## 2 System Model  
A UWOC-MIMO system with \(M\) laser diodes (LDs) at the transmitter side and \(N\) photodetectors at the receiver side was employed with rectangular mQAM and Gray coding, as shown in Figure 1. At the transmitter side, the input data bits are first modulated by an electrical mQAM modulator, which drives the LDs. The light wave propagates through the oceanic turbulence channel with pointing errors to the receiver side. At the receiver side, the avalanche photodiode (APD) converts an optical signal into an electrical signal. Finally, the output signal is obtained after the maximal-ratio combining (MRC) and the mQAM demodulator. For the sake of generality, we employed the following assumptions.  
1. The UWOC turbulence channel is memoryless, stationary, and ergodic, as well as independent and identically distributed. Its channel state information (CSI) is known by the transmitters and receivers;
2. Since the optical communication link length is much longer than the optical wavelength, the systemignoresspatial correlations. Moreover, thenoise at received branches is independent and uncorrelated.  
It was assumed that \(h(t)\) is a stationary random process for the system's existing pointing errors and optical signal fading caused by ocean turbulence over the period entailed by each QAM symbol. Thus, the output signal of the APD receiver can be obtained by [18]:  
\[r(t)=\Re\overline{g}P_{\mathrm{s}}m_{0}h_{\mathrm{c}}h_{\mathrm{l}}h(t)s(t)+n (t), \tag{1}\]  
where \(\Re\) is the APD's responsivity, \(\overline{g}\) is its average gain, \(P_{\mathrm{S}}\) is the transmitted optical power per symbol, and \(m_{0}\) is the modulation index indicating the number of bits per transmitted QAM symbol. \(s(t)\) is the output signal of the electrical QAM modulator, \(h_{\mathrm{c}}\) is the channel attenuation, \(h_{\mathrm{l}}\) is the geometric loss, and \(n(t)\) is the receiver noise.  
In the UWOC system, the effect of the optical signal scintillation on APD shot noise leads to uncertainty in the shot noise variance, and the thermal noise in the high-sensitivity APD is very low. The ocean turbulence links with pointing errors can be modeled as a slow-fading channel. It is thus possible to consider the samples for the channel fading process at a given time \(h=h(t=t_{0})\). As a result, the APD shot noise present can be largely taken into account and modeled as zero-mean Gaussian white noise with the variance conditioned on \(h\); i.e., \(\sigma_{\mathrm{sh}}^{2}=2q\overline{g}^{2}\mathrm{F_{A}}\Re h_{\mathrm{c}}h_ {\mathrm{l}}h_{\mathrm{l}}h_{\mathrm{S}}m_{0}B\)[18], where \(q\) denotes the electron charge, \(\mathrm{F_{A}}\) is the excess noise factor of the APD, and \(B\) indicates the symbol's effective noise bandwidth. Therefore, the received instantaneous signal-to-noise ratio (SNR) contributed by the sub-channel between the \(m\)-th light beam and \(n\)-th photodetector in this UWOC-MIMO system can be expressed as  
\[\gamma_{mn}=\frac{\Re P_{\mathrm{s}}m_{0}h_{\mathrm{c}}h_{\mathrm{l}}}{2 MN_{ \mathrm{f}}\mathrm{F_{A}}\mathrm{B}}h_{mn}=\overline{\gamma}h_{mn},\ m=1,\ 2,\ \ldots,\ M,\ n=1,\ 2,\ \ldots,\ N \tag{2}\]  
where \(\overline{\gamma}=\Re P_{\mathrm{s}}m_{0}h_{\mathrm{c}}h_{\mathrm{l}}/\left( 2 MN_{\mathrm{f}}\mathrm{F_{A}}\mathrm{B}\right)\) is the average SNR and \(h_{mn}\) represents the instantaneous fading value from the \(m\)-th light beam to \(n\)-th photodetector. The output SNR of the UWOC-MIMO system with MRC can be indicated as follows [19]:  
\[\gamma_{\mathrm{MRC}}=\sum_{n=1}^{N}\sum_{m=1}^{M}\gamma_{mm}=\overline{\gamma }\sum_{n=1}^{N}\sum_{m=1}^{M}h_{mm}. \tag{3}\]  
Figure 1: Schematic diagram of the UWOC-MIMO system.