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# A 200 mV, 1 kHz sine wave processed by both SPICE and state-space Rangemasters. Vcc = 9 V, fs = 176.4 kHz.

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Iterative solvers are required for the discrete-time simulation of nonlinear behaviour in analogue distortion circuits. Unfortunately, these methods are often computationally too expensive for real-time simulation. Two methods are presented which attempt to reduce the expense of iterative solvers. This is achieved by applying information that is de...

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## Citations

... One of the most commonly used iterative solvers in VA modelling is the Newton-Raphson method [11], [19], [25], [26]. Newton's method finds an approximation to the roots of a function, so in order to solve (2.13), let: ...

... Starting from some initial guess, the next iterate (i.e approximation to v n ) can be expressed by: [19] v ...

... A modification can be made to the plain Newton's method to prevent divergence, by ensuring the residual decreases in magnitude at each iterate. This can be achieved by reducing the step size by a factor of 2 until the current residual is less than the previous [19]: ...

Aliasing is an inherent problem in virtual analogue modelling when simulating nonlinear
systems such as guitar amplifiers and distortion effects units. Such systems introduce
harmonics into the signal, which in the discrete-time domain can exceed the Nyquist
frequency, resulting in unpleasant aliasing distortion. Recent research has shown that
aliasing can be significantly reduced by using the antiderivatives of the nonlinear
function, and that this method can be applied to systems with state as well as
memoryless nonlinearities. In this work, the application of antiderivative antialiasing
in the state-space modelling of several nonlinear circuits will be outlined in detail.
Existing literature has focused on one-port nonlinearities, so in this work a method
for two-port nonlinearities is proposed and demonstrated by example. Furthermore, a
second order antialiasing method for state-space models is presented. The antialiasing
methods were found to significantly improve the signal to noise ratio and reduce aliasing
at low oversampling rates. In the case of scalar nonlinearities, the methods introduced
no notable extra computational cost, but for two-port nonlinearities the processing
time increased with the order of antialiasing. Finally, the suitability of antiderivative
antialiasing in a real-time context was demonstrated through the development of a
virtual analogue guitar effects plug-in.

... Depending on the nonlinear solver and the initial parameter set, this can drastically influence the computational load of the digital model. Although Holmes et al. described a method for improving the nonlinear solver [116], the computational load is still high, especially for complex circuits with multiple nonlinearities. Additionally Holters proposed a method to automatically decompose the large statespace matrices into smaller ones to be able to solve the system quicker [105]. ...

Digital systems gain more and more popularity in todays music industry. Musicians and producers are using digital systems because of their advantages over analog electronics. They require less physical space, are cheaper to produce and are not prone to aging circuit components or temperature variations. Furthermore, they always produce the same output signal for a defined input sequence. However, musicians like vintage equipment. Old guitar amplifiers or legendary recording equipment are sold at very high prices. Therefore, it is desirable to create digital models of analog music electronics which can be used in modern digital environments.
This work presents an approach for recreating nonlinear audio circuits using system identification techniques. Measurements of the input- and output-signals from the analog reference devices are used to adjust a digital model treating the reference device as a ‘black-box’. With this technique the schematic of the reference device does not need to be known and no circuit elements have to be measured to recreate the analog device.
An appropriate block-based model is chosen, depending on the type of reference system. Then the parameters of the digital model are adjusted with an optimization method according to the measured input- and output-signals. The performance of the optimized digital model is evaluated with objective scores and listening tests.
Two types of nonlinear reference systems are examined in this work. The first type of reference systems are dynamic range compressors like the ‘MXR Dynacomp’, the ‘Aguilar TLC’, or the ‘UREI 1176LN’. A block-based model describing a generic dynamic range compression system is chosen and an automated routine is developed to adjust it. The adapted digital models are evaluated with objective scores and a listening test is
performed for the UREI 1176LN studio compressor.
The second type of nonlinear systems are distortion systems like e.g. amplifiers for electric guitars. This work presents novel modeling approaches for different kinds of distortion systems from basic distortion circuits which can be found in distortion pedals for guitars to (vintage) guitar amplifiers like the ‘Marshall JCM900’, or the ‘Fender Bassman’. The linear blocks of the digital model are measured and used in the model while the nonlinear blocks are adapted with parameter optimization methods like the Levenberg–Marquardt method. The quality of the adjusted models is evaluated with objective scores and listening tests.
The adjusted digital models give convincing results and can be implemented as real-time digital versions of their analog counterparts. This enables the musician to safe a snapshot of a certain sound and recall it anytime with a digital system like a VST plug-in or as a program on a dedicated hardware.

... Therefore, there is always a possibility that a large number of iterations will be required, significantly increasing computation time. Holmes et al. present a root-finding method that allows for a set number of iterations through the use of analytic inverses of the nonlinear equations to be solved [23]. ...

This paper presents in detail the state-space approach to virtual analog modeling. A variety of different numerical methods are derived and implemented so that their performance may be compared. Four different guitar effects circuits are analyzed and simulated, and the detailed MATLAB code for each is provided.

... Many papers have been written about vacuum-tube guitar amplifiers modeling [1] [6], and about the particularities of linear and non-linear distortion effects suited for guitar [2][3][4] [5]. More generally, works such as James J. Clark's "Advanced programming techniques for modular synthesizers" book, are not focused on guitar but discuss thoroughly the different approaches for achieving a distortion effect. ...

The ANR project WASABI will last 42 months and consists in developing a 2 million songs database with interactive WebAudio enhanced client applications. Client applications target composers, music schools, sound engineering schools, musicologists, music streaming services and journalists. In this paper, we present a virtual pedal board (a set of chainable audio effects on the form of "pedals"), and a guitar tube amplifier simulation for guitarists, that will be associated with songs from the WASABI database. Music schools and music engineering schools are interested in such tools that can be run in a Web page, without the need to install any further software. Take a classic rock song: isolate the guitar solo, study it, then mute it and play guitar real-time along the other tracks using an online guitar amplifier that reproduces the real guitar amp model used in the song, with its signature sound, proper dynamic and frequency response. Add some audio effects such as a reverberation, a delay, a flanger, etc. in order to reproduce Pink Floyd's guitar sound or Eddie Van Halen famous "Brown Sound". Learn interactively, guitar in hands, how to fine tune a compressor effect, or how to shape the sound of a tube guitar amp, how to get a "modern metal" or a "Jimi Hendrix" sound, using only your Web browser.

... To understand the cost of increasing the complexity of the BJT model, computational requirements of each model were compared. The nonlinear equation of the circuit models was solved using damped Newton's method as described in [16], which uses an inner iterative loop to aid in convergence. This provides three metrics: time needed for one second of simulation, average iterations, and average sub-iterations. ...

The Ebers-Moll model has been widely used to represent Bipolar Junction Transistors (BJTs) in Virtual Analogue (VA) circuits. An investigation into the validity of this model is presented in which the Ebers-Moll model is compared to BJT models of higher complexity , introducing the Gummel-Poon model to the VA field. A comparison is performed using two complementary approaches: on fit to measurements taken directly from BJTs, and on application to physical circuit models. Targeted parameter extraction strategies are proposed for each model. There are two case studies , both famous vintage guitar effects featuring germanium BJTs. Results demonstrate the effects of incorporating additional complexity into the component model, weighing the trade-off between differences in the output and computational cost.

... However, getting close to the sound of a real guitar amplifier is a real challenge that Chris Wilson's examples did not address. Many papers have been written about vacuum-tube guitar amplifiers modeling [1] [6], and about the particularities of linear and non-linear distortion effects suited for guitar [2][3][4] [5]. More generally, works such as James J. Clark's "Advanced programming techniques for modular synthesizers" book, are not focused on guitar but discuss thoroughly the different approaches for achieving a distortion effect. ...

We propose to present a tube guitar amplifier simulation we’ve been designing using the Web Audio API with the aim to faithfully reproduce the main parts of the Marshall JCM 800 amplifier schematics. Each stage of the real amp has been recreated (preamp, tone stack, reverb, power amp and speaker simulation). We’ve also added an extra multiband EQ. This “classic rock” amp simulation we’ve been building has been used in real gigs and can be favorably compared with some native amp simulation both in terms of latency, sound quality, dynamics and comfort of the guitar play. The amp is open source1 and can be tested online2, even without a real guitar plugged-in. It comes with an audio player, dry guitar samples and a wave generator that can be used as inputs. Figure 1 shows the current GUI, with some optional frequency analyzers and oscilloscopes that we’ve been using to probe the signal at different stages of the simulation. One purpose was to evaluate the limits of the Web Audio API and see if it was possible to design a web-based guitar amp simulator that could compete with native simulations.

... Many papers have been written about vacuum-tube guitar amplifiers modeling [1] [6], and about the particularities of linear and non-linear distortion effects suited for guitar [2][3][4] [5]. Some works such as James J. Clark "Advanced programming techniques 3 https://www.w3.org/TR/mediacapture-streams/ for modular synthesizers" book, are not focused on guitar but cover in deep the different approaches for achieving a distortion effect on a signal [9]. ...

This paper presents a tube guitar amplifier simulation made with the WebAudio API, that reproduces the main parts of the Marshall JCM 800 amplifier schematics. Each stage of the real amp has been recreated (preamp, tone stack, reverb, power amp and speaker simulation, and we added an extra multiband EQ). The “classic rock” amp simulation we built has been used in real gigs and can be compared with some native amp simulation both in terms of latency, sound quality, dynamics and comfort of the guitar play. Unfortunately, as of today, low latency can be achieved only with certain configurations, due to audio driver limitations of current browsers on certain operating systems. The paper discusses the latency problems encountered with WebAudio, common traps, current limitations, and proposes some solutions.

... A critical issue encountered when stochastically selecting parameter sets for the Dallas Rangemaster is that the simulation can fail. This happens when the nonlinear solver does not converge to the root of the equation [19]. To counteract this, failing parameter sets are regenerated using the stochastic technique used to generate the initial population. ...

In this work we explore optimising parameters of a physical circuit model relative to input/output measurements, using the Dallas Rangemaster Treble Booster as a case study. A hybrid meta-heuristic/gradient descent algorithm is implemented, where the initial parameter sets for the optimisation are informed by nominal values from schematics and datasheets. Sensitivity analysis is used to screen parameters, which informs a study of the optimisation algorithm against model complexity by fixing parameters. The results of the optimisation show a significant increase in the accuracy of model behaviour, but also highlight several key issues regarding the recovery of parameters.

... Depending on the nonlinear solver and the initial parameter set, this can drastically influence the computational load of the digital model. Although Holmes et al. described a method for improving the nonlinear solver in [9], the computational effort is still high, especially for complex circuits with multiple nonlinearities. ...

This paper describes black-box modeling of distortion circuits. The analyzed distortion circuits all originate from guitar effect pedals, which are widely used to enrich the sound of an electric guitar with harmonics. The proposed method employs a block-oriented model which consists of a linear block (filter) and a non-linear block. In this study the nonlinear block is represented by an extended parametric input/output mapping function. Three distortion circuits with different nonlinear elements are analyzed and modeled. The linear and nonlinear parts of the circuit are analyzed and modeled separately. The Levenberg–Marquardt algorithm is used for iterative optimization of the nonlinear parts of the circuits. Some circuits could not be modeled with high accuracy, but the proposed model has shown to be a versatile and flexible tool when modeling distortion circuits.

... Besides using the same z (0) for every time step, a common strategy is to use the solution found for the previous time step, assuming only small inter-sample differences to occur inx andū. In [6], a more elaborate scheme is proposed where (1) is simplified such that an explicit solution can be found analytically which is then used to compute z (0) . However, none of these approaches guarantee to deliver a z (0) sufficiently close to the correct solution, and therefore the Newton solver may need a large number of iterations, or worse, not converge at all. ...

... However, none of these approaches guarantee to deliver a z (0) sufficiently close to the correct solution, and therefore the Newton solver may need a large number of iterations, or worse, not converge at all. To help convergence, different approaches like damped Newton iteration [7], homotopy [4], or heuristical modification of the Newton step [6] have been successfully employed. However, the total number of required iterations may still be prohibitively large, especially for more complex circuits. ...

In the digital simulation of non-linear audio effect circuits, the arising non-linear equation generally poses the main challenge for a computationally cheap implementation. For any but the simplest circuits, using an iterative solver at execution time will be too slow, while exhaustive look-up tables quickly grow intolerably large. To better cope with the situation, in this paper we propose to store solutions non-uniformly sampled from the parameter space to enable an iterative solver to quickly converge when being started from the closest initial solution. Efficient look-up of this closest solution is realized by using a k-d tree. The method is supported by a step to reduce the dimension of the parameter space and a linear extrapolation from the closest solution stored to the actually needed parameter vector.