Paper Archive

Nonlinear systems play an important role in musical signal processing, but their digital implementation suffers from the occurrence of aliasing distortion. Consequently, various aliasing reduction methods have been proposed in the literature. In this work, a novel approach is examined that uses samples of a signal’s derivative in addition to the signal’s samples themselves. This allows some aliasing reduction, but is usually insufficient on its own. However, it can elegantly and fruitfully be combined with antiderivative antialiasing to obtain an effective method. Unfortunately, it still compares unfavorably to oversampled antiderivative antialiasing. It may therefore be regarded as a negative result, but it may hopefully still form a basis for further developments.

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Memoryless waveshapers are commonly used in audio signal processing. In discrete time, they suffer from well-known aliasing artifacts. We present a method for applying antiderivative antialising (ADAA), which mitigates aliasing, to any waveshaping function that can be represented as a piecewise polynomial. Specifically, we treat the special case of a piecewise linear waveshaper. Furthermore, we introduce a method for for replacing the sharp corners and jump discontinuities in any piecewise linear waveshaper with smoothed polynomial approximations, whose derivatives match the adjacent line segments up to a specified order. This piecewise polynomial can again be antialiased as a special case of the general piecewise polynomial. Especially when combined with light oversampling, these techniques are effective at reducing aliasing and the proposed method for rounding corners in piecewise linear waveshapers can also create more “realistic” analog-style waveshapers than standard piecewise linear functions.

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Antiderivative Antialiasing (ADAA), has become a pivotal method for reducing aliasing when dealing with nonlinear function at audio rate. However, its implementation requires analytical computation of the antiderivative of the nonlinear function, which in practical cases can be challenging without a symbolic solver. Moreover, when the nonlinear function is given by measurements it must be approximated to get a symbolic description. In this paper, we propose a simple approach to ADAA for practical applications that employs numerical integration of lookup tables (LUTs) to approximate the antiderivative. This method eliminates the need for closed-form solutions, streamlining the ADAA implementation process in industrial applications. We analyze the trade-offs of this approach, highlighting its computational efficiency and ease of implementation while discussing the potential impact of numerical integration errors on aliasing performance. Experiments are conducted with static nonlinearities (tanh, a simple wavefolder and the Buchla 259 wavefolding circuit) and a stateful nonlinear system (the diode clipper).

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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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In recent years, virtual analog modeling with neural networks experienced an increase in interest and popularity. Many different modeling approaches have been developed and successfully applied. In this paper we do not propose a novel model architecture, but rather address the problem of aliasing distortion introduced from nonlinearities of the modeled analog circuit. In particular, we propose to apply the general idea of antiderivative antialiasing to a state-trajectory network (STN). Applying antiderivative antialiasing to a stateful system in general leads to an integral of a multivariate function that can only be solved numerically, which is too costly for real-time application. However, an adapted STN can be trained to approximate the solution while being computationally efficient. It is shown that this approach can decrease aliasing distortion in the audioband significantly while only moderately oversampling the network in training and inference.

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Neural networks have become invaluable for general audio processing tasks, such as virtual analog modeling of nonlinear audio equipment. For sequence modeling tasks in particular, recurrent neural networks (RNNs) have gained widespread adoption in recent years. Their general applicability and effectiveness stems partly from their inherent nonlinearity, which makes them prone to aliasing. Recent work has explored mitigating aliasing by oversampling the network—an approach whose effectiveness is directly linked with the incurred computational costs. This work explores an alternative route by extending the antiderivative antialiasing technique to explicit, computable RNNs. Detailed applications to the Gated Recurrent Unit and Long Short-Term Memory cell are shown as case studies. The proposed technique is evaluated on multiple pre-trained guitar amplifier models, assessing its impact on the amount of aliasing and model tonality. The method is shown to reduce the models’ tendency to alias considerably across all considered sample rates while only affecting their tonality moderately, without requiring high oversampling factors. The results of this study can be used to improve sound quality in neural audio processing tasks that employ a suitable class of RNNs. Additional materials are provided in the accompanying webpage.

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Nonlinear systems, like e.g. guitar distortion effects, play an important role in musical signal processing. One major problem encountered in digital nonlinear systems is aliasing distortion. Consequently, various aliasing reduction methods have been proposed in the literature. One of these is based on using the antiderivative of the nonlinearity and has proven effective, but is limited to memoryless systems. In this work, it is extended to a class of stateful systems which includes but is not limited to systems with a single one-port nonlinearity. Two examples from the realm of virtual analog modeling show its applicability to and effectiveness for commonly encountered guitar distortion effect circuits.

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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 continuoustime 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.

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The recently proposed antiderivative antialiasing (ADAA) technique for stateful systems involves two key features: 1) replacing a nonlinearity in a physical model or virtual analog simulation with an antialiased nonlinear system involving antiderivatives of the nonlinearity and time delays and 2) introducing a digital filter in cascade with each original delay in the system. Both of these features introduce the same delay, which is compensated by adjusting the sampling period. The result is a simulation with reduced aliasing distortion. In this paper, we study ADAA using equivalent circuits, answering the question: “Which electrical circuit, discretized using the bilinear transform, yields the ADAA system?” This gives us a new way of looking at the stability of ADAA and how introducing extra filtering distorts a system’s response. We focus on the Wave Digital Filter (WDF) version of this technique.

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Aliasing is an inherent problem in nonlinear digital audio processing which results in undesirable audible artefacts. Antiderivative antialiasing has proved to be an effective approach to mitigate aliasing distortion, and is based on continuous-time convolution of a linearly interpolated distorted signal with antialiasing filter kernels. However, the performance of this method is determined by the properties of interpolation filter. In this work, cubic interpolation kernels for antiderivative antialiasing are considered. For memoryless nonlinearities, aliasing reduction is improved employing cubic interpolation. For stateful systems, numerical simulation and stability analysis with respect to different interpolation kernels remain in favour of linear interpolation.

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