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## 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 {bi}1≤i≤N−1 is non-null. We can easily reduce this to a relation where a0 = 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(kM) and of N + 1 values of the input signal x(k),…, x(kN). 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 ...

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