New application of non-binary Galois fields Fourier transform: digital analog of convolution theorem

Abstract

It is shown that the use of the representation of digital signals varying in the restricted amplitude range through elements of Galois fields and the Galois field Fourier transform makes it possible to obtain an analogue of the convolution theorem. It is shown that the theorem makes it possible to analyze digital linear systems in same way that is used to analyze linear systems described by functions that take real or complex values (analog signals). In particular, it is possibile to construct a digital analogue of the transfer function for any linear system that has the property of invariance with respect to the time shift. It is shown that the result obtained has a fairly wide application, in particular, it is of interest for systems in which signal processing methods are combined with the use of neural networks.

Algebraic fields and rings as a digital signal processing tool

“…In a similar way, one can construct extensions of Galois fields, which, as shown in [11], [23], allow one to construct analogs of Rademacher functions, which are direct analogs of harmonic functions. The restriction is that the number of clock cycles on which the complete basis formed from analogues of the Rademacher functions proposed in [11], [23] should be exactly equal to the number of nonzero elements of the used Galois field. Consequently, the transition to fields containing a greater number of elements is also of practical interest.…”

“…Let us show how exactly one can construct analogs of the Rademacher functions proposed in [11], [23], but containing a greater number of elements using the concrete example-Galois field 𝐺𝐹 (5). All Galois fields 𝐺𝐹 (5) are isomorphic.…”

“…Then the algebraic extension of the field 𝐺𝐹(5) (in this case, the field 𝐺𝐹(5 2 )) will contain exactly 25 elements representable in the form (2), and in this formula a and b should be understood as elements of the original field 𝐺𝐹(5), and under i is the root of (3). In accordance with the methodology proposed in [11], [23], the set of generalized Rademacher functions forming a complete basis on a set of 24 clock cycles can be constructed based on the fact that for any element 𝜁 of an arbitrary Galois field containing 𝑛 + 1 elements,…”

“…let us form next sequences: 23 , 𝜃 23•2 , 𝜃 23•3 , … , 𝜃 23•23 ) (5) where 𝜃 = 1 − 𝑖, as follows from (5), there are exactly 23 such sequences, more generally, 𝑛 − 1.…”

“…Accordingly, the second element of the sequence 𝑤 𝑘 ̃ is the inverse element to the second element of the sequence 𝑤 𝑘 , which is uniquely determined; the same is true for the other elements by the construction of these sequences. Further, in an arbitrary Galois field, there is a general theorem for the sum of powers, used, among other things, in [11], [23]:…”