Group Delay

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

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

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.