Question 1 a) For any linear phase filter, prove that if z is a zero, then...

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Question 1 a) For any linear phase filter, prove that if z is a zero, then so must 1 be. Hint: Using the properties of the z-transform, write hin]th[N - n ] in the z-domain, and substitute z z0 b) For any Type III or Type IV filter, prove that z 1 is a zero. e) For any Type II filter, prove that z -1 is a zero. d) In light of the above, find the zeros of the filter 1 0n3 h 0otherwise Looking at the zeros, argue whether this is a lowpass, highpass, bandpass, or bandstop filter. Hint: Since all coefficients of H(z) are real, its complex roots must come in conjugate pairs. Combine this fact with a and c. e) Repeat part d for hn 1) 0 <n<3 otherwise Question 1 a) For any linear phase filter, prove that if z is a zero, then so must 1 be. Hint: Using the properties of the z-transform, write hin]th[N - n ] in the z-domain, and substitute z z0 b) For any Type III or Type IV filter, prove that z 1 is a zero. e) For any Type II filter, prove that z -1 is a zero. d) In light of the above, find the zeros of the filter 1 0n3 h 0otherwise Looking at the zeros, argue whether this is a lowpass, highpass, bandpass, or bandstop filter. Hint: Since all coefficients of H(z) are real, its complex roots must come in conjugate pairs. Combine this fact with a and c. e) Repeat part d for hn 1) 0 <n<3 otherwise