# Linear Precoding Designs for MIMO VLC Using Multi-Color LEDs under Multiple Lighting Constraints
## 3 Problem Formulation
These color tolerance regions are graphically shown in the CIE 1931 chromaticity diagram as Figure 2[25]. Based on [PERSON]'s laws of color mixture [26], the chromaticity coordinate of mixed color light \((\hat{x},\hat{y})\) can be calculated by multiple monochromatic coordinates \((x_{k},y_{k})\) as:  
\[(\hat{x},\hat{y})=\left(\frac{\mathbf{a}^{T}\boldsymbol{\gamma}}{\mathbf{b}^{T} \boldsymbol{\gamma}},\frac{\mathbf{1}^{T}\boldsymbol{\gamma}}{\mathbf{b}^{T} \boldsymbol{\gamma}}\right)=\left(\frac{\sum_{k=1}^{N}\frac{\tau_{k}}{y_{k}} \gamma_{k}}{\sum_{k=1}^{N}\frac{1}{y_{k}}\gamma_{k}},\frac{\sum_{k=1}^{N}\gamma _{k}}{\sum_{k=1}^{N}\frac{1}{y_{k}}\gamma_{k}}\right), \tag{16}\]  
where \(\mathbf{a}=\left[\frac{\tau_{1}}{y_{1}},\frac{\tau_{2}}{y_{2}},\cdots,\frac{ \tau_{N}}{y_{N}}\right]^{T}\) and \(\mathbf{b}=\left[\frac{1}{y_{1}},\frac{1}{y_{2}},\cdots,\frac{1}{y_{N}} \right]^{T}\) are coefficient vectors and \(\boldsymbol{\gamma}=\left[\gamma_{1},\gamma_{2},\cdots,\gamma_{N}\right]^{T}\) denotes the average optical power percentage vector of \(N\) colors. Thus, the average optical power percentage \(\boldsymbol{\gamma}_{j}\) of the \(j\)-th LED can be expressed as:  
\[\boldsymbol{\gamma}_{j}=\mathbf{E}_{j}\mathbf{L}/L_{t}. \tag{17}\]  
Based on Equation (11), there is a transformation relationship between \(\boldsymbol{\gamma}_{j}\) and the DC bias vector \(\mathbf{d}\) as follows:  
\[L_{t}\boldsymbol{\gamma}_{j}\circ\mathbf{c}=\mathbf{E}_{j}\mathbf{d}. \tag{18}\]  
Therefore, the constraints of the mixed white light chromaticity \((\hat{x},\hat{y})\) in a specific quadrangle can be given by:  
\[\left\{ \begin{array}{l}\hat{y}\leq k_{11}\hat{x}+k_{12},\\ \hat{y}\leq k_{21}\hat{x}+k_{22},\\ \hat{y}\geq k_{31}\hat{x}+k_{32},\\ \hat{y}\geq k_{41}\hat{x}+k_{42}.\end{array} \right. \tag{19}\]  
where \(k_{pq}\) is a coefficient constant determined by the quadrangle vertices in chromaticity diagrams, which is given in [23], \(p=1,2,3,4\), \(q=1,2\). After plugging Equation (16) into the chromaticity Constraint (19), we have:  
\[\left\{ \begin{array}{l}\mathbf{1}^{T}\boldsymbol{\gamma}_{j}\leq k_{11} \mathbf{a}^{T}\boldsymbol{\gamma}_{j}+k_{12}\mathbf{b}^{T}\boldsymbol{\gamma} _{j},\\ \mathbf{1}^{T}\boldsymbol{\gamma}_{j}\leq k_{21}\mathbf{a}^{T}\boldsymbol{ \gamma}_{j}+k_{22}\mathbf{b}^{T}\boldsymbol{\gamma}_{j},\\ \mathbf{1}^{T}\boldsymbol{\gamma}_{j}\geq k_{31}\mathbf{a}^{T}\boldsymbol{ \gamma}_{j}+k_{32}\mathbf{b}^{T}\boldsymbol{\gamma}_{j},\\ \mathbf{1}^{T}\boldsymbol{\gamma}_{j}\geq k_{41}\mathbf{a}^{T}\boldsymbol{ \gamma}_{j}+k_{42}\mathbf{b}^{T}\boldsymbol{\gamma}_{j}.\end{array} \right. \tag{20}\]  
Figure 2: Graphical representation of the chromaticity tolerance regions of LED products on the CIE 1931 chromaticity diagram. To simplify the expression of Constraint (20), we denote:  
\[\left\{ \begin{array}{l}\mathbf{g}_{1}=-k_{11}\mathbf{a}-k_{12}\mathbf{b}+\mathbf{ 1}_{N\times 1},\\ \mathbf{g}_{2}=-k_{21}\mathbf{a}-k_{22}\mathbf{b}+\mathbf{1}_{N\times 1},\\ \mathbf{g}_{3}=k_{31}\mathbf{a}+k_{32}\mathbf{b}-\mathbf{1}_{N\times 1},\\ \mathbf{g}_{4}=k_{41}\mathbf{a}+k_{42}\mathbf{b}-\mathbf{1}_{N\times 1}.\end{array} \right. \tag{21}\]  
Then, we define \(\mathbf{G}=\left[\mathbf{g}_{1},\mathbf{g}_{2},\mathbf{g}_{3},\mathbf{g}_{4} \right]^{T}\), and \(\mathbf{G}\) is an \(4\times N\) matrix. Finally, the chromaticity constraint can be expressed as:  
\[\mathbf{G}\boldsymbol{\gamma}_{j}\leq\mathbf{0},\,\forall j=1,2,\cdots,N_{t}. \tag{22}\]  
where \(\mathbf{0}\) is an \(N\times 1\) zero vector. Considering the above three lighting constraints in this MIMO VLC system using multi-color LEDs, the main problem of our paper is formulated as follows:  
\[