# Energy Efficient Transceiver Design for NOMA VLC Downlinks with Finite-Alphabet Inputs
## 2 System Model  
As shown in Figure 2, we consider a downlink indoor NOMA VLC system with one LED and \(M\) users, where the transmitter LED is located on the ceiling and the \(M\) users are distributed in the same receiving plane. The LED serves \(M\) users simultaneously by modulating the optical intensity according to the transmitted data. In addition, all users are equipped with a single photodiode (PD) that performs direct detection to extract the useful signal. As an initial research, we consider that the transmitted signals of this NOMA VLC system are drawn from unipolar pulse amplitude modulation (PAM) constellations, motivated by the fact that the actual transmitted signals in practical digital communication systems are drawn from finite-alphabet sets and the requirement that only nonnegative real-valued signals are allowed in VLC systems.  
Unlike the conventional NOMA, which is based on SIC receiver, our proposed NOMA scheme is based on the joint detection receiver. The concept of our proposed NOMA VLC is illustrated by Figure 3, where the unipolar signals for different users are superposed in the power domain at the transmitter side and the joint detection scheme is carried out at the receiver side. According to the this NOMA VLC principle, the transmitted signals of the \(M\) users, which are superposed in the power domain, are given as follows.  
Figure 1: Illustration of the conventional NOMA principle.  
\[x=\sum_{i=1}^{M}p_{i}s_{i} \tag{1}\]  
where \(s_{i}\in\mathcal{S}_{i}=\left\{m\right\}_{m=0}^{m=2\xi_{i}-1}\) is the signal intended for the \(i\)-th user and \(p_{i}\) denotes the corresponding power allocation coefficient. In addition, due to the normalized total power constraint, the power allocation coefficients should satisfy \(\sum_{i=1}^{M}p_{i}(2^{K_{i}}-1)=1\). Thus, at the receiver side, the received signal at the \(m\)-th user can be represented as:  
\[y_{m}=h_{m}\sum_{i=1}^{M}p_{i}s_{i}+\zeta_{m} \tag{2}\]  
where \(\zeta_{m}\) represents additive white Gaussian noise (AWGN) with zero mean and variance \(\sigma_{m}^{2}\). \(h_{m}\) denotes the channel coefficient between the LED transmitter and \(m\)-th user. Since the power of the non-line-of-sight (NLOS) signals is much weaker than that of the line-of-sight (LOS) signal for VLC channels, we assume that LOS links between the transmitter and the receivers [20,21]. Then, \(h_{m}\) is determined by:  
\[h_{m}=\left\{\begin{array}{ll}\frac{(\tau+1)A}{2\pi d_{m}^{2}}\cos^{\tau}( \phi_{m})\cos(\psi_{m}),&0<\psi_{m}<\Psi,\\ 0,&\psi_{m}>\Psi.\end{array}\right. \tag{3}\]  
where \(d_{m}\) depicts the distance between the LED transmitter and the \(m\)-th user's photodiode (PD) receiver, \(A\) denotes the effective PD detector area and \(\Psi\) is the field-of-view angle of the PD receiver. In addition, \(\phi_{m}\) is the angle of emergence with respect to the transmitter axis; \(\psi_{m}\) is the angle of incidence with respect to the receiver axis; and \(\tau=\frac{-ln2}{ln(\cos\Phi_{\frac{1}{2}})}\) with \(\Phi_{\frac{1}{2}}\) defining the half-power angle of the LED transmitter.  
Due to the low mobility of the VLC users, the channel coefficients can be easily obtained by channel estimation techniques. Thus, in this paper, we assume that the channel coefficients are available at both the transmitter and the receivers. Without loss of generality, we assume the channel coefficients have the following order:  
\[h_{1}\geq h_{2}\geq\cdots\geq h_{m}\geq\cdots\geq h_{M} \tag{4}\]  
According to the principle of NOMA that the users with worse channel conditions are allocated more power and the users with better channel conditions are allocated less power, we can get:  
\[p_{1}\leq p_{2}\leq\cdots\leq p_{m}\leq\cdots\leq p_{M} \tag{5}\]  
In addition, the maximum likelihood (ML) detector is considered to get optimal BER performance.  
Figure 2: Downlink NOMA system model.