# Linear Precoding Designs for MIMO VLC Using Multi-Color LEDs under Multiple Lighting Constraints
## 3 Problem Formulation  
With the MMSE equalizer \(\mathbf{W}\) employed in this MIMO VLC system, the mean-squared-error correlation matrix could be expressed as:  
\[\mathbf{R}_{E}=\mathbb{E}\left[\left(\mathbf{\hat{x}}-\mathbf{x}\right) \left(\mathbf{\hat{x}}-\mathbf{x}\right)^{T}\right].\]  
(1) Total power constraint  
In a MIMO VLC system using multi-color LEDs, to offer a constant white light illumination, the total power of multiple color chips in each multi-color LED needs to be equal and invariable. Here, we define \(N\times NN_{t}\) selection matrices \(\mathbf{E}_{j}=\mathbf{I}_{N}\otimes\mathbf{e}_{j}^{T}\), where \(\otimes\) denotes the Kronecker product and \(\mathbf{e}_{j}\) represents the \(j\)-th column of the identity matrix \(\mathbf{I}_{N_{t}}\), \(j=1,2,\cdots,N_{t}\). Therefore, the total power constraint can be written as:  
\[\left|\mathbf{E}_{j}\mathbf{L}\right|_{1}=L_{t},\;\forall j=1,2,\cdots,N_{t}. \tag{10}\]  
where \(\left|\cdot\right|_{1}\) denotes the one-norm and \(L_{t}\) is the total optical power of each white light LED. \(\mathbf{L}\) is the \(NN_{t}\times 1\) optical power vector of multiple color streams, \(\mathbf{L}=\left[\mathbf{L}_{\mathrm{c}_{1}}^{T},\mathbf{L}_{\mathrm{c}_{2}}^ {T},\cdots,\mathbf{L}_{\mathrm{c}_{N}}^{T}\right]^{T}\), where the optical power vector of the \(k\)-th color is \(\mathbf{L}_{\mathrm{c}^{k}}=\left[L_{\mathrm{c}_{1}^{k}},L_{\mathrm{c}_{2}^{k }},\cdots,L_{\mathrm{c}_{N_{t}}^{k}}\right]^{T}\). The transformation relationship between \(\mathbf{L}\) and the DC bias vector \(\mathbf{d}\) can be written as:  
\[\mathbf{d}=\mathbf{L}\circ\left(\mathbf{c}\otimes\mathbf{1}_{N_{t}\times 1} \right). \tag{11}\]  
where the notation \(\circ\) denotes the Hadamard product (i.e., \(\left(A\circ B\right)_{ij}=A_{ij}\times B_{ij}\)). \(\mathbf{c}=\left[c_{1},c_{2},\cdots,c_{N}\right]^{T}\) is the luminance flux-forward current conversion coefficient vector. (2) Amplitude constraint  
To guarantee the nonnegativity of transmitted intensity signal \(\mathbf{s}\), we need the minimum value of \(\mathbf{F}\mathbf{x}+\mathbf{d}\) to be larger than \(\mathbf{0}\). As the modulated PAM signal vector is limited in the range of \(\left[-\Delta,\Delta\right]\), we have:  
\[\mathbf{F}\mathbf{x}\leq\mathrm{abs}\left(\mathbf{F}\right)\Delta. \tag{12}\]  
Therefore, the nonnegative constraint can be satisfied by letting:  
\[\mathbf{d}\geq\mathrm{abs}\left(\mathbf{F}\right)\Delta, \tag{13}\]  
where \(\Delta=\left[\Delta,\Delta,\cdots,\Delta\right]_{N\times 1}^{T}\otimes \mathbf{1}_{N_{t}\times 1}\). The transmitted signal would suffer clipping distortion when the amplitude exceeds the maximum electric amplitude, so each electric stream of multi-color LEDs has a maximum forward current limitation. We have the constant coefficient vector \(\mathbf{c}\) to convert the luminous flux to the forward current. Assuming that all the chips with the same color of different LEDs keep the same photoelectric property, there is:  
\[\mathrm{abs}\left(\mathbf{F}\right)\Delta\leq\mathbf{A}_{Dm}\otimes\mathbf{1} _{N_{t}\times 1}-\mathbf{d}, \tag{14}\]  
where \(\mathbf{A}_{Dm}\) is the \(N\times 1\) maximum forward current vector, \(\mathbf{A}_{Dm}=\left[A_{Dm_{1}},A_{Dm_{2}},\cdots,A_{Dm_{N}}\right]^{T}\). Thus, combining Constraints (13) and (14), we can rewrite the amplitude constraint as:  
\[\mathrm{abs}\left(\mathbf{F}\right)\Delta\leq\min\left(\mathbf{d},\mathbf{A}_{ Dm}\otimes\mathbf{1}_{N_{t}\times 1}-\mathbf{d}\right). \tag{15}\]  
(3) Chromaticity constraint  
Chromaticity is the basic characteristic of the color perceived by human eyes. In the ANSI C78.376-2001 [22], seven-step MacAdam ellipses are used to define chromaticity tolerances of perceptible color differences for fluorescent lamps. As the chromaticity specifications in that standard have been modified to meet the needs of LED products [23], chromaticity coordinates are located in quadrangles rather than ellipses being specified to describe chromaticity tolerances approximately.