{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:20:55Z","timestamp":1760145655278,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T00:00:00Z","timestamp":1722902400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Science and Higher Education of the Russian Federation as part of the World-class Research Center program: Advanced Digital Technologies","award":["075-15-2022-311"],"award-info":[{"award-number":["075-15-2022-311"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>One of the ways to increase the volume of transmitted information is to increase the bit rate above the Nyquist barrier. However, an increase in bit rate in the case of FTN (Faster-Than-Nyquist) signals leads to an increase in energy costs for receiving information on channels with limited bandwidth, for example, in Digital Video Broadcasting satellite systems like DVB-S2\/S2X. It is possible to minimize energy losses by using the processing algorithm \u201cmaximum likelihood sequence estimation\u201d. However, the computational complexity of this algorithm is extremely high, which limits its use, especially in terrestrial mobile satellite terminals. We propose a new bit-by-bit decision feedback algorithm with maximum likelihood ratio estimation of subsequent symbols in the observation interval. This algorithm provides minimal energy costs comparable to the method \u201cmaximum likelihood sequence estimation\u201d at speeds 2\u20133 times higher than the Nyquist barrier. At the same time, the complexity is two orders of magnitude less. It is shown by simulation for a channel with additive noise that energy losses in relation to the potential bit error rate (BER) are less than 4.5 dB. In the presence of Rayleigh fading, the application of the proposed algorithm makes it possible to provide the processing of FTN signals for double bit rates in urban areas with energy costs equal to 12 dB when using an equalizer. We give numerical estimations of the increase in computational complexity for the proposed processing algorithm. It is shown that an increase in the bit rate by 1.5 times leads to an increase in the computational complexity by more than an order of magnitude. The same conclusion can be reformulated in another form: for the proposed algorithm, each decibel of energy gain is achieved by increasing the number of computational operations by 1.5\u00d7105. It is experimentally shown that additional energy losses due to non-ideal phase and timing synchronization are no more than 1 dB when the proposed algorithm is applied in a fading channel. The energy costs in fading channels relative to a stationary channel for twice the Nyquist rate are equal to 13.8 dB when using an equalizer.<\/jats:p>","DOI":"10.3390\/sym16081001","type":"journal-article","created":{"date-parts":[[2024,8,6]],"date-time":"2024-08-06T11:54:19Z","timestamp":1722945259000},"page":"1001","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Energy Efficiency for Faster-than-Nyquist Data Transmission Using Processing Algorithms with Decision Feedback"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9143-942X","authenticated-orcid":false,"given":"Wenjing","family":"Shang","sequence":"first","affiliation":[{"name":"School of Physics and Electronic Information, Yantai University, Yantai 264005, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6983-1388","authenticated-orcid":false,"given":"Ilya","family":"Lavrenyuk","sequence":"additional","affiliation":[{"name":"Institute of Electronics and Telecommunications, Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya, 29, 195251 St. Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7006-9634","authenticated-orcid":false,"given":"Sergey","family":"Makarov","sequence":"additional","affiliation":[{"name":"Institute of Electronics and Telecommunications, Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya, 29, 195251 St. Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3401-2718","authenticated-orcid":false,"given":"Anna","family":"Ovsyannikova","sequence":"additional","affiliation":[{"name":"Institute of Electronics and Telecommunications, Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya, 29, 195251 St. Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3398-3616","authenticated-orcid":false,"given":"Sergey","family":"Zavjalov","sequence":"additional","affiliation":[{"name":"Institute of Electronics and Telecommunications, Higher School of Applied Physics and Space Technologies, Peter the Great St. Petersburg Polytechnic University (SPbPU), Polytechnicheskaya, 29, 195251 St. Petersburg, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dingfeng","family":"Yu","sequence":"additional","affiliation":[{"name":"Hubei Key Laboratory of Marine Electromagnetic Detection and Control, Wuhan 430064, China"},{"name":"Wuhan Second Ship Design and Research Institute, Wuhan 430064, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Xue","sequence":"additional","affiliation":[{"name":"Yantai Research Institute of Harbin Engineering University, Yantai 264000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1109\/JPROC.2012.2233451","article-title":"Faster-Than-Nyquist Signaling","volume":"101","author":"Anderson","year":"2013","journal-title":"Proc. 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