Stefan Bilbao

Chin-Yun Yu, Christopher Mitcheltree, Alistair Carson et al.

Infinite impulse response filters are an essential building block of many time-varying audio systems, such as audio effects and synthesisers. However, their recursive structure impedes end-to-end training of these systems using automatic differentiation. Although non-recursive filter approximations like frequency sampling and frame-based processing have been proposed and widely used in previous works, they cannot accurately reflect the gradient of the original system. We alleviate this difficulty by re-expressing a time-varying all-pole filter to backpropagate the gradients through itself, so the filter implementation is not bound to the technical limitations of automatic differentiation frameworks. This implementation can be employed within audio systems containing filters with poles for efficient gradient evaluation. We demonstrate its training efficiency and expressive capabilities for modelling real-world dynamic audio systems on a phaser, time-varying subtractive synthesiser, and compressor. We make our code and audio samples available and provide the trained audio effect and synth models in a VST plugin at https://diffapf.github.io/web/.

Alistair Carson, Alec Wright, Stefan Bilbao

Modulation effects such as phasers, flangers and chorus effects are heavily used in conjunction with the electric guitar. Machine learning based emulation of analog modulation units has been investigated in recent years, but most methods have either been limited to one class of effect or suffer from a high computational cost or latency compared to canonical digital implementations. Here, we build on previous work and present a framework for modelling flanger, chorus and phaser effects based on differentiable digital signal processing. The model is trained in the time-frequency domain, but at inference operates in the time-domain, requiring zero latency. We investigate the challenges associated with gradient-based optimisation of such effects, and show that low-frequency weighting of loss functions avoids convergence to local minima when learning delay times. We show that when trained against analog effects units, sound output from the model is in some cases perceptually indistinguishable from the reference, but challenges still remain for effects with long delay times and feedback.

Victor Zheleznov, Stefan Bilbao, Alec Wright et al.

Modal methods are a long-standing approach to physical modelling synthesis. Extensions to nonlinear problems are possible, leading to coupled nonlinear systems of ordinary differential equations. Recent work in scalar auxiliary variable techniques has enabled construction of explicit and stable numerical solvers for such systems. On the other hand, neural ordinary differential equations have been successful in modelling nonlinear systems from data. In this work, we examine how scalar auxiliary variable techniques can be combined with neural ordinary differential equations to yield a stable differentiable model capable of learning nonlinear dynamics. The proposed approach leverages the analytical solution for linear vibration of the system's modes so that physical parameters of a system remain easily accessible after the training without the need for a parameter encoder in the model architecture. Compared to our previous work that used multilayer perceptrons to parametrise nonlinear dynamics, we employ gradient networks that allow an interpretation in terms of a closed-form and non-negative potential required by scalar auxiliary variable techniques. As a proof of concept, we generate synthetic data for the nonlinear transverse vibration of a string and show that the model can be trained to reproduce the nonlinear dynamics of the system. Sound examples are presented.

Alistair Carson, Vesa Välimäki, Alec Wright et al.

Neural networks have become ubiquitous in audio effects modelling, especially for guitar amplifiers and distortion pedals. One limitation of such models is that the sample rate of the training data is implicitly encoded in the model weights and therefore not readily adjustable at inference. Recent work explored modifications to recurrent neural network architecture to approximate a sample rate independent system, enabling audio processing at a rate that differs from the original training rate. This method works well for integer oversampling and can reduce aliasing caused by nonlinear activation functions. For small fractional changes in sample rate, fractional delay filters can be used to approximate sample rate independence, but in some cases this method fails entirely. Here, we explore the use of real-time signal resampling at the input and output of the neural network as an alternative solution. We investigate several resampling filter designs and show that a two-stage design consisting of a half-band IIR filter cascaded with a Kaiser window FIR filter can give similar or better results to the previously proposed model adjustment method with many fewer filtering operations per sample and less than one millisecond of latency at typical audio rates. Furthermore, we investigate interpolation and decimation filters for the task of integer oversampling and show that cascaded half-band IIR and FIR designs can be used in conjunction with the model adjustment method to reduce aliasing in a range of distortion effect models.

Victor Zheleznov, Stefan Bilbao, Alec Wright et al.

Modal synthesis methods are a long-standing approach for modelling distributed musical systems. In some cases extensions are possible in order to handle geometric nonlinearities. One such case is the high-amplitude vibration of a string, where geometric nonlinear effects lead to perceptually important effects including pitch glides and a dependence of brightness on striking amplitude. A modal decomposition leads to a coupled nonlinear system of ordinary differential equations. Recent work in applied machine learning approaches (in particular neural ordinary differential equations) has been used to model lumped dynamic systems such as electronic circuits automatically from data. In this work, we examine how modal decomposition can be combined with neural ordinary differential equations for modelling distributed musical systems. The proposed model leverages the analytical solution for linear vibration of system's modes and employs a neural network to account for nonlinear dynamic behaviour. Physical parameters of a system remain easily accessible after the training without the need for a parameter encoder in the network architecture. As an initial proof of concept, we generate synthetic data for a nonlinear transverse string and show that the model can be trained to reproduce the nonlinear dynamics of the system. Sound examples are presented.

Alistair Carson, Cassia Valentini-Botinhao, Simon King et al.

Machine learning approaches to modelling analog audio effects have seen intensive investigation in recent years, particularly in the context of non-linear time-invariant effects such as guitar amplifiers. For modulation effects such as phasers, however, new challenges emerge due to the presence of the low-frequency oscillator which controls the slowly time-varying nature of the effect. Existing approaches have either required foreknowledge of this control signal, or have been non-causal in implementation. This work presents a differentiable digital signal processing approach to modelling phaser effects in which the underlying control signal and time-varying spectral response of the effect are jointly learned. The proposed model processes audio in short frames to implement a time-varying filter in the frequency domain, with a transfer function based on typical analog phaser circuit topology. We show that the model can be trained to emulate an analog reference device, while retaining interpretable and adjustable parameters. The frame duration is an important hyper-parameter of the proposed model, so an investigation was carried out into its effect on model accuracy. The optimal frame length depends on both the rate and transient decay-time of the target effect, but the frame length can be altered at inference time without a significant change in accuracy.