Infinite impulse response

IIR (infinite impulse response) is a property of signal processing systems. Systems with that property are known as IIR systems or if we are dealing with electronic filter systems 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.

Recursive filters are signal processing filters which re-use one or more output(s) of the filter as inputs. This feedback results in an unending impulse response characterized by either exponentially growing, decaying, or sinusoidal signal output components.

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 dependant on that of their analog counterparts which is because there are plenty of recources, works and straightforward design methods concerning analog feedback filter design while there are hardly 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

$y(n)=b_{0}x(n)+b_{1}x(n-1)+\cdots +b_{P}x(n-P)+a_{1}y(n-1)+a_{2}y(n-2)+\cdots +a_{Q}y(n-Q)$

where $P$ is the forward filter order, $b_{i}$ are the forward filter coefficients, $Q$ is the feedback filter order, $a_{i}$ are the feedback filter coefficients, $x(n)$ is the input signal and $y(n)$ is the output signal. A more condense form of the difference equation is

$y(n)=\sum _{i=0}^{P}b_{i}x(n-i)+\sum _{k=1}^{Q}a_{k}y(n-k)$

To find the impulse response we set

$x(n)=\delta (n)$

where $\delta (n)$ is the Kronecker delta impulse. The impulse response for an IIR filter follows as

$h(n)=\sum _{i=0}^{P}b_{i}\delta (n-i)+\sum _{k=1}^{Q}a_{k}h(n-k)$

The Z-transform of the impulse response yields the transfer function of the IIR filter

$H(z)=Z\{h(n)\}=\sum _{n=-\infty }^{\infty }h(n)z^{-n}$

We note that $Z\{\delta (n)\}=1$ then with the definition of the impulse response and the time shift property of the Z-transform follows

$H(z)=\sum _{i=0}^{P}b_{i}z^{-i}+\sum _{k=1}^{Q}a_{k}z^{-k}H(z)$

Stating all $H(z)$ on the left hand side

$H(z)-\sum _{k=1}^{Q}a_{k}z^{-k}H(z)=\sum _{i=0}^{P}b(i)z^{-i}$

Isolating $H(z)$ on the left hand side leads to the desired format of the transfer function

$H(z)={\frac {\sum _{i=0}^{P}b_{i}z^{-i}}{1-\sum _{k=1}^{Q}a_{k}z^{-k}}}$

Block diagram

A typical block diagram of an IIR filter looks like the following. The $T$ block is a unit delay. The coefficients and number of feedback/feedfoward paths is 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 all poles of the transfer function to have an absolute value smaller than one. In other words, all poles must be located within a unit circle in the $z$-plane. To find the poles of the transfer function we have to extend it with ${\frac {z^{O}}{z^{O}}}$ where $O=max(P,Q)$ and arrive at

$H(z)={\frac {\sum _{i=0}^{P}b_{i}z^{O-i}}{z^{O}-\sum _{k=1}^{Q}a_{k}z^{O-k}}}$

The poles of the IIR filter transfer function are the zeros of the denominator polynomial of the transfer function. The poles are evaluated as

$0=z^{O}-\sum _{k=1}^{Q}a_{k}z^{O-k}$

Clearly, if $a_{k}\neq 0$ then the poles are not located on the origin of the z-plane. This is in contrast to the FIR filter where all poles are located on the origin of z-plane.

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

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

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

• The fifth module of the BORES Signal Processing DSP course - Introduction to DSP