Choice of Delay Lengths

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

Mean Free Path

$${\overline d} = 4\frac{V}{S}\qquad\hbox{(mean free path)}$$

where $V$ is the total volume of the room, and $S$ is total surface area enclosing the room.

Regarding each delay line as a mean-free-path delay, the mean free path length, in samples, is the average delay-line length:

$$\frac{{\overline d}}{cT} = \frac{1}{N} \sum_{i=1}^N M_i$$

where $c$ = sound speed and $T$ = sampling period.

This is only a lower bound because many reflections are diffuse in real rooms, especially at high frequencies (one plane-wave reflection scatters in many directions)

Mode Density Requirement

FDN order = sum of delay lengths:

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

• 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 that 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 an exact implementation. Thus, recursive filtering is about 7 times more efficient by this rule of thumb.