V-Vector Algebra and Volterra Filters

Publisher Summary This chapter describes an algebraic structure that is usefully applied to the representation of the input–output relationship of the class of polynomial filters known as “discrete Volterra filters.” Such filters are essentially based on the truncated discrete Volterra series expansion obtained by suitably sampling the continuous Volterra series expansion, which is widely applied for representation and analysis of continuous nonlinear systems. Volterra series expansions form the basis of the theory of polynomial nonlinear systems (or filters), including Volterra filters. The Volterra series expansions for both continuous and discrete systems are introduced and their main properties are reviewed in the chapter. The main elements of V-vector algebra are introduced, together with their relevant properties. In principle, V-vectors can be defined as nonrectangular matrices, and V-matrices represent appropriate collections of V-vectors, replacing the vectors and the matrices of linear algebra. The basic operations between V-vectors and V-matrices are defined, and the concepts of inverse, transposed, and triangular matrices of linear algebra are adapted to V-vector algebra.