In this case, the value of the received symbol is re-evaluated depending on the values of subsequent received symbols falling within the observation interval. In other words, the evaluations of previous and subsequent symbols in the message are used to make a decision about a given symbol received. In [18], the authors propose two methods for symbol sequence estimation in a received sequence: namely SSSSE (Sequential Character Sequence Estimation) and SSSgbKSE (Sequential Character Sequence Estimation with go-back-K Sequence Estimation). Simulation results show that the proposed methods are suitable for low intersymbol interference scenarios, allowing up to 11% increases in the data rate. However, the authors do not consider the possibility of further increasing the symbol rate beyond this value. The SSSSE and SSSgbKSE algorithms show a promising direction for low-computational-complexity FTN signal processing. An interesting algorithm for processing FTN signals with a high level of intersymbol interference was proposed in [24; 25]. This algorithm implements the rule of a decision-directed successive interference cancellation (DDSIC) scheme with Minimum Mean Square Error (MMSE) equalization for FTN signaling. FTN signals after equalization and decision making are regenerated and then subtracted from the original signals at the receiver input to estimate the level of intersymbol interference. This value is used for the next iteration of interference suppression. Based on the simulation results, the BER performance for the DDSIC scheme is better than those described in [18]. The advantage of this algorithm is its low computational complexity. In [25], the authors propose a practical methodology for reducing intersymbol interference in an FTN signal sequence within linear equalization. The framework is based on a low-complexity linear equalizer and serial interference cancellation techniques in the frequency domain. This approach requires additional computational operations in the frequency domain and is characterized by high computational complexity. Let us consider the energy costs for FTN signals with a binary alphabet as a quantitative illustration of the energy costs in case of increasing the data rate to 42% above the Nyquist barrier. Table 1 shows the signal-to-noise ratios in dB at the input of the receiving device for various processing algorithms. From this table, we can see that the energy costs are in the range of signal-to-noise ratios of 7-12 dB for \(BER=10^{-3}\) for various algorithms even with a relatively small increase in the bit rate. These values will increase with increases in the data rate. The indicated signal-to-noise ratio values can be used as quantitative guidelines when obtaining new algorithms for processing FTN signals. Thus, we can identify general trends in improving methods and algorithms for signal processing with an increase in data rates above the Nyquist barrier. The greatest attention is focused on algorithms for bit-by-bit iterative processing with decision feedback, which provide sufficiently high BER performance for receiving FTN signals along with significantly lower computational complexity than with MLSE. Taking these trends into account, we formulated research objectives:  
* Obtaining an iterative algorithm with decision feedback, using the generalized maximum likelihood criterion, that performs additional functions of estimating the values of subsequent symbols falling within the observation interval, providing a minimum number of computational procedures when determining the choice of maximum numbers in the sequence. * Determining the BER performance of FTN signals by simulation modeling at data rates higher than the Nyquist barrier by 2-3 times, with an assessment of energy costs for transmission channels with additive noise and channels with signal fading. * Determining the computational complexity of the algorithm to find the relationship between the increase in the number of addition and multiplication operations and the increase in the data transfer speed.