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Property of many linear time invariant (LTI) systems

Infinite impulse response (IIR) is a property applying to many linear time invariant systems that are distinguished by having an impulse response 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 for some finite , 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.

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 coefficients with (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.

The main advantage digital IIR filters have over FIR filters is their efficiency in implementation, in order to meet a specification in terms of passband, stopband, ripple, and/or roll off. Such a set of specifications can be accomplished with a lower order (Q in the above formulae) IIR filter than would be required for an FIR filter meeting the same requirements. If implemented in a signal processor, this implies a correspondingly fewer number of calculations per time step; the computational savings is often of a rather large factor. On the other hand, FIR filters can be easier to design, for instance, to match a particular frequency response requirement. This is particularly true when the requirement is not one of the usual cases (high pass, low pass, notch, etc.) which have been studied and optimized for analog filters. Also FIR filters can be easily made to be linear phase (constant group delay vs frequency)—a property that is not easily met using IIR filters and then only as an approximation (for instance with the Bessel filter). Another issue regarding digital IIR filters is the potential for limit cycle behavior when idle, due to the feedback system in conjunction with quantization.