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Group Delay

A more commonly encountered representation of filter phase response is called the group delay, defined by

$\displaystyle \zbox {D(\omega) \isdefs - \frac{d}{d\omega} \Theta(\omega).}
\qquad\hbox{(Group Delay)}
$

For linear phase responses, i.e., $ \Theta(\omega) = -\alpha\omega$ for some constant $ \alpha$ , the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to $ \alpha$ samples when $ \omega\in[-\pi,\pi]$ ). If the phase response is nonlinear, then the relative phases of the sinusoidal signal components are generally altered by the filter. A nonlinear phase response normally causes a ``smearing'' of attack transients such as in percussive sounds. Another term for this type of phase distortion is phase dispersion. This can be seen below in §7.6.5.

An example of a linear phase response is that of the simplest lowpass filter, $ \Theta(\omega) = -\omega T/2 \,\,\Rightarrow\,\,
P(\omega)=D(\omega)=T/2$ . Thus, both the phase delay and the group delay of the simplest lowpass filter are equal to half a sample at every frequency.

For any reasonably smooth phase function, the group delay $ D(\omega)$ may be interpreted as the time delay of the amplitude envelope of a sinusoid at frequency $ \omega$ [63]. The bandwidth of the amplitude envelope in this interpretation must be restricted to a frequency interval over which the phase response is approximately linear. We derive this result in the next subsection.

Thus, the name ``group delay'' for $ D(\omega)$ refers to the fact that it specifies the delay experienced by a narrow-band ``group'' of sinusoidal components which have frequencies within a narrow frequency interval about $ \omega$ . The width of this interval is limited to that over which $ D(\omega)$ is approximately constant.



Subsections
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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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