Suppose we write a narrowband signal centered at frequency
as

(8.5)

where is defined as the carrier frequency (in
radians per sample), and is some narrowband amplitude
modulation signal. The modulation can be complex-valued to
represent either phase or amplitude modulation or both. By
``narrowband,'' 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

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. (7.6) gives

where we also used the definition of phase delay,
, in the last step. 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. (7.5) 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.