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Zita-Rev1 Delay-Line Filters

In zita-rev1, the damping filter for each delay line consists of a low-shelf filter $ H_l(z)$ [452],4.19in series with a unique first-order lowpass filter $ H_h(z)$ that sets the high-frequency $ t_{60}$ to be half that of the middle-band at a particular frequency $ f_h$ (specified as ``HF Damping'' in the GUI). Since the filter $ H_h$ is constrained to be a lowpass, $ t_{60}(f)<t_{60}(f_h)$ for $ f>f_h$ , i.e., the decay time gets shorter at higher frequencies.

Viewing the resulting damping filter $ H_d(z)=H_l(z)H_h(z)$ as a three-band filter bank3.7.5), let $ g_0$ and $ g_m$ denote the desired band gains at dc and ``middle frequencies'', respectively.4.20 Then the low shelf may be set for a desired dc-gain of $ g_0/g_m$ , and its input (or output) signal multiplied by $ g_m$ to obtain the resulting filter

$\displaystyle H_l(z) \eqsp g_m + (g_0-g_m)\frac{1-p_l}{2}\frac{1+z^{-1}}{1-p_lz^{-1}},
$

where $ p_l$ denotes the (real) first-order lowpass pole, given by [452]

$\displaystyle p_l \isdefs \frac{1-\pi f_1T}{1+\pi f_1T},
$

where $ f_1$ specifies (in Hz) the crossover point between ``low'' and ``middle'' frequencies, and $ T$ denotes the sampling interval as usual.

The lowpass filter $ H_h(z)$ is also first order, and to provide half the middle-band $ t_{60}$ at the beginning of the ``high'' band, the lowpass should ``break'' to a gain of $ g_m$ at the ``HF Damping'' frequency $ f_h$ specified in the GUI. A unity-dc-gain one-pole lowpass has the form [452]

$\displaystyle H_h(z) = \frac{1-p_h}{1-p_hz^{-1}},
$

where the pole $ p_h$ must be found to give a gain of $ g_m$ at frequency $ f_h$ :

$\displaystyle \left\vert H_h\left(e^{j2\pi f_hT}\right)\right\vert \eqsp
\left\vert\frac{1-p_h}{1-p_he^{-j2\pi f_hT}}\right\vert \eqsp g_m
$

Squaring and normalizing yields a quadratic equation of the form $ p_h^2 + b\,p_h +1=0$ . Solving for $ p_h$ using the quadratic formula yields

$\displaystyle p_h \eqsp -\frac{b}{2} - \sqrt{\left(\frac{b}{2}\right)^2 - 1},
$

where

$\displaystyle -\frac{b}{2} \eqsp \frac{1-g_m^2\cos(2\pi f_h T)}{1-g_m^2} > 1,
$

and the unstable solution $ -b/2 + \sqrt{(b/2)^2 - 1} > 1$ is discarded. To ensure $ \vert g_m\vert<1$ , the GUI must limit the middle-band $ t_{60}$ to finite values. (The upper limit is presently $ 8$ seconds for both low and middle frequencies.)


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA