You have  designed an IIR low-pass filter with an input-output relation given by  the difference equation

(i) y[n] = 0.5y[n − 1] + x[n] + x[n − 1]       n ≥ 0

where x[n] is the input and y[n] the output. You are told that by changing the difference equation to

(ii) y[n] = − 0.5y[n − 1] + x[n] − x[n − 1]   n ≥ 0

you obtain a high-pass filter.

(a) From the eigen function property find the frequency response of the  two filters at ω = 0, π/2 and π radians. Use the MATLAB functions  freqzand abs to compute the magnitude responses of the two filters. Plot them to verify that the filters are low-pass and high-pass.

(b) Call H₁ (e^jω) the frequency response of the first filter and  H₂(e^jω) the frequency response of the second filter. Show that H₂(e^jω) = H₁(e^{j(π − ω)}) and relate the impulse response h₂[n] to  h₁ [n].

(c) Use the MATLAB function zplane to find and plot the poles and  zeros of the filters and determine the relation between the poles  and zeros of the two filters.