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.

The collision interaction mechanism between a vibrating string and a non-resonant obstacle is at the heart of many musical instruments. This paper focuses on the identification of perceptually salient auditory features related to this phenomenon. The objective is to design a signal-based synthesis process, with an eye towards developing intuitive control strategies. To this end, a database of synthesized sounds is assembled through physics-based emulation of a string/obstacle collision, in order to characterize the effect of collisions on time-frequency content. The investigation of this database reveals characteristic time-frequency patterns related to the position of the obstacle during the interaction. In particular, a frequency shift of certain modes is apparent for strong interactions, which, alongside the generation of new frequency components, leads to increased perceived roughness and inharmonicity. These observations enable the design of a real-time compatible signal-based sound synthesis process, with a mapping of synthesis parameters linked to the perceived location of the obstacle. The accuracy of the signal model with respect to the physical model sound output and recorded sounds was evaluated through listening tests: time-frequency patterns reproduced by the signal model enabled listeners to precisely recognize the transverse location of the obstacle.

In any acoustics simulation setting relying on computation over a spatial grid, interpolation of the acoustic field is essential in order to accurately model source and receiver positions. Most available approaches to 3D interpolation, such as those used in computer graphics or medical imaging, are based on polynomial or windowed-sinc designs. In this short contribution, it is shown that highly accurate optimised designs are available if particular features of acoustic wave propagation and numerical scheme design are incorporated: performance can be tuned to an acoustic wavenumber range of interest, taking into account numerical dispersion artefacts, and the interdependence of the solution to the acoustic wave equation at neighbouring time steps can be further exploited, leading to extremely compact locally-defined interpolation designs. Numerical results are presented.

The simulation of nonlinear components is central to virtual analog simulation. In audio effects, circuit elements often include devices such as diodes and transistors, mostly operating in the strongly nonlinear regime. Mathematical models are of the form of systems of nonlinear ordinary differential equations (ODEs), and traditional integrators, such as the trapezoid and midpoint methods, can be employed as solvers. These methods are fully implicit, and require the solution of a nonlinear algebraic system at each time step, introducing further complications regarding the existence and uniqueness of the solution, as well as the choice of halting conditions for the iterative root finder. On the other hand, fast explicit methods such as Forward Euler, or explicit Runge-Kutta methods, are prone to unstable behaviour at standard audio sample rates, even at moderate amplitudes. For these reasons, in this work a family of linearly-implicit schemes is presented. These schemes take the form of a perturbation expansion, making the construction of higher-order schemes possible. Compared with classic implicit designs, the proposed methods have the advantage of efficiency, since the update is computed in a single iteration, through the solution of a linear system of equations. Furthermore, the existence and uniqueness of the update are proven by simple inspection of the update matrix. Compared to classic explicit designs, the proposed schemes display stable behaviour at standard audio sample rates. In the case of a single scalar ODE, sufficient conditions for numerical stability can be derived, imposing constraints on the choice of the sampling rate. Several theoretical results are provided, as well as numerical examples for typical stiff equations used in virtual analog modelling.

This paper presents an application of the port Hamiltonian formalism to the nonlinear simulation of the OTA-based Korg35 filter circuit and the Moog 4-pole ladder filter circuit. Lyapunov analysis is used with their state-space representations to guarantee zero-input stability over the range of parameters consistent with the actual circuits. A zero-input stable non-iterative discrete-time scheme based on a discrete gradient and a change of state variables is shown along with numerical simulations. Simulations show behavior consistent with the actual operation of the circuits, e.g., self-oscillation, and are found to be stable and have lower computational cost compared to iterative methods.

In this work, a number of numerical schemes are presented in the context of virtual-analog simulation. The schemes are linearly-implicit in character, and hence directly solvable without iterative methods. Schemes of increasing order of accuracy are constructed, and convergence and stability conditions are proven formally. The schemes are able to handle stiff problems very efficiently, because of their fast update, and can be run at higher sample rates to reduce aliasing. The cases of the diode clipper and ring modulator are investigated in detail, including several numerical examples.

For physical modelling sound synthesis, many techniques are available; time-stepping methods (e.g., finite-difference time-domain (FDTD) methods) have an advantage of flexibility and generality in terms of the type of systems they can model. These methods do, however, lack the capability of easily handling smooth parameter changes while retaining optimal simulation quality and stability, something other techniques are better suited for. In this paper, we propose an efficient method to smoothly add and remove grid points from a FDTD simulation under sub-audio rate parameter variations. This allows for dynamic parameter changes in physical models of musical instruments. An instrument such as the trombone can now be modelled using FDTD methods, as well as physically impossible instruments where parameters such as e.g. material density or its geometry can be made time-varying. Results show that the method does not produce (visible) artifacts and stability analysis is ongoing.

In this paper, a complete simulation of a trombone using finite-difference time-domain (FDTD) methods is proposed. In particular, we propose the use of a novel method to dynamically vary the number of grid points associated to the FDTD method, to simulate the fact that the physical dimension of the trombone's resonator dynamically varies over time. We describe the different elements of the model and present the results of a real-time simulation.

The measurement of directivity for sound sources that are not electroacoustic transducers is fundamentally limited because the source cannot be driven with arbitrary signals. A consequence is that directivity can only be measured at a sparse set of frequencies-for example, at the stable partial oscillations of a steady tone played by a musical instrument or from the human voice. This limitation prevents the data from being used in certain applications such as time-domain room acoustic simulations where the directivity needs to be available at all frequencies in the frequency range of interest. We demonstrate in this article that imposing the signature of the directivity that is obtained at a given distance on a spherical wave allows for all interpolation that is required for obtaining a complete spherical harmonic representation of the source's directivity, i.e., a representation that is viable at any frequency, in any direction, and at any distance. Our approach is inspired by the far-field signature of exterior sound fields. It is not capable of incorporating the phase of the directivity directly. We argue based on directivity measurement data of musical instruments that the phase of such measurement data is too unreliable or too ambiguous to be useful. We incorporate numerically-derived directivity into the example application of finite difference time domain simulation of the acoustic field, which has not been possible previously.

Approaches for incorporating sound source directivity into wave-based room acoustic simulations using a spherical harmonic representation have been presented recently. Normally, the directivity is measured or prescribed on a spherical surface centered at the nominal source position. In wave-based simulations, this directivity can be represented through a locally-defined driving term acting at the source location. In practice, the directivity of real-world sound sources like musical instruments or industrial machinery can only be measured approximately in terms of spatial resolution and accuracy. We show that the measurement data can be augmented such that the impairments due to the limitations of the measurement accuracy are mitigated. We revisit the previously proposed approach of only using the angle-dependent magnitude of the measured directivity together with a spherical-wave propagation model and demonstrate its potential by means of numerical simulations based on two case studies.

Volumetric time-domain simulation methods, such as the finite difference time domain method, allow for a fine-grained representation of the dynamics of the acoustic field. A key feature of such methods is complete access to the computed field, normally represented over a Cartesian grid. Simple solutions to the problem of extracting spatially encoded signals, necessary in virtual acoustics applications, result. In this letter, a simple time-domain representation of spatially encoded spherical harmonic signals is written directly in terms of spatial derivatives of the acoustic field at the receiver location. In a discrete setting, encoded signals may be obtained, at very low computational cost and latency, using local approximations with minimal number of grid points, and avoiding large convolutions and frequency-domain block processing of previous approaches. Numerical results illustrating receiver directivity and computed time-domain responses are presented, as well as numerical solution drift associated with repeated time integration.

Volumetric wave-based acoustic simulation relies on the complete solution to the three-dimensional wave equation over a spatial grid. Detailed modeling of sources, however, requires interpolation over the grid, which is complicated by the directional character of the source itself. In this paper, a new model of point sources of arbitrary directivity and location with respect to an underlying grid is presented. The model is framed in the spatio-temporal domain directly through the differentiation of Dirac distributions, leading to a spatial Fourier-based approximation strategy. Various approximants are presented, of both separable and nonseparable type, which allow for optimization over a specified wavenumber range. Such approximants are then employed in a finite difference time domain setting, yielding numerical results for sources of various types, which are then compared against exact solutions.

Aliasing is a commonly encountered problem in audio signal processing, particularly when memoryless nonlinearities are simulated in discrete time. A conventional remedy is to operate at an oversampled rate. A new aliasing reduction method is proposed here for discrete-time memoryless nonlinearities, which is suitable for operation at reduced oversampling rates. The method employs higher order antiderivatives of the nonlinear function used. The first-order form of the new method is equivalent to a technique proposed recently by Parker et al. Higher order extensions offer considerable improvement over the first antiderivative method, in terms of the signal-to-noise ratio. The proposed methods can be implemented with fewer operations than oversampling and are applicable to discrete-time modeling of a wide range of nonlinear analog systems.

Time-domain finite difference (FDTD) methods are popular tools for three-dimensional (3-D) room acoustics modeling, but numerical dispersion is an inherent problem that can place limitations on the usable bandwidth of a given scheme. Compact explicit 27-point schemes and “large-star” schemes with high-order spatial differences offer improvements to the simplest scheme, but are ultimately limited by their second-order accuracy in time. In this paper, we use modified equation methods to derive FDTD schemes with high orders of accuracy in both space and time, resulting in significant improvements in numerical dispersion as compared to the aforementioned schemes. In comparison to such schemes, the high-order accurate schemes presented in this paper use significantly less memory and fewer operations when low error tolerances in numerical phase velocities are critical, leading to higher usable bandwidths for auralization purposes. Simulation results are also presented, demonstrating improved approximations to modal frequencies of a shoe-box room and free-space propagation of a bandlimited pulse.

In room acoustics simulation and virtualization applications, accurate wall termination is a perceptually crucial feature. It is particularly important in the setting of wave-based modeling of 3D spaces, using methods such as the finite difference time domain method or finite volume time domain method. In this paper, general locally reactive impedance boundary conditions are incorporated into a 3D finite volume time domain formulation, which may be specialized to the various types of finite difference time domain method under fitted boundary termination. Energy methods are used to determine stability conditions for general room geometries, under a large family of nontrivial wall impedances, for finite volume methods over unstructured grids. Simulation results are presented, highlighting in particular the need for unstructured or fitted cells at the room boundary in the case of the accurate simulation of frequency-dependent room mode decay times.

Due to recent increases in computing power, room acoustics simulation in 3D using time stepping schemes is becoming a viable alternative to standard methods based on ray tracing and the image source method. Finite Difference Time Domain (FDTD) methods, operating over regular grids, are perhaps the best known among such methods, which simulate the acoustic field in its entirety over the problem domain. In a realistic room acoustics setting, working over a regular grid is attractive from a computational standpoint, but is complicated by geometrical considerations, particularly when the geometry does not conform neatly to the grid, and those of boundary conditions which emulate the properties of real wall materials. Both such features may be dealt with through an appeal to methods operating over unstructured grids, such as finite volume methods, which reduce to FDTD when employed over regular grids. Through numerical energy analysis, such methods lead to direct stability conditions for complex problems, including convenient geometrical conditions at irregular boundaries. Simulation results are presented.

Finite difference time-domain simulation methods in acoustics applications have seen increased interest recently. The simplest scheme exhibits various weaknesses, such as numerical dispersion and anisotropy. More general parameterized families of schemes are explored here, with a view towards reducing such numerical artefacts through optimization. Numerical results are presented.

In seeking to model realistic room acoustics, direct numerical simulation can be employed. This paper presents 3D Finite Difference Time Domain schemes that incorporate losses at boundaries and due to the viscosity of air. These models operate within a virtual room designed on a detailed floor plan. The schemes are computed at 44.1kHz, using large-scale data sets containing up to 100 million points each. A performance comparison is made between serial computation in C, and parallel computation using CUDA on GPUs, showing up to 80 times speed-ups. Testing on two different Nvidia Tesla cards shows the benefits of the latest FERMI architecture for double precision floating-point computation.

The synthesis of sound based on physical models of 2-D percussion instruments is problematic and has been approached only infrequently in the literature. Beyond the computational expense inherent to the simulation of 2-D systems, a deeper difficulty is in dealing with the strong nonlinearity exhibited by thin structures when struck-this nonlinearity leads to phenomena which are not captured, even approximately, by a linear model, and nearly all synthesis work is based on the assumption that the distributed resonating component of a musical instrument is linear. Perceptually, the effects of the vibration of a thin structure at high amplitudes can be heard as crashes, pitch glides, and the slow buildup of high-frequency energy characteristic of gongs. A large family of instruments may be described, approximately, as circular thin shells, of approximately spherical geometry, in which case a tractable PDE description, described here, is available. Time-domain finite-difference schemes, in radial coordinates, are a suitable method for synthesis. Stability conditions, numerical boundary conditions both at the edge and center, and implementation details are discussed, and simulation results are presented, highlighting the various perceptual effects mentioned above.

The digital emulation of analog audio effects and synthesis components, through the simulation of lumped circuit components has seen a large amount of activity in recent years; electromechanical effects have seen rather less, primarily because they employ distributed mechanical components, which are not easily dealt with in a rigorous manner using typical audio processing constructs such as delay lines and digital filters. Spring reverberation is an example of such a system-a spring exhibits complex, highly dispersive behavior, including coupling between different types of wave propagation (longitudinal and transverse). Standard numerical techniques, such as finite difference schemes are a good match to such a problem, but require specialized design and analysis techniques in the context of audio processing. A model of helical spring vibration is introduced, along with a family of finite difference schemes suitable for time domain simulation. Various topics are covered, including numerical stability conditions, tuning of the scheme to the response of the model system, numerical boundary conditions and connection to an excitation and readout, implementation details, as well as computational requirements. Simulation results are presented, and full energy-based stability analysis appears in an Appendix.

In this paper, a simple family of explicit two-step finite difference methods for solving the classical equation of motion of a thin plate is examined. This family depends on several free parameters, and special attention is paid to the stability properties of these schemes, computational issues, and, in particular, the reduction of directional numerical dispersion. Numerical results, employing frequency warping techniques, are presented

In this article, a digital plate reverberation algorithm is presented, based on a direct numerical simulation of the equations of motion of a thin linear plate of Kirchhoff type. While such an algorithm will be more expensive, computationally, than digital filter-based algorithms, the resulting algorithm allows far more flexible control on the part of the user, in that the defining parameters have physical significance (i.e., they are related directly to material and geometry of the plate itself). A partial differential equation model is presented, followed by a discussion of a finite difference scheme, which is then specialized to the case of plate reverberation; numerical simulation results are presented

From a modal synthesis point of view, digital waveguides allow the efficient synthesis of harmonically spaced components for a fixed number of arithmetic operations. They are thus well suited to audio synthesis based on one-dimensional physical models (such as strings or tubes) that possess nearly harmonic spectra. In higher dimensions, however, modal frequencies do not fall in harmonic series. In this letter, we show how digital waveguide structures may be extracted from higher dimensional problems, yielding algorithms more efficient than modal synthesis, with reference to the solution of the two-dimensional wave equation defined over a square region. Numerical results are presented.

The computational requirements of finite difference schemes for the solution of the wave equation for physical modelling can be huge. Field programmable gate arrays (FPGAs) provide an ideal platform for performing highly parallel DSP computations, but the challenge is to be able to implement complex systems quickly and efficiently on FPGA platforms. The paper presents a system level design approach based on a dataflow model of computation using a particular finite difference scheme for the solution of a 2+1D wave equation. The results suggest that 84000 nodes could be accommodated on a single Virtex II FPGA.

Time-varying fractional delays are applied, for example, in physics based modeling of musical instruments, particularly for string and wind instruments. While Lagrange interpolation and allpass filters are used routinely in such sound synthesis models, they are found somewhat problematic, for example, in plucked string simulation when the length of the string is varied due to glissando or vibrato. There can be problems with signal energy levels and aliasing. We study two variable delay filter designs that have a physically realistic energetic behavior and keep undesirable side effects, such as aliasing, in control. The first one is sliding termination point simulation with energy correction and the second one is based on controllable wave digital filter delay lines.

In many audio applications, digital all-pass filters are of central importance; a key property of such filters is energy (l/sub 2/ norm) preservation. In audio effect and sound synthesis algorithms, it is desirable to have filters that behave as all-passes with time-varying characteristics, but direct generalizations of time-invariant designs can lose the important norm-preserving property; for fast parameter variation, large gain increases are possible. We here call attention to some simple time-varying filter structures, based on wave digital filter designs, that do preserve signal energy and that reduce to simple first- and second-order all-pass filters in the time-invariant case.

The timbre of certain stringed instruments is strongly dependent on large-amplitude vibration, in which case linear models (such as the 1D wave equation), which are often used for sound synthesis purposes, are unsatisfactory. We discuss here a nonlinear generalization of the wave equation, sometimes called the Kirchhoff-carrier equation, which models large amplitude vibration through a modulation of string tension. In particular, we look at a finite difference scheme for the Kirchhoff-carrier equation which is both efficient, and has excellent stability properties (this is often difficult to ensure for nonlinear difference schemes). The key to this stability property is the close attention paid to the energetic behavior of the model and its analogue in the finite difference scheme; such a difference scheme is capable of discrete energy conservation to machine precision. Implementation details are discussed, and simulation results are presented.

In this paper, some simple families of explicit two-step finite difference methods for solving the wave equation in two and three spatial dimensions are examined. These schemes depend on several free parameters, and can be associated with so-called interpolated digital waveguide meshes. Special attention is paid to the stability properties of these schemes (in particular the bounds on the space-step/time-step ratio) and their relationship with the passivity condition on the related digital waveguide networks. Boundary conditions are also discussed. An analysis of the directional numerical dispersion properties of these schemes is provided, and minimally directionally-dispersive interpolated digital waveguide meshes are constructed.

The optimal window, the time limited sequence whose energy is most concentrated in a finite frequency interval, is related to a particular discrete prolate spheroidal sequence. The optimal window is actually a family of windows with many degrees of freedom. The Kaiser (1974) window is an approximation to this optimal window. Kaiser used this approximation because the standard method employed to compute the optimal window is numerically ill-conditioned. We show the actual optimal window can be efficiently computed by using an alternative formulation of the discrete prolate spheroidal sequences. We then give a set of design formulas to generate the optimal window for the desired window length, mainlobe width, and relative peak sidelobe height.