Editing Linear filter (section)

===IIR transfer functions===

Since classical analog filters are IIR filters, there has been a long history of studying the range of possible transfer functions implementing various of the above desired filter responses in continuous time systems. Using [[Bilinear transform|transform]]s it is possible to convert these continuous time frequency responses to ones that are implemented in discrete time, for use in digital IIR filters. The complexity of any such filter is given by the ''order'' N, which describes the order of the [[rational function]] describing the frequency response. The order N is of particular importance in analog filters, because an N<sup>th</sup> order electronic filter requires N reactive elements (capacitors and/or inductors) to implement. If a filter is implemented using, for instance, [[biquad]] stages using [[op-amp]]s, N/2 stages are needed. In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation (in some sense) to the desired response using a smaller N, as we shall now illustrate.

Below are the frequency responses of several standard filter functions that approximate a desired response, optimized according to some criterion. These are all fifth-order low-pass filters, designed for a cutoff frequency of .5 in normalized units. Frequency responses are shown for the  [[Butterworth filter|Butterworth]], [[Chebyshev filter|Chebyshev]], [[chebyshev filter#Type II Chebyshev filters|inverse Chebyshev]], and [[elliptic filter]]s.

[[Image:Filters order5.svg|500px|center]]

As is clear from the image, the elliptic filter is sharper than the others, but at the expense of [[ripple (filters)|ripples]] in both its passband and stopband. The Butterworth filter has the poorest transition but has a more even response, avoiding ripples in either the passband or stopband. A [[Bessel filter]] (not shown) has an even poorer transition in the frequency domain, but maintains the best phase fidelity of a waveform. Different applications emphasize different design requirements, leading to different choices among these (and other) optimizations, or requiring a filter of a higher order.

[[Image:Sallen-Key Lowpass General.svg|thumb|300px|right|Low-pass filter implemented with a Sallen&ndash;Key topology]]