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 [317]. As discussed in §2.3, this type of transformation can be obtained using the substitution

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

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

Solving for $G$ , the propagation attenuation per-sample, gives

$$G(e^{j\omega T}) =\! (0.001)^{\frac{1}{n_{60}(\omega)}} \eqsp 10^{-3/n_{60}} \eqsp \left(e^{\mbox{ln}(10)}\right)^{-3/n_{60}} \eqsp e^{-3\,\mbox{ln}(10)/n_{60}}$$

$$=\! e^{-T/\tau(\omega)} \protect$$ (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.,

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

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