Let
denote the desired reverberation time at radian frequency
, and let
denote the transfer function of the lowpass
filter to be placed in series with delay line
. The problem we
consider now is how to design these filters to yield the desired
reverberation time. We will specify an ideal amplitude
response for
based on the desired reverberation time at each
frequency, and then use conventional filter-design methods to obtain a
low-order approximation to this ideal specification.
Since losses will be introduced by the substitution
, we need to find its effect on the pole radii of the lossless
prototype. Let
The radius we desire for a pole at frequency is that which
gives us the desired
:
Now that we have specified the ideal delay-line filter
,
any number of filter-design methods can be used to find a low-order
which provides a good approximation to satisfying Eq. (2.3).
Examples include the functions invfreqz and stmcb in
Matlab. Since the variation in reverberation time is typically very
smooth with respect to
, the filters
can be very low
order.