Differential evolution-based optimal power allocation scheme for NOMA-VLC systems

Zanyang Dong; Tao Shang; Qian Li; Tang Tang

doi:10.1364/OE.397761

1. Introduction

Driven by the service requirements of astoundingly enhanced data rate and ubiquitous “Internet of Things” in 5G, even 6G communication systems, many emerging technologies such as ultra-dense networks, massive multiple-input multiple-output, mmWave, terahertz, and visible light communication (VLC) have drawn great attention in both academia and industry [1]. As for VLC, lots of inherent advantages determine its significance in complementing radio frequency (RF) communications, such as high capacity, enhanced confidentiality, anti-electromagnetic interference, unlicensed spectrum, and green energy saving [2].

In VLC, along with many other technologies such as orthogonal frequency division multiplexing, parallel communication schemes, advanced modulation schemes, equalization and so on, multiple access has been widely studied to guarantee simultaneous network access of multiple users [3]. Recently, non-orthogonal multiple access (NOMA), which allows to share the entire time-frequency resources among multiplexing users and superimposes the signals of different users in the power domain, has been proposed as a promising technology in 5G cellular systems compared to conventional orthogonal multiple access (OMA) [4–6]. Fortunately, NOMA proves to be ideally suited to VLC systems and abundant research achievements have been obtained [7–10]. As a crucial issue in both RF- and VLC- based NOMA systems, power allocation scheme has attracted much attention. In NOMA cellular systems, fixed power allocation (FPA) and fractional transmit power allocation are two basic and well investigated schemes [5,11]. The former allocates preset values to the scheduled users as the power ratios, while the latter calculates the user power ratios according to their channel qualities. Likewise, in NOMA-VLC systems, authors of [12] investigate the system performance based on FPA scheme. In [13] and [14], gain ratio power allocation (GRPA) and normalized gain difference power allocation are proposed to compensate for channel differences among multiplexed users.

Furthermore, power allocation optimizations concerning user fairness, as well as sum rate, have been addressed in many prior works. Most of them consider proportional fairness [11,15,16]. In [11] and [15], for NOMA cellular systems, iterative water-filling based power allocation and tree-searching based transmission power allocation are respectively proposed to maximize the sum of ratios of the instantaneous user rate to its average rate. In [16], for NOMA-VLC systems, the sum logarithmic utility function of user rate is maximized to guarantee the user fairness. Yet there are also some works advocating user fairness in the max-min sense [17,18]. In [17], for NOMA cellular systems, the authors formulate a quasi-concave optimization problem to maximize the minimum user rate, and prove that all the multiplexed users get the same data rate when the optimal solution is achieved. In [18], for NOMA-VLC systems, the maximization of minimum user rate is also pursued by gradient projection algorithm under quality of service (QoS) constraint. In addition, the work in [19] provides another fairness index with all the multiplexed users achieving a capacity at least as good as OMA by limiting the power ratio within a certain range. Nonetheless, in these works, the numeric scope of normalized user rates is indeterminate, i.e., vague fairness, except those regarding max-min fairness where the normalized rates of multiplexed users are all equal to 1. However, the max-min fairness only corresponds to the special case where the scope is monotropic and cannot be flexibly adjusted. Moreover, proportional fairness aims to maximize the long-term averaged user rates, thus targeting long-term fairness. However, from the short-term perspective, proportional fairness can suffer from severe rate fluctuations over time.

In this paper, we set the gap between minimum rate and maximum rate among all users as the constraint to guarantee determinate numeric scope of normalized user rates, under which we formulate an optimization problem to maximize sum rate, yielding user fairness-guaranteed sum rate maximization. In other words, we define user fairness as the ratio of minimum rate to maximum rate among all users. Such definition of user fairness makes it explicit in a certain scenario and flexible as a constraint. Moreover, this optimization problem can guarantee real-time user fairness due to fast convergence towards our preset user fairness requirement in each scheduling frame. Considering administrators’ subjective preference between sum rate and user fairness, we formulate another relative optimization problem, i.e., sum rate-guaranteed user fairness maximization, wherein the threshold guaranteeing sum rate can also be flexibly adjusted according to the actual requirement. In addition, the two proposed schemes can be applied to any system with any scheduling and user-pairing approach, therefore all users will have equal opportunity to be scheduled, thus guaranteeing fair channel access ratios from a time-sharing perspective as well. In the following, in order to solve the two non-convex problems, we propose two corresponding power allocation algorithms through classified discussion and mathematical derivation, and verify the superiority of them over conventional schemes through numerical simulation.

2. System model

For illustrative purposes, we assume a single optical attocell consisting of one optical access point (AP) and many randomly distributed users, wherein M users are multiplexed in one scheduling frame, i.e., ${U_1}, \cdots ,{U_M}$.$M = 1$ corresponds to OMA scheme while $M \ge 2$ corresponds to NOMA. Moreover, larger M means enhanced spectral efficiency and increased connectivity due to user diversity gain, and yet more sophisticated successive interference cancellation (SIC) receivers with higher computational power and delay. Without loss of generality, we sort all the multiplexed users in an ascending order of channel gain, i.e., ${h_1} \le \cdots \le {h_k} \le \cdots \le {h_M},k \in {\cal M} = \{{1,2, \cdots ,M} \}$. Specifically, with the assumption that the light source follows the generalized Lambertian radiation pattern and each photodiode (PD) faces vertically upwards, the channel gain for ${\textrm{U}_k}$ would be ${h_k} = C(m + 1){H^{m + 1}}/{(r_k^2 + {H^2})^{(m + 3)/2}}$, where $m$ denotes AP Lambertian order, $H$ denotes the vertical distance between the AP and the receiving plane of these users, ${r_k}$ denotes the horizontal separation of ${\textrm{U}_k}$ from the AP, and $C = A{R_p}{T_s}({\psi _k})g({\psi _k})/(2\pi )$, where $A$ and ${R_p}$ denote the receiving area and the responsivity of PD, ${\psi _k}$ denotes the angle of incidence at ${\textrm{U}_k}$, ${T_s}({\psi _k})$ and $g({\psi _k})$ denote the gain of optical filter and concentrator. Let $n$ denote the refractive index in optical concentrator, and ${\psi _{FOV}}$ denote the field of view (FOV) of PD.$g({\psi _k})$ is given by ${n^2}/{\sin ^2}{\psi _{FOV}}$ if ${\psi _k} \le {\psi _{FOV}}$, otherwise it is 0.

Focusing on the downlink of a multi-user VLC system, we present the data transmission process based on NOMA. Let ${\cal S} = \{{{s_i}} , {i \in {\cal M}} \}$ denote the messages intended for all users correspondingly, then the superposed signal of ${\cal S}$ through power-domain superposition with associated electrical power values $\{{{P_i}} , {i \in {\cal M}} \}$ and addition of a direct current bias ${I_{DC}}$ would be:

(1)$$x = \sum\nolimits_{1 \le i \le M} {\sqrt {{P_i}} } {s_i} + {I_{DC}}.$$

Note that ${P_i} \ge {P_{i + 1}},i \in {\cal I} = \{{1, \cdots ,M - 1} \}$. Moreover, let ${P_{elec}}$, ${a_i}$ be total electrical power for ${\cal S}$ and power allocation coefficient for ${s_i}$, respectively, hence, ${P_i} = a_i^2{P_{elec}}$, $\sum\nolimits_{\textrm{1} \le i \le M} {a_i^2\textrm{ = 1}}$, $a_1^2 \ge \cdots \ge a_M^2$. At user ${U_k}$ side, considering VLC channel state and removing the direct current term, the received signal ${y_k}$ can be calculated by:

(2)$${y_k} = {h_k}(\sum\nolimits_{1 \le i \le M} {\sqrt {{P_i}} } {s_i}) + {z_k},$$

where ${z_k}$ denotes the additive white Gaussian noise whose power spectral density denoted by ${N_0}$. According to the SIC process, user ${U_k}$ can decode ${s_1}, \cdots ,{s_k}$ in sequence. When decoding ${s_i}$ with $i \le k$, the interference from users with lower order than $i$ can be removed completely while the interference from others must be treated as noise. After all, the SINR for user ${U_k}$ can be obtained and the corresponding achievable data rate can be calculated by:

(3)$${R_k} = \left\{ \begin{array}{l} \beta {\log_2}{\kern 1pt} [1 + {({h_k}{a_k})^2}/(\sum\nolimits_{k + 1 \le j \le M} {{{({h_k}{a_j})}^2} + 1/\rho } )],{\kern 1pt} \;k \in {\cal I},\\ \beta {\log_2}{\kern 1pt} [1 + \rho {({h_k}{a_k})^2}],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} \;\;\;\;\;\;{\kern 1pt} {\kern 1pt} {\kern 1pt} k = M, \end{array} \right.$$

where $\rho = {P_{elec}}/({N_0}B)$, $\beta = B/2$, $B$ denotes the channel bandwidth, and the scaling factor 1/2 results from the constraint of real-valued signal, i.e., Hermitian symmetry.

3. Optimal power allocation

For the first time, we set the numeric scope of normalized user rates as $[\eta ,1]$ and define user fairness as $\eta$, which equals to the gap between minimum rate and maximum rate among all users, i.e., $\textrm{min(}{\cal R}\textrm{)/max(}{\cal R}\textrm{)}$ with ${\cal R}\textrm{ = }\{{{R_k}} , {k \in {\cal M}} \}$. On this basis, we formulate two optimization problems according to different application requirements.

3.1 User fairness-guaranteed max-sum rate criterion

By optimizing the power allocation coefficients (positive values), i.e., ${\cal A} = \{{{a_i}} , {i \in {\cal M}} \}$, we aim to maximize the sum rate of all users while satisfying the target threshold of $\eta$. Mathematically, the corresponding optimization problem can be formulated as follows:

(4)$$\begin{array}{l} \mathop {\max }\limits_{\cal A} {R_{\textrm{sum}}},\;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s.t.\;\;\sum\nolimits_{1 \le i \le M} {a_i^2} = 1;\;\;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\textrm{constraint}\;1)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_1} \ge \cdots \ge {a_M} \ge 0;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\textrm{constraint}\;2)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} {R_k} \ge {{\tilde{R}}_k},\;\;{\kern 1pt} k \in {\cal M};\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} {\kern 1pt} (\textrm{constraint}\;3)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} \eta \ge {C_1},\;\;{C_1}\;\textrm{is}\;\textrm{a}\;\textrm{constant,}\;\textrm{and}\;{C_1} \in [0,1].{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\textrm{constraint}\;4){\kern 1pt} \end{array}$$

${\tilde{R}_k}$ denotes the target rate of user ${U_k}$, which is determined by its QoS requirement. The constant ${C_1}$ is the target threshold which can be flexibly set according to practical application. Obviously, the optimization problem (4) is non-convex due to the non-convexity of the objective function and constraint 4. The problem (4) can be classified and discussed according to different ${C_1}$. Towards this direction, we state propositions 1, 2 and 3. Note that when $M = \textrm{2}$, the power allocation coefficients and consequently the sum rate should be uniquely obtained by solving the following equations:

(5)$${R_\textrm{1}}\textrm{/}{R_\textrm{2}}\textrm{ = }\eta \;\textrm{and}\;a_1^2 + a_2^2 = 1,$$

without the need to perform optimization.

Proposition 1: Through a sequence of algebraic expressions and size comparisons, we could determine whether the sum rate maximization problem (4) is feasible without constraint 4. If not, a system outage would occur. If feasible, the optimal solution without constraint 4 $\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$ can be obtained during the above process. Then, when ${C_1} \le \sigma$, where $\sigma$ equals to $\min \{{{{\tilde{R}}_1}, \cdots ,{{\tilde{R}}_{M - 1}},\beta {{\log }_2}{\kern 1pt} [1 + \rho {{({h_M}a_M^\ast )}^2}]} \}/\max \{{{{\tilde{R}}_1}, \cdots ,{{\tilde{R}}_{M - 1}},\beta {{\log }_2}{\kern 1pt} [1 + \rho {{({h_M}a_M^\ast )}^2}]} \}$, the optimal solution to (4) under all constraints is $\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$.

Proof: see Appendix A.

Proposition 2: When ${C_1} = 1$, that is, the rates of all users are the same (i.e., ${R_i} = v,i \in {\cal M}$), $v$ is the root of a nonlinear equation, and $v \in [0,\beta {\log_2}(1 + \rho h_M^2)]$. By approximately solving $v$ through a bisection procedure, the solution to (4) without constraint 3 can be obtained subsequently through a sequence of algebraic expressions. If $v \ge \max \{{{{\tilde{R}}_k},k \in {\cal M}} \}$, the above solution would be the final solution to (4) under all constraints. Otherwise, the problem (4) is not feasible and a system outage would occur.

Proof: see Appendix B.

Proposition 3: In other cases of ${C_1}$($\sigma < {C_1} < 1$), when the optimal solution is reached, it’s unwarranted that the user with the best channel condition achieves a rate of ${R^ \ast }$ while all the other users achieve that of ${C_1}{R^ \ast }$(condition 1), or the user with the worst channel condition achieves a rate of ${C_1}{R^ \ast }$ while all the other users achieve that of ${R^ \ast }$(condition 2). That’s to say, the problem cannot be simplified to maximization of an auxiliary variable ${R^ \ast }$.

Proof: see Appendix C.

Therefore, when the problem (4) is feasible without constraint 4 and $\sigma < {C_1} < 1$, we adopt differential evolution (DE) based heuristic algorithm to realize the optimization. Moreover, there are four highlights worthy of note during our use of DE algorithm. First, note that DE is mainly used for optimization of continuous variable, hence, we introduce auxiliary variables, i.e., power allocation factor $\alpha$, and apply variable transformation concerning constraints 1 and 2. Specifically, ${\alpha _{i(i + 1)}}$ is defined as ${\alpha _{i(i + 1)}} = a_{i + 1}^2/a_i^2,i \in {\cal I}$, and constraints 1 and 2 are equivalent to the following:

(6)$$\left\{ {\begin{array}{{c}} {a_j^2 = 1/(1 + \sum\nolimits_{1 \le k \le M - 1} {(\prod\nolimits_{1 \le i \le k} {{\alpha_{i(i + 1)}}} } )),\;\;\;\;j = 1,}\\ {a_j^2 = \prod\nolimits_{1 \le i \le j - 1} {{\alpha_{i(i + 1)}}} /(1 + \sum\nolimits_{1 \le k \le M - 1} {(\prod\nolimits_{1 \le i \le k} {{\alpha_{i(i + 1)}}} } )),\;j = 2, \cdots M, } \end{array}} \right.$$

(7)$${\alpha _{i(i + 1)}}\textrm{ is}\;\textrm{continuous, }{\alpha _{i(i + 1)}} \in [0,1],\;i \in {\cal I}.$$

Second, the initial population $pop[NP][M - 1]$ is randomly generated according to Eq. (7). $NP$ denotes the population size, and each individual corresponds to one set of $\{{{\alpha_{i(i + 1)}},i \in {\cal I}} \}$, which contains $M - 1$ elements. In this process, we determine whether each individual satisfies constraint 4 according to Eqs. (3) and (6). For the individual which does not satisfy constraint 4, assuming the $u$-th user and the $w$-th user achieve minimum rate and maximum rate among all users, respectively, we adjust the individual iteratively until it is satisfactory, and the adjustment rule is as follows:

(8)$$\begin{array}{l} a{_u^{ \ast 2}} = a_u^2 + ( a_w^2 - a_u^2)/16,\;if\;u - w = 1;{\kern 1pt} \;or\;a_u^2 + a_w^2/8,\;\textrm{else}\,{\textrm{if}}\;u = 1\;\textrm{and}\;w = M;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} or\;a_u^2 + \min (a_w^2/8,(a_{u - 1}^2 - a_u^2)/8),\;\textrm{else}\,{\textrm{if}}\;w = M;\;or\;a_u^2 + (a_w^2 - a_{w + 1}^2)/8,\;\textrm{else}if\;u = 1;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} or\;a_u^2 + \min (a_{u - 1}^2 - a_u^2,a_w^2 - a_{w + 1}^2)/8,\;\textrm{else}\textrm{.}{\kern 1pt} \\ a{_w^ {\ast 2}} = a_u^2 + a_w^2 - a{_u^ {\ast 2}}. \end{array}$$

where, $a_{u - 1}^{}$, $a_u^{}$, $a_w^{}$, and $a_{w + 1}^{}$ denote the power allocation coefficients before adjustment for users ${U_{u - 1}}$, ${\textrm{U}_u}$, ${\textrm{U}_w}$, and ${\textrm{U}_{w + 1}}$, respectively;$a_u^\ast $ and $a_w^\ast $ denote the power allocation coefficients after adjustment for users ${\textrm{U}_u}$ and ${\textrm{U}_w}$, respectively. Third, following the normal mutation and crossover, which involve the scaling factor $F$ and the crossover probability $Cr$ respectively, we perform the selection operation with additional decision conditions. Specifically, we select the better individual according to the following steps: 1) the individual which satisfies constraint 3 is the better one; 2) if both the individuals satisfy constraint 3, or neither of the individuals satisfies constraint 3, the individual which satisfies constraint 4 is the better one; 3) if the two individuals are neck and neck in the previous steps, the one which can lead to higher sum rate is the better one. Fourth, after $G$ generations of evolution, we choose the individuals that satisfy both constraint 3 and constraint 4 from the final population, and form a set $\Upsilon $. If $\Upsilon = \emptyset$, the problem (4) is not feasible and a system outage would occur. Otherwise, the optimal solution to problem (4) would be the element of $\Upsilon $ that leads to the highest sum rate.

The pseudocode of algorithm 1 is as follows.

1: Init.$M$, $H$, $m$, $A$, ${R_p}$, ${\psi _{FOV}}$, $n$, ${P_{elec}}$, $B$, ${N_0}$, ${\tilde{R}_1}, \cdots ,{\tilde{R}_M}$, ${r_1}, \cdots ,{r_M}$, $NP$, $G$, $F$, $Cr$, ${C_1}$

2: Calculate ${h_1}, \cdots ,{h_M}$, and then $a_1^{\ast }, \cdots ,a_M^{\ast }$

3: If not ($0 < {(a_k^{\ast })^2} < 1,\;\forall k \in {\cal M}$ and $\beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}] \ge {\tilde{R}_M}$) then

4: System outage

5: else

6: If (${C_1} \le \sigma$) then

7: Return$\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$

8: elseif (${C_1} = 1$) then

9: Solve $v$ through a bisection procedure

10: If ($v \ge \max \{{{{\tilde{R}}_k},k \in {\cal M}} \}$) then

11: Return$\{{{a_1}, \cdots ,{a_M}} \}$ calculated by Eqs. (16) and (17)

12: else

13: System outage

14: else

15: for ($i = 1$ to $NP$) do

16: $pop(i,:) = rand(1,M - 1)$; Calculate $\eta$ induced by $pop(i,:)$

17: If ($\eta < {C_1}$) then

18: Adjust $pop(i,:)$ according to Eq. (8)

19: for ($gen = 1$ to $G$) do

20: for ($i = 1$ to $NP$) do

21: $Hpop(i,:) = Mutation\;\{ pop\}$;$Vpop(i,:) = Crossover\;\{ pop(i,:),Hpop(i,:)\}$

22: Judge which one of $pop(i,:)$ and $Vpop(i,:)$ is the better individual

23: If ($Vpop(i,:)$ is better) then

24: $pop(i,:) \leftarrow Vpop(i,:)$

25: $flag = 0$, $Rsum = zeros(1,NP)$

26: for ($i = 1$ to $NP$) do

27: If ($pop(i,:)$ satisfies constraints 3 and 4) then

28: $flag \leftarrow flag + 1$;$Rsum(i) \leftarrow$ the sum rate induced by $pop(i,:)$

29: If ($flag = 0$) then

30: System outage

31: else

32: Find the maximum $Rsum({i^\ast })$ in $Rsum$

33: Return $\{{{a_1}, \cdots ,{a_M}} \}$ calculated by $pop({i^\ast },:)$ according to Eq. (6)

3.2 Sum rate-guaranteed max-user fairness criterion

Mathematically, the corresponding optimization problem can be formulated as follows:

(9)$$\begin{array}{l} \mathop {\max }\limits_{\cal A} \eta ,\;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} s.t.\;\;\sum\nolimits_{1 \le i \le M} {a_i^2} = 1;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\textrm{constraint}\;1)\;\;\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {a_1} \ge \cdots \ge {a_M} \ge 0;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (\textrm{constraint}\;2)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} {R_k} \ge {{\tilde{R}}_k},\;\;{\kern 1pt} k \in {\cal M};\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} (\textrm{constraint}\;3)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{R_{\textrm{sum}}} \ge {C_2}.{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\kern 1pt} (\textrm{constraint}\;5) \end{array}$$

The constant ${C_2}$ is set artificially to adjust the target threshold of sum rate. Similarly, according to constraints 1, 2, and 3, $a_1^{\ast }, \cdots ,a_M^{\ast }$ can be calculated in sequence. Then, the solution can be discussed in the following cases: 1) if it does not hold that $0 < {(a_k^{\ast })^2} < 1,\;\forall k \in {\cal M}$ and $\beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}] \ge {\tilde{R}_M}$, the problem (9) is not feasible and there occurs a system outage; 2) if case 1 is invalid, i.e., the problem (9) is feasible without constraint 5, the maximum sum rate would be $R_{\textrm{sum}}^{(\max )} = \sum\nolimits_{1 \le k \le M - 1} {{{\tilde{R}}_k}} + \beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}]$, as mentioned in section 3.1. Hence, when ${C_2} > R_{\textrm{sum}}^{(\max )}$, the problem (9) is not feasible under all constraints and a system outage would occur; 3) if both case 1 and case 2 are invalid, i.e., the problem (9) is feasible under all constraints, when $Mv \ge {C_2}$ and $v \ge \max \{{{{\tilde{R}}_k},k \in {\cal M}} \}$, where $v$ can be obtained as in section 3.1, the user fairness $\eta$ can reach the maximum of 1, and the optimal solution to problem (9) can be obtained through a sequence of algebraic expressions as mentioned in section 3.1; 4) in other cases, DE based heuristic algorithm is also adopted, and there are three differences with this round of usage. First, in population initialization process, if an individual does not satisfy constraint 5, assuming the $u$-th user possesses the lowest order among the ones whose power allocation coefficients are the smallest, and the $w$-th user possesses the highest order among the ones whose power allocation coefficients are the biggest, we adjust the individual iteratively following another rule expressed as:

(10)$$\begin{array}{l} a{_u^ {\ast 2}} = a_u^2 + (a_w^2 - a_u^2)/4,\;if\;u - w = 1;\;or\;a_u^2 + \min (a_{u - 1}^2 - a_u^2,a_w^2 - a_{w + 1}^2),\;else,\\ a{_w^ {\ast 2}} = a_u^2 + a_w^2 - a{_u^ {\ast 2}}, \end{array}$$

where, $a_{u - 1}^{}$, $a_u^{}$, $a_w^{}$, $a_{w + 1}^{}$, $a_u^\ast $, and $a_w^\ast $ are defined above. Second, during the selection operation of each iteration, we select the better individual according to the following steps: 1) the individual which satisfies constraint 3 is the better one; 2) if both the individuals satisfy constraint 3, or neither of the individuals satisfies constraint 3, the individual which satisfies constraint 5 is the better one; 3) if the two individuals are neck and neck in the previous steps, the one which can lead to higher user fairness is the better one. Third, after $G$ generations of evolution, we choose the individuals that satisfy both constraint 3 and constraint 5 from the final population, and form a set $\Lambda $. If $\Lambda = \emptyset$, the problem (9) is not feasible and a system outage would occur. Otherwise, the optimal solution to problem (9) would be the element of $\Lambda $ that leads to the highest user fairness.

The pseudocode of algorithm 2 is as follows.

1: Init.$M$, $H$, $m$, $A$, ${R_p}$, ${\psi _{FOV}}$, $n$, ${P_{elec}}$, $B$, ${N_0}$, ${\tilde{R}_1}, \cdots ,{\tilde{R}_M}$, ${r_1}, \cdots ,{r_M}$, $NP$, $G$, $F$, $Cr$, ${C_1}$

2: Calculate ${h_1}, \cdots ,{h_M}$, and then $a_1^{\ast }, \cdots ,a_M^{\ast }$

3: If not ($0 < {(a_k^{\ast })^2} < 1,\;\forall k \in {\cal M}$ and $\beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}] \ge {\tilde{R}_M}$) then

4: System outage

5: else

6: If (${C_2} > R_{\textrm{sum}}^{(\max )}$) then

7: System outage

8: else

9: Solve $v$ through a bisection procedure

10: If ($Mv \ge {C_2}$ and $v \ge \max \{{{{\tilde{R}}_k},k \in {\cal M}} \}$) then

11: Return$\{{{a_1}, \cdots ,{a_M}} \}$ calculated by Eqs. (16) and (17)

12: else

13: for ($i = 1$ to $NP$) do

14: $pop(i,:) = rand(1,M - 1)$; Calculate ${R_{\textrm{sum}}}$ induced by $pop(i,:)$

15: If (${R_{\textrm{sum}}} < {C_2}$) then

16: Adjust $pop(i,:)$ according to Eq. (10)

17: for ($gen = 1$ to $G$) do

18: for ($i = 1$ to $NP$) do

19: $Hpop(i,:) = Mutation\;\{ pop\}$;$Vpop(i,:) = Crossover\;\{ pop(i,:),Hpop(i,:)\}$

20: Judge which one of $pop(i,:)$ and $Vpop(i,:)$ is the better individual

21: If ($Vpop(i,:)$ is better) then

22: $pop(i,:) \leftarrow Vpop(i,:)$

23: $flag = 0$, $\eta = zeros(1,NP)$

24: for ($i = 1$ to $NP$) do

25: If ($pop(i,:)$ satisfies constraints 3 and 5) then

26: $flag \leftarrow flag + 1$;$\eta \leftarrow$ the user fairness induced by $pop(i,:)$

27: If ($flag = 0$) then

28: System outage

29: else

30: Find the maximum $\eta ({i^\ast })$ in $\eta$

31: Return$\{{{a_1}, \cdots ,{a_M}} \}$ calculated by $pop({i^\ast },:)$ according to Eq. (6)

4. Simulation and discussion

In order to verify the superiority of our proposed algorithms over conventional schemes, we conduct a simulation analysis and the setup are parameterized as: AP Lambertian order $m = 1$, PD physical area $A = 1\;\textrm{c}{\textrm{m}^2}$, PD responsivity ${R_p} = 0.48\;\textrm{A/W}$, PD FOV ${\psi _{FOV}} = {60^ \circ }$, optical filter gain ${T_s}({\psi _k}) = 1$, refractive index $n = 1.5$, $H = 3\;\textrm{m}$, ${P_{\textrm{elec}}} = 0.25\;\textrm{W}$, $B = 20\textrm{ MHz}$, ${N_0} = {10^{ - 21}}\textrm{ }{\textrm{A}^\textrm{2}}\textrm{/Hz}$. Wherein, for FPA scheme, we set ${\alpha _{\textrm{FPA}}}\textrm{ = }{\alpha _{i(i + 1)}},i \in {\cal I}$ to 0.3 [16].

First, we release constraint 3 to verify the enhancement of sum rate and user fairness by algorithm 1 and algorithm 2 respectively. Assuming user number ranges from 2 to 10, we randomly select the positions of all the users involved to conduct the simulation. To be more persuasive, we simulate 100 times for each user number and take the average value. For each configuration, i.e., given user number and users’ positions, we first calculate the sum rate and user fairness achieved by OMA, NOMA with FPA. Then, we set user fairness achieved by NOMA with FPA as the threshold ${C_1}$ to execute algorithm 1, and obtain the sum rate by NOMA with algorithm 1. The sum rates by the above three schemes are shown in Fig. 1(a). Likewise, Fig. 1(b) shows the sum rates by OMA, NOMA with GRPA, and NOMA with algorithm 1, wherein the threshold ${C_1}$ equals to user fairness achieved by NOMA with GRPA during the execution of algorithm 1. Apparently, the sum rate by OMA fluctuates within a narrow range, while the sum rates by NOMA with algorithm 1 and GRPA are much higher and increase with increasing $M$. However, the sum rate by NOMA with FPA reaches the maximum at $M = \textrm{3}$, and goes below that by OMA when $M \ge \textrm{6}$, which confirms that NOMA is efficient in multiplexing a few number of users based on FPA [13]. In general, when $M = \textrm{2}$, our proposed algorithm 1 is equivalent to FPA and GRPA due to the uniqueness of the solution, as mentioned above. When $M \ge \textrm{3}$, the algorithm 1 can always achieve the best sum rate performance and the performance gain increases with increasing $M$. Moreover, the gain of algorithm 1 over FPA grows faster compared to that over GRPA. The reason is that GRPA can compensate for channel differences among users and improve the user fairness, which makes the constraint more rigorous during the execution of algorithm 1 and the optimization space smaller. Specifically, the gain of algorithm 1 in sum rate is 2.69% and 2.83% compared to FPA and GRPA for $M\textrm{ = 3}$, while the gain respectively reaches 176.79% and 6.37% for $M\textrm{ = 10}$. In addition, in practical NOMA systems, the maximum number of multiplexed users per frame is normally limited to 3 due to the limited processing capacity of SIC receivers. We therefore further calculate the cumulative distribution functions (CDFs) of sum rates achieved for 100 configurations when $M\textrm{ = 3}$, as is shown in Figs. 2(a) and 2(b). Specifically, Fig. 2(a) corresponds to Fig. 1(a), and Fig. 2(b) corresponds to Fig. 1(b). It proves again that NOMA with algorithm 1 has the highest rate, followed by NOMA with conventional schemes, and OMA in a descending order. The percentage of sum rate higher than $\textrm{6} \times \textrm{1}{\textrm{0}^\textrm{7}}$ Mbps is 57%, 52% and 12% by NOMA with algorithm 1, NOMA with FPA and OMA respectively when ${C_1}$ equals to user fairness achieved by NOMA with FPA, while the percentage is 64%, 58% and 24% by algorithm 1, GRPA and OMA respectively when ${C_1}$ equals to user fairness achieved by NOMA with GRPA.

Fig. 1. Achievable sum rates under user fairness constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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Fig. 2. CDFs of sum rates for $M\textrm{ = 3}$ under user fairness constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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For each configuration, we first calculate the sum rate and user fairness achieved by OMA, NOMA with FPA. Then, we set the sum rate achieved by NOMA with FPA as the threshold ${C_2}$ to execute algorithm 2, and obtain the user fairness by NOMA with algorithm 2. The user fairness by the above three schemes are shown in Fig. 3(a). Likewise, Fig. 3(b) shows the user fairness by OMA, NOMA with GRPA, and NOMA with algorithm 2, wherein the threshold ${C_2}$ equals to the sum rate achieved by NOMA with GRPA during the execution of algorithm 2. Furthermore, the CDFs of user fairness for $M\textrm{ = 2}$ and $M\textrm{ = 3}$ are presented in Figs. 4(a) and 4(b). A similar performance difference can be observed that NOMA with algorithm 2 has the best user fairness performance, followed by NOMA with conventional schemes, and OMA in a descending order, except $M \ge \textrm{7}$ and FPA is adopted. The performance gain of algorithm 2 over other schemes generally increases with increasing $M$. Note that algorithm 2 can still work when $M = \textrm{2}$. Specifically, compared to FPA and GRPA, the gain of algorithm 2 in user fairness is 4.87% and 4.54%, 19.04% and 36.54% for $M\textrm{ = 2}$ and $M\textrm{ = 3}$, respectively. When ${C_2}$ equals to the sum rate achieved by NOMA with FPA, by algorithm 2, FPA and OMA, the percentage of user fairness higher than 0.7 is 13%, 9% and 4%, and 19%, 10% and 0% for $M\textrm{ = 2}$ and $M\textrm{ = 3}$, respectively. When ${C_2}$ equals to the sum rate achieved by NOMA with GRPA, by algorithm 2, GRPA and OMA, the percentage of user fairness higher than 0.9 is 35%, 25% and 0%, and 37%, 3% and 0% for $M\textrm{ = 2}$ and $M\textrm{ = 3}$, respectively.

Fig. 3. Achievable user fairness under sum rate constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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Fig. 4. CDFs of user fairness for $M\textrm{ = 2}$ and $M\textrm{ = 3}$ under sum rate constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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Then, considering the QoS requirement of each user and adopting the Monte Carlo method, we verify the enhancement of coverage probability by proposed algorithms compared to conventional schemes. Coverage probability denotes the probability that all users can achieve reliable detection without outage. Moreover, we assume $M\textrm{ = 3}$, and ${\tilde{R}_k} = R,\;\;{\kern 1pt} \forall k \in {\cal M}$, which ranges from 6 Mbps to 18 Mbps. Under user fairness constraint of no less than that achieved by FPA or GRPA respectively, the coverage probability of NOMA with algorithm 1, NOMA with corresponding conventional scheme, and OMA are presented in Figs. 5(a) and 5(b). Under sum rate constraint of no less than that achieved by FPA or GRPA respectively, the coverage probability of NOMA with algorithm 2, NOMA with corresponding conventional scheme, and OMA are presented in Figs. 6(a) and 6(b). As is shown, NOMA with proposed algorithms has the highest coverage probability, followed by NOMA with conventional schemes, and OMA in a descending order. Furthermore, comparing Figs. 5(a) and 6(a), or Figs. 5(b) and 6(b), we can conclude that the coverage probability gain of algorithm 1 over conventional schemes is higher than that of algorithm 2. The reason for that lies in the following aspects: 1) we conduct the simulation by setting the user fairness or sum rate achieved by conventional schemes themselves as the threshold, i.e., constraints 4 and 5 impose no restrictions on conventional schemes, thus the coverage probabilities achieved by one conventional scheme for Fig. 5 and Fig. 6 are almost the same; 2) for other power allocation schemes, a lower coverage probability would be achieved under sum rate constraint compared to that under user fairness constraint, i.e., the coverage probability achieved by algorithm 2 is lower than that by algorithm 1. Precisely because of the second aspect mentioned above and the low spectrum efficiency of OMA, the coverage probabilities achieved by OMA for Fig. 6(a) and Fig. 6(b) are declined to almost 0.

Fig. 5. Coverage probability vs. target rate of each user for $M\textrm{ = 3}$ under user fairness constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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Fig. 6. Coverage probability vs. target rate of each user for $M\textrm{ = 3}$ under sum rate constraint of no less than that achieved by: (a) FPA; (b) GRPA.

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5. Conclusion

In this paper, considering the practical application and denoting the lower bound of normalized user rates as explicit user fairness, we formulate two optimization problems concerning power allocation in NOMA-VLC systems, namely, user fairness-guaranteed sum rate maximization and sum rate-guaranteed user fairness maximization. Then, we propose two corresponding power allocation algorithms through classified discussion and mathematical derivation, wherein DE based heuristic algorithm is involved. Finally, we verify the superiority of our proposed algorithms over conventional schemes through numerical simulation.

Appendix

Appendix A

Proof: First, under constraints 1 and 2, the sum rate ${R_{\textrm{sum}}}$ can be expressed as:

(11)$$\begin{array}{l} {R_{\textrm{sum}}} = \beta [{\log _2}(1 + h_1^2a_1^2/[h_1^2(\sum\nolimits_{2 \le j \le M} {a_j^2)} + 1/\rho ]) + \cdots + {\log _2}(1 + \rho h_M^2a_M^2)]\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \beta {\log _2}([\frac{{h_1^2 + 1/\rho }}{{h_1^2(\sum\nolimits_{2 \le j \le M} {a_j^2)} + 1/\rho }}] \times \cdots \times (\frac{{h_M^2a_M^2 + 1/\rho }}{{1/\rho }}))\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \beta {\log _2}[(\frac{{h_1^2 + 1/\rho }}{{1/\rho }}) \times [\frac{{h_2^2(\sum\nolimits_{2 \le j \le M} {a_j^2} ) + 1/\rho }}{{h_1^2(\sum\nolimits_{2 \le j \le M} {a_j^2} ) + 1/\rho }}] \times \cdots \times (\frac{{h_M^2a_M^2 + 1/\rho }}{{h_{M - 1}^2a_M^2 + 1/\rho }})]\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \beta {\log _2}[(\rho h_1^2 + 1)[\frac{{h_2^2}}{{h_1^2}} + \frac{{1 - h_2^2/h_1^2}}{{\rho h_1^2(\sum\nolimits_{2 \le j \le M} {a_j^2} ) + 1}}] \times \cdots \times (\frac{{h_M^2}}{{h_{M - 1}^2}} + \frac{{1 - h_M^2/h_{M - 1}^2}}{{\rho h_{M - 1}^2a_M^2 + 1}})]. \end{array}$$

Since ${h_i} \le {h_{i + 1}}$ with $i \in {\cal I}$, a conclusion can be drawn from Eq. (11) that lager $\sum\nolimits_{2 \le j \le M} {a_j^2}$, $\sum\nolimits_{3 \le j \le M} {a_j^2}$,…, and $a_M^2$ yields larger sum rate. As a corollary, in order to maximize the sum rate, $a_1^2$, $a_2^2$,…, and $a_{M - 1}^2$ should be reduced in sequence as much as possible.

According to ${R_1} \ge {\tilde{R}_1}$ and Eq. (3), it can be derived that

(12)$$a_1^2 \ge \frac{{SIN{R_1}[1 + 1/(\rho h_1^2)]}}{{1\textrm{ + }SIN{R_1}}},$$

where $SIN{R_k} = {2^{{{\tilde{R}}_k}/\beta }} - 1,\;k \in {\cal M}$. Considering the corollary mentioned above, we have the optimal ${a_1}$, $a_1^{\ast }$, that satisfies the inequality constraint in Eq. (12) with equality, i.e., $a_1^{\ast } = \sqrt {SIN{R_1}[1 + 1/(\rho h_1^2)]/(1\textrm{ + }SIN{R_1})}$. Likewise, according to constraint 1, it holds that $\sum\nolimits_{k + 1 \le j \le M} {a_j^2} = 1 - \sum\nolimits_{j < k} {a_j^2} - a_k^2$. Then, we substitute it into constraint 3 for $k \in \{{2, \cdots ,M - 2} \}$, and derive that

(13)$$a_k^2 \ge \frac{{SIN{R_k}[1 + 1/(\rho h_k^2) - \sum\nolimits_{j < k} {a_j^2} ]}}{{1\textrm{ + }SIN{R_k}}},\;k \in \{{2, \cdots ,M - 1} \}.$$

According to the corollary, it can be further obtained that

(14)$$\begin{array}{l} a_k^{\ast } = \sqrt {\frac{{SIN{R_k}[1 + 1/(\rho h_k^2) - \sum\nolimits_{j < k} {a_j^2} ]}}{{1\textrm{ + }SIN{R_k}}}} ,\;k \in \{{2, \cdots ,M - 1} \}, \\ a_M^{\ast } = \sqrt {1 - \sum\nolimits_{1 \le k \le M - 1} {{{(a_k^{\ast })}^2}} } . \end{array}$$

After all, $a_1^{\ast }, \cdots ,a_M^{\ast }$ can be calculated in sequence. At this time, we need to determine whether $0 < {(a_k^{\ast })^2} < 1,\;\forall k \in {\cal M}$ and $\beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}] \ge {\tilde{R}_M}$. If they are not be satisfied simultaneously, the problem (4) is not feasible even without constraint 4, and there occurs a system outage. Otherwise, $\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$ is the optimal solution without constraint 4.

Then, when $\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$ is adopted as the power allocation coefficient set, it holds that ${R_1} = {\tilde{R}_1}, \cdots ,{R_{M - 1}} = {\tilde{R}_{M - 1}},{R_M} = \beta {\log _2}{\kern 1pt} [1 + \rho {({h_M}a_M^\ast )^2}]$. In this case, $\eta$ equals to

(15)$$\sigma = \frac{{\min \{{{{\tilde{R}}_1}, \cdots ,{{\tilde{R}}_{M - 1}},\beta {{\log }_2}{\kern 1pt} [1 + \rho {{({h_M}a_M^\ast )}^2}]} \}}}{{\max \{{{{\tilde{R}}_1}, \cdots ,{{\tilde{R}}_{M - 1}},\beta {{\log }_2}{\kern 1pt} [1 + \rho {{({h_M}a_M^\ast )}^2}]} \}}}.$$

Obviously, when ${C_1} \le \sigma$, the constraint 4 is also satisfied. In conclusion, $\{{a_1^{\ast }, \cdots ,a_M^{\ast }} \}$ is the optimal solution to problem (4) under all constraints, completing the proof.

Appendix B

Proof: First, according to ${R_M} = v$, i.e., $\beta {\log _2}(1 + \rho h_M^2a_M^2) = v$, we can obtain:

(16)$$a_M^2 = ({2^{v/\beta }} - 1)/(\rho h_M^2).$$

In the same way, according to ${R_{M - 1}} = v$, ${R_{M - 2}} = v$,…, ${R_1} = v$, we can express $a_{M - 1}^2$, $a_{M - 2}^2$,…, $a_1^2$ in terms of $v$ in sequence, and the recursion relation is as follows:

(17)$$a_i^2 = ({2^{v/\beta }} - 1)(\sum\nolimits_{i + 1 \le j \le M} {a_i^2} + 1/h_i^2)/\rho ,\;\;\;i \in {\cal I}.\;$$

Then, based on constraint 1, a nonlinear equation can be given:

(18)$$f(v) = \sum\nolimits_{1 \le i \le M} {a_i^2(v)} - 1 = 0.$$

Based on Eqs. (16) and (17), we can see that larger $v$ yields larger $a_i^2,i \in {\cal M}$. Hence, $f(v)$ is a monotonicity increasing function of v. Moreover, $f(0) = \sum\nolimits_{1 \le i \le M} {a_i^2(0)} - 1 = 0 - 1 ={-} 1 < 0$, and $f(\beta lo{g_2}(1 + \rho h_M^2)) = \sum\nolimits_{1 \le i \le M - 1} {a_i^2(\beta lo{g_2}(1 + } \rho h_M^2)) + 1 - 1 > 0$. Based on the intermediate value theorem, the root $v$ of Eq. (18) can be approximately obtained through a bisection procedure in the range of 0 to $\beta lo{g_2}(1 + \rho h_M^2)$. Once $v$ is obtained, the solution to problem (4) without constraint 3 can be given by Eqs. (16) and (17). At this juncture, ${R_k} = v,\;k \in {\cal M}$. Taking constraint 3 into account, we need to determine whether $v \ge \max \{{{{\tilde{R}}_k},k \in {\cal M}} \}$. If it holds, then constraint 3 is also satisfied, that is, the above solution would be the final solution to (4) under all constraints. Otherwise, the problem (4) is not feasible and a system outage would occur. Completing the proof

Appendix C

Proof: We take $M = 3$ for example and adopt the method of reduction to absurdity. First, assume that when condition 1 is satisfied and the auxiliary variable ${R^ \ast }$ has been maximized, the objective function can get the maximum $(1 + 2{C_1}){R^ \ast }$. At this time, considering Eq. (3), we can derive the following:

(19)$$\left\{ \begin{array}{l} h_1^2a_1^2/(h_1^2(1 - a_1^2) + 1/\rho ) = h_2^2a_2^2/(h_2^2a_3^2 + 1/\rho ) = {2^{{C_1}{R^ \ast }/\beta }} - 1,\\ \rho {({h_3}{a_3})^2} = {2^{{R^ \ast }/\beta }} - 1.{\kern 1pt} \end{array} \right.$$

If ${h_1} < {h_2} \simeq {h_3}$, then:

(20)$$\left\{ {\begin{array}{{c}} {a_3^2 = ({2^{{R^ \ast }/\beta }} - 1)/(\rho h_3^2),}\\ {a_2^2 = {2^{{R^ \ast }/\beta }}({2^{{C_1}{R^ \ast }/\beta }} - 1)/(\rho h_3^2),}\\ {a_1^2 = [h_1^2({2^{(1 + 2{C_1}){R^ \ast }/\beta }} - {2^{{C_1}{R^ \ast }/\beta }} - {2^{(1 + {C_1}){R^ \ast }/\beta }} + 1) + h_3^2({2^{{C_1}{R^ \ast }/\beta }} - 1)]/(\rho h_1^2h_3^2).} \end{array}} \right.$$

Combing with $\sum\nolimits_{1 \le i \le M} {a_i^2} = 1$, it can be derived that:

(21)$${2^{(1 + 2{C_1}){R^ \ast }/\beta }} + {2^{{C_1}{R^ \ast }/\beta }}(h_3^2/h_1^2 - 1) = h_3^2/h_1^2 + \rho h_3^2.$$

We choose the following case as the benchmark:${R_1} = {C_1}R_{ben}^ \ast $, ${R_2} = {R_3} = R_{ben}^ \ast $, and consequently ${R_{\textrm{sum}}} = (2 + {C_1})R_{ben}^ \ast $. At this time, it can be derived that:

(22)$$\begin{array}{l} \left\{ \begin{array}{l} h_1^2a_1^2/(h_1^2(1 - a_1^2) + 1/\rho ) = {2^{{C_1}R_{ben}^ \ast{/}\beta }} - 1\\ h_2^2a_2^2/(h_2^2a_3^2 + 1/\rho ) = \rho {({h_3}{a_3})^2} = {2^{R_{ben}^ \ast{/}\beta }} - 1 \end{array} \right.\\ \buildrel {{\Delta _1}} \over \longrightarrow \left\{ {\begin{array}{{c}} {a_3^2 = ({2^{R_{ben}^ \ast{/}\beta }} - 1)/(\rho h_3^2)}\\ {a_2^2 = {2^{R_{ben}^ \ast{/}\beta }}({2^{R_{ben}^ \ast{/}\beta }} - 1)/(\rho h_3^2)}\\ {a_1^2 = [h_1^2({2^{(2 + {C_1})R_{ben}^ \ast{/}\beta }} - {2^{{C_1}R_{ben}^ \ast{/}\beta }} - {2^{2R_{ben}^ \ast{/}\beta }} + 1) + h_3^2({2^{{C_1}R_{ben}^ \ast{/}\beta }} - 1)]/(\rho h_1^2h_3^2)} \end{array}} \right.\\ \buildrel {{\Delta _2}} \over \longrightarrow {2^{(2 + {C_1})R_{ben}^ \ast{/}\beta }} + {2^{{C_1}R_{ben}^ \ast{/}\beta }}(h_3^2/h_1^2 - 1) = h_3^2/h_1^2 + \rho h_3^2, \end{array}$$

where, ${\Delta _1}$ is due to ${h_2} \simeq {h_3}$ and ${\Delta _2}$ is due to $\sum\nolimits_{1 \le i \le M} {a_i^2} = 1$. Combining Eqs. (21) and (22), we can conclude that

(23)$${2^{(1 + 2{C_1}){R^ \ast }/\beta }} + {2^{{C_1}{R^ \ast }/\beta }}(h_3^2/h_1^2 - 1) = {2^{(2 + {C_1})R_{ben}^ \ast{/}\beta }} + {2^{{C_1}R_{ben}^ \ast{/}\beta }}(h_3^2/h_1^2 - 1).$$

We further classify and discuss the relationship between $(1 + 2{C_1}){R^ \ast }$ and $(2 + {C_1})R_{ben}^ \ast $ as follows:

(1) If ${R^ \ast } \le R_{ben}^ \ast $, since ${C_1} \in (\sigma ,1)$, $(1 + 2{C_1}){R^ \ast } < (2 + {C_1})R_{ben}^ \ast $.
(2) If ${R^ \ast } > R_{ben}^ \ast $, since ${C_1} > 0$, $\beta > 0$, and $h_3^2 > h_1^2$, it can be derived that $(24)$$\begin{array}{l} {2^{{C_1}{R^ \ast }/\beta }}(h_3^2/h_1^2 - 1) > {2^{{C_1}R_{ben}^ \ast{/}\beta }}(h_3^2/h_1^2 - 1)\buildrel {{\Delta _3}} \over \longrightarrow {2^{(1 + 2{C_1}){R^ \ast }/\beta }} < {2^{(2 + {C_1})R_{ben}^ \ast{/}\beta }}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \buildrel {{\Delta _4}} \over \longrightarrow (1 + 2{C_1}){R^ \ast } < (2 + {C_1})R_{ben}^ \ast , \end{array}$$$

${\Delta _3}$ is due to Eq. (23), and ${\Delta _4}$ denotes the operations of taking logarithm of both sides and then multiplying them by $\beta$. After all, it always holds that $(1 + 2{C_1}){R^ \ast } < (2 + {C_1})R_{ben}^ \ast $, which is contrary to the initial assumption.

Likewise, we assume that when condition 2 is satisfied and the auxiliary variable ${R^ \ast }$ has been maximized, the objective function can get the maximum. Under the extreme hypothesis that ${h_1} \simeq {h_2} < {h_3}$, we can always find another special case where the sum rate is larger than that induced under condition 2. Apparently, when the optimal solution is reached, the relationship among user rates depends on their respective channel gains, and cannot be simplified as condition 1 or 2. Note that we choose extreme relationship among channel gains just for reduction to absurdity, and the complexity and uncertainty of this relationship can further support proposition 3. Moreover, when $M > 3$, the proposition can be proven in the same manner, completing the proof.

Funding

National Natural Science Foundation of China (61172080, 61771357).

Disclosures

The authors declare no conflicts of interest.

References

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Differential evolution-based optimal power allocation scheme for NOMA-VLC systems

Abstract

1. Introduction

2. System model

3. Optimal power allocation

3.1 User fairness-guaranteed max-sum rate criterion

3.2 Sum rate-guaranteed max-user fairness criterion

4. Simulation and discussion

5. Conclusion

Appendix

Appendix A

Appendix B

Appendix C

Funding

Disclosures

References

Cited By

Figures (6)

Equations (24)

Optics Express