## 3 TR-FrFT Method for LFM Signal Detection in Underwater Multi-Path Environment
### TR-FrFT Method  
As described in Section 2.3, in an underwater multi-path environment, the result of received signal is extended in FrFT domain. The false peaks will confuse the detector. Considering time-reversal processing has the characteristics of energy focusing and compression, we propose a novel method based on TR-FrFT to detect LFM signal in this complex environment. A schematic illustration of the novel method is shown in Figure 2.  
The TR method is presented in Section 3.1. Substituting (14) into (20), \(z(t)\) can be re-written as  
\[z(t)=a\sum_{i=1}^{M}a_{i}^{2}s(-t)+a\sum_{i=1}^{M}\sum_{i=1}^{M}a_{i}a_{i}s(-t- \tau_{ii})+a\sum_{i=1}^{M}a_{i}^{2}n_{1}(-t-\tau_{i})+n_{2}(t), \tag{22}\]  
where \(\tau_{ii}=\tau_{l}-\tau_{i}\) is the delay difference between the \(l\)-th path and \(i\)-th path. Substituting (22) into (1) with \(x(t)\) set to \(z(t)\), the FrFT of \(z(t)\) is  
\[ \begin{split} Z_{p}(u)&=\{F^{p}[z(t)]\}(u)\\ &=a\sum_{i=1}^{M}a_{i}^{2}S_{p}(-u)+a\sum_{i=1}^{M}\sum_{\begin{subarray} {c}l=1\\ i\
eq l\end{subarray}}^{M}a_{i}a_{l}B_{\tau_{l}}S_{p}(-u+u_{\tau_{l}})+a\sum_{ i=1}^{M}a_{i}^{2}C_{\tau_{l}}N_{1p}(u+u_{\tau_{l}})+N_{2p}(u),\end{split}  \tag{23}\]  
with  
\[ \begin{split} B_{\tau_{l}}&=e^{j\pi x_{l}^{2}\sin \alpha\cos\alpha-j\mathbb{Z}\pi u_{\tau_{l}}\sin\alpha},\quad i=1,2,\ldots,M, \quad l=1,2,\ldots,M,\\ C_{\tau_{l}}&=e^{j\pi\tau_{l}^{2}\sin\alpha\cos \alpha-j\mathbb{Z}\pi u_{\tau_{l}}\sin\alpha},\quad i=1,2,\ldots,M,\quad l=1,2,\ldots,M,\\ u_{\tau_{l}}&=\tau_{l}\cos\alpha,\quad i=1,2,\ldots,M, \quad l=1,2,\ldots,M,\end{split}  \tag{24}\]  
and \(Z_{p}(u)\) is the FrFT form of the emitted signal \(z(t)\) with transform order \(p\). \(N_{2p}(u)\) is the FrFT form of the additive noise \(n_{2}(t)\) with transform order \(p\). There are four terms on the right hand side of (23). The first term is the focusing signal with the coherent superposition of each paths. In the second term, the delay of each paths are different in the FrFT domain resulting in non-coherent superposition. The last two are about noise in the FrFT domain. Thus, processed by TR-FrFT, the optimal order FrFT of \(z(t)\) is focusing. We can use the method shown in Algorithm 1 to search the optimal transform order \(p_{opt}\). Once the optimal transform order \(p_{opt}\) is found, the largest peak will appear in the optimal order fractional Fourier domain. Figure 2: Schematic illustration of the novel method based on TR-FrFT used to detect LFM signal in an underwater multi-path environment. Figure 4: The simulated channel generated by Bellhop. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & **Amplitude** & **Relative Time** & **Number of Sea** & **Number of Seafloor** \\  & & **Delay (ms)** & **Surface Reflections** & **Reflections** \\ \hline \(1\) & \(5.403\times 10^{-4}\) & 51.759 & 2 & 1 \\ \(2\) & \(1.105\times 10^{-3}\) & 28.946 & 1 & 1 \\ \(3\) & \(1.460\times 10^{-3}\) & 13.267 & 1 & 0 \\ \(4\) & \(1.044\times 10^{-3}\) & 10.639 & 0 & 0 \\ \(5\) & \(1.605\times 10^{-3}\) & 20.451 & 0 & 1 \\ \(6\) & \(7.529\times 10^{-4}\) & 37.115 & 1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Acoustic channel parameters. Figure 3: Sound speed profile. Calculated by Algorithm 1, the optimal order of \(y(t)\) is 1.0141. When the SNR\({}_{8}\) defined in Section 4.3 is 0 dB, the waveform of the received signal is illustrated in Figure 5. The result processed by the optimal order FrFT is shown in Figure 6. We can find that, due to the influence of multi-path effect, the result in time domain is expended, and the corresponding result processed by the optimal order FrFT has multiple peaks. The difference between the maximum spurious peak and the main peak is 1.07 dB. The presence of spurious peaks will affect the decision of the detector. We now use the TR-FrFT method processing the received data.