Editing Linear phase (section)

== Definition ==
A filter is called a linear phase filter if the phase component of the frequency response is a linear function of frequency.  For a continuous-time application, the frequency response of the filter is the [[Fourier transform]] of the filter's [[impulse response]], and a linear phase version has the form:

:<math>H(\omega) = A(\omega)\ e^{-j \omega \tau},</math>

where:
*A(ω) is a real-valued function.
*<math>\tau</math> is the group delay.

For a discrete-time application, the [[discrete-time Fourier transform]] of the linear phase impulse response has the form:

:<math>H_{2\pi}(\omega) = A(\omega)\ e^{-j \omega k/2},</math>

where:
*A(ω) is a real-valued function with 2π periodicity.
*k is an integer, and k/2 is the group delay in units of samples.

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

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

where the <math>\widehat H</math> notation distinguishes the Z-transform from the Fourier transform.