First-Order Delay-Filter Design

In [218], simple first-order filters are proposed for the delay-line filters $H_i(z)$:

$$H_i(z) = g_i \frac{1-a_i}{1-a_iz^{-1}}$$

where $g_i$ is set to give a desired reverberation time at dc, and $a_i$ determines the reverberation time at high frequencies. Note that the dc response of the filter is $H_i(1)=g_i$. From Eq.$\,$(2.5), we obtain

$$g_i = 10^{-3 M_i T / t_{60}(0)}$$

A calculation detailed in [218] arrives at

$$a_i = \frac{\mbox{ln}(10)}{4}\log_{10}(g_i)\left(1-\frac{1}{\alpha^2}\right)$$

where

$$\alpha \isdef \frac{t_{60}(\pi/T)}{t_{60}(0)} \protect$$ (3.6)

denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.