Editing Finite impulse response (section)

==Frequency response==

The filter's effect on the sequence <math>x[n]</math> is described in the frequency domain by the [[convolution theorem]]''':'''

:<math>\underbrace{\mathcal{F}\{x*h\}}_{Y(\omega)} = \underbrace{\mathcal{F}\{x\}}_{X(\omega)} \cdot \underbrace{\mathcal{F}\{h\}}_{H(\omega)}</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>y[n] = x[n]*h[n]= \mathcal{F}^{-1}\big\{X(\omega)\cdot H(\omega)\big\},</math>

where operators <math>\mathcal{F}</math> and <math>\mathcal{F}^{-1}</math> respectively denote the [[discrete-time Fourier transform]] (DTFT) and its inverse. Therefore, the complex-valued, multiplicative function <math>H(\omega)</math> is the filter's [[frequency response]].  It is defined by a [[Fourier series]]''':'''

:<math>H_{2\pi}(\omega)\ \triangleq \sum_{n=-\infty}^{\infty} h[n]\cdot \left({e^{i \omega}}\right)^{-n} = \sum_{n=0}^{N}b_n\cdot \left({e^{i \omega}}\right)^{-n},</math>

where the added subscript denotes <math>2\pi</math>-periodicity. Here <math>\omega</math> represents frequency in [[Normalized frequency (digital signal processing)|normalized units]] (''radians per sample''). The function <math>H_{2\pi}(2\pi f')</math> has a periodicity of <math>1</math> with <math>f'</math> in units of ''cycles per sample'', which is favored by many filter design applications.{{efn-ua
|An exception is MATLAB, which prefers a periodicity of <math>2,</math> because the Nyquist frequency in units of ''half-cycles/sample'' is <math>1,</math>, a convenient choice for plotting software that displays the interval from 0 to the Nyquist frequency.
}}&nbsp; The value <math>\omega = \pi</math>, called [[Nyquist frequency]], corresponds to <math>f' = \tfrac{1}{2}.</math>&nbsp;  When the sequence  <math>x[n]</math> has a known sampling-rate <math>f_s</math> (in ''samples per second''), ordinary frequency is related to normalized frequency by <math>f = f'\cdot f_s = \tfrac{\omega}{2\pi}\cdot f_s</math> ''cycles per second'' ([[Hz]]).  Conversely, if one wants to design a filter for ordinary frequencies <math>f_1,</math> <math>f_2,</math> etc., using an application that expects ''cycles per sample'', one would enter <math>f_1/f_s,</math>&nbsp; <math>f_2/f_s,</math>&nbsp; etc.

<math>H_{2\pi}(\omega)</math> can also be expressed in terms of the [[Discrete-time_Fourier_transform#Relationship_to_the_Z-transform|Z-transform]] of the filter impulse response:

:<math>
\widehat H(z)\ \triangleq \sum_{n=-\infty}^{\infty} h[n]\cdot z^{-n}.
</math>

:<math>H_{2\pi}(\omega) = \left. \widehat H(z) \, \right|_{z = e^{j \omega}} = \widehat H(e^{j \omega}).</math>