Paper Archive

Despite the prevalence of modern audio technology, vacuum tube amplifiers continue to play a vital role in the music industry. For this reason, over the years, many different digital techniques have been introduced for accomplishing their emulation. In this paper, we propose a novel quadric surface model for tube simulations able to overcome the Cardarilli model in terms of efficiency whilst retaining comparable accuracy when grid current is negligible. After showing the model capability to well outline tubes starting from measurement data, we perform an efficiency comparison by implementing the considered tube models as nonlinear 3-port elements in the Wave Digital domain. We do this by taking into account the typical common-cathode gain stage employed in vacuum tube guitar amplifiers. The proposed model turns out to be characterized by a speedup of 4.6× with respect to the Cardarilli model, proving thus to be promising for real-time Virtual Analog applications.

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Several digital signal processing approaches, generally referred to as Virtual Analog (VA) modeling, are currently under development for the software emulation of analog audio circuitry. The main purpose of VA modeling is to faithfully reproduce the behavior of real-world audio gear, e.g., distortion effects, synthesizers or amplifiers, using efficient algorithms. In this paper, however, we provide a preliminary discussion about how VA modeling can be exploited to infer the input signal of an analog audio system, given the output signal and the parameters of the circuit. In particular, we show how an inversion theorem known in circuit theory, and based on nullors, can be used for this purpose. As recent advances in Wave Digital Filter (WDF) theory allow us to implement circuits with nullors in a systematic fashion, WDFs prove to be useful tools for inverse VA modeling. WDF realizations of a nonlinear audio system and its inverse are presented as an example of application.

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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 oversampling. 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 nonlinear 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 statespace 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.

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Virtual Analog (VA) modeling is the practice of digitally emulating analog audio gear. Over the past few years, with the purpose of recreating the alleged distinctive sound of audio equipment and musicians, many different guitar pedals have been emulated by means of the VA paradigm but little attention has been given to phasers. Phasers process the spectrum of the input signal with time-varying notches by means of shifting stages typically realized with a network of transistors, whose nonlinear equations are, in general, demanding to be solved. In this paper, we take as a reference the famous MXR Phase 90 guitar pedal, and we propose an efficient time-varying model of its Junction Field-Effect Transistors (JFETs) based on a channel resistance approximation. We then employ such a model in the Wave Digital domain to emulate in real-time the guitar pedal, obtaining an implementation characterized by low computational cost and good accuracy.

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The recently developed extension of Wave Digital Filters based on vector wave variables has broadened the class of circuits with linear two-port elements that can be modeled in a modular and explicit fashion in the Wave Digital (WD) domain. In this paper, we apply the vector definition of wave variables to nonlinear twoport elements. In particular, we present two vector WD models of a Bipolar Junction Transistor (BJT) using characteristic equations derived from an extended Ebers-Moll model. One, implicit, is based on a modified Newton-Raphson method; the other, explicit, is based on a neural network trained in the WD domain and it is shown to allow fully explicit implementation of circuits with a single BJT, which can be executed in real time.

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Neural networks have been applied within the Wave Digital Filter (WDF) framework as data-driven models for nonlinear multi-port circuit elements. Conventionally, these models are trained on wave variables obtained by sampling the current-voltage characteristic of the considered nonlinear element before being incorporated into the circuit WDF implementation. However, isolating multi-port elements for this process can be challenging, as their nonlinear behavior often depends on dynamic effects that emerge from interactions with the surrounding circuit. In this paper, we propose a novel approach for training neural models of nonlinear multi-port elements directly within a circuit’s Wave Digital (WD) discretetime implementation, relying solely on circuit input-output voltage measurements. Exploiting the differentiability of WD simulations, we embed the neural network into the simulation process and optimize its parameters using gradient-based methods by minimizing a loss function defined over the circuit output voltage. Experimental results demonstrate the effectiveness of the proposed approach in accurately capturing the nonlinear circuit behavior, while preserving the interpretability and modularity of WDFs.

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