Many tools are available to study linear systems (either for modelling, simulation, identification or control). But in the nonlinear case, many problems remain open. Volterra series give an input-output representation, approximated to within a given error, of any time-invariant continuous nonlinear system with fading memory (this excludes hysteresis effects, regime changes, self-oscillations or chaos). Technically, they correspond to a series expansion sorted by homogeneity order with respect to the input: each homogeneous term is characterized by a convolutive kernel -- often called generalized impulse response -- whose set provides a "complete signature" of the modelled system. These kernels allow to model and simulate systems, but also to identify them from measurements, which is an important issue in the study of nonlinear systems.
This thesis addresses the identification of nonlinear systems that can be represented with Volterra series, and its application to audio systems. The works presented are based on the development of a preliminary step that consists in separating the series' terms to improve identification. Compared to existing homogeneous order separation methods, which are based on amplitude relationships between test signals, the approach chosen in this thesis is to exploit phase relationships between signals to obtain a robust method.
This is first obtained in the theoretical case of complex excitation signals. From this idea, several methods suited to the use of real signals are developed. This leads to define new signals categories that describes the output of a Volterra series, sorting nonlinear contributions according to their phase properties. The proposed separation methods are applied and tested on a guitar pedal effect. Then, specific identification methods for the new types of signals are presented Finally, a method for estimating the parameters of a polynomial nonlinear state-space representation is developed. This is applied to an electrodynamic loudspeaker whose nonlinear characteristics are studied.
Damien Bouvier will defend his doctoral carried out in the Sound Systems and Signals: Audio/Acoustics, InstruMents team (STMS - IRCAM/CNRS/Sorbonne Université/Ministère de la Culture).
The jury will be composed:
Françoise Lamnabhi-Lagarrigue, CNRS
Johan Schoukens - Vrije, Universiteit Brussel
Béatrice Laroche, INRA
Roland Badeau, Télécom ParisTech
Marc Rébillat, ENSAM
Benoît Fabre, Sorbonne Université
Thomas Hélie, CNRS
David Roze, CNRS