Non-Linear Digital Implementation of the Moog Ladder Filter

Antti Huovilainen, Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology. Proc. 7th Int. Conference on Digital Audio Effects (DAFx-04), Naples, October 2004, 4 pp. Read in full. Funded by the Academy of Finland (project #104934).

Brought in by the 2026-08-12 research pass for growth edge 1’s successor — the circuit half of the analog argument, which meyer-moran-adda-loop and digital-filter-audibility leave untouched because both test conversion rather than topology.

Why it is the right subject and the wrong instrument

The paper opens on exactly the claim this spoke tracks: existing digital Moog filters “are linear and some people feel that they sound ‘digital’ and lack the ‘warmth’ that is charasteristic of analog filters — especially the Moog filter.” Huovilainen’s answer is to derive the filter from the circuit instead of approximating its response.

It is not the bench comparison the edge asked for. There is no measurement of a real ladder filter, no THD figure, no listening test, and no comparison against hardware — the validation is analytical. Recorded as a partial advance for that reason. What it does supply is the mechanism the argument has been missing, which turns out to be more useful than another verdict on audibility.

Where “warmth” is located

Moog’s 1965 circuit (17th AES Convention) is a transistor ladder: four stages, two transistors and a capacitor each, using the base-to-emitter resistance of bipolar transistors as voltage-controlled RC sections, with the transistors also buffering the stages from one another.

The nonlinearity is not a flaw or a flavour added afterwards. It falls out of the differential pair, whose differential current is a tanh of the base-emitter voltage difference over the thermal voltage Vt. Because the stages are buffered — beta assumed infinite, base current zero, the Early effect negligible — each stage depends only on its own state and the current arriving from the previous one, so the whole ladder discretizes as a cascade of first-order IIR sections with embedded nonlinearities.

The sentence that matters for the purism argument: “there are no extra coefficients that would need to be tuned by ear — the ‘warmth’ is determined by the input amplitude.” On this account the analog character of a ladder filter is a specific, derivable amplitude-dependent saturation, not an irreducible property of being analog. It is an argument that the circuit is what matters, made by someone modelling the circuit.

What the digital version costs

Three honest concessions, each a place the model and the circuit differ:

• Oversampling is mandatory. The embedded nonlinearity requires it, and it also brings the Euler solution closer to the ideal one. Higher-order solvers (Runge-Kutta) would need the equation evaluated between samples — equivalent to a higher input than output rate — which also breaks the resonance feedback path.
• The unit delay detunes the resonance. In the analog filter each stage shifts phase 45° at cutoff, giving 180° across four stages, which with inversion makes the feedback positive at cutoff. Digitally, the feedback path’s unit delay adds 4·pstage + 180·fc/Fs, so resonance drifts away from the cutoff frequency and the attenuation at resonance is no longer exactly 3 dB per stage (12 dB total). The feedback needed for a given resonance therefore varies with frequency.
• Tuning is a small-signal accident. For inputs where tanh is nearly linear, the stage reduces to an ordinary one-pole lowpass, and tuning depends only on control current, capacitance and sample rate — which is why the exact component values do not matter and the coefficient can be computed as g = 1 − e^(−2π·fc/Fs).

For the argument in this spoke

It cuts against purism and for the purists’ premise at once, which is why the page keeps both halves. The premise it grants: the character lives in the circuit’s nonlinearity, exactly where analog-vs-digital-pedals puts the overdrive case. The conclusion it denies: that the nonlinearity is unmodellable — it is one hyperbolic tangent per differential pair, derived from the schematic.

What no source here can yet say is whether the derived model and the circuit are discriminable in a listening test, or how far apart they measure on a bench. See analog-vs-digital and synthesis.

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analog-vs-digital · analog-vs-digital-pedals · meyer-moran-adda-loop · digital-filter-audibility · filtopia · polyuanalog · synthesis