Gradient-based Optimisation of Modulation Effects
Alistair Carson, Alec Wright, Stefan Bilbao
eess.AS, cs.LG, cs.SD
2026-01-08
Modelling phaser, flanger and chorus FX with differentiable DSP: frequency-domain training, zero-latency inference, and low-frequency weighting that fixes delay-time learning.
Modulation effects widely used with electric guitar, namely phaser, flanger and chorus, all use a low-frequency oscillator (LFO) to wiggle the phase or delay time of a signal. Digitally recreating an analog effect unit (a pedal) is called virtual analog modelling. Existing routes each have a flaw: circuit simulation is accurate but specific to one device; black-box neural networks are general but heavy and high-latency; RNN-based models misjudge the longer delay times of flangers. Differentiable digital signal processing (DDSP) is a middle path, combining automatic differentiation with known signal-processing priors. This paper extends a prior DDSP phaser method to flanger and chorus, and crucially trains in the frequency domain while inferring in the time domain at zero latency.
The hard part is learning delay times by gradient descent. The loss surface for a delay D has its global minimum at D, but the basin is extremely narrow: if the initial estimate is more than about one sample off, the gradient leads to a local minimum and the wrong answer. The authors show that low-frequency weighting of the loss (or a low-pass input signal) widens the convex region around the global minimum to about N' samples, making gradient descent workable. For all-pass coefficient estimation (the phaser case), low-frequency weighting is less decisive and full-band optimization in fact works on the analog phaser.
In training the model is frequency-domain: the input is windowed into frames (1024, 2048, 4096 points), FFT'd, and each frame predicts a time-invariant frequency response hm multiplied with the input frame to approximate the target STFT. The LFO is a learnable lookup table (M entries), each passed through a small MLP (one hidden layer, width 16, tanh); at inference, wavetable synthesis extends it to arbitrary length. The frequency response is assembled from biquad filters, a comb filter, and an all-pass cascade, with two variants for flanger/chorus (linear-phase delay) and phaser (all-pass cascade), and two feedback-loop switch positions for the biquad. Biquads use a state-variable form with FLAMO constraints keeping poles inside the unit circle for stability. The loss is computed in the frequency domain with an optional pre-emphasis filter; training is 15k steps with Adam, and each experiment runs 30 random seeds reporting the median and 95% CI.
A toy problem first (recovering a known flanger/phaser configuration): the flanger needs a triangular kernel as input or pre-emphasis, otherwise it barely beats plugging in no model at all; the phaser also works with full-band AP-chirp input.
On real devices:
The perceptual test (MUSHRA-style, 12 participants, 10 after screening) is the headline:
| effect | model rating | reference rating | verdict |
| BF-2-A flanger | 92.9 | 98.8 | no significant difference |
| SV-1-B chorus | 95.0 | 98.4 | no significant difference |
| SS-A phaser | 98.9 | 100 | no significant difference |
| SS-B phaser | 98.3 | 98.8 | no significant difference |
| BF-2-B flanger (feedback) | 42.0 | 100 | different, only "fair" band |
Four models are statistically indistinguishable from the analog reference; only the long-delay-plus-feedback BF-2-B clearly falls behind, consistent with the objective metric.
For anyone modelling audio effects, two things are directly portable: the frequency-domain-train, time-domain-infer pattern that gives zero latency and suits real-time use, and the low-frequency-spectral-weighting trick that fixes the local-minimum problem of learning delay times by gradient. The second is really a general optimization insight: when the loss surface is too narrow in the parameter direction you care about, low-passing it widens the convex region enough for gradient descent to work. For guitar and audio-plugin developers this grey-box DDSP is lighter than black-box networks and more general than circuit simulation, with code, weights and audio examples released.
Effects with long delay times and feedback (BF-2-B) remain hard; the model systematically underestimates feedback, and adding another filter in the feedback path was tried without success. The LUT-based LFO overfits to the training duration and frame rate, and inference relies on wavetable synthesis, which assumes a periodic LFO and breaks for complex non-periodic modulation. The biquad is an infinite-impulse-response filter, so some in-frame time-aliasing can remain; the authors assume its decay is short relative to the delay. The all-pass-coefficient part gives inconsistent low-frequency-weighting results, so choosing the input signal is partly empirical. The perceptual test uses the uneffected input as its anchor, a weak baseline, with only 10 listeners. Frequency-domain training is itself a proxy, and its mismatch with time-domain inference is not fully quantified.