Resonant Filters

These are the dynamically-controlled resonant filters: state-variable and ladder topologies whose cutoff can be modulated every sample (by an envelope or LFO) while staying stable and free of zipper noise, the property a direct-form Biquad lacks. There are three, trading cost against accuracy and character:

The Biquad filters are excellent, but for a different job. A biquad is a second-order direct-form filter whose coefficients are derived from the RBJ cookbook assuming a fixed, time-invariant transfer function. It is the right tool for static work: EQ, mixing, and fixed-cutoff filtering, and in particular the peaking and shelving shapes with gain that the filters here do not provide. Configured once, it is cheap and accurate.

The trouble is modulation. A direct-form biquad stores past input and output samples, and those only correspond to the coefficients that produced them. Change the coefficients to sweep the cutoff, as a synth filter driven by an envelope or LFO must, and the charged state no longer matches the new transfer function: the output gets zipper noise and transient clicks, and at high Q it can ring or briefly blow up. Retuning is also not cheap, a sin, a cos, and several divides per change.

The resonant filters here are built for exactly that case. Their state is integrator outputs with a consistent physical meaning, so the cutoff can change every sample and the filter stays stable and free of zipper noise, even at high resonance and into self-oscillation. Retuning costs a single transcendental and at most a couple of divides (see Cost). On top of that, the State Variable Filter yields several responses from one tick, and the State Variable Filter and Moog Ladder Filter give musical resonance and, for the ladder, self-oscillation, none of which a biquad is meant to do.

Rule of thumb: reach for a Biquad for static EQ and fixed-cutoff filtering (especially shelving and peaking EQ); reach for these when the filter is modulated, a synth voice, an auto-wah, a filter sweep.

Approximate cost, to help choose. Tick counts the multiplies in the per-sample processing loop (adds and subtracts are not counted). Retune is the cost of changing the cutoff or Q: it is paid only when they change, so it is free for a static filter and per-sample for a swept one.

The retune transcendental (sin / tan) uses Q’s fast-math approximation, so a swept cutoff stays cheap; the divides are the dominant retune cost for the svf and the ladder under per-sample modulation.

Copyright (c) 2014-2026 Joel de Guzman. All rights reserved. Distributed under the Boost Software License, Version 1.0