A more commonly encountered representation of filter phase response is called the group delay, defined by
For linear phase responses, i.e.,
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.