Wave Digital Filter (WDF) theory has been widely used to design nonlinear digital filters that behave like reference analog circuits, especially in the field of Virtual Analog modeling, i.e., the digital emulation of audio circuits. WDF principles allow us to implement circuits with one nonlinearity in an explicit fashion, by properly setting the free parameters that are introduced in the Wave Digital (WD) domain. Although this property does not extend to the WDF realization of circuits with multiple nonlinearities, which instead requires the use of iterative solvers, a proper setting of the free parameters is beneficial also in these cases, in terms of both robustness and efficiency of the implementation. In particular, recent research shows that a common optimal policy for the setting of free parameters can be adopted when either a WD fixed-point method or a WD Newton-Raphson (NR) method is used to solve the same nonlinear circuit. In this work, we present a class of WD extended fixed-point solvers with order from zero to infinity that generalizes the aforementioned iterative methods; the zero-order case corresponds to the existent WD fixed-point method, while the infinite-order case to WD NR. The proposed solvers do not require to compute inverse Jacobian matrices and are characterized by superlinear speed of convergence. Moreover, we show that the very same policy of free parameter setting can be applied independently of the order, causing, in any case, an increase of robustness and convergence speed.

A family of iterative root-finding methods for nonlinear discrete-time systems of equations is presented, with a formulation that puts it in between the Fixed-Point (FP) and Newton-Raphson (NR) methods. Applicability of this family is allowed, provided that the Jacobian matrix of the nonlinear system has a spectral radius less than one. By varying the order of a matrix geometric sum that approximates the inverse Jacobian matrix, root-finding at any iteration can be steered toward the FP or conversely toward the NR method, becoming identical to either of them if the order is equal to zero or infinitely large, respectively. Since the methods in this family do not need the solution of a linear system at each iteration as required by NR, their computational cost makes them palatable for the online digital implementation of nonlinear models. As an example of application, a Virtual Analog model of the voltage-controlled filter onboard a popular music synthesizer is tested, showing that for some orders of the aforementioned geometric sum the proposed methods perform better than FP and NR in terms of computational cost, while exhibiting the same accuracy.

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.

For their numerical properties and speed of convergence, Newton-Raphson methods are frequently used to compute nonlinear audio electronic circuit models in the digital domain. These methods are traditionally employed regardless of preliminary considerations about their applicability, primarily because of a lack of flexible mathematical tools making the convergence analysis an easy task. We define the basin delimiter, a tool that can be applied to the case when the nonlinear circuit is modeled by a delay-free loop network. This tool is derived from a known convergence theorem providing a sufficient condition for quadratic speed of convergence of the method. After substituting the nonlinear characteristics with equivalent linear filters that compute Newton-Raphson on the existing network, through the basin delimiter, we figure out constraints guaranteeing quadratic convergence speed in the diode clipper. Further application to a ring modulator circuit does not lead to comparably useful constraints for quadratic convergence; however, also in this circuit, the basin delimiter has a magnitude roughly proportional to the number of iterations needed by the solver to find a solution. Together, such case studies foster refinement and generalization of this tool as a speed predictor, with potential application to the design of virtual analogue systems for real-time digital audio effects.

An iterative method is proposed for the explicit computation of discrete-time nonlinear filter networks containing delay-free loops. The method relies on a fixed-point search of the signal values at every temporal step. The formal as well as numerical properties of fixed-point solvers delimit its applicability: On the one hand, the method allows for a reliable prediction of the frequency rates where the simulation is stable, while, on the other hand, its straightforward applicability is counterbalanced by low speed of convergence. Especially in presence of specific nonlinear characteristics, the use of a fixed-point search is limited if the real-time constraint holds. For this reason, the method becomes useful especially during the digital model prototyping stage, as exemplified while revisiting a previous discrete-time realization of the voltage-controlled filter aboard the EMS VCS3 analog synthesizer. Further tests conducted on a digital ring modulator model support the above considerations.

A decimated version of a broad family of Finite-Difference Time-Domain schemes is derived by an algebraic rearrangement of their matrix formulation, allowing for computing the grid nodes at half the temporal steps compared to the original scheme. This rearrangement can ask for solving a generalized matrix inversion problem. The decimated scheme generates solutions having comparable accuracy to that exhibited by the original simulations. However, the broader applicability of the proposed technique requires to solve currently unanswered theoretical issues of spatial grid decimation, as well as to make extensive tests using large matrices.