An IIR  filter is characterized by the following difference equation  y[n] = 0.5y[n − 1] + x[n] − 2x[n − 1], n ≥ 0, where x[n] is the input and  y[n] the output of the filter. Let H(z) be the transfer function of the filter.

(a) The given filter is LTI, as such the eigenfunction property applies.  Obtain the magnitude response H(e^jω) of the filter using the eigen-function property.

(b) Compute the magnitude response |H(e^jω)| at discrete frequencies  ω = 0, π/2, and π radians. Show that the magnitude response is  constant for 0 ≤ ω ≤ π and as such this is an all-pass filter.

(c) Use the MATLAB function freqz to compute the frequency  response (magnitude and phase) of this filter and to plot them.

(d) Determine the transfer function H(z) = Y(z)/X(z), find its pole and  zero and indicate how they are related.