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Delay-Line Damping Filter Design

Let $ t_{60}(\omega)$ denote the desired reverberation time at radian frequency $ \omega $ , and let $ H_i(z)$ denote the transfer function of the lowpass filter to be placed in series with the $ i$ th delay line which is $ M_i$ samples long. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for $ H_i(z)$ based on the desired reverberation time at each frequency, and then use conventional filter-design methods to obtain a low-order approximation to this ideal specification.

In accordance with Eq.$ \,$ (3.6), the lowpass filter $ H_i(z)$ in series with a length $ M_i$ delay line should approximate

$\displaystyle H_i(z) = G^{M_i}(z)

which implies

$\displaystyle \left\vert H_i(e^{j\omega T})\right\vert^{\frac{t_{60}(\omega)}{M_iT}} = 0.001.

Taking $ 20\log_{10}$ of both sides gives

$\displaystyle 20 \log_{10}\left\vert H_i(e^{j\omega T})\right\vert = -60 \frac{M_i T}{t_{60}(\omega)}. \protect$ (4.9)

This is the same formula derived by Jot [218] using a somewhat different approach.

Now that we have specified the ideal delay-line filter $ H_i(e^{j\omega T})$ in terms of its amplitude response in dB, any number of filter-design methods can be used to find a low-order $ H_i(z)$ which provides a good approximation to satisfying Eq.$ \,$ (3.9). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to $ \omega $ , the filters $ H_i(z)$ can be very low order.

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2017-05-16 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University