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Piecewise linear system modeling based on a continuous threshold decomposition

The continuous threshold decomposition is a segmentation operator used to split a signal into a set of multilevel components. This decomposition method can be used to represent continuous multivariate piecewise linear (PWL) functions and, therefore, can be employed to describe PWL systems defined over a rectangular lattice. The resulting filters are canonical and have a multichannel structure that can be exploited for the development of rapidly convergent algorithms. The optimum design of the class of PWL filters introduced in this paper can be postulated as a least squares problem whose variables separate into a linear and a nonlinear part. Based on this feature, parameter estimation algorithms are developed. First, a block data processing algorithm that combines linear least-squares with grid localization through recursive partitioning is introduced. Second, a time-adaptive method based on the combination of an RLS algorithm for coefficient updating and a signed gradient descent module for threshold adaptation is proposed and analyzed. A system identification problem for wave propagation through a nonlinear multilayer channel serves as a comparative example where the concepts introduced are tested against the linear, Volterra, and neural network alternatives.