Phase Delay and Group Delay¶

We use the notation of the overview. Let $\phi(\omega)$ be the phase response of a filter, defined in The Transfer Function and Frequency Response. The phase response records a shift at each frequency; two derived quantities express that shift as a time.

Definition: phase delay

The phase delay at frequency $\omega$ is $$ \tau_\phi(\omega) = -\frac{\phi(\omega)}{\omega} . $$ It is the time by which the filter retards a pure sinusoid of frequency $\omega$.

Definition: group delay

The group delay is $$ \tau_g(\omega) = -\frac{d\phi}{d\omega}(\omega) . $$ It is the time by which the filter retards the envelope of a narrowband signal centered at $\omega$.

When the group delay is not constant in $\omega$, distinct frequency components of a transient emerge at distinct times, and the signal is said to be dispersed. The filter classes treated in Linear-Phase and Minimum-Phase Filters are distinguished by the behavior of their group delay.

Example: a pure delay

The filter that reproduces its input shifted by $t_0$ has impulse response $h(t) = \delta(t - t_0)$ and frequency response $H(i\omega) = e^{-i\omega t_0}$, so $\phi(\omega) = -\omega t_0$. Both the phase delay and the group delay equal $t_0$ at every frequency: the filter delays the whole signal uniformly and distorts nothing.

A pulse and its copy delayed by the group delay, undistorted

In discrete time

The group delay of a filter given by its taps is computed from the sampled phase response; it is expressed in samples, and multiplying by $1/f_s$ converts it to seconds. A pure delay of $m$ samples has taps equal to the unit impulse shifted to index $m$.

A pure delay of two samples has constant group delay equal to two:

import numpy as np
from scipy.signal import group_delay

h = np.array([0.0, 0.0, 1.0])            # unit impulse delayed by two samples
w, gd = group_delay((h, 1), w=512)
assert np.allclose(gd, 2.0)

References¶

Group delay and phase delay (Wikipedia).