Nonorthogonal Multiple Access for Visible Light Communication IoT Networks

In this study, we investigated the nonorthogonal multiple access (NOMA) for visible light communication (VLC) Internet of Things (IoT) networks and provided a promising system design for 5G and beyond 5G applications. Specifically, we studied the capacity region of a practical uplink NOMA for multiple IoT devices with discrete and continuous inputs, respectively. For discrete inputs, we proposed an entropy approximation method to approach the channel capacity and obtain the discrete inner and outer bounds. For the continuous inputs, we derived the inner and outer bounds in closed forms. Based on these results, we further investigated the optimal receiver beamforming design for the multiple access channel (MAC) of VLC IoT networks to maximize the minimum uplink rate under receiver power constraints. By exploiting the structure of the achievable rate expressions, we showed that the optimal beamformers are the generalized eigenvectors corresponding to the largest generalized eigenvalues. Numerical results show the tightness of the proposed capacity regions and the superiority of the proposed beamformers for VLC IoT networks.


Introduction
As the wireless data traffic exponentially increased in 5G, traditional radio frequency-(RF-) based Internet of Things (IoT) network suffers from a limited data rate and network capacity due to the shortage of RF spectra and massive IoT devices. With its vast unlicensed bandwidth, visible light communication (VLC) is a promising complementary solution to meet the growing wireless traffic demands for IoT networks [1,2]. By exploiting the widespread deployment of the light-emitting diodes (LEDs) as transmitters, VLC has attracted an increasing interest due to its dual functionality: communication and illumination [3][4][5]. Besides a wider spectrum, VLC has other inherent advantages such as high spatial reuse, high energy efficiency, no electromagnetic radiation, and inherent security [6,7].
Thus far, traditional IoT networks have generally utilized orthogonal multiple access (OMA) techniques such as frequency division multiple access (FDMA) and time division multiple access (TDMA). In OMA, the resources are allocated orthogonally to multiple users, and it cannot provide sufficient resource reuse. In contrast, the nonorthogonal multiple access (NOMA) technique exploits the power domain for multiple access and is able to serve multiple users at the same time frequency-code resource [8][9][10], which has recently been included into the 3GPP long-term evolution advanced standard [11][12][13] and is widely recognized as a promising candidate for the MAC scheme in 5G-enabled IoT applications.
Recently, uplink NOMA has received significant research attention [14][15][16][17][18][19][20]. Based on the theory of the Poisson cluster process, the authors in [14] have provided a framework to analyze the rate coverage probability. In [15], the optimal user pairing was investigated for various uplink NOMA scenarios. In [16], the joint subchannel assignment and power allocation problem were investigated. In [17], an interference balance power control scheme was derived. By using stochastic geometry, a signal alignment-based framework was developed in [18] for both multiple-input, multiple-output-(MIMO-) NOMA downlink and uplink transmissions. In [19], a theoretical framework was proposed to analyze the outage probability and the average achievable rate in NOMA downlink and uplink multicell wireless systems. In [20], a phase predistorted joint detection method was proposed to reduce the bit error ratio (BER) for uplink NOMA in VLC systems. Most of the aforementioned research works focused on the RF uplink NOMA [14][15][16][17][18][19], while the VLC uplink NOMA is not well discussed [20]. Until now, the achievable rate expression of VLC uplink NOMA is still unknown, which makes it great difficult to undertake the optimal NOMA beamforming design for VLC IoT networks.
Different from the RF communications, VLC generally adopts intensity modulation and direct detection (IM/DD), where the messages are modulated to the intensity of the signals. Therefore, the transmitted VLC signals are real and nonnegative, which differ from the RF complex-valued signals. Additionally, due to the eye safety standards and physical limitations, both the peak and average amplitudes of VLC signals are restricted. Hence, the classic Shannon capacity formula with Gaussian input [21] cannot quantify the capacity of VLC IoT networks.
This study is aimed at providing a solution to the abovementioned issues in the area of VLC IoT networks. First, we investigated the capacity region of MAC in VLC IoT networks. Then, we further studied the optimal beamforming design in a practical NOMA uplink. The main contributions of this study are summarized as follows: (i) Due to the peak optical power constraint, the optimal input is discrete [22]. Thus, we supposed that the input follows a discrete distribution and develops both the inner and outer bounds of the capacity region of uplink NOMA in VLC IoT networks. Specifically, finding the capacity region was formulated as an entropy maximization problem which is a mixed discrete optimization problem. To overcome the challenge, we proposed an entropy maximization approximation method and obtained the capacity bounds (ii) Based on the continuous inputs, a closed-form expression for the achievable rate of uplink NOMA of VLC IoT networks is presented. Specifically, with the continuous inputs, the channel capacity of uplink NOMA in VLC IoT networks can be approximated as a differential entropy maximization problem. The corresponding optimal continuous distributions were ABG distributions, and we obtained both the inner and outer bounds in closed forms. To the best of our knowledge, the proposed inner and outer bounds are the first theoretical bounds of the channel capacity region for uplink NOMA of VLC IoT networks (iii) Finally, based on the obtained results of NOMA, we further studied the optimal receiver beamforming design for VLC IoT. Specifically, we first extended the ABG inner bound to a single-input, multiple-output (SIMO) uplink NOMA case and then maximized the minimum uplink rate of multiple users under receiver power constraints. By exploiting the structure of the achievable rate expression, we equivalently reformulated this problem as a generalized eigenvalue maximization problem, and the optimal beamformers are the generalized eigenvectors corresponding to the largest generalized eigenvalues The rest of this paper is organized as follows. In Section 2, the capacity regions of the discrete and continuous distribution for uplink NOMA of VLC IoT networks are presented. In Section 3, the achievable rate of multi-LED and optimal beamforming design derived for uplink NOMA of VLC IoT networks is described. In Section 4, the simulation results of the capacity regions and optimal beamforming design in NOMA VLC IoT networks are presented. Finally, the conclusions are presented in Section 5.

Capacity Region of Uplink NOMA for VLC IOT Networks
As shown in Figure 1, N single-LED users (IoT devices) simultaneously transmit its own information to a single-PD base station (BS) over the same channel. Let s i be the message of the ith user, where js i j ≤ A i , Efs i g = 0, and Efs 2 i g = ε i . The transmitted signal of the ith user is given by where p i is the transmit power of the ith user and b i is the direct current (DC) bias of the ith user. To ensure that the transmitted signal is nonnegative, the DC bias needs to sat- As the received signal power is dominated by the power from the line-of-sight (LOS) link [23,24], the diffuse link can be neglected. Thus, the channel gain between user i and the BS is given by [25] where m is the Lambertian index of the LED, which depends on the semiangle ϕ 1/2 by m = − log 2/ðlog ðcos ðϕ 1/2 ÞÞÞ; d i denotes the distance between user i and the receiver; ϕ is the angle of irradiance; ψ i is the angle of incidence; ψ FOV is the field of vision (FOV) semiangle of the receiver; and A R denotes the effective area of the PD. The received signal at the BS is given by where z ∼ N ð0, σ 2 Þ represents the sum of contributions from the shot noise and the thermal noise [26,27]. At the BS, the 2 Wireless Communications and Mobile Computing multiple received signals may cause interference to each other. To mitigate the interference, the BS applies SIC to decode and remove the partial interference. Without the loss of generality, we assume that the terms The BS adopts the SIC technique to decode the received signals in a descending order [28][29][30][31], i.e., from s 1 to s N . Specifically, when the BS decodes s i , it first decodes the signal intended for user s k with the order k ≤ i and then subtracts it from y.
Thus far, the capacity region of uplink NOMA for VLC IoT networks has been an open problem, which is a major barrier for signal processing in VLC IoT networks. To overcome the challenge, we derived both the inner and outer bounds of the channel capacity region for uplink NOMA of VLC IoT networks.
2.1. Capacity Region with Discrete Inputs. As in the previous section, we assumed that the signal s i is a discrete random variable with M i real values fa i,m g 1≤m≤M i . Specifically, the signal s i satisfies where a i,m denotes the mth point for signal s i and p i,m denotes the corresponding probability.
Based on (5), the discrete inner bound uplink NOMA of VLC can be obtained by maximizing the entropy hðŷ i Þ, i.e., As the noiseẑ k,i follows the Gaussian distribution with zero mean and σ 2 /K variance, the probability density function (PDF) f Y ðyÞ is given by Thus, the entropy hðŷ i Þ maximization problem is given by Problem (8) is a mixed discrete and nonconvex problem that is generally difficult to solve.
To handle Problem (8), we first defined some vectors as follows: Based on the above mentioned definitions in (9), we where ⊙ denotes the Hadamard product. Note that given both M i and a i , Problem (10), is convex with respect to p i , which can be solved efficiently using the available interiorpoint algorithms [32,33]. Without the loss of generality, we assumed the space among the M i points fa i,m g M i m=1 is equally placed in the range Note that when M i is larger than the optimal values M * i , redundant points exist in fa i,m g M i m=1 . However, the effects of the redundant points can be eliminated by optimizing the PDF p i . Thus, for a sufficiently large M i , the maximum entropy hðŷ i Þ can be approximated by solving Problem (10) under condition (11). In summary, the proposed entropy hðŷ i Þ approximation method is listed in Algorithm 1.
Let h * ðŷ i Þ denote the entropy hðŷ i Þ computed by Algorithm 1. Substituting h * ðŷ i Þ to, we obtained the discrete inner bound of NOMA VLC as Let ℛ inner dis denote the achievable rate region of NOMA VLC IoT networks bounded by (12), which is given by 2.1.2. Outer Bound with Discrete Inputs. In this subsection, the outer bound of the channel capacity region of the NOMA VLC IoT network with discrete input is proposed. The upper bound of the achievable rate R i is given by where the inequality (14a) follows the EPI [21] and hðxÞ ≤ 1 /2 log 2πe var ðxÞ and h * ðŷ i Þ is calculated by Algorithm ℛ outer dis denotes the channel capacity region of NOMA VLC IoT networks bounded, which is given by 2.2. Capacity Region with Continuous Inputs. Although the discrete inner and outer bounds are obtained, they are not in closed-forms, which is the main obstacle in determining the capacity region. To this end, we assumed that the input signal s i follows a continuous distribution and derived the ABG inner bound of NOMA VLC IoT networks in closedform expressions.

ABG Inner
Bound. Let f ðs i Þ denote the pdf of s i which satisfies the following peak optical power (Equation (16a)), average optical power (Equation (16b)), and electrical power constraints (Equation (16c)), i.e., Then, for 1 ≤ i ≤ N, the lower bound of the achievable rate, R i , is given by = max where the inequality (17c) is true due to the EPI [21] and hðQÞ ≤ 1/2 log 2πe var ðQÞ for a random variable with variance, var ðQÞ. The equality (17d) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy and is given by where the parameters α i , β i , and γ i are the solutions of the following equations: For k = N, the upper bound R k is given by Let ℛ inner con denote the achievable rate region of NOMA VLC, which is given by where the inequality (23b) follows the EPI [21] and hðQÞ ≤ 1/2 log 2πe var ðxÞ and the equality (23c) holds because s i follows the ABG distribution [34]. Let ℛ outer con denote the channel capacity region of NOMA VLC IoT networks bounded by (23c), which is given by

Optimal Beamforming Design for Uplink NOMA of VLC IOT Networks
In this section, we further considered a single-input, multiple-output (SIMO) uplink NOMA for a VLC IoT network as illustrated in Figure 2, which includes N single Wireless Communications and Mobile Computing LED users (IoT devices) and a L PDs BS. Let s i denote the transmitted message from user i, where the definition of s i is similar to that in the previous SISO scenario. Thus, the received signal at BS can be expressed as where g i ∈ ℝ L×1 denotes the channel vector between user i and BS, p i ε i ≥ 0 is the transmitted power of user i, and z ∼ N ð0, σ 2 IÞ represents the additive white Gaussian noise vector.
For message s i , the BS invokes a linear receive beamformer w i ∈ ℝ L to the received signals y as follows: Without the loss of generality, we assumed that the terms fkg T i ffiffiffiffiffiffiffi p i ε i p kg N i=1 satisfy a descending order, i.e., kg T Then, the BS adopts the SIC technique to decode the received signals in a descending order [35], i.e., from s 1 to s N . Specifically, let R i denote the achievable rate of decoding message s i , 1 ≤ i ≤ N.
When 1 ≤ i < N, R i is given by = max ≥ max where the inequality (27d) holds due to the EPI [21] and hðQÞ ≤ ð1/2Þ log 2πe var ðQÞ for a random variable with variance, var ðQÞ. The equality (27e) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy. For k = N, R N is given by Therefore, for 1 ≤ i ≤ N, the expression of the lower bound R i can be expressed as where Γ i is an indicator function as follows: Based on the explicit achievable rate expression in (29), we investigated the optimal receiver beamformers' design to maximize the minimum achievable rates which satisfies the

Numerical Results
In this section, the performance of the capacity region and the optimal beamforming design for uplink NOMA in VLC IoT networks are evaluated using numerical results.
In the following, we present the performance of the discrete inner and outer bounds and the ABG inner and outer bounds of the capacity region for the MAC NOMA in VLC IoT networks. Assume that g 1 = 1, g 2 = 1/2, A ≜ A 1 = A 2 , and ε ≜ ε 1 = ε 2 . Let φ ≜ A 2 /ε denote the amplitude-tovariance ratio, and define SNR as ≜ ε/σ 2 . Moreover, both the uniform inner and outer bounds of the capacity region of MAC NOMA of VLC IoT networks are also presented    [36][37][38]. Figures 3(a)-3(c) show the inner and outer bounds for the channel capacity region of uplink NOMA in VLC IoT networks with SNR = 10 dB, φ = 3, 6, and 8, respectively. Figure 3(a) shows that the ABG inner bound is identical to the uniform inner bound. A similar case is observed for the outer bound; this is because the ABG has a uniform distribution for φ = 3. Moreover, the inner bound with discrete inputs is larger than the ABG inner bound, while the outer bound with discrete inputs cannot dominate the ABG outer bound. Figures 3(b) and 3(c) show that the inner bound with discrete inputs is the highest among the three types of inner bounds, while the ABG outer bound is the lowest among the three types of outer bounds for φ = 6 and 8. Comparing  Figures 3(a)-3(c), it can be seen that as the value of φ increases, the gap between the inner bound with discrete input and the ABG inner bound decreases, and the gap between the ABG inner bound and the ABG outer bound also decreases. Figure 4 compares the sum rates r 1 + r 2 (bits/sec/Hz) of the discrete inner and outer bounds, the ABG inner and outer bounds, and the uniform and inner bounds against SNR(dB) with ϕ = 6. As shown in Figure 4, the sum rate of each bound increases as the SNR gets larger, and the ABG inner bound is higher than those with the discrete inputs and the uniform inner bound, while the ABG outer bound is lower than the one with discrete inputs and the uniform outer bound. Additionally, we can observe that the gap between the proposed ABG inner bound and ABG outer bound increases as the SNR increases. Figure 5 shows the achievable rates of three users with respect to the transmit power. We can see that the rate of each user increases as the rate of the transmit power increases. Furthermore, the rate for each user of the proposed method is larger than that of the users of the non-SIC methods.

Conclusions
In this paper, we investigated the NOMA transmission for VLC IoT networks. Specifically, the channel capacity region of the practical NOMA VLC IoT networks was established with discrete and continuous inputs, respectively. To the best of our knowledge, the proposed inner and outer bounds are the first theoretical bounds of the channel capacity region for uplink NOMA of VLC IoT networks. Furthermore, we developed the optimal receiver NOMA beamforming design for VLC IoT networks and showed that the optimal beamformers are the generalized eigenvectors corresponding to the largest generalized eigenvalues.

Data Availability
The data of the numerical results can be obtained by emailing the author (mashuai001@cumt.edu.cn).

Conflicts of Interest
The authors declare that they have no conflicts of interest.