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Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Achieving Desired Reverberation Times

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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 (§2.3,§B.7.15), we want the high-frequency poles to be more damped than the low-frequency poles [314]. As discussed in §2.3, this type of transformation can be obtained using the substitution

$\displaystyle z^{-1}\leftarrow G(z)z^{-1}, \protect$ (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).4.14Thus, 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$) [451], 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

\begin{eqnarray*} e^{-T/\tau(\omega)} &\!=\!& 1 - \frac{T}{\tau(\omega)} + \frac... ... \frac{3\mbox{ln}(10)}{n_{60}} \approxs 1 - \frac{6.91}{n_{60}}. \end{eqnarray*}


Subsections
Previous: Prime Power Delay-Line Lengths
Next: Conformal Map Interpretation of Damping Substitution

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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