Achieving Desired Reverberation Times

The lossless prototype reverberator has all of its poles on the unit circle in the $z$ plane, and its reverberation time is infinity. To set the reverberation time to a desired value, we need to move the poles slightly inside the unit circle. Furthermore, we want the high-frequency poles to be more damped than the low-frequency poles. As discussed in §1.3, this type of transformation can be obtained using the substitution

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

where $G(z)$ denotes the filtering per sample in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).^3.11Thus, to set the reverberation time in an FDN, we need to find the $G(z)$ which moves the poles where desired, and then design lowpass filters $H_i(z)\approx G^{M_i}(z)$ which will be placed at the output (or input) of each delay line.

An important design principle introduced by Jot [201] is that all pole radii in the reverberator should vary smoothly with frequency. To see why this is desired, consider momentarily the frequency-independent case in which we desire the same reverberation time at all frequencies. In this case, it is ideal for all of the poles to have this reverberation time. Otherwise, the late decay of the impulse response will be dominated by the poles having the largest magnitude, and it will be ``thinner'' than it was at the beginning of the response when all poles were contributing equally to the output. Only when all poles have the same magnitude will the late response maintain the same modal density throughout the decay.