## 3 Performance Assessment  
### Simulation  
#### 3.1.1 Channel Model  
This subsection introduces channel model used in simulation. The channel model was constructed based on the multipath geometry [1,34]. The frequency response of the shallow water channel can be expressed as:  
\[H\left(\omega\right)=\sum_{p=1}^{p}c_{p}e^{-\
icefrac{{\
u}}{{\tau_{p}}}} \tag{12}\]  
where \(P\) denotes the number of multipath arrivals, \(c_{p}\) denotes the gain of the \(p\)-th propagation path, and \(\tau_{p}\) denotes the delay of the \(p\)-th propagation path. Both \(c_{p}\) and \(\tau_{p}\) are computed from the length of the \(p\)-th propagation path \(l_{p}\). The path delay \(\tau_{p}\) is computed as \(\tau_{p}=l_{p}/c\), where \(c\) denotes the sound speed. The phase of path gain \(c_{p}\) is calculated as \(\angle c_{p}=-2\pi f_{c}\tau_{p}\), where \(f_{c}\) denotes carrier frequency. The magnitude of \(c_{p}\) is computed as \(\left|c_{p}\right|=\Gamma_{p}/\sqrt{A\left(l_{p}\right)}\), where \(\Gamma_{p}\) denotes the magnitude of cumulative reflection coefficient along the \(p\)-th propagation path and \(A\left(l_{p}\right)\) denotes the nominal acoustic propagation loss of the \(p\)-th propagation path. The magnitude of cumulative reflection coefficient \(\Gamma_{p}\) is computed as \(\Gamma_{p}=\left|v_{s}\right|^{n_{s}}\left|v_{b}\right|^{n_{s}}\), where \(\left|v_{s}\right|\) denotes the magnitude of the surface reflection coefficient, \(\left|v_{b}\right|\) denotes the magnitudes of the bottom reflection coefficient, \(n_{s}\) denotes the number of the surface reflection, and \(n_{b}\) denotes the number of the bottom reflection. The propagation loss \(A\left(l_{p}\right)\) is computed as \(A\left(l_{p}\right)=l_{p}^{k}\left[a\left(f_{c}\right)\right]^{\
icefrac{{\gamma }}{{\tau_{p}}}}\), where \(k\) denotes the spreading factor and \(a\left(f_{c}\right)\) denotes the absorption coefficient. The coefficient \(a\left(f_{c}\right)\) is computed in dB/km as \(10\log a\left(f_{c}\right)=0.11f_{c}^{2}/\left(1+f_{c}^{2}\right)+44f_{c}^{2} \left(4100+f_{c}^{2}\right)+2.75\times 10^{-4}\,f_{c}^{2}+0.0003\)  when \(f_{c}\) is given in kHz.  
The parameters for building the shallow water channel model were set as follows. The channel was 75 m in depth, and its bottom was supposed as flat. The transmit sensor and receive sensor were individually placed at 40 m and 30 m below the sea surface, and the range between them was 3000 m. The paths which have undergone bottom reflection twice at most were considered in channel modeling. The sound speed, \(c\), was set to 1500 m/s. The surface reflection coefficient \(\left|v_{s}\right|\), and the bottom reflection coefficient \(\left|v_{b}\right|\) were set to 1 and 0.8, respectively. The spreading factor \(k\)  was set to 2. The entire communication band was 7.5-15.5 kHz. The frequency of the carrier, \(f_{c}\), was 11.5 kHz when the acoustic channel response corresponding to the SC modulation was computed. When the channel response corresponding to the first subcarrier of FMT modulation was computed, the frequency values of the carrier were set to 8 kHz, 7.75 kHz and 7.625 kHz for \(M=8\), \(M=16\), and \(M=32\), respectively. The channel responses for other subcarriers of FMT for \(M=8\), \(M=16\), and \(M=32\)  were computed based on Equation (6).  
The other parameters were set as follows. Binary phase shift keying (BPSK) was used to encode information symbols. The number of information symbols is 6400, and 800 symbols ahead were usedas training symbols. The linear equalizers were used for post-process to avoid error propagation. The RRC shaping filter with the roll-off factor \(\alpha=0.5\) was selected as the transmit filter.