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

For
linear phase responses,
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
![$ \omega\in[-\pi,\pi]$](http://www.dsprelated.com/josimages_new/filters/img899.png)
). 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

as
 |
(8.6) |
where

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.,
for some

. The modulation
bandwidth is thus
bounded by

.
Using the above
frequency-domain expansion of

,

can be
written as
which we may view as a scaled superposition of
sinusoidal components
of the form
with

near 0. Let us now pass the frequency component

through an
LTI filter 
having
frequency response
to get
![$\displaystyle y_\omega(n) = \left[G(\omega_c+\omega)A_m(\omega)\right] e^{j[(\omega_c +\omega) n + \Theta(\omega_c+\omega)]}. \protect$](http://www.dsprelated.com/josimages_new/filters/img912.png) |
(8.7) |
Assuming the
phase response

is approximately linear
over the narrow frequency interval
![$ \omega\in[\omega_c-\epsilon,\omega_c+\epsilon]$](http://www.dsprelated.com/josimages_new/filters/img913.png)
, we can write
where

is the
filter group delay at

.
Making this substitution in Eq.

(
7.7) gives
where we also used the definition of
phase delay,

, in the last step. In this expression we
can already see that the carrier
sinusoid is delayed by the phase
delay, while the amplitude-
envelope frequency-component is delayed by
the group delay. Integrating over

to recombine the
sinusoidal components (
i.e., using a
Fourier superposition integral for

) gives
where

denotes a
zero-phase filtering of the
amplitude
envelope 
by

. We see that the amplitude
modulation is delayed by

while the carrier wave is
delayed by

.
We have shown that, for narrowband signals expressed as in
Eq.

(
7.6) as a modulation envelope times a sinusoidal carrier, the
carrier wave is delayed by the filter phase delay, while the
modulation is delayed by the filter group delay, provided that the
filter phase response is approximately linear over the narrowband
frequency interval.
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