## 3 FTN System Model Design  
_Transmission System Model Based on DMD-JD FTN Signal Detection Algorithm_  
Figure 1 shows the principle block diagram of the communication system using the DMD-JD FTN detection algorithm based on the data-driven Deep-Bi-LSTM structure in the multipath channels. Among them, the ISI actively introduced by FTN at the transmitting end is obtained by using \(\tau\)T as the sampling interval in the up-samplingink of the shaping filter. It can be seen that the overall process of FTN signaling generation is roughly the same as the signal generation process of a traditional communication system. In the data preprocessing stage, targeted preprocessing will be done due to different modulation methods. This allows the neural network to extract effective features from the signal data to eliminate the introduced ISI and its nonlinear aliasing after passing through the channel. For example, taking QPSK (4 QAM) modulation as an example, it is assumed that the transmitting end sends a transmission bit sequence with aength of \(N\), which corresponds to a receiving end symbol sequence with aength of \(N/2\) after down-sampling. At the receiving end, the received symbol sequence is divided according to the real part and the imaginary part, and the data set is constructed according to the principle of one-to-one pairing with the transmitted symbols. What needs to be emphasized is the role of the channel equalization module. In the purely data-driven situation, due to the time-varying characteristics of the multipath channel, it will be difficult to match the training data and the test data, which willead to a sharp deterioration in the performance of the test results and it is difficult to achieve the desired performance. The increase of the channel equalization module can alleviate the model training problem caused by the data mismatch. In addition, the introduction of the equalization module can also achieve better generalization performance in various multipath channel switching scenarios, thereby improving the adaptability and robustness of the system. Consider the binary phase-shift keying (BPSK) modulated FTN signaling system, assuming that the information symbol is denoted as \(a(d)\). Where \(d\) and \(D\) respectively represent the symbol index and the number of symbols in each information symbol. After FTN shaping and filtering, the modulated base-band continuous signal can be expressed as  
\[s(t)=\sum_{d=0}^{D-1}a(d)p(t-d\tau T),0<\tau<1 \tag{1}\]  
\(p(t)\) is a pulse-shaped filter with T as the orthogonal shift interval, theength of the filter is L, and \(\tau\) is the time-domain compression factor. Sampling with \(\Delta T\) as the interval, we can obtain  
\[s(n)=\sum_{d=0}^{D-1}a(d)p\Big{(}n-dN_{f}\Big{)},0\leq n\leq(D-1)\times N_{f}+L-1 \tag{2}\]  
where \(\tau T=N_{f}\Delta T\), \(\tau=N_{f}/N_{s}\). \(N_{f}\) is the pulse-shaped filter shift interval, and \(N_{s}\) is the up-sampling rate of the pulse-shaped filter, which satisfies as  
\[\sum_{0}^{L-1}p(n)p^{*}(n-dN_{s})=\left\{ \begin{array}{ll}1,d=0\\ 0,d\
eq 0\end{array} \right. \tag{3}\]  
After going through the multipath channel \(h(t)\), our signal can be expressed as  
\[y(n)=s(n)\otimes h(n)+w(n) \tag{4}\]  
where \(h(n)\) is the channel impulse response, \(w(n)\) is the white noise generated by the AWGN channel with variance \(\sigma^{2}\), and \(\otimes\) represents the cyclic convolution operation. Performing DFT on \(y(n)\) can transform the time domain convolution into the form of the frequency-domain product. After passing through the multipath channel, perform IDFT on the received signal to get the received signal at the receiving end \(y(d)\). \[y(d)=\sum_{d^{\prime}=0}^{D-1}a\big{(}d^{\prime}\big{)}g\big{(}d-d^{\prime} \big{)}+\int_{-\infty}^{+\infty}w(n)p^{*}\Big{(}n-dN_{f}\Big{)}dn \tag{5}\]  
The second term in Equation (5) is the colored noise after Gaussian white noise and matched filter convolution.