A lossless prototype reverberator, as in Fig.
3.10 when

,
has all of its
poles on the unit circle in the

plane, and its
reverberation time is infinity. To set the reverberation time to a
desired value, we need to move the poles slightly inside the unit
circle. Furthermore, due to air absorption
(§
2.3,§
B.7.15), we want the high-frequency
poles to be more damped than the low-frequency poles
[
314]. As discussed in §
2.3, this type
of transformation can be obtained using the substitution

 |
(4.5) |
where

denotes the
filtering per sample in the
propagation medium (a
lowpass filter with gain not exceeding 1 at all
frequencies).
4.14Thus, to set the
FDN reverberation time to

at frequency

,
we want propagation through

samples to result in attenuation
by
dB,
i.e.,
![$\displaystyle \left[G(e^{j\omega T})\right]^{n_{60}(\omega)} \eqsp 0.001. \protect$](http://www.dsprelated.com/josimages_new/pasp/img818.png) |
(4.6) |
Solving for

, the propagation attenuation per-sample, gives
The last form comes from

ln

, where

denotes the
time constant of decay (time to decay by

)
[
451],
i.e.,
ln |
(4.8) |
Series expanding

and assuming

samples (

seconds) provides the practically useful approximation
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
![$ H'(z)\isdeftext H[z/G(z)]$](http://www.dsprelated.com/josimages_new/pasp/img837.png)
:
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
![$ z'_k\approx p_kG(p_k)=G[\exp(j\omega_k
T)]\exp(j\omega_k T)$](http://www.dsprelated.com/josimages_new/pasp/img844.png)
. In other words, we can think of the pole

as moving from

to near
![$ G[\exp(j\omega_k
T)]\exp(j\omega_k T)$](http://www.dsprelated.com/josimages_new/pasp/img847.png)
, 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.
In an
FDN, such as the one shown in Fig.
3.10, delays

appear
in long
delay-line chains

. Therefore, the filter needed at
the output (or input) of the

th delay line is

(replace

by

in Fig.
3.10).
4.15 However, because

is so close to

in magnitude, and because it varies so weakly
across the frequency axis, we can design a much lower-order filter

that provides the desired attenuation
versus frequency to within
psychoacoustic resolution. In fact,
perfectly nice reverberators can be designed in which

is
merely
first order for each

[
314,
217].
Next Section: Delay-Line Damping Filter DesignPrevious Section: Choice of Delay Lengths