Damping Filters for Reverberation Delay Lines

In an FDN, such as the one shown in Fig.3.10, delays $ z^{-1}$ appear in long delay-line chains $ z^{-M_i}$. Therefore, the filter needed at the output (or input) of the $ i$th delay line is $ G^{M_i}(z)$ (replace $ g_i$ by $ G^{M_i}(z)$ in Fig.3.10).4.15 However, because $ G(e^{j\omega T})$ is so close to $ 1$ in magnitude, and because it varies so weakly across the frequency axis, we can design a much lower-order filter $ H_i(z)\approx G^{M_i}(z)$ that provides the desired attenuation versus frequency to within psychoacoustic resolution. In fact, perfectly nice reverberators can be designed in which $ H_i(z)$ is merely first order for each $ i$ [314,217].

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First-Order Delay-Filter Design
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Conformal Map Interpretation of Damping Substitution