## 8 Conclusions  
We proposed a new bit-by-bit algorithm, (9), for processing FTN signals designed for bit rates above the Nyquist barrier. The advantage of this algorithm is BER performance close to MLSE with minimal computational complexity. Algorithm (9) is based on an iterative algorithm with decision feedback. It is complemented by an algorithm for receiving subsequent signals falling within the observation interval, which is optimal according to the generalized maximum likelihood criterion, providing a minimum number of computational  
Figure 13: Experimental BER performance for proposed algorithm (9) with increased bit rate in multipath channel. procedures when determining the choice of maximum numbers in the sequence. It is shown that the computational complexity of the proposed algorithm is more than two orders less than the complexity of the MLSE algorithm. The energy costs for receiving FTN signals at bit rates above the Nyquist barrier for transmission channels with additive noise (AWGN) and fading channels for proposed algorithm (9) are given in Table 7. This table contains the results of simulation and the results of experimental research. In this paper, we give a numerical estimation of the increase in computational complexity for proposed algorithm (9) at bit rates exceeding the Nyquist barrier. An increase in the bit rate by 1.5 times (from \(R=1.42/T\) to \(R=2/T\)) leads to an increase in energy losses from 1.2 dB to 3.7 dB in the area of \(BER=10^{-4}\) and under conditions of fixed computational complexity. If we fix the possible energy losses, for example, at a value of 2 dB, then an increase in the bit rate by 1.5 times (from \(R=1.11/T\) to \(R=1.66/T\)) leads to an increase in computational complexity by more than an order, from \(0.1\times 10^{5}\) to \(3.3\times 10^{5}\) operations. The same conclusion can be reformulated in another form: for proposed algorithm (9), each decibel of energy gain is achieved by increasing the number of computational operations by \(1.5\times 10^{5}\) operations. The application of proposed algorithm (9) in the DVB standard of satellite broadcasting systems S2/S2X can provide increases in the bit rate without significant degradation in the BER with minimal computational complexity of the demodulator. This is especially important for mobile terminals of satellite broadcasting ground stations. Potential future directions for this research are related to the optimization of pulse shapes for different channels to achieve additional increases in spectral and energy efficiency without increasing computational complexity. Conceptualization, S.M., W.S. and I.L. ; methodology, S.M. and D.Y. ; software, I.L., W.S., A.O. and S.Z. ; validation, I.L., A.O. and S.Z. ; formal analysis, S.M. and W.X. ; writing--original draft preparation, S.M., I.L., A.O., W.S. and S.Z. ; writing--review and editing, S.M., W.X., A.O. and S.Z. ; project administration, S.Z. and D.Y. ; funding acquisition, S.Z. and D.Y. All authors have read and agreed to the published version of the manuscript. This research was partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of the World-class Research Center program: Advanced Digital Technologies (contract No. 075-15-2022-311 dated 20 April 2022). Data Availability Statement: Data is contained within the article. Conflicts of Interest: The authors declare no conflicts of interest.