A discrete-time system with system function H(Z) and impulse response h[n] has frequency response
where 0 < θc < 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(ejω) 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).
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