2024-11-24, C. V. Pines
Suppose we want to solve equations of the form x = a + b \exp\left(cx\right)
where all values are real and bc < 0. We can use the Lambert W
function, defined as W_k(z) \exp(W_k(z)) = z, to obtain an exact solution
for x as follows:
\begin{split}
a + b \exp\left(cx\right) &= x \\
\exp(cx) &= b^{-1}\left(x - a\right) \\
\exp(-cx)(x-a) &= b \\
\exp(-cx)(-cx + ac) &= -bc \\
\exp(-cx + ac)(-cx + ac) &= -bc \exp(ac) \\
-cx + ac &= W_0\left(-bc \exp(ac)\right) \\
x &= a - c^{-1} W_0\left(-bc \exp(ac)\right)
\end{split}
In particular, the restriction that bc < 0 ensures that exactly one
solution exists and the principal branch W_0 is sufficient.
Often, it is more convenient to represent solutions in terms of the Wright omega
function, defined as \omega(x) = W_0(\exp(x)).
This is easily done:
\begin{split}
x &= a - c^{-1} W_0\left(-bc \exp(ac)\right) \\
&= a - c^{-1} W_0\left(\exp(ac + \ln(-bc))\right) \\
&= a - c^{-1} \omega\left(ac + \ln(-bc)\right)
\end{split}
When formulating transfer functions for circuits with semiconducting elements, it is often more
convenient to express equations in the alternate form \alpha x + \beta = \exp(\gamma x + \delta) where
\alpha \gamma < 0.
By simple substitution of a = -\beta \alpha^{-1}, b = \alpha^{-1} \exp(\delta), c = \gamma we find:
\begin{split}
x &= -\tfrac{\beta }{\alpha } -\gamma^{-1} W_0\left(-\tfrac{\gamma}{\alpha } \exp\left(\delta - \beta \tfrac{\gamma}{\alpha }\right)\right) \\
&= -\tfrac{\beta }{\alpha } -\gamma^{-1} \omega\left(\delta - \beta \tfrac{\gamma}{\alpha } + \ln\left(-\tfrac{\gamma}{\alpha }\right)\right)
\end{split}
Generally, for audio circuits expressible in this form,
\alpha and \gamma represent
constants or control-rate values, while \beta and \delta
represent audio-rate signals. This alternate formulation makes it easier to perform multi-rate processing optimizations.
