A discrete-time system with system function H(Z) and impulse response h[n] has frequency...

A discrete-time system with system function H(Z) and impulse response h[n] has frequency response

where 0 < θ_c < π.This filter is transformed into a new filter by the transformation Z = −z²; i.e.,

(a) Obtain a relationship between the frequency variable θ for the original lowpass system H(Z) and the frequency variable ω for the new system H₁(z).

(b) Sketch and carefully label the frequency response H₁(e^jω) for the new filter.

(c) Obtain a relationship expressing h₁[n] in terms of h[n].

(d) Assume that H(Z) can be realized by the set of difference equations

where x[n] is the input and y[n] is the output of the system. Determine a set of difference equations that will realize the transformed system H₁(z) = H(−z²).