# Nonorthogonal Multiple Access for Visible Light Communication IoT Networks

Authors: Chun Du; Shuai Ma; Yang He; Songtao Lu; Hang Li; Han Zhang; Shiyin Li
Journal: Wireless Communications and Mobile Computing
 (2020)

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## Abstract

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.


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## Body

## 1. 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 [3–5]. 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 [8–10], which has recently been included into the 3GPP long-term evolution advanced standard [11–13] 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 [14–20]. 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 [14–19], 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 eigenvaluesThe rest of this paper is organized as follows. In Section2, 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.


## 2. Capacity Region of Uplink NOMA for VLC IOT Networks

As shown in Figure1, N single-LED users (IoT devices) simultaneously transmit its own information to a single-PD base station (BS) over the same channel. Let si be the message of the ith user, where si≤Ai, Esi=0, and Esi2=εi.Figure 1
Schematic of an uplink NOMA of VLC IoT network.The transmitted signal of theith user is given by
(1)xi=pisi+bi,where pi is the transmit power of the ith user and bi is the direct current (DC) bias of the ith user. To ensure that the transmitted signal is nonnegative, the DC bias needs to satisfy bi≥piAi.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]
(2)gi=m+1AR2πdκ2cosmϕkcosψk,ifψk≤ψFOV;0,otherwise,where m is the Lambertian index of the LED, which depends on the semiangle ϕ1/2 by m=−log2/logcosϕ1/2; di denotes the distance between user i and the receiver; ϕ is the angle of irradiance; ψi is the angle of incidence; ψFOV is the field of vision (FOV) semiangle of the receiver; and AR denotes the effective area of the PD.The received signal at the BS is given by(3)y=∑i=1Ngipisi+bi+z,where z∼N0,σ2 represents the sum of contributions from the shot noise and the thermal noise [26, 27]. At the BS, the 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 gi2εii=1N satisfy a descending order, i.e., g12ε1≥g22ε2≥⋯≥gN2εN. The BS adopts the SIC technique to decode the received signals in a descending order [28–31], i.e., from s1 to sN. Specifically, when the BS decodes si, it first decodes the signal intended for user sk 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 signalsi is a discrete random variable with Mi real values ai,m1≤m≤Mi.Specifically, the signalsi satisfies
(4a)Prsi=ai,m=pi,m,m=1,⋯,Mi,(4b)Esi=∑m=1Mipi,mai,m=0,(4c)Esi2=∑m=1Mipi,mai,m2=εi,(4d)∑m=1Mipi,m=1,−Ai≤ai,m≤Ai,k=1,⋯,Mi,where ai,m denotes the mth point for signal si and pi,m denotes the corresponding probability.


#### 2.1.1. Inner Bound with Discrete Inputs

LetRi denote the capacity of user i, where 1≤i≤N, the capacity of Ri can be written as
(5)Ri=maxPsiIy;sis1,⋯,si−1,=maxPsihys1,⋯,si−1−hys1,⋯,si,=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,≥maxPsi12log2∑k=iN22hy^k−12log22πe∑j=i+1Ngj2pjεj+σ2,where y^k=gkpksk+z^k, ∑k=iKz^k=z, due to the entropy power inequality (EPI) [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ.Based on (5), the discrete inner bound uplink NOMA of VLC can be obtained by maximizing the entropy hy^i, i.e.,
(6)hy^i=−∫−∞∞fY^iylog2fY^iydy.As the noisez^k,i follows the Gaussian distribution with zero mean and σ2/K variance, the probability density function (PDF) fYy is given by
(7)fY^iy=K2πσ∑m=1Mipi,me−Ky−gipiai,m2/2σ2.Thus, the entropyhy^i maximization problem is given by
(8)minMi,ai,m,pi,m∫−∞∞fY^iylog2fY^iydys.t.4a,4b,4c,4d.Algorithm 1. Entropyhy^i approximation method.
1 initialization: Setn=0, h0=0; set c1 as the stopping parameter, choose Mi≥2;
2 letn=n+1, and compute ai according to (11);
3 compute the entropyhn=hy^i by solving Problem (10);
4 ifhn−hn−1≤c1, stop, and output hn; otherwise, Mi=Mi+1, and go to step 2.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:
(9)ai≜ai,1,⋯,ai,MiT,pi≜pi,1,⋯,pi,MiT,qi≜K2πσe−Ky−gipiai,12/2σ2,⋯,e−Ky−gipiai,Mi2/2σ2T.Based on the above mentioned definitions in (9), we reformulated Problem (8) as
(10)minMiai,pi∫−∞∞piqilog2piqidy,s.t.piTai=0,piTai⊙ai=εi,piT1Mi=1,pi≥0,where ⊙ denotes the Hadamard product. Note that given both Mi and ai, Problem (10), is convex with respect to pi, which can be solved efficiently using the available interior-point algorithms [32, 33].Without the loss of generality, we assumed the space among theMi points ai,mm=1Mi is equally placed in the range of −Ai,Ai, i.e.,
(11)ai,m=2AiMi−1mi−1−Ai.Note that whenMi is larger than the optimal values Mi∗, redundant points exist in ai,mm=1Mi. However, the effects of the redundant points can be eliminated by optimizing the PDF pi. Thus, for a sufficiently large Mi, the maximum entropy hy^i can be approximated by solving Problem (10) under condition (11). In summary, the proposed entropy hy^i approximation method is listed in Algorithm 1.Leth∗y^i denote the entropy hy^i computed by Algorithm 1. Substituting h∗y^i to, we obtained the discrete inner bound of NOMA VLC as
(12)Ri≥12log2∑k=1N22h∗y^k2πe∑j=i+1Ngj2pjεj+σ2.Letℛdisinner denote the achievable rate region of NOMA VLC IoT networks bounded by (12), which is given by
(13)ℛdisinner≜r1,⋯,rNri∈ℝ,ri≥12log2∑k=iN22h∗y^k2πe∑j=i+1Ngj2pjσ2,i=1,⋯,N.


#### 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 rateRi is given by
(14a)Ri=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(14b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hy^i,(14c)=12log22πeσ2+2πe∑k=iNgk2pkεk∑j=i+1N22hy^j,where the inequality (14a) follows the EPI [21] and hx≤1/2log2πevarx and h∗y^i is calculated by Algorithm ℛdisouter denotes the channel capacity region of NOMA VLC IoT networks bounded, which is given by
(15)ℛdisouter≜r1,⋯,rNri∈ℝ,ri≤12log22πeσ2+2πe∑j=iNgj2pjεj∑k=i+1N22h∗y^k,i=1,⋯,N.


### 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 signalsi follows a continuous distribution and derived the ABG inner bound of NOMA VLC IoT networks in closed-form expressions.


#### 2.2.1. ABG Inner Bound

Letfsi denote the pdf of si which satisfies the following peak optical power (Equation (16a)), average optical power (Equation (16b)), and electrical power constraints (Equation (16c)), i.e.,
(16a)∫−AiAifsidsi=1,(16b)∫−AiAisifsidsi=0,(16c)∫−AiAisi2fsidsi=εi.Then, for1≤i≤N, the lower bound of the achievable rate, Ri, is given by
(17a)Ri=maxfsiIy;sis1,⋯,si−1,(17b)=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(17c)≥maxfsi12log2∑k=iN22hsk+log2gk2pk+22hz−12log22πevar∑j=i+1Ngjpjsj+z,(17d)=12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,where the inequality (17c) is true due to the EPI [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ. The equality (17d) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy and is given by
(18)fisi=e−1−αi−βisi−γisi2,−Ai≤si≤Ai;0,otherwise,where the parameters αi, βi, and γi are the solutions of the following equations:
(19)TiAi−Ti−Ai=e1+αi,βieAiβi−γiAi−e1+αi−e−Aiβi+γiAi=0,eAiβi−γiAiβi−2γiAie−2Aiβi−βi−2γiAi+βi2+2γie1+αi=4γi2εie1+αi,(20)TiX=πeβi2/4γierfβi+2γiX/2γi2γi.Fork=N, the upper bound Rk is given by
(21)RN=12log21+pNgN2e1+2αN+γNεN2πσ2Letℛconinner denote the achievable rate region of NOMA VLC, which is given by
(22)ℛconinner≜r1,⋯,rKri∈ℝ,ri≥12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,i=1,⋯,K.


#### 2.2.2. ABG Outer Bound

Finally, we developed the ABG outer bound of uplink NOMA of VLC IoT networks in closed-form expressions for the continuous input. The upper bound of the maximum achievable rate,Ri, is given by
(23a)Ri=maxfsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(23b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hsj+log2gj2pj+22hz,(23c)=12log22πσ2+2π∑k=iNgk2pkεk∑j=i+1Npjgj2e1+2αj+γjεj+2πσ2,where the inequality (23b) follows the EPI [21] and hQ≤1/2log2πevarx and the equality (23c) holds because si follows the ABG distribution [34].Letℛconouter denote the channel capacity region of NOMA VLC IoT networks bounded by (23c), which is given by
(24)ℛconouter≜r1,⋯,rNri∈ℝ,ri≤12log22πσ2+2π∑j=iNgj2pjεj∑k=i+1Npkgk2e1+2αk+γkεkj+2πσ2,i=1,⋯,K.


## 2.1. Capacity Region with Discrete Inputs

As in the previous section, we assumed that the signalsi is a discrete random variable with Mi real values ai,m1≤m≤Mi.Specifically, the signalsi satisfies
(4a)Prsi=ai,m=pi,m,m=1,⋯,Mi,(4b)Esi=∑m=1Mipi,mai,m=0,(4c)Esi2=∑m=1Mipi,mai,m2=εi,(4d)∑m=1Mipi,m=1,−Ai≤ai,m≤Ai,k=1,⋯,Mi,where ai,m denotes the mth point for signal si and pi,m denotes the corresponding probability.


### 2.1.1. Inner Bound with Discrete Inputs

LetRi denote the capacity of user i, where 1≤i≤N, the capacity of Ri can be written as
(5)Ri=maxPsiIy;sis1,⋯,si−1,=maxPsihys1,⋯,si−1−hys1,⋯,si,=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,≥maxPsi12log2∑k=iN22hy^k−12log22πe∑j=i+1Ngj2pjεj+σ2,where y^k=gkpksk+z^k, ∑k=iKz^k=z, due to the entropy power inequality (EPI) [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ.Based on (5), the discrete inner bound uplink NOMA of VLC can be obtained by maximizing the entropy hy^i, i.e.,
(6)hy^i=−∫−∞∞fY^iylog2fY^iydy.As the noisez^k,i follows the Gaussian distribution with zero mean and σ2/K variance, the probability density function (PDF) fYy is given by
(7)fY^iy=K2πσ∑m=1Mipi,me−Ky−gipiai,m2/2σ2.Thus, the entropyhy^i maximization problem is given by
(8)minMi,ai,m,pi,m∫−∞∞fY^iylog2fY^iydys.t.4a,4b,4c,4d.Algorithm 1. Entropyhy^i approximation method.
1 initialization: Setn=0, h0=0; set c1 as the stopping parameter, choose Mi≥2;
2 letn=n+1, and compute ai according to (11);
3 compute the entropyhn=hy^i by solving Problem (10);
4 ifhn−hn−1≤c1, stop, and output hn; otherwise, Mi=Mi+1, and go to step 2.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:
(9)ai≜ai,1,⋯,ai,MiT,pi≜pi,1,⋯,pi,MiT,qi≜K2πσe−Ky−gipiai,12/2σ2,⋯,e−Ky−gipiai,Mi2/2σ2T.Based on the above mentioned definitions in (9), we reformulated Problem (8) as
(10)minMiai,pi∫−∞∞piqilog2piqidy,s.t.piTai=0,piTai⊙ai=εi,piT1Mi=1,pi≥0,where ⊙ denotes the Hadamard product. Note that given both Mi and ai, Problem (10), is convex with respect to pi, which can be solved efficiently using the available interior-point algorithms [32, 33].Without the loss of generality, we assumed the space among theMi points ai,mm=1Mi is equally placed in the range of −Ai,Ai, i.e.,
(11)ai,m=2AiMi−1mi−1−Ai.Note that whenMi is larger than the optimal values Mi∗, redundant points exist in ai,mm=1Mi. However, the effects of the redundant points can be eliminated by optimizing the PDF pi. Thus, for a sufficiently large Mi, the maximum entropy hy^i can be approximated by solving Problem (10) under condition (11). In summary, the proposed entropy hy^i approximation method is listed in Algorithm 1.Leth∗y^i denote the entropy hy^i computed by Algorithm 1. Substituting h∗y^i to, we obtained the discrete inner bound of NOMA VLC as
(12)Ri≥12log2∑k=1N22h∗y^k2πe∑j=i+1Ngj2pjεj+σ2.Letℛdisinner denote the achievable rate region of NOMA VLC IoT networks bounded by (12), which is given by
(13)ℛdisinner≜r1,⋯,rNri∈ℝ,ri≥12log2∑k=iN22h∗y^k2πe∑j=i+1Ngj2pjσ2,i=1,⋯,N.


### 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 rateRi is given by
(14a)Ri=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(14b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hy^i,(14c)=12log22πeσ2+2πe∑k=iNgk2pkεk∑j=i+1N22hy^j,where the inequality (14a) follows the EPI [21] and hx≤1/2log2πevarx and h∗y^i is calculated by Algorithm ℛdisouter denotes the channel capacity region of NOMA VLC IoT networks bounded, which is given by
(15)ℛdisouter≜r1,⋯,rNri∈ℝ,ri≤12log22πeσ2+2πe∑j=iNgj2pjεj∑k=i+1N22h∗y^k,i=1,⋯,N.


## 2.1.1. Inner Bound with Discrete Inputs

LetRi denote the capacity of user i, where 1≤i≤N, the capacity of Ri can be written as
(5)Ri=maxPsiIy;sis1,⋯,si−1,=maxPsihys1,⋯,si−1−hys1,⋯,si,=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,≥maxPsi12log2∑k=iN22hy^k−12log22πe∑j=i+1Ngj2pjεj+σ2,where y^k=gkpksk+z^k, ∑k=iKz^k=z, due to the entropy power inequality (EPI) [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ.Based on (5), the discrete inner bound uplink NOMA of VLC can be obtained by maximizing the entropy hy^i, i.e.,
(6)hy^i=−∫−∞∞fY^iylog2fY^iydy.As the noisez^k,i follows the Gaussian distribution with zero mean and σ2/K variance, the probability density function (PDF) fYy is given by
(7)fY^iy=K2πσ∑m=1Mipi,me−Ky−gipiai,m2/2σ2.Thus, the entropyhy^i maximization problem is given by
(8)minMi,ai,m,pi,m∫−∞∞fY^iylog2fY^iydys.t.4a,4b,4c,4d.Algorithm 1. Entropyhy^i approximation method.
1 initialization: Setn=0, h0=0; set c1 as the stopping parameter, choose Mi≥2;
2 letn=n+1, and compute ai according to (11);
3 compute the entropyhn=hy^i by solving Problem (10);
4 ifhn−hn−1≤c1, stop, and output hn; otherwise, Mi=Mi+1, and go to step 2.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:
(9)ai≜ai,1,⋯,ai,MiT,pi≜pi,1,⋯,pi,MiT,qi≜K2πσe−Ky−gipiai,12/2σ2,⋯,e−Ky−gipiai,Mi2/2σ2T.Based on the above mentioned definitions in (9), we reformulated Problem (8) as
(10)minMiai,pi∫−∞∞piqilog2piqidy,s.t.piTai=0,piTai⊙ai=εi,piT1Mi=1,pi≥0,where ⊙ denotes the Hadamard product. Note that given both Mi and ai, Problem (10), is convex with respect to pi, which can be solved efficiently using the available interior-point algorithms [32, 33].Without the loss of generality, we assumed the space among theMi points ai,mm=1Mi is equally placed in the range of −Ai,Ai, i.e.,
(11)ai,m=2AiMi−1mi−1−Ai.Note that whenMi is larger than the optimal values Mi∗, redundant points exist in ai,mm=1Mi. However, the effects of the redundant points can be eliminated by optimizing the PDF pi. Thus, for a sufficiently large Mi, the maximum entropy hy^i can be approximated by solving Problem (10) under condition (11). In summary, the proposed entropy hy^i approximation method is listed in Algorithm 1.Leth∗y^i denote the entropy hy^i computed by Algorithm 1. Substituting h∗y^i to, we obtained the discrete inner bound of NOMA VLC as
(12)Ri≥12log2∑k=1N22h∗y^k2πe∑j=i+1Ngj2pjεj+σ2.Letℛdisinner denote the achievable rate region of NOMA VLC IoT networks bounded by (12), which is given by
(13)ℛdisinner≜r1,⋯,rNri∈ℝ,ri≥12log2∑k=iN22h∗y^k2πe∑j=i+1Ngj2pjσ2,i=1,⋯,N.


## 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 rateRi is given by
(14a)Ri=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(14b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hy^i,(14c)=12log22πeσ2+2πe∑k=iNgk2pkεk∑j=i+1N22hy^j,where the inequality (14a) follows the EPI [21] and hx≤1/2log2πevarx and h∗y^i is calculated by Algorithm ℛdisouter denotes the channel capacity region of NOMA VLC IoT networks bounded, which is given by
(15)ℛdisouter≜r1,⋯,rNri∈ℝ,ri≤12log22πeσ2+2πe∑j=iNgj2pjεj∑k=i+1N22h∗y^k,i=1,⋯,N.


## 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 signalsi follows a continuous distribution and derived the ABG inner bound of NOMA VLC IoT networks in closed-form expressions.


### 2.2.1. ABG Inner Bound

Letfsi denote the pdf of si which satisfies the following peak optical power (Equation (16a)), average optical power (Equation (16b)), and electrical power constraints (Equation (16c)), i.e.,
(16a)∫−AiAifsidsi=1,(16b)∫−AiAisifsidsi=0,(16c)∫−AiAisi2fsidsi=εi.Then, for1≤i≤N, the lower bound of the achievable rate, Ri, is given by
(17a)Ri=maxfsiIy;sis1,⋯,si−1,(17b)=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(17c)≥maxfsi12log2∑k=iN22hsk+log2gk2pk+22hz−12log22πevar∑j=i+1Ngjpjsj+z,(17d)=12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,where the inequality (17c) is true due to the EPI [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ. The equality (17d) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy and is given by
(18)fisi=e−1−αi−βisi−γisi2,−Ai≤si≤Ai;0,otherwise,where the parameters αi, βi, and γi are the solutions of the following equations:
(19)TiAi−Ti−Ai=e1+αi,βieAiβi−γiAi−e1+αi−e−Aiβi+γiAi=0,eAiβi−γiAiβi−2γiAie−2Aiβi−βi−2γiAi+βi2+2γie1+αi=4γi2εie1+αi,(20)TiX=πeβi2/4γierfβi+2γiX/2γi2γi.Fork=N, the upper bound Rk is given by
(21)RN=12log21+pNgN2e1+2αN+γNεN2πσ2Letℛconinner denote the achievable rate region of NOMA VLC, which is given by
(22)ℛconinner≜r1,⋯,rKri∈ℝ,ri≥12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,i=1,⋯,K.


### 2.2.2. ABG Outer Bound

Finally, we developed the ABG outer bound of uplink NOMA of VLC IoT networks in closed-form expressions for the continuous input. The upper bound of the maximum achievable rate,Ri, is given by
(23a)Ri=maxfsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(23b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hsj+log2gj2pj+22hz,(23c)=12log22πσ2+2π∑k=iNgk2pkεk∑j=i+1Npjgj2e1+2αj+γjεj+2πσ2,where the inequality (23b) follows the EPI [21] and hQ≤1/2log2πevarx and the equality (23c) holds because si follows the ABG distribution [34].Letℛconouter denote the channel capacity region of NOMA VLC IoT networks bounded by (23c), which is given by
(24)ℛconouter≜r1,⋯,rNri∈ℝ,ri≤12log22πσ2+2π∑j=iNgj2pjεj∑k=i+1Npkgk2e1+2αk+γkεkj+2πσ2,i=1,⋯,K.


## 2.2.1. ABG Inner Bound

Letfsi denote the pdf of si which satisfies the following peak optical power (Equation (16a)), average optical power (Equation (16b)), and electrical power constraints (Equation (16c)), i.e.,
(16a)∫−AiAifsidsi=1,(16b)∫−AiAisifsidsi=0,(16c)∫−AiAisi2fsidsi=εi.Then, for1≤i≤N, the lower bound of the achievable rate, Ri, is given by
(17a)Ri=maxfsiIy;sis1,⋯,si−1,(17b)=maxPsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(17c)≥maxfsi12log2∑k=iN22hsk+log2gk2pk+22hz−12log22πevar∑j=i+1Ngjpjsj+z,(17d)=12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,where the inequality (17c) is true due to the EPI [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ. The equality (17d) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy and is given by
(18)fisi=e−1−αi−βisi−γisi2,−Ai≤si≤Ai;0,otherwise,where the parameters αi, βi, and γi are the solutions of the following equations:
(19)TiAi−Ti−Ai=e1+αi,βieAiβi−γiAi−e1+αi−e−Aiβi+γiAi=0,eAiβi−γiAiβi−2γiAie−2Aiβi−βi−2γiAi+βi2+2γie1+αi=4γi2εie1+αi,(20)TiX=πeβi2/4γierfβi+2γiX/2γi2γi.Fork=N, the upper bound Rk is given by
(21)RN=12log21+pNgN2e1+2αN+γNεN2πσ2Letℛconinner denote the achievable rate region of NOMA VLC, which is given by
(22)ℛconinner≜r1,⋯,rKri∈ℝ,ri≥12log2∑k=iNpkgk2e1+2αk+γkεk+2πσ22π∑j=i+1Npjgj2εj+2πσ2,i=1,⋯,K.


## 2.2.2. ABG Outer Bound

Finally, we developed the ABG outer bound of uplink NOMA of VLC IoT networks in closed-form expressions for the continuous input. The upper bound of the maximum achievable rate,Ri, is given by
(23a)Ri=maxfsih∑k=iNgkpksk+z−h∑j=i+1Ngjpjsj+z,(23b)≤12log22πevar∑k=iNgkpksk+z−maxfsi12log2∑j=i+1N22hsj+log2gj2pj+22hz,(23c)=12log22πσ2+2π∑k=iNgk2pkεk∑j=i+1Npjgj2e1+2αj+γjεj+2πσ2,where the inequality (23b) follows the EPI [21] and hQ≤1/2log2πevarx and the equality (23c) holds because si follows the ABG distribution [34].Letℛconouter denote the channel capacity region of NOMA VLC IoT networks bounded by (23c), which is given by
(24)ℛconouter≜r1,⋯,rNri∈ℝ,ri≤12log22πσ2+2π∑j=iNgj2pjεj∑k=i+1Npkgk2e1+2αk+γkεkj+2πσ2,i=1,⋯,K.


## 3. 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 Figure2, which includes N single LED users (IoT devices) and a L PDs BS. Let si denote the transmitted message from user i, where the definition of si is similar to that in the previous SISO scenario. Thus, the received signal at BS can be expressed as
(25)y=∑k=1Ngkpksk+z,where gi∈ℝL×1 denotes the channel vector between user i and BS, piεi≥0 is the transmitted power of user i, and z∼N0,σ2I represents the additive white Gaussian noise vector.Figure 2
Schematic of SIMO NOMA of a VLC IoT network.For messagesi, the BS invokes a linear receive beamformer wi∈ℝL to the received signals y as follows:
(26)y^i=wiTy=∑k=1NwiTgkpksk+wiTz,i=1,⋯,N.Without the loss of generality, we assumed that the termsgiTpiεii=1N satisfy a descending order, i.e., g1Tp1ε1≥g2Tp2ε2≥⋯≥gNTpNεN. Then, the BS adopts the SIC technique to decode the received signals in a descending order [35], i.e., from s1 to sN. Specifically, let Ri denote the achievable rate of decoding message si, 1≤i≤N.When1≤i<N, Ri is given by
(27a)Ri=maxfsiIy^;sis1,⋯,si−1,(27b)=maxfsihy^s1,⋯,si−1−hy^s1,⋯,si,(27c)=maxfsih∑j=iNwiTgjpjsj+wiTz−h∑m=i+1NwkTgmpmsm+wiTz,(27d)≥maxfsi12log2∑j=iN22hsj+log2wiTgj2pj+22hwiTz−12log22πevar∑m=i+1NwiTgm2+pmsm+wi2σ2,(27e)=12log2∑j=iNwiTgj2pje1+2αj+γjεj+2πwi2σ22π∑m=i+1NwiTgm2pmεm+2πwi2σ2,where the inequality (27d) holds due to the EPI [21] and hQ≤1/2log2πevarQ for a random variable with variance, varQ. The equality (27e) holds because the corresponding input distribution (termed ABG distribution) [34] maximizes the differential entropy. For k=N, RN is given by
(28)RN=12log21+wNTgN2pNe1+2αN+γNεN2πσ2.Therefore, for1≤i≤N, the expression of the lower bound Ri can be expressed as
(29)Ri=12log2∑j=iNwiTgj2pje1+2αj+γjεj+2πwi2σ22πΓi∑m=i+1NwiTgm2pmεm+2πwi2σ2,where Γi is an indicator function as follows:
(30)Γi≜1,∀i≠N,0,i=N.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 power constraints as follows:
(31)maxwi12log2∑j=iNwiTgj2pje1+2αj+γjεj+2πwi2σ22πΓi∑m=i+1NwiTgm2pmεm+2πwi2σ2s.t.wi2≤1,1≤i≤N.Note that Problem (31) is nonconvex which is hard to solve. To deal with this difficulty, we first define some variables as follows:
(32)p^i=pie1+2αi+γiεi,Gi≜0,⋯,p^igi,⋯,p^NgN,G¯i≜2π0,⋯,Γipi+1εi+1gi,⋯,ΓipNεNgN,c≜2πσ2.With the introduced variables in (32), we can equivalently rewrite Problem (31) to a concise form as follows:
(33)maxwi12log2wiTGiGiTwi+cwiTwiwiTG¯iG¯iTwi+cwiTwi,s.t.wi2≤1,1≤i≤N,which is a quadratically constrained quadratic problem (QCQP). As the logarithmic function is monotonically increasing, Problem (33) can be further reformulated as follows:
(34)maxwiwiTGiGiT+cIwiwiTGiGiT+cIwis.t.wi2≤1,1≤i≤N.Letλi,max denote the largest generalized eigenvalue of matrix Ai and matrix Bi, where Ai≜GiGiT+cI and Bi≜G¯iG¯iT+cI.Furthermore, letwi,max denote the generalized eigenvector corresponding to the largest eigenvalue λi,max, which satisfies Aiwi,max=λi,maxBiwi,max. Thus, for the NOMA VLC uplink, the optimal beamformer wi of Problem (34) is given by
(35)wiopt=wi,maxwi,max,and the maximum achievable rate of message si is log2λi,max.


## 4. 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 thatg1=1, g2=1/2, A≜A1=A2, and ε≜ε1=ε2. Let φ≜A2/ε denote the amplitude-to-variance ratio, and define SNR as ≜ε/σ2. Moreover, both the uniform inner and outer bounds of the capacity region of MAC NOMA of VLC IoT networks are also presented for comparison, where the input signals follow a uniform distribution [36–38].Figures3(a)–3(c) show the inner and outer bounds for the channel capacity region of uplink NOMA in VLC IoT networks with SNR=10dB, 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 3
Outer and inner bounds of the capacity region for MAC NOMA in VLC IoT networks with (a)φ=3, (b) φ=6, and (c) φ=8, respectively.
(a)
(b)
(c)Figure4 compares the sum rates r1+r2 (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 4
Sum ratesr1+r2 (bits/sec/Hz) of the discrete inner and outer bounds, ABG inner and outer bounds, and uniform inner and outer bounds versus SNR(dB) with φ=6.Figure5 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.Figure 5
Maximum rates of users versus transmit power.


## 5. 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.



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*Source: 5791436-2020-05-05.xml*