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 the
th delay line which is
samples long. 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.

In accordance with Eq.(3.6), the lowpass filter in series with a length delay line should approximate

which implies

Taking of both sides gives

This is the same formula derived by Jot [218] using a somewhat different approach.

Now that we have specified the ideal delay-line filter
in
terms of its amplitude response in dB, any number of filter-design
methods can be used to find a low-order
which provides a good
approximation to satisfying Eq.(3.9). 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.

- First-Order Delay-Filter Design
- Orthogonalized First-Order Delay-Filter Design
- Multiband Delay-Filter Design

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