# Performance of Time Reversal Based Underwater Target Detection in Shallow Water
## 2 Target Peak Enhancement Using Time Reversal Processing
### Detection Performance Comparison
#### 2.3.2 Time Reversal-Based Detection  
It is difficult to obtain the pdf of the received signal for the retransmitted time-reversed signal in closed form. Thus, the probability of detection and the probability of false alarm are obtained by using the numerical method.  
As mentioned in the earlier section, the initial return signal has the K-distributed intensity after matched filtering. Therefore, the retransmitted signal \(s_{tr}^{k}\) has the K-distribution. If the time-varying property of the channel is weak, passing the retransmitted signal through the channel results in the autocorrelation of the channel. However, the real channel is time-varying while the properties do not change suddenly compared to the previous channel [12; 28]. This deviation is modeled by Gaussian noise. Therefore, the pdf of the current channel becomes the convolution between the previous channel's K-distributed pdf \(f_{K_{1}}(x)\) and the Gaussian pdf \(f_{G}(x)\), where RV \(K_{1}\) represents the intensity of the previously received signal (i.e., the previous clutter channel) and RV \(G\) is the deviation. Thus, the intensity of the received signal \(R_{2}\) for the retransmitted signal is the correlation of \(K_{2}=K_{1}+G\) and \(K_{1}\), that is, \(R_{2}=K_{2}K_{1}^{*}\), which implies the case that there is no target, the null hypothesis \(H_{0}\). For the case that a target exists, the alternative hypothesis \(H_{1}\), the intensity of the received signal includes the target signal intensity \(P_{t}\).  
Figure 2 shows the probabilities of detection for the conventional and proposed schemes for various false alarm probabilities when the target-to-clutter ratio (TCR) varies from \(-15\) dB to \(10\) dB. Note that the shape parameter \(\alpha=1\) and the scale parameter \(\lambda=1\). As observed in the figure, the proposed method is better than the conventional one by about \(5\) dB at a probability of detection of \(0.9\). TCR is measured by power spectral densities of the clutter and target signals, and is defined as follows:  
\[TCR=\frac{\sum\limits_{BW}\left|H_{tg}(\omega)\right|^{2}}{\sum\limits_{BW} \left|H_{c}(\omega)\right|^{2}}\,, \tag{12}\]  
where \(BW\) represents the band width that contains significant power.  
In order to investigate the effect of the shape parameter \(\alpha\) on the probability of detection, the probability of detection is numerically estimated as a function of the normalized target signal power with respect to the scale parameter; that is, \(S_{0}=P_{t}/\lambda\) for the probability of false alarm \(P_{FA}=0.005\). The shape parameter \(\alpha\) is set to \(1\), \(5\), \(10\) and \(20\) while \(\lambda=1\). Figure 2(a),b demonstrate the probabilities of detection for the time reversal-based detection and the conventional detection, respectively. As observed in the figures, the probability of detection becomes worse as the shape parameter becomes larger (\(\alpha=20\)), which converges to the Rayleigh distribution.  
Figure 3: Numerically obtained probabilities of detection for \(\alpha=1\), 5, 10, and 20 when \(\lambda=1\) and \(P_{FA}=0.005\). (**a**) Time-reversal-based detection; (**b**) Conventional detection.  
Figure 2: Probabilities of detection numerically obtained by the proposed and conventional methods for \(P_{FA}=0.1\), 0.05, 0.01, and 0.005 with \(\alpha=1\) and \(\lambda=1\). (**a**) \(P_{FA}=0.1\); (**b**) \(P_{FA}=0.05\); (**c**) \(P_{FA}=0.01\); (**d**) \(P_{FA}=0.005\). \(P_{FA}\): false alarm probability ; TCR: target-to-clutter ratio.