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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.

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Orthogonalized First-Order Delay-Filter Design
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Damping Filters for Reverberation Delay Lines