## 3 System Model  
This section introduces the theoretical basis of the photon-counting MIMO system; the system model of each technology will be further explained below. As presented in Figure 1, the considered photon-counting MIMO downlink system has \(N_{t}\) light-emitting diodes (LEDs) in the base station (BS) that transmit \(N_{s}\) data streams collaboratively to the \(N_{r}\) photo detector (PD). Assume that the number of data streams and PDs is identical (\(N_{s}=N_{r}\)) and that the data streams are uncorrelated. The precoded signal is \(\mathbf{W}\)s, where \(\boldsymbol{s}=\left[s_{1},\cdots,s_{N_{t}}\right]^{T}\) represents the data streams, and \(\mathbf{W}\in\mathbb{R}^{N_{t}\times N_{s}}\) is the precoding matrix. To ensure the non-negativity of the signals, a direct current (DC) bias \(\boldsymbol{b}=\left[b_{1},\cdots,b_{N_{t}}\right]^{T}\) is appended to the transmit symbols through bias-tree circuits. Therefore, the input signal of the LED is \(\boldsymbol{x}=\mathbf{W}\boldsymbol{s}+\boldsymbol{b}\), driving the LEDs to transmit the signal through the FSO turbulent fading channels. Figure 1: System block diagram of the UV NLOS MIMO. Free-space light transmits data to the PD through an optical medium via LEDs. FSO systems encounter several challenges during transmition, such as shadows when the lighting location is blocked from an area, which leads to information loss. The channel irradiance between the \(i\)-th PD and the \(j\)-th LED is denoted by \(g_{i,j}\), which obeys the gamma-gamma distribution with the PDF:  
\[\Pr\big{(}g_{i,j}\big{)}=\frac{2(\alpha\beta)^{\frac{(\alpha+\beta)}{2}}}{ \gamma(\alpha)\gamma(\beta)}g_{i,j}^{\frac{(\alpha+\beta)}{2}-1}K_{\alpha- \beta}\Big{(}2\sqrt{\alpha\beta g_{i,j}}\Big{)},\ g_{i,j}>0, \tag{1}\]  
where \(\gamma(\cdot)\) is the standard gamma function, and \(\alpha\), \(\beta\) are the distribution shaping parameters satisfying:  
\[\alpha=\Bigg{[}exp\Bigg{(}\frac{0.49\chi^{2}}{\big{(}1+0.18d^{2}+0.56\chi^{12/5 }\big{)}^{7/6}}\bigg{)}-1\Bigg{]}^{-1}, \tag{2}\]  
\[\beta=\Bigg{[}exp\Bigg{(}\frac{0.51\chi^{2}\Big{(}1+0.69\chi^{12/5}\Big{)}^{-5 /6}}{\big{(}1+0.9d^{2}+0.62d^{2}\chi^{12/5}\big{)}^{5/6}}\bigg{)}-1\Bigg{]}^{- 1}, \tag{3}\]  
where \(d=\big{(}kD^{2}/4L\big{)}^{1/2}\), \(\chi^{2}=0.5C_{n}^{2}k^{7/6}L^{11/6}\), \(k=2\pi/\lambda\) is the light wave factor; \(\lambda\) represents the wavelength; \(D\) is the diameter of the aperture of the condenser at the receiver; \(L\) is the distance of the link; and \(C_{n}^{2}\) represents the index value of the refractive structure parameter and is a function of altitude \(h\). The \(i\)-th PD's channel matrix is \(g_{i}=[g_{1,1},\cdots,g_{1,N_{i}}]\); hence, the channel matrix between the user and the BS is denoted by  
\[\mathbf{G}=\Big{[}g_{1}^{T},\cdots,g_{N_{r}}^{T}\Big{]}^{T}. \tag{4}\]  
The photon-counting process model is adopted at the receiver. The random variable corresponding to the number of photons detected by the \(j\)-th PD, denoted by \(z_{j}\), follows the Poisson distribution with the PDF:  
\[\Pr\big{(}z_{j}|\mathbf{s}\big{)}=\frac{\Big{[}g_{j}(\mathbf{Ws}+\mathbf{b})+n _{b}\Big{]}^{z_{j}}}{z_{j}!}e^{-\big{[}g_{j}(\mathbf{Ws}+\mathbf{b})+n_{b} \big{]}}. \tag{5}\]  
In (5), \(n_{b}=\eta\frac{P_{b}T}{h\eta}\) is the number of noise photons generated by background radiation, where \(\eta\) is the PD efficiency, \(P_{b}\) is the background radiation power, \(T\) is the duration of a symbol, \(h\) is [PERSON]'s constant, and \(f\) is the center frequency. After removing the DC bias, the received signal follows:  
\[\Pr\big{(}y_{j}=r_{j}|\mathbf{s}\big{)}=\frac{\Big{(}g_{j}\mathbf{Ws}+n_{b} \Big{)}^{r_{j}}}{r_{j}!}e^{-\big{(}g_{j}\mathbf{Ws}+n_{b}\big{)}}. \tag{6}\]  
The received signal from the transmitter is typically subject to various forms of interference and distortion during propagating through the channel. This results in a degraded signal at the receiver, which requires further promotion in order to extract useful information.