Digital Signal Processing with Examples in MATLAB®, 2nd Edition

6

IIR Filter Design

6.1    Introduction

In Chapter 4, we saw that the transfer function of a linear recursive or infinite impulse response (IIR) filter is a ratio of polynomials in z, rather than just a single polynomial that describes a finite impulse response (FIR) filter. The direct-form algorithm, transfer function, and gain of an IIR filter were given in (4.10) through (4.12):

$\begin{array}{|lll|}\hline \text{Algorithm:}\hfill & \hfill y_k& ={\displaystyle \sum _{n=0}^{N-1}{b}_n{x}_{k-n}}-{\displaystyle \sum _{m=1}^{M-1}{a}_m{y}_{k-m}}\hfill \\ \text{Transfer}\text{function:}\hfill & \hfill H\left(z\right)& =\frac{Y\left(z\right)}{X\left(z\right)}=\frac{b_0+b_1{z}^{-1}+\cdots +b_{N-1}{z}^{-\left(N-1\right)}}{1+a_1{z}^{-1}+\cdots +a_{M-1}{z}^{-\left(M-1\right)}}\hfill \\ \text{Filter}\text{gain:}\hfill & \hfill H\left(e^{j\text{ω}T}\right)& =\frac{b_0+b_1{e}^{j\text{ω}T}+\cdots +b_{N-1}{e}^{-j\left(N-1\right)\text{ω}T}}{1+a_1{e}^{-j\text{ω}T}+\cdots +a_{M-1}{e}^{-j\left(M-1\right)\text{ω}T}}\hfill \\ \hline\end{array}$

(6.1)

Thus, the IIR transfer function has poles as well as (usually) zeros on the z-plane, and, as stated in (4.39), the poles must be inside the unit circle for stability. ...