Question

2. Consider the DT LTI system defined by the impulse response h[n]-i[n]-?[n-1]. The input to this system is the signal rn: (a) Sketch hn and n (b) Determine the output of the system, y[n], using convolution. Sketch y[n (c) Determine the DTFTs H(ei) and X(e). Make fully-labeled sketches of the magn tudes of these DTFTs. (d) Recall that the discrete Fourier transform (DFT) is simply defined as samples of the discrete-time Fourier transform (DTFT). Compute the 4-point (N = 4) DFT's of h[n] and n. Call these DFTs HAk] and X[k]. Note that you while you can find HA[k] and Xi[k] using the DFT analysis formula, you should also be able to compute the values by taking samples of the DTFT expressions you found in part (c) (e) Suppose that you define Yk] as follows: i.e., Y4[k] is the element-wise muliplication of H4k and X4k]. Determine and sketch y[n], which is the inverse DFT of Yk. Note, you should be able to write y4In in terms of the signal yn you found in part (b). For n 0,..,N -1, is ya n yn? (f) Repeat the exercise in parts (d) and (e) using 3-point DFT's (N 3). How do your results change?

So sorry for the long question, I am able to do a) and b) but not sure about the rest

Answer 1

Given impulse response h(n) = delta (n) - delta(n - 1) and i/p to the system x(n) x(n) = delta(n) + 2 delta(n) + delta (n - 2) a) sketch h(n) and x(n) b) determine o/p of y(n) using construction y(n) sketch h(n) = {1, - 1} x(n) = {1,2,1} by using row column method y(n) = {1,1, - 1, - 1} y(n0 = delta (n) + delta (n - 1) - delta(n - 2) - delta(n - 3) y(n) = delta(n) + delta(n - 1) - delta (n - 2) - delta (n - 3)