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.

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.

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The question of the nature of the distributed memory of neural networks is considered. Since the memory capacity of a neural network depends on the presence of feedback in its structure this question requires further study. It is shown that the neural networks without feedbacks can be exhaustively described based on analogy with the algorithms of noiseproof coding. For such networks the use of the term "memory" is not justified at all. Moreover, functioning of such networks obeys the analog of Shannon formula, first obtained in this paper. This formula allows to specify in advance the number of images that a neural network can recognize for a given code distance between them. It is shown that in the case of artificial neural networks with negative feedback it is really justified to talk about a distributed memory network. It is also shown that in this case the boundary between distributed memory of a neural network and information storage mechanisms in such elements as RS-triggers is diffuse. For the given example a specific formula is obtained, which connects the number of possible states of outputs of the network (and, hence, the capacity of its memory) with the number of its elements.

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.