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Virtual Analog Modeling of Guitar Effects Circuits

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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.
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... These transistors are less reliable than their silicon counterparts but are preferred by guitarists and pedal builders as they produce a smoother, warmer distortion. The Rangemaster (or similar) has been widely studied in existing VA literature [19], [34]- [37], but to the author's knowledge, never in the context of antialiasing. The transistor is a 2-port nonlinearity. ...
... The nonlinear behaviour of BJT transistors can be described via the Ebers-Moll model, which relates the base-emitter voltage, v be , and base-collector voltage v bc to the current at the base, i b , and the current at the collector, i c [34], [37]: ...
... The full derivation of the state-space model for the Rangemaster is not included in this document, so the reader can refer to [37] for detailed circuit analysis. Taking the state variables as: ...
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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.
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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.
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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 derived from the specific form of the nonlin-earity. The approach is first explained through the modelling of an asymmetrical diode clipper, and further exemplified by application to the Dallas Rangemaster Treble Booster guitar pedal, which provides an initial perspective of the performance on systems with multiple nonlinearities.
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