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 complexvalued 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
frequencydomain 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

(8.7) 
Assuming the
phase response
is approximately linear
over the narrow frequency interval
, 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 frequencycomponent 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
zerophase 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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