Editing Linear filter (section)

===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]].