0.12 Iir filters

An IIR filter describes a system with input $x\left(n\right)$ and output $y\left(n\right)$ , related by the following expression

Since the current output $y\left(n\right)$ depends on the input as well as on $N$ previous output values, the output of an IIR filter might not be zero well after $x\left(n\right)$ becomes zero (hence the name “Infinite”). Typically IIR filters are described by a rational transfer function of the form

where

and $h\left(n\right)$ is the infinite impulse response of the filter. Its frequency response is given by

Substituting [link] into [link] we obtain

Given a desired frequency response $D\left(\omega \right)$ , the $l_2$ IIR design problem consists of solving the following problem

for the $M+N+1$ real filter coefficients $a_n,b_n$ with $\omega \in \Omega$ (where $\Omega$ is the set of frequencies for which the approximation is done). A discrete version of [link] is given by

where ${\omega }_k$ are the $L$ frequency samples over which the approximation is made. Clearly, [link] is a nonlinear least squares optimization problem with respect to the filter coefficients.