### Delay-Line Damping Filter Design

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

This is the same formula derived by Jot [217] 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

The first-order case is very simple while enabling separate control of low-frequency and high-frequency reverberation times. For simplicity, let's specify and , denoting the desired decay-time at dc () and half the sampling rate ( ). Then we have determined the coefficients of a one-pole filter:

where denotes the th delay-line length in seconds. These two equations are readily solved to yield

The truncated series approximation

#### Orthogonalized First-Order Delay-Filter Design

In [217], first-order delay-line filters of the form

denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.

^{4.16}

#### Multiband Delay-Filter Design

In §3.7.5, we derived first-order FDN delay-line filters which
can independently set the reverberation time at dc and at half the
sampling rate. However, perceptual studies indicate that
reverberation time should be independently adjustable in at least
three frequency bands [217]. To provide this degree
of control (and more), one can implement a multiband delay-line filter
using a general-purpose *filter bank*
[370,500]. The output, say, of each delay
line is split into bands, where is recommended, and then,
from Eq.(3.6), the gain in the th band for a length
delay-line can be set to

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Spectral Coloration Equalizer

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Achieving Desired Reverberation Times