# 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):

$\begin{array}{|lll|}\hline \text{Algorithm:}\hfill & \hfill {y}_{k}& ={\displaystyle \sum _{n=0}^{N-1}{b}_{n}{x}_{k-n}}-{\displaystyle \sum _{m=1}^{M-1}{a}_{m}{y}_{k-m}}\hfill \\ \text{Transfer}\text{\hspace{0.17em}}\text{function:}\hfill & \hfill H\left(z\right)& =\frac{Y\left(z\right)}{X\left(z\right)}=\frac{{b}_{0}+{b}_{1}{z}^{-1}+\cdots +{b}_{N-1}{z}^{-\left(N-1\right)}}{1+{a}_{1}{z}^{-1}+\cdots +{a}_{M-1}{z}^{-\left(M-1\right)}}\hfill \\ \text{Filter}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{gain:}\hfill & \hfill H\left({e}^{j\text{\omega}T}\right)& =\frac{{b}_{0}+{b}_{1}{e}^{j\text{\omega}T}+\cdots +{b}_{N-1}{e}^{-j\left(N-1\right)\text{\omega}T}}{1+{a}_{1}{e}^{-j\text{\omega}T}+\cdots +{a}_{M-1}{e}^{-j\left(M-1\right)\text{\omega}T}}\hfill \\ \hline\end{array}$ |
(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. ...

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