Freeverb Allpass Approximation

In Eq.(3.2) we defined the allpass notation $\hbox{AP}_{N}^{\,g}$ by

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

A look at allpass.h reveals that Freeverb implements

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

As a result, each of the four Freeverb ``allpass'' sections is really a feedback comb-filter $\hbox{FBCF}_{N}^{\,g}$ in series with a feedforward comb-filter $\hbox{FFCF}_{N}^{\,-1,1+g}$ , where (cf. §2.6)

$$\begin{eqnarray*} \hbox{FBCF}_{N}^{\,g} &\isdef & \frac{1}{1 - g\,z^{-N}}\\ [5pt] \hbox{FFCF}_{N}^{\,-1,1+g} &\isdef & -1 + (1+g)z^{-N}. \end{eqnarray*}$$

A true allpass is obtained only for $g=(\sqrt{5}-1)/2\approx 0.618$ (reciprocal of the ``golden ratio''). The default value used in Freeverb (see revmodel.cpp) is $0.5$ . A detailed discussion of feedforward and feedback comb filters appears in §2.6, and corresponding Schroeder allpass filters are described in §2.8.