####
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* [

326] 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 [

449] 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
[

217] 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.

**Next Section:** Damping
Filters for Reverberation Delay Lines**Previous Section:** Prime
Power Delay-Line Lengths