An infinite impulse response (IIR) filter has an impulse response which lasts forever. An IIR has infinite duration because feedback of the outputs is used to calculate other output values. Unlike the finite impulse response (FIR) filter, the IIR filter must have an initial condition for each feedback value.

Contents

1 Transfer function 2 Block Diagram 3 Stability 4 Example 5 Truncated IIR filter 6 See also 7 External links

Transfer function

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

<math> y(n) = \left[b(0) x(n) + b(1) x(n-1) + \cdots + b(P) x(n-P)\right] + \left[a(1) y(n-1) + a(2) y(n-2) + \cdots + a(Q) y(n-Q)\right]<math>

<math> y(n) = \sum_{k=0}^P b(k)x(n-k) + \sum_{k=1}^Q a(k) y(n-k)<math>

and after taking the z-transform

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

See Z-transform#LCCD equation for more details.

Block Diagram

A typical block diagram of an IIR filter looks like the following. The "T" block is a unit delay.

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Iir-filter-wiki.png
Image:Iir-filter-wiki.png

Stability

From the region of convergence of the Z-transform, the stability of an IIR filter can be evaluated. If all poles are within the unit circle (i.e., <math>|z_p| < 1<math>) then the filter is stable.

By definition of the region of convergence, then if the system is stable then <math>\sum_{k=-\infty}^{\infty} |h(k)| < \infty<math> will hold true. Otherwise, if the system is unstable then the sum will be <math>\infty<math>.

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.

Example

Let the transfer function of a filter H be

<math>H(z) = \frac{A(z)}{B(z)} = \frac{1}{1 - a z^{-1}}<math> with ROC <math>a < |z|<math> and <math>0 < a < 1<math>

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

<math>h(n) = a^{n} u(n)<math>

which is non-zero for <math>n >= 0<math>.

Truncated IIR filter

Sometimes the coefficients of an IIR filter are truncated such that the filter is implementable. In the previous example, the time function <math>h(n) = a^{n} u(n)<math> has infinite length. This filter is not implementable in a digital computer because of computers do not have infinite memory. Depending on the value of a, the impulse response can be truncated to a certain length with negligible effects. For example, for <math>a = 0.5<math> then <math>h(20) = 9.5367e-7<math> and values of <math>n \ge 20<math> could be dropped with minimal effects on the output due to the relatively small addition these coefficients contribute.

See also

• Electronic filter
• Finite impulse response
• System analysis

External links

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