## 3 TRM Technique
### Influence of Multipath Structure on TRM  
The focusing effect of TRM is related to the multipath structure of UWA channels. Taking the single-element TRM as an example, and the CIR of the UWA channel is assumed to be  
\[h(t)=A_{1}\delta(t)+\sum\limits_{m=2}^{L}A_{m}\cdot\delta(t-\tau_{m}) \tag{12}\]  
After TRM processing, the matched output of CIR is  
\[\begin{array}{ll}h(t)\otimes h(-t)&=\sum\limits_{m=1}^{L}A_{m}^{2}\cdot \delta(t)+\sum\limits_{m=2}^{L}A_{1}A_{m}[\delta(t-\tau_{m})+\delta(t+\tau_{m}) ]\\ &+\sum\limits_{m=2}^{L}\sum\limits_{n=m+1}^{L}A_{m}A_{n}[\delta(t-(\tau_{m}- \tau_{n}))+\delta(t+(\tau_{m}-\tau_{n}))].\end{array} \tag{13}\]  
Therefore, after TRM treatment, the maximum amplitude is increased to  
\[A_{\max}=\sum\limits_{m=1}^{L}A_{m}^{2}. \tag{14}\]  
It can be seen from (14) that the number and magnitude of multipath in the UWA channel will affect the magnitude of the maximum of the focusing peak. Next, the sidelobe structure of the time-reversal channel is analyzed. The unilateral sidelobe interference is \[B+C=\sum_{m=2}^{L}A_{1}A_{m}\delta(t-\tau_{m})+\sum_{m=2}^{L}\sum_{n=m+1}^{L}A_{m} A_{n}\delta(t+(\tau_{m}-\tau_{n})). \tag{15}\]  
Next, taking six paths of a multipath channel shown in Figure 5 as an example, the effect of multipath delay on the performance of single-element TRM is analyzed.  
When the number of multipath \(L\) is 6,  
\[\begin{array}{ll}B&=\sum\limits_{m=2}^{6}A_{1}A_{m}\delta(t-\tau_{m})\\ &=A_{1}A_{2}\delta(t-\tau_{2})+A_{1}A_{3}\delta(t-\tau_{3})+A_{1}A_{4}\delta(t -\tau_{4})+A_{1}A_{5}\delta(t-\tau_{5})+A_{1}A_{6}\delta(t-\tau_{6})\\ &=A_{1}A_{2}\delta(t-\Delta t_{1})+A_{1}A_{3}\delta(t-(\Delta t_{1}+\Delta t_{ 2}))+A_{1}A_{4}\delta(t-(\Delta t_{1}+\Delta t_{2}+\Delta t_{3}))\\ &+A_{1}A_{5}\delta(t-(\Delta t_{1}+\Delta t_{2}+\Delta t_{3}+\Delta t_{4}))+A_ {1}A_{6}\delta(t-(\Delta t_{1}+\Delta t_{2}+\Delta t_{3}+\Delta t_{4}+\Delta t _{5})),\end{array} \tag{16}\]  
and  
\[\begin{array}{ll}C&=\sum\limits_{m=2}^{6}\sum\limits_{n=m+1}^{6}A_{m}A_{n} \delta(t+(\tau_{m}-\tau_{n}))\\ &=A_{2}A_{3}\delta(t-\Delta t_{2})+A_{2}A_{4}\delta(t-(\Delta t_{2}+\Delta t_{ 3}))+A_{2}A_{5}\delta(t-(\Delta t_{2}+\Delta t_{3}+\Delta t_{4}))\\ &+A_{2}A_{6}\delta(t-(\Delta t_{2}+\Delta t_{3}+\Delta t_{4}+\Delta t_{5}))+A_ {3}A_{4}\delta(t-\Delta t_{3})+A_{3}A_{5}\delta(t-(\Delta t_{3}+\Delta t_{4}) )\\ &+A_{3}A_{6}\delta(t-(\Delta t_{3}+\Delta t_{4}+\Delta t_{5}))+A_{4}A_{5} \delta(t-\Delta t_{4})+A_{4}A_{6}\delta(t-(\Delta t_{4}+\Delta t_{5}))\\ &+A_{5}A_{6}\delta(t-\Delta t_{5}).\end{array} \tag{17}\]  
If the structure of the multipath channel is assumed as shown in Figure 6a, where \(\Delta t_{1}\
eq\Delta t_{2}\
eq\Delta t_{3}\
eq\Delta t_{4}\
eq\Delta t_{5}\), the delays of each item in \(C\) are different and independent, and they cannot be superimposed on each item in \(B\). Therefore, in such multipath channel, only the energy of the direct path becomes larger after TRM treatment, and the primary-secondary peak ratio of the channel is significantly improved.  
Figure 5: Diagram of the multipath channel.  
Figure 6: Multipath channel with unequal time delay differences. (**a**) Original channel; (**b**) matched channel.  
If the structure of the multipath channel is assumed as shown in Figure 7a, where \(\Delta t_{1}=\Delta t_{2}=\Delta t_{3}=\Delta t_{4}=\Delta t_{5}\), each item in \(C\) is added to each item in \(B\), which will increase the energy of all paths. The primary-secondary peak ratio of this matched channel is equal to the original channel. Therefore, TRM is not suitable for similar structure channels.