Question

What is the system function of the second filter other than comb filter in the realization of FIR filter?

(a) \(\sum_{k=0}^M \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

(b) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1+e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

(c) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

(d) None of the mentioned

I have been asked this question in unit test.

The above asked question is from Structures for FIR Systems topic in division Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

Correct option is (c) \(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)

Easiest explanation: The system function H(z) which is characterized by the set of frequency samples is obtained as

H(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}}z^{-1}}\)

We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)

Here H1(z) represents the all-zero filter or comb filter, and the system function of the other filter is given by the equation

H2(z)=\(\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}\)