window ii. L = 80, N=1000, Rectangular window iii. L = 200, N=1000, Rectangular window iv. L=500, N=1000, Rectangular window (b) Comment on what happens as L is increased. Specifically, at which value of L are you able to see two distinct sinusoids? (c) Repeat (a-b) but use a Hamming window. (d) Compare the results in (a)-(c). What are the tradeoffs in using the Hamming window instead of the rectangular window? (e) Suppose that the input signal was x ( t ) = 100 cos (2 π 1500 t ) + cos (2 π 2500 t ) (2) Repeat a-iv and c-iv using this new input signal and plot the DFT magnitude vs. F for both cases. What problem exists in both cases? What window would you use to try to solve this problem? Apply this new window in MATLAB and 2
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