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High-Frequency-Damping Lowpass

High-Frequency Damping Lowpass:

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

For $ t_{60}$ at ``HF Damping'' frequency $ f_h$ to be half of middle-band $ t_{60}$ (gain $ g_m$ ), we require

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

Squaring and normalizing yields a quadratic equation:

$\displaystyle p_h^2 + b\,p_h +1=0
$

Solving for $ p_h$ using the quadratic formula yields

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

where

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

Discard unstable solution $ -b/2 + \sqrt{(b/2)^2 - 1} > 1$

To ensure $ \vert g_m\vert<1$ , GUI keeps middle-band $ t_{60}$ finite


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``Artificial Reverberation and Spatialization'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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