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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Artificial Reverberation
       FDN Reverberation
          Delay-Line Damping Filter Design
             First-Order Delay-Filter Design

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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... ...(\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{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.

Previous: Delay-Line Damping Filter Design
Next: Orthogonalized First-Order Delay-Filter Design

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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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