Performance Analysis of Short Packets in NOMA VLC Systems

In this paper, we propose a theoretical framework for analyzing the performance of the short-packet communication (SPC) in a downlink non-orthogonal multiple access (NOMA) visible light communication (VLC) system in which one light emitting diode (LED) communicates with two single-photodiode users. The analytical expression of the block error rate (BLER) is approximated by Gaussian-Chebyshev quadrature method, based on which the reliability, throughput, and latency expressions are deduced. Further, we jointly optimize power allocation coefficients and transmission rates to maximize sum throughput of the SPC-NOMA VLC system. The numerical results show that (i) our proposed system well satisfies the stringent requirements of the ultra-reliable and low latency communication (URLLC) at signal-to-noise ratio (SNR) larger than 130 dB; (ii) SPC-NOMA VLC systems outperform SPC- orthogonal multiple access (OMA) VLC systems in terms of reliability, latency, and throughput; (iii) the impacts of block-length, power allocation coefficients, transmission rates, and LED semi-angle are examined.


I. INTRODUCTION
Driven by the ever-increasing penetration of the digitization era in the globe, wireless data traffic with high network capacity demand is expected to substantially increase [1]. Ongoing research on future wireless networks that could plausibly satisfy future client data needs has consistently intensified [2], [3], [4]. Along with the fifth-generation (5G) technologies, visible light communication (VLC) has attracted considerable attention for high-speed and short-range wireless communications due to its many inherent advantages such as license-free spectrum, low-cost front-ends, high security, and strong immunity to electromagnetic interference [3], [4]. To be considered as a high-capacity wireless broadband technology, the developed VLC systems can effectively support low-latency communications that are pivotal to ensure realtime functionality in interactive communications of machines in industrial automation applications [5] and support multiple users with simultaneous network access [6].
5G technology is expected to satisfy the ultra-reliable (i.e., > 99.999%) and low latency (sub-millisecond) communication (URLLC) requirement for the real time applications and Internet-of-Things (IoTs) ones (e.g., telesurgery, fac-tory automation, machine-to-machine communication, and intelligent transportation). In [7], the domain knowledge of communications and networking was integrated into emerging deep learning techniques to guarantee the quality of services (QoS) constraints, and improved the performance of networks in terms of delay, reliability, and jitter, since transmission delay and reliability are two key parameters for the URLLC applications. To end this, the short packet communication (SPC) was introduced as a potential method in 5G and beyond [8], [9]. The authors of [10] showed that the Shannon capacity of the infinite block-length cannot be applied for the finite block-length communication in SPC. Alternatively, the achievable rate of the SPC was represented in term of the Shannon capacity, block-length, and block error rate (BLER) of the model, and the length of the block was longer than that of the actual data [11], [12]. The benefits of the SPC have been widely studied on different communication systems, e.g., quasi-static multiple-input multiple-output [13], latencycritical packet scheduling [14]. However, to the best of our knowledge, the application and investigation of SPC for VLC systems to reduce transmission latency are still questioned.
Second, the conventional orthogonal multiple access VOLUME 4, 2016 (OMA) (e.g., frequency division multiple access (FDMA), time division multiple access (TDMA), and code division multiple access) has limitations for massive connectivity due to the rareness of resources (i.e., time slots, frequencies, and bandwidth) [11], [12]. As a solution of this gap, nonorthogonal multiple access (NOMA) has recently been developed as an auspicious candidate technology for ability to serve multiple users simultaneously in the same resource block (i.e., time slot, subcarrier, and spreading code) in power domain [15]. In NOMA, higher power coefficients are allocated to the message signal of a poor-channel user, and lower power coefficients are allocated to the message signal of a good-channel user. Therefore, the performance of NOMA systems has been studied in various contexts, such as broadcast channels [16], full-duplex communications [17], physical layer security [18], multiple-input multiple-output systems [19], cognitive radio [20], and radio-frequency energy harvesting [21]. By utilizing NOMA, higher power was allocated to the message signal for the poor-channel users to ensure the achievable target rates at these users, thus balancing the network throughput and user fairness [20]. The necessary and sufficient conditions for the power ranges for each user in the two-user NOMA system were determined with the error floor limitation of the symbol constellation detection of successive interference cancellation (SIC) operation in [23]. Moreover, the authors of [24] proposed a framework for power ranges and optimum power allocation for NOMA systems with an arbitrary number of users. Also, the implementation of NOMA in the wireless communication provided higher flexibility and more efficient use of spectrum and energy, and overloading spectrum caused by multiplexing users was solved [25]. The implement of NOMA in VLC systems can provide efficient resource utilization to support a huge number of high-speed devices in indoor environments. Few studies about the implementation of NOMA in VLC systems have confirmed that NOMA is a potential candidate for high-speed VLC systems. For instance, the authors of [26] proposed the NOMA based directional light fidelity (LiFi) system in which visible light was used in the downlink and infrared light was used in uplink to improve energy efficiency and support for high-speed IoT devices. The performance of NOMA and orthogonal FDMA was compared in a VLC system with illumination constraints, and its outperformance was shown in [25]. A prior theoretical study on the performance evaluation of NOMA in a multi-user VLC system was presented in [27]. The authors of [28] proposed a power allocation algorithm based on the user level and subcarrierlevel power allocation to obtain the maximum sum of data rate and the minimum subcarrier loss rate of the OFDM based NOMA VLC system. It is highlighted that due to the fundamental differences between the VLC channel and the radio frequency (RF) channel, applying NOMA to VLC required careful reconsiderations [27], [29].
To overcome the barriers and accommodate demand on URLLC as well as provide massive connectivity, the combination of NOMA and SPC has been studied in the field of RF systems [12], [13], [16]. The SPC can be employed in VLC systems by utilizing Hermitian symmetry and adding direct current (DC) bias. However, to the best of our knowledge, few researches had been studied to explore the effectiveness of the NOMA VLC system. The aforementioned works have mainly discussed on the achievable rate of the NOMA VLC systems while the reliability and latency were not mentioned [6], [27], [29]. This brings a new exploration of a coupled SPC-NOMA scheme for the VLC system of multiple users and URLLC. This paper is the first study on the SPC-NOMA VLC with the analyses for the reliability, latency, and throughput. The contributions of this paper are summarized as below.
• We, for the first time, propose the SPC in NOMA VLC systems to increase reliability and reduce latency. The results of the proposed SPC-NOMA VLC system show that the reliability increases slowly at low SNR; however, it increases quickly at the medium and high SNR. • The analytical expression of the BLER are approximated by Gaussian-Chebyshev quadrature method, and then the approximate expressions for reliability, latency, and throughput are also carried out. • SPC and long-packet communication (LPC) in NOMA VLC systems are evaluated in terms of reliability and latency. The satisfaction of the SPC-NOMA VLC system for URLLC applications is presented in the comparison to the LPC-NOMA VLC system. • The performance of the SPC-NOMA VLC system and SPC-OMA VLC system is analyzed in terms of reliability, latency, and throughput. • The effect of choosing the LEDs with various semiangles on the sum throughput is evaluated for the SPC-NOMA VLC system. • The optimal values of transmission rates and power allocation coefficients to maximize the sum throughput of the SPC-NOMA VLC system are determined by the numerical search method.
The rest of this paper is organised as follows. The preliminaries of the SPC in the RF system, and both NOMA and OMA in the VLC system are introduced in Section II. In Section III, the system model is described, and the performance evaluation of the SPC in both NOMA VLC and OMA VLC systems and the optimization design for the throughput of the SPC-NOMA VLC is presented. Section IV shows the simulation results and the comparison of the LPC to the SPC in the NOMA VLC system, and the comparison of the SPC-NOMA VLC to the OMA VLC schemes. Finally, Section V is conclusion of the paper. For clarity, we summarize the main symbols and notations in Table 1.

A. SPC IN RF SYSTEMS
In the RF system, the message signal x s of the SPC is directly transmitted from the source to the destination. The signal at where |h| is magnitude of the channel gain, P is the electrical power of the message signal, and µ D is the additive white Gaussian noise (AWGN) with the variance σ 2 . In the SPC, each packet is transmitted in message signals. Each packet contains the information bits and addition bits (or metadata) for correct functioning [11]. The BLER of the detection at the receiver is approximated by [8,Eq. (59)], [30,Eq. (11)] where γ = P |h| 2 /σ 2 is the received signal-to-noise ratio (SNR), N is the block-length (N ≥ 100), R = k/N is the transmission rate, k is the number of information bits in a packet, C (γ) = log 2 (1 + γ) is the Shannon capacity, and V (γ) The average BLER is calculated bȳ where E{.} is expectation, and E{X} = ∞ −∞ xf X (x) dx.

B. NOMA VLC
The principle of NOMA schemes is based on the channel quality, shown in Fig. 1. We consider that two users are scheduled in one cluster and served in a resource block 1 . We define that the far user U 1 has lower channel gain, and the near user U 2 has higher channel gain. The message signals for users is superposed in the power domain. The poorer the user's channel is, the higher power coefficient is allocated. After Hermitian symmetry and adding DC bias [6], [27], [29], the superposed signal at one LED is given by where V DC is the DC bias added for non-negative unipolar signal, s 1 and s 2 are the intended message signal for U 1 and U 2 , and a 1 and a 2 are the power coefficients allocated to the message signals of U 1 and U 2 . Total power allocation coefficients must satisfy: a 1 + a 2 = 1 and a 1 ≥ a 2 . At U i (i ∈ 1, 2), DC bias is eliminated, and the received signal is given by where µ i is the zero-mean real-valued AWGN with the variance σ 2 i and h i is the channel gain between the LED and U i . After optical-to-electrical conversion, the signal processing is presented as following.
At U 1 , the SIC is not performed, and the intended signal is directly decoded. The SINR is given by where γ Tx = P σ 2 i is the transmitted SNR. At U 2 , the SIC is performed to remove the interference of the message for U 1 , and the message for U 1 is treated as the interference. The SINR of the intended message for U 2 at U 1 is given by The SINR of the own signal for U 2 is given by where δ 21 ∈ [0, 1] is the imperfect SIC factor 2 . In this paper, for the goal of evaluating optimal system performance, we consider δ 21 = 0.

C. OMA VLC
In this paper, we consider that TDMA is adopted to serve users in the orthogonal time resource. The transmitted signal for U i at one LED is given by 1 When there are more than two users in the system, users are sorted and paired, and the hybrid NOMA/OMA is used to support multiple user pairs. The power domain NOMA multiplexes two users of each pair. User pairs are multiplexed by the OMA scheme. When the number of users is odd, users are sorted and paired, and the remaining unpairing one is supported by the separate power and resource block [31]. 2 In practice, due to the imperfect SIC circuit design and error propagation, the operation of SIC may result in the residual interference, which leads to the degradation of system performance. Equation (9) assumes that there is no interference at U i caused by other users in the OMA VLC system. After removing DC bias, the received signal at U i is presented by Then, the received SNR at U i is given by

D. DISTRIBUTION FUNCTION OF THE RECEIVED SNR
The power allocation strategy for all users at the transmitter is based on the knowledge of the channel quality. In the twouser NOMA system, the channel gain quality is assumed to be ordered as The positions of receivers are assumed to be uniformly distributed within the circular area under the LED. The cell radius is denoted by r c . The vertical distance between the LED and users on the circle plane is denoted by L. In the polar coordinate system, the position of U i is presented by where r i is horizontal distance from the LED to the receiver and θ i is the polar angle from the polar axis. The channel gain between U i and the LED is given by where Lambertian radiation pattern is given by m = − ln (2)/ ln (cos (Φ 1/2 )); Φ 1/2 is the semi-angle of the LED; Ψ FOV is the field-of-view (FOV) semi-angle of the photodiode (PD) at each receiver; A is detection area of the PD; R p is the responsivity of the PD; d i is the Euclidean distance between the receiver and the LED; ψ i is the angle of irradiance; ψ i is the angle of incidence; T (ψ i ) is the gain of optical filter at the receiver; and g(ψ i ) is the gain of the optical concentrator at the receiver, given by where n is the refractive index of the optical concentrator, n ∈ [1, 2]. From Fig. 1, the Euclidian distance between the LED and U i , the angle of irradiance, and the angle of incidence are calculated by d i = r 2 i + L 2 , cos (ϕ i ) = L/ r 2 i + L 2 , and cos ψ i = L/ d 2 i + L 2 , respectively. With the uniform distribution, the probability density function (PDF) of a location in a circle is given by f ri (r) = 2r/r 2 c . According to [29], the unordered PDF of received SNR at U i is obtained by using the method of transformation and the change of variable method [32,Chap. 6] as The unordered CDF of the received SNR is given as Since the channel quality is ordered for the power allocation, the ordered statistics [29], [32,Chap. 6], [33] are considered in this paper. The ordered CDF of the received SINR at U i is presented by where ω = 1

III. SYSTEM MODEL AND PERFORMANCE EVALUATION A. SYSTEM MODEL
In this work, we consider the transmission in a downlink broadcast channel for the indoor environment. To effectively present the system performance, we choose a general VLC system consisting of one LED and two users. The system Metadata Data Data Long data packet model is depicted in Fig. 2. The LED is located on the ceiling to serve two single-photodiode users. The channel is static or quasi-static, so that the channel state is not changed during the transmission. We assume that the channel state information (CSI) is perfectly known at the LED and users. The users are uniformly distributed in a circle area. At the receiver side, the received signals include both line-of-sight (LOS) components and diffuse components caused by the reflections on the interior surfaces, such as the ceiling, floor, and walls. The authors of [6] have reported that the power of the LOS components is much higher than that of the non-LOS components. Therefore, we consider the LOS link from the LEDs to users. In the OMA scheme, TDMA is chosen to serve two users in orthogonal time resource. In the NOMA scheme, two users are served simultaneously, and perfect SIC is used at the poorer channel user for the signal detection. We denote the far user as the U 1 with the channel gain h 1 , and the near user is denoted by U 2 with the channel gain h 2 , (h 1 < h 2 ). The noise is real value AWGN with zero mean and unit variance σ 2 = 1.

B. THE SPC IN VLC SYSTEMS
In VLC systems, the transmitted signals are required to be real and non-negative for the intensity modulation and direct detection (IM/DD). When employing the SPC in VLC systems, Hermitian symmetry is used to get real signal, and then the DC bias is added at the LED to guarantee nonnegative signals for transmission. The message signal at the LED is presented by After removing DC bias at the receiver, the received signal is given as y = where µ 0 is zero mean real-valued AWGN with σ 2 0 . After removing DC bias at the receiver, the SNR at U i is given by Due to the utilization of Hermitian symmetry, the scaling factor 0.5 is multiplied with the Shannon capacity C (γ i ) and channel dispersion V (γ i ) in the expression of the BLER of the SPC in the VLC system. The BLER of the SPC in the VLC system at U i (i ∈ {1, 2}) is approximated by where N i , k i , and R i are block-length, number of information bits allocated to U i , and transmission rate at U i , respectively. The average BLER of the SPC in the VLC system is approximated bȳ Since the received SNR is limited in the circle area under the LED, then γ ∈ [γ min , γ max ]. The average BLER is approximated bȳ Using partial integral, (23) is expressed bȳ where F γi (x) is cumulative distribution function (CDF) of the variable x, and ζ(x) is the first order derivative of the BLER expression, and dQ(f (x))

VOLUME 4, 2016
It is complicated to derive the close-form expression of the remaining integral in (24). The average BLER is approximated by using the Gaussian-Chebyshev quadrature method as [34, whereV is the complexity-accuracy trade-off parameter, ν i = γi,max−γi,min 2 , and ϑ i = γi,max+γi,min 2 . Reliability at U i is the probability that the packet is correctly detected, given by Throughput at U i is the number of correctly determined information bits at the receiver per transmission, presented by Latency at U i is the delay in transmissions, given by where T c is the duration of a block.

C. THE SPC IN NOMA VLC SYSTEMS
In NOMA transmission strategy, the allocated block-length for U 1 and U 2 is N 1 = N 2 = N , since the LED transmits signal to two users simultaneously with the different power allocation. At the receiver side, the SIC is performed at U 2 to remove the interference of the U 1 message in the received signal, and then U 2 decodes its message. At U 1 , the message of U 2 is treated as the interference, and U 1 directly decodes its own message in the received signal.

D. THE SPC IN OMA VLC SYSTEMS
In OMA systems, since two users are assigned to the different orthogonal resources, there is no interference at U 1 (or U 2 ) caused by U 2 (or U 1 ). The power allocation to each user is based on the channel quality. N 1 and N 2 are the allocated block-length for U 1 and U 2 , and N 1 + N 2 = N . At U i (i ∈ {1, 2}), the SNR is given by Since (17) and (40), the ordered CDF of the received SNR at U i is also presented as (41) The average BLER at U i is approximated bȳ

E. OPTIMAL SUM THROUGHPUT DESIGN OF THE SPC-NOMA VLC SYSTEM
In the SPC-NOMA VLC system, the transmitted signals to users are superimposed in the power domain, and the LED serves both U 1 and U 2 simultaneously in the blocklength N . It is assumed that N 1 = N 2 = N in NOMA systems. The goal is to obtain maximum sum throughput of the SPC-NOMA VLC system. The optimization problem is formulated as where T = T 1 + T 2 . According to (21) and (28), the sum throughput T is formulated as . The transmission rates R i and power allocation coefficients a i for U i are determined at the LED, and the transmission rate of U 1 is independent to that of U 2 . We consider the equality a 1 + a 2 = 1 in (43c), where total power is consumed to transmit signal to all users. According to (44), there are three variables in this optimization problem. We need to determine the power allocation and transmission rates to obtain the maximum sum throughput T subject to the constraints in (44). (21) is monotonically decreasing with SNR/SINR.

Proposition 1: The error probability in
Proof: Please see Appendix A. According to (21) and (28), the effective throughput at each user depends on its power allocation. Following the NOMA principle, there is the trade-off between throughput and power allocation to each user. Since the Q-function is used in the throughput calculation, the optimal power allocation cannot be obtained in closed form. Therefore, the one-dimensional search is used to find the solution close to the optimal solution, where R 1 and R 2 are fixed values.
Proposition 2: The error probability in (21) is monotonically increasing with the transmission rate R i .
Proof: Please see Appendix B. Proposition 3: The transmission rate at each user is limited in the range [0, 2.56].
Proof: The transmission rate R is given by R = k/N . According to [35], the length of the information bit k of a packet is less than 256 (32 bytes) to get error probability of 10 −5 . In SPC, the block-length N is longer than 100. Therefore, R i ≤ 2.56. (43d) is verified.
Lemma 1: The sum throughput T does not monotonically increase with R 1 but is concave with respect to R 1 .
Proof: Please see Appendix C. Lemma 2: The sum throughput T does not monotonically increase with R 1 but is concave with respect to R 2 .
Proof: Please see Appendix D.
From Appendix C and Appendix D, there exists the optimal transmission rates by solving ∂T ∂R1 = 0 and ∂T ∂R2 = 0. According (54) and (58), the optimal solutions cannot be derived in closed form. Due to three variables a 1 , R 1 , and R 2 , we first find the optimal power allocation a 2 by onedimension search, and then we use two-dimensional search to find the maximum sum throughput with respect to R 1 and R 2 based on the optimal power allocation coefficients derived by one-dimensional search. The optimal results are presented in the next section. We summarize the proposed algorithm to find the optimal solution in Algorithm 1.
The accuracy of the proposed method is dependent on the given step size τ . The complexity of the search method is O(N I ), where N I is the size of the input values.

IV. NUMERICAL RESULTS
In this section, we present numerical results and utilize Monte Carlo simulations to verify the analysis in Section III. The benefits of the SPC in terms of reliability and latency are shown in the comparison between SPC and LPC in NOMA VLC systems. To demonstrate the effectiveness of the proposed system, we compare the performance of the SPC in NOMA and OMA VLC systems, and discuss the effect of the semi-angles of the LEDs to the system throughput of the SPC in the NOMA VLC system. The parameters of the channel gains are shown in Table 2. In the simulation, we choose the block-length N 1 = N 2 = 200, k 1 = k 2 = 80 bits for NOMA scheme, and N 1 = N 2 = 100, k 1 = k 2 = 80 bits for OMA scheme. In comparison of the SPC-NOMA VLC to VOLUME 4, 2016 Algorithm 1: Algorithm to find optimal power allocation and transmission rates Input: Initialize vector a 2 = 0.05 : τ : 0.5, with step size τ = 0.05, optimal power allocation a opt 2 = 0, and maximum sum throughput Set T max ← 0, R opt 1 = 0, R opt 2 = 0, R 1 = 0.05 : τ : 2.5, and R 2 = 0.05 : τ : 2.5. Calculate T ← T (R 1 , R 2 ) with optimal power allocation a opt 2 and a opt 1 = 1 − a opt 2 by (44).  In Fig. 3, we investigate the effects of the power allocation on the BLER performance of users in both NOMA and OMA systems. The analytical results match well with simulation results at different power allocation strategies. In SPC-OMA system, the BLER of U 1 is always lower than that of U 2 , since there is no interference caused by another user. The BLER decreases slowly at low SNR (110-120 dB) in both two systems. At medium SNR (120-130 dB), the BLER of the SPC-NOMA system decreases quicker than that of the SPC-OMA sytem, while the BLER of the SPC-NOMA system is less than 10 −5 at 130 dB. With a 1 = 0.6 and a 2 = 0.4, the SPC-OMA system can get the BLER less than 10 −5 , while the BLER of the SPC-NOMA system cannot be less than 10 −5 . When the power allocation for the far user increases, the BLER of the SPC-NOMA system is improved, since the interference at the near user caused by the far user in NOMA system decreases. The SPC-NOMA system can satisfy the error probability at 131 dB, while the SPC-OMA system requires 135 dB at all users. From this, the use of SPC in NOMA system requires less power than that of SPC in OMA system while increasing power to far user. Moreover, the performance of the NOMA system is enhanced when the power allocation is boosted for the far user. In Fig. 4, we compare reliability and latency of the short packet (400 bits) and long packet (800 bits and 1000 bits) in NOMA VLC system. The reliability and latency are calculated as the function of the total block-length at γ Tx = 130 dB. The reliability of short messages is higher than that of the long ones. The SPC can satisfy the reliability requirement with more than 1500 block-length while the LPC uses more than 2500 block-length to satisfy the reliability requirement. The latency of the short messages is lower than that of long ones. The use of the SPC can provide the latency in submillisecond while the latency of the system using the LPC is more than one millisecond. The use of the SPC can satisfy the requirements of both reliability and latency.
In Fig. 5, the reliability for U 1 is higher than that of U 2 at both NOMA and OMA systems in the range of the medium SNR. The reliability of U 2 is higher than that of U 1 at the high SNR. It shows that U 2 only obtains the high reliability when the transmitted power is high. The required SNR to fulfil the reliability of the NOMA system is lower than that of the OMA system, i.e., 130 dB for the NOMA system, and 135 dB for the OMA system. As shown in (21), the reliability of the system not only depends on the transmitted  SNR but also depends on the transmission rate. In the SPC-NOMA system, the LED serves two users simultaneously with N 1 = N 2 = N while the LED in the SPC-OMA VLC system serves two users in orthogonal time slots with N 1 = N 2 = N/2. The transmission rate of each user in the SPC-NOMA VLC system is always lower than that of the SPC-OMA VLC system. The SPC-NOMA VLC system provides higher reliability. In Fig. 6, the latency of SPC-NOMA users is compared to SPC-OMA users. The latency at U 2 in both NOMA and OMA VLC systems is quite similar. The latency at U 1 in NOMA system is approximately half of that at U 1 in the OMA system. The SPC-NOMA VLC system reduces latency for the users. Especially, the NOMA system fulfils the latency requirement at lower SNR than the OMA system, i.e., 116 dB for the NOMA system, and 119 dB for the OMA system. of the NOMA system is higher than that of the OMA system. The NOMA system achieves maximum sum throughput at 130 dB while the OMA system achieves the maximum sum throughput at 134 dB. Since SIC is performed at U 2 in the NOMA system, the throughput of U 2 is higher than that of U 1 at the low SNR. At the high SNR, the far user has higher throughput. The throughput at each user depends on transmitted power, transmission rate, block-length for each user, and total block-length. All users in the NOMA system have the same block-length while the OMA system uses the block-length N to serve all users with the different allocation of total block-length for each user. The throughput of users in the NOMA schemes is higher than that of users in the OMA scheme. Fig. 8 depicts sum throughput of the SPC in NOMA and OMA VLC schemes with various semi-angles of the LED. As shown in (13), the channel gain depends on Lambertian radiation pattern which is subject to the semi-angle of the LED. Since the sum throughput of the system depends on the channel gain, the semi-angle of the LED relates to the sum throughput. At γ Tx = 120 dB, the sum throughput of the SPC-NOMA VLC system can achieve maximum at 50 • semi-angle, while that of the SPC-OMA system can achieve maximum at 40 • semi-angle. At γ Tx = 130 dB, 35 • semiangle LED can provide the maximum sum throughput of the SPC-NOMA VLC system while the optimal semi-angle of the LED for the SPC-OMA VLC system is 45 • semi-angle. The optimal semi-angle of the LED depends on the value of SNR. When we increase the value of SNR, the LED with lower semi-angle can be useful.
In Fig. 9, the optimal power allocation can be derived by using one-dimension search while keeping constant transmission rates of two users. The sum throughput increases in the range of power allocation from 0 to 0.25, and then it decreases. When the power allocation for U 2 decreases, the throughput at U 2 decreases. After obtaining optimal power  allocation coefficients, the maximum sum throughputs of two users are derived by two-dimensions search. The changing values are transmission rates of two users. The result is shown in Fig. 10. Fig. 10 depicts the impact of transmission rates on the sum throughputs. As shown in (32), (35), (38), and (39), the throughput of U 1 is independent to the throughput of the U 2 . The throughput of the U 2 not only depends on R 2 but also depends on R 1 . Fig. 10 shows that the sum throughput increases with R 1 in the range R 1 ∈ [0, 0.5], and then decreases with R 1 > 0.5. The sum throughput increases when R 2 increases in the range [0, 1.2] and then decreases with R 2 > 1.2. Therefore, Lemma 1 and Lemma 2 are verified. The sum throughput increases to maximum point, and then it decreases with transmission rates R 1 and R 2 . It presents the trade-off between transmission rates and sum throughput of the system. According to (21), (28) and Proposition 2, the error probability is small with small values of transmission rates (R 1 < 0.5 and R 2 < 1.2), but the sum throughput is not high. Since the error probability is increasing with the transmission rates, the sum throughput decreases with the increasing of transmission rates (R 1 > 0.5 and R 2 > 1.2). Thus, this approach demonstrates that there exists a unique parameter-space position that obtains the maximum of the sum throughput.

V. CONCLUSION
In this paper, we proposed the SPC in a downlink NOMA VLC system with perfect SIC. The SPC-NOMA VLC system provides higher reliability and lower latency in comparison to the LPC-NOMA VLC systems. Moreover, the results show that the SPC-NOMA VLC system outperforms the SPC-OMA VLC system in terms of reliability, latency, and throughput. We also find out the optimal transmission rates and optimal power allocation strategy to maximize the sum throughput of the SPC-NOMA VLC system with two users. The SPC in VLC systems can support the various URLLC applications in factory automation and intelligent transportation systems.

APPENDIX A PROOF OF PROPOSITION 1
The first partial derivative of ε i with γ i is given as where ∂Ξ(γi,Ri) ∂γi is the partial derivative of Ξ (γ i , R i ) with respect to γ i , given by It is not clear whether the partial derivative of Ξ (γ i , R i ) with respect to γ i is positive or negative. Based on (46), we define a function, H(x) = log 2 x x 2 −1 . We find the range of value of H(x) with x ≥ 1 since γ i + 1 > 1. We calculate the first derivative of H(x) with respect to x, given by where h(x) = 1 ln 2 1 − 1 x − 2x log 2 x. Since (x 2 − 1) 2 > 0 with x > 1, the sign of H(x) is the same as h(x). The first derivative of h(x) is given by Since h ′ (x) < 0 with x > 1, h(x) is a decreasing function with respect to x > 1, and h(x) < h(1) = 0. h(x) is negative with x > 1, and then H ′ (x) < 0 with x > 1, which means that H(x) is also a decreasing function with respect to x > 1.

APPENDIX B PROOF OF PROPOSITION 2
The first partial derivative of ε i with R i is given as (51) Then, ∂ ∂Ri Q (Ξ (γ i , R i )) in (50) can be expressed as Since Θ (γ i ) ≥ 0, ∂ ∂Ri Q (Ξ (γ i , R i )) ≥ 0. It can be concluded that ε i is an increasing function with respect to R i .

APPENDIX C PROOF OF LEMMA 1
To determine the optimal value of R 1 which maximizes T , we examine the monotonicity and concavity of T with respect to R 1 . The first and second derivatives of T are examined as below.
(54) Since the value of (54) is not always positive or negative, T is not an increasing or decreasing function with respect to R 1 . Next, we examine the monotonicity of T with respect to R 1 .

APPENDIX D PROOF OF LEMMA 2
To find out the existence of the optimal transmission rate R 2 that maximizes T , the monotonicity and concavity are examined by deriving the first and second derivatives of T with respect to R 2 , presented as following. From (44), the first derivative of T with respect to R 2 is given by × (1 − Q (Ξ (γ 21 , R 1 ))) .
(57) By substituting (54) into (57), we have Since the value ∂T ∂R2 is not always positive or negative, T is not an increasing or decreasing function with respect to R 2 . Then, we examine the second derivative of T with respect to R 2 . From (58), we have ×(1 − Q (Ξ (γ 21 , R 1 ))).