# Nonlinear filter

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In signal processing, a **nonlinear** (or **non-linear**) **filter** is a filter whose output is not a linear function of its input. That is, if the filter outputs signals *R* and *S* for two input signals *r* and *s* separately, but does not always output *αR* + *βS* when the input is a linear combination *αr* + *βs*.

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Both continuous-domain and discrete-domain filters may be nonlinear. A simple example of the former would be an electrical device whose output voltage *R*(*t*) at any moment is the square of the input voltage *r*(*t*); or which is the input clipped to a fixed range [*a*,*b*], namely *R*(*t*) = max(*a*, min(*b*, *r*(*t*))). An important example of the latter is the running-median filter, such that every output sample *R*_{i} is the median of the last three input samples *r*_{i}, *r*_{i−1}, *r*_{i−2}. Like linear filters, nonlinear filters may be shift invariant or not.

Non-linear filters have many applications, especially in the removal of certain types of noise that are not additive. For example, the median filter is widely used to remove spike noise — that affects only a small percentage of the samples, possibly by very large amounts. Indeed, all radio receivers use non-linear filters to convert kilo- to gigahertz signals to the audio frequency range; and all digital signal processing depends on non-linear filters (analog-to-digital converters) to transform analog signals to binary numbers.

However, nonlinear filters are considerably harder to use and design than linear ones, because the most powerful mathematical tools of signal analysis (such as the impulse response and the frequency response) cannot be used on them. Thus, for example, linear filters are often used to remove noise and distortion that was created by nonlinear processes, simply because the proper non-linear filter would be too hard to design and construct.

From the foregoing, we can know that the nonlinear filters have quite different behavior compared to linear filters. The most important characteristic is that, for nonlinear filters, the filter output or response of the filter does not obey the principles outlined earlier, particularly scaling and shift invariance. Furthermore, a nonlinear filter can produce results that vary in a non-intuitive manner.