## 5 Results and Analysis
### The One-Block Approach  
In this subsection, the channel is estimated only once for PPC processing in each data packet. The channel is estimated using training symbols, which are specified symbols in the beginning of communications. Following PPC processing, ISI is removed by the adaptive channel equalizers. Figure 9 shows scatter plots of soft output estimates \(\left\{\hat{I}_{*}\right\}\) of different receiver structures, where the LS method is used to estimate the channel for PPC processing. The output SNR given by (1-MSE)/MSE, and the bit error rate (BERs) are given in the legends. Obviously, McDFE achieves the best performance with an output SNR of 9.2 dB, PPC-McDFE approximates the performance of McDFE with an output SNR of 8.5 dB, and PPC-DFE achieves the worst performance with an output SNR of 3.8 dB and a BER of 2.0e-3. As shown in Section 3, the difference between PPC-McDFE and PPC-DFE is the multichannel combining scheme. The results of 15 periods are shown in Figure 10, where the symbol rate is 2 kilo-symbol/s. PPC-DFE achieves the worst performance for both BPSK and QPSK, and obviously it fails in several periods for QPSK. It is apparent that the performance of PPC-McDFE consistently follows that of  
Figure 8: Performance of PPC-DFE using different shaping pulses. (**a**) BPSK; (**b**) QPSK. LFM is used (\(\Box\)), and RRC is used (\(\circ\)). Figure 9: Scatter plot of estimated 8-QAM symbols using different receiver structures. (**a**) McDFE; (**b**) PPC-McDFE; (**c**) PPC-DFE. McDFE. For BPSK shown in Figure 10(a), PPC-McDFE overtakes PPC-DFE with a maximum 6.6 dB output SNR (the 3 rd period) and a minimum 2.7 dB output SNR (the 11 th period). Figure 11 shows the spatial coherence measured in the 3 rd and 11 th periods, respectively. The spatial coherence between the \(k\)th and \(m\)th channel is calculated by:  
\[\psi\left(k,m\right)=\frac{\left|r_{k}\left(-t\right)\otimes r_{m}\left(t \right)\right|_{\max}}{\sqrt{\left|r_{k}\left(-t\right)\otimes r_{k}\left(t \right)\right|_{\max}\left|r_{m}\left(-t\right)\otimes r_{m}\left(t\right) \right|_{\max}}} \tag{10}\]  
where \(\left|r_{k}\left(-t\right)\otimes r_{m}\left(t\right)\right|_{\max}\) denotes the maximum absolute value of the correlation between \(r_{k}\left(t\right)\) and \(r_{m}\left(t\right),\) and \(r_{k}\left(t\right)\) is the received signal of the \(k\)th hydrophone. Interchannel correlations shown in Figure 11(a) are stronger than those shown in Figure 11(b), where the time elapse between these two periods is 1,760 s. For instance, the correlation coefficient between Ch #3 and Ch #4 is 0.84 in Figure 11(a), and it is 0.23 in Figure 11(b). Furthermore, in Figure 11(a), it is interesting that there are stronger correlations among the signals received by the vertical array (Ch #1-8) than those among the signals received by the horizontal array (Ch #9-12). Figure 11: Spatial coherence in different periods. (**a**) The 3 rd period; (**b**) The 11 th period. Figure 10: Performance in terms of output SNR for different modulations. (**a**) BPSK; (**b**) QPSK. McDFE (\(\Box\)), PPC-McDFE (\(\circ\)), and PPC-DFE (\(\lozenge\)). The strength of interchannel correlations correlates with the performance difference between PPC-McDFE and PPC-DFE, which is shown in Figure 10(a). Since it is difficult to predict the time variant spatial coherence in a real oceanic environment [10], it is advantageous to obtain spatial gain using the adaptive multichannel combining, especially in the scenario of a small number of receivers. Thus it is preferable to use McDFE in a channel of short time spread, while PPC-McDFE is suggested in a channel of long time spread. As shown in Figure 4, the channel impulse response is sparse. This property can be exploited by the MP algorithm. The conventional LS method obtains values for the taps that should be zero in sparse channels, and the MP algorithm only estimates dominant arrivals. For both methods, it is required that the time window is long enough to include all time-spanned arrivals that cause ISI.