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Orthogonalized First-Order Delay-Filter Design

In [217], first-order delay-line filters of the form

$\displaystyle H_i(z) \eqsp g'_i \frac{1-p_i}{1-p_iz^{-1}}
$

are proposed. Clearly $ g_i=g'_i\cdot(1-p_i)$. This form has the advantage that the dc gain is always $ H_i(1)=g'_i$ for all (stable) values of $ p_i$. Therefore, we can set $ g'_i$ to give a desired reverberation time at dc, and not have to change it when $ p_i$ is varied to modulate the high-frequency decay rate. As in the previous section, from Eq.$ \,$(3.9), we obtain

$\displaystyle g'_i \eqsp 10^{-3 M_i T / t_{60}(0)}.
$

A calculation given in [217] arrives at

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

where

$\displaystyle \alpha \isdef \frac{t_{60}(\pi/T)}{t_{60}(0)} \protect$ (4.10)

denotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.4.16


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