# Performance Optimization of Underwater Communication Links at Different Ranges for AIS Relay to AUV
## Appendix A Doppler Estimation  
In UAC, the Doppler deeply affects the received signal. Unlike radio communications, in UAC, Doppler not only creates a shift of the carrier frequency but also a time compression/dilatation [35]. Doppler \(d\) for a constant speed \(v\) between transmitter and receiver can be approximated by:  
\[d=1-\frac{v}{c} \tag{1}\]  
The method described in [35; 36] allows one to jointly estimate the Doppler, detect, and synchronize the signal. This method uses a pilot signal and a bank of matched filters. Each filter in this filter bank is a unique time compression/dilatation of a pilot signal. Hence, each filter is matched to a pilot signal affected by one possible Doppler shift.  
Theses filters can be created by resampling the pilot signal. Let \(s_{0}(t)\) be the OFDM pilot signal:  
\[s_{0}(t)=exp(j2\pi f_{0}t)\sum_{(n=0)}^{(N-1)}q_{0}^{(n)}exp(j2\pi nBt)g(t) \tag{2}\]  
where \(N\) is the number of subcarriers, \(q_{0}^{(n)}\) is the \(n^{\text{th}}\) symbol, \(B\) is the subcarrier spacing, \(g(t)\) is the pulse-shaping filter and \(f_{0}\) is the non-centered carrier frequency. Once affected by a Doppler shift \(d=1-\frac{v}{c}\), it becomes:  
\[s_{0}(t,v)=exp(j2\pi f_{0}td)\sum_{n=0}^{N-1}q_{0}^{(n)}exp(j2\pi nBtd)g(td) \tag{3}\]  
Matched filters of \(s_{0}(t,v)\) are created.  
By correlating the filter bank with the received samples, Doppler can be estimated as equal to the compression/dilatation of the filter with the maximum correlation peak. Figure 11 shows the simulated result of the correlation between an OFDM signal affected by a Doppler shift and a bank of filters.  
In Figure 11, \(R(u,v)\) is the result of the cross-correlation between the filters and the received signal. It depends on the delay and the relative speed \(v\) used to create the filters. \(R(u,v)\) can be defined as followed:  
\[R(u,v)=\int s_{0}(t-u_{0},v_{0})s_{0}^{*}((t-u),v)\,\mathrm{d}t \tag{4}\]  
where \(u_{0}\) and \(v_{0}\) are the delay and Doppler speed affecting the sent pilot signal \(s_{0}(t)\).  
In Figure 11, a high correlation peak between the received signal and the filter simulating a 3 m\(\cdot\)s\({}^{-1}\) Doppler shift is present. Two lobes can be seen. They are the result of the cyclic prefix correlation. The estimated Doppler shift is the one used to create the filter giving the highest correlation peak.  
Once estimated, Doppler is compensated by resampling the signal. This maximum correlation peak can be used for synchronization but also for detection by comparing it to a threshold defined by CA-CFAR (Cell Averaging Constant False Alarm Rate) method [37]. The pilot signal is generated in a similar way as the data signal and is, therefore, a MC-SS symbol with \(M=1\). To increase its difference with the data signal, a distinctive scrambling sequence is used for the pilot signal. The frame structure is a pilot signal followed by MC-SS symbols. Using this method, the range of Doppler speeds that can be estimated depends on the filters used and the Doppler speed they simulate. The modem proposed here can estimate a Doppler speed \(v\) up to \(\pm 10\) m\(\cdot\)s\({}^{-1}\).