Construction of generalized Rademacher functions in terms of ternary logic: solving the problem of visibility of using Galois fields for digital signal processing
Abstract
Using generalized Rademacher functions, constructed as a sequence of elements of Galois fields , and intended to find the spectral representation of signals with n levels, the expediency of using the concept of "logical imaginary unit" is substantiated. These functions form a complete basis on the interval corresponding to 3^n -1 discrete time intervals and for n=1 passing into the classical Rademacher functions. The advantage of such spectra obtained using Galois Fields Fourier Transform is that the range of variation of the spectrum amplitudes remains the same as the range of variation of the original signal, which is modeled on discrete time functions taking values in the Galois field.
References
Sergey I. Babaev, Alexander Bastrychkin, Boris V. Kostrov, N.V. Lukina, A.A. Vyugina, E.P. Koroleva, “Aspects of Binary Images Spectral Analysis,” 2019 8th Mediterranean Conference on Embedded Computing (MECO), July 2019. DOI: 10.1109/MECO.2019.8760003
Zulfikar M. Yusuf, Shuja A. Abbasi, A. R. M., “Alamoud. FPGA based processing of digital signals using Walsh analysis,” International Conference on Electrical, Control and Computer Engineering 2011 (InECCE), 21-22 June 2011. DOI: 10.1109/INECCE.2011.5953922.
Lubomyr Petryshyn. “Applying of the Walsh functions systems in navigation digital data processing,” 2016 4th International Conference on Methods and Systems of Navigation and Motion Control (MSNMC), 18-20 Oct. 2016. DOI: 10.1109/MSNMC.2016.7783166
D. E. Dutkay, and G. Picioroaga, "Generalized Walsh bases and applications,” Acta applicandae mathematicae, vol. 133(1), pp. 1-18, 2014.
R. H. Wang and W. Dan, "Improved Haar and Walsh functions over triangular domains,” Journal of the Franklin Institute, vol. 347(9), pp. 1782-1794, 2010.
Zulfikar, Shuja A. Abbasi, A.R.M. Alamoud, “A Novel Complete Set of Walsh and Inverse Walsh Transforms for Signal Processing,” 2011 International Conference on Communication Systems and Network Technologies, June 2011. DOI: 10.1109/CSNT.2011.108
R.M. Aron, M. Lacruz, R.A. Ryan, A.M. Tonge, "The generalized Rademacher functions,” Note di Matematica, vol. 12, pp.15-25, 1992.
H. Mussa, Jonathan D. Tyzack, R. Glen, “Note on the Rademacher-Walsh Polynomial Basis Functions,” Journal of Mathematics Research, 2013. DOI:10.5539/JMR.V5N1P114
Poolakkaparambil, M., Mathew, J., Jabir, A. M., & Mohanty, S. P., “An investigation of concurrent error detection over binary Galois fields in CNTFET and QCA technologies,” IEEE computer society annual symposium on VLSI, 2012, pp. 141-146.
Pruss, T., Kalla, P., & Enescu, F., “Equivalence verification of large Galois field arithmetic circuits using word-level abstraction via Gröbner bases,” In Proceedings of the 51st Annual Design Automation Conference, 2014, pp. 1-6.
Xiusheng Liu., et al., "Galois LCD codes over finite fields," Elsevier, Finite Fields and Their Applications, vol. 49, pp. 227-242, Jan 2018.
Shivashankar S., et al., "A Galois field-based texture representation for face recognition," International Journal of Applied Engineering Research, vol. 13(18), pp. 13460-13465, 2018.
Shu Lin, et al., "Iterative soft-decision decoding of Reed-Solomon codes of prime lengths, " IEEE International Symposium on Information Theory (ISIT), pp. 341-345, Aug 2017.
Liu, P., et al., "Parameter identification of Reed-Solomon codes based on probability statistics and Galois field Fourier transform, " IEEE Access, vol.7, pp. 33619-33630, Mar 2019.
Huang, Q., et al., "Low-complexity encoding of quasi-cyclic codes based on Galois Fourier transform," IEEE Transactions on Communications, vol. 62(6), pp. 1757-1767, Apr 2014.
Wu, G., et al., "Blind recognition of BCH code based on Galois field Fourier transform," 2015 International Conference on Wireless Communications & Signal Processing (WCSP), IEEE, pp. 1-4, Oct 2015.
Lukasiewicz J., "On Three-Valued Logic // Jan Lukasiewicz. Selected Works," Ed. by L. Borkowski. Amsterdam: North-Holland, pp. 87–88, 1970.
Karpenko, A., & Tomova, N., "Bochvar's three-valued logic and literal paralogics: Their lattice and functional equivalence," Logic and Logical Philosophy, 26(2), pp. 207-235, 2017.
Schumann, A., "Logical Determinacy versus Logical Contingency. The Case of Łukasiewicz’s Three-valued Logic," Studia Humana, 8(2), pp. 8-15, Dec 2019.
Aireti Abulikemu, Abudurusuli Aosiman, Maliyamuguli Maimaiti, Tuerhongjiang Abudukelimu, “The Common Problem of Reasoning Based on Multi-Valued-Logic: An Example of Artificial Neural Network,” CSSE 2020: Proceedings of the 2020 3rd International Conference on Computer Science and Software Engineering, 2019, pp. 186–190. DOI: https://doi.org/10.1145/3403746.3403926
Suleimenov, I. E., et al., "Artificial Intelligence: what is it?," In Proceedings of the 2020 6th International Conference on Computer and Technology Applications, pp. 22-25, Apr 2015.
Suleimenov, I. E., et al., "Dialectical understanding of information in the context of the artificial intelligence problems," In IOP Conference Series: Materials Science and Engineering, Vol. 630, No. 1, p. 012007, Oct 2019.
Anil Kumar Singh, “Error detection and correction by Hamming code,” 2016 International Conference on Global Trends in Signal Processing, Information Computing and Communication (ICGTSPICC), Dec. 2016.
DOI: 10.1109/ICGTSPICC.2016.7955265
Rawnaq A. Habeeb, “Coding-Decoding Ternary Logic,” Journal of Electrical and Electronic Engineering, 2014. DOI:10.33762/EEEJ.2014.93015
A. Soltani, Saeed Mohammadi, “A Novel Three Value Logic for Computing Purposes,” International Journal of Information Engineering and Electronic Business, 2013. DOI:10.7763/IJIEE.2013.V3.341
Giniyatullin V.M., Arslanov I.G., Bogdanova P.D., Gabitov R.N., Salikhova M.A., “Methods Of Implementation Of Ternary Logic Functions,” Software for innovative information technologies, pp. 239-254, 2014.
Vitulyova, Y. S., Bakirov, A. S., Shaltykova, D. B., & Suleimenov, I. E., “Prerequisites for the analysis of the neural networks functioning in terms of projective geometry,” In IOP Conference Series: Materials Science and Engineering, Vol. 946, No. 1, p. 012001, 2020.
Akhat S Bakirov, Ibragim E Suleimenov, “On the possibility of implementing artificial intelligence systems based on error-correcting code algorithms,” Journal of Theoretical and Applied Information Technology, 99(1) pp 83-99, 2021 DOI: http://www.jatit.org/volumes/Vol99No1/8Vol99No1.pdf
Suleimenov, I., Bakirov, A., & Moldakhan, I., “Formalization of Ternary Logic for Application to Digital Signal Processing,” In Energy Management of Municipal Transportation Facilities and Transport, pp. 26-35, Springer, Cham, 2019.
Morris Kline, “Mathematics: The Loss of Certainty,” Oxford University Press, 1980.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 International Journal of Electronics and Telecommunications
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
1. License
The non-commercial use of the article will be governed by the Creative Commons Attribution license as currently displayed on https://creativecommons.org/licenses/by/4.0/.
2. Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. The undersigned also warrants that the manuscript (or its essential substance) has not been published other than as an abstract or doctorate thesis and has not been submitted for consideration elsewhere, for print, electronic or digital publication.
3. User Rights
Under the Creative Commons Attribution license, the author(s) and users are free to share (copy, distribute and transmit the contribution) under the following conditions: 1. they must attribute the contribution in the manner specified by the author or licensor, 2. they may alter, transform, or build upon this work, 3. they may use this contribution for commercial purposes.
4. Rights of Authors
Authors retain the following rights:
- copyright, and other proprietary rights relating to the article, such as patent rights,
- the right to use the substance of the article in own future works, including lectures and books,
- the right to reproduce the article for own purposes, provided the copies are not offered for sale,
- the right to self-archive the article
- the right to supervision over the integrity of the content of the work and its fair use.
5. Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
6. Termination
This agreement can be terminated by the author or the Journal Owner upon two months’ notice where the other party has materially breached this agreement and failed to remedy such breach within a month of being given the terminating party’s notice requesting such breach to be remedied. No breach or violation of this agreement will cause this agreement or any license granted in it to terminate automatically or affect the definition of the Journal Owner. The author and the Journal Owner may agree to terminate this agreement at any time. This agreement or any license granted in it cannot be terminated otherwise than in accordance with this section 6. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
7. Royalties
This agreement entitles the author to no royalties or other fees. To such extent as legally permissible, the author waives his or her right to collect royalties relative to the article in respect of any use of the article by the Journal Owner or its sublicensee.
8. Miscellaneous
The Journal Owner will publish the article (or have it published) in the Journal if the article’s editorial process is successfully completed and the Journal Owner or its sublicensee has become obligated to have the article published. Where such obligation depends on the payment of a fee, it shall not be deemed to exist until such time as that fee is paid. The Journal Owner may conform the article to a style of punctuation, spelling, capitalization and usage that it deems appropriate. The Journal Owner will be allowed to sublicense the rights that are licensed to it under this agreement. This agreement will be governed by the laws of Poland.
By signing this License, Author(s) warrant(s) that they have the full power to enter into this agreement. This License shall remain in effect throughout the term of copyright in the Work and may not be revoked without the express written consent of both parties.
