Infinite impulse response

An infinite impulse response (IIR) filter, as suggested by its name, has an impulse response which lasts forever.

An IIR filter is typically characterized by its order, which is the number of feedback stages required. An infinite impulse response filter having <math>P<math> feed-forward stages and <math>Q<math> feedback stages has the following form:

<math> y_n = b_0 x_n + b_1 x_{n-1} + \cdots + b_P x_{n-P} + a_1 y_{n-1} + a_2 y_{n-2} + \cdots + a_Q y_{n-Q} <math>

The z-transform of the above <math>Q<math>^th-order filter is given as

<math> H(z) = \frac{\sum_{k=0}^P b_k z^{-k}} {1 - \sum_{k=1}^Q a_k z^{-k}} <math>

If the poles of <math>H(z)<math> are located inside the unit circle then the filter is stable. If any pole is outside the unit circle, the filter will be unstable. If a filter is unstable, its impulse response will be unbounded.

Infinite impulse response filters are sometimes prefered to finite impulse response filters because an IIR filter can achieve a much sharper transition region roll-off than an FIR filter of the same order.