1 Introduction
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Due to the complex mathematical expansion of a NOMA-VLC throughput target, this paper provides an alternative lower bound as an approximate target to evaluate the performance of the VLC user data rate for NOMA mathematical convenience.
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To demonstrate the accuracy of lower bound substitution in NOMA-VLC networks, details such as asymptotic properties and performance boundary are analyzed. We discover that the substitutability of our alternative bound is feasible for VLC channels.
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This paper presents a GRPA strategy based on VLC channel gains. Considering the indoor VLC network environment, the home is a typical application scenario, which involves a limited room size and few users. For two- and three-user cases, we derive an analytic proof that shows that our GRPA strategy is better than the reported strategy in previous papers [11, 26].
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For a great number of users, a performance analysis between the previously reported strategy and our strategy is performed by numerical simulations. Experimental results show that the average user data rate of NOMA is greater than that of OMA, and our GRPA strategy outperforms that previous strategy with the number of users not limited to three.
2 Methodology
2.1 Channel gain of VLC
2.2 GRPA of NOMA in VLC
2.3 User data rate bound
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Several studies [9, 10, 15] have continued to use the Shannon capacity formula, which may not be accessible without the maximizing input distribution. However, the bound can be referred to as the achievable bound. The signal to interference plus noise ratio (SINR) for the k-th user in (9) is usually used to measure the VLC user data rate [9, 23, 32, 33], where N0 is a constant noise power spectral density and B is a constant VLC bandwidth.$$\begin{array}{@{}rcl@{}} SIN{R_{k}} = \frac{{{{\left({\gamma \cdot {h_{k}} \cdot {a_{k}} \cdot PT} \right)}^{2}}}}{{\sum\limits_{i = k + 1}^{K} {{{\left({\gamma \cdot {h_{k}} \cdot {a_{i}} \cdot PT} \right)}^{2}} + {N_{0}}B}}},k < K \end{array} $$(9)
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On the other hand, no exact closed-form expression for the VLC channel capacity has yet been established due to constraints of the input VLC signals. Instead, the use of several tight capacity bounds [7, 25, 30, 31, 34] is more practical to approach the channel capacity in VLC performance analysis. Due to the different constraints imposed on the VLC system, the expression of the lower bound is different.
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NOMA utilizes the whole available bandwidth, whose efficiency can be normalized. We can regard the bandwidth as a common factor that can be extracted and treat \(\frac {2}{{\pi e}}\) as a constant η.
2.4 Motivation of GRPA strategy design
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Power law strategy (PLS) [11, 26]: Previous studies’ factor strategy is discussed in (15).$$ {a_{k}} = \left\{ {\begin{array}{ll} {\frac{1}{{1 + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}^{2}} + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}^{2}} \cdot {{\left({\frac{{{h_{1}}}}{{{h_{3}}}}} \right)}^{3}} \cdots + \prod\limits_{i = 2}^{K} {{{\left({\frac{{{h_{1}}}}{{{h_{i}}}}} \right)}^{i}}}}}}&{,k = 1}\\ {\frac{{\prod\limits_{i = 2}^{K} {{{\left({\frac{{{h_{1}}}}{{{h_{i}}}}} \right)}^{i}}}}}{{1 + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}^{2}} + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}^{2}} \cdot {{\left({\frac{{{h_{1}}}}{{{h_{3}}}}} \right)}^{3}} \cdots + \prod\limits_{i = 2}^{K} {{{\left({\frac{{{h_{1}}}}{{{h_{i}}}}} \right)}^{i}}}}}}&{,2 \le k \le K} \end{array}} \right. $$(15)
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Our strategy (OurS): This paper’s GRPA factor strategy is discussed in (16). Using h k to compute a k , this paper presents a revision for degrading the power counting of \(\frac {{{h_{1}}}}{{{h_{k}}}}\) as our strategy.$$ {a_{k}} = \left\{ {\begin{array}{ll} {\frac{1}{{1 + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}} + \cdots + \left({\frac{{{h_{1}}}}{{{h_{k}}}}} \right)}}}&{,k = 1}\\ {\frac{{{{\left({\frac{{{h_{1}}}}{{{h_{k}}}}} \right)}}}}{{1 + {{\left({\frac{{{h_{1}}}}{{{h_{2}}}}} \right)}} + \cdots + {{\left({\frac{{{h_{1}}}}{{{h_{k}}}}} \right)}}}}}&{,2 \le k \le K} \end{array}} \right. $$(16)
3 Performance evaluation of GRPA strategy
3.1 NOMA-VLC GRPA in the two-user case
3.1.1 Two-user basic throughput model
3.1.2 Our alternative bound in the two-user model
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When the number of user K increases, the limited cell coverage becomes crowded. If we choose any two users from the total K users, two users’ radii may be approximate and two channel gains may be converging; thus, (23) can be inferred.$$ \underset{K \to + \infty,\left({{r_{1}} - {r_{2}}} \right) \to 0}{\lim} \left({{S_{2}} - {S'_{2}}} \right) = 0 $$(23)
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For the case in which the number of users K is two, we analyze the performance difference between two targets in the worst communication situation to demonstrate the compact boundary of our target in (24).$$ \begin{array}{l} \underset{K = 2,{r_{1}} > {r_{2}}}{\lim} \left({{S_{2}} - {{S'}_{2}}} \right)\\ \le \underset{K = 2,{r_{1}} > {r_{2}}}{\lim} \frac{{{{\left({{\gamma^{2}}{a_{1}}} \right)}^{2}}{a_{2}}{{\left({\rho \left({m + 1} \right){L^{m + 1}}} \right)}^{3}}}}{{{{\left({{N_{0}}B} \right)}^{\frac{3}{2}}}}} \cdot \left({\frac{1}{{\Omega_{1} \cdot \Omega_{2}^{2}}} - \frac{1}{{\Omega_{1}^{3}}}} \right)\\ \le \left[ {1 - {{\left({\frac{{r_{2}^{2} + {L^{2}}}}{{r_{1}^{2} + {L^{2}}}}} \right)}^{m + 3}}} \right] \cdot {S_{2}} \end{array} $$(24)
3.1.3 Strategy comparison of two-user model
3.2 NOMA-VLC GRPA in the three-user case
3.2.1 Three-user GRPA model
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Power law strategy: The three-user factor forms are given by (34), and the target is indicated by (35).$$ {\begin{aligned} {a_{1}} &= \frac{{h_{2}^{2}h_{3}^{2}}}{{h_{1}^{2}h_{3}^{3} + h_{2}^{2}h_{3}^{3} + h_{1}^{5}}},{a_{2}} = \frac{{h_{1}^{2}h_{3}^{2}}}{{h_{1}^{2}h_{3}^{3} + h_{2}^{2}h_{3}^{3} + h_{1}^{5}}},\\ {a_{3}} &= \frac{{h_{1}^{5}}}{{h_{1}^{2}h_{3}^{3} + h_{2}^{2}h_{3}^{3} + h_{1}^{5}}} \end{aligned}} $$(34)$$ {{}\begin{aligned} PL\_{S^{\prime\prime}_{3}} = \frac{{h_{1}^{2}h_{2}^{4}h_{3}^{6} + h_{1}^{4}h_{2}^{2}h_{3}^{6} + h_{1}^{10}h_{3}^{2}}}{{{{\left({h_{1}^{2}h_{2}^{3} + h_{2}^{2}h_{3}^{3} + h_{1}^{5}} \right)}^{2}}}} \end{aligned}} $$(35)
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Our strategy: The three-user factor forms are given by (36), and the target is indicated by (37).$$ {\begin{aligned} {a_{1}} &=\! \frac{{h_{2}h_{3}}}{{h_{1}h_{2} + h_{2}h_{3} + h_{1}h_{3}}},{a_{2}} =\! \frac{{h_{1}h_{3}}}{{h_{1}h_{2} + h_{2}h_{3} + h_{1}h_{3}}},\\{a_{3}} &=\! \frac{{h_{1}h_{2}}}{{h_{1}h_{2} + h_{2}h_{3} + h_{1}h_{3}}} \end{aligned}} $$(36)$$ {{}\begin{aligned} Our\_{S^{\prime\prime}_{3}} = \frac{{h_{1}^{2}h_{2}^{2}h_{3}^{2}}}{{\frac{1}{3}{{\left({h_{1} h_{2} + h_{2} h_{3} + h_{1} h_{3}} \right)}^{2}}}} \end{aligned}} $$(37)
3.2.2 Comparison between power law strategy and our strategy
3.2.2.1 (1) General NOMA-VLC user scenario: inequality condition
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If \({Q_{(3)}^{2}} \ge 0\), the difference \(({Q_{Our\_{S^{\prime \prime }_{3}}}^{2}} - {Q_{PL\_{S^{\prime \prime }_{3}}}^{2}})\) should be positive.
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If \({Q_{(3)}^{2}} < 0\), due to the relationship x8θ>x2θy4θz2θ, we can establish (47) to remain the positive.
3.2.2.2 (2) Special NOMA-VLC user scenario: local equality condition
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x=y>z: In (50), it is obvious that \({{part}_{1}^{Our}} > {{part}_{1}^{PLS}},{{part}_{2}^{Our}} > {{part}_{2}^{PLS}}\) and \({{part}_{3}^{Our}} > {{part}_{3}^{PLS}}\) because of (49) and θ>0.$$ {\begin{aligned} \frac{{Our\_{S^{\prime\prime}_{3}}}}{{PL\_{S^{\prime\prime}_{3}}}} &= \frac{{3{{\left({2{x^{5\theta}} + {x^{2\theta}}{z^{3\theta}}} \right)}^{2}}}}{{{{\left({2{x^{\theta}} + {z^{\theta}}} \right)}^{2}} \cdot \left({{x^{8\theta}} + {x^{6\theta}}{z^{2\theta}} + {x^{4\theta}}{z^{4\theta}}} \right)}}\\ &= \frac{{\underbrace {\left({12{x^{10\theta}}} \right)}_{{part}_{1}^{Our}} + \underbrace {\left({9{x^{7\theta}}{z^{3\theta}}} \right)}_{{part}_{2}^{Our}}}}{{\underbrace {\left({4{x^{10\theta}} + 4{x^{8\theta}}{z^{2\theta}} + 4{x^{9\theta}}{z^{\theta}}} \right)}_{{part}_{1}^{PLS}} + \underbrace {\left({4{x^{7\theta}}{z^{3\theta}} + 5{x^{6\theta}}{z^{4\theta}}} \right)}_{{part}_{2}^{PLS}}}}\\ & \quad + \frac{{\underbrace {\left({2{x^{4\theta}}{z^{6\theta}}} \right)}_{{part}_{3}^{Our}}{\mathrm{+}}\underbrace {\left({3{x^{7\theta}}{z^{3\theta}} + {x^{4\theta}}{z^{6\theta}}} \right)}_{{part}_{4}^{Our}}}}{{\underbrace {\left({{x^{4\theta}}{z^{6\theta}} + {z^{10\theta}}} \right)}_{{part}_{3}^{PLS}}{\mathrm{+}}\underbrace {\left({4{x^{5\theta}}{z^{5\theta}}} \right)}_{{part}_{4}^{PLS}}}} \end{aligned}} $$(50)$$ {\begin{aligned} {{part}_{4}^{Our} - {part}_{4}^{PLS}} &= 3{x^{7\theta}}{z^{3\theta}} + {x^{4\theta}}{z^{6\theta}} - 4{x^{5\theta}}{z^{5\theta}}\\ &= {x^{4\theta}}{z^{3\theta}}\left({{x^{2\theta}} - {z^{2\theta}} + 3{x^{\theta}}{z^{\theta}} + 2{x^{2\theta}}} \right)\left({{x^{\theta}} - {z^{\theta}}} \right)\\ &> 0 \end{aligned}} $$(51)To demonstrate the hypothesis that \({Our\_{S^{\prime \prime }_{3}}} > {PL\_S^{\prime \prime }}\), the condition holding \({{part}_{4}^{Our}} > {{part}_{4}^{PLS}}\) should be valid. Due to the validity of (51), \({{part}_{4}^{Our}} > {{part}_{4}^{PLS}}\) is true, and our strategy is better than the power law strategy in this special case.
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x>y=z: In (52), it is obvious that \({{part}_{5}^{Our}} = {{part}_{5}^{PLS}}\), \({{part}_{6}^{Our}} > {{part}_{6}^{PLS}}\), \({{part}_{7}^{Our}} > {{part}_{7}^{PLS}}\), and \({{part}_{8}^{Our}} > {{part}_{8}^{PLS}}\). The relationship between \({{part}_{9}^{Our}}\) and \({{part}_{9}^{PLS}}\) can be given by (53). Hence, in this special case, our strategy is better than the power law strategy.$$ {\begin{aligned} \frac{{Our\_{S^{\prime\prime}_{3}}}}{{PL\_{S^{\prime\prime}_{3}}}} &= \frac{{3{{\left({{x^{5\theta}} + {x^{3\theta}}{y^{2\theta}} + {y^{5\theta}}} \right)}^{2}}}}{{{{\left({{x^{\theta}} + 2{y^{\theta}}} \right)}^{2}} \cdot \left({{x^{8\theta}} + 2{y^{8\theta}}} \right)}}\\ &= \frac{{\underbrace {\left({{x^{10\theta}}{\mathrm{+}}4{x^{8\theta}}{y^{2\theta}}} \right)}_{{part}_{5}^{Our}} + \underbrace {\left({3{y^{10\theta}} + 5{x^{3\theta}}{y^{7\theta}}} \right)}_{{part}_{6}^{Our}} + \underbrace {\left({{x^{3\theta}}{y^{7\theta}} + {x^{6\theta}}{y^{4\theta}}} \right)}_{{part}_{7}^{Our}}}}{{\underbrace {\left({{x^{10\theta}}{\mathrm{+}}4{x^{8\theta}}{y^{2\theta}}} \right)}_{{part}_{5}^{PLS}} + \underbrace {\left({8{y^{10\theta}}} \right)}_{{part}_{6}^{PLS}} + \underbrace {\left({2{x^{2\theta}}{y^{8\theta}}} \right)}_{{part}_{7}^{PLS}}}}\\ & \quad + \frac{{\underbrace {\left({6{x^{5\theta}}{y^{5\theta}} + 2{x^{6\theta}}{y^{4\theta}}} \right)}_{{part}_{8}^{Our}} + \underbrace {\left({2{x^{10\theta}} + 2{x^{8\theta}}{y^{2\theta}}} \right)}_{{part}_{9}^{Our}}}}{{\underbrace {\left({8{x^{\theta}}{y^{9\theta}}} \right)}_{{part}_{8}^{PLS}} + \underbrace {\left({4{x^{9\theta}}{y^{\theta}}} \right)}_{{part}_{9}^{PLS}}}} \end{aligned}} $$(52)$$\begin{array}{*{20}l} {part}_{9}^{Our} - {part}_{9}^{PLS} &= 2{x^{10\theta}} + 2{x^{8\theta}}{y^{2\theta}} - 4{x^{9\theta}}{y^{\theta}} \\ &= 2{\left({{x^{\theta}} - {y^{\theta}}} \right)^{2}} \cdot {x^{8\theta}} > 0 \end{array} $$(53)
4 Results and discussion
4.1 The SNR distribution in a single VLC cell
Symbol | Name | Value |
---|---|---|
γ
| Optical-electrical conversion efficiency | 0.5(A/W) |
a1,a2,⋯,a
K
| GRPA factor | 0<a
K
<⋯<a2<a1<1 |
Ψ
FOV
| Angle field of vision | 60° |
Φ
1/2
| Semi-angle | 60° |
A
| Area of PD | 10−4(m2) |
T
filter
| Gain of an optical filter | 1 |
n
| Refractive index | 1.5 |
g(ψ
k
=Ψ
FOV
) | Gain of an optical concentrator | 3 |
ρ
| A·T
filter
·g(ψ
k
)/2π | ρ≈5×10−5 |
m=−1/log2(cos(Φ1/2=60°)) | Order of Lambertian emission | 1 |
L
| Height of the LED above receivers | 2.5(m) |
N
0
| Noise power spectral density | 10−19(A2/Hz) |
B
| Bandwidth of a single LED | 20(MHz) |
P
T
| Transmitted power of a single LED | 2(W) |
R
O
O
M
| Room size | 6×6(m2) |
4.2 Comparisons of two- and three-user cases
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Φ1/2 is the unique independent variable of Lambertian emission order m.
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Log2(∙) will not transform the increasing monotonicity of \({\sum \limits _{k = 1}^{K} {{{\left ({\gamma {h_{k}}{a_{k}}} \right)}^{2}}}}\) into the decreasing monotonicity.
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If a user located in the coverage of a VLC AP follows a uniform distribution, in polar coordinates, the radius of a point or user is determined from the density \(f(r) = \frac {{2r_{k}}}{{{{\left ({{r_{\max }}} \right)}^{2}}}}\), where r k is the radius of the k-th user and rmax is the maximum radius of available received coverage. Focusing on the two-user case and Fig. 2, we can choose two users from a sufficient number of users and rmax=3, which is related to room size. The expectation average radius \(\bar r_{k}\) of two users is approximately equal to the centroid of a sufficient number of users, as indicated by (54).$$\begin{array}{@{}rcl@{}} \bar r_{k}^{} = \frac{{\int_{0}^{{r_{\max}}} {f\left({{r_{k}}} \right) \cdot {r_{k}}d{r_{k}}}}}{{\int_{0}^{{r_{\max}}} {f\left({{r_{k}}} \right)d{r_{k}}}}} = \frac{2}{3}{r_{\max}} \end{array} $$(54)
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Given ρ, r k , fixed Ψ FOV , and the definite values of power allocation factors, we focus on the relationship between Φ1/2 and the two-user data rate in NOMA-VLC networks, as indicated by (55).
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The parameters of our alternative target forms are the same as those of the original target in Fig. 3.