Question

1. a) Compute the impulse response of the following filters. If the impulse response is infinite, feel free to stop once a pattern becomes 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 starts at 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 FIR filters; which ones? For each one, construct plots (using a computer) of the DFT of the impulse response magnitude |H[m]|. (To make it concrete, zero-pad the impulse response to N = 512 samples.) What can you infer about these filters from the resulting plots?

d) For all of the systems given above (either FIR or IIR), compute (sample) their z-transform by using scipy.signal.freqz, and plot the resulting magnitude responses |H(z)|. For the FIR systems, how do the plots compare to your plots in question 3? For the IIR systems, what can you learn about the systems from the frequency response 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 or low-pass, and whether or not it is stable.

Answer 1