## 4 UWOC-MIMO System Performance Analysis  
### System Average BER Analysis  
Assuming that each symbol of the mQAM signal is sent with an equal probability, each QAM symbol will be included by the \(M_{\rm I}\) dimensions in-phase and \(M_{\rm Q}\) dimensions quadrature-phase symbols, respectively, with \(m=M_{\rm I}\times M_{\rm Q}\). The system instantaneous BER is the sum of the error probability in the quadrature component and the error probability in the in-phase component. The quadrature-to-in-phase decision distance ratio \(r_{0}=d_{\rm Q}/d_{\rm I}\). \(d_{\rm Q}\) and \(d_{\rm I}\) denote the decision distance in the quadrature and in-phase components, respectively. Without loss of generality, assuming that \(r_{0}=1\), the UWOC-MIMO system instantaneous BER can be expressed as [7; 32]:  
\[ \begin{split} P_{\rm e,inst}=\frac{1}{m_{0}M_{\rm I}}\sum_{m_{ \rm I}=1}^{\log_{2}M_{\rm I}}\sum_{i=0}^{M_{\rm I}}\sum_{i=0}^{M_{\rm I}} \sum_{m_{\rm Q}=1}^{M_{\rm I}}\sum_{j=0}^{\log_{2}M_{\rm Q}}\sum_{j=0}^{M_{ \rm Q}(1-2^{-m_{\rm Q}})-1}2C_{\rm f}Q\bigg{(}D_{j}\sqrt{2\gamma}\bigg{)}\\ \end{split}  \tag{18}\]  
where \(m_{0}\) is the modulation index, and \(m_{0}=\log_{2}(M_{\rm I}\times M_{\rm Q})\). \(\gamma\) is the SNR received by the MIMO system, which is the SNR at the output of the MRC combiner, as shown in Equation (4). \({\rm Q}(\cdot)\) is the Gaussian Q function. \(C_{i}\), \(C_{j}\), \(D_{i}\), and \(D_{j}\) are represented as follows [7; 32]:  
\[ \begin{split} C_{i}=(-1)^{\lfloor\frac{i2^{m}-1}{M_{\rm I}} \rfloor}\Big{(}2^{m_{\rm I}-1}-\Big{\lfloor}\frac{i\cdot 2^{m_{\rm I}-1}}{M_{\rm I}}+\frac{1}{2} \Big{\rfloor}\Big{)},C_{j}=(-1)^{\lfloor\frac{i2^{m_{\rm Q}-1}}{M_{\rm Q}} \rfloor}\Big{(}2^{m_{\rm Q}-1}-\Big{\lfloor}\frac{i2^{m_{\rm Q}-1}}{M_{\rm Q} }+\frac{1}{2}\Big{\rfloor}\Big{)},\\ D_{i}=(2i+1)\sqrt{\frac{3m_{0}}{(M_{\rm I}^{2}-1)+r_{0}^{2}(M_{ \rm Q}^{2}-1)}},D_{j}=(2j+1)\sqrt{\frac{3r_{0}^{2}m_{0}}{(M_{\rm I}^{2}-1)+r_{0 }^{2}(M_{\rm Q}^{2}-1)}}.\end{split}  \tag{19}\]  
For each optical link of the MIMO system, assuming that the channel fading caused by the ocean turbulence and pointing errors is independent and identically distributed, the joint PDF of the channel fading vector \({\bf h}=(h_{\rm I1},h_{\rm I2},\ldots,h_{MN})^{\rm T}\), and \(f_{h}({\bf h})=\prod\limits_{m=1}^{M}\prod\limits_{m=1}^{N}f_{h}(h_{mn})\). Therefore, the system average BER, which represents the mean of the system BER for the joint fading channel, can be expressed as:  
\[P_{\rm e,MIMO}=\int_{0}^{\infty}P_{\rm e,inst}f_{h}({\bf h})d{\bf h}, \tag{20}\]  
Inserting Equation (3) into Equation (18), using the approximation of \({\rm Q}(x)\), \({\rm Q}(x)\approx\exp\big{(}-x^{2}/2\big{)}/12+\exp\big{(}-2x^{2}/3\big{)}/4\), and then employing (07.34.21.0011.01) from [29] and letting \(A_{x}=A_{0}h_{c}h_{l}\), the average BER in the UWOC-MIMO system with MRC can be deduced as:  
\[P_{\rm e,MIMO} \approx \left\{\left[\left(\frac{1}{3}+\left(\frac{3}{4}\right)^{MN\xi^{2}} \right)\left(\overset{\log_{2}M_{1}}{\sum\limits_{m_{i}=1}^{\infty}}\sum \limits_{i=0}^{\infty}\frac{(-1)^{n}}{\beta_{n}}\text{Re}\bigg{\{}\frac{M_{ \gamma_{\rm nacc}}(-(A_{\rm E}+2\pi jn)/2\gamma_{\rm th})}{(A_{\rm E}+2\pi jn)/2 \gamma_{\rm th}}\bigg{\}}+E(A_{\rm E},N_{\rm E},Q),\right.\right. \tag{24}\]  
where \(\beta_{n}=\left\{ \begin{array}{ll}2,\ n=0\\ 1,\ n=1,2,\cdots,N_{\rm E}\end{array} \right.,\)\(\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left.\left. \cdot\cdot\right.\right.\right.\right.\right.\right.\right.\right.\right.\right.\) indicates taking the real part, and the chosen values for \(A_{\rm E}\), \(N_{\rm E}\), and \(Q\) can be found in [6]. \(E(A_{\rm E},N_{\rm E},Q)\) is the overall error term. \(M_{\gamma_{\rm nacc}}(s)\) is the moment-generating function (MGF) of \(\gamma_{\rm MRC}\) at the output of the MRC combiner. \(\gamma_{\rm th}\) denotes the outage SNR threshold.