Derivation of Group Delay as Modulation Delay
Suppose we write a narrowband signal centered at frequency
as
where







Using the above frequency-domain expansion of ,
can be
written as
![$\displaystyle x(n) \eqsp a_m(n) e^{j\omega_c n} \eqsp
\left[\frac{1}{2\pi} \int_{-\epsilon}^{\epsilon} A_m(\omega)e^{j\omega n} d\omega\right] e^{j\omega_c n},
$](http://www.dsprelated.com/josimages_new/filters/img908.png)





Assuming the phase response

![$ \omega\in[\omega_c-\epsilon,\omega_c+\epsilon]$](http://www.dsprelated.com/josimages_new/filters/img913.png)




![\begin{eqnarray*}
y_\omega(n)
&=& \left[G(\omega_c+\omega)A_m(\omega)\right]
e^...
...\right]
e^{j\omega[n-D(\omega_c)]} e^{j\omega_c[n-P(\omega_c)]},
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img916.png)
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
![\begin{eqnarray*}
y(n) &=& \frac{1}{2\pi}\int_{\omega} y_\omega(n) d\omega \\
&...
...)]}\\
&=& a^f[n-D(\omega_c)] \cdot e^{j\omega_c[n-P(\omega_c)]}
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/filters/img918.png)
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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