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# The Online Encyclopedia and Dictionary   ## Encyclopedia ## Dictionary ## Quotes  # Digital filter

In electronics, a digital filter is an electronic filter (usually linear), in discrete time, that is implemented through digital electronic computation of digital signals.

One form of a typical linear digital filter, expressed as a transform in the Z-domain, is $H(z) = \frac{B(z)}{A(z)} = \frac{{b_{0}+b_{1}z^{-1}+b_{2}z^{-2} + \cdots + b_{N}z^{-N}}}{{1+a_{1}z^{-1}+a_{2}z^{-2} + \cdots +a_{M}z^{-M}}}$

where M is the order of the filter. See Z-transform#LCCD equation for further discussion of this transfer function.

This form is for an infinite impulse response filter, but if the denominator is unity then this form is for a finite impulse response filter.

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## State-space filters

Another form of a digital filter is that of a state space model. A well used state-space filter is the Kalman filter published by Rudolf Kalman in 1960.

One advantage of digital filters is the stability of their parameters. A digital computer is much less susceptible to environmental conditions unlike analog circuits. With digital computers becoming cheaper and cheaper, the advantages of digital signal processing are outweighing complex analog filters. Although, for proper sampling an anti-aliasing filter is needed, which is usually a low-pass filter or band-pass filter.

## Optimum filter

The best filter that can be applied is the so called Optimum filter . It gives the best Signal-to-noise ratio (SNR), but it needs to know in advance the average shape of the signal. This filter could not be implemented in hardware, but could only be applied by software after the data has been collected.  