Filters

Infinite impulse response
Infinite impulse response (IIR) is a property applying to many s. Common examples of linear time-invariant systems are most and s. Systems with this property are known as IIR systems or  IIR filters, and are distinguished by having an  which does not become exactly zero past a certain point, but continues indefinitely. This is in contrast to a (FIR) in which the impulse response  h(t) does become exactly zero at times t > T for some finite T, thus being of finite duration.

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, s (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  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  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 $$a_i$$ coefficients with $$i > 0$$ (feedback terms) are zero and the filter has no finite.

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. 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, , or. Thus digital IIR filters can be based on well-known solutions for analog filters such as the, , and , inheriting the characteristics of those solutions.

Causal filter
In, a causal filter is a. The word causal indicates that the filter output depends only on past and present inputs. A whose output also depends on future inputs is non-causal, whereas a filter whose output depends only on future inputs is anti-causal. Systems (including filters) that are realizable (i.e. that operate in ) must be causal because such systems cannot act on a future input. In effect that means the output sample that best represents the input at time $$t,$$ comes out slightly later. A common design practice for s is to create a realizable filter by shortening and/or time-shifting a non-causal impulse response. If shortening is necessary, it is often accomplished as the product of the impulse-response with a.

An example of an anti-causal filter is a filter, which can be defined as a, anti-causal filter whose inverse is also stable and anti-causal.