Some advantages of non-binary Galois fields for digital signal processing

Abstract

It is shown that the convenient processing facilities of digital signals that varying in a finite range of amplitudes are non-binary Galois fields, the numbers of which elements are equal to prime numbers. Within choosing a sampling interval which corresponding to such a Galois field, it becomes possible to construct a Galois field Fourier transform, a distinctive feature of which is the exact correspondence with the ranges of variation of the amplitudes of the original signal and its digital spectrum. This favorably distinguishes the Galois Field Fourier Transform of the proposed type from the spectra, which calculated using, for example, the Walsh basis. It is also shown, that Galois Field Fourier Transforms of the proposed type have the same properties as the Fourier transform associated with the expansion in terms of the basis of harmonic functions. In particular, an analogue of the classical correlation, which connected the signal spectrum and its derivative, was obtained. On this basis proved, that the using of the proposed type of Galois fields makes it possible to develop a complete analogue of the transfer function apparatus, but only for signals presented in digital form.

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

“…Building of non-binary Galois field Fourier transform is based on the following considerations [17]. Discrete-time functions taking values in the Galois field GF(p) can serve as a model for any signals that can be converted to digital form.…”

“…and the fact that +1 − = 0 and − 1 = 0 for any element ζ of an arbitrary Galois field containing = + 1 elements. [17], we can define sequences of the form;…”

“…This is illustrated in Figure 1-4. Figure 1 shows the dependence of the number of deaths per week, during 16 weeks of 2020, in the state of Georgia, USA, used as an example of a model signal, in Figure 2 the same dependence, but reduced to a coarse scale corresponding to the Galois field (17), and in Figure 3 is its diagram corresponding to its spectrum in terms of this field. For comparison, Figure 4 shows the spectrum obtained using the classical Walsh function (the comparison is legitimate, since in both cases the sequences of 16 signal values from time are considered de facto).…”

“…In [17], it was proposed the following approach for processing of any digital signals that actually occur in practice: they all change in a finite range of amplitudes, and, therefore, the number of digital signal levels is also finite. Hence, it becomes possible to use Galois field, i.e.…”

“…In [17] it was shown that for digital signal processing it is often advisable to use non-binary Galois field, choosing the number of their elements equal to the number of signal levels. Of course, in practice, scales corresponding to binary logic (and, accordingly, binary Galois field) are currently used, but this is nothing more than a matter of agreement.…”