Suppose that we are given a continuous-time lowpass filter with frequency response H c (j) such that

Suppose that we are given a continuous-time lowpass filter with frequency response H_c(jω) such that 

1− δ₁ ≤ | H_c(jΩ) | ≤ 1 + δ₁,        |Ω| ≤ Ω_p,

|H_c(j Ω)| ≤ δ₂,               |Ω| ≥ Ω_s.

 A set of discrete-time lowpass filters can be obtained from H_c(s) by using the bilinear transformation, i.e.,

H(z) = H_c(s)|_{s = (2/Td)[(1 – z – 1)/(1 + z – 1)]}’

 with T_d variable.

(a) Assuming that Ω_p is fixed, find the value of T_d such that the corresponding passband cutoff frequency for the discrete-time system is ω_p = π/2.

(b) With Ω_p fixed, sketch ω_p as a function of 0 < T_d < ∞.

(c) With both Ω_p and Ω_s fixed, sketch the transition region ∆ω = (ω_s – ω_p) as a function of 0 < T_d < ∞.