Conformal Map Interpretation of Damping Substitution

The relation
[Eq.(3.7)] can
be written down directly from
[Eq.(3.5)] by interpreting Eq.(3.5) as an approximate
*conformal map* [329] which takes each pole
,
say, from the unit circle to the point
.
Thus, the new pole radius is approximately
,
where the approximation is valid when
is approximately constant
between the new pole location and the unit circle. To see this,
consider the partial fraction expansion [452] of a proper
th-order lossless transfer function
mapped to
:

where denotes the th original pole on the unit circle. Then has a pole at , which must be solved iteratively for , in general, since can be a complicated function of . However, if , which is typically true when damping digital waveguides for music applications, then . In other words, we can think of the pole as moving from to near , provided it doesn't move too far compared with the near-constant behavior of . Another way to say it is that we need to be approximately the same at the new pole location and its initial location on the unit circle in the lossless prototype.

Happily, while we may not know precisely where our poles have moved as a result of introducing the per-sample damping filter , the relation [Eq.(3.6)] remains exact at every frequency by construction, as it is based only on the physical interpretation of each unit delay as a propagation delay for a plane wave across one sampling interval , during which (zero-phase) filtering by is assumed (§2.3). More generally, we can design minimum-phase filters for which , and neglect the resulting phase dispersion.

In summary, we see that replacing by everywhere in the FDN lossless prototype (or any lossless LTI system for that matter) serves to move its poles away from the unit circle in the plane onto some contour inside the unit circle that provides the desired decay time at each frequency.

A general design guideline for artificial reverberation applications
[218] is that *all pole radii in the
reverberator should vary smoothly with frequency*. This translates
to
having a *smooth frequency response*. To see why this
is desired, consider momentarily the frequency-independent case in
which we desire the same reverberation time at all frequencies
(Fig.3.10 with real
, as drawn). In this case, it is
ideal for all of the poles to have this decay time. Otherwise, the
late decay of the impulse response will be dominated by the poles
having the largest magnitude, and it will be ``thinner'' than it was
at the beginning of the response when all poles were contributing to
the output. Only when all poles have the same magnitude will the late
response maintain the same modal density throughout the decay.

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