Using the Relationship between the Theory of Algebraic Fields and Number Theory for Developing Promising Methods of Digital Signal Processing

Abstract

An analogue of the Dirac δ-function is proposed, constructed for the case of Galois fields, and providing the possibility of conveniently reducing operations of multivalued logics to algebraic ones, including for the purposes of digital signal processing. It is shown that this function admits a convenient representation in the form of a binomial polynomial, whose coefficients take a constant value equal to one. The verification of the obtained results is carried out by the method of comparison with the proof of Wilson's theorem obtained using the theory of algebraic fields. An analogue of Wilson's theorem is obtained for the case of the binomial coefficients mentioned above and its visual illustration is given using an analogue of Pascal's triangle constructed for the case of simple Galois fields.