Consider a second-order discrete-time system represented by the following difference equation: y[n] 2r cos( 0 ) y[n 1] + r 2 y[n

Consider a second-order discrete-time system represented by the following difference  equation:

y[n] − 2r cos(ω₀) y[n − 1] + r²y[n − 2] = x[n]          n ≥ 0

where r > 0 and 0 ≤ ω₀ ≤ 2π, y[n] is the output and x[n]the input.

(a) Find the transfer function H(z) of this system.

(b) Determine the values of ω₀ and of r that make the system stable. Use the MATLAB function zplane to plot the poles and zeros for r = 0.5 and ω₀ = π/2 radians.

(c) Let ω₀ = π/2, find the corresponding impulse response h[n] of the  system. For what other value of ω₀ would get the same impulse  response?