# Indoor MIMO-VLC Using Angle Diversity Transmitters
## 2 System Model  
In this section, we consider a general indoor MIMO-VLC system model, and its geometric setup is illustrated in Figure 1. An ADT array consisting of \(N_{t}\) ADTs are installed on the ceiling and used as transmitters to transmit data. The PD array composed of \(N_{r}\) PDs is located at the user plane. Usually, the light received by the PD consists of two parts: one is the LOS component, and the other is the non-line-of-sight (NLOS) component. Previous studies have shown that the power of the LOS component is usually much larger than that of the NLOS component [9]. Therefore, we only consider the influence of LOS component on the MIMO-VLC system performance in the following.  
Figure 1: Geometric setup of a general MIMO-VLC system.  
Assuming that each LED follows the Lambertian radiation pattern, the LOS channel gain from LED to PD can be expressed as follows [25]:  
\[h=\frac{(\mu+1)S_{d}}{2\pi L^{2}}\text{cos}^{\mu}(\alpha)\delta\tau cos(\beta), \tag{1}\]  
where \(\mu=-\text{ln}2/\text{ln}(\text{cos}(\Psi))\) is the Lambertian emission order and \(\Psi\) is the semi-angle at half power of the LED; \(S_{d}\) is the active area of the PD; \(L\) is the distance between LED and PD; the emission angle and the incident angle are expressed by \(\alpha\) and \(\beta\), respectively; \(\delta\) is the gain of optical filter; and \(\tau\) is the gain of lens. Note that \(h\) will be zero when the incident angle is larger than the FOV of the PD or the emission angle is larger than the semi-angle of the LED.  
For a typical MIMO-VLC system equipped with \(N_{t}\) LED and \(N_{r}\) PDs, letting \(\mathbf{x}=[x_{1},x_{2},\cdots,x_{N_{t}}]^{T}\) denote the transmitted signal vector, \(\mathbf{H}\) represent the \(N_{r}\times N_{t}\) MIMO channel matrix and \(\mathbf{w}=[w_{1},w_{2},\cdots,w_{N_{t}}]^{T}\) be the additive noise vector, the received signal vector \(\mathbf{y}=[y_{1},y_{2},\cdots,y_{N_{r}}]^{T}\) can be expressed by  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{w}, \tag{2}\]  
where the channel matrix of \(N_{r}\times N_{t}\) MIMO-VLC system can be given by  
\[\mathbf{H}=\left[\begin{array}{ccc}h_{11}&\cdots&h_{1N_{t}}\\ \vdots&\ddots&\vdots\\ h_{N_{t},1}&\cdots&h_{N_{t}N_{t}}\end{array}\right], \tag{3}\]  
where \(h_{lim}\) (\(i=1,2,...,N_{r};m=1,2,...,N_{t}\)) denotes the channel gain between the \(i\)-th PD and the \(m\)-th LED.  
It can be seen from (2) that the output signal from each PD is a function of the input signal \(x_{m}\) and the additive noise \(w_{i}\). In order to recover the transmitted signal at the receiver side, zero-forcing (ZF) based MIMO demultiplexing is adopted because of its simplicity and low computational complexity. Assuming \(\mathbf{H}\) is full-column-rank (\(N_{r}\geq N_{t}\)), the transmitted signal vector \(\mathbf{x}\) can be obtained by  
\[\mathbf{\hat{x}}=\mathbf{\tilde{H}}\mathbf{y}=\mathbf{x}+\mathbf{\tilde{H}} \mathbf{w}, \tag{4}\]  
where \(\mathbf{\tilde{H}}\) denotes the pseudo inverse of \(\mathbf{H}\), which can be calculated by  
\[\mathbf{\tilde{H}}=(\mathbf{H}^{*}\mathbf{H})^{-1}\mathbf{H}^{*}, \tag{5}\]  
where \(\mathbf{H}^{*}\) denotes the conjugated transpose of \(\mathbf{H}\).  
The additive noise includes thermal noise and shot noise [25], which can be modeled as a real-valued additive white Gaussian noise (AWGN). The noise power can be calculated by \(P_{w}=N_{0}B\), where \(N_{0}\) and \(B\) denote the noise power spectral density (PSD) and the signal bandwidth, respectively.