Infinite impulse response

:"IIR redirects here. For the conference company IIR, see" Informa.Infinite impulse response (IIR) is a property of signal processing systems. Systems with that property are known as "IIR systems" or when dealing with electronic filter systems as "IIR filters". They have an impulse response function which is non-zero over an infinite length of time. This is in contrast to finite impulse response filters (FIR) which have fixed-duration impulse responses. The simplest analog IIR filter is an RC filter made up of a single resistor (R) feeding into a node shared with a single capacitor (C). This filter has an exponential impulse response characterized by an RC time constant.

IIR filters may be implemented as either analog or digital filters. In digital IIR filters, the output feedback is immediately apparent in the equations defining the output. Note that unlike with FIR filters, in designing IIR filters it is necessary to carefully consider "time zero" case in which the outputs of the filter have not yet been clearly defined.

Design of digital IIR filters is heavily dependent on that of their analog counterparts because there are plenty of resources, works and straightforward design methods concerning analog feedback filter design while there are hardly any for digital IIR filters. As a result, mostly, if a digital IIR filter is going to be implemented, first, an analog filter (e.g. Chebyshev filter, Butterworth filter, Elliptic filter) is designed and then it is converted to digital by applying discretization techniques such as Bilinear transform or Impulse invariance.

In practice, electrical engineers find IIR filters to be "fast" and "cheap", but with "poorer bandpass filtering and stability characteristics" than FIR filters.

Example IIR filters include the Chebyshev filter, Butterworth filter, and the Bessel filter.

In the following subsections we focus on discrete time IIR filters which can be implemented in Digital Signal Processors.

Discussion

We start the discussion by stating the difference equation which defines how the input signal is related to the output signal:

:

where:
*$P$ is the feedforward filter order
*$b\_\{i\}$ are the feedforward filter coefficients
*$Q$ is the feedback filter order
*$a\_\{i\}$ are the feedback filter coefficients
*$x\; [n]$ is the input signal
*$y\; [n]$ is the output signal.

A more condensed form of the difference equation is:

:$y\; [n]\; =\; sum\_\{i=0\}^P\; b\_\{i\}x\; [n-i]\; -\; sum\_\{j=1\}^Q\; a\_\{j\}\; y\; [n-j]$

which, when rearranged, becomes:

:$sum\_\{j=0\}^Q\; a\_\{j\}\; y\; [n-j]\; =\; sum\_\{i=0\}^P\; b\_\{i\}x\; [n-i]$

if we let $a\_0\; =\; 1$.

To find the transfer function of the filter, we first take the Z-transform of each side of the above equation, where we use the time-shift property to obtain:

:$sum\_\{j=0\}^Q\; a\_\{j\}\; z^\{-j\}\; Y(z)\; =\; sum\_\{i=0\}^P\; b\_\{i\}\; z^\{-i\}\; X(z)$

We define the transfer function to be:

:

Description of block diagram

A typical block diagram of an IIR filter looks like the following. The $z^\{-1\}$ block is a unit delay. The coefficients and number of feedback/feedforward paths are implementation-dependent.

Stability

The transfer function allows us to judge whether or not a system is bounded-input, bounded-output (BIBO) stable. To be specific, the BIBO stability criteria requires the ROC of the system include the unit circle. For example, for a causal system, all 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 $z$-plane.

The poles are defined as the values of $z$ which make the denominator of $H(z)$ equal to 0:

:$0\; =\; sum\_\{j=0\}^Q\; a\_\{j\}\; z^\{-j\}$

Clearly, if $a\_\{j\}\; e\; 0$ then the poles are not located at the origin of the z-plane. This is in contrast to the 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 FIR filter of the same order.

Example

Let the transfer function of a filter "H" be

:$H(z)\; =\; frac\{B(z)\}\{A(z)\}\; =\; frac\{1\}\{1\; -\; a\; z^\{-1$ with ROC and

which has a pole at "a", is stable and causal.The time-domain impulse response is

:$h(n)\; =\; a^\{n\}\; u(n)$

which is non-zero for $n\; >=\; 0$.

See also

*Electronic filter
*Finite impulse response
*System analysis

External links

* [http://www.bores.com/courses/intro/iir/index.htm The fifth module of the BORES Signal Processing DSP course - Introduction to DSP]