Editing Digital filter (section)

==Types of digital filters==
There are various ways to characterize filters; for example:
* A ''linear'' filter is a [[linear transformation]] of input samples; other filters are ''[[nonlinear]]''. Linear filters satisfy the [[superposition principle]], i.e. if an input is a weighted linear combination of different signals, the output is a similarly weighted linear combination of the corresponding output signals.
* A ''causal'' filter uses only previous samples of the input or output signals; while a ''non-causal'' filter uses future input samples. A non-causal filter can usually be changed into a causal filter by adding a delay to it.
* A ''time-invariant'' filter has constant properties over time; other filters such as [[adaptive filter]]s change in time.
* A ''stable'' filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An ''unstable'' filter can produce an output that grows without bounds, with bounded or even zero input.
* A [[finite impulse response]] (FIR) filter uses only the input signals, while an [[infinite impulse response]] (IIR) filter uses both the input signal and previous samples of the output signal. FIR filters are always stable, while IIR filters may be unstable.

A filter can be represented by a [[block diagram]], which can then be used to derive a sample processing [[algorithm]] to implement the filter with hardware instructions. A filter may also be described as a [[difference equation]], a collection of [[zeros and poles]] or an [[impulse response]] or [[step response]].

Some digital filters are based on the [[fast Fourier transform]], a mathematical algorithm that quickly extracts the [[frequency spectrum]] of a signal, allowing the spectrum to be manipulated (such as to create very high order band-pass filters) before converting the modified spectrum back into a time-series signal with an inverse FFT operation. These filters give O(n log n) computational costs whereas conventional digital filters tend to be O(n<sup>2</sup>).

Another form of a digital filter is that of a [[state space (controls)|state-space]] model. A well used state-space filter is the [[Kalman filter]] published by [[Rudolf Kálmán]] in 1960.

Traditional linear filters are usually based on attenuation. Alternatively nonlinear filters can be designed, including energy transfer filters,<ref name="SAB1">Billings S.A. "Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains". Wiley, 2013</ref> which allow the user to move energy in a designed way so that unwanted noise or effects can be moved to new frequency bands either lower or higher in frequency, spread over a range of frequencies, split, or focused. Energy transfer filters complement traditional filter designs and introduce many more degrees of freedom in filter design. Digital energy transfer filters are relatively easy to design and to implement and exploit nonlinear dynamics.