## 3 Proposed System
### System Model  
The MIMO-VLC system [22] is generally comprised of \(M_{t}\) transmitting LEDs and \(M_{r}\) receiving PDs for communication. The amount of transmitting LED and receiving PDs are the same during the communication (i.e., \(M_{t}=M_{r}=M\)). Therefore, the \(M_{t}\) independent data streams are transmitted simultaneously to the respective PDs. In the transmitter side, the time domain signal is acquired by using the Inverse Fast Fourier Transform (IFFT), parallel to serial conversion, and Cyclic Prefix (CP) insertion. The transmitted signals are mapped by using the QAM with the modulation order 64. Subsequently, the BBD precoding matrix is added to transmitter data as well as the digital to analog converter used in the transmitter to convert the digital signals into analog signals. Similarly, the analog to digital converter is used for converting the analog signals into digital signals. In this MIMO-VLC system, the transmitted data is recovered through the removal of CP, serial to parallel conversion, Fast Fourier Transform (FFT) and QAM demodulation. The received signal at the PD is specified in Equation (1).  
\[z=\textbf{H}y+\textbf{D}\textbf{C}+m \tag{1}\]  
where, the received and input signal vectors through the indoor MIMO-VLC communication system are \(z\) and \(y\), respectively. The DC bias matrix is represented as \(\textbf{D}\textbf{C}\) which is added in the transmitted signal to keep the signal as positive. The noise vector is represented as \(m\) and the MIMO channel is represented as the square matrix \(\textbf{H}\). The square matrix is represented in the following Equation (2).  
\[\textbf{H}=\left[\begin{array}{cccc}h_{11_{4}}&h_{12_{2}}&\dots&h_{1M_{j}} \\ h_{21_{7}}&h_{22_{d}}&\dots&h_{2M_{j}}\\ \vdots&\vdots&\vdots&\vdots\\ h_{M1_{j}}&h_{M2_{j}}&\dots&h_{MM_{j}}\\ \end{array}\right] \tag{2}\]  
where, the desired and interference channel paths are represented as \(h_{jk_{d}}\) and \(h_{jk_{i}}\), respectively. The collected signal in \(j\)th is given in Equation (3).  
\[z_{j}=\gamma Ph_{j}^{t}y+m_{j} \tag{3}\]  
where the detector responsivity is denoted as \(\gamma\); the transmitting power of LED is \(P\); the \(j\)th row in \(H\) is \(h_{j}^{t}\); \(y\) represents the transmitted signal and the receiver noise is denoted as \(m_{j}\). Equation (4) depicts the receiver noise that occurred while transmitting the data through the MIMO-VLC transmitter.  
\[\sigma_{j}^{2}=\sigma_{sh_{j}}^{2}+\sigma_{th_{j}}^{2} \tag{4}\]  
where, the thermal and shot noise are represented as \(\sigma_{th_{j}}^{2}\) and \(\sigma_{sh_{j}}^{2}\), respectively. Equation (5) defines the variance of the shot noise.  
\[\sigma_{sh_{j}}^{2}=2 qB\left(\gamma P\sum_{k=1}^{M}h_{jk}y_{k}+l_{b}l_{2}\right) \tag{5}\]  
where, the electronic charge is represented as \(q\); bandwidth is \(B\); the background current is \(l_{b}\) and the bandwidth factor of noise is \(l_{2}\). Additionally, the thermal noise is caused in the receiver circuitry of transimpedance and the thermal noise is depicted in Equation (6).  
\[\sigma_{th_{j}}^{2}=\frac{8\pi kT}{G}\eta A_{j}l_{2}B^{2}+\frac{16\pi^{2}kT}{ g}\eta^{2}A_{j}{}^{2}I_{3}B^{3} \tag{6}\]  
where, Boltzmann's constant is specified as \(k\); the absolute temperature is \(T\); the gain of open-loop voltage is represented as \(G\); the area of PD is denoted as \(A_{j}\); the PD's fixed capacitance is specified as \(\eta\)the channel noise factor of the Field Effect Transistor (FET) is \(\Gamma\); the transconductance of the FET is \(g\) and the weighting function is \(I_{3}=0.0868\), which is based on the optical pulse shape of the input.  
By considering the aforesaid measures, a parallel signal-to-interference-plus-noise ratio (SINR) in \(j\)th PD is expressed as in Equation (7).  
\[\text{SINR}_{j}=\frac{\left(\gamma Ph_{jj}\right)^{2}}{\left(\gamma^{2}P^{2} \sum_{k=1,k\
eq j}^{M}h_{jk}^{2}\right)+\sigma_{j}^{2}} \tag{7}\]