## Abstract

Orthogonal frequency division multiplexing non-orthogonal multiple access (OFDM-NOMA) is a promising multi-user access scheme in indoor visible light communication (VLC) systems. In this paper, we propose three novel joint subcarrier and power allocation algorithms in OFDM-NOMA-VLC to improve the throughputs and/or user fairness. These three proposed algorithms address the requirements in fairness-throughput-balanced (FTB), fairness-first (FF), and throughput-first (TF) scenarios, respectively. All of them improve the objective function in the previous joint allocation algorithm and ensure the fairness of users in terms of their overall throughput, rather than that in every subcarrier that is tight and redundant. The designs using the proposed algorithms also exhibit reduced peak-to-average power ratios (PAPRs) and so the average signal power can be better used to further enhance the throughput when the system is limited by the signal peak power. Simulations verify that the proposed algorithms are superior to the previous joint subcarrier and power allocation algorithm as well as the conventional fixed power allocation (FPA) and gain ratio power allocation (GRPA) algorithms. The performance improvement of the proposed algorithms is particularly greater for a larger number of multiplexed users per subcarrier or a larger number of OFDM subcarriers, under which the PAPRs of designs using conventional algorithms are higher. When the total user number, the number of the multiplexed users per subcarrier, and the number of subcarriers are 5, 2, and 16, respectively, the throughputs of the three proposed algorithms are 62.17%, 53.35%, and 67.25% higher than the conventional joint allocation algorithm, while the user fairness is improved by 4.64%, 7.87%, and degraded by 20.71%, respectively. Therefore, the three proposed algorithms can address the requirements in FTB, FF, and TF scenarios, respectively.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Visible light communication (VLC) is suitable for indoor multi-user communications due to its high data rate, excellent security performance, and low power consumption [1,2]. In this application, the unique optical power constraint due to signal non-negativity and human eye’s safety is a problem that must be considered. The user fairness and the total throughputs of the system also need to meet the requirements of different scenarios. Recently, as a promising multi-user access technology, non-orthogonal multiple access (NOMA) is utilized in the VLC system to enhance the capacity or fairness [3]. Unlike the orthogonal multiple access (OMA) scheme, multiple users in NOMA are multiplexed in the same frequency band or time slot with different power levels, and successive interference cancellation (SIC) is used at the receiver to decode the signal of each user. Due to the power-domain multiplexing, power allocation between users is essential in NOMA as it may significantly impact the throughput and user fairness. There have been some numerical and experimental works on the power allocation in NOMA VLC [4–13]. In [4], the max-min power allocation method was used to ensure fairness among users in the multi-cell VLC network. In [5], the logarithmic utility function was used in the power allocation algorithm to balance the throughput and user fairness. The authors in [6] defined the user fairness from a new perspective and proposed an algorithm to achieve a flexible trade-off between the user fairness and total throughput. However, in their designed resource allocation strategy, some practical factors such as the optical peak power limitation in the actual VLC system were not considered. More importantly, all of the above methods aim at single-carrier NOMA systems and are not optimal for orthogonal frequency division multiplexing (OFDM) NOMA, whose power allocation should be optimized at both the subcarrier level and the user level. Recently, numerical and experimental investigations were conducted on the OFDM-NOMA-VLC [7–13]. In [7], a phase pre-distortion method was proposed to improve the performance of the NOMA-VLC system. The OFDM-NOMA-VLC uplink and downlink system, as well as the influence of power allocation and channel estimation, was experimentally investigated in [8]. The performance advantage of non-Hermitian symmetric (NHS) inverse-fast-Fourier-transform (IFFT)/FFT based OFDM-NOMA VLC was experimentally verified in [9] while that of offset-QAM/OFDM-NOMA VLC was achieved in [10]. The work in [11] experimentally demonstrated the performance advantage of combining multiple inputs multiple outputs (MIMO) and OQAM-OFDM-NOMA over the conventional MIMO scheme. However, previous studies [7–11] still used the simple fixed power allocation (FPA) algorithm, which does not fully consider the channel characteristics of different users on different subcarriers. Authors of [12] investigated an enhanced power allocation (EPA) algorithm for OFDM-NOMA-VLC and proved that the throughput could be improved compared to the conventional algorithms. However, it only considered the power allocation for different users within each subcarrier and did not consider the power allocation between subcarriers. In [13], a normalized gain difference power allocation algorithm was proposed for MIMO-NOMA-VLC to achieve higher sum rate and lower complexity.

In addition to power allocation, subcarrier allocation is also an essential issue in the practical application of the OFDM-NOMA-VLC system. In the existing OFDM-NOMA-VLC works, it is assumed that any subcarrier would multiplex all users, so there is no need for subcarrier allocation. However, in a practical OFDM-NOMA system, the multiplexed users on each subcarrier should be limited, considering the complexity of SIC at the receiver [14]. In OFDM-NOMA, the number of multiplexed users in each subcarrier may be less than the number of total users, while the user fairness can still be maintained using subcarrier allocation. In [15], a software-based NOMA-VLC system with dynamic adjustment of power and carrier allocation was developed but it did not study the specific subcarrier allocation algorithm. In [16], a subcarrier and power allocation scheme for asymmetric clipped optical (ACO) OFDM-NOMA for the uplink underwater VLC system was proposed to save energy and support large-scale device connections. However, this scheme cannot be applied to DC-biased OFDM-NOMA in indoor VLC systems where the peak amplitude of the electrical signal should be limited to ensure the non-negativity of the optical signal and the safety of human eyes.

In [17,18], we proposed an effective algorithm for joint subcarriers and power allocation while considering the practical requirements in indoor VLC systems such as severe high-frequency fading induced by the limited bandwidth and the signal peak power limitation due to the eyes’ safety and the non-negativity of the optical signal. This method, named as Algorithm 1 in this paper, employs the logarithmic utility function on the capacity of each user in each subcarrier to balance the throughput and user fairness in every subcarrier. It is found that this design is tight and redundant because in practice, it is only required to ensure the fairness of the overall throughput between users. In this paper, we propose three improved algorithms, namely Algorithms 2, 3, and 4, for applications in the fairness-throughput-balanced (FTB), fairness-first (FF), and throughput-first (TF) scenarios, respectively. These algorithms improve the objective function and only require the user fairness in terms of their overall throughput, instead of that in every subcarrier. The advantages of Algorithms 2–4 are more prominent for a larger number of multiplexed users per subcarrier and a larger number of OFDM subcarriers because they can result in designs with reduced peak-to-average-power ratios (PAPRs) and so can make better use of the average signal power under the limit of the signal peak power. Simulations with different total user numbers (3, 4, and 5), different numbers of multiplexed users per subcarrier (1, 2, 3, 4, and 5), and different OFDM subcarriers (8, 10, 12, 14, and 16) verify that the proposed Algorithms 2–4 are superior to Algorithm 1 as well as the conventional FPA and gain ratio power allocation (GRPA) in all cases. It is also verified that the proposed algorithms have a greater performance gain due to lower PAPRs when the system is limited by the optical signal peak power. When the total user number, the number of the multiplexed users per subcarrier, and the number of subcarriers are 5, 2, and 16, respectively, the throughputs of the three proposed algorithms are 62.17%, 53.35%, and 67.27% higher than the conventional algorithms, while the user fairness is improved by 4.64%, 7.87%, and degraded by 20.71%, respectively. Therefore, the three proposed algorithms can address the requirements in FTB, FF, and TF scenarios, respectively.

## 2. System model

Figure 1 shows the model of the OFDM-NOMA VLC system. Assuming the user index as *m = *1*, …, M* and the channel response of the user *m* as *h _{m}*, we can obtain the frequency-domain channel gain of the user

*m*on the

*n*

^{th}subcarrier in the OFDM-NOMA signal,

*H*, by taking the Fourier transform of

_{m,n}*h*, where the subcarrier index

_{m}*n*= 1,…,

*N*. Due to the SIC used in the NOMA system, we should design the decoding order of users properly for each subcarrier. In general, because a user is able to decode all users with the channel worse than itself [14], the best decoding order is that the user with a better channel should be decoded later. Therefore, the multiplexed users on each subcarrier are re-arranged in the ascending order of the channel gain, resulting in $|{H_{{\pi _n}}}{_{{(1)}_{,n}}}|\le |{H_{{\pi _n}}}{_{{(2)}_{,n}}}|\le \ldots \le |{H_{{{\pi _n}}}}{_{{(k)}_{,n}}}|\le \ldots \le |{H_{{\pi _n}}}{_{{({K_n})}_{,n}}}|$, where

*π*(

_{n}*k*) and ${H_{{{\pi _n}}}}{_{{(k)}_{,n}}}$,

*k*= 1,…,

*K*, represent the

_{n}*k*

^{th}decoded user and its channel response on the subcarrier

*n*, respectively. Note that the decoding order can be different for different subcarrier

*n*due to the frequency selective effect of the VLC system. In the previous VLC works, the number of multiplexed users on each subcarrier

*K*is commonly the same as

_{n}*M*, i.e., all users are multiplexed on every subcarrier. However, in the practical system,

*K*should be limited regardless of

_{n}*M*considering the complexity of the receiver for SIC. For example,

*K*is limited to 2 in a 5-user system, while different users can be allocated to different subcarriers to achieve the optimal throughput and user fairness. In this case, subcarrier allocation is critical. This issue was not considered in conventional VLC works where

_{n}*K*=

_{n}*M*and only power allocation is required. In [17,18], joint subcarrier and power allocation was studies in the OFDM-NOMA-VLC system. However, as will be shown later, this algorithm balances the throughput and user fairness in every subcarrier that is however unnecessary, and also results in a high PAPR that hinders the full use of average signal power under the limitation of signal peak power. In this paper, we will propose three new joint allocation algorithms to improve the performance. In the following discussions, we assume

*K*is the same for all subcarriers and is defined as

_{n}*K*for simplicity. At the transmitter, different users are superposed in the power domain, and the multiplexed signal of the subcarrier

*n*is:

*k*

^{th}decoded user on the subcarrier

*n*. The transmitted OFDM signal with

*N*subcarriers is:

In VLC using intensity modulation, the optical power is proportional to the electrical amplitude. Therefore, *x*(*t*) ≥ 0 and *β*·max(*x*(*t*))≤*D* are required for the non-negative optical power and the limited optical peak power for eyes’ safety, respectively, where *β* and *D* are the electrical-to-optical conversion coefficient and the power limit. At the receiver, assuming user *m* is the one multiplexed on subcarrier *n*, its achievable rate on the subcarrier *n* is:

*m*on the subcarrier

*n*.

*σ*and

^{2}*W*are the additive white Gaussian noise and the bandwidth of the subcarrier

_{n}*n*, respectively. In Eq. (3), the user

*m*regards the signals of the users decoded after itself as the interference. In contrast to previous VLC works, which focus on the maximal throughput, both throughput and user fairness are considered in this paper. The user fairness is assessed by the Jain’s fairness index [14]: where

*R*denotes the total throughput of the user

_{m}*m*:

*p*= 0 if the user

_{m,n }*m*is not multiplexed on the subcarrier

*n*.

## 3. Principles of the proposed algorithms

We will first review our previously proposed Algorithm 1 and then introduce Algorithms 2–4 with improved objective functions to enhance the performance. In all algorithms, we adopt a three-step procedure since both power and subcarrier allocations are required. At the first step, we pre-allocate the power under the assumption that every subcarrier multiplexes all *M* users. This is achieved by maximizing an objective function under practical constraints in the OFDM-NOMA VLC system. At the second step, subcarrier allocation is performed using the power allocation obtained at the first step. Specifically, the throughput of each user on each subcarrier is calculated. Then, the *K* users with the highest throughput on a subcarrier are selected as the multiplexed ones on this subcarrier. At the last step, we allocate the power using an objective function and the allocated subcarriers obtained at Step 2. Algorithms 2–4 address different requirements of throughput and user fairness in different scenrarios, and all of them only require the fairness of user’s overall throughput rather than that in every subcarrier.

#### 3.1 Conventional *Algorithm 1*

The first step of Algorithm 1 [17,18] solves the following optimization problem:

At this step, every subcarrier multiplexes all *M* users in order to pre-allocate the power. In Eq. (6a), the log utility function is utilized directly on the capacity of the *k*^{th} decoded user in subcarrier *n* to balance the throughput and fairness of all users in every subcarrier. The frequency-selective effect including the LED bandwidth and the fading in the transmission is considered in the channel response $H_{m,n}^2$. Equation (6b) limits the total average electrical power and is derived from Eq. (2) under the assumption that different users ${s_{{\pi _n}(k),n}}$ are independent. Note that the average electrical power here represents the signal power excluding the DC. Equation (6c) is a limit of optical signal peak power due to the non-negativity of the optical VLC signal and the safety of the human’s eyes. *C* can be derived from Eq. (2) and is equal to min{*I _{DC}*/|

*s*|

*, (*

_{max}*D/β*-

*I*)/|

_{DC}*s*|

*}, where |*

_{max}*s*|

*is the maximal absolute value of ${s_{{\pi _n}(k),n}}$. Equation (6d) ensures the non-negative power. Equation (6e) is the optical power limit for each user. ${C_m} = (C{(\sum\nolimits_{n = 1}^N {H_{m,n}^2} )^{ - \alpha }})/({\sum\nolimits_{m = 1}^M {(\sum\nolimits_{n = 1}^N {H_{m,n}^2} )} ^{ - \alpha }})$ where*

_{max}*α*is an adjustment factor and the right term means that users with worse channel are allocated more power to maintain the user fairness. Note that Eq. (6e) is not required in the proposed Algorithms 2-4, as shown later.

At the second step, based on the optimal *p _{m,n}* obtained at the first step, the achievable rate of each user on each subcarrier can be calculated using Eq. (3). For each subcarrier, we select the

*K*users with the largest rates as the ones to be multiplexed on that subcarrier. The set of subcarriers allocated to the user

*m*is defined as

*Ω*. For example,

_{m}*Ω*= {1, 2, 4} means that user

_{m}*m*is allocated to subcarriers 1, 2, and 4.

At the last step, we update the parameters in Eq. (6) by changing all summation range of *k* from [1, *M*] to [1, *K*] using the allocated subcarriers at Step 2. Then we optimize the power subject to Eqs. (6b–6d). Note that Eq. (6e) is not required at Step 3.

#### 3.2 Proposed *Algorithm 2*

By carefully observing the objective function Eq. (6a) in Algorithm 1, we can find that this function employs the log utility function directly on the capacity of the *k*^{th} decoded user in subcarrier *n*, and then sum over all users and all subcarriers. Physically, it means that the throughput and user fairness should be balanced for all multiplexed users in every subcarrier. However, in practical systems, it is only required to ensure the fairness of the overall throughputs between users. Therefore, we propose to change Eq. (6) as follows:

In Eq. (7a), we firstly sum the throughput of user *m* over all subcarriers, then take the log utility function, and finally sum over all users. Physically, Eq. (7a) tries to ensure the fairness of user’s overall throughput over all subcarriers rather than that in every subcarrier. Equations (7b–7d) are combined with Eq. (7a) for power pre-allocation. Note that different from Algorithm 1, Eq. (6e) is not required in Algorithm 2 as will be verified in simulations. The second step is the same as that in Algorithm 1. At the last step, for each user *m*, we update the parameters in Eq. (7) by changing all summation range of *n* from [1, *N*] to *n*∈*Ω _{m}*. Then we optimize the power in Eq. (7a) subject to Eqs. (7b–7d).

#### 3.3 Proposed *Algorithm 3*

In some cases, fairness between users is more important. Therefore, we propose Algorithm 3 based on the max-min algorithm. The first two steps are the same as those of Algorithm 2. The third step of Algorithm 3 can be expressed as follows:

Equation (8) can be solved using optimization algorithms such as genetic or Whale algorithm. In each iteration, the objective function is *R _{m}* for the user

*m*that has the minimal throughput.

#### 3.4 Proposed *Algorithm 4*

In some scenarios, the total throughput of the OFDM-NOMA-VLC system is the primary consideration. We thus propose Algorithm 4 to address the requirement in these cases. The first two steps of Algorithm 4 are the same as those in Algorithm 2. The third step maximizes the total throughput of all users over all subcarriers. The formula is described as follows:

*P*is the power limit of the user

_{max,m}*m*. Note that Eq. (9a) maximizes the total throughput without considering the fairness. Thus, we add Eq. (9b) to ensure certain fairness so as not to allocate all the power to the user with the best channel.

#### 3.5 Methods to solve the problems in Algorithms 1-4

Several optimization algorithms can be used to solve the non-convex problems in Algorithms 1–4. In this paper, we will investigate two algorithms: the traditional genetic algorithm (GA) [19] and the recently proposed Whale optimization algorithm (WOA) [20]. Because the optimization algorithms are only the mathematical tool to solve Eqs. (6–9) and their principle can be found in the literature [19,20], we do not give the details here. Based on their principles, we can calculate the complexity. Take Algorithm 2 as an example, the required multiplications using GA and WOA are obtained as follows: 1). At the first step of Algorithm 2, the required multiplications using GA and WOA to solve Eq. (7) are *T _{GA}*

_{1}·

*Q*·

*M*·

*N*·(4 + 3/2·

*a*+3·

_{c}*a*) and

_{m}*T*

_{WOA}_{1}·

*Q*·

*M*·

*N*·(4+

*a*+2

_{b}*a*(1-

_{e}*a*) + 2(1-

_{b}*a*)(1-

_{b}*a*)), respectively, where

_{e}*T*

_{GA}_{1}/

*T*

_{WOA}_{1}is the iteration number of GA/WOA at Step 1,

*Q*is the population size,

*a*and

_{c}*a*are the probabilities of crossover and mutation in GA,

_{m}*a*,

_{b}*a*(1-

_{e}*a*) and (1-

_{b}*a*)(1-

_{b}*a*) are the probabilities to adopt Bubble-net attacking, Encircling prey and searching for prey in WOA. 2). Step 2 of Algorithm 2 can use the results at Step 1 without additional complexity. 3). The complexity at Step 3 is similar to that at Step 1 but

_{e}*M*should be changed to

*K*. In this paper,

*a*,

_{c}*a*,

_{m}*a*, and

_{b}*a*are set as 0.5. Therefore, the total required multiplications of Algorithm 2 using GA and WOA are (25/4)·

_{e}*Q*·

*N*·(

*T*

_{GA}_{1}·

*M*+

*T*

_{GA}_{3}·

*K*) and (22/4)·

*Q*·

*N*·(

*T*

_{WOA}_{1}·

*M*+

*T*

_{WOA}_{3}·

*K*), respectively. The complexity for other algorithms can be calculated in a similar way. Although the complexities of GA and WOA do not differ much in their expressions, we will show that WOA converges much more quickly than GA so that it can greatly reduce the complexity by using smaller

*T*

_{WOA}_{1}/

*T*

_{WOA}_{3}. However, GA has a better global searching capability and can obtain a more optimal solution than WOA. In practice, GA is better for fixed or slowly moving users while WOA is suitable for mobile users with real-time resource allocation.

## 4. Simulation setup and results

The VLC simulation model is the same as Fig. 1. We considered the LOS and NLOS channel responses up to the 5^{th} order, which were derived from [12] and obtained using the Monte Carlo method [21]. The cases with *M* of 3, 4, and 5 were studied. In the 5-user case, the users were located at (0.5, 0.5, 0.85), (4, 4, 0.85), (2.5, 2.5, 0.85), (3.625, 1.375, 0.85), (1.75, 1.75, 0.85) all in meters, respectively. The channel conditions in an ascending order are user 1, user 2, user 4, user 5 and user 3. For *M*=3, the 4^{th} and 5^{th} users were not used, and for *M*=4, the 5^{th} user was not used. The number of subcarriers *N* was 8 unless otherwise stated. The number of multiplexed users per subcarrier *K* was set as 2 unless otherwise stated. The adjustment factor *α* in Algorithm 1 was 1*.* In Algorithm 4, the single user power constraint *P _{max},_{m}* was set as

*P*/

_{max}*M*. Detailed parameters are summarized in Table 1.

We firstly investigate the advantage of the proposed algorithms over conventional methods. Figure 2(a) & (b) depict the total throughputs and the Jain’s fairness index for Algorithms 1–4 and the conventional FPA and GRPA algorithms. Note that the conventional FPA and GRPA algorithms assume *K *= *M*, and no subcarrier allocation is employed. To facilitate the comparison, we first adopt Steps 1–2 in Algorithm 1 to allocate subcarriers and then use FPA or GRPA for power allocation. These two algorithms are thus denoted as “Improved FPA” and “Improved GRPA” in the figure. All algorithms have the same average electrical power and optical peak power constraints as in Table 1. The results show that in conventional algorithms, Algorithm 1 exhibits higher throughput and better Jain’s fairness index than the improved FPA and GRPA. The proposed Algorithms 2–4 enhance the capacity with the throughputs all above 90 Mb/s as the number of users varies from 3 to 5. In contrast, the throughput of Algorithm 1 ranges from 80 to 90 Mb/s. This is because Algorithms 2–4 allocates the subcarrier and power by ensuring the fairness in terms of the overall throughput of each user rather than that in every subcarrier. Algorithm 2 provides the optimal solution to balance the total throughput and user fairness. Algorithm 3 can obtain the best user fairness but at the expense of reduced total throughput compared to Algorithm 2. However, its throughput is still higher than that of Algorithm 1. On the other hand, Algorithm 4 achieves the best total throughput in all algorithms but has performance degradation on the user fairness. Therefore, the proposed Algorithms 3 and 4 can meet the FF and TF scenarios, respectively. It is also seen that the advantage of the proposed algorithms is prominent when the number of users is small. This is because users 1–3 have a large difference in channel conditions while users 4–5 have channel conditions between those of users 1–3. Therefore, Algorithm 1 has a higher loss of total throughput in order to ensure the user fairness in every subcarrier when only users 1–3 are employed in the system. When the number of users is 3, the throughputs of the proposed Algorithms 2, 3, and 4 are 20.0%, 17.9%, and 22.5% higher than that of Algorithm 1, respectively.

In order to further explore the reason for the performance improvement of Algorithms 2–4 over Algorithm 1, Fig. 3 shows (a) the overall throughputs of users 1–5 using different algorithms and (b) the Jain’s fairness index for different subcarriers using Algorithms 1–4. Algorithm is abbreviated as “Algo-” in the figure. From Fig. 3(a), we can obtain that the throughputs of different users in Algorithms 1–3 are more even, resulting in better fairness than other algorithms. In particular, the throughputs of all users in Algorithm 3 are almost the same, verifying that Algorithm 3 can achieve the best user fairness. On the other hand, the throughput distributions over users in Algorithms 1–4 have the following characteristics: In Algorithm 1, the users with poorer channel conditions (users 1–2) have high throughputs; In Algorithm 2, the throughputs of users with better channel conditions (users 3–5) are higher; In Algorithm 3, the throughputs of all users are the same; and in Algorithm 4, the throughput of the user with the best channel (user 3) is the highest while that of the user with the worst channel (user 1) is the lowest. Note that concentrating the resources on users with a better channel condition results in a higher total throughput. The throughput distributions over users qualitatively explain why the total throughputs of Algorithms 1–4 in Fig. 2(a) are Algorithm 4, Algorithm 2, Algorithm 3 and Algorithm 1 in the descending order. Figure 3(b) shows that Algorithm 1 exhibits the best user fairness on every subcarrier. Algorithms 2–4 do not guarantee the fairness on the subcarrier level and only target the fairness of the overall throughput between users. Note that the fairness for every subcarrier is unnecessary in practice and Algorithms 2–4 can thus achieve higher throughput and/or better user fairness by relaxing this requirement.

Figures 2,3 are based on the GA to solve the non-convex problems. Figure 4(a) shows the total throughput of Algorithm 2 versus the number of iterations using GA or WOA. It is seen that WOA converges much more quickly and can achieve the saturated performance with less than 10 iterations, which is in contrast to GA that requires ∼ 50 iterations. However, GA can achieve a higher throughput than WOA when the number of iterations is large, due to the superior global searching capability. Figure 4(b) depicts the throughputs of Algorithms 1–4 using GA or WOA. It is confirmed that GA is better than WOA in terms of the performance while the improvement of the proposed Algorithms 2–4 is valid regardless of the optimization algorithms. In practical applications, GA is more suitable for fixed or slowly moving users while WOA is better for mobile users that require real-time resource allocations with a low latency. In the following, we adopt GA in order to obtain better solutions for the algorithms.

As aforementioned in Section 3, the constraint of Eq. (6e) plays a vital role in Algorithm 1, but it is not required in Algorithms 2–4. Figure 5 depicts the total throughputs and the Jain’s fairness index versus *M* for different algorithms with and without Eq. (6e). It is verified that when Eq. (6e) is not used in Algorithm 1, the throughput can be improved but the user fairness is degraded significantly. The reason is that some users may be allocated with very few subcarriers, leading to severe degradation in the user fairness. In contrast, Eq. (6e) has little influence on the subcarrier allocation of Algorithms 2–4, and both throughput and fairness performance can be maintained even without Eq. (6e). This is because Algorithms 2–4 target the fairness of users in terms of the overall throughput instead of that in every subcarrier. Consequently, each user can concentrate its power on some subcarriers without considering the fairness within these subcarriers. This facilitates both subcarrier and power allocation. Therefore, Algorithms 2–4 without Eq. (6e) can alleviate the problem of allocating some users with very few subcarriers, and even when it happens, they can concentrate the power on these subcarriers to maintain the fairness of the overall throughput. On the other hand, when Eq. (6e) is not used in Algorithm 1, the power of each user is not concentrated enough on subcarriers so that some users may be allocated with more subcarriers while some may be allocated with very few subcarriers. Because power allocation should also ensure user fairness in every subcarrier, the overall throughputs of users allocated with more/less subcarriers would be higher/lower and so the user fairness is degraded.

In the above analysis, we have proved that the proposed algorithms have better performance than conventional algorithms under the constraints of both average electrical signal power *P _{max}* and optical signal peak power

*C*. As shown later,

*P*and

_{max}*C*in Table 1 can be simultaneously reached for Algorithms 1–4 when

*K*and

*N*are 2 and 8, respectively. However, there are cases that the actual electrical power consumption cannot reach the power constraint

*P*due to the limitation of the optical signal peak power

_{max}*C*, or vice verse. The first case matches practical applications the most because the signal peak power is intrinsically limited by the DC bias or the linear operation region of the LED/LD while the average electrical signal power is generally more flexible given a sufficient output power/gain of the driving amplifier. Note that the electrical/optical power here is the AC signal power excluding the DC and is not equivalent to the illumination power of LED/LD that is mainly determined by the DC. In practice, the average electrical signal power should be maximized to achieve the highest throughputs provided that the signal peak voltage is within the linear region and does not exceed the DC bias. In fact, the relation between the average electrical signal power and the optical signal peak power implies the PAPR. In the following, we will show that the proposed Algorithms 2–4 result in designs with lower PAPRs than those using conventional algorithms when

*K*or

*N*increases. Consequently, Algorithms 2–4 can make better use of the average electrical signal power when the system is mainly limited by the optical signal peak power and thus further improve the performance for a larger

*K*or a larger

*N*.

We first study the performance of different algorithms when the number of multiplexed users per subcarrier changes. Figure 6 depicts the total throughputs and user fairness versus *K* by using different algorithms. Improved FPA and GRPA in the figure employ Algorithm 1 for subcarrier allocation and the conventional FPA and GRPA for power allocation. It is seen that for *K* = 1 where the system degenerates to OMA, the user fairness of all algorithms is limited because there is no sufficient freedom to evenly allocate users over subcarriers. It is also observed that the total throughput of Algorithm 4 for *K*=1 is much lower than that in the case of *K*=2. This is because the power limit for each user *P _{max}*

_{,m}in Eq. (9b) is set as

*P*/

_{max}*M*, which limits the power of users with better channel conditions and so the total throughput. For

*K*> 1 that corresponds to NOMA, the total throughputs of FPA and GRPA decrease as

*K*increases. As shown later, one reason is that the PAPR increases with

*K*by using FPA and GRPA, and under the limit of optical signal peak power

*C*, the actual electrical signal power consumption is reduced. In addition, these two conventional algorithms use the fixed power allocation rule that diverges more from the optimal solution as

*K*increases. Therefore, both the total throughputs and user fairness are reduced. On the other hand, the total throughput of Algorithm 1 also decreases significantly as

*K*increases. This is because this algorithm needs to ensure the user fairness in every subcarrier, which also results in designs with higher PAPRs for larger

*K*and so reduces the actual electrical power consumption under the limit

*C*. In contrast, the proposed Algorithms 2–4 exhibit better and stable performance as

*K*changes, because they have more freedom to allocate subcarriers and powers to make full use of the electrical signal power under the limit

*C*. Regardless of

*K*, Algorithms 2–4 always have the best balance between the total throughput and user fairness, the best fairness, and the best total throughput, respectively. This confirms the suitability of the proposed algorithms to address the requirement of corresponding application scenarios. When

*K*is 5, the throughputs of the proposed Algorithms 2–4 are 70.7%, 67.8%, and 80.2% higher than that of Algorithm 1, respectively.

In order to better understand the results in Fig. 6, Fig. 7(a) shows the actual electrical signal power consumption under the limit of *P _{max}* and

*C*. For

*K*=1, the PAPRs of all algorithms are low. The system is limited by

*P*in this case and the actual power consumption is the same as

_{max}*P*. However, the PAPRs of Algorithm 1, FPA and GRPA increase as

_{max}*K*increases. The system becomes limited by

*C*instead of

*P*for large

_{max}*K*and the actual electrical power consumptions decrease, resulting in reduced throughputs in Fig. 6(a). Note that although Algorithm 1 has less utilized electrical power than conventional FPA and GRPA, it has a better power allocation rule and so in Fig. 6, it can still achieve a similar total throughput as the conventional FPA and GRPA. On the other hand, Algorithms 2–4 can make full use of the electrical power

*P*regardless of

_{max}*K*, thus ensuring the stability of the total throughputs. We further qualitatively investigate why the actual electrical power consumptions of different algorithms vary. Figure 7(b) shows the Jain’s fairness index of the allocated power

*p*obtained by different algorithms. It is seen that when

_{m,n}*K*is 2, the fairness indices of

*p*in GRPA and Algorithm 1 are higher, that is

_{m,n}*p*is more evenly distributed over subcarriers, while the corresponding consumed electrical powers of GRPA and Algorithm 1 in Fig. 7(a) are smaller. Similarly, for

_{m,n}*K*of 5, the fairness index of

*p*in Algorithm 1 is the highest and the corresponding electrical power consumption in Fig. 7(a) is the lowest. This phenomenon can be explained by analyzing Eq. (7b) and Eq. (7c). Due to the square root relationship between the average electrical power and the optical signal peak power, it can be derived that for a fixed

_{m,n}*C*, the electrical power consumption

*P*is minimized when

*p*are the same for all

_{m,n}*m*and

*n*. That is, the PAPR is the highest when the power is evenly distributed over subcarriers and users or the fairness index is the highest.

Figure 8(a)-(b) shows the total throughput versus *P _{max}* when the number of multiplexed users

*K*is 2 and 5, respectively. When

*P*is small, the system is limited by the average electrical signal power. As

_{max}*P*increases, the total throughputs of all algorithms increase. However, the throughput cannot be increased infinitely due to the constraint of optical signal peak power

_{max}*C*, resulting in performance saturation. When

*K*=2, i.e. in Fig. 8(a), the performances of the conventional Algorithm 1, FPA and GRPA saturate at

*P*= 0.02 W while those of Algorithms 2–4 saturate at

_{max }*P*of 0.025∼0.03 W. This implies that the system using Algorithms 2–4 is still limited by

_{max}*P*instead of

_{max}*C*when

*P*= 0.02 W. It is also observed that even when

_{max }*P*= 0.01 W at which all algorithms are limited by the average electrical signal power

_{max }*P*, the proposed Algorithms 2–4 are still better than conventional FPA, GRPA and Algorithm 1 due to the superior allocation rule of subcarriers and powers. On the other hand, in Fig. 8(b), the performances of improved FPA, improved GRPA, and Algorithm 1 saturate more quickly, because their PAPRs increase for large

_{max}*K*and so the system is easily limited by the optical signal peak power

*C*. In contrast, the proposed Algorithms 2–4 can keep low PAPRs. When

*P*= 0.02 W, the throughputs do not saturate, implying that the system is still limited by

_{max }*P*so that all available electrical power can be utilized, as depicted in Fig. 7(a). The improvement of Algorithms 2–4 over conventional algorithms in this case is from not only the superior allocation rule but also better use of electrical signal power. It is also deduced that the performance improvement of the proposed Algorithms 2–4 in Figs. 2–5 can be further enhanced if the constraint

_{max}*P*is discarded.

_{max}Then we study the performance of different algorithms when the number of subcarriers changes. Figure 9 shows (a) the total throughput and (b) the Jain’s fairness index as a function of the number of subcarriers *N*. It is seen that as *N* increases, the throughputs of Algorithm 1, the improved FPA and the improved GRPA decrease in a similar way as in Fig. 6. This is also due to the reduced electrical signal power consumption as will be verified later. In contrast, the proposed Algorithms 2–4 maintain stable throughputs and user fairness and so their performance advantages over conventional algorithms become more prominent for a larger *N*. When *N* is 16, the throughputs of Algorithms 2–4 are 62.17%, 53.35%, and 67.25% higher than that of Algorithm 1, while the user fairness is improved by 4.64%, 7.87%, and degraded by 20.71%, respectively.

We further explore the reason why the total throughputs of Algorithm 1, the improved FPA and improved GRPA in Fig. 9(a) decrease with the increase of *N*. Figure 10(a) shows the actual electrical power consumption versus *N* under the limit of *P _{max}* and

*C*. It can be seen that the actual electrical signal power consumption of Algorithm 1, FPA and GRPA decreases with the increase of

*N*due to higher PAPRs. On the other hand, the proposed Algorithms 2–4 show more stable actual electrical power consumption. Figure 10(b) depicts the total throughputs versus

*P*for different algorithms when the number of subcarriers is 12. It can be seen that the throughputs of conventional algorithms become saturated when

_{max}*P*is around 0.01 W because they have higher PAPRs and are easily limited by the peak power

_{max}*C*. In contrast, the proposed Algorithms 2–4 reach the saturation region at a higher

*P*and so exhibit better performance.

_{max}## 5. Conclusion

We have proposed three novel joint subcarrier and power allocation algorithms to improve the throughputs and/or user fairness in OFDM-NOMA-VLC systems. The proposed Algorithms 2–4 address the requirements in FTB, FF, and TF scenarios, respectively. All the proposed algorithms have improved objective function to eliminate the tight and redundant constraint of guaranteeing the user fairness in every subcarrier in the previous joint allocation algorithm and can also make better use of the electrical signal power under the limit of optical signal peak power by reducing the PAPR. Simulations verify that the proposed Algorithms 2–4 are superior to Algorithm 1 and the conventional FPA and GRPA algorithms. The improvement is particularly prominent when the system is limited by the optical signal peak power. When the total user number, the number of the multiplexed users per subcarrier, and the number of subcarriers are 5, 2, and 16, respectively, the throughputs of the three proposed algorithms are 62.17%, 53.35%, and 67.25% higher than the conventional joint algorithm, while the user fairness is improved by 4.64%, 7.87%, and degraded by 20.71%, respectively. Therefore, the proposed Algorithms 2–4 can address the requirements in FTB, FF, and TF scenarios, respectively.

## Funding

Science and Technology Planning Project of Guangdong Province (2019A050503003); National Natural Science Foundation of China (61971199); Natural Science Foundation of Guangdong Province (2021A1515012309); Fundamental Research Funds for the Central Universities (D2193110).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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