Editing Filter design (section)

==== Digital filters ====

[[Digital filter]]s are classified into one of two basic forms, according to how they respond to a [[Kronecker delta|unit impulse]]:

*[[Finite impulse response]], or '''FIR''', filters express each output sample as a weighted sum of the last ''N'' input samples, where ''N'' is the order of the filter.  FIR filters are normally non-recursive, meaning they do not use feedback and as such are inherently stable. A [[moving average]] filter or [[CIC filter]] are examples of FIR filters that are normally recursive (that use feedback). If the FIR coefficients are symmetrical (often the case), then such a filter is [[linear phase]], so it [[Group delay|delays]] signals of all frequencies equally which is important in many applications. It is also straightforward to avoid overflow in an FIR filter. The main disadvantage is that they may require significantly more [[Instructions per second|processing]] and [[computer memory|memory]] resources than cleverly designed IIR variants. FIR filters are generally easier to design than IIR filters - the [[Parks-McClellan filter design algorithm]] (based on the [[Remez algorithm]]) is one suitable method for designing quite good filters semi-automatically. (See [[#Methodology|Methodology]].)
*[[Infinite impulse response]], or '''IIR''', filters are the digital counterpart to analog filters. Such a filter contains internal state, and the output and the next internal state are determined by a [[linear combination]] of the previous inputs and outputs (in other words, they use [[feedback]], which FIR filters normally do not). In theory, the impulse response of such a filter never dies out completely, hence the name IIR, though in practice, this is not true given the finite resolution of computer arithmetic. IIR filters normally require less [[computing]] resources than an FIR filter of similar performance. However, due to the feedback, high order IIR filters may have problems with [[instability]], [[arithmetic overflow]], and [[limit cycle]]s, and require careful design to avoid such pitfalls. Additionally, since the [[Phase (waves)|phase shift]] is inherently a non-linear function of frequency, the time delay through such a filter is frequency-dependent, which can be a problem in many situations. 2nd order IIR filters are often called '[[Digital biquad filter|biquads]]' and a common implementation of higher order filters is to cascade biquads. A useful reference for computing biquad coefficients is the [https://www.w3.org/TR/audio-eq-cookbook/ RBJ Audio EQ Cookbook].