# Linear Precoding Designs for MIMO VLC Using Multi-Color LEDs under Multiple Lighting Constraints
## 2 System Model  
We consider \(N_{t}\)\(N\)-color LED transmitters and \(N_{t}\) receivers (each receiver consists of \(N\)-color filters and \(N\) PDs) to form an \(NN_{t}\times NN_{r}\) MIMO VLC system. The system model is shown in Figure 1.  
At the transmitter side, the source data \(\mathbf{x}\) are an \(NN_{t}\times 1\) multi-level pulse amplitude modulation (PAM) vector with zero mean, \(\mathbf{x}=\left[\mathbf{x}_{c^{1}}^{T},\mathbf{x}_{c^{2}}^{T},\cdots,\mathbf{ x}_{c^{N}}^{T}\right]^{T}\), where the \(k\)-th color signal vector is \(\mathbf{x}_{c^{k}}=\left[x_{c^{k}_{1}},x_{c^{k}_{2}}^{T},\cdots,x_{c^{k}_{N_{t }}}\right]^{T}\). The \(M\)-level PAM signal is formed in the range of \([-\Delta,\Delta]\), \(M=2^{K}\), where \(K\) is the number of bits per transmitted symbol. We modulate the source data with precoder \(\mathbf{F}\) and then add an \(NN_{t}\times 1\) positive offset vector \(\mathbf{d}\) to each stream to guarantee the non-negativeness of the intensity of transmitted signals. Therefore, the transmitted signals can be written as:  
\[\mathbf{s}=\mathbf{F}\mathbf{x}+\mathbf{d}. \tag{1}\]  
We first consider the channel matrix in a single color band \(c\). For the \(k\)-th color, the channel gain between the \(j\)-th LED and \(i\)-th PD in the discrete-time channel matrix \(\mathbf{H}_{c^{k}}\) is determined by [24]:  
\[h_{c^{k}_{ij}}=\left\{ \begin{array}{ll}\frac{(m+1)A_{R}c^{k}}{2\pi D_{ij}^{2 }}\text{cos}(\phi)^{m}\text{cos}\left(\psi\right),&0\leq\phi\leq\Psi_{c},\\ 0,&\phi>\Psi_{c},\end{array} \right. \tag{2}\]  
where \(m\) is the order of Lambertian emission given by \(m=-\frac{\text{ln}2}{\text{ln}(\text{cos}\Phi_{1/2})}\), in which \(\Phi_{1/2}\) is the half power angle of the LED, \(A_{R}\) is the detector area of the receiver, \(\delta\) is the receiver responsibility, \(D_{ij}\) is the distance between the \(j\)-th LED chip and the \(i\)-th PD, \(\phi\) is the angle of irradiance and \(\psi\) is the angle of incidence. As has been studied in [3], the diffuse light component in an indoor scenario is much lower than the weakest line-of-sight (LOS) component received, so here, we only take the LOS components of visible light into account. Due to the overlapping spectrum of multiple colors and imperfect optical filters in receivers, the multi-color crosstalk in the overall channel matrix \(\mathbf{H}\) needs to be considered. As the overlap of the color spectrum usually occurs between two adjacent colors, for simplicity, only the crosstalk between  
Figure 1: The system model of MIMO VLC using multi-color LEDs. adjacent colors is measured in the channel matrix \(\mathbf{H}\)[20]. The overall channel matrix in the MIMO VLC system using multi-color LEDs is given by:  
\[\mathbf{H}=\left[ \begin{array}{cccc}\left(1-\eta\right)\mathbf{H}_{c^{1}}& \eta\mathbf{H}_{c^{2}}&\cdots&\mathbf{0}\\ \eta\mathbf{H}_{c^{1}}&\left(1-2\eta\right)\mathbf{H}_{c^{2}}&\cdots&\mathbf{0 }\\ \vdots&\vdots&\ddots&\eta\mathbf{H}_{c^{N}}\\ \mathbf{0}&\mathbf{0}&\eta\mathbf{H}_{c^{N-1}}&\left(1-\eta\right)\mathbf{H}_ {c^{N}}\end{array} \right], \tag{3}\]  
where \(\mathbf{H}\in\mathbb{R}^{NN_{r}\times NN_{l}}\) is the electrical-optical-electrical (EOE) channel matrix, \(\eta\) is defined as the crosstalk interference ratio and \(\eta\in\left[0,0.5\right)\). At the receiver side, the \(NN_{r}\times 1\) received signal vector is sorted by \(N\) colors written as \(\mathbf{y}=\left[\mathbf{y}_{c^{1}}^{T},\mathbf{y}_{c^{2}}^{T},\cdots, \mathbf{y}_{c^{N}}^{T}\right]^{T}\), where the \(k\)-th color received signal vector is \(\mathbf{y}_{c^{k}}=\left[y_{c^{k}_{1}}^{k}y_{c^{k}_{2}}^{k},\cdots,y_{c^{k}_{ N_{r}}}^{k}\right]^{T}\). Therefore, the received signal \(\mathbf{y}\) can be expressed as:  
\[\mathbf{y}=\mathbf{H}\mathbf{s}+\mathbf{z}=\mathbf{H}\mathbf{F}\mathbf{x}+ \mathbf{H}\mathbf{d}+\mathbf{z}. \tag{4}\]  
where \(\mathbf{z}\) is the \(NN_{r}\times 1\) additive white Gaussian noise (AWGN) vector with zero mean and noise variance of \(\sigma\), \(z_{i}^{k}\in\mathcal{N}\left(0,\sigma^{2}\right)\).