Reverberation Time and Energy Equalization

• Reverberation time is controlled by lowpass filters implemented within each feedback channel (the $\mathbf{G}$ matrix discussed above).

• A lowpass filter in series with a length $M_{i}$ delay line should approximate $H_{i}(z) = G^{M_{i}}(z)$, where $G(z)$ is the ideal per-sample decay filter. In terms of a desired $t_{60}$, this implies
  $$\left\vert H_{i}(e^{j \omega T_s}) \right\vert^{\frac{t_{60}(\omega)}{M_{i}T_s}} = 0.001.$$

• Jot proposes first-order filters of the form:
  $$H_{i}(z) = g_{i} \frac{1 - a_{i}}{1 - a_{i}z^{-1}},$$
  where $g_{i}$ is set to give a desired reverberation time at dc and $a_{i}$ determines the reverberation time at high frequencies.

• From the expression above, we find
  $$g_{i} = 10^{-3 M_{i}T_s/t_{60}(0)}$$
  and from (Jot and Chaigne, 1991)
  $$a_{i} = \frac{\ln(10)}{4} \log_{10}(g_{i}) \left( 1 - \frac{1}{\alpha^2} \right),$$
  where
  $$\alpha \ensuremath{\stackrel{\Delta}{=}}\frac{t_{60}(\pi/T_s)}{t_{60}(0)}.$$

• Finally, overall energy through the FDN can be equalized (to account for the effects of the loop filters) by applying a tonal correction filter at the output. For the first-order loop filters described here, a good equalization filter is given by
  $$E(z) = \frac{1 - b z^{-1}}{1 - b},$$
  where
  $$b = \frac{1 - \alpha}{1 + \alpha}.$$