Moog Ladder Filter

Overview

The moog_ladder is a 4-pole (24 dB/octave) lowpass ladder filter with resonance, the classic "fat" subtractive-synthesizer voice. It is four topology-preserving one-pole lowpasses in series wrapped in a global resonance feedback loop, after Vadim Zavalishin’s The Art of VA Filter Design and Will Pirkle’s Designing Software Synthesizer Plug-Ins in C++.

Moog ladder block diagram
Figure 1. Building blocks: four one-pole lowpass stages and a resonance feedback inverted back into the summer. An optional tanh sits in the feedback for the saturating Moog tone.

  • Each one-pole stage has the gain g = tan(pi * fc / fs).

  • k = 4r scales the lowpass output, and × −1 subtracts it from the input.

  • In the nonlinear mode a tanh saturates the feedback.

In the default linear mode the instantaneous (zero-delay) feedback is resolved analytically, so, like State Variable Filter, the filter stays stable and free of zipper noise when the cutoff is modulated every sample.

Parametrization is exact: the cutoff coefficient is the bilinear prewarp g = tan(pi * fc / fs), and the resonance is a normalized r in [0, 1] (internally k = 4r). The filter reaches self-oscillation exactly at r = 1. As resonance rises the passband thins, the characteristic ladder behavior.

Moog ladder magnitude response
Figure 2. The 24 dB/oct lowpass at rising resonance: the passband thins and a peak climbs toward self-oscillation as r approaches 1.

The optional nonlinear mode runs a tanh saturation in the feedback path for the saturating Moog character. That path uses the previous output for the nonlinearity (a one-sample delay in the saturation only; the four stages stay TPT), the standard inexpensive nonlinear ladder; it shifts tuning slightly at very high resonance. The linear mode has no such approximation.

It is the most expensive of the resonant filters: about ten multiplies per sample, plus two divides when the cutoff or resonance changes. See Resonant Filters for a comparison.

moog_ladder is Ladder Filter with a transistor ladder’s cell, which is where the topology, the pole count and the resonance scaling are documented. Its sibling OTA Ladder Filter is the same shape with the variable gain cell the Curtis chips used, which distorts in the second harmonic rather than the odd ones.

For a 12 dB/oct multimode filter (lowpass, bandpass, highpass, …​), see State Variable Filter; for the cheaper Chamberlin state-variable filter, see Chamberlin Filter.

Include

#include <q/fx/ladder.hpp>

Declaration

   struct moog_ladder
   {
                     moog_ladder(
                        frequency f, float sps
                      , float reso = 0.0f
                      , bool nonlinear = false
                     );

      float          operator()(float x);   // 24 dB/oct lowpass

      void           cutoff(frequency f, float sps);
      void           resonance(float r);     // normalized [0, 1]
      void           nonlinear(bool on);

      moog_ladder&   operator=(float y);
   };

Expressions

Notation

ld, a, b

Objects of type moog_ladder.

f

Object of type frequency (the cutoff).

sps

Floating point value representing samples per second.

r

Normalized resonance, a float in [0, 1].

nl

bool, nonlinear (tanh) mode.

x

Input sample.

y

A float.

Constructors and Assignment

Expression Semantics

moog_ladder(f, sps, r, nl)

Construct with cutoff f, samples per second sps, resonance r, and nonlinear flag nl.

moog_ladder(f, sps, r)

As above, linear (nl = false).

moog_ladder(f, sps)

As above with no resonance (r = 0).

moog_ladder(b)

Copy construct from b.

a = b

Assign b to a.

a = y

Clear the filter state to y.
C++ brace initialization may also be used.

Function Call

Expression Semantics Return Type

ld(x)

Process the input sample x and return the 24 dB/oct lowpass output.

float

Mutators

Expression Semantics Return Type

ld.cutoff(f, sps)

Set the cutoff to frequency f. Cheap enough to call every sample.

void

ld.resonance(r)

Set the resonance from a normalized r in [0, 1]; r = 1 self-oscillates.

void

ld.nonlinear(nl)

Enable or disable the tanh feedback saturation.

void

Example

q::moog_ladder filter{1_kHz, sps, 0.7f};      // cutoff, sps, resonance

// ... in the processing loop, per sample:
filter.cutoff(q::frequency{cutoff_hz}, sps);  // sweep the cutoff
auto out = filter(in);                         // 24 dB/oct lowpass