Nonlinear System Identification: An Effective Framework Based on the Karhunen–LoÈve Transform

This paper proposes, on the basis of a rigorous mathematical formulation, a general framework that is able to define a large class of nonlinear system identifiers. This framework exploits all those relationships that intrinsically characterize a limited set of realizations, obtained by an ensemble of output signals and their parameterized inputs, by means of the separation property of the Karhunen-Loeve transform. The generality and the flexibility of the approximating mappings (ranging from traditional approximation techniques to multiresolution decompositions and neural networks) allow the design of a large number of distinct identifiers each displaying a number of properties such as linearity with respect to the parameters, noise rejection, low computational complexity of the approximation procedure. Exhaustive experimentation on specific case studies reports high identification performance for four distinct identifiers based on polynomials, splines, wavelets and radial basis functions. Several comparisons show how these identifiers almost always have higher performance than that obtained by current best practices, as well as very good accuracy, optimal noise rejection, and fast algorithmic elaboration. As an example of a real application, the identification of a voice communication channel, comprising a digital enhanced cordless telecommunications (DECT) cordless phone for wireless communications and a telephone line, is reported and discussed.