Question

Which of the following is the equivalent representation of the denominator of the system function of a general IIR filter?

(a) \(\prod_{k=1}^N (1+p_k z^{-1})\)

(b) \(\prod_{k=1}^N (1+p_k z^{-k})\)

(c) \(\prod_{k=1}^N (1-p_k z^{-k})\)

(d) \(\prod_{k=1}^N (1-p_k z^{-1})\)

The question was asked during an interview for a job.

This intriguing question originated from Quantization of Filter Coefficients topic in chapter Digital Filters Design of Digital Signal Processing

Answer 1

Right answer is (d) \(\prod_{k=1}^N (1-p_k z^{-1})\)

Easiest explanation: We know that the system function of a general IIR filter is given by the equation

H(z)=\(\frac{\sum_{k=0}^M b_k z^{-k}}{1+\sum_{k=1}^N a_k z^{-k}}\)

The denominator of H(z) may be expressed in the form

D(z)=\(1+\sum_{k=1}^N a_k z^{-k}=\prod_{k=1}^N (1-p_k z^{-1})\)

where pk are the poles of H(z).