An infinite impulse response (IIR) system is a type of a [[discrete-time LTI system]] where the output depends both on a finite number of input and output samples. It is also known as an IIR filter. A [[discrete-time LTI system]], [[causal system|causal]], with input $x(n)$ and output $y(n)$, can be characterized by a [[difference equation]] in the form: $ y(n) = \frac{1}{a_{0}} \left(\sum_{k=0}^{M} b_k x(n-k) -\sum_{k=1}^{N} a_k y(n-k) \right) $ where $a_{0}$ is often $1$. The system is an infinite impulse response system if there is at least another non-zero $a_{k}$ parameter. To compute the output signal it is necessary to know the initial conditions of the filter. ![[fdir1.svg|500]]