The input-output relationships of two systems, H 1 and H 2 , are described by (a) For each system, find and sketch the magnitude and phase responses. What types of filters (highpass, lowpass, etc.)...

The input-output relationships of two systems, H₁
and H₂, are described by

(a) For each system, find and sketch

the magnitude and phase responses.

What types of filters (highpass, lowpass,

etc.) are H1 and H2?

(b) Find the responses of these filters to the sinusoids x_a[n] = cos(0.01πn) and x_b[n] = cos(0.99πn). In general, show that the gain of filter H₁
at Ω₀
is the same as the gain of filter H₂
at π−Ω₀; that is, show that |H₁(Ω₀)| = |H₂(π− Ω₀)|.