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 (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 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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