Suppose we write a narrowband signal centered at frequency as
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
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.