# Indoor MIMO-VLC Using Angle Diversity Transmitters
## 3 MIMO-VLC Using ADTs
### LOS Channel Gain Using ADT  
Figure 3 illustrates the placement of four ADTs in the ceiling by considering an indoor \(N_{r}\times N_{t}\) MIMO-VLC system with \(N_{t}=4\), where the coordinates of the four center LEDs in the four ADTs are \((2-d,2-d,z)\), \((2-d,2+d,z)\), \((2+d,2+d,z)\) and \((2+d,2-d,z)\) with unit meter. Hence, the spacing between two adjacent ADTs is \(2d\). As the coordinates of the center LED in each ADT are known, the LOS channel gain of the center LED in the \(m\)-th (\(m=1,2,\ldots,N_{t}\)) ADT to the \(i\)-th (\(i=1,2,\ldots,N_{r}\)) PD, i.e., \(h_{i,m0}\), can be calculated by using (1). However, the LOS channel gains of the side LEDs in each ADT to each PD cannot be calculated directly utilizing (1) due to the distinctive positions and orientations of the side LEDs in the same ADT. In order to calculate the LOS channel gains of the side LEDs in the ADT, the coordinates and emission angle of each side LED should be determined first. Let \((x_{m0},y_{m0},z_{m0})\) and \((x_{mn},y_{mn},z_{mn})\) denote the coordinates of positions of the center LED and the \(n\)-th (\(n=1,2,\ldots,N-1\)) side LED in the \(m\)-th ADT, respectively. Given the azimuth angle \(\omega_{n}\) and the inclination angle \(\theta\) of the \(n\)-th side LED, according to Figure 2b, the coordinates of the \(n\)-th side LED in the \(m\)-th ADT can be obtained as follows:  
\[\left\{ \begin{array}{l}x_{mn}=x_{m0}+\left[l+r(1+\cos\theta)\right]\cos \omega_{n}\\ y_{mn}=y_{m0}+\left[l+r(1+\cos\theta)\right]\sin\omega_{n}\\ z_{mn}=z_{m0}+r\sin\theta\end{array} \right.. \tag{7}\]  
Figure 4 shows the geometry for the calculation of the LOS channel gain of the side LEDs in the ADT, where the positions of the \(n\)-th side LED in the \(m\)-th ADT and the \(i\)-th PD are \((x_{mn},y_{mn},z_{mn})\) and \((\dot{x}_{i},\dot{y}_{i},\dot{z}_{i})\), respectively. Hence, the corresponding incident angle \(\beta_{i,mn}\) can be given by  
\[\cos\beta_{i,mn}=\frac{z_{mn}-\dot{z}_{i}}{\sqrt{\left(\dot{x}_{i}-x_{mn} \right)^{2}+\left(\dot{y}_{i}-y_{mn}\right)^{2}+\left(\dot{z}_{i}-z_{mn}\right) ^{2}}}. \tag{8}\]  
Figure 2: (**a**) Top view and (**b**) side view of the ADT. Only the center LED and the \(n\)-th side LED are shown for illustration. As the azimuth angle \(\omega_{n}\) and the inclination angle \(\theta\) of the side LED change, the emission angle \(\alpha_{i,mn}\) from the \(n\)-th side LED in the \(m\)-th ADT to the \(i\)-th PD changes accordingly. Let \(\mathbf{v}_{n}\) denote the normal vector of the plane of the side LED and \(\mathbf{v}_{i,mn}\) represent the vector from the side LED to the PD; hence, we have:  
\[\cos\alpha_{i,mn}=\frac{(\mathbf{v}_{i,mn},\mathbf{v}_{n})}{||\mathbf{v}_{i,mn }||\cdot||\mathbf{v}_{n}||}, \tag{9}\]  
where \(||\cdot||\) and \((\cdot,\cdot)\) denote the norm of vector and inner product of two vectors, respectively. Given the azimuth angle \(\omega_{n}\) and the inclination angle \(\theta\) of the side LED, the vector \(\mathbf{v}_{n}\) of the plane of the side LED can be calculated by  
\[\mathbf{v}_{n}=(\sin\theta\cos\omega_{n},\sin\theta\sin\omega_{n},-\cos\theta). \tag{10}\]  
Figure 4: Geometry for LOS channel gain calculation of the side LEDs in the ADT. Figure 3: Placement of four ADTs in the ceiling.