## 2 Signal and System Models  
The system is deployed in a room with dimensions 5 \(\times\)5 \(\times\)5 m\({}^{3}\) (width, length and height). All the walls are white, have a constant reflection coefficient throughout and are modeled as a first order Lambertian source. There are sixteen dimmable light emitting arrays (LEAs) as transmitters located on the ceiling. Each LEA is a uniform circular array (UCA) consisting of \(M\) LED bulbs. A UCA geometry is used due to its symmetrical configuration which enables the phased array antenna to scan azimuthally with minimal changes in its beam width and sidelobe levels. The array factor (_AF_) at a far-field of \(\left(\theta_{0},\phi_{0}\right)\) is given by:  
\[AF\left(\theta,\varphi_{m}\right)=\sum_{m=1}^{M}a_{m}\exp\left(jlr\sin\theta \cos\left(\varphi_{0}-\varphi_{m}\right)\right) \tag{1}\]  
where \(a_{m}\) and \(\phi_{m}\) are the amplitude excitation and the azimuth angle of the \(m^{\mathrm{th}}\) element, and \(l=2\pi/\lambda\). A far-field assumes that the target is farther from the source than this nominal separation, and the source can be modeled or measured as an emitting point. The \"5 times\" rule of thumb is valid for a source with circular shape and Lambertian emission. This rule states that for a distance of five times the source diameter, the error from using the inverse-square law is 1%. Therefore, the far-field condition can be assumed for this application. The reader can be assumed to be located in the far field of LEDs [15]. The zenith angle is represented by \(\theta\) as shown in Figure 1. The UCA is designed optimally to produce zero-order Bessel-like or Bessel-Gauss patterns. A Bessel beam is also named as nondiffracting beam [16,17]. In addition to the typical rectangular grid shape, the LEAs' placement is also deployed in an hexagonal lattice alignment as proposed in [18], both are illustrated in Figures 2 and 3. The LED reader, e.g., photodiode or image sensor, which can be plugged into a smartphone, is held by a tracking person at the height of 1.20 m, _i.e._, 1.80 m below the ceiling. Each dimmable LED bulb can be not only remotely controlled with an app on the smartphone, but it can also be linked to keypads, motion sensors or door sensors, to ensure that the lights are used when people are in the room.  
Figure 1: Propagation model of optical wireless channels.  
Figure 2: LEAs’ placement on the ceiling in a typical rectangular grid.  
For simplicity, the discrete form for our system is used. At time instant \(k\), the system can be modeled as a linear time-invariant system having the impulse response \(h[k]\) together with additive white Gaussian noise (AWGN) \(W[k]\) representing the noise at the optical receiver, e.g., for avalanche photo diode receiver \(W[k]\) including both shot noise and thermal noise with zero mean and variance \(\sigma_{\mathrm{w}}^{{}^{2}}\) as:  
\[P_{\mathrm{R}}[k]=F\sum_{i=1}^{M}\bigl{(}P_{\mathrm{T}_{i}}[k]w_{i}*h_{i}[k] \bigr{)}+W[k] \tag{2}\]  
where \(P_{\mathrm{T}_{i}}[k]\) and \(P_{\mathrm{R}}[k]\) are the instantaneous transmitted optical power on the \(i^{\mathrm{th}}\) LED in an LEA and the instantaneous optical power or the intensity on the photodetector, respectively. Before each LED in an LEA emits the optical signal, the RF signal is multiplied with a weight vector \(\mathrm{w}=[w_{0}\), \(w_{I}\),..., \(w_{m}]\) through a dimming control using pulse width modulation (PWM) as explained in the next subsection. The photodetector responsivity or optical/electrical conversion efficiency is denoted by \(F\). The symbol * denotes a convolution integral.