Editing Filter design (section)

== Theoretical basis ==

Parts of the design problem relate to the fact that certain requirements are described in the frequency domain while others are expressed in the time domain and that these may conflict. For example, it is not possible to obtain a filter which has both an arbitrary impulse response and arbitrary frequency function. Other effects which refer to relations between the time and frequency domain are

* The uncertainty principle between the time and frequency domains
* The variance extension theorem
* The asymptotic behaviour of one domain versus discontinuities in the other

=== The uncertainty principle ===

As stated by the [[Gabor limit]], an uncertainty principle, the product of the width of the frequency function and the width of the impulse response cannot be smaller than a specific constant. This implies that if a specific frequency function is requested, corresponding to a specific frequency width, the minimum width of the filter in the signal domain is set. Vice versa, if the maximum width of the response is given, this determines the smallest possible width in the frequency. This is a typical example of contradictory requirements where the filter design process may try to find a useful compromise.

=== The variance extension theorem ===

Let <math>\sigma^{2}_{s}</math> be the variance of the input signal and let <math>\sigma^{2}_{f}</math> be the variance of the filter. The variance of the filter response, <math>\sigma^{2}_{r}</math>, is then given by

: <math>\sigma^{2}_{r}</math> = <math>\sigma^{2}_{s}</math> + <math>\sigma^{2}_{f}</math>

This means that <math>\sigma_{r} > \sigma_{f}</math> and implies that the localization of various features such as pulses or steps in the filter response is limited by the filter width in the signal domain. If a precise localization is requested, we need a filter of small width in the signal domain and, via the uncertainty principle, its width in the frequency domain cannot be arbitrary small.

=== Discontinuities versus asymptotic behaviour ===

Let ''f(t)'' be a function and let <math>F(\omega)</math> be its Fourier transform. There is a theorem which states that if the first derivative of ''F'' which is discontinuous has order <math>n \geq 0</math>, then ''f'' has an asymptotic decay like <math>t^{-n-1}</math>.

A consequence of this theorem is that the frequency function of a filter should be as smooth as possible to allow its impulse response to have a fast decay, and thereby a short width.