# Digital filter

In electronics, a digital filter is any electronic filter that works by performing digital math operations on an intermediate form of a signal. This is in contrast with older analog filters which work entirely in the analog realm and must rely on physical networks of electronic components (such as resistors, capacitors, transistors, etc.) to achieve a desired filtering effect.

Digital filters can achieve virtually any filtering effect that can be expressed as a mathematical algorithm. The two primary limitations of digital filters are their speed (the filter can't operate any faster than the computer at the heart of the filter), and their cost. However as the cost of integrated circuits have continued to drop over time following Moore's Law, digital filters have become increasingly commonplace and are now an essential element of many everyday objects such as radios, cellphones, and stereo receivers.

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Digital filters can easily achieve performance metrics far beyond what is (even theoretically) possible with analog filters. It is not particularly difficult, for example, to create a 1000Hz low-pass filter which can achieve near-perfect transmission of a 999Hz input while entirely blocking a 1001Hz signal. Analog filters cannot split apart such closely spaced signals.

Also, for complex multi-stage filtering operations, digital filters have the potential to attain much better signal to noise ratios than analog filters. This is because whereas at each intermediate stage the analog filter adds more noise to the signal, the digital filter performs noiseless math operations at each intermediate step in the transform. The primary source of noise in a digital filter is found in the initial analog to digital conversion step, where in addition to any circuit noise introduced, the signal is subject to an unavoidable quantization error due to the finite resolution of the digital representation of the signal.

Note also that digital filters will become confounded if presented with an input signal which contains any substantial subcomponent with a frequency exceeding half the sampling rate of the filter (cf. Nyquist sampling theorem). Thus a small anti-aliasing filter is always placed ahead of the analog to digital conversion circuitry to prevent these high-frequency components from interfering with the sampling.

## Types of digital filters

Many 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 pass-band filters) before converting the modified spectrum back into a time-series signal.

Another form of a typical linear digital filter, expressed as a transform in the Z-domain, is

[itex]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}}}[itex]

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.

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.

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