#### 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
:

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