## 2 System Model
### Receiver  
After transmission through underwater acoustic channels, the signal received by the \(\beta\)-th receive element is  
\[y_{\beta}(kT_{\rm c})=\sum_{\rho=-\infty}^{+\infty}\sum_{\alpha=1}^{2}\sum_{i =0}^{M-1}\sum_{n=-\infty}^{+\infty}s^{a,i}(nT)g_{\rm t}(kT_{\rm c}-\rho T_{\rm c }-nT)h_{\beta}^{\alpha}(\rho T_{\rm c})e^{i\frac{2\pi}{M}i(k-\rho)}+w_{\beta} (kT_{\rm c}) \tag{4}\]  
where \(h_{\beta}^{\alpha}(\rho T_{\rm c})\) denotes the discrete time-domain response of the channel between the \(\alpha\)-th transmit element and the \(\beta\)-th receive element, and \(w_{\beta}(kT_{\rm c})\) represents channel noise received by the \(\beta\)-th receive element. Referring to Figure 2, at the receiver, the received signal \(y_{\beta}(kT_{\rm c})\) is processed by FMT demodulation, and the \(m\)-th output signal is  
\[ \begin{array}{l}y_{\beta}^{m}(nT)=\sum_{l=-\infty}^{+\infty}\sum_{\alpha=1} ^{2}\sum_{k=-\infty}^{+\infty}\sum_{\rho=-\infty}^{+\infty}g_{\rm t}(kT_{\rm c }-\rho T_{\rm c}-IT)h_{\beta}^{\alpha}(\rho T_{\rm c})e^{-i\frac{2\pi}{M}im \rho}g_{\rm r}(nT-kT_{\rm c})s^{a,m}(IT)\\ +\sum_{\begin{array}{c}i=0\\ i\
eq m\end{array}}^{M-1}\sum_{l=-\infty}^{+\infty}\sum_{\alpha=1}^{2}\sum_{k =-\infty}^{+\infty}\sum_{\rho=-\infty}^{+\infty}g_{\rm t}(kT_{\rm c}-\rho T_{ \rm c}-IT)h_{\beta}^{\alpha}(\rho T_{\rm c})e^{-i\frac{2\pi}{M}i\rho}g_{\rm r }(nT-kT_{\rm c})e^{i\frac{2\pi}{M}(i-m)k}s^{a,i}(IT)\\ \qquad i\
eq m\\ +\sum_{k=-\infty}^{+\infty}w_{\beta}(kT_{\rm c})g_{\rm r}(nT-kT_{\rm c})e^{-i \frac{2\pi}{M}mk}\end{array}  \tag{5}\]  
where \(g_{\rm r}(nT_{\rm c})\) refers to the discrete time response of the receive filter. Figure 1: Block diagram for transmit structure. Observing Equation (5), the second signal component on the right side is the ICI. In FMT, benefitting from non-overlapping subbands with high containment, ICI is a minor concern even with the Doppler spread caused by water motion. This paper's main concern is joint suppression of residual ISI and CCI, as well as noise in MIMO-FMT TR acoustic communications. The effect of ICI is not taken into consideration. When ICI is neglected, \(y_{\beta}^{m}(nT)\) can be simplified as  
\[ \begin{array}{ll}y_{\beta}^{m}(nT)&=\sum\limits_{l=-\infty}^{+\infty}\sum \limits_{\alpha=1}^{2}\sum\limits_{k=-\infty}^{+\infty}\sum\limits_{\rho=- \infty}^{+\infty}g_{\rm t}(kT_{\rm c}-\rho T_{\rm c}-lT)h_{\beta}^{\alpha}( \rho T_{\rm c})e^{-j\frac{2\pi}{M}m\rho}g_{\rm r}(nT-kT_{\rm c})s^{\alpha,m}( lT)\\ &+\sum\limits_{k=-\infty}^{+\infty}w_{\beta}(kT_{\rm c})g_{\rm r}(nT-kT_{\rm c })e^{-j\frac{2\pi}{M}mk}\end{array}  \tag{6}\]  
It can be observed from Equation (6) that the discrete time response of the \(m\)-th subchannel between the \(\alpha\)-th transmit element and the \(\beta\)-th receive element is  
\[h_{\beta}^{\alpha,m}(nT_{\rm c})=h_{\beta}^{\alpha}(nT_{\rm c})e^{-j\frac{2 \pi}{M}mn} \tag{7}\]  
Equation (7) indicates that the responses of subchannels are different and, therefore, the output signals of FMT demodulation should be divided into \(M\) groups \(\left\{y_{1}^{m}(nT),y_{2}^{m}(nT)\right\}\), \(m=0,\cdots M-1\) that are processed by TR individually. The procedure of the \(m\)-th TR processing is shown in Figure 3, where \(\left[H_{\beta}^{\alpha,m}(\omega)\right]^{*}\) denotes the frequency domain response of the filter matched to the frequency domain response of the \(m\)-th subchannel between the \(\alpha\)-th transmit element and the \(\beta\)-th receive element \(H_{\beta}^{\alpha,m}(\omega)\). Figure 2: Block diagram for receive structure. The \(\alpha\)-th output signal of the \(m\)-th TR processing is  
\[r^{\alpha,m}(nT)=\sum\limits_{l=-\infty}^{+\infty}\sum\limits_{\gamma=1}^{2}q^{ (\alpha,\gamma),(m,m)}(lT,nT)s^{\gamma,m}(lT)+\eta^{\alpha,m}(nT) \tag{8}\]  
where \(q^{(\alpha,\gamma),(m,m)}(lT,nT)\) and \(\eta^{\alpha,m}(nT)\) denote the composite channel response and noise after TR processing, and the expressions are as follows:  
\[