Editing Linear filter (section)

== Frequency response ==

The frequency response or [[transfer function]] <math>|H(\omega)|</math> of a filter can be obtained if the impulse response is known, or directly through analysis using [[Laplace transform]]s, or in discrete-time systems the [[Z-transform]]. The frequency response also includes the phase as a function of frequency, however in many cases the phase response is of little or no interest. FIR filters can be made to have zero phase, but with IIR filters that is generally impossible. With most IIR transfer functions there are related transfer functions having a frequency response with the same magnitude but a different phase; in most cases the so-called [[minimum phase]] transfer function is preferred.

Filters in the time domain are most often requested to follow a specified frequency response. Then, a mathematical procedure finds a filter transfer function that can be realized (within some constraints), and approximates the desired response to within some criterion. Common filter response specifications are described as follows:

*A [[low-pass filter]] passes low frequencies while blocking higher frequencies.
*A [[high-pass filter]] passes high frequencies.
*A [[band-pass filter]] passes a band (range) of frequencies.
*A [[band-stop filter]] passes high and low frequencies outside of a specified band.
*A [[notch filter]] has a null response at a particular frequency. This function may be combined with one of the above responses.
*An [[all-pass filter]] passes all frequencies equally well, but alters the [[Group delay and phase delay|group delay]] and phase relationship among them.
*An equalization filter is not designed to fully pass or block any frequency, but instead to gradually vary the amplitude response as a function of frequency: filters used as [[pre-emphasis]] filters, [[Equalization (audio)|equalizer]]s, or [[tone control]]s are good examples.

===FIR transfer functions===

Meeting a frequency response requirement with an FIR filter uses relatively straightforward procedures. In the most basic form, the desired frequency response itself can be sampled with a resolution of <math>\Delta f</math> and Fourier transformed to the time domain. This obtains the filter coefficients ''h<sub>i</sub>'', which implements a zero phase FIR filter that matches the frequency response at the sampled frequencies used. To better match a desired response,  <math>\Delta f</math> must be reduced. However the duration of the filter's impulse response, and the number of terms that must be summed for each output value (according to the above discrete time convolution) is given by  <math>N=1/(\Delta f \, T)</math> where ''T'' is the [[sampling period]] of the discrete time system (N-1 is also termed the ''order'' of an FIR filter). Thus the complexity of a digital filter and the computing time involved, grows inversely with <math>\Delta f</math>, placing a higher cost on filter functions that better approximate the desired behavior. For the same reason, filter functions whose critical response is at lower frequencies (compared to the [[sampling frequency]] ''1/T'') require a higher order, more computationally intensive FIR filter. An IIR filter can thus be much more efficient in such cases.

Elsewhere the reader may find further discussion of design methods for [[FIR filter#Filter design|practical FIR filter design]].

===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]]