# Design and Implementation of a Multi-Function Hydrophone for Underwater Acoustic Application
## 3 Results and Analysis
### Experimental Results in Mode 2  
In this subsection, the results of mode 2 are described. The experiment was also conducted in Wuyuan Bay, Xiamen, China. The depths of the hydrophones at 3 m, 6 m, and 9 m were chosen. Three multi-function hydrophones were connected to a router, and the router was connected to a gateway called \"Bulltin\". The \"Bulltin\" transmitted the wired signals to WIFI, and a host computer on the shore received the signals. There are two threads in the host computer, the first thread retrieves raw acoustic signal and saves it into a cache, and the second thread acts as a receiver, which decodes the transmitted symbol online. For single-carrier underwater acoustic communication, the common channel equalization approaches in the time domain are the channel estimation-based decision  
Figure 11: Diagram of the received signal. (**a**) Time domain diagram. (**b**) The spectrum diagram. Figure 10: The diagram of deployment and the sound speed profile. (**a**) The diagram of deployment. (**b**) The sound speed profile. feedback equalizer (CE-DFE), and the direct adaptive decision feedback equalizer (DA-DFE). The DA-DFE requires long training sequences to achieve convergence. Compared to DA-DFE, the CE-DFE approach directly measures the channel via shorter training sequences and calculates the channel equalizer's filters based on channel estimation [25]. It needs less controlling parameters to achieve faster convergence and obtains lower output error. Considering the merit of CE-DFE, a receiver based on CE-DFE was used in the host computer. The structure of CE-DFE is shown in Figure 12. In Figure 12, notations \(\mathbf{y}_{1}\), \(\mathbf{y}_{2}\), and \(\mathbf{y}_{3}\) are the received signals from three multi-function hydrophones, \(\mathbf{x}_{s}\) is the soft output of CE-DFE, and \(\hat{\mathbf{x}}\) is the hard output of CE-DFE. The soft output of CE-DFE can be written as  
\[\mathbf{x}_{s}(i)=\mathbf{f}^{H}\mathbf{r}, \tag{3}\]  
where \(\mathbf{f}\) is defined as \(\mathbf{f}=(\mathbf{g}_{ff_{1}}^{H},\mathbf{g}_{ff_{2}}^{H},\mathbf{g}_{ff_{3 }}^{H},\mathbf{g}_{ff_{3}}^{H})^{H}\), \(\mathbf{r}\) is defined as \(\mathbf{r}\)=\((\mathbf{y}_{1}^{H},\mathbf{y}_{2}^{H},\mathbf{y}_{3}^{H},\hat{\mathbf{x}}^{H})^ {H}\). Then the optimal filter can be obtained by solving the cost function  
\[\hat{\mathbf{f}}=\operatorname*{arg\,min}_{\mathbf{f}}\|\mathbf{f}^{H}\mathbf{ r}-\mathbf{s}(i)\|_{2}^{2}. \tag{4}\]  
Finally, the filters can be obtained  
\[\mathbf{g}_{ff}=(\mathbf{g}\mathbf{g}^{H}+\mathbf{D})^{-1}\mathbf{g} \tag{5}\]  
\[\mathbf{g}_{fb}=-\mathbf{F}^{H}(\mathbf{g}\mathbf{g}^{H}+\mathbf{D})^{-1} \mathbf{g} \tag{6}\]  
In (5), \(\mathbf{g}_{ff}\) is defined as \(\mathbf{g}_{ff}=(\mathbf{g}_{ff_{1}}^{H},\mathbf{g}_{ff_{2}}^{H},\mathbf{g}_ {ff_{3}}^{H})^{H}\). Please refer to [25] for more details about the derivation of (5), (6), \(\mathbf{F}\), and \(\mathbf{g}\). Some parameters in this experiment are shown in Table 2. Based on [26], the longer the length of the feedback filter, the better communication performance it achieves. And the length of the feedback filter should be less than the channel length. Considering the computational complexity and our previous work in [25; 27], the length of the feedforward filter is set three times of channel length, and the length of the feedback filter is set approximate to the channel length. The discrete channel length is 100 corresponding to 25 ms, and the lengths of the feedforward filter and feedback filter are 75 ms and 25 ms, respectively. In this mode, the quadrature phase shift keying (QPSK) constellation mapping is utilized. Before channel estimation and channel equalization, the Doppler is estimated and compensated using match filtering [27]. In the CE-DFE receiver, the channel is estimated via training sequences or symbols after the decision, and the algorithm for channel estimation is the least square. The filter coefficients are obtained by (5) and (6).