# Channel and Bit Adaptive Power Control Strategy for Uplink NOMA VLC Systems
## 2 System Model  
As shown in Figure 1, in this paper, we consider a multiple access VLC system with \(N\) LED transmitters and one PD receiver, which is equivalent to an uplink NOMA VLC system. The \(N\) LED transmitters transmit their messages simultaneously to the PD receiver by modulating the optical intensity according to the transmitted data and the PD receiver performs direct detection to extract each LED's signal. Since the transmitted signals in realistic communication systems are drawn from finite-alphabet constellations, as well as the fact the transmitted signals in VLC systems must be non-negative and real-valued, in this paper, we consider unipolar pulse amplitude modulation (UPAM) for each LED's transmitted signal. We consider that the information-carrying signal from \(i\)-th LED to PD denoted by \(s_{i}\) is randomly, equally-likely and independently chosen from a \(2^{K_{i}}\)-ary UPAM constellation, i.e., \(s_{i}\in\mathcal{S}_{i}=\{k\}_{k=0}^{k=2^{K_{i}}-1}\), where \(K_{i}\) is the modulation order of UPAM. Then, let \(\gamma_{i}\) denote the power control factor (\(0<\gamma_{i}\leqslant 1\)) and thus, the received signal \(y\) at the PD receiver can be described as  
Figure 1: Multiple access system model. \[y=\sum_{i=1}^{N}h_{i}\gamma_{i}s_{i}+\zeta \tag{1}\]  
The receiver noise \(\zeta\) is mainly due to the shot noise and the thermal noise in VLC. The shot noise is mainly caused by the ambient light and the main source of ambient noise are sunlight and artificial light such as that of incandescent and fluorescent lamps [22; 23]. Note that in the case where the blue light is used at the receiver for signal detection by narrow spectral filtering, the influence of ambient light is considerably reduced [24]. If the ambient light is negligible, then the dominant noise source is the receiver preamplifier thermal noise, which is signal independent and Gaussian [1; 24]. Thus, in this paper, the noise \(\zeta\) is approximately modeled as additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma^{2}\). The parameter \(h_{i}\) denotes the channel coefficient between the PD receiver and the \(i\)-th LED transmitter. For VLC channels, the power of non-line-of-sight signals is much weaker than that of the line-of-sight (LOS) signals [25; 26]. Thus, usually only LOS links are considered in VLC systems [16; 17; 20; 26]. Then, \(h_{i}\) is determined by [1; 25]  
\[h_{i}=\left\{ \begin{array}{ll}\frac{(m+1)A}{2\pi d_{i}^{2}}\cos^{m}\phi_{i}T _{s}(\psi_{i})g(\psi_{i})\cos\psi_{i},&0<\psi_{i}<\Psi,\\ 0,&\psi_{i}>\Psi.\end{array} \right. \tag{2}\]  
where \(A\) is the effective detector area of PD, \(d_{i}\) depicts the distance between the PD receiver and the \(i\)-th LED transmitter, \(T_{s}(\psi_{i})\) is the gain of the optical filter, and \(\Psi\) denotes the field-of-view angle of the PD receiver. In addition, \(\phi_{i}\) is the angle of emergence, \(\psi_{i}\) is the angle of incidence, \(m=\frac{-\ln 2}{\ln(\cos\Phi_{1}^{2})}\) with \(\Phi_{\frac{1}{2}}\) being defined the LED transmitter's half-power angle, and \(g(\psi_{i})\) represents the gain of the optical concentrator given by  
\[g(\psi_{i})=\left\{ \begin{array}{ll}\frac{n^{2}}{\sin^{2}\Psi},&0<\psi_{i}< \Psi,\\ 0,&\psi_{i}>\Psi.\end{array} \right. \tag{3}\]  
with \(n\) being the corresponding refractive index. In this paper, we assume that the PD receiver receives all the users' signals at the same time and treat all the signals as useful signal components instead of noises. In addition, due to the relatively slow movement of people and fixed objects within a room, the channel can effectively be considered as time invariant and the channel state information can be easily obtained by training symbols and sent back to the transmitter periodically through uplink [24; 27; 28]. Therefore, in this paper, we ignore the influence of the asymmetry of the optical wireless links [29; 30; 31; 32] and assume that the channel coefficients are fully available at the LED transmitters and the PD receiver.