## 2 FSO Channel Model
### Pointing Error Model  
Compared to MISO based FSO systems, the MIMO based FSO systems are assumed to have positioning sensors on each receiver side. The detected misalignments from each receiver are gathered and averaged to minimize the increased alignment error due to multiple transmissions. For \(N\) receivers to detect the misalignment with \(M\) transmitted beams, the transceivers are arranged to form a regular polygon with an equal distance between transceivers \(d\) to achieve the independent channel. We assumed that the transceivers are fixed in the same device with equivalent instantaneous pointing jitter, and the transmitted beams are directed to the center of the receiver arrangement. The detected misalignment for the \(n\)-th receiver is equivalent to the detected misalignment at the center of the receiver distribution with the translation by displacement of the receiver. The centroid of received multiple beams can be rewritten as the combination of the centroid \(\Delta x_{mn}\) and the received power \(I^{mn}\) of the \(m\)-th transmitted beam at the \(n\)-th receiver as described in [18]. Then, the aggregated horizontal centroid for the \(n\)-th receiver in MIMO transmissions can be expressed as  
\[\Delta x_{ac,n}=x_{n}+\frac{\sum_{m=1}^{M}\Delta x_{mn}\times I^{mn}}{\sum_{m =1}^{M}I^{mn}}, \tag{4}\]  
where \(\Delta x_{ac,n}\) and \(x_{n}\) are the aggregated centroid and displacement of the \(n\)-th receiver. The averaged centroid for \(N\) receivers is  
\[\Delta x_{ac}=\frac{1}{N}\sum_{n=1}^{N}\Delta x_{ac,n}=\frac{1}{N}\sum_{n=1}^ {N}x_{n}+\frac{1}{N}\sum_{n=1}^{N}\left(\frac{\sum_{m=1}^{M}\Delta x_{mn} \times I^{mn}}{\sum_{m=1}^{M}I^{mn}}\right). \tag{5}\]  
The receivers are also distributed with equal distance \(d\) to achieve the independent channel as the transmitter distribution. Thus, the sum of the receiver positions becomes the sum of the sines and cosines, which equals zero.  
Note that the atmospheric channels for all diversity channels are independent of the distance between the transmitters or receivers and larger than the coherence length. Therefore, the detected misalignments for the \(N\) receivers are also independent. The sum of the log-normal random variable of received powers could be approximated to a single log-normal random variable using [PERSON]'s method [20]. Then, the averaged centroid in Equation (4) can be approximated as  
\[\Delta x_{ac}=\frac{1}{N}\sum_{n=1}^{N}\Delta x_{ac,n}=\frac{1}{N}\sum_{n=1}^ {N}S, \tag{6}\]  
where \(S\) is the Gaussian distributed statistical misalignment model with a zero mean and the variance \(\sigma_{c,n}^{2}\) of aggregated centroid \(\Delta x_{ac,n}\) as described in our previous work [18]. Since the sum of the Gaussian random variables is also the Gaussian random variable and the detected centroids for each receiver are independent, the mean of the averaged centroid is zero, and the variance of the overall detected misalignment error can be expressed as  
\[\sigma_{c}^{2}=\sum_{n=1}^{N}\frac{\sigma_{c,n}^{2}}{N^{2}}\approx\frac{ \left(e^{\sigma_{Z}^{2}}-1\right)d^{2}}{8\left(e^{\sigma_{Z}^{2}}-1+M\right)N \sin^{2}(\pi/M)}. \tag{7}\]  
Note that the distance between the transmitter and the center of the transmitter arrangement is \(d/(2\sin(\pi/M))\). The detected misalignment error is independent of the pointing jitter \(\sigma_{s}^{2}\). Thus, the variance of the combined pointing error with increased misalignment and pointing jitter becomes \(\sigma_{c}^{2}+\sigma_{s}^{2}\), and the pointing error model follows the Rayleigh distribution.