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 mentionedI 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

What is the system function of the second filter other than comb filte

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

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