# Effective Receiver Design for MIMO Visible Light Communication with Quadrichromatic LEDs
## 2 System Model  
In this paper, a MIMO-VLC system with \(N_{t}\) QLEDs and \(N_{r}\) receivers (each receiver has four PDs and RAGB filters) are utilized to transmit data information as a \(4N_{t}\times 4N_{r}\) MIMO system shown with modulation order \(M\) in Figure 1. The transmitted symbol \(\mathbf{x}_{m}\) is a vector of size \(4N_{t}\times 1\), where \(\mathbf{x}_{m}\in\mathcal{S},m=1,2,...,M\). The constellation \(\mathcal{S}=\{\mathbf{x}_{1},...,\mathbf{x}_{M}\}\) where \(M\) is the order of modulation with the transmission rate is \(R=\log_{2}M\) bps. Moreover, vector \(\mathbf{x}_{m}\) can be expressed as \(\mathbf{x}_{m}=\left[\mathbf{x}_{m,R}^{T},\mathbf{x}_{m,A}^{T},\mathbf{x}_{ m,G}^{T},\mathbf{x}_{m,B}^{T}\right]^{T}\) that denotes the mean current of the corresponding red, amber, green, and blue LED chips of all \(N_{t}\) RAGB-LEDs for the \(m\)-th sending symbol. Moreover, the modulated signal vector of size \(N_{t}\times 1\) corresponding to each color is \(\mathbf{x}_{m,c}=\left[x_{m,c}^{1},x_{m,c^{\prime}}^{2},...,x_{m,c}^{N_{t}} \right]^{T}\), where \(c\) denotes the LED color, \(c\in\{\)R,A,G,B\(\}\). In this paper, each QLED is assumed to employ the general CSK constellation that satisfies the maximum total power [7]. The nonnegative signals also need to satisfy the maximum amplitude value constraint to avoid the clipping distortion and the chromaticity constraint [5]. Moreover, since the design of the transmit signal is not our focus, some issues such as the nonlinear, flickering, or synchronization effect are assumed to be perfect [5; 6; 7; 10]. Meanwhile, a data stream is mapped to a symbol vector chosen from the above constellation \(\mathcal{S}\). After transmitting through an optical channel, the received signal vector can be given by  
\[\mathbf{r}_{m}=\mathbf{H}\mathbf{x}_{m}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{n}\) is the noise vector with shot noise and thermal noise. The noise is assumed as the zero-mean additive white Gaussian (AWGN) noise with a covariance matrix \(\sigma^{2}\mathbf{I}_{4N_{r}}\)[3]. \(\mathbf{H}\) denotes the channel matrix between the transmitters and the receivers with optical filters. We consider herein the crosstalk between adjacent colors and the \(4N_{t}\times 4N_{r}\) mutli-color channel matrix \(\mathbf{H}\) in [5] can be expressed by two matrices as  
\[\mathbf{H} = \begin{bmatrix}\left(1-\epsilon\right)\mathbf{H}^{\mathrm{R}}& \epsilon\mathbf{H}^{\mathrm{A}}&\mathbf{0}&\mathbf{0}\\ \epsilon\mathbf{H}^{\mathrm{R}}&\left(1-2\epsilon\right)\mathbf{H}^{\mathrm{A }}&\epsilon\mathbf{H}^{\mathrm{G}}&\mathbf{0}\\ \mathbf{0}&\epsilon\mathbf{H}^{\mathrm{A}}&\left(1-2\epsilon\right)\mathbf{H }^{\mathrm{G}}&\epsilon\mathbf{H}^{\mathrm{B}}\\ \mathbf{0}&\mathbf{0}&\epsilon\mathbf{H}^{\mathrm{G}}&\left(1-\epsilon\right) \mathbf{H}^{\mathrm{B}}\end{bmatrix}  \tag{2}\] \[=\hat{\mathbf{H}}\circ\mathbf{W},\]  
Figure 1: System model. with \(4N_{t}\times 4N_{r}\) size matrices \(\mathbf{W}= \begin{bmatrix}\mathbf{1}-\epsilon&\epsilon&\mathbf{0}&\mathbf{0}\\ \epsilon&\mathbf{1}-2\epsilon&\epsilon&\mathbf{0}\\ \mathbf{0}&\epsilon&\mathbf{1}-2\epsilon&\epsilon\\ \mathbf{0}&\mathbf{0}&\epsilon&\mathbf{1}-\epsilon\\ \end{bmatrix} \), \(\hat{\mathbf{H}}= \begin{bmatrix}\mathbf{H}^{\mathrm{R}}&\mathbf{H}^{\mathrm{A}} &\mathbf{H}^{\mathrm{G}}&\mathbf{H}^{\mathrm{B}}\\ \mathbf{H}^{\mathrm{R}}&\mathbf{H}^{\mathrm{A}}&\mathbf{H}^{\mathrm{G}}& \mathbf{H}^{\mathrm{B}}\\ \mathbf{H}^{\mathrm{R}}&\mathbf{H}^{\mathrm{A}}&\mathbf{H}^{\mathrm{G}}& \mathbf{H}^{\mathrm{B}}\\ \end{bmatrix} \), where \(\epsilon\in[0,0.5)\) denotes the crosstalk interference ratio. Furthermore, the \(N_{t}\times N_{r}\) submatrix \(\mathbf{H}^{\mathrm{c}}\) denotes the channel coefficients between the \(N_{t}\) QLEDs and \(N_{r}\) receivers, which can be written as  
\[\mathbf{H}^{\mathrm{c}}= \begin{bmatrix}h_{11}^{\mathrm{c}}&\cdots&h_{1N_{t}}^{ \mathrm{c}}\\ \vdots&\ddots&\vdots\\ h_{N_{t},1}^{\mathrm{c}}&\cdots&h_{N_{t},N_{t}}^{\mathrm{c}}\\ \end{bmatrix}