Group Delay

A more commonly encountered representation of filter phase response is called the group delay, defined by

$\displaystyle \zbox {D(\omega) \isdefs - \frac{d}{d\omega} \Theta(\omega).}
\qquad\hbox{(Group Delay)}

For linear phase responses, i.e., $ \Theta(\omega) = -\alpha\omega$ for some constant $ \alpha$, the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to $ \alpha$ samples when $ \omega\in[-\pi,\pi]$). If the phase response is nonlinear, then the relative phases of the sinusoidal signal components are generally altered by the filter. A nonlinear phase response normally causes a ``smearing'' of attack transients such as in percussive sounds. Another term for this type of phase distortion is phase dispersion. This can be seen below in §7.6.5.

An example of a linear phase response is that of the simplest lowpass filter, $ \Theta(\omega) = -\omega T/2 \,\,\Rightarrow\,\,
P(\omega)=D(\omega)=T/2$. Thus, both the phase delay and the group delay of the simplest lowpass filter are equal to half a sample at every frequency.

For any reasonably smooth phase function, the group delay $ D(\omega)$ may be interpreted as the time delay of the amplitude envelope of a sinusoid at frequency $ \omega$ [63]. The bandwidth of the amplitude envelope in this interpretation must be restricted to a frequency interval over which the phase response is approximately linear. We derive this result in the next subsection.

Thus, the name ``group delay'' for $ D(\omega)$ refers to the fact that it specifies the delay experienced by a narrow-band ``group'' of sinusoidal components which have frequencies within a narrow frequency interval about $ \omega$. The width of this interval is limited to that over which $ D(\omega)$ is approximately constant.

Derivation of Group Delay as Modulation Delay

Suppose we write a narrowband signal centered at frequency $ \omega_c$ as

$\displaystyle x(n) = a_m(n) e^{j\omega_c n} \protect$ (8.6)

where $ \omega_c$ is defined as the carrier frequency (in radians per sample), and $ a_m(n)$ is some ``lowpass'' amplitude modulation signal. The modulation $ a_m$ can be complex-valued to represent either phase or amplitude modulation or both. By ``lowpass,'' we mean that the spectrum of $ a_m$ is concentrated near dc, i.e.,

$\displaystyle a_m(n)
\isdefs \frac{1}{2\pi} \int_{-\pi}^{\pi} A_m(\omega)e^{j\...
\frac{1}{2\pi} \int_{-\epsilon}^{\epsilon} A_m(\omega)e^{j\omega n} d\omega,

for some $ \left\vert\epsilon\right\vert\ll\pi$. The modulation bandwidth is thus bounded by $ 2\epsilon\ll\pi$.

Using the above frequency-domain expansion of $ a_m(n)$, $ x(n)$ 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},

which we may view as a scaled superposition of sinusoidal components of the form

$\displaystyle x_\omega(n) \isdefs A_m(\omega)e^{j\omega n} e^{j\omega_c n}
= A_m(\omega)e^{j(\omega+\omega_c) n}

with $ \omega$ near 0. Let us now pass the frequency component $ x_\omega(n)$ through an LTI filter $ H(z)$ having frequency response

$\displaystyle H(e^{j\omega}) = G(\omega) e^{j\Theta(\omega)}

to get

$\displaystyle y_\omega(n) = \left[G(\omega_c+\omega)A_m(\omega)\right] e^{j[(\omega_c +\omega) n + \Theta(\omega_c+\omega)]}. \protect$ (8.7)

Assuming the phase response $ \Theta(\omega)$ is approximately linear over the narrow frequency interval $ \omega\in[\omega_c-\epsilon,\omega_c+\epsilon]$, we can write

$\displaystyle \Theta(\omega_c+\omega)\;\approx\;
\Theta(\omega_c) + \Theta^\prime(\omega_c)\omega
\isdefs \Theta(\omega_c) - D(\omega_c)\omega,

where $ D(\omega_c)$ is the filter group delay at $ \omega_c$. Making this substitution in Eq.$ \,$(7.7) gives

&=& \left[G(\omega_c+\omega)A_m(\omega)\right]
e^{j\omega[n-D(\omega_c)]} e^{j\omega_c[n-P(\omega_c)]},

where we also used the definition of phase delay, $ P(\omega_c) =
-\Theta(\omega_c)/\omega_c$, 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 $ \omega$ to recombine the sinusoidal components (i.e., using a Fourier superposition integral for $ y$) gives

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)]}

where $ a^f(n)$ denotes a zero-phase filtering of the amplitude envelope $ a(n)$ by $ G(\omega+\omega_c)$. We see that the amplitude modulation is delayed by $ D(\omega_c)$ while the carrier wave is delayed by $ P(\omega_c)$.

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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Group Delay Examples in Matlab
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Phase Unwrapping