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History of FDNs for Artificial Reverberation

Feedback delay networks were first suggested for artificial reverberation by Gerzon [157], who proposed an ``orthogonal matrix feedback reverberation unit''. He noted that individual feedback comb filters yielded poor quality, but that several such filters could sound good when cross-coupled. An ``orthogonal matrix feedback'' around a parallel bank of delay lines was suggested as a means of obtaining maximally rich cross-coupling. He was especially concerned with good stereo spreading of the reverberation at a time when most artificial reverberators sought merely to decorrelate the reverberation in each output channel.

Later, and apparently independently, Stautner and Puckette [475] suggested a specific four-channel FDN reverberator and gave general stability conditions for the FDN. They proposed the feedback matrix

$\displaystyle \mathbf{A}= g\frac{1}{\sqrt{2}}
\left[\begin{array}{rrrr}
0 & 1 & 1 & 0 \\
-1 & 0 & 0 & -1\\
1 & 0 & 0 & -1\\
0 & 1 & -1 & 0
\end{array}\right]
$

which can be understood as a permutation (with one row sign inversion) of a $ 4\times4$ Hadamard matrix (see §3.7.2).

More recently, Jot [218,217] developed a systematic FDN design methodology allowing largely independent setting of reverberation time in different frequency bands. Using Jot's methodology, FDN reverberators can be polished to a high degree of quality, and they are presently considered to be among the best choices for high-quality artificial reverberation.

Jot's early work was concerned only with single-input, single-output (SISO) reverberators. Later work [219] with Jullien and others at IRCAM was concerned also with spatializing the reverberation.

Figure 3.10: Feedback Delay Network (FDN) structure proposed for artificial reverberation by Jot.
\includegraphics[width=\twidth]{eps/FDNJot}

An example FDN reverberator using three delay lines is shown in Fig.3.10. It can be seen as an FDN (introduced in §2.7), plus an additional low-order filter $ E(z)$ applied to the non-direct signal. This filter is called a ``tonal correction'' filter by Jot, and it serves to equalize modal energy irrespective of the reverberation time in each band. In other words, if the decay time is made very short in some band, $ E(z)$ will have a large gain in that band so that the total energy in the band's impulse-response is unchanged. This is another example of orthogonalization of reverberation parameters: In this case, adjustments in reverberation time, in any frequency band, do not alter total signal energy in the impulse response in that band.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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