Substation’s “LP4” filter uses a unique
realtime DSP implementation of a famous (ly expensive) analog ladder lowpass filter.
In particular, the implementation allows for controlled and frequency-compensated
resonance and highly accurate frequency tracking. In fact, it can be used as an oscillator.
The algorithm is described below.
For input signal x[n], cutoff frequency
f_c[n] in Hz, and resonance
r[n] \in [0, 1], update the filter state using
\begin{align*}
D_0[n] &= -c\ g[n] \tanh^\star\left(\frac{x[n] + r^\star[n]\ S_3[n-1]}{c}\right) - g[n]\ S_0[n-1] \\
S_0[n] &= S_0[n-1] + D_0[n] + D_0[n -1] \\\\
D_1[n] &= g[n]\ S_0 [n] - g[n]\ S_1[n-1] \\
S_1[n] &= S_1[n-1] + D_1[n] + D_1[n -1] \\\\
D_2[n] &= g[n]\ S_1 [n] - g[n]\ S_2[n-1] \\
S_2[n] &= S_2[n-1] + D_2[n] + D_2[n -1] \\\\
D_3[n] &= -c\ g[n] \tanh^\star\left(\frac{S_3[n-1]}{c}\right) + g[n]\ S_2[n] \\
S_3[n] &= S_3[n-1] + D_3[n] + D_3[n -1]
\end{align*}
where
\begin{align*}
g[n] &= \frac{\pi f_c[n]}{f_s} \\
r^\star[n] &= r[n] \left(4 - \frac{f_c[n]}{7500}\right).
\end{align*}
\tanh^\star(\cdot) is a piecewise rational
approximation to \tanh(\cdot),
\tanh^\star(x)=
\begin{cases}
-1 & \text{if } x \leq -3\\
\frac{27x + x^3}{27 + 9x^2} & \text{if } x \in (-3, 3) \\
1 & \text{if } x \geq 3
\end{cases}
and c is the filter drive; for
x[n] \in [-1, 1],
c = 0.8 is suggested. I have not tested
modulating c, though it would probably
sound okay with enough oversampling and slew limiting.
The filter update section should be oversampled by at least 6× when running at
audio rates (remember to adjust f_s accordingly).
Multiple filter outputs are available:
\begin{align*}
y_{LP1}[n] &= -(1 + r[n])\ S_0[n] \\
y_{LP2}[n] &= -(1 + r[n])\ S_1[n] \\
y_{LP3}[n] &= -(1 + r[n])\ S_2[n] \\
y_{LP4}[n] &= -(1 + r[n])\ S_3[n] \\
y_{BP2}[n] &= S_0[n] - S_1[n] \\
y_{BP4}[n] &= 2 S_1[n] - 4 S_2[n] + 2 S_3[n] \\
\end{align*}
Highpass outputs are possible, but of low quality.
