# Optimization of Bit Allocation for Spatial Multiplexing in MIMO VLC System with Smartphones
## 2 System Model
### VLC Modulation  
The transmitter is designed in a smartphone, as shown in Figure 2. Unlike the optical components used in traditional VLC systems, such as LEDs, PDs, etc., the smart device consists of an LED array display as the transmitter and a camera as the receiver. These optical components can be assembled as many transceivers due to their use of high-resolution devices. Consequently, the massive MIMO can be realized with these optical components, while the issue of ICI can be overcome.  
SMP is one of the spatial diversity schemes for MIMO techniques with the same numbers of transmitters and receivers, such that both \(N_{\rm t}\) and \(N_{\rm r}\) are \(N\). Each pair of transmitter and receiver can provide one channel for a data stream, as shown in Figure 3. In this case, four data streams are simultaneously launched. In [19], the spectrum benefits of SMP and SM with the same modulation order \(M\) can be obtained by (1) and (2), respectively.  
Figure 1: Diagram of MIMO VLC system.  
Figure 2: The paradigm of four transmitters in a \(4\times 4\) MIMO VLC system.  
With a fixed number of channels and pulse-amplitude modulation (PAM) levels, SMP provides higher spectral efficiency compared to SM.  
\[N\times log_{2}(M) \tag{1}\]  
\[log_{2}(N)+log_{2}(M) \tag{2}\]  
An optical channel with IM/DD is assumed. The signal \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{N}]\) is transmitted in the channel and the received data are formulated as:  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{W} \tag{3}\]  
where \(\mathbf{H}\) is the channel matrix, and \(\mathbf{W}\) is the sum of thermal noise and ambient shot light noise, which is independent of the transmitted signal, as indicated in [20]. Therefore, the entries of \(\mathbf{W}\) are composed of a zero-mean additive white Gaussian noise (AWGN) in real numbers with the following variance:  
\[\sigma_{\mathbf{W}}^{2}=_{\mathit{ent}}^{2}+\sigma_{\mathit{shot}}^{2} \tag{4}\]  
where \(\sigma_{\mathit{thermal}}^{2}\) and \(\sigma_{\mathit{shot}}^{2}\) are variances of thermal and shot noises, respectively. The channel matrix \(\mathbf{H}\) can be expressed as:  
\[\mathbf{H}=\begin{bmatrix}h_{1,1}&\cdots&h_{1N}\\ \vdots&\ddots&\vdots\\ h_{N,1}&\cdots&h_{N,N}\end{bmatrix}\ldots \tag{5}\]  
where \(\mathrm{h}_{i,j}\) is the wireless channel transmission coefficient between the \(j\)-th transmitter and the \(i\)-th receiver. MIMO systems usually use the system model defined by (6) to indicate the channel gain. Light of sight (LOS) paths can be modeled with a generalized Lambertian radiant intensity [21]. In general, the energy of the reflected light can be omitted compared with the LOS light, and the system performance mainly depends on the optical signals in LOS [22]. The gain \(h_{i,j}\) in channel matrix \(\mathbf{H}\) is:  
\[h_{i,j}=\begin{cases}\frac{(k+1)A}{2\pi d^{2}}\cos^{k}(\varphi)\cos(\psi)&0 \leq\varphi\leq\psi_{\frac{1}{2}}\\ 0&\psi>\psi_{\frac{1}{2}}\end{cases} \tag{6}\]  
As shown in the Figure 4, \(\varphi\) is the angle of emergence to the emitter axis, and \(\psi\) is the angle of incidence to the receiver axis. The parameter \(k\) can be calculated with logarithmic terms as:  
\[\frac{-ln(2)}{ln(\cos(\Phi_{1/2}))} \tag{7}\]  
Figure 3: The paradigm of SMP with \(N=4\) and 2-PAM (pulse-amplitude modulation).  
\(\Phi_{1/2}\) is the half angle of the transmitter, \(\Psi_{1/2}\) is the half angle of receiver's field of view (FOV), \(A\) is the detection area of the receiver, and the channel coefficient \(h_{i,j}\) depends on the position of the \(j\)-th transmitter and the \(i\)-th receiver. If the receiver and transmitter are not within the other's coverage angle, then the gain is 0. Eventually, the channel matrix \(\mathbf{H}\) considered in this paper can be measured in a physical environment. In the SMP system, the \(M\)-ary PAM with grayscale mapping is employed, and each intensity can be expressed in (8). The \(M\)-level PAM with grayscale intensity equation is:  
\[I_{m}=\frac{I}{M}m,m=1,\ldots M \tag{8}\]  
where \(I\) is the maximum grayscale in the channel.