1. Compute the 8-point DFT of the following four vectors, BY HAND, and simplify.

[1, 0, 0, 0, 0, 0, 0, 0]' , [0, 0, 0, 0, 0, 0, 0, 1]' , [0, 0, 0, 1, 0, 1, 0, 0]' , [1, 1, 1, 1, 1, 1, 1, 1]' .

Express each answer as a vector with 8 complex entries. Compare with the answers you get using

MATLAB commands "t" and "it" and explain any discrepancies.

2. For the six filters listed below, do the following:

a. find the Z-transform of the filter (the transfer function H(z).),

b. identify the zeros and poles (if any) of the function H(z),

c. find the frequency response of the filter, as a function of normalized frequency ω:H(e^2Πiω),

d. plot the amplitude response of the filter |H(e^2Πiω|,

e. identify whether the filter is low pass, high pass, or band pass,

f. compute and plot the impulse response of the filter.

You should think carefully about how to present your answers in a neat, organized manner.3. Find and plot the phase response for each of the above six filters. (You may need to do this numerically with MATLAB, for some of them.)

4. Find the filter coefficients h = (: : : , h 1, h0, h1, h2, : : :) for a filter with frequency response:

Note: the calculation involves an integral of ! times a trig function, solve using integration by parts.

5. Use MATLAB to make overlapping plots of 4 window functions, Hann, Hamming, Taylor, and Blackman, each with 31 points. Then plot the amplitude response in decibels using plot(pow2db(abs(t( However, use more than 31 points in the amplitude response. Are they dierent?