## 2 System Model with Channel Crosstalk  
For the indoor VLC system with intensity modulation and direct detection (IM/DD), a multi-color LED with \(N\) chips is employed as transmitters and \(N\) photodiodes (PDs) are employed as receivers. Thus, the information bits can be divided into \(N\) parts where each LED chip transmits the optical signals on one color. According to [14], the optical power can be regarded as the combination of two parts where the direct current (DC) power corresponding to the average intensity is responsible for illumination and the alternating current (AC) power is used to provide the variation for communication. In a sense, the source of the DC power is the DC bias, and the AC power is the modulation signal for the VLC system. Thus, the optical power of the transmitted signal \(\mathbf{P}\) can be expressed as:  
\[\mathbf{P}=\mathbf{p}+\mathbf{P}, \tag{1}\]where \(\mathbf{\hat{P}}=[\hat{P}_{1},\hat{P}_{2},\cdots,\hat{P}_{N}]^{T}\) denotes the average transmitted optical power of the multi-color LED and \(\mathbf{p}=[p_{1},p_{2},\cdots,p_{N}]^{T}\) denotes the instantaneous AC power of the modulated optical signal utilizing pulse amplitude modulation (PAM). To exploit the dynamic range of LEDs fully, the M-level PAM signal can be applied, which is normalized in the range of \([-\Delta_{i},\Delta_{i}]\), where \(\Delta_{i}\) is a nonnegative value with \(i\in\{1,2,\cdots,N\}\). Typically, the line-of-sight (LOS) path can be assumed for the indoor VLC system where the receivers remain still and the reflected light is much weaker than the direct light [16]. A multi-color LED can be modeled as a Lambertian emitter, where the optical channel gain \(h_{ij}\) between the \(i^{\text{th}}\) color LED chip and \(j^{\text{th}}\) PD can be expressed as [1]:  
\[h_{ij}=\left\{ \begin{array}{cc}\frac{A_{r}(m_{0}+1)}{2\pi d_{ij}^{2}}cos^{m_ {0}}(\phi)cos(\psi),&0\leq\psi\leq\Psi_{C}\\ 0,&\psi>\Psi_{C}\end{array} \right. \tag{2}\]  
where \(\phi\) is the angle of irradiance and \(\psi\) is the angle of incidence. \(m_{0}\) is defined as the order of the Lambertian emission with \(m_{0}=\frac{-ln2}{ln(cos\Phi_{1/2})}\) where \(\Phi_{1/2}\) is the semi-angle at half-power of the LED. \(\Psi_{C}\) denotes the field-of-view (FOV) of the PD. \(d_{ij}\) depicts the distance between the \(i^{\text{th}}\) color LED chip, and \(j^{\text{th}}\) PD. \(A_{r}\) denotes the effective detector area of the PD. Accordingly, the optical channel model is shown in Figure 1.  
Based on the analysis of [7], the spectral power distribution (SPD) for the multi-color LED \(S(\lambda)\) with a peak wavelength \(\lambda_{0}\) and half spectral width \(\Delta\lambda_{0.5}\) is modeled as:  
\[S(\lambda)=\frac{g(\lambda,\lambda_{0},\Delta\lambda_{0.5})+2g^{5}(\lambda, \lambda_{0},\Delta\lambda_{0.5})}{3}, \tag{3}\]  
where \(g(\lambda,\lambda_{0},\Delta\lambda_{0.5})=exp\{-[(\lambda-\lambda_{0})/ \Delta\lambda_{0.5}]^{2}\}\). The spectral model of a typical red, amber, green, blue LED (RAGB-LED) is shown in Figure 2 at temperature of 300 K, where the optical channel crosstalk will be generated due to the overlapping spectra of the multi-color LED and the imperfection of the optical filters. Obviously, the received optical signal contains the desired signal and interference signal through color filters for the practical multi-color LED-based VLC system. Figure 3 shows the  
Figure 1: Optical channel model. VLC system model considering the channel crosstalk problem, which implies that the conventional WDM-VLC system model is inappropriate. It can be seen that only the two adjacent color bands have the crosstalk problem due to their close wavelength. Thus, the color crosstalk matrix **T** can be described as [17]:  
\[\textbf{T}=\left[ \begin{array}{ccccc}1-\epsilon&\epsilon&0&\cdots&0\\ \epsilon&1-2\epsilon&\epsilon&\ddots&\vdots\\ 0&\ddots&\ddots&\ddots&0\\ \vdots&\ddots&\epsilon&1-2\epsilon&\epsilon\\ 0&\cdots&0&\epsilon&1-\epsilon\end{array} \right]_{N\times N} \tag{4}\]  
where \(e\in[0,0.5)\) denotes the crosstalk coefficient.