The modulation-based frequency transformation of the DTFT is applicable to IIR filters. It is obvious in the case of FIR filters, but requires a few more steps in the case of IIR filters. In fact, if we have that the transfer function of the prototype IIR low-pass filter is H(z) = B(z)/A(z), with impulse response h[n], let the transformed filter be Ĥ(z) = Z(2h[n] cos(ω_on)) for some frequency ω₀.

(a) Find the transfer function Ĥ(z) in terms of H(z)

(b) Consider an IIR low-pass filter H(z) = 1/(1 − 0.5z⁻¹). If ω₀ = π/2 determine Ĥ(z).

(c) How would you obtain a high-pass filter from H(z) given in the previous item? Use MATLAB to plot the resulting filters here and in the past item.