On the Best Quadratic Approximation of Nonlinear Systems

We present a new approach to the approximation of the input--output map of a nonlinear system which transforms stochastic signals. The proposed approach is based on the best approximation technique with a specific second degree operator acting on the observed input signal. We show that the obtained approximant minimizes the mean squared error between a desired output signal and the output signal of the approximating system. The strict justification of the proposed approach is provided including theorems on the error representation associated with the approximation process. Three broad extensions of the method are outlined. The simulations demonstrate the efficiency of the proposed technique with an example of its application to image processing. Index Terms--Best approximation, input--output map, nonlinear systems.