Infinite impulse response (IIR) is a fundamental property applying to many linear time-invariant systems that are distinguished by having an impulse response h ( t ) {\displaystyle h(t)} that does not become exactly zero past a certain point but continues indefinitely. This is in contrast to a finite impulse response (FIR) system, in which the impulse response does become exactly zero at times t > T {\displaystyle t>T} for some finite T {\displaystyle T} , thus being of finite duration. Common examples of linear time-invariant systems are most electronic and digital filters. Systems with this property are known as IIR systems or IIR filters. In practice, the impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction. For instance, analog electronic filters composed of resistors, capacitors, and/or inductors (and perhaps linear amplifiers) are generally IIR filters. On the other hand, discrete-time filters (usually digital filters) based on a tapped delay line employing no feedback are necessarily FIR filters. The capacitors (or inductors) in the analog filter have a "memory" and their internal state never completely relaxes following an impulse (assuming the classical model of capacitors and inductors where quantum effects are ignored). But in the latter case, after an impulse has reached the end of the tapped delay line, the system has no further memory of that impulse and has returned to its initial state; its impulse response beyond that point is exactly zero.
Implementation and design Although almost all analog electronic filters are IIR, digital filters may be either IIR or FIR. The presence of feedback in the topology of a discrete-time filter (such as the block diagram shown below) generally creates an IIR response. The z domain transfer function of an IIR filter contains a non-trivial denominator, describing those feedback terms. The transfer function of an FIR filter, on the other hand, has only a numerator as expressed in the general form derived below. All of the a i {\displaystyle a_{i}} coefficients with i > 0 {\displaystyle i>0} (feedback terms) are zero and the filter has no finite poles. The transfer functions pertaining to IIR analog electronic filters have been extensively studied and optimized for their amplitude and phase characteristics. These continuous-time filter functions are described in the Laplace domain. Desired solutions can be transferred to the case of discrete-time filters whose transfer functions are expressed in the z domain, through the use of certain mathematical techniques such as the bilinear transform, impulse invariance, or pole–zero matching method. Thus digital IIR filters can be based on well-known solutions for analog filters such as the Chebyshev filter, Butterworth filter, and elliptic filter, inheriting the characteristics of those solutions.
Transfer function derivation Digital filters are often described and implemented in terms of the difference equation that defines how the output signal is related to the input signal:
y [ n ]
= b 0 x [ n ] + b 1 x [ n − 1 ] + ⋯ + b P x [ n − P ]
+ a 1 y [ n − 1 ] + a 2 y [ n − 2 ] + ⋯ + a Q y [ n − Q ] {\displaystyle {\begin{aligned}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]\end{aligned}}}
where:
P {\displaystyle \ P} is the feedforward filter order
b i {\displaystyle \ b_{i}} are the feedforward filter coefficients
Q {\displaystyle \ Q} is the feedback filter order
a i {\displaystyle \ a_{i}} are the feedback filter coefficients
x [ n ] {\displaystyle \ x[n]} is the input signal
y [ n ] {\displaystyle \ y[n]} is the output signal. A more condensed form of the difference equation is:
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