Fast Approximation of the Lambert W Function for Virtual Analog Modelling

Authors
Stefano D'Angelo1, Leonardo Gabrielli2, Luca Turchet3
Status
Published in Proceedings of the 22nd International Conference on Digital Audio Effects (DAFx-19), Birmingham, UK, September 2019
1
Independent researcher, Agropoli, Italy
2
Università Politecnica delle Marche, Ancona, Italy
3
Università degli Studi di Trento, Trento, Italy
BibTeX
@inproceedings{dangelo2019omega,
  title={Fast Approximation of the Lambert W Function for Virtual Analog Modelling},
  author={D'Angelo, Stefano and Gabrielli, Leonardo and Turchet, Luca},
  booktitle={Proc. 22\textsuperscript{nd} Intl. Conf. Digital Audio Effects (DAFx-19)},
  month={September},
  year={2019},
  address={Birmingham, UK}
}

Abstract

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.

Implementation