## 3 TR-FrFT Method for LFM Signal Detection in Underwater Multi-Path Environment
### Detection Performance Comparison  
In this subsection, we compare the detection performance of TR-FrFT detector and FrFT detector in both simulated data and lake experiments data. To generate the performance curves as a function of signal to noise ratio (SNR). We add numerically generated noise to the simulated data and real data. The \(\text{SNR}_{\text{s}}\) corresponding to simulated data is defined by  
\[\begin{split}\text{SNR}_{\text{s}}&=10\log\biggl{(} \frac{P\{s(t)*h_{1}(t)\}}{P\{n_{1}(t)\}}\biggr{)}\\ &=10\log\biggl{(}\frac{\frac{1}{T}\int_{T}|s(t)*h_{1}(t)|^{2}dt} {\sigma_{n}^{2}}\biggr{)},\end{split} \tag{29}\]  
where \(T\) is the observation time of received signal; \(\sigma_{n}^{2}\) is the power of \(n_{1}(t)\).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **The Difference Between Main and Aide Lobes** \\ \hline FrFT & 2.6 dB \\ TR-FrFT & 10.2 dB \\ \hline \hline \end{tabular}
\end{table}
Table 6: The comparison results of FrFT and TR-FrFT with lake experiments data.  
Figure 12: The waveform of received signal processed by TR with lake experiments data.  
Figure 13: The result of received signal processed by TR-FrFT with lake experiments data.  
For real data, (29) is no longer suitable for calculating SNR, because it is hard to separate the detected signal and mixed noise. The \(\text{SNR}_{\text{r}}\), which is an approximate calculation of \(\text{SNR}_{\text{s}}\), corresponding to real lake experiments data, is defined by  
\[\text{SNR}_{\text{r}}=10\log\left(\frac{P_{\text{s}}}{\sigma_{n}^{2}}\right), \tag{30}\]  
where \(P_{\text{s}}\) is the power of the experimental measured LFM signal.  
In order to obtain the performance curve of TR-FrFT detector for a given probability of false alarm \(P_{\text{F}}\), we need to select the threshold \(\eta\). We generate 6000 independent Monte Carlo trials and compute the test statistic given by (27). Then, we sort the results of test statistics in ascend order. The threshold is selected to result in a given \(P_{\text{F}}\). After that, we compute the \(P_{D}\) by the chosen threshold. To do so, We generate 6000 new independent Monte Carlo trials containing both signal and noise. Compare each test statistics with the chosen threshold and count the numbers that the test statistic exceeds the threshold. The detection probability \(P_{D}\) is calculated as a ratio of A over B. A is the number of test statistics that exceed the threshold. B is the number of new experiments.  
With the same arguments, we can obtain the performance curves of FrFT and TR-FrFT detectors. In order to make the simulation environment close to the real environment, we add the Doppler frequency shift parameter to calculate the detection performance curve. Figures 14-16 depict, for \(P_{\text{F}}=0.1\), the detection performance of simulated data for FrFT detector and TR-FrFT detector. The Doppler frequency shifts are set to 0 Hz, 40 Hz and 80 Hz in these three simulated experiments. Figure 17 shows the detection performance of lake experiment data. Observe that the proposed TR-FrFT detector has the best performance compared to the FrFT detector in both simulated and lake experiments data.  
Figure 14: Detection probability vs. SNR for FrFT and TR-FrFT with the simulated data. The false alarm rate is \(P_{\text{F}}=0.1\). The Doppler frequency shift is 0 Hz.  
Figure 16: Detection probability vs. SNR for FrFT and TR-FrFT with the simulated data. The false alarm rate is \(P_{F}=0.1\). The Doppler frequency shift is 80 Hz.  
Figure 15: Detection probability vs. SNR for FrFT and TR-FrFT with the simulated data. The false alarm rate is \(P_{F}=0.1\). The Doppler frequency shift is 40 Hz.