## 2 System Model  
In the multi-color LED based VLC system with intensity modulation and direct detection (IM/DD), we consider \(N\)-color LED as transmitters and \(N\) photodiodes (PDs) as receivers. Thus, the data are divided into \(N\) parts to form CMIMO channels for multiple parallel transmission, and the system model is shown in Figure 1. At the transmitter side, the source data vector can be sorted by \(\mathbf{d}=[d_{1},d_{2},\cdots,d_{N}]^{T}\). Utilizing pulse amplitude modulation (PAM), the M-level PAM source data are normalized in the range of \([-1,1]\) where \(M\) is the modulation order. Thus, the transmitted signal \(\mathbf{s}\) can be expressed as  
\[\mathbf{s}=\boldsymbol{\gamma}\circ\mathbf{d}+\mathbf{I}_{DC}, \tag{1}\]where amplifiers are utilized to fully exploit the dynamic range of LEDs with amplification coefficient denoted as \(\mathbf{\gamma}=[\gamma_{1},\gamma_{2},\cdots,\gamma_{N}]^{T}\), and \(\circ\) is the Hadamard product. To ensure the non-negativity of the input signal, direct current (DC)-bias vector \(\mathbf{I}_{DC}=[I_{DC}^{1},I_{DC}^{2},\cdots,I_{DC}^{N}]^{T}\) should be added. Since \(E[\mathbf{\gamma}\circ\mathbf{d}]=\mathbf{0}\), the average illumination level is uniquely determined by DC bias \(\mathbf{I}_{DC}\) where \(E[\cdot]\) is the expectation operator [16]. Meanwhile, the multi-color light should be guaranteed to be mixed into white light in the free space. In general, multi-color LEDs can be modeled as Lambertian emitters. According to the literature [17], the effect of reflected light can be ignored compared with the direct light where the reflected light is much weaker than the direct light. Thus, line of sight (LOS) path can be considered for the indoor VLC system. Accordingly, the channel gain \(h_{ij}\) between \(j\)th color LED chip and \(i\)th receiver PD is determined by [1]  
\[h_{ij}=\left\{ \begin{array}{cc}\frac{\mu(m+1)A}{2\pi d_{ij}^{2}}cos^{m}(\phi )T_{s}(\psi)g(\psi)cos(\psi),&0\leq\psi\leq\Psi_{C}\\ 0,&\psi>\Psi_{C}\end{array} \right. \tag{2}\]  
where \(A\) is the effective detector area of the PD, \(d_{ij}\) denotes the distance between \(j\)th color LED chip and \(i\)th PD, \(\mu\) is the receiver responsivity, \(\phi\) and \(\psi\) represent the angle of irradiance and incidence respectively. \(m\) is the order of Lambertian emission which is given by the semi-angle at half-power of the LED \(\Phi_{1/2}\) as \(m=\frac{-ln2}{ln(cos\Phi_{1/2})}\). \(T_{s}(\psi)\) is the gain of the optical filter, \(g(\psi)\) is the gain of an optical concentrator, and \(\Psi_{C}\) denotes field-of-view (FOV) for the PD. According to Ref. [7], the crosstalk problem inherently exists in the multi-color LED based VLC system for the overlapping spectra, and the spectral power distribution (SPD) \(S(\lambda)\) can be modeled as  
\[\left\{ \begin{array}{cc}S(\lambda)=\frac{g(\lambda,\lambda_{0},\Delta \lambda_{0.5})+2g^{5}(\lambda,\lambda_{0},\Delta\lambda_{0.5})}{3},\\ g(\lambda,\lambda_{0},\Delta\lambda_{0.5})=exp\{-[(\lambda-\lambda_{0})/ \Delta\lambda_{0.5}]^{2}\},\end{array} \right. \tag{3}\]  
where \(\lambda_{0}\) and \(\Delta\lambda_{0.5}\) denote the peak wavelength and half spectral width respectively. Accordingly, the spectral model for the QLED with RAGB colors at temperature of 300 K is shown in Figure 2. We can see that the overlapping spectra of the multi-color LED may cause color crosstalk problem and the realistic optical filters cannot separate the interference light completely. Meanwhile, we assume  
Figure 1: CMIMO model for the multi-color light-emitting diodes (LED) based visible light communication (VLC) system. the color crosstalk problem only takes place between the two adjacent color bands due to their close wavelength [18]. Therefore, the color crosstalk matrix \(\mathbf{T}\) is given as  
\[\mathbf{T}=\left[ \begin{array}{cccccc}1-\tau&\tau&0&\cdots&0\\ \tau&1-2\tau&\tau&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\tau&1-2\tau&\tau\\ 0&\cdots&0&\tau&1-\tau\end{array}