# Estimate of Passive Time Reversal Communication Performance in Shallow Water
## 4 Results
### Performance Analysis of Passive Time Reversal Communication  
Figure 5a,b show examples of the normalized \(q\)-functions estimated using a single channel (\(N=1\)) and the summation for four channels (\(N=4\)), respectively. These values were estimated using the data measured in the source-receiver range of 109 m at site 2. For the case of \(N=1\), the \(q\)-function has many side lobes, of which the highest magnitude corresponds to \(\sim\)0.3 of the main-lobe level. The non-negligible side lobes spread over 30 symbol durations from the main lobe. The parameter \(E_{q}\) was estimated to be 0.5 for this case. In contrast, in the case of \(N=4\), the side lobe of \(q\)-function was suppressed, the magnitudes of most side lobes were less than 0.1 of the main-lobe level, and the estimated \(E_{q}\) was 0.85.  
Figure 4: Channel intensity impulse responses measured at a source depth of 30 m and a receiver depth of 15 m and their eigenray tracing outputs for source-receiver ranges of (**a**) 109 m; (**b**) 497 m and (**c**) 1078 m at site 2. The seabed-grazing angle is noted for the B path.  
Figure 6a,b show comparisons of the estimates of \(E_{q}\) with the BER and the output SNR, respectively. The symbol color and shape indicate the number of channel summations, source-receiver range and the site. This information is listed in the box at the lower right of Figure 6. The output SNR can be estimated by the reciprocal of the mean-squared error [19; 26], which is given by  
\[SNR_{o}\ =\ \frac{1}{E\Big{|}d(n)-\hat{d}(n)\Big{|}^{2}}, \tag{3}\]  
Here, \(E\) denotes the expectation, and \(d(n)\) and \(\hat{d}(n)\) are the original data symbols and their estimates. For both sites, overall the BER and the output SNR improved as the \(E_{q}\) increased. It is interesting to note that the best communication performance was achieved at site 1 (having an acoustically slow bottom) when \(N=4\), and in this case the value of \(E_{q}\) was highest, showing \(E_{q}>0.85\). On the other hand, site 2 (having an acoustically fast bottom) showed a limit to communication performance improvement due to the residual ISI of the \(q\)-function, and the \(E_{q}\) values were less than 0.83 even in case of \(N=4\). Overall, the performance improved as \(N\) increased. Also, as the source-receiver range increased, the performance improved except for ranges of 500 and 1000 m in site 1, in which the performance at a range of 500 m was better than that at a range of 1000 m. For every  
Figure 5: Right-hand side of the \(q\)-function obtained using (**a**) a single receiver and (**b**) four receivers.  
Figure 6: Comparisons of the \(E_{q}\) to (**a**) the BER and (**b**) the output SNR for sites 1 and 2; Comparison between the BER and the side-lobe to main-lobe ratio is shown in (**c**).  
case, the \(E_{q}\) parameter showed a strong correlation with the BER and the output SNR. A criterion for the best communication performance may be proposed using the parameter \(E_{q}\). For example, if a BER of 1% is chosen as a tolerance for communication, the \(E_{q}\) value is about 0.67 in our case, and a BER of 0.1% results in an \(E_{q}\) of about 0.78. However, it may change if other techniques (such as adaptive time reversal or additional equalization) are applied to the time reversal process. Finally, Figure 6c shows a comparison between the BER and the maximum side-lobe level normalized to main-lobe level. Overall, the BER also improved as the side-lobe level decreased. In reference 18, the side-lobe to main-lobe ratio of 0.1 was roughly suggested as a criterion to minimize BER. However, large variation of BER is observed in the case of side-lobe to main-lobe ratio less than 0.1, which implies that the parameter \(E_{q}\) is more suitable as a parameter to describe the performance of time reversal communication.