#### Derivation of Group Delay as Modulation Delay

Suppose we write a narrowband signal centered at frequency as

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.*,

Using the above frequency-domain expansion of , can be written as

Assuming the phase response is approximately linear over the narrow frequency interval , we can write

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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