24
D. Rocchesso: Sound Processing
precursors
phase delay
group delay
filter. Before defining this concept more precisely we illustrate what happens
to the wave shape that is impressed by a hammer to the string in the piano.
The string behaves like a nonlinear-phase filter, and the dispersion of the fre-
quency components becomes increasingly more evident while the wave shape
propagates away from the hammer along the string. Fig. 4 illustrates the string
displacement signal as it is produced by a physical model (see chapter 5 for
details) of the hammer-string system. The initial wave shape progressively loses
its initial form. In particular, the fact that high frequencies are subject to a
smaller propagation delay than low frequencies is visible in the form of little
precursors, i.e., small high-frequency oscillations that precede the return of the
main components of the wave shape. Such an effect can be experienced with
an aerial ropeway like those that are found in isolated mountain houses. If we
shake the rope energetically and keep our hand on it, after a few seconds we
perceive small oscillations preceding a strong echo.
time
-1.0
1.0
0.02
.11
Figure 4: Struck string: string displacement at the bridge termination
The effects of the phase response of a filter can be better formalized by
introducing two mathematical definitions: the phase delay and the group delay.
The phase delay is defined as
ph
= -
H()
,
(13)
i.e., at any frequency, it is given by the phase response divided by the frequency
itself. In practice, given the phase-response curve, the phase delay at one point
is obtained as the slope of the straight line that connects that point with the
origin. The group delay is defined in differential terms as
gr
= -
d H()
d
.
(14)
Therefore, the group delay at one point of the phase-response curve, is equal
to the slope of the curve. The fig. 5 illustrates the difference between phase
delay and group delay. It is clear that, if the phase is linear, the two delays are
equal and coincident with the slope of the straight line that represents the phase
response.