A Wave Digital Newton-Raphson Method for Virtual Analog Modeling of Audio Circuits with Multiple One-Port Nonlinearities

Abstract

The digital implementation of a nonlinear audio circuit often employs the Newton-Raphson (NR) method for solving the corresponding system of implicit ordinary differential equations in the discrete-time domain. Although its quadratic convergence speed makes NR attractive for real-time audio applications, quadratic convergence is not always guaranteed, since it depends on initial conditions, and also divergence might occur. For this reason, especially in the context of Virtual Analog modeling, techniques for increasing the robustness of NR are in order. Among the various approaches, the Wave Digital (WD) formalism recently showed potential to rethink traditional circuit simulation methods. In this manuscript, we discuss an original formulation of the NR method in the WD domain for the solution of audio circuits with multiple one-port nonlinearities. We provide an in-depth theoretical analysis of the proposed iterative method and we show how its quadratic convergence strongly depends on the free parameters (called port resistances) introduced when modeling the reference circuit in the WD domain. In particular, we demonstrate that the size of the basin where the WD NR solver can be initialized to converge on a solution with quadratic speed is a function of the free parameters. We also show that by setting each port resistance value as close as possible to the derivative w.r.t. current of the nonlinear element v-i characteristic we keep the basin size large. We finally implement an audio ring modulator circuit with four diodes in order to test the proposed iterative method.

Multidomain modeling of nonlinear electromagnetic circuits using wave digital filters

“…We have compared its performance to the one of MathWorks Simscape and LTspice obtaining promising results and we expect that such a performance gain would be more pronounced dealing with larger nonlinear circuits 12 . The results discussed in this manuscript encourage the development of new general‐purpose simulators based on WD principles that take advantage of the latest findings in the WDF literature 21,47 …”

“…12 The results discussed in this manuscript encourage the development of new general-purpose simulators based on WD principles that take advantage of the latest findings in the WDF literature. 21,47 As far as future works are concerned, it is worth noticing that most of the HSIM operations can be easily parallelized, since each nonlinearity is locally handled; therefore, the investigation of a multithreaded implementation of HSIM is promising. Another possible development is to model reluctances with alternative piecewise representation of nonlinear functions that are differentiable.…”

“…In other studies, 25,47 it has also been shown that, by expressing f l , u as a function of a l , u , the reflection coefficient can be rewritten as the derivative of f l , u with respect to a l , u , i.e., , which depends also on Z l , u . It is possible then to minimize the instantaneous dependence of b l , u on a l , u by minimizing .…”

“…Therefore, the only DFLs left to be solved are those involving all the pairs of nonlinear elements that cannot be adapted. For the SIM algorithm, it has been shown that by setting the free parameters Z l , u equal to the slope of the tangent at the current operating point on the element v − i characteristic, the number of iterations needed to reach convergence is minimized 25,47 . Although we cannot ensure HSIM convergence for each value of Z l , u , we can suppose that, by following the same approach, the speed of converge can be increased.…”

Non-Iterative Simulation Methods for Virtual Analog Modelling

“…An exception is the case of circuits including a single scalar nonlinear element, which may be resolved explicitly in the wave domain by placing the nonlinearity at the "root" of the WDF network [15], [6]. For typical VA systems, the iterative search for zeros is a computational bottleneck and furthermore is serial in nature, though in some cases the number of Newton-Raphson iterations may be kept low through a judicious selection of the free parameters in a nonlinear WDF [16]. Beyond this basic computational cost, numerous theoretical and computational difficulties emerge: iterative methods typically operate at a variable cost at each time step (depending on the number of iterations required in order to achieve a predefined error); the user is faced with new choices regarding appropriate tolerance thresholds and the maximum number of iterations; and existence and uniqueness of numerically-computed solutions is generally not ensured [17].…”