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An algebraic approach to implementation of generalized polynomial filters

Methods of modern algebra have appeared to be extremely useful in system theory and digital signal processing. New classes of linear filters and fast linear convolution algorithms have been developed based on the algebraic approach. In this paper, we apply this approach to the description of a class of polynomial (Volterra) filters of signals and fields defined over finite groups. Since the polynomial filters can be considered as a direct extension of linear filters, it is reasonable to apply the algebraic methods to a nonlinear case too. After a brief introduction to the abstract signal theory, a matrix representation of generalized polynomial filter is presented. Finally, we discuss the construction of fast nonlinear convolutions algorithms on the basis of their linear counterparts.