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Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Zita-Rev1
             Zita-Rev1 Delay-Line Filters

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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)$ [449],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 bank (§3.7.5), 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 [449]

$\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 [449]

$\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.)

Previous: Zita-Rev1
Next: Further Extensions

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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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