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Achieving Desired Reverberation Times

A lossless prototype reverberator, as in Fig.3.10 when $ g_i=1$ , has all of its poles on the unit circle in the $ z$ plane, and its reverberation time is infinity. To set the reverberation time to a desired value, we need to move the poles slightly inside the unit circle. Furthermore, due to air absorption (§2.3B.7.15), we want the high-frequency poles to be more damped than the low-frequency poles [317]. As discussed in §2.3, this type of transformation can be obtained using the substitution

$\displaystyle z^{-1}\rightarrow G(z)z^{-1}, \protect$ (4.5)

where $ G(z)$ denotes the filtering per sample in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).4.15Thus, to set the FDN reverberation time to $ t_{60}(\omega)\isdeftext n_{60}(\omega)T$ at frequency $ \omega $ , we want propagation through $ n_{60}$ samples to result in attenuation by $ 60$ dB, i.e.,

$\displaystyle \left[G(e^{j\omega T})\right]^{n_{60}(\omega)} \eqsp 0.001. \protect$ (4.6)

Solving for $ G$ , the propagation attenuation per-sample, gives
$\displaystyle G(e^{j\omega T})$ $\displaystyle =\!$ $\displaystyle (0.001)^{\frac{1}{n_{60}(\omega)}}
\eqsp 10^{-3/n_{60}}
\eqsp \left(e^{\mbox{ln}(10)}\right)^{-3/n_{60}} \eqsp e^{-3\,\mbox{ln}(10)/n_{60}}$  
  $\displaystyle =\!$ $\displaystyle e^{-T/\tau(\omega)}
\protect$ (4.7)

The last form comes from $ t_{60}=3$ln$ (10)\tau\approx 6.91\tau$ , where $ \tau $ denotes the time constant of decay (time to decay by $ 1/e$ ) [454], i.e.,

$\displaystyle e^{-t_{60}/\tau}=0.001 \;\;\Leftrightarrow\;\; t_{60}\eqsp -3$ln$\displaystyle (10)\tau. \protect$ (4.8)

Series expanding $ e^{-T/\tau(\omega)}$ and assuming $ n_{60}(\omega)\gg 7$ samples ( $ \tau(\omega)\gg T$ seconds) provides the practically useful approximation

&\!=\!& 1 - \frac{T}{\tau(\omega)} + \frac{1}{2!}\left[\frac{T}{\tau(\omega)}\right]^2 - \cdots
\approxs 1 - \frac{3\mbox{ln}(10)}{n_{60}}
\approxs 1 - \frac{6.91}{n_{60}}.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University