## 3 Antenna Selection Algorithm Based on Pearson Coefficient
For the \(k\)-th partition, the correlation coefficient between any LED combination and the fingerprint database PD can be expressed as  
\[\rho(LED_{m,k},f_{n,k})=\frac{\text{cov}(R_{m,k},S_{n,k})}{\sigma R_{m,k} \cdot\sigma S_{n,k}}=\frac{\sum\limits_{m=1}^{M_{k}}\sum\limits_{n=1}^{N_{k}} R_{m,k}\cdot S_{n,k}^{(m)}-\sum\limits_{n=1}^{N_{k}}R_{m,k}\cdot\overline{S}_{n,k}}{ \sqrt{\sum\limits_{m=1}^{M_{k}}\sum\limits_{n=1}^{N_{k}}\left(R_{m,k}\right)^ {2}-R_{m,k}}^{2}\cdot\sqrt{\sum\limits_{m=1}^{M_{k}}\sum\limits_{n=1}^{N_{k}} \left(\overline{S}_{n,k}\right)^{2}-\sum\limits_{n=1}^{N_{k}}\left(\overline{S }_{n,k}\right)^{2}}} \tag{11}\]where \(\overline{S}_{n,k}\) denotes the average illuminance of the LED combination received by the \(n\)-th PD in the \(k\)-th partition fingerprint database, and \(m\) LEDs with high similarity are used as the transmitting antenna combination. Taking 4 m \(\times\) 4 m \(\times\) 3 m scene as the spatial model, the specific light source layout is shown in Figure 3. The coordinates of LED lights are the following: (0.5, 0.5, 3), (0.5, 1, 3), (0.5, 1, 3), (1, 0.5, 3), (1, 1, 3), (1, 1.5, 3), (2.5, 0.5, 3), (2.5, 1, 3), (2.5, 1, 3), (3, 0.5, 3), (3, 1, 3), (3, 1.5, 3), (0.5, 2.5, 3), (0.5, 3, 3), (0.5, 3, 3), (0.5, 3, 3), (0.5, 3, 3), (1, 2.5, 3), (1, 3, 3), (1, 3, 5, 3), (2.5, 2.5, 3), (2.5, 3), (2.5, 3, 3), (2.5, 3, 3), (2.5, 3, 3), (2.5, 3, 3), (3, 2.5, 3), (3, 2.5, 3), (3, 3, 3), (3, 3.5, 3). According to the VLC communication, link model, it is assumed that there are six LEDs in the \(k\)-th partition fingerprint library, and their central luminous intensity is 21 (lx); then, according to Equation (10), the LED information in the first area can be expressed as:  
\[LED_{m,k}=\left\{ \begin{array}{l}\{21,(0.5,0.5,3)\}\ \{21,(0.5,1,3)\}\ \{21,(0.5,1.5,3)\}\ \{21,(0.5,1.5,3)\}\\ \{21,(1,0.5,3)\}\ \{21,(1,1,3)\}\ \{21,(1,1.5,3)\}\end{array} \right\} \tag{12}\]  
Using the visible-light communication model, the received illuminance of each PD can be obtained. Assuming that there are two PDs, and the coordinates are (0.8, 0.8, 0) and (0.8, 1.6, 0), there are 12 possibilities for receiving illuminance. According to Equation (9), the partition fingerprint database is expressed as  
\[f_{n,k}=\left\{ \begin{array}{l}\left\{\begin{array}{l}6.87_{1}^{(1)},(0.8, 0.8,0)\end{array}\right\}\left\{\begin{array}{l}7.01_{1}^{(2)},(0.8,0.8,0) \end{array}\right\}\left\{\begin{array}{l}5.94_{1}^{(3)},(0.8,0.8,0) \end{array}\right\}\\ \{7.42_{1}^{(4)},(0.8,0.8,0)\}\left\{\begin{array}{l}7.14_{1}^{(5)},(0.8,0.8,0 )\end{array}\right\}\left\{\begin{array}{l}6.04_{1}^{(6)},(0.8,0.8,0) \end{array}\right\}\\ \{4.66_{2}^{(1)},(0.8,1.6,0)\}\left\{\begin{array}{l}6.21_{2}^{(2)},(0.8,1.6,0)\end{array}\right\}\left\{\begin{array}{l}7.09_{2}^{(3)},(0.8,1.6,0) \end{array}\right\}\\ \{4.74_{2}^{(4)},(0.8,1.6,0)\}\left\{\begin{array}{l}6.33_{2}^{(5)},(0.8,1.6,0)\end{array}\right\}\{7.23_{2}^{(6)},(0.8,1.6,0)\}\end{array} \right\} \tag{13}\]  
where \(\overline{S}_{n,k}\) represents the average illumination value at \(t_{n}\) = 3. Because six LEDs and two PD receivers are known, in the OGSMP-MIMO system, i.e., the transmitting antenna \(N_{t}\) = 6 and the receiving antenna \(N_{r}\) = 2, assuming that the transmitting antenna \(N_{a}\) = 2 is activated, there are a total of 15 antenna combinations in OGSMP-MIMO; however, the actual antenna combination of OGSMP-MIMO must satisfy the integer power of 2. Therefore, the actual antenna combination is selecting 8 of the 15 types. According to Equations (11)-(13), the set of Pearson coefficients can be obtained as  
\[\rho=\left\{ \begin{array}{l}\rho_{1,2}=0.9852,\rho_{1,3}=0.9842,\rho_{1,4}= 0.9721,\rho_{1,5}=0.9845,\rho_{1,6}=0.9835\\ \rho_{2,3}=0.9962,\rho_{2,4}=0.9831,\rho_{2,5}=0.9975,\rho_{2,6}=0.9961,\rho_{ 3,4}=0.9818\\ \rho_{3,5}=0.9961,\rho_{3,6}=0.9947,\rho_{4,5}=0.9828,\rho_{4,6}=0.9815,\rho_{ 5,6}=0.9961\end{array} \right\} \tag{14}\]  
Figure 3: The layout of light sources in 4 m \(\times\) 4 m \(\times\) 3 m scene.