Multiband Delay-Filter Design

In §3.7.5, we derived first-order FDN delay-line filters which can independently set the reverberation time at dc and at half the sampling rate. However, perceptual studies indicate that reverberation time should be independently adjustable in at least three frequency bands [218]. To provide this degree of control (and more), one can implement a multiband delay-line filter using a general-purpose filter bank [373,504]. The output, say, of each delay line is split into $K$ bands, where $K\ge 3$ is recommended, and then, from Eq.(3.6), the gain in the $k$ th band for a length $M_i$ delay-line can be set to

$$G^{M_i}(e^{j\omega_kT}) = 10^{-\frac{3M_i}{n_{60}(\omega_k)}} = e^{-\frac{3\,\mbox{ln}(10)M_i}{n_{60}(\omega_k)}} \approx 1-\frac{3\,\mbox{ln}(10)\,M_i}{n_{60}(\omega_k)} \approx 1-\frac{6.91\,M_i}{n_{60}(\omega_k)}$$

to produce the desired decay-time in that band, where $n_{60}(\omega)=t_{60}(\omega)/T$ denotes the desired 60-dB decay time in samples. Faust implementations of FDN reverberator along these lines are described in §3.7.9 below.