Let the filter H(z) be the cascade of a causal filter with transfer function G(z) and an anti-causal filter with transfer function G(z^ˆ’1), so that

H(z) = G(z) G(z^ˆ’1)

(a) Suppose that G(z)is an FIR filter with transfer function

Find the frequency response H(e^jω) and determine its phase.

(b) Determine the impulse response of the filter H(z). Is H(z) a causal  filter? If not, would delaying its impulse response make it causal?  Explain. What would be the transfer function of the causal filter?

(c) Use MATLAB to verify the unwrapped phase of H(z) you  obtained analytically, and to plot the poles and zeros of H(z).

(d) How would you use the MATLAB function convto find the  impulse response of H(z).

(e) Suppose then that G(z) = 1/(1 ˆ’ 0.5z^ˆ’1), find the filter H(z) = G(z) G(z^ˆ’1). Is this filter zero-phase? If so, where are its  poles and zeros located? If you think of filter H(z) as causal, is it  BIBO stable?

Transcribed Image Text:

G(z) = -(1+2z- +z-2) 3