Editing Infinite impulse response (section)

== Stability ==
The transfer function allows one to judge whether or not a system is [[Bounded-input, bounded-output stability|bounded-input, bounded-output (BIBO) stable]]. To be specific, the BIBO stability criterion requires that the [[Radius of convergence|ROC]] of the system includes the unit circle. For example, for a causal system, all [[Pole_(complex_analysis)|poles]] of the transfer function have to have an absolute value smaller than one. In other words, all poles must be located within a unit circle in the <math>z</math>-plane.

The poles are defined as the values of <math>z</math> which make the denominator of <math>H(z)</math> equal to 0:

:<math>\ 0 = \sum_{j=0}^Q a_{j} z^{-j}</math>

Clearly, if <math>a_{j}\ne 0</math> then the poles are not located at the origin of the <math>z</math>-plane. This is in contrast to the [[Finite Impulse Response|FIR]] filter where all poles are located at the origin, and is therefore always stable.

IIR filters are sometimes preferred over FIR filters because an IIR filter can achieve a much sharper transition region [[roll-off]] than an FIR filter of the same order.