Frequency-Dependent Damping

For frequency-dependent damping, consider the mapping

$$\zbox{z^{-1}\leftarrow G(z)\,z^{-1}}$$

where $G(z)$ is a lowpass filter satisfying $\left\vert G(e^{j\omega T})\right\vert\leq 1$ , $\forall\omega$

• Neglecting phase in the loss filter $G(z)$ , the substitution $z^{-1}\leftarrow G(z)\,z^{-1}$ only affects the pole radius, not angle

• $G(z)$ = per-sample filter in the propagation medium

• Schroeder (1961):

  The reverberation times of the individual modes must be equal or nearly equal so that different frequency components of the sound decay with equal rates $\Rightarrow$

  • All pole radii in the reverberator should vary smoothly with frequency
  • Otherwise, late decay will be dominated by largest pole(s)