## 2 Description of the FTN Signaling System  
Signal \(s_{r}(t)\) with the arbitrary form of amplitude pulse \(a(t)\) with maximum value \(A_{0}\) and carrier frequency \(f_{0}\) is described by the following expression:  
\[s_{r}(t)=A_{0}a(t)d_{r}^{(0)}cos(2\pi f_{0}t), \tag{1}\]  
where \(d_{r}^{(0)}\) is the value of the modulation symbol at time interval \([0,T]\), which is equal to a transferring time of one information bit. A random sequence of \(N\) signals (1) for an arbitrary value of symbol rate \(R=1/\xi T\) can be written as  
\[y(t)=A_{0}\sum_{n=0}^{N-1}a(t-\xi nT)d_{r}^{(n)}cos(2\pi f_{0}t). \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & \(BER=10^{-3}\) & \(BER=10^{-4}\) \\ \hline MLSE [18; 19; 20; 21] & 7.35 & 9.01 \\ \hline SSSSE [18] & 11.53 & - \\ \hline SSSgbKSE [18] & 9.92 & - \\ \hline DDSIC [24] & 8.00 & 10.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The signal-to-noise ratios in dB at the input of the receiving device for various processing algorithms. Random sequence (2) provides the transfer of binary data at rate \(R=1/\xi T,\ 0<\xi<\ 1\). The amplitude pulse \(a(t)\) determines the main spectral and correlation properties of random sequence (2). For signal duration \(T_{s}=LT,L>1\), intersymbol interference appears even at a transmission rate below the Nyquist barrier. This interference increases significantly when transmitting at a rate that exceeds the Nyquist barrier. The appearance of significant intersymbol interference leads to a degradation in correlation properties and BER performance. Let us consider a generalized block diagram of the data transmission system based on FTN signals (Figure 1). A sequence of binary symbols arrives at the transmitter input from the message source at rate \(R=1/\xi T,\ 0<\xi<1\). These binary symbols are converted into modulation symbols of the message in the \"Mapper\" block. For example, for binary phase modulation, \(d_{r}^{(0)}=\pm 1,\ r=1,2\). A sequence of FTN signals on the basis of the amplitude pulse \(a(t)\) is generated in the \"FTN modulator\". The structure of this device is described in detail in [8]. The symbol rate for forming FTN signals is determined by the \"Symbol rate \(R\)\" block. Sequence of FTN signals (2) goes to block \"Channel\" from the transmitter output. Two types of channels are used as a transmission channel for simulation and for experimental studies: a channel with constant parameters and additive Gaussian noise (AWGN) \(n(t)\) with an average power spectral density \(N_{0}/2\) and a channel with multipath propagation. The random process \(x(t)=y(t)+n(t)\) arrives at the input of the receiver from the output of the transmission channel. The realization of \(x(t)\) is processed in accordance with one or another processing algorithm in the block \"FTN demodulator\". The choice of the reception algorithm is determined by the acceptable energy costs and the computational complexity of the implementation. We propose new bit-by-bit iterative processing algorithms with decision feedback, providing sufficiently high BER performance for FTN signals with limited computational complexity. A possible algorithm for processing the evaluation of the current symbol can be divided into two parts:  
Figure 1: Uncoded FTN signaling transceiver architecture. * The first part is the reception of a given symbol, taking into account decisions about previous accepted symbols (decision feedback), which is implemented with minimal computational complexity;
* The second part is the optimal processing of subsequent symbols falling within the observation interval. Note that we cannot separate these parts of the algorithm. In this case, we make an analytical generation of the algorithm using the generalized maximum likelihood criterion. This general algorithm performs decision feedback processing and estimation of the values of subsequent symbols that fall within the observation interval.