Digital Filters
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envelope
wave packets
H( )
ph
gr
Figure 5: Phase delay and group delay
The difference between local slope and slope to the origin is crucial to un-
derstand the physical meaning of the two delays. The phase delay at a certain
frequency point is the delay that a single frequency component is subject to
when it passes through the filter, and the quantity (13) is, indeed, a delay in
samples. Vice versa, in order to interpret the group delay let us consider a local
approximation of the phase response by the tangent line at one point. Locally,
propagation can be considered linear and, therefore, a signal having frequency
components focused around that point has a time-domain envelope that is de-
layed by an amount proportional to the slope of the tangent. For instance, two
sinusoids at slightly different frequencies are subject to beats and the beat fre-
quency is the difference of the frequency components (see fig. 6). Therefore,
beats are a frequency local phenomenon, only dependent on the relative dis-
tance between the components rather than on their absolute positions. If we
are interested in knowing how the beat pattern is delayed by a filter, we should
consider local variations in the phase curve. In other words, we should consider
the group delay.
0
0.1
0.2
0.3
-2
-1
0
1
2
s
beats
Figure 6: Beats between a sine wave at 100 Hz and a sine wave at 110 Hz
In telecommunications the group delay is often the most significant between
the two delays, since messages are sent via wave packets localized in a narrow
frequency band, and preservation of the shape of such packets is important.
Vice versa, in sound processing it is more meaningful to consider the set of
frequency components in the audio range as a whole, and the phase delay is