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Wavefolders are a particular class of nonlinear waveshaping circuits, and a staple of the “West Coast” tradition of analog sound synthesis. In this paper, we present analyses of two popular wavefolding circuits—the Lockhart and Serge wavefolders—and show that they achieve a very similar audio effect. We digitally model the input–output relationship...

## Contexts in source publication

**Context 1**

... factor accounts for the difference between physical parameters I s and η in each circuit. Figure 9 shows a comparison of the transfer functions for the Lockhart (R L = 7.5 kΩ) and Serge wavefolders implemented using the parameter values in Tables 2 and 4, respectively. From this figure, we can observe that the only significant difference between both transfer functions is in their sharpness at the folding points. ...

**Context 2**

... although the Lockhart wavefolder was originally designed to operate as a standalone unit, it can be adapted into a series topology with relative ease. Here, we propose using the wavefolding structure shown in Figure 19 to expand the synthesis capabilities of the Lockhart wavefolder. This design, while not based on any existing circuit, is comparable to that of the Yusynth Metalizer which also utilizes four Lockhart circuits in series [55]. ...

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## Citations

... From [34], a model for the ring modulator can be written as in (37). with M = 5 and N = 4. ...

... An analysis of the Korg35 filter is given in [35]. In this model, one has M = 2, N = 1 and, under a linear transformation, the model can be written in form (37), with ...

... While in these examples Matlab's lambertw was used, this generally requires some kind of iterative solver, such as Fritsch's iteration, see e.g. [36], [37]. A separate iterative method is required in order to evaluate the Lambert function, in addition to any iterative solver required in order to perform an update, and remains as a fixed cost, and is independent of the choice of numerical method (i.e. ...

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.

... The additional introduced delay, however, becomes problematic in systems having feedback paths. For this reason ADAA has been applied almost exclusively to memoryless systems [29,30,32]. An extension to stateful systems has been proposed by Holters in [33] and consists of a global parameter modification to the coefficient matrices of the state-space formulation to compensate for the additional delay introduced in the system by the ADAA filter. ...

... Equations (31) and (32) are then used to solve g Wv (v, a (l) ) = 0 for v with the Newton-Raphson method. Finally, the application of ADAA follows the procedure detailed in SEC. ...

The Wave Digital Filter (WDF) formalism is becoming a popular approach for the digital emulation of audio circuits. Nonlinear WDFs, like other kinds of discrete-time nonlinear filters used in Virtual Analog modeling applications, are often affected by aliasing distortion. Recently formalized Antiderivative Antialiasing (ADAA) methods are capable of significant aliasing reduction even with low oversampling factors. This paper discusses different strategies to integrate pth-order ADAA methods into stateful WDFs with a single one-port or multiport nonlinearity while preserving the modularity property typical of traditional WDFs. The effectiveness of the proposed approach is verified by applying the discussed ADAA techniques to three nonlinear audio circuits containing diode-based nonlinearities and a BJT transistor.

... Over the past decades, there has been a growing interest in developing faithful digital models of classic analog audio circuits, such as sound effect processors and synthesizers [1][2][3][4][5][6][7][8][9][10][11]. The broad class of digital signal processing methods aimed at emulating analog audio circuits is referred to as Virtual Analog (VA) modeling [1,2]. ...

The Scattering Iterative Method (SIM) is a recently developed fixed-point method relying on Wave Digital (WD) principles for the discrete-time simulation of electrical networks containing multiple one-port nonlinearities. Due to its robustness and efficiency, SIM proved itself to be suitable for the digital emulation of nonlinear audio circuits in Virtual Analog applications. The existent SIM formalization uses voltage wave variables. In this paper, we extend such a formalization to accommodate circuit descriptions based on generalized wave variables, including voltage, current, and power-normalized waves, as particular cases. A SIM-based WD implementation of a passive audio compressor employing the newly introduced generalized wave framework is presented, along with an analysis of the SIM convergence speed considering different types of waves and two different initialization strategies.

... In synthesizers, converters of sinusoidal waveforms to triangle and square waves use different and more complicated circuitry. See [EPPB17b,GEPP18] for more information about "west coast" waveshaping audio synthesis. ...

This work addresses the real-time simulation of nonlinear audio circuits. In this thesis, we use the port-Hamiltonian (pH) formalism to guarantee power balance and passivity. Moreover, we adopt a continuous-time functional framework to represent "virtual analog" signals and propose to approximate solutions by projection over time frames. As a main result, we establish a sufficient condition on projectors to obtain time-continuous power-balanced trajectories. Our goal is twofold: first, to manage frequency-bandwidth expansion due to nonlinearities, we consider numerical engines processing signals that are not bandlimited but, instead, have a "finite rate of innovation"; second, to get back to the bandlimited domain, we design "virtual analog-to-digital converters". Several numerical methods are built to be power-balanced, high-order accurate, with a controllable regularity order. Their properties are studied: existence and uniqueness, accuracy order and dispersion, but also, frequency resolution beyond the Nyquist frequency, aliasing rejection, reproducing and Peano kernels. This approach reveals bridges between numerical analysis, signal processing and generalised sampling theory, by relating accuracy, polynomial reproduction, bandwidth, Legendre filterbanks, etc. A systematic framework to transform schematics into equations and simulations is detailed. It is applied to representative audio circuits (for the UVI company), featuring both ordinary and differential-algebraic equations. Special work is devoted to pH modelling of operational amplifiers. Finally, we revisit pH modelling within the framework of Geometric Algebra, opening perspectives for structure encoding.

... Also, due to the non-inverting amplifier used, we must have |vf| ≥ |Kv2|, with equality when the diodes conduct enough to be approximated by a short circuit. Similar to the circuit analyses in [23] and [24], the current due to the forward-biased diode can be considered to dominate the saturation current of the reverse-biased diode. This, along with |vf| ≥ |Kv2| gives ...

... Although it is expected that unbounded growth would occur for α ≥ 8, ill-conditioning effects appear from α ≥ αmin ≈ 7.7 even though double-precision floats are used for calculations. The Lambert-W function is computed using Fritsch's iterations as in [23] and [31]. A MATLAB implementation of Fritsch's iterations can be found on the website accompanying [23]. ...

... The Lambert-W function is computed using Fritsch's iterations as in [23] and [31]. A MATLAB implementation of Fritsch's iterations can be found on the website accompanying [23]. Processing a 12.8 second input audio sample takes about 2.0 seconds. ...

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.

... Nowadays, methods to design band-limited oscillators [1] and linear filters are well established [2]. Current trends in this research field are towards implementing stable timevarying structures [3] and accurate nonlinear devices [4,5]. ...

Nonlinear digital circuits and waveshaping are active areas of study, specifically for what concerns numerical and aliasing issues. In the past, an effective method was proposed to discretize nonlinear static functions with reduced aliasing based on the antiderivative of the nonlinear function. Such a method is based on the continuous-time convolution with an FIR antialiasing filter kernel, such as a rectangular kernel. These kernels, however, are far from optimal for the reduction of aliasing. In this paper we introduce the use of arbitrary IIR rational transfer functions that allow a closer approximation of the ideal antialiasing filter, required in the fictitious continuous-time domain before sampling the nonlinear function output. These allow a higher degree of aliasing reduction and can be flexibly adjusted to balance performance and computational cost.

... The situation is similar for analog synthesizers, such as in the "West Coast" tradition of analog sound synthesis, which relies on wavefolders for sound generation. The core of a wavefolder circuit consists of an NPN BJT and a PNP BJT, which can be digitized by means of large-signal equivalent circuits [16]. It should be noted that these "virtual analog" models have been achieved by assuming symmetric nonlinear (i.e., with odd-order harmonics) transfer functions. ...

... The additional introduced delay, however, becomes problematic in systems having feedback paths. For this reason, ADAA has been applied almost exclusively to memoryless systems [26,27,29]. An extension to stateful systems has been proposed by Holters in [30], and consists of a global parameter modification to the coefficient matrices of the state-space formulation, to compensate for the additional delay introduced in the system by the ADAA filter. ...

A major problem in the emulation of discrete-time nonlinear systems , such as those encountered in Virtual Analog modeling, is aliasing distortion. A trivial approach to reduce aliasing is over-sampling. However, this solution may be too computationally demanding for real-time applications. More advanced techniques to suppress aliased components are arbitrary-order Antiderivative Antialiasing (ADAA) methods that approximate the reference non-linear function using a combination of its antiderivatives of different orders. While in its original formulation it is applied only to memoryless systems, recently, the applicability of first-order ADAA has been extended to stateful systems employing their state-space description. This paper presents an alternative formulation that successfully applies arbitrary-order ADAA methods to Wave Digital Filter models of dynamic circuits with one nonlinear element. It is shown that the proposed approach allows us to design ADAA models of the nonlinear elements in a fully local and modular fashion, independently of the considered reference circuit. Further peculiar features of the proposed approach, along with two examples of applications, are discussed.

... Convolution is also used to emulate other linear systems, such as the tonality of guitar cabinets. A review of techniques for the digital emulation of tube-based amplifiers have can be found in References [135,136]. ...

Audio effects are an essential tool that the field of music production relies upon. The ability to intentionally manipulate and modify a piece of sound has opened up considerable opportunities for music making. The evolution of technology has often driven new audio tools and effects, from early architectural acoustics through electromechanical and electronic devices to the digitisation of music production studios. Throughout time, music has constantly borrowed ideas and technological advancements from all other fields and contributed back to the innovative technology. This is defined as transsectorial innovation and fundamentally underpins the technological developments of audio effects. The development and evolution of audio effect technology is discussed, highlighting major technical breakthroughs and the impact of available audio effects.

... When only a single exponential term is present or predominant, it is possible to utilize the Lambert W function [12,13,14] to analytically solve these equations [15,16,17,18,19,20,21]. When applicable, such an approach brings remarkable advantages. ...

... In particular, if a and c have the same sign, the argument of W () is negative and there are either two (possibly coincident) or no solutions, otherwise the argument is positive, there is one solution, and W (x) = W0(x). In circuit modelling, often a and c have opposite sign and are either constants or control-rate expressions, while b or d contain audio-rate components [17,18,19,20,21]. Therefore, under the previous formulation, not only one needs to compute W (), but also the exponential in its argument. ...

When modelling circuits one has often to deal with equations containing both a linear and an exponential part. If only a single exponential term is present or predominant, exact or approximate closed-form solutions can be found in terms of the Lambert W function. In this paper, we propose reformulating such expressions in terms of the Wright Omega function when specific conditions are met that are customary in practical cases of interest. This eliminates the need to compute an exponential term at audio rate. Moreover, we propose simple and real-time suitable approximations of the Omega function. We apply our approach to a static and a dynamic nonlinear system, obtaining digital models that have high accuracy, low computational cost, and are stable in all conditions, making the proposed method suitable for virtual analog modelling of circuits containing semiconductor devices.