Energy Efﬁcient Transceiver Design for NOMA VLC Downlinks with Finite-Alphabet Inputs

: Non-orthogonal multiple access (NOMA) is considered as a promising technique in visible light communication (VLC) systems for performance enhancement. In order to improve the system’s energy efﬁciency, we carry out transceiver design for an NOMA VLC system. We develop an optimal power allocation scheme for the system based on ﬁnite-alphabet inputs, and we propose a joint detection and decoding (JDD) algorithm for signal detection. Comprehensive simulation results demonstrate that our proposed NOMA strategy can achieve signiﬁcant performance gains over the time division multiple access (TDMA) scheme, and our designed JDD algorithm outperforms the conventional successive interference cancellation (SIC) algorithm for NOMA.


Introduction
As an adjunct or alternative to radio frequency (RF) communications and as a promising technology for 5G systems, visible light communications (VLC) has become a very important area of research [1][2][3][4][5][6][7][8] due to its advantage of low cost, ultra-high bandwidth, freedom from spectral licensing issues, etc.
Considering the realistic application, the VLC systems are generally expected to support multiple users, and multiple access strategies are essential for VLC systems. Well-known orthogonal multiple access (OMA) methods such as code division multiple access (CDMA), time division multiple access (TDMA) and orthogonal frequency division multiple access (OFDMA) have been widely used to support multiuser communications. However, since different users are allocated orthogonal resources in either the code, time or frequency domain, these methods cannot provide sufficient resource reuse. In contrast to these OMA methods, a novel multiple access strategy called non-orthogonal multiple access (NOMA), which has been proposed as a promising solution for the 5G wireless networks [9][10][11][12], allows multiple users simultaneously to utilize the entire available code, time and frequency resources by multiplexing users in the power domain using superposition coding at the transmitter and successive interference cancellation (SIC) at the receiver. In VLC systems, NOMA has recently drawn great attention as an efficient solution to the limited bandwidth of light emitting diodes (LEDs) [13][14][15]. In [13], the performance of the NOMA VLC downlink was investigated in terms of coverage probability and ergodic sum rate, and it was verified that NOMA can achieve higher system capacity than TDMA. In [14], the authors proposed a grain ratio power allocation strategy to ensure efficiency and fairness, and it was shown that the system sum rate can be enhanced by tuning the semi-angle of LEDs and the field of view (FOV) of photodiodes (PDs). An optimal power control algorithm was proposed in [15] by using the dual method to get maximized sum logarithmic user rate. However, the above studies were mainly committed to system capacity, sum rate and user fairness, ignoring the bit error rate (BER), which is an important index to evaluate the reliability of wireless communication systems. In addition, all the above studies utilized SIC at the receiver. As illustrated by Figure 1, the SIC-based receiver first detects one signal from the received composite signal by treating the other signals present in the same composite signal as noise. Then, the detected signal is subtracted from the received composite signal so as to detect the other signals. Since the SIC-based receiver treated the other signals as noise when detecting one signal from the same received composite signal,the SIC receiver may lead to error propagation and degrade system performance. Motivated by the aforementioned considerations, in this paper, we consider the energy-efficient transceiver design for NOMA VLC downlinks from a BER performance perspective. We develop an optimal power allocation strategy with finite-alphabet inputs for the transmitter and propose a joint detection and decoding (JDD) algorithm for the receiver. In addition, since TDMA is commonly used in the multi-user VLC system and it is easy to implement [16][17][18][19], in this paper, we choose TDMA for comparison. Comprehensive computer simulations demonstrate that our proposed NOMA strategy can achieve significant performance gains over the time division multiple access (TDMA) scheme, and our designed JDD algorithm outperforms the conventional SIC algorithm for NOMA.
The rest of this paper is organized as follows. Section 2 introduces the indoor NOMA VLC system model. Section 3 presents an optimal power allocation strategy for the transmitter and develops an efficient joint detection and decoding algorithm for the receiver. Section 4 compares the BER performance of our proposed NOMA and TDMA, as well as compares the BER performance of our proposed JDD and conventional SIC. Section 5 summarizes this paper.

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.
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 ∑ M i=1 p i (2 K i − 1) = 1. Thus, at the receiver side, the received signal at the m-th user can be represented as: where ζ m represents additive white Gaussian noise (AWGN) with zero mean and variance σ 2 m . 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: 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 Ψ is the field-of-view angle of the PD receiver.
In addition, φ m is the angle of emergence with respect to the transmitter axis; ψ m is the angle of incidence with respect to the receiver axis; and τ = −ln2 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: 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: In addition, the maximum likelihood (ML) detector is considered to get optimal BER performance.

Transceiver Design for NOMA VLC
In this section, to design an energy-efficient transceiver, we first develop an optimal power allocation strategy with unipolar PAMinputs by maximizing the minimum Euclidean distance (MED) of the received signals for signal transmission. Then, thanks to the nice structure of the received signal, we will propose a low complexity JDD algorithm for signal detection.

Optimal Power Allocation
This subsection aims at proposing an optimal power allocation scheme to improve the system's energy efficiency.
As we know, under high signal-to-noise ratio (SNR) conditions, the error performance of the ML detection is determined by the MED of the received signals. To optimize the performance of the ML detector, our main task is to maximize the MED between any distinct received signals under the power constraint. Hence, the corresponding problem can be stated as follows.

Problem 1.
For the given channel coefficient h i and any given positive integer K i , find power allocation coefficients p i , i = 1, 2, · · · , M, such that the MED: The optimal solution to Problem 1 is determined in the following theorem.
Theorem 1. Letp i , i = 1, 2, · · · , M denote the optimal power allocation coefficient, Then,p i andD can be determined in the following.
The proof of Theorem 1 is postponed to Appendix A.

Joint Detection and Decoding Receiver Design
Intended for the optimal power allocation scheme we have proposed in the last subsection, in this subsection, we are committed to developing an efficient demodulation and decoding algorithm.
is a 2 ∑ M j=1 K j -ary equally-spaced PAM constellation. In this context, the following low-complexity JDD algorithm can be utilized for signal detection. Algorithm 1. (Joint detection and decoding (JDD)) Given the received signals y m denoted by (2), the optimal ML estimate of the sum signal g m ∈ G m can be attained by: Then, the estimate signalŝ mi of s i at user m is given as follows.
From Algorithm 1, we can observe that our proposed JDD algorithm treats all the signals as useful information, which is different from SIC, which treats other signals as noise when detecting. In addition, we can get that the overall complexity of our proposed JDD algorithm is O(∑ i−1 j=1 K j ), and it is lower than the traditional ML-based SIC algorithm with complexity O(2 ∑ i j=1 K j ) [22].

Simulations
In this section, we evaluate the performance of our proposed transceiver design for the NOMA VLC system by comprehensive computer simulations.
As illustrated by Figure 2, we carry out the simulations in a 4.0 m × 4.0 m × 3.0 m room for an indoor environment; the transmitter LED is located at (2.0 m, 2.0 m, 3.0 m); and all the users are located at a plane with a height of 0.85 m, which is denoted by user plane. Some other key system parameters are shown in Table 1. The channel coefficient of each channel realization can be computed based on (3) and normalized by being divided by the maximum channel coefficient. The SNR is defined by 1 σ 2 for the normalized peak optical power. In addition, to make all the comparisons fair, the ML demodulator is adopted by all schemes. More details of our computer simulations are given as follows. We first compare the average BER performance of our proposed NOMA scheme and the conventionally used TDMA. The transmitted signals of the two schemes are shown in Table 2, and we consider that all the users are randomly distributed on the user plane, as illustrated in Figure 2. We randomly select 20 user groups to simulate their average BER performance, and the simulation results for two, three or four users per user group are shown in Figures 4-6, respectively. From these figures, we can get that our proposed NOMA scheme significantly outperforms the conventionally used TDMA. For example, the average gain of our proposed NOMA scheme over TDMA at the error rate of 10 −5 is about 7 dB for M = 2, K 1 = 1, K 2 = 3. The main reason for this is that the error performance is dominated by the MED of the received constellation, and the ratio of MED of our proposed NOMA and TDMA is determined by: Then, for M = 2, K 1 = 1, K 2 = 3, the obtained gain of our proposed NOMA over TDMA is given by 20 log 10 = 20 log 10 2 2×1 +2 2×3 −2 2(2 1+3 −1) ≈ 6.85 dB, which is consistent with the simulation results in Figure 4. In addition, we simulate the average BER performance of our proposed NOMA scheme with different transmitter LED position, and the simulation results are illustrated in Figure 7. From this figure, we can see that no matter where the transmitter LED is, our proposed design has a good performance over TDMA.
Furthermore, we also compare the average BER performance of our proposed JDD algorithm and the conventional SIC algorithm. As illustrated by Figure 4, our proposed JDD algorithm outperforms the SIC algorithm. For example, the obtained gain of the JDD algorithm compared to the SIC algorithm is about 5 dB for M = 2, K 1 = 2, K 2 = 3.
As illustrated in the above simulations, our proposed transceiver design for NOMA significantly outperforms SIC-based NOMA and TDMA. Using our proposed NOMA-JDD scheme, we can achieve higher rates of data transmission than using TDMA or NOMA-SIC with the same BER and SNR.

Conclusions
In this paper, we have designed the transceiver for an indoor NOMA VLC system to enhance the system's energy efficiency. For the transmitter, we have proposed an optimal power allocation scheme by maximizing the MED of the received signal, and for the receiver, we have developed a low-complexity and reliable JDD algorithm. The simulation results have demonstrated that our proposed NOMA scheme significant outperforms TDMA, and our designed JDD algorithm can achieve higher detection accuracy than the conventionally used SIC algorithm.

Conflicts of Interest:
The authors declare no conflicts of interest.

Appendix A
Proof of Theorem 1. We need to establish the following two propositions to prove Theorem 1.

Proof of Proposition A1:
Since D > 0, we can rearrange all the 2 ∑ M j=1 K j elements of G M in a strictly increasing order such that it is more clear to see that if we reduce the power of S i such that g k = kD, then min g =g,g,g∈G M |g − , and the consumed peak power of each user forG M is not larger than that for G M . In addition, it holds that p i ≥p i for 1 ≤ i ≤ M. Then, according to Proposition A1 in [23], if we let X i = {h Mpi k i : for 2 ≤ i ≤ M, i.e., h Mp1 = D and h Mpu = D × 2 ∑ i−1 j=1 K j for 2 ≤ i ≤ U. Therefore, it indeed holds that h M p 1 ≥ h Mp1 = D and h M p i ≥ h Mpi = D × 2 ∑ i−1 j=1 K j for 2 ≤ i ≤ M, and the proof of Proposition A1 is complete.
Proposition A2. For any given minimum Euclidean distance D, if h M p 1 ≥ D, h M p i ≥ D × 2 ∑ i−1 j=1 K j for 2 ≤ i ≤ M and ∑ M i=1 p i (2 K i − 1) = 1, then we have D ≤D.
Proof of Proposition A2: To prove it by contradiction, we assume that under the condition that h M p 1 ≥ D, h M p i ≥ D × 2 ∑ i−1 j=1 K j for 2 ≤ i ≤ M and ∑ M i=1 p i (2 K i − 1) = 1, there exists at least one Dsuch that D >D. Then, we can get that h M p 1 >D and h M p i >D × 2 ∑ i−1 j=1 K j for 2 ≤ i ≤ M, and thus, we have: for i = 2, 3, · · · , M. Therefore, the proof of Theorem 1 is complete.