1. a) Compute the impulse response of the following filters. Ifthe impulse response is infinite, feel free to stop once a patternbecomes apparent.

i. y[n] = 2/3 · x[n − 1] − 1/3 x[n − 2]

ii. y[n] = x[n − 1] − x[n − 2] − 1/3 y[n − 2]

b) What are the feed-back (b[k]) and feed-forward (a[k])coefficients of the following filters? You may assume that a startsat delay of k = 1 (i.e., does not include the gain factor).

i. y[n] = 1/2 · x[n] + 1/2 · x[n − 2] + y[n − 1] − y[n − 2] +y[n − 3]

ii. y[n] = ∑ 4 k=0 (−1)^k · x[n − k]

c) Two of the systems given in questions (1) and (2) are FIRfilters; which ones? For each one, construct plots (using acomputer) of the DFT of the impulse response magnitude |H[m]|. (Tomake it concrete, zero-pad the impulse response to N = 512samples.) What can you infer about these filters from the resultingplots?

d) For all of the systems given above (either FIR or IIR),compute (sample) their z-transform by using scipy.signal.freqz, andplot the resulting magnitude responses |H(z)|. For the FIR systems,how do the plots compare to your plots in question 3? For the IIRsystems, what can you learn about the systems from the frequencyresponse curves? What do these curves not tell you?

e) Compute the poles and zeros of the filters given in questions(1) and (2). For each filter, determine if it is high-pass orlow-pass, and whether or not it is stable.