Consider a continuous-time lowpass filter Hc(s) with passband and stopband specification...

Consider a continuous-time lowpass filter Hc(s) with passband and stopband specifications

This filter is transformed to a lowpass discrete-time filter H₁(z) by the transformation

and the same continuous-time filter is transformed to a highpass discrete-time filter by the transformation

(a) Determine a relationship between the passband cutoff frequency Ωp of the continuous time lowpass filter and the passband cutoff frequency ω_p1 of the discrete-time lowpass filter.

(b) Determine a relationship between the passband cutoff frequency Ωp of the continuous time lowpass filter and the passband cutoff frequency ω_p2 of the discrete-time highpass filter.

(c) Determine a relationship between the passband cutoff frequency ω_p1 of the discrete time lowpass filter and the passband cutoff frequency ω_p2 of the discrete-time highpass filter.

(d) The network in Figure P7.28 depicts an implementation of the discrete-time lowpass filter with system function H₁(z). The coefficients A, B, C, and D are real. How should these coefficients be modified to obtain a network that implements the discrete-time highpass filter with system function H₂(z)?