## 6.1. Introduction to infinite impulse response filters

Infinite impulse response (IIR) filters are recursive mode filters that are characterized by the following difference equation:

where at least one of the coefficients {*b*_{i}}_{1≤i≤N−1} is non-null. We can easily reduce this to a relation where *a*_{0} = 1. From here, we will assume that this hypothesis is satisfied.

Equation (6.1) is verified for all the values of *k*. We thus have:

By reinjecting equation (6.2) of *y*(*k* − 1) in the difference equation in (6.1), we see that *y*(*k*) depends on the preceding values of the output *y*(*k* − 2),…, *y*(*k* − *M*) and of *N* + 1 values of the input signal *x*(*k*),…, *x*(*k* −*N*). By repeating this step to infinity, we express the output *y*(*k*) as a linear combination of an infinity of terms of the input signal *x*(*k*). The filter is therefore an IIR filter.

The z-transform on equation (6.2) helps us obtain the transfer function of the filter, which is a rational fraction in z:

The division following the increasing powers of the numerator by the denominator then leads to an infinite sum of terms.

If the order of a finite impulse response (FIR) filter is between 25 and 400, that of an equivalent IIR filter is generally lower and ...