# Unified Performance Analysis of MIMO Mixed RF/FSO Relaying System
## 2 System and Channel Model  
The block diagram of the system is shown in Figure 1. The channel from source to relay link is Rayleigh distributed and the FSO link is characterized by Gamma-Gamma (IT) turbulence. The system uses DF relay to convert electrical signals to optical signals. For the RF link, it consists of \(N_{t}\) transmitting antennas and \(N_{r}\) antennas for reception. Binary phase shift keying (BPSK) modulation is adopted in the RF link. The received signal vector can be expressed as \(\mathbf{Y}=\mathbf{H}\mathbf{x}+\mathbf{n}\). where \(\mathbf{H}= \begin{bmatrix}h_{11}&\cdots&h_{1N_{t}}\\ \vdots&\ddots&\vdots\\ h_{N_{t}1}&\cdots&h_{N_{t}N_{t}}\end{bmatrix} \) is the channel coefficients matrix, which consists of \(N_{t}\times N_{r}\) statistically independent elements. \(\mathbf{n}\) is the noise vector, which is assumed to be Additive White Gaussian Noise (AWGN). This link is assumed to be characterized by Rayleigh fading. \(\mathbf{x}=c\mathbf{v}^{\mathbf{T}}=c[p_{1}\cdots v_{N_{t}}]^{\mathbf{T}}\) is the transmitted signal vector. \(c\) is the transmitted symbol and \(\mathbf{v}\) is the transmit weighting vector. The source to relay MIMO RF link adopts the maximal ratio transmission (MRT) [11]. At the relay, the corresponding CDF of \(\gamma_{RF}\) is  
\[F_{\gamma_{RF}}(\gamma)=1-\exp\left(-\frac{\gamma}{\bar{\gamma}_{RF}}\right) \sum\limits_{u=0}^{N_{t}N_{t}-1}\frac{1}{u!}\left(\frac{\gamma}{\bar{\gamma}_{ RF}}\right)^{u}. \tag{1}\]  
We adopt a DF relaying based system. The relay decodes the received RF signal and modulates the signal into an optical signal. For the FSO link, we assumed that the relay is equipped M photo aperture transmitters and the destination is equipped with N photodetectors. The transmitter adopts repetition coding (RC), and an equal gain combining (EGC) technique is employed at the receiver. The relay node uses the subcarrier intensity modulation (SIM) technique for converting the input signal into the optical signal [12]. A DC bias is added to the optical signal by the laser diode to avoid the negative valued signal. At the receiver, the DC bias is filtered out. We assume that the distance of photodetectors is greater than the coherence distance to ensure that the received signals are uncorrelated. First, we consider the case where the pointing error is not negligible. The received signal at the \(j=1,\cdots,N\) th antenna can be expressed as:  
\[y_{j}=\eta\sum\limits_{i=1}^{M}{I_{i,j}x+n_{j}} \tag{2}\]  
where \(\eta\) denotes the optical-to-electrical conversion efficiency, \(x\) denotes the transmitted signal and \(n_{j}\) is the Additive White Gaussian Noise (AWGN) with zero mean and variance \(\sigma^{2}\). Let \(\mathbb{E}\left[\left|x\right|^{2}\right]=E_{x}\), using the equal gain combining (EGC) technique, the total received SNR can be expressed as:  
\[\gamma=\frac{\eta^{2}E_{x}\left(\sum\limits_{i=1}^{M}{\sum\limits_{j=1}^{N}{I_ {i,j}}}\right)^{2}}{M^{2}N^{2}\sigma^{2}}=\frac{\bar{\gamma}\left(\sum\limits_ {i=1}^{M}{\sum\limits_{j=1}^{N}{I_{i,j}}}\right)^{2}}{M^{2}N^{2}}=\frac{\bar{ \gamma}}{M^{2}N^{2}}{I_{\mathrm{T}}^{2}} \tag{3}\]  
where, the average SNR is defined as \(\bar{\gamma}=\frac{\bar{\gamma}^{2}E_{x}}{\sigma^{2}}\), and \({I_{\mathrm{T}}}=\sum\limits_{i=1}^{M}{\sum\limits_{j=1}^{N}{I_{i,j}}}\). For the pointing errors, the independent identical Gaussian distributions for the elevation and the horizontal displacement (swap) are considered. The PDF of the pointing  
Figure 1: Block diagram of the MIMO mixed RF/FSO relaying system. error existing on the i-th transmitting antenna received by the j-th receiving antenna is given by:  
\[f\left(h_{i,j}^{p}\right)=\frac{\phi^{2}}{A_{0}^{\phi^{2}}}\left(h_{i,j}^{p} \right)^{\phi^{2}-1},0\leq h_{i,j}^{p}\leq A_{0} \tag{4}\]  
where \(\phi\)=\(\frac{\omega_{zeq}}{2\sigma_{z}}\) is the ratio between the equivalent beam radius \(\omega_{zeq}\) at the receiver and the pointing error displacement standard deviation \(\sigma_{z}\) (jitter) at the receiver.