### Achieving Desired Reverberation Times

A lossless prototype reverberator, as in Fig.3.10 when , has all of its poles on the unit circle in the 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, due to air absorption (§2.3,§B.7.15), we want the high-frequency poles to be more damped than the low-frequency poles [314]. As discussed in §2.3, this type of transformation can be obtained using the substitutionwhere denotes the

*filtering per sample*in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).

^{4.14}Thus, to set the FDN reverberation time to at frequency , we want propagation through samples to result in attenuation by dB,

*i.e.*,

Solving for , the propagation attenuation per-sample, gives

The last form comes from ln, where denotes the time constant of decay (time to decay by ) [451],

*i.e.*,

Series expanding and assuming samples ( seconds) provides the practically useful approximation

#### Conformal Map Interpretation of Damping Substitution

The relation [Eq.(3.7)] can be written down directly from [Eq.(3.5)] by interpreting Eq.(3.5) as an approximate*conformal map*[326] which takes each pole , say, from the unit circle to the point . Thus, the new pole radius is approximately , where the approximation is valid when is approximately constant between the new pole location and the unit circle. To see this, consider the partial fraction expansion [449] of a proper th-order lossless transfer function mapped to :

*all pole radii in the reverberator should vary smoothly with frequency*. This translates to having a

*smooth frequency response*. To see why this is desired, consider momentarily the frequency-independent case in which we desire the same reverberation time at all frequencies (Fig.3.10 with real , as drawn). In this case, it is ideal for all of the poles to have this decay 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 to the output. Only when all poles have the same magnitude will the late response maintain the same modal density throughout the decay.

#### Damping Filters for Reverberation Delay Lines

In an FDN, such as the one shown in Fig.3.10, delays appear in long delay-line chains . Therefore, the filter needed at the output (or input) of the th delay line is (replace by in Fig.3.10).^{4.15}However, because is so close to in magnitude, and because it varies so weakly across the frequency axis, we can design a much lower-order filter that provides the desired attenuation versus frequency to within psychoacoustic resolution. In fact, perfectly nice reverberators can be designed in which is merely

*first order*for each [314,217].

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Delay-Line Damping Filter Design

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Choice of Delay Lengths