Homework Hints & Clarifications

Homework 6

Problem 2: The sampling frequency is Omega = 2*pi/T, where T = ln 2. The frequency of the input signal is Omega/10. Note that, for example, A*delta(Omega-Omega_0) in the frequency domain is associated with a power of |A/(2*pi)|^2. And remember that when a continuous-time signal is sampled, the replicas in the frequency domain are scaled by 1/T.
Problem 7: The wording of the problem is a little confusing, but all the answers should be in terms of x[n] and the autocorrelation of x[n].
Problem 7(c): You don't have to do the autocorrelation limit in this part.
Problem 7(d): You don't have to sketch the parameters. You do not have to do the mean-squared value in this part.
Problem 11: You may assume that the output is zero mean, as well as uncorrelated.

Homework 5

Problem 1 & 2: You should be able to compute all the parameters analytically. Use the equations in the Kaiser Window and Multiband sections of Lecture 7 (reader). You should use Matlab just to plot the frequency response. Some of you are having problems plotting the impulse response because of discontinuities. One way of doing it in Matlab is:

% Ideal Impulse response
h = sin(w2*(n - M/2+eps))./(pi*(n-M/2+eps)) - sin(w1*(n - M/2+eps))./(pi*(n-M/2+eps));

% kaiser window
window_Kaiser = kaiser(M,Beta);
h_t = h'.*window_Kaiser;

H_k = freqz(h_t,1,512);

Note that w1 and w2 are the cutoff frequencies of the filter (that you calculated) and "eps" is used to avoid the discontinuity at n=0 ("eps" is the smallest number closest to zero in Matlab).
Problem 3: There are no restrictions on the order N of the filters. Only the criteria need to be satisfied. To give you some idea of the order of the filters: (a) N=27; (b) N=34; (c) N=152; (d) N=140.
Problem 3(b): Don't spend too much time trying to meet the exact specifications with yulewalk. If your results are reasonable, you'll get full credit. In general, when specifying the frequency response avoid abrupt transitions from passband to stopband. Play with the slope of the transition and try to get it to be as smooth as possible.

Homework 4

Problem 1: Compute the Fourier transform of g(t) and sketch a diagram. It will help to assume that T_H << T.
Problem 3: a) What are the properties of the DFT of a real signal? b) Look at the equations for the DFT and IDFT
Problem 4: Split the DFT of f[n] into the three sums by using the proper indexing in "n".

Homework 3

Problem 6: Use the definitions and properties (e.g., aliasing, location of poles, and zeros) of impulse invariance and bilinear transformation. Wherever appropriate use the equations for these methods. No need to assume bandlimitedness. For the impulse invariance method, the zeros are not mapped, regardless of aliasing.
Problem 6(e): With regards to impulse invariance, consider one of the transfer functions to be bandlimited, and the other one to be not bandlimited (e.g., rect). Think about what will happen if you multiply the transfer functions first and then sample, as opposed to sampling first and then multiplying the sampled versions. Will the end result be the same? Use similar approach for 6(g).
Problem 7: For a continuous time filter, if the system is causal, then for stability its poles should be on the left half of the s-plane. (However, if the filter is anti-causal, then for stability the poles should be on the right half of the s-plane.)
Problems 8 & 9: You may find the following Matlab functions useful: buttord, buttap, cheb2ord, cheb2ap, ellipord, ellipap, lp2lp, bilinear, freqs, freqz.