Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search


First-Order Delay-Filter Design

The first-order case is very simple while enabling separate control of low-frequency and high-frequency reverberation times. For simplicity, let's specify $ t_{60}(0)$ and $ t_{60}(\pi/T)$ , denoting the desired decay-time at dc ($ \omega=0$ ) and half the sampling rate ( $ \omega=\pi/T$ ). Then we have determined the coefficients of a one-pole filter:

$\displaystyle H_i(z) = \frac{g_i}{1-p_iz^{-1}}
$

The dc gain of this filter is $ H_i(1)=g_i/(1-p_i)$ , while the gain at $ \omega=\pi/T$ is $ H_i(-1)=g_i/(1+p_i)$ . From Eq.$ \,$ (3.9) (or Eq.$ \,$ (3.8)), we obtain two equations in two unknowns:

\begin{eqnarray*}
\frac{g_i}{1-p_i} &=& 10^{-3 M_i T / t_{60}(0)}
\eqsp e^{-M_iT/\tau(0)} \isdefs R_0^{M_i}\\ [5pt]
\frac{g_i}{1+p_i} &=& 10^{-3 M_i T / t_{60}(\pi/T)}
\eqsp e^{-M_iT/\tau(\pi/T)} \isdefs R_\pi^{M_i}\\ [5pt]
\end{eqnarray*}

where $ D_i\isdeftext M_iT$ denotes the $ i$ th delay-line length in seconds. These two equations are readily solved to yield

\begin{eqnarray*}
p_i &=& \frac{R_0^{M_i}-R_\pi^{M_i}}{R_0^{M_i}+R_\pi^{M_i}}\\ [5pt]
g_i &=& \frac{2R_0^{M_i}R_\pi^{M_i}}{R_0^{M_i}+R_\pi^{M_i}}
\end{eqnarray*}

The truncated series approximation

$\displaystyle R_\omega^{M_i} \isdefs e^{-\frac{M_iT}{\tau(\omega)}}
\approxs 1 - \frac{M_iT}{\tau(\omega)}
\approxs 1 - \frac{6.91\,M_iT}{t_{60}(\omega)}
\isdefs 1 - \frac{6.91\,M_i}{n_{60}(\omega)}
$

has been found to work well in practical FDN reverberators.


Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA