Algebraic fields and rings as a digital signal processing tool

Dinara Kutlimuratovna Matrassulova (Almaty University of Power Engineering and Telecommunications after Gumarbek Daukeyev)
Yelizaveta Sergeevna Vitulyova (Almaty University of Power Engineering and Telecommunications after Gumarbek Daukeyev)
Sergey Vladimirovich Konshin (Almaty University of Power Engineering and Telecommunications after Gumarbek Daukeyev)
Ibragim Esenovich Suleimenov (National Engineering Academy of the Republic of Kazakhstan,)

Indonesian Journal of Electrical Engineering and Computer Science
Vol 29, No 1: January 2023

It is shown that algebraic fields and rings can become a very promising tool for digital signal processing. This is mainly due to the fact that any digital signals change in a finite range of amplitudes and, therefore, there are only a finite set of levels that can correspond to the amplitudes of a signal reduced to a discrete form. This allows you to establish a one-to-one correspondence between the set of levels and such algebraic structures as fields, rings, etc. This means that a function that takes values in any of the algebraic structures containing a finite set of elements can serve as a model of a signal reduced to a discrete form. A special case of such a signal model are functions that take values in Galois fields. It is shown that, along with Galois fields, in certain cases, algebraic rings contain zero divisors can be used to construct signal models. This representation is convenient because in this case it becomes possible to independently operate with the digits of the number that enumerates the signal levels. A simple and intuitive method for constructing rings is proposed, based on an analogy with the method of algebraic extensions.