User:KYN/Filter (signal processing)

In signal processing the concept of a signal is usually taken to describe a function which

1. carries information which can be encoded be means of a well-defined method,
2. has variables which can be given a well-defined interpretation, typically as time or space.

In this context a filter is mapping on the set of signals which preserves the signal character such that the resulting function

1. carries information which can be encoded with the same method as the original signal,
2. is a function of the same type of variables as the original function.

The specific condition that the information encoding and function variables should be invariant to the filter mapping excludes certain types of mappings on functions. For example, a Fourier transform of a time-dependent signal results in a new function which contains the same information as the original signal but with another encoding, and with frequency-dependent variables. Also, a modulation of a time-dependent signal results in a new time-dependent signal but with a different encoding.

A third characterization which can be given to a filter is that it is local. This property implies that mapping from the original signal, $s$, to the resulting signal, $r$, is such that the function value $r(t_{0})$ depends only (or mainly) on $s(t)$ for t in a neighbourhood around $t_{0}$.

The preceding definition of a filter is not universally established, but it sufficiently general to include how the term is used in signal processing and in certain areas of mathematics and computer science. It also means that the concept of a filter is highly context dependent. For example, a specific mapping on functions may be a filter or not depending on the character of the set of signals which it is applied to. It can also be further generalized, for example by considering signals as stochatic processes rather than physically measurable entities.