Question

What is the system function of all-zero filter or comb filter?

(a) \(\frac{1}{M}(1+z^{-M} e^{j2πα})\)

(b) \(\frac{1}{M}(1+z^M e^{j2πα})\)

(c) \(\frac{1}{M}(1-z^M e^{j2πα})\)

(d) \(\frac{1}{M}(1-z^{-M} e^{j2πα})\)

The question was asked in examination.

I need to ask this question from Structures for FIR Systems topic in chapter Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

The correct answer is (d) \(\frac{1}{M}(1-z^{-M} e^{j2πα})\)

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

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

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 whose system function is given by the equation

H1(z)=\(\frac{1}{M}(1-z^{-M} e^{j2πα})\).