Newton–Raphson Solution of Nonlinear Delay-Free Loop Filter Networks

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.