Question

Where does the poles of the system function of the second filter locate?

(a) e^j2π(k+α)M

(b) e^j2π(k+α)/M

(c) e^j2π(k-α)/M

(d) e^jπ(k+α)/M

This question was addressed to me in semester exam.

Enquiry is from Structures for FIR Systems in chapter Discrete Time Systems Implementation of Digital Signal Processing

Answer 1

Right option is (b) e^j2π(k+α)/M

For explanation I would say: The system function of the second filter in the cascade of an FIR realization by frequency sampling method is given by

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

We obtain the poles of the above system function by equating the denominator of the above equation to zero.

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

=>z=pk=\(e^{\frac{j2π(k+α)}{M}}\), k=0,1….M-1