Stable systems for nonlinear discrete sound synthesis with the functional transformation method

Abstract

Discrete systems for digital sound synthesis derived with the functional transformation method (FTM) from physical models have recently been presented. For linear systems, the FTM solves the partial differential equation (PDE) describing the vibrating structure analytically. The algorithms obtained after discretization of the analytical solution preserve the inherent physical stability of the continuous system. For nonlinear physical models, as they occur in real musical instruments, the direct application of the FTM leads to an implicit continuous equation. This paper shows that discretization results in an explicit solution. Furthermore, the stability problems occurring after discretization of the nonlinear system are fixed by instantaneous energy considerations. For the example of a slapped transversal oscillating tightened string with frequency dependent loss terms an efficient algorithm is derived. It can also be applied to other types of interaction between a resonating body and its environment.