## 3 TRM Technique  
### Theory of TRM  
TRM has a good space-time focusing performance and has attracted wide attention in UWA communications [19; 20; 21; 22]. The theoretical basis of TRM is the reciprocity of the acoustic field. It can match UWA channels automatically, thus suppressing inter-symbol interference and channel fading [23]. If the signal is transmitted from a distant acoustic source, then the received signal in the receiver shows a complex structure, which contains the information of the UWA channel. When the received signal is time-reversed and retransmitted to the ocean, all multipath signals reach the acoustic source at the same time, thus realizing the energy aggregation of multipath signals.  
The single-element TRM is composed of only one hydrophone array. Compared with the TRM arrays, it simplifies the complexity of the equipment and therefore open wide range of applications in UWA communications. Now the focusing gain of single-element TRM is analyzed concretely. It is assumed that the transmitted detection signal is \(p(t)\), and the channel impulse response (CIR) of the UWA channel is \(h(t)\). After passing through the UWA channel, the detection signal arriving at the receiver is  
\[p_{r}(t)=p(t)\otimes h(t)+n(t)=\sum_{i=1}^{L}\left[Ap(t-\tau_{i})+n_{i}(t)\right] \tag{6}\]  
Assuming that the noise \(n_{i}(t)\) of each path is independent of each other, and the SNRs are the same, the noise of each path satisfies  
\[E[n_{i}(t)n_{j}(t)]=0\quad(i\
eq j) \tag{7}\]  
The SNRs of each path is  
\[SNR_{in}=\frac{A_{1}^{2}}{\sigma_{1}^{2}}=\frac{A_{2}^{2}}{\sigma_{2}^{2}}= \frac{A_{3}^{2}}{\sigma_{3}^{2}}=\ldots=\frac{A_{L}^{2}}{\sigma_{L}^{2}}. \tag{8}\]  
where \(\sigma_{i}^{2}\) is the variance of the noise superimposed on the signal in the path \(i\). When the received signal is processed by TRM, the components of each multipath signal are superposed in the same phase at the same time. Then, the reconstructed received signal can be expressed as  
\[p_{r}^{\prime}(t)=\sum_{i=1}^{L}A_{i}p(t)+\sum_{i=1}^{L}n_{i}(t) \tag{9}\]  
Figure 4: BER curves of four systems with approximately equal communication rates.  
Therefore, the SNR of the time-reversed signal can be deducted as  
\[\begin{array}{ll}\mathit{SNR}_{TRM}&=\,(\sum\limits_{i=1}^{L}A_{i})^{2}\bigg{/} \sum\limits_{i=1}^{L}\sigma_{i}^{2}\,=\,(\sum\limits_{i=1}^{L}A_{i})^{2}\bigg{/} \bigg{(}\frac{\sigma_{1}^{2}}{A_{1}^{2}}\sum\limits_{i=1}^{L}A_{i}^{2}\,)\\ &=\,\frac{A_{1}^{2}}{\sigma_{1}^{2}}(\sum\limits_{i=1}^{L}A_{i})^{2}\bigg{/} \sum\limits_{i=1}^{L}A_{i}^{2}\,=\,\frac{A_{1}^{2}}{\sigma_{1}^{2}}(\sum \limits_{i=1}^{L}A_{i}^{2}+\sum\limits_{i=1}^{L}\quad\sum\limits_{\begin{array} []{c}j=1\\ j\
eq i\end{array}}^{L}A_{i}A_{j})\bigg{/}\sum\limits_{i=1}^{L}A_{i}^{2}\\ &=\,\frac{A_{1}^{2}}{\sigma_{1}^{2}}[1+\,(\sum\limits_{i=1}^{L}\quad\sum \limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{L}A_{i}A_{j})\bigg{/}\sum\limits_{i=1}^{L}A_{i}A_{j}) \bigg{/}\sum\limits_{i=1}^{L}A_{i}^{2}\\ &j\
eq i\end{array} \tag{10}\]  
In decibels, it can be written as  
\[\mathit{SNR}_{TRM}=\mathit{SNR}_{in}+10\lg[1+(\sum\limits_{i=1}^{L}\quad\sum \limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{L}A_{i}A_{j})\bigg{/}\sum\limits_{i=1}^{L}A_{i}^{2}\,] (11)  
It can be seen that after TRM treatment, the multipath diversity in the received signal is realized, that is, the SNR of the time-reversed signal is larger than that of the original received signal. \(10\lg[1+(\sum\limits_{i=1}^{L}\quad\sum\limits_{\begin{array}{c}j=1\\ j\
eq i\end{array}}^{L}A_{i}A_{j})\bigg{/}\sum\limits_{i=1}^{L}A_{i}^{2}\,]\) in (11) is the TRM focusing gain, which is closely related to the \(j\
eq i\)  
number and amplitude of multipath signals. In UWA communications, the sparser the channel, the greater the focusing gain obtained by using single-element TRM.