## Appendix A  
The likelihood ratio functional for FTN signals (1), (4) and (5) for a specific \(i\)-th combination of the preceding and \(q\)-th combination of subsequent symbols of the channel alphabet in the case of an arbitrary observation interval can be written in the following form:  
\[\Lambda_{tr}^{(i,q)}=\frac{\exp\biggl{(}\frac{2\mu}{\text{N}_{0}}\int_{0}^{LT}x( t)[s_{l}(t)+y_{-}(t,i)+y_{+}(t,q)]dt-\frac{\mu^{2}}{\text{N}_{0}}\int_{0}^{LT}[s_{l} (t)+y_{-}(t,i)+y_{+}(t,q)]^{2}dt\biggr{)}}{\exp\biggl{(}\frac{2\mu}{\text{N}_{ 0}}\int_{0}^{LT}x(t)[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]dt-\frac{\mu^{2}}{\text{N}_ {0}}\int_{0}^{LT}[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]^{2}dt\biggr{)}} \tag{1}\]  
This functional is written for a specific i-th combination of the preceding and q-th combination of subsequent symbols. A decision is made in favor of the symbol \(l\) if the value of the likelihood ratio functional (1) is positive and in favor of the symbol \(r\) if it is negative for a binary alphabet and opposite signals. Then, (1) can be rewritten. The symbol \(l\) is registered if the following inequality is satisfied:  
\[\frac{\exp\biggl{(}\frac{2\mu}{\text{N}_{0}}\int_{0}^{LT}x(t)[s_{l}(t)+y_{-}( t,i)+y_{+}(t,q)]dt-\frac{\mu^{2}}{\text{N}_{0}}\int_{0}^{LT}[s_{l}(t)+y_{-}(t,i)+y_{ +}(t,q)]^{2}dt\biggr{)}}{\exp\biggl{(}\frac{2\mu}{\text{N}_{0}}\int_{0}^{LT}x (t)[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]dt-\frac{\mu^{2}}{\text{N}_{0}}\int_{0}^{LT }[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]^{2}dt\biggr{)}}>1 \tag{2}\]  
The optimal Bayesian decision strategy involves averaging the numerator and denominator of this expression over all combinations of previous and subsequent characters. This process has high computational complexity (large number of addition and multiplication operations). The computational complexity can be reduced by applying another criterion for the optimal strategy, namely the generalized maximum likelihood criterion. This will significantly simplify the computational decision-making procedure without noticeablelosses in reception noise immunity. We obtained this algorithm based on the use of iterative signal processing with decision feedback. We assume that all previous characters were received correctly. Then, the functions \(y(t,i)\) (4) are known, which partially compensate for intersymbol interference in (3). \[ \begin{split}\underset{(i,q)}{\text{max}}\,\exp& \bigg{(}\frac{2\mu}{N_{0}}\int_{0}^{LT}x(t)[s_{l}(t)+y_{-}(t,i)+y_{+}(t,q)] dt\\ &-\frac{\mu^{2}}{N_{0}}\int_{0}^{LT}[s_{l}(t)+y_{-}(t,i)+y_{+}(t, q)]^{2}dt\bigg{)}\gtrless\\ \underset{(i,q)}{\text{max}}\,\exp&\bigg{(}\frac{2 \mu}{N_{0}}\int_{0}^{LT}x(t)[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]dt\\ &-\frac{\mu^{2}}{N_{0}}\int_{0}^{LT}[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]^{2}dt\bigg{)}.\end{split}  \tag{10}\]  
Note that the choice of the maximum value from the previous symbols can be replaced by ideal decision feedback. Then, the maximum must be selected only from subsequent symbols \(q\):  
\[ \begin{split}\underset{(q)}{\text{max}}\,\exp& \bigg{(}\frac{2\mu}{N_{0}}\int_{0}^{LT}x(t)[s_{l}(t)+y_{-}(t,i)+y_{+}(t,q)]dt \\ &-\frac{\mu^{2}}{N_{0}}\int_{0}^{LT}[s_{l}(t)+y_{-}(t,i)+y_{+}(t, q)]^{2}dt\bigg{)}\gtrless\\ \underset{(q)}{\text{max}}\,\exp&\bigg{(}\frac{2 \mu}{N_{0}}\int_{0}^{LT}x(t)[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]dt\\ &-\frac{\mu^{2}}{N_{0}}\int_{0}^{LT}[s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]^{2}dt\bigg{)}.\end{split}  \tag{11}\]  
Given the monotonicity of exponential functions, (11) can be expressed as follows:  
\[ \begin{split}\underset{(q)}{\text{max}}& \bigg{(}\int_{0}^{LT}x(t)[s_{l}(t)+y_{-}(t,i)+y_{+}(t,q)]dt\\ -\frac{\mu}{2}&\int_{0}^{LT}[s_{l}(t)+y_{-}(t,i)+y_{ +}(t,q)]^{2}dt\bigg{)}\gtrless\\ \underset{(q)}{\text{max}}&\bigg{(}\int_{0}^{LT}x(t) [s_{r}(t)+y_{-}(t,i)+y_{+}(t,q)]dt\\ -\frac{\mu}{2}&\int_{0}^{LT}[s_{r}(t)+y_{-}(t,i)+y_{ +}(t,q)]^{2}dt\bigg{)}.\end{split}  \tag{12}\]  
The expression (12) can be simplified as  
\[