Interpretation of the category of “complex” in terms of dialectical positivism

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.

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“…based on (32), each element of 𝐴 can be associated with a matrix defined by the next formula as (33).…”

Section: Resultsmentioning

confidence: 99%

Algebraic fields and rings as a digital signal processing tool

Matrassulova

Vitulyova

Konshin

et al. 2022

IJEECS

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“…A corresponding example was presented in [29], where it was shown that there are conditions under which any voting council is converted into a kind of analogue of a neural network. This example is informative because it allows one to prove that under certain conditions, a decision on a particular issue is made not by a set of Council members, but by a neural network formed from them, which itself has a certain distributed memory.…”

Section: Discussionmentioning

confidence: 99%

Distributed memory of neural networks and the problem of the intelligence`s essence

et al. 2022

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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.

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“…Direct calculation shows that the spectrum in Figure 7 is the product of the spectra in Figure 6 and 3. The resulting formula (25) makes it possible to use a direct analogue of the transfer function for digital periodic signals. Such an analogue is the set of quantities (⃗⃗ , ), appearing in (25).…”

Section: Description Of Linear Systems In Terms Of Generalized Rademacher Functionsmentioning

confidence: 99%

“…The resulting formula (25) makes it possible to use a direct analogue of the transfer function for digital periodic signals. Such an analogue is the set of quantities (⃗⃗ , ), appearing in (25). As in classical radio engineering, the description of any linear operation that has the property of invariance with respect to a shift in time is reduced to an indication of its transfer function.…”

Section: Description Of Linear Systems In Terms Of Generalized Rademacher Functionsmentioning

confidence: 99%

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

Vitulyova

Matrassulova

Сулейменов

2021

IJEECS

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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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Interpretation of the category of “complex” in terms of dialectical positivism

Cited by 17 publications

References 11 publications

Algebraic fields and rings as a digital signal processing tool

Algebraic fields and rings as a digital signal processing tool

Distributed memory of neural networks and the problem of the intelligence`s essence

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

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