### Group Delay

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

*i.e.*, for some constant , the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to samples when ). 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, . 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 may be interpreted as the

*time delay of the amplitude envelope*of a sinusoid at frequency [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 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 . The width of this interval is limited to that over which is approximately constant.

#### Derivation of Group Delay as Modulation Delay

Suppose we write a narrowband signal centered at frequency aswhere is defined as the

*carrier frequency*(in radians per sample), and is some ``lowpass''

*amplitude modulation*signal. The modulation can be complex-valued to represent either phase or amplitude modulation or both. By ``lowpass,'' we mean that the spectrum of is concentrated near dc,

*i.e.*,

Assuming the phase response is approximately linear over the narrow frequency interval , we can write

*i.e.*, using a Fourier superposition integral for ) gives

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Group Delay Examples in Matlab

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