Orthogonalized First-Order Delay-Filter Design

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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

for all (stable) values of

. Therefore, we can set

to give a desired reverberation time at dc, and not have to change it when

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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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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