## 3 TR-FrFT Method for LFM Signal Detection in Underwater Multi-Path Environment
### Lake Experiments  
The LFM signal detection experiments were carried out in Danjiangkou Reservoir, Xichuan County, Nanyang City, Henan Province on 15 July 2019. The original picture of experimental work performed is shown in Figure 9. The depth of the experimental water is about 40 m. In the lake experiments, we used two ships, named Ship A and Ship B respectively, which are 100 m apart. The longitude and latitude of these ships are shown in Figure 9. Ship A carries a transceiver with the functions of transmitting and receiving signals, and the depth of the transceiver is 4 m. Ship B carries a target simulator, which is used to simulate the underwater target echo signal. The depth of the target simulator is 4 m. The parameters of transceiver and target simulator are shown in Table 4 and 5 respectively. The transceiver emits the LFM signal, having the same parameters as described in Section 4.1. The target simulator records the LFM signal and sends it back. The transceiver receives the simulated echo signal. The waveform of the received echo signal is shown in Figure 10 in the latest revised version. FrFT is performed on the received signal, and the result is shown in Figure 11.  
Figure 8: The result of received signal processed by TR-FrFT.  
Figure 7: The waveform of received signal processed by TR.  
Figure 11 shows that, in the underwater multi-path environment, the received signal is expanded, both in the time domain and the fractional Fourier domain. It is consistent to the simulation results in Figure 6. The difference shown in Table 6 between the maximum spurious peak and the main peak is 2.6 dB of FrFT from Figure 11.  
We use TR to process the echo signal. The transceiver emits time reversed signal. The target simulator records the TR signal and sends it back. The transceiver receives the new simulated echo signal. The waveform of the received new echo signal is shown in Figure 9. The original picture of experimental work performed.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline Length, width, and height & 15 cm \(\times\) 15 cm \(\times\) 60 cm \\ Source level & 192 dB \(\pm\) 1.5 dB (18 kHz\(\sim\)24 kHz) dB \\ Receiving sensitivity & \(-\)185 dB \\ Passband & 19 kHz\(\sim\)23 kHz (\(\pm\)1 dB) \\ Stopband & 15 kHz\(\sim\)19 Hz (\(\geq\)20 dB), 23 kHz\(\sim\)27 kHz (\(\geq\)20 dB) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters of transceiver.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **The Difference Between Main and Side Lobes** \\ \hline FrFT & 1.07 dB \\ TR-FrFT & 9.82 dB \\ \hline \hline \end{tabular}
\end{table}
Table 3: The comparison results of FrFT and TR-FrFT with simulated data.  
Figure 9: The original picture of experimental work performed.  
Figure 12. The TR-FrFT results are shown in Figure 13. Comparing Figure 12 with Figure 10, it can be seen that, the expanded received signal in the time domain is focused after being processed by TR, while in the time frequency domain, the signal has little change before and after processed by TR. The reason for this phenomenon is that the received signal has a high signal-to-noise ratio, which makes the difference not prominent. However, we can find a significant change in the optimal order fractional Fourier domain, which can be seen in Figure 13. The results of the received signal processed by TR-FrFT indicate that the main peak is enhanced and the false peaks are suppressed. The difference between the maximum spurious peak and the main peak is improved 10.2 dB. The difference is improved 7.6 dB comparing to FrFT. The TR-FrFT method has the ability to resist multi-path effects and achieve signal focusing in the fractional Fourier domain.  
Figure 11. The result of received signal processed by FrFT with lake experiments data.  
Figure 10. The waveform of received signal with lake experiments data.