# Performance of Time Reversal Based Underwater Target Detection in Shallow Water
## 2 Target Peak Enhancement Using Time Reversal Processing  
### Concept of the Time Reversal Scheme  
In this section, the idea of the time reversal processing is introduced under the time-varying channel \(h(t,\tau)\). This channel represents the signal traveling path from the source to resolution cell of the sonar and back to the receiver of the sonar. This channel can be assumed to be time-varying since the channel is affected by the clutter in the resolution cell. Furthermore, for the case where the target is placed in the resolution cell, the channel is a superposition of the time-varying component due to the clutter and the time-invariant one from the target. Time reversal processing has the spatial-temporal focusing property by retransmitting a time-reversed waveform of the received signal [27]. This process can be conceptually described as follows. The Fourier transform \(R(t,\omega)\) of the received signal \(r(t,\tau)\) through the channel with its impulse response \(h(t,\tau)=\mathcal{F}^{-1}\{H(t,\omega)\}\) at time \(t\) is given by  
\[R(t,\omega)=H(t,\omega)S(\omega)\,, \tag{1}\]  
where \(S(\omega)\) is the Fourier transform of a transmission signal \(s(\tau)\). Then, the time-reversed transmission signal \(S_{tr}(t,\omega)\) at time \(t\) is given by  
\[S_{tr}(t,\omega)=[R(t,\omega)S^{*}(\omega)]^{*}=R^{*}(t,\omega)S(\omega)\,, \tag{2}\]  
where the superscript \(*\) denotes complex conjugation. The received signal \(R_{tr}(t_{o},\omega)\) for the time-reversed transmission signal \(S_{tr}(t,\omega)\) at time \(t_{o}\
eq t\) is as follows:  
\[ \begin{array}{rcl}R_{tr}(t_{o},\omega)&=&H(t_{o},\omega)S_{tr}(t,\omega)=H( t_{o},\omega)R^{*}(t,\omega)S(\omega)\\ &=&H(t_{o},\omega)H^{*}(t,\omega)S^{*}(\omega)S(\omega)=H(t_{o},\omega)H^{*}( t,\omega)|S(\omega)|^{2}\,.\end{array}  \tag{3}\]  
This shows that the waveform \(R_{tr}(t_{o},\omega)\) contains the time varying spectrum of \(H(t_{o},\omega)H^{*}(t,\omega)\). When the channel is time-invariant \(H(t_{o},\omega)=H(t,\omega)\), the received signal \(R_{tr}(t_{o},\omega)\) contains the power spectrum \(\left|H(t,\omega)\right|^{2}\), \(\forall t=t_{o}\), which means a peak. In this work, the case of detecting a cylindrical object bottoming at seafloor in shallow water is considered as shown in Figure 1. A test of two hypotheses can be applied to this investigation. In the null hypothesis, H\({}_{0}\), the channel of shallow water is characterized only by clutter. In the alternative hypothesis H\({}_{1}\), a cylindrical target is also present. This problem can be written as follows:  
\[ \begin{array}{rcl}\text{H}_{0}&:&H(t,\omega)=H_{c}(t,\omega)\\ \text{H}_{1}&:&H(t,\omega)=H_{c}(t,\omega)+H_{lg}(t,\omega)\end{array} \,, \tag{4}\]  
Figure 1: Schematic diagram of the implementation of the time-reversed retransmission signal. where \(H_{c}(t,\omega)\) represents the time-varying component due to the clutter while \(H_{tg}(t,\omega)\) represents the component due to the target. Note that the time-varying characteristics of the clutter and target channels are different in shallow water since the target is bottoming at the seabed. The reverberation by clutter in shallow water has the time-varying power level [12], but the scattering by a cylindrical object is dependent on the size of the cylinder and its angles with respect to the source because the target is stationary. Therefore, the target channel \(H_{tg}(t,\omega)\) is independent of time \(t\). That is, \(H_{tg}(t,\omega)=H_{tg}(\omega)\), which allows the hypotheses in Equation (4) to be rewritten as follows:  
\[ \begin{array}{ll}\mathrm{H_{0}}&:\ R_{tr}(t_{o},\omega)=H_{c}(t_{o},\omega) H_{c}^{*}(t,\omega)|S(\omega)|^{2}\\ \mathrm{H_{1}}&:\ R_{tr}(t_{o},\omega)=\{H_{c}(t_{o},\omega)+H_{tg}(\omega)\} \{H_{c}(t,\omega)+H_{tg}(\omega)\}^{*}|S(\omega)|^{2}\end{array} .