arXiv subjects

Stefan Bilbao

Publications and source records attributed to Stefan Bilbao.

15 recordsLinked to original sources

arXiv2026

Explicit and Stable Pseudospectral Time-Domain Method for the Föppl-von Kármán Equations

Modal synthesis is a widely-used technique for simulation of musical instrument dynamics. In the linear case, a modal decomposition leads to an uncoupled system of damped and forced harmonic oscillators which can be efficiently solved by standard time-stepping methods. However, extensions to nonlinear problems are challenging due to the presence of products of modal expansions in the governing equations. In the case of the Föppl-von Kármán plate, the nonlinear coupling between the modes is described by a fourth-order tensor and is prohibitively expensive to evaluate in the modal domain. In this work, we propose a pseudospectral method in which the products are evaluated on a grid in the spatial domain while spatial derivatives are computed exactly in the modal domain. Discrete sine and cosine transforms between the modal and spatial domains are used to impose simply supported boundary conditions for the plate. Finally, we prove non-negativity of the nonlinear potential energy of the system and employ a scalar auxiliary variable technique for explicit and stable time integration in the modal domain. As a result, we reduce the computational cost of modal synthesis while preserving its advantages like a precise control over the simulated frequency range. Sound examples are presented.

cs.SD↗

arXiv2026

Gradient-based Optimisation of Modulation Effects

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.

eess.AS↗

arXiv2026

Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics

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.

cs.SD↗

arXiv2025

Resampling Filter Design for Multirate Neural Audio Effect Processing

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.

eess.AS↗

arXiv2025

Anti-aliasing of neural distortion effects via model fine tuning

Neural networks have become ubiquitous with guitar distortion effects modelling in recent years. Despite their ability to yield perceptually convincing models, they are susceptible to frequency aliasing when driven by high frequency and high gain inputs. Nonlinear activation functions create both the desired harmonic distortion and unwanted aliasing distortion as the bandwidth of the signal is expanded beyond the Nyquist frequency. Here, we present a method for reducing aliasing in neural models via a teacher-student fine tuning approach, where the teacher is a pre-trained model with its weights frozen, and the student is a copy of this with learnable parameters. The student is fine-tuned against an aliasing-free dataset generated by passing sinusoids through the original model and removing non-harmonic components from the output spectra. Our results show that this method significantly suppresses aliasing for both long-short-term-memory networks (LSTM) and temporal convolutional networks (TCN). In the majority of our case studies, the reduction in aliasing was greater than that achieved by two times oversampling. One side-effect of the proposed method is that harmonic distortion components are also affected. This adverse effect was found to be model-dependent, with the LSTM models giving the best balance between anti-aliasing and preserving the perceived similarity to an analog reference device.

eess.AS↗

arXiv2025

Learning Nonlinear Dynamics in Physical Modelling Synthesis using Neural Ordinary Differential Equations

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.

cs.SD↗

arXiv2025

Interpolation Filter Design for Sample Rate Independent Audio Effect RNNs

Recurrent neural networks (RNNs) are effective at emulating the non-linear, stateful behavior of analog guitar amplifiers and distortion effects. Unlike the case of direct circuit simulation, RNNs have a fixed sample rate encoded in their model weights, making the sample rate non-adjustable during inference. Recent work has proposed increasing the sample rate of RNNs at inference (oversampling) by increasing the feedback delay length in samples, using a fractional delay filter for non-integer conversions. Here, we investigate the task of lowering the sample rate at inference (undersampling), and propose using an extrapolation filter to approximate the required fractional signal advance. We consider two filter design methods and analyse the impact of filter order on audio quality. Our results show that the correct choice of filter can give high quality results for both oversampling and undersampling; however, in some cases the sample rate adjustment leads to unwanted artefacts in the output signal. We analyse these failure cases through linearised stability analysis, showing that they result from instability around a fixed point. This approach enables an informed prediction of suitable interpolation filters for a given RNN model before runtime.

eess.AS↗

arXiv2024

Differentiable All-pole Filters for Time-varying Audio Systems

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/.

eess.AS↗

arXiv2024

Sample Rate Independent Recurrent Neural Networks for Audio Effects Processing

In recent years, machine learning approaches to modelling guitar amplifiers and effects pedals have been widely investigated and have become standard practice in some consumer products. In particular, recurrent neural networks (RNNs) are a popular choice for modelling non-linear devices such as vacuum tube amplifiers and distortion circuitry. One limitation of such models is that they are trained on audio at a specific sample rate and therefore give unreliable results when operating at another rate. Here, we investigate several methods of modifying RNN structures to make them approximately sample rate independent, with a focus on oversampling. In the case of integer oversampling, we demonstrate that a previously proposed delay-based approach provides high fidelity sample rate conversion whilst additionally reducing aliasing. For non-integer sample rate adjustment, we propose two novel methods and show that one of these, based on cubic Lagrange interpolation of a delay-line, provides a significant improvement over existing methods. To our knowledge, this work provides the first in-depth study into this problem.

eess.AS↗

arXiv2023

Differentiable Grey-box Modelling of Phaser Effects using Frame-based Spectral Processing

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.

eess.AS↗

arXiv2022

Explicit Exactly Energy-conserving Methods for Hamiltonian Systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Stormer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam, as well as for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates.

math.NA↗

arXiv2021

Simulation of the Geometrically Exact Nonlinear String via Energy Quadratisation

String vibration represents an active field of research in acoustics. Small-amplitude vibration is often assumed, leading to simplified physical models that can be simulated efficiently. However, the inclusion of nonlinear phenomena due to larger string stretchings is necessary to capture important features, and efficient numerical algorithms are currently lacking in this context. Of the available techniques, many lead to schemes which may only be solved iteratively, resulting in high computational cost, and the additional concerns of existence and uniqueness of solutions. Slow and fast waves are present concurrently in the transverse and longitudinal directions of motion, adding further complications concerning numerical dispersion. This work presents a linearly-implicit scheme for the simulation of the geometrically exact nonlinear string model. The scheme conserves a numerical energy, expressed as the sum of quadratic terms only, and including an auxiliary state variable yielding the nonlinear effects. This scheme allows to treat the transverse and longitudinal waves separately, using a mixed finite difference/modal scheme for the two directions of motion, thus allowing to accurately resolve the wave speeds at reference sample rates. Numerical experiments are presented throughout.

math.NA↗

arXiv2021

Lyapunov Function for the Nonlinear Moog Voltage Controlled Filter

In this short report, a new Lyapunov function for the Moog voltage-controlled filter is demonstrated, under zero-input conditions, and under nonlinear autonomous conditions (i.e. when parameters are not time-varying). The new definition allows for a proof of stability over the entire allowable range of parameters (cutoff frequency and resonance), and can be used as a starting point for Hamiltonian-based numerical simulation methods.

math.DS↗

arXiv2019

Conservative Numerical Methods for Nonlinear String Dynamics: Non-planar Vibration

Nonlinear string vibration, in particular the case of nonplanar motion, has been an area of intense study for many years. Numerical simulation methods, essential for the comparison between measured data and theory, have received somewhat less attention. In this article, various numerical schemes for nonlinear nonplanar string dynamics are presented, with an emphasis on discrete conservation of energy and angular momentum. Simple numerical stability conditions may be arrived at, even under strongly nonlinear conditions, by employing these conservation properties. Full implementation details and various numerical examples are presented, and several topics, including a discussion of numerical loss models and spectral methods, are dealt with in the Appendix.

math.AP↗

arXiv2014

Numerical Modeling of Collisions in Musical Instruments

Collisions play an important role in many aspects of the physics of musical instruments. The striking action of a hammer or mallet in keyboard and percussion instruments is perhaps the most important example, but others include reed-beating effects in wind instruments, the string/neck interaction in fretted instruments such as the guitar as well as in the sitar and the wire/membrane interaction in the snare drum. From a simulation perspective, whether the eventual goal is the validation of musical instrument models or sound synthesis, such highly nonlinear problems pose various difficulties, not the least of which is the risk of numerical instability. In this article, a novel finite difference time domain simulation framework for such collision problems is developed, where numerical stability follows from strict numerical energy conservation or dissipation, and where a a power law formulation for collisions is employed, as a potential function within a Hamiltonian formulation. The power law serves both as a model of deformable collision, and as a mathematical penalty under perfectly rigid, non-deformable collision. This formulation solves a major problem underlying previous work, where a Hamiltonian framework was not employed for collisions, and thus stability was not ensured. Various numerical examples, illustrating the unifying features of such methods across a wide variety of systems in musical acoustics are presented, including numerical stability and energy conservation/dissipation, bounds on spurious penetration in the case of rigid collisions, as well as various aspects of musical instrument physics.

math.NA↗