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).^4.16 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 $i$ [317,218].