Question

Consider the DT LTI system defined by the mpulse response h[n] = ?[n] The input to this system is the signal rn: ?[n-1) (a) Sketch h[n] and r[n] (b) Determine the output of the systern, ylnj, using convolution. Sketch y[n] (c) Determine the DTFTs H(e) and X(e. Make fully-labeled sketches of the magni- 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) DFTs of h[n and r[n]. Call these DFTs HA[k] and X4[k]. Note that you while you can find H4[k] and X4 kl 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 Y4[k as follows: i.e., Y4[k] is the element-wise multiplication of HA[k and X4[k]. Determine and sketch y4n], which is the inverse DFT of Y4[k]. Note, you should be able to write y4In in terms of the signal y[n] you found in part (b). For n-0,...,N -1, is ynn? (f) Repeat the exercise in parts (d) and (e) using 3-point DFTs (N- 3). How do your results change?

Answer 1

yin? an ter jus 〔e 2 Sin 4 久 Cy -ス

Show Data Cursor Figure 2: pl 2 3.5 1.5 2.5 0.5 F 1.5 -0.5 0.5 1.5 F .2 -2 -2 2

l t point DFT 2T (2) Ht(k)는 4-petntPFT.of h1n) re