Use MATLAB to design a Butterworth second-order low-pass discrete filter H(Z) with half-power frequency hp =

Use MATLAB to design a Butterworth second-order low-pass discrete filter H(Z) with half-power frequency θ_hp = π/2, and dc gain of 1. Consider this low-pass filter a prototype that can be used to obtain other filters. Implement using MATLAB the frequency transformations Z⁻¹ = N(z)/D(z) using the convolution property to multiply polynomials to obtain:

(a) A high-pass filter with a half-power frequency ω_hp = π/3 from the low-pass filter.

(b) A band-pass filter with ω₁ = π/2 and ω₂ = 3π/4 from the low-pass filter.

(c) Plot the magnitude of the low-pass, high-pass and band-pass filters. Give the corresponding transfer functions for the low-pass as well as the high-pass and the band-pass filters.