Volterra Series and State Transformation for Real-Time Simulations of Audio Circuits Including Saturations: Application to the Moog Ladder Filter

Volterra series are known to be efficient to represent weakly nonlinear systems and the first distortions. Their truncated versions allow one to derive realizations (in the sense of system theory) leading to networks composed of linear filters, sums, and instantaneous products of signals, without instantaneous feedback loops. Nevertheless, if saturation phenomena arise, truncating the series at low order is not sufficient and the convergence can also be lost. In this paper, the case of the Moog ladder filter is investigated. Low-cost simulations based on realizations of Volterra series are given. Their limitations with respect to the amplitude of input signals are exhibited. Methods to increase the validity range and to improve the efficiency of Volterra series expansions are detailed on a single stage of the filter. In particular, changes of states based on the difference between the original state and predictors (parameterized by a tunable delay T ) yield satisfying results. The digital simulation of this system preserves the properties mentioned above. It includes two delay lines (where the delay T can be chosen to be one sample) and nonlinear static functions given by the method.