Damping Filter Design

The damping filter $G_i(z)$ associated with the delay line of length $M_i$ in the FDN can be written in principle as

$$G_i(z) = G_T^{M_i}(z) L_i(z)$$

where $G_T(z)$ is the lowpass filter corresponding to one sample of wave propagation through air, and $L_i(z)$ is a lowpass corresponding to absorbing/scattering boundary reflections along the (hypothetical) $i$ th propagation path.

Define

$$\begin{eqnarray*} t_{60}(\omega)&=& \mbox{desired reverberation time at frequency $\omega$}\\ [10pt] p_k &=& e^{j\omega_kT} = \mbox{$k$th pole of the lossless prototype} \end{eqnarray*}$$

We can introduce frequency-independent damping with the (conformal map) substitution

$$z^{-1}\leftarrow g\,z^{-1}$$

• This $z$ -plane mapping pulls all poles in the $z$ plane from the unit circle to the circle of radius $g$

• Pole $p_k=e^{j\omega_kT}$ moves to $\tilde{p}_k=g\,e^{j\omega_kT}$