# Extended Spatial-Index LED Modulation for Optical MIMO-OFDM Wireless Communication
## 2 System Model
### Indoor Optical Wireless Channel Model  
In a typical VLC system, the optical signals emitted from the LEDs are conveyed over an optical MIMO channel. Because of some negative attributes of the VLC channels (i.e., reflection, diffusion, interference, movement, and noise), the channel causes loss, fading, and distortion of the transmitted signal [35]. This loss must be as small as possible since it affects the transmission rate directly. Therefore, it is important to understand the response of different VLC channels well. Evaluating the channel impulse response (CIR) of the VLC systems is a difficult job as its received power is associated with the optical channel parameters [31].  
Since the visible light signal at any point consists of a line of sight (LOS) signal as well as a contribution from number of reflections by walls or objects in the room (i.e., Non-LOS), to evaluate the total gain of indoor VLC channel, the VLC channel model in [36] was exploited in [31; 32] as it is focused on the limited number of reflections (i.e., up to 2 nd order).  
In general, the channel gains of optical link between certain LED and the PD can be expressed as [28; 31]  
\[h_{r,t}=\frac{(m+1)A_{r}R_{p}}{2\pi d_{r,t}^{2}}cos^{m}(\phi_{r,t})cos(\psi_{r,t}),\qquad 0\leq\psi_{r,t}\leq\Psi_{1/2}, \tag{19}\]  
where \(m=-ln(2)/ln(cos(\Phi_{1/2}))\) is the order of Lambertian emission expressing directivity of the source beam, \(\Phi_{1/2}\) is the semi-angle of the LED, \(A_{r}\) is the PD area, \(R_{p}\) is the responsivity of PD and it always takes as unity, \(\phi_{r,t}\), \(\psi_{r,t}\) are the angles of irradiance and incidence, respectively, \(d_{r,t}\) is the distance between the LED\({}_{t}\), and the PD\({}_{r}\) and \(\Psi_{1/2}\) is the semi-angle of PD (i.e., field of view (FOV)).  
Moreover, the LED maximum transmitted optical power can be expressed as  
\[P_{t}=\lim_{0\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}X(t)dt, \tag{20}\]  
where \(X(t)\) is the instantaneous transmitted optical power.  
Thus, the average received optical power \(P_{r}\) at the PDs can be estimated by  
\[P_{r}=h_{r,t}P_{t}. \tag{21}\]  
The channel model (19) was limited to the assumption of a Lambertian source and did not consider the effect of reflections. Thus, the physical channel modeling approach in [37] has been introduced, whereas the VLC channels were modeled based on the ray-tracing based approaches that take into account precisely effects of propagation environment, light sources, and detectors (i.e., objects, FOV, etc.).  
However, the same setup configurations that were used in [31] are considered in this paper, where a \(4\times 4\) MIMO-VLC scheme was used for ease of demonstration as shown in Figure 2. The physical VLC channel parameters of the simulation environment are summarized in Table 3, which contains the coordinates of the luminaries and PDs for three different configurations as shown in Figure 2. In configuration (config) A, the PDs are located in the center of the table with 0.1 m distance between each, while, in configuration B, the PDs are located in the corner of the table with 0.8 m distance between each. Both luminaries and PDs are assumed in a vertical position in aforesaid configurations; in contrast, in configuration C, the PDs are located in the center of the table with distance 0.1 m between each other, but rotated by \(45^{\circ}\) in the \(xy\)-plane and titled by \(60^{\circ}\) in the \(yz\)-plane, where it can increase the LOS link strength. These parameters are used to obtain the CIR between each transmitter and receiver