Choice of Delay Lengths $M_i$

• Delay line lengths $M_i$ are typically mutually prime (Schroeder)
• For sufficiently high mode density, $\sum_i M_i$ must be sufficiently large.
  • No ``ringing tones'' in the late impulse response
  • No ``flutter''

Mode Density Requirement

FDN order = sum of delay lengths:

$$M \mathrel{\stackrel{\mathrm{\Delta}}{=}}\sum_{i=1}^N M_i\qquad\hbox{(FDN order)}$$

• Order = number of poles
• All $M$ poles are on the unit circle in the lossless prototype
• If uniformly distributed, mode density =
  $$\frac{M}{f_s}=MT \quad\hbox{modes per Hz}$$

• Schroeder suggests 0.15 modes per Hz
  (when $t_{60}=1$ second)
• Generalizing:
  $$M \geq 0.15\, t_{60}\, f_s$$

• Example: For $f_s= 50$ kHz and $t_{60}=1$ second, $M\geq 7500$

• Note that $M=t_{60}\,f_s$ is the length of the FIR filter giving a perceptually exact implementation. Thus, recursive filtering is about 7 times more efficient by this rule of thumb.