Since the formulation of the reconstruction theorem by Takens [10] it has been clear that a nonlinear predictor of a dynamical system may be directly derived from a systems time series. The method has been investigated extensively and with good success for the prediction of time series of nonlinear systems. Especially the combination of reconstruction techniques and neural networks has shown good results [12].
In our work we extend the ideas of predicting nonlinear systems by the more demanding task of building system models, which are able to resynthesize the systems time series. In the case of chaotic or strange attractors the resynthesis of identical time series is known to be impossible. However, the modeling of the underlying attractor leads to the possibility to resynthesis time series which are consistent with the system dynamics. Moreover, the models may be used for the analysis of the system dynamics, for example the estimation of the Lyapunov exponents [6]. In the following we investigate the modeling of music and speech signals, where the system dynamics are known to be instationary. Therefore, we develop an extension of the modeling approach, such that we are able to handle instationary systems.
In the following, we first give a short review concerning the state space reconstruction from time series by delay coordinate vectors, a method that has been introduced by Takens [10] and later extended by Sauer et al. [9]. Then we explain the structure of the neural networks we used in the experiments and the enhancements necessary to be able to model instationary dynamics. As an example we apply the neural models to a saxophone tone and a speech signal and demonstrate that the signals may be resynthesized using the neural models. Furthermore, we discuss some of the problems and outline further developments of the application.