Kurt James Werner

Modeling nonlinear wave digital elements using the Lambert function

A large class of transcendental equations involving exponentials can be made explicit using the Lambert W function. In the last fifteen years, this powerful mathematical tool has been extensively used to find closed-form expressions for currents or voltages in circuits containing diodes. Until now almost all the studies about the W function in circuit analysis concern the Kirchhoff (K) domain, while only few works in the literature describe explicit models for diode circuits in the Wave Digital (WD) domain. However explicit models of NonLinear Elements (NLEs) in the WD domain are particularly desirable, especially in order to avoid the use of iterative algorithms. This paper explores the range of action of the W function in the WD domain; it describes a procedure to search for explicit wave mappings, for both one-port and multi-port NLEs containing diodes. WD models, describing an arbitrary number of different parallel and anti-parallel diodes, a transformerless ring modulator and some BJT amplifier configurations, are derived. In particular, an extended version of the BJT Ebers-Moll model, suitable for implementing feedback between terminals, is introduced.

A general and explicit formulation for wave digital filters with multiple/multiport nonlinearities and complicated topologies

A general, explicit, and novel formulation for Wave Digital Filters (WDFs) with multiple/multiport nonlinearities is presented. It confronts graph-theoretic views of WDF structures (SPQR tree) with techniques from the nonlinear loop resolution literature (the K method) and a novel method for deriving scattering matrices (WDF adaptors) of arbitrarily complex topologies. It accommodates any number of memoryless nonlinearities with any number of ports each without requiring simplifying assumptions. A case study on the first clipping stage of the Big Muff Pi distortion pedal is presented. This circuit, with its multiple nonlinearities, multiport nonlinearity, and complicated topology, poses an intractable problem for state-of-the-art WDF methods. Hence, successful simulation of this circuit demonstrates the novelty and robustness of our framework.

Multi-port nonlinearities in Wave Digital Structures

Wave Digital Structures (WDS), with their inherent stability and robustness, would be particularly suitable for nonlinear (NL) circuit modeling in Virtual Analog applications, if it were possible for them to accommodate multi-port nonlinearities. In this work we present a method for modeling a rather general class of multi-port NL elements in the WD domain, which are obtained as the interconnection of linear and NL resistive bipoles. The method is based on a Piece-Wise Linear approximation of the individual nonlinearities that constitute the multi-port element. This method advances the state of the art as it enables the modeling of arbitrary interconnections between outer ports of the nonlinearity and individual ports of the local NL bipoles. As an example of application of the method, we show a WD implementation of a transformer-less ring modulator.

Modeling a class of multi-port nonlinearities in wave digital structures

Wave Digital Structures (WDS) are particularly interesting for applications of interactive modeling of nonlinear (NL) elements in the context of Virtual Analog modeling. NL circuits, however, often include multiple nonlinearities or multi-port nonlinearities, which cannot readily be accommodated by traditional WDS. In this work we present a novel method for modeling in the WD domain a class of multi-port NL elements that are obtained as the interconnection of linear and NL resistive bipoles. Our technique is based on a Piece-Wise Linear approximation of the individual bipoles that constitute the multi-port element. The method generalizes the existing solutions that are available in the literature as it enables the modeling of arbitrary interconnections between outer ports of the nonlinearity and individual ports of the local NL bipoles.

Wave digital filter modeling of circuits with operational amplifiers

We extend the Wave Digital Filter (WDF) approach to simulate reference circuits that involve operational amplifiers (op-amps). We handle both nullor-based ideal op-amp models and controlled-source-based linear op-amp macromodels in circuits with arbitrary topologies using recent derivations for complicated scattering matrices. The presented methods greatly increase the class of appropriate circuits for virtual analog modeling, and readily extend to circuits with any number of op-amps. Although op-amps are essential to many circuits and deviations from ideal can be important, previous WDF research applies only to the limited case of circuits with ideal op-amps, in differential amplifier topology, with no global feedback.

Optimizing differentiated discretization for audio circuits beyond driving point transfer functions

One goal of Virtual Analog modeling of audio circuits is to produce digital models whose behavior matches analog prototypes as closely as possible. Discretization methods provide a systematic approach to generate such models but they introduce frequency response error, such as frequency warping for the trapezoidal method. Recent work showed how using different discretization methods for each reactive element could reduce such error for driving point transfer functions. It further provided a procedure to optimize that error according to a chosen metric through joint selection of the discretization parameters. Here, we extend that approach to the general case of transfer functions with one input and an arbitrary number of outputs expressed as linear combinations of the network variables, and we consider error metrics based on the L2 and the L1 norms. To demonstrate the validity of our approach, we apply the optimization procedure for the response of a Hammond organ vibrato/chorus ladder filter, a 19-output, 36th order filter, where each output frequency response presents many features spread across its passband.

Recent extensions to topological and nonlinear aspects of wave digital filter theory

The Wave Digital Filter approach can be used to create computational models of lumped reference systems, including rectilinear mechanical, rotational mechanical, acoustical, and electronic systems. When they are applicable, Wave Digital Filters enjoy properties including modularity, accuracy, and guaranteed incremental passivity that make them attractive in the context of musical instrument and audio effect simulation. However, the class of reference systems that can be modeled with Wave Digital Filters has historically been restricted to systems with simple topologies (which can be decomposed entirely into series and parallel connections) and which contain only a single “nonadaptable” element (e.g., a diode, ideal source, or switch). This talk details recent Wave Digital Filter advances from Stanford Universitys Center for Computer Research in Music and Acoustics (CCRMA), which broaden the applicability of Wave Digital Filters to reference systems with any topology and an arbitrary number of nonlinearit...

An energetic interpretation of nonlinear wave digital filter lookup table error (invited paper)

In this paper, we study error in wave digital filter nonlinearities from an energetic perspective. By ensuring that power dissipation corresponding to this error is non-negative, we respect the basic wave digital filter premise that errors should not correspond to an increase in system energy. In particular, this has implications for the formation of lookup tables and the choice of lookup table interpolation method. We give recommendations for both based on the sign and second derivative of the wave-domain function which is to be tabulated. These recommendations are used to study interpolation of a diode characteristic.