Freeverb Allpass Approximation

Schroeder Diffusion Allpass

$$\hbox{AP}_{N}^{\,g} \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{-g + z^{-N}}{1 - g z^{-N}}$$

Freeverb implements

$$\hbox{AP}_{N}^{\,g} \approx \frac{-1 + (1+g)z^{-N}}{1 - g z^{-N}}$$

• Each Freeverb ``allpass'' is more precisely a feedback comb-filter $\hbox{FBCF}_{N}^{\,g}$ in series with a feedforward comb-filter $\hbox{FFCF}_{N}^{\,-1,1+g}$ , where
  $$\begin{eqnarray*} \hbox{FBCF}_{N}^{\,g} &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& \frac{1}{1 - g\,z^{-N}}\\ [5pt] \hbox{FFCF}_{N}^{\,-1,1+g} &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& -1 + (1+g)z^{-N}. \end{eqnarray*}$$
  

• A true allpass is obtained at $g=(\sqrt{5}-1)/2\approx 0.618$ (reciprocal of ``golden ratio'')
• Freeverb default is $g=0.5$