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D. Rocchesso: Sound Processing
just noticeable difference
frequency-dependent
absorption
lossy delay line
Fig. 3 shows that the delay of the allpass filter is approximately constant
only in a narrow frequency range. We can reasonably assume that such range, for
positive values of c smaller than one, extends from 0 to F
s
/5. With F
s
= 50kHz
we see that at F
s
/5 = 10kHz we have an error of about 0.05 time samples.
In a note at that frequency produced by a feedback delay line, such an error
produces a pitch deviation smaller than 1%. For lower fundamental frequencies,
such as those found in actual musical instruments, the error is smaller than
the just noticeable difference measured with slow pitch modulations (see the
appendix C).
If the first-order filter represents an elegant and efficient solution to the
problem of tuning a delay line, it has also the relevant side effect of detuning
the upper partials, due to the marked phase nonlinearity. Such detuning can be
tolerated in most cases, but has to be taken into account in some other contexts.
If a phase response closer to linear is needed, we can use higher-order allpass
filters [51]. In some cases, especially in sound synthesis by physical modeling,
a specific inharmonic distribution of resonances has to be approximated. This
can be obtained by designing allpass filters that approximate a given phase
response along the whole frequency axis. In these cases the problem of tuning
is superseded by the most difficult problem of accurate partial positioning [83].
With allpass interpolators it is more complicated to handle continuous delay
length variations, since the recursive structure of the filter does not show an
obvious way of transferring memory cells from and to the delay line, as it was
in the case of the FIR interpolator, which is constructed on the delay line by
a certain number of taps. Indeed, the glissando can be implemented with the
allpass filter by adding a new cell to the delay line whenever the filter coefficient
becomes one and, at the same time, zeroing out the filter state variable and
the coefficient. What is really more complicated with allpass filters is to handle
sudden variations of the delay length, as they are found, for instance, when a
finger hole is opened in a wind instrument. In this case, the recursive nature of
allpass filters causes annoying transients in the output signal. Ad hoc structures
have been devised to cancel these transients [51].
3.3
The Non-Recursive Comb Filter
Sounds, propagating in the air, come into contact with surfaces and objects of
various kinds and this interaction produces physical phenomena such as reflec-
tion, refraction, and diffraction. A simple and very important phenomenon is
the reflection of sound about a planar surface. Due to a reflection such as this, a
listener receives two delayed copies of the same signal. If the delay is larger than
about a hundred milliseconds, the second copy is perceived as a distinguished
echo, while if the delay is smaller than about ten milliseconds, the effect of a
single reflection is perceived as a spectral coloration.
A simple model of single reflection can be constructed starting from the
basic blocks described in this and in the preceding chapters. It is constructed as
an m-samples delay line, with the incidental fractional part of m obtained by
FIR interpolation or allpass filtering, cascaded with an attenuation coefficient
g, possibly replaced by a filter if a frequency-dependent absorption has to be
simulated. The output of this lossy delay line is summed to the direct signal.
Let us analyze the structure in the case that m is integer and g is a positive