Multi-user access in wireless optical communication system

Ruijie Li; Anhong Dang

doi:10.1364/OE.26.022658

1. Introduction

Driven by the tablets, smartphones, and real-time bandwidth-intensive applications, wireless traffic will increase by over a factor of 100: from under 3 exabytes in 2010 to over 190 exabytes by 2018 [1]. Wireless optical communication (WOC) has been attracted much interest in both academia and industry as a high-speed wireless communication technology, due to the advantages of quick and easy deployment, high bandwidth qualities, and high security [2,3]. More importantly, the features of no license fees and no government regulations become significant advantages as the radio frequency (RF) spectrum is heavily congested [4–7].

As a wireless technology, it is imperative that WOC can support multiple users to access the network. Generally, the multiple access techniques include orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA). The OMA is widely utilized and analyzed, like time division multiple access (TDMA), frequency division multiple access (FDMA), and code division multiple access (CDMA). In [8], the authors proposed an optical CDMA (OCDMA) network by assigning the fast frequency hopping-based codes in the hybrid optical system. However, for OCDMA techniques, the optical orthogonal codes (OOCs) are considered, which limits the resource reuse due to the orthogonal property of OOCs [9]. The TDMA is analyzed in [10], where the central node intents to serve “only” one user at each time instant. However, as mentioned in this paper, it is quite hard to achieve the tradeoff between the system throughput and the fairness. The space and time division multiple access technique is proposed by J. Liu et al., in which the space dimension is utilized [11]. However, for OMA, only one user can be accessed in each orthogonal resource block, which cannot provide sufficient resource reuse. On the other hand, NOMA technique has been proposed in [12], and has been considered as a reasonable candidate for future wireless access. Until now, NOMA has been widely analyzed in RF, and researches show that NOMA can fully exploit the capacity and outperforms OMA. However, the features of WOC link are quite different with RF link, one is the non-broadcast nature of optical beam and the other is the vulnerable turbulence channel. Hence, performance of NOMA in WOC is still unknown, and we cannot make the conclusion that NOMA outperforms OMA in WOC without thorough research. Note that so far, references [13] and [14] considered NOMA in WOC. Reference [14] proposed the dynamic NOMA, which is for minimizing the outage probability. However, it suffers from some limitations. Firstly, it only presented the results of two users. Although it may be extended to the case in which the arbitrary users are considered, the scheme would be quite complicated, and cannot be widely utilized in practical system. Secondly, there is no power control scheme, then, the utilization of the scheme is limited. The users’ performance is just determined by the channel fading. If the channel fading is approximately equal, users’ message cannot be decoded perfectly. On the other side, if the link distance between user 1 and central node is much smaller than the link distance between user 2 and central node, the central node always decodes the message of user 1 priority, and the fairness of the users cannot be guaranteed. Moreover, the [13] and [14] just focus on the outage probability of NOMA and cannot present a comprehensive insight into the NOMA. Hence, features of NOMA in WOC are still unknown, and the reasons why we utilize NOMA instead of OMA in WOC are still unknown. In this paper, we intent to explain the reasons and present thorough research of NOMA in WOC.

Motivated by the previous works, and considering the limitations of the existing schemes, we investigate the performance of NOMA in WOC. As mentioned above, the features of WOC are quite different with RF. Then, the application scenario of NOMA in WOC is different with that in RF. In this paper, we consider a multipoint-to-point WOC system, consisting of several users and a central node, shown in Fig. 1(a), which is quite different with the one in [8–11]. The system model is reasonable, and the system model provides a solution for the uplink scenario of multi-user access, while [10] focuses on the downlink scenario. In this model, the message of users is multiplexed in power domain, and users communicate with the central node in the same orthogonal resources. Because each user experiences different optical link, the channel fading is distinct. The randomization of atmospheric induced intensity fading is no longer treated as the nuisance, which needs to be eliminated, but as a source of power diversity that allows the central node to split users in power domain though successive interference cancellation (SIC). Meanwhile, a power back-off scheme is proposed to enlarge power diversity. The performances of the system are evaluated through outage probability, bit error rate (BER) and ergodic capacity. The resultsshow that NOMA outperforms OMA, and the performance gain increases with the increase of turbulence strength. Moreover, the user pairing scheme in turbulence channel is presented.

Fig. 1 Diagram of the NOMA in WOC system. (a) System model of the proposed system. (b) A scenario of the proposed system model.

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This paper is organized as follows. The system and channel models are introduced in Section 2. In Section 3, performance analyses for NOMA in WOC system are shown. The simulation results are depicted in Section 4. Finally, conclusions are drawn.

2. System and channel models

As shown in Fig. 1(a), we consider a multipoint-to-point WOC system consisting of K users, indexed by {1,2,…,K}, and a central node. The system model is reasonable, and one of the possible application scenarios is shown in Fig. 1(b). Figure 1(b) presents an integration of the RF and WOC networks, where the users in different network cell (NC) intent to transmit their signal to the backbone node through the base station (BS). The system model investigated in this paper can provide an effective solution of the multiple access protocol between BS and backbone node. In this model, we assume that each user is equipped with one telescope for transmitting their own message, and at the central node the telescope with wide field angle is used for receiving the signals from all users. The users transmit their own message on the same time-frequency resources. At the central node, the telescope receives the signal simultaneously. The central node decodes the signal by utilizing the power diversity of the received power with the help of SIC. In the following, we assume that users to be served quasi-static, therefore, their channel state information (CSI) is not outdated until the next channel estimation, which means both the user and the central node have the perfect CSI. Note that the assumption is reasonable because of the slow fading characteristic of turbulence channel. Particularly, intensity modulation/direct detection (IM/DD) is adopted. Without loss of generality, assuming that all the users are ordered based on their channel gain, $g_{1} \geq \dots \geq g_{k} \dots \geq g_{K}$ , where g_k is the channel fading of the k-th user. The channel gain is considered to be a product of two factors, i.e. g_k = L_kI_k, where L_k is the deterministic propagation loss and I_k is the intensity scintillation caused by turbulence. In the next subsections, we firstly give the system model and then present the channel model.

2.1 System model

At the transmitter side, the users utilize intensity modulator to imprint the message onto the optical carrier, and then transmit the optical signal into turbulence channel through telescope. The transmitted message of k-th users is P_kx_k, where P_k is the transmitted power and x_k is the transmitted message. At the receiver side, the optical signal from all the users are collected by the telescope and focused on the photodetector. Then, we can express the received electrical signal at the central node as

y = \sum_{k = 1}^{K} g_{k} ξ P_{k} x_{k} + n,

where n is the additive white Gaussian noise (AWGN) with zero mean and variance of σ²,

ξ

is the optical-to-electrical conversion coefficient, which is considered as 1 for convenience in the following derivations. After receiving the message, SIC is adopted to decode the message. SIC requires diverse arrived power to distinguish multiplexed users. The randomization of atmospheric induced intensity fading is a source of power diversity that allows the central node to split users in power domain. To maintain the power diversity, besides the random channel fading, a power back-off scheme is required. In [15], the authors proposed the power control scheme by applying back-off to users with longer distance. However, in WOC link, the one with smaller path loss not always maintains large scintillation fading. Then, the scheme in [15] is not suitable for WOC. A revised power control scheme in WOC is required. Firstly, we express the transmitted power control for point-to-point WOC as

P = P_{a i m} / L,

where P_aim is the targeted arrived power and L is the power loss caused by path loss. Based on the above result, in NOMA system, the transmitted power of k-th user is expressed as

P_{k} = \frac{P_{a i m}}{L_{k} 10^{\frac{(k - 1) ς}{10}}} = a_{k} P_{a i m},

where ζ is the power back-off step in dB. Note that the power control scheme contains two parts. The first part, 1/L_k, is for counteracting the path loss, which achieves similar objective as in point-to-point WOC, and can guarantee the fairness of users. The second part is for enlarging the power diversity between users by gradually degrading the arrived power with a step of ζ dB, which maintains the power diversity besides the randomization of channel fading. In this way, the randomization of channel fading is fully exploited.

Based on [Eq. (46), [16]], the achievable data rate for k-th user can be expressed as

R_{k} = {\begin{cases} {[\frac{1}{2} \log (1 + \frac{ρ {(μ_{k} g_{k} a_{k})}^{2}}{ρ \sum_{i = k + 1}^{K} {(μ_{i} g_{i} a_{i})}^{2} + A}) - ε_{ϕ}]}^{+} \geq {\tilde{R}}_{k}, k < K \\ {[\frac{1}{2} \log (1 + \frac{ρ {(μ_{K} g_{K} a_{K})}^{2}}{A}) - ε_{ϕ}]}^{+} \geq {\tilde{R}}_{K}, \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} k = K \end{cases}

where

{\tilde{R}}_{k}

is the targeted data rate of k-th user,

ρ = P_{a i m}^{2} / σ^{2}

with A = 9(1 + ε_μ)²,

ε_{ϕ} = 0.016

and ε_μ = 0.0015. [x]⁺ denotes max{0,x}, and μ_k∈[0,0.5] represents the ratio between expectation of the received power and the maximum received power. If the inequation in (4) is satisfied, we assume that perfect SIC can be performed in the central node without error propagations.

2.2 Channel model

The channel gain of WOC is considered to be a product of two factors, i.e. g_k = L_kI_k. Path loss L_k is determined by the exponential Beers-Lambert law as $L_{k} = e^{- Φ \times d_{k}}$ , where d_k is the link distance between k-th user and central node, and Φ is the atmospheric attenuation coefficient. $Φ = 3.91 / V (km) \times {(λ (nm) / 550)}^{- q}$ , where V is the visibility in kilometers, λ is the wavelength of laser in nanometers, and q is a parameter related to the visibility [17]. For the intensity scintillation, I_k, gamma-gamma probability density function (PDF) is the most widely accepted model for describing the distribution of intensity scintillation, which is [18]

\begin{matrix} f_{I} (I) = \frac{2 {(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} I^{(α + β) / 2 - 1} K_{α - β} (2 \sqrt{α β I}), & I > 0 \end{matrix},

where Γ(⋅) is the gamma function, K_α-β(⋅) is the (α-β)th-order modified Bessel function, α and β are determined by Rytov variance (

σ_{R}^{2}

) related with the atmospheric conditions [18]

α = {[exp (\frac{0.49 σ_{R}^{2}}{{(1 + 1.11 σ_{R}^{12 / 5})}^{7 / 6}}) - 1]}^{- 1},

β = {[exp (\frac{0.51 σ_{R}^{2}}{{(1 + 0.69 σ_{R}^{12 / 5})}^{5 / 6}}) - 1]}^{- 1} .

Denote $h_{k} = I_{k}^{2}$ . The PDF of the unordered channel fading can be calculated using the “change of variable” method, therefore, the PDF of h_k can be expressed as

f_{h_{k}} (h) = \frac{{(α β)}^{(α + β) / 2}}{Γ (α) Γ (β)} {(\sqrt{h})}^{(α + β) / 2 - 2} K_{α - β} (2 \sqrt{α β \sqrt{h}}) .

Integral (8), the cumulative distribution function (CDF) of the unordered variable h_k can be expressed as

F_{h_{k}} (y) = \frac{{(α β)}^{\frac{α + β}{2}} y^{\frac{α + β}{4}}}{4 π Γ (α) Γ (β)} G_{1, 5}^{4, 1} (\frac{1}{16} {(α β)}^{2} y | \begin{matrix} 1 - (α + β) / 4 \\ \frac{α - β}{4}, \frac{α - β + 2}{4}, \frac{β - α}{4}, \frac{2 - α + β}{4}, - \frac{α + β}{4} \end{matrix}),

where

G_{p, q}^{m, n}

(y|·) is the Meijer’s G-function [19]. Based on the proposed power control scheme in (3), the path loss induced fading is counteracted. Then, only the intensity scintillation involves into the order statistics. Using order statistics [20], the PDF of the ordered channel fading h_k is

{f^{'}}_{h_{k}} (h) = \frac{K!}{(k - 1)! (K - k)!} {F_{h_{k}} (h)}^{K - k} {1 - F_{h_{k}} (h)}^{k - 1} f_{h_{k}} (h) .

The joint density of all k-order statistics is [20]

{f^{'}}_{h_{1}, \dots, h_{k}} (h_{1}, \dots, h_{k}) = k! \prod_{i = 1}^{k} f_{h_{i}} (h_{i}), h_{1} \geq \dots \geq h_{k} .

3. Performance analysis

In this section, the performance of the proposed NOMA system is evaluated from outage probability and BER. In order to present the superiority of NOMA in WOC, the ergodic sum rate is analyzed, where NOMA shows its special characteristics in WOC link. With the special features of WOC in mind, the user pairing scheme is also presented.

3.1 Outage probability

In this case, each user has a targeted data rate, determined by the quality of service (QoS), rather than the channel condition. When (4) is satisfied, the sum rate of the network is $\sum_{k} {\tilde{R}}_{k}$ . Outage probability is of interest. Based on (4), the outage probability can be expressed as [15]

P_{k}^{o u t} = 1 - P (E_{1}^{c} \cap \dots \cap E_{k}^{c}) = 1 - \prod_{i = 1}^{k} P (E_{i}^{c}),

where

E_{k} ≜ {R_{k} < {\tilde{R}}_{k}}

, and

E_{k}^{c}

is the complementary set of E_k. When k<K,

\begin{array}{l} P (E_{k}^{c}) = P {{[\frac{1}{2} \log (1 + \frac{{(μ_{k} g_{k} P_{k})}^{2}}{\sum_{i = k + 1}^{K} {(μ_{i} g_{i} P_{i})}^{2} + A σ^{2}}) - ε_{ϕ}]}^{+} \geq {\tilde{R}}_{k}} \\ \overset{(a)}{=} 1 - P {h_{k} \leq \frac{A ϕ_{k} + ϕ_{k} ρ \sum_{i = k + 1}^{K} L_{i}^{2} μ_{i}^{2} h_{i} a_{i}^{2}}{ρ μ_{k}^{2} a_{k}^{2} L_{k}^{2}}} = 1 - \int \dots \int_{y_{k + 1}}^{υ} {f^{'}}_{h_{k}, \dots, h_{K}} (y_{k}, \dots, y_{K}) d y_{k} \dots d y_{K}, \end{array}

where

υ = (A ϕ_{k} + ϕ_{k} ρ \sum_{i = k + 1}^{K} L_{i}^{2} μ_{i}^{2} h_{i} a_{i}^{2}) / (ρ μ_{k}^{2} a_{k}^{2} L_{k}^{2})

,

ϕ_{k} = e^{2 ({\tilde{R}}_{k} + ε_{ϕ})} - 1

, and

ρ = P_{a i m}^{2} / σ^{2}

. Step (a) is obtained by assuming

h_{k} / (υ ϕ_{k}) > \exp (2 ε_{ϕ}) - 1

. When k = K, (13) is changed to

P (E_{K}^{c}) = 1 - \int_{0}^{ψ_{K}} K {[1 - F_{h_{k}} (y)]}^{K - 1} f_{h_{k}} (y) d y = {[1 - F_{h_{k}} (ψ_{K})]}^{K},

where

ψ_{K} = A ϕ_{K} / ρ μ_{K}^{2} L_{K}^{2} a_{K}^{2}

. Substituting (13) and (14) into (12), the outage probability of k-th user can be derived.

3.2 BER performance

We assume on-off keying (OOK) modulation, which is the most widely utilized technique in WOC link. The conditional BER of WOC link is given by the following theorem.

Theorem 1: At the backbone node, the user’s message is decoded successively with SIC. The conditional BER of k-th user can be expressed as

\begin{array}{l} P_{e} (C_{k} | h) = \sum_{e_{k - 1} = 0, 1, - 1} \dots \sum_{e_{1} = 0, 1, - 1} P_{e} (C_{k} | e_{k - 1}, \dots, e_{1}, h) \\ \times P_{e} (e_{k - 1} | e_{k - 2}, \dots, e_{1}, h) P_{e} (e_{2} | e_{1}, h) P_{e} (e_{1} | h), \end{array}

where C_k expresses the event that the bit error occurs, h is (h₁, h₂, …, h_K).

e_{k} = x_{k} - {\hat{x}}_{k}

denotes the error in detecting k-th user. The error probabilities of all the previous detection,

P_{e} (C_{k} | e_{k - 1}, \dots, e_{1}, h)

, can be expressed as

P_{e} (e_{k} | e_{k - 1}, \dots, e_{1}, h) = {\begin{cases} \frac{1}{2} P_{e} (D_{k} | e_{k - 1}, \dots, e_{1}, h), e_{k} = 1, \\ 1 - P_{e} (C_{k} | e_{k - 1}, \dots, e_{1}, h), e_{k} = 0, \\ \frac{1}{2} P_{e} (E_{k} | e_{k - 1}, \dots, e_{1}, h), e_{k} = - 1, \end{cases}

where D_k is the event that error occurs when the transmitted message is ‘1’, and E_k is the event that error occurs when the transmitted message is ‘0’, then,

P_{e} (C_{k} | e_{k - 1}, \dots, e_{1}, h) = \frac{1}{2} [P_{e} (D_{k} | e_{k - 1}, \dots, e_{1}, h) + P_{e} (E_{k} | e_{k - 1}, \dots, e_{1}, h)],

where

P_{e} (D_{k} | e_{k - 1}, \dots, e_{1}, h) = \frac{1}{2^{K - k + 1}} \sum_{x_{k}} e r f c (γ_{k}^{1}),

and

P_{e} (E_{k} | e_{k - 1}, \dots, e_{1}, h) = \frac{1}{2^{K - k + 1}} \sum_{x_{k}} e r f c (γ_{k}^{2}),

and

\begin{array}{l} γ_{k}^{1} = \frac{(a_{k} g_{k} P_{a i m} + F_{k} - τ_{k})}{\sqrt{2} σ}, \\ γ_{k}^{2} = \frac{(τ_{k} - F_{k})}{\sqrt{2} σ}, \end{array}

where τ_k is the threshold when detecting the message of k-th user with maximum likelihood (ML) detection, shown in (27).

Proof: The proof is provided in Appendix A.

Theorem 1 presents the conditional BER of k-th user. It can be concluded from (15) that the BER of k-th user is influenced by the detection of i-th (i<k) user. Because of the multiple integration, we present the conditional BER rather than the average BER over channels. However, based on the conditional BER, the average BER can be easily obtained through numerical simulation.

3.3 Ergodic sum rate

In this case, user’s QoS depends on their own channel conditions, i.e. $R_{k} = {\tilde{R}}_{k}$ . Therefore, in this case, the inequation in (4) always holds. There is no outage and perfect SIC can be done. Ergodic sum rate is the ideal metric rather than outage probability, because it is always zero. Through the analysis of ergodic sum rate of WOC link, we intent to explain why NOMA technique is utilized in this paper, and present the unique features of NOMA in WOC. Ergodic sum rate can be expressed as

\begin{array}{l} R_{s u m}^{N O M A} = E {\sum_{k = 1}^{K} R_{k}} \overset{(b)}{=} E {\frac{1}{2} \log (1 + \frac{ρ \sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2}}{A}) - K ε_{ϕ}} \\ \overset{(c)}{\approx} \frac{1}{2} E {\log (\frac{ρ \sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2}}{A})} = \frac{1}{2} E {\log (\sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2})} + \frac{1}{2} \log ρ - \frac{1}{2} \log A, \end{array}

where E{·} is the expectation over channel fading. Step (b) is obtained by substituting (4) into it, and step (c) is derived at large ρ. It can be seen in (21) that the first term is determined by the channel condition and power allocation coefficient, and the third term is a constant. Only the second term is relative to the ρ. Therefore, the ergodic sum rate of NOMA is proportional to logρ. Based on the above equation, the following Propositions are presented.

Proposition 1: In terms of ergodic sum rate, NOMA outperforms OMA for ρ→∞.

Proof: Based on (21) and (30) (in Appendix B), the proposition can be easily proved with the help of Jensen’s inequality.

Proposition 1 presents that NOMA outperforms OMA at large ρ. Although WOC has some special characters (non-broadcast nature of optical beam and the vulnerable turbulence channel), NOMA still performs better. In RF, NOMA outperforms OMA in the whole range of ρ, however, NOMA outperforms OMA at large ρ. Notice that the gain comes from SIC, which increases the complexity of system. It is quite important to obtain the trade-off between the performance and complexity.

Proposition 2: In terms of ergodic sum rate, the performance gain of NOMA, $R_{g a i n} = R_{s u m}^{N O M A} - R_{s u m}^{O M A}$ , increases with the increase of turbulence strength.

Proof: The proof is provided in Appendix B.

Besides the influence of ρ, the turbulence strength is another important factor in WOC. Proposition 2 intents to explain the influence of turbulence strength. Proposition 2 presents that the performance gain of NOMA is larger in strong turbulence channel. As we all know, the increase of turbulence strength causes the performance loss. However, the NOMA technique can neutralize the performance loss. The reason behind the proposition is that the randomization of fading level increases with the increase of turbulence strength, which helps to increase the power diversity. The increased power diversity can neutralize the performance loss caused by enlarging turbulence strength. Combining the two propositions, we can easily make the conclusion that NOMA technique is more suitable for WOC system, especially in strong turbulence channel.

3.4 User pairing

The performance gain of NOMA comes from increase of systematic complexity due to SIC. The increase of the total number of users in the same NOMA group results into the complexity of the practical system. Generally, the hybrid scheme, consisting of NOMA and OMA, is an alternative method. The pairing criterion is quite important for the hybrid scheme. In this subsection, we aim at analysing the impact of user pairing on the system performance, and then present the criterion for user pairing. For giving the straightforward but insightful results, we consider a group size of two. Moreover, the results can be easily extended to the case in which the arbitrary users are paired. We analyse the effects from two aspects. Firstly, we derive the power conditions that the NOMA throughput gains are guaranteed. Secondly, the influence of user pairing on ergodic sum rate is studied.

Assume that the i-th user and j-th user are paired together, and i-th user is decoded priority. Without loss of generality, for OMA system, the degree of freedom allocated to two users is 0.5. Therefore, the achieved data rate for each user is

R_{k}^{O M A} = 0.5 {[\frac{1}{2} \log (1 + \frac{2 ρ {(μ_{k} g_{k} a_{k})}^{2}}{A}) - ε_{ϕ}]}^{+},

where k = {i, j}. Based on (4) and (22), the power condition is given by the following theorem, with the condition

R_{k} > R_{k}^{O M A}

,

\forall k

.

Theorem 2: Assume that the i-th user and j-th (i<j) user are paired in the same NOMA group, in which the i-th user is decoded before j-th user. For j-th user, the performance of NOMA is always better than that of OMA. However, for i-th user, to guarantee that NOMA outperforms OMA, j-th user’s power has to meet a necessary condition, which is $a_{j} < \sqrt{A \sqrt{(1 + B)} - A / 2 ρ {(μ_{j} g_{j})}^{2}}$ , where $B = 2 ρ {(μ_{i} g_{i} a_{i})}^{2} / A$ .

Proof: The proof is provided in Appendix C.

Theorem 2 presents the criterion for two users, and it is quite important for i-th user that the power diversity is big enough. Another important feature revealed in the Theorem 2 is that the performance of user with smaller received power (j-th user) can be enhanced by utilizing NOMA. This theorem is the basic condition for user pairing. The users in the same group have to satisfy the power condition given in Theorem 2. Besides the impact of pairing scheme on data rate of individual user, the pairing scheme also has great effect on the ergodic sum rate. Theorem 3 presents the influence to the ergodic sum rate at large ρ.

Theorem 3: Assuming that the i-th user and j-th (i<j) user are paired in the same NOMA group (i-th user is priority decoded), the ergodic sum rates of NOMA and OMA satisfy the following equations.

For $\forall i < K$ ,

\lim_{ρ \to \infty} E {R_{i + 1} + R_{j} - R_{i + 1}^{O M A} - R_{j}^{O M A}} < \lim_{ρ \to \infty} E {R_{i} + R_{j} - R_{i}^{O M A} - R_{j}^{O M A}} .

For $\forall j < K$ ,where $R_{i}$ and ${R^{'}}_{i}$ are the date rate of i-th user when it is pairing with j-th user and (j + 1)-th user, respectively.

\lim_{ρ \to \infty} E {R_{i} + R_{j} - R_{i}^{O M A} - R_{j}^{O M A}} < \lim_{ρ \to \infty} E {{R^{'}}_{i} + R_{j + 1} - R_{i}^{O M A} - R_{j + 1}^{O M A}} .

Proof: The proof is provided in Appendix D.

Theorem 3 demonstrates how the ergodic sum rate varies if i and j changes. We can make the conclusion that the optimal sum rate gain is achieved when the two users with the most distinctive channel are paired together. Theorem 3 presents the criterion for the optimization of ergodic sum rate. Note that both Theorem 2 and Theorem 3 should be considered for user pairing.

4. Simulation results

In this section, the performance of NOMA is evaluated, where OMA is used as benchmark. Both theoretical and Monte Carlo simulation results are presented, denoted as “Theo.” and “Simu.”, respectively. Two-user system is considered and the link distances between usersand central node are 1 km and 3 km, respectively. The wavelength of the laser is 1550 nm. Clear visibility of 16 km with $σ_{R}^{2}$ = {0.1, 1} is considered. In Fig. 2, the outage probability for two users are depicted when the power back-off are 2 dB, 3 dB, and 5 dB, respectively. Figure 2(a) shows the results when Rytov variance ( $σ_{R}^{2}$ ) is 0.1, and the targeted data rates for two users are ${\tilde{R}}_{1} =0 .5$ and ${\tilde{R}}_{2} =0 .5$ . For user 1, the outage performance of 5 dB outperforms that of 2 dB and 3 dB power back-off because large power back-off helps to decrease the interference from the second user. However, user 2 holds the opposite results, due to the essence of the back-off scheme is reducing the power of second user. Although better performance of the first user helps to decode user 2, the arrived signal-to-noise ratio (SNR) of second user is too small when the back-off power is large enough. Figure 2(b) presents the similar results as Fig. 2(a), when the Rytov variance is 1. Comparing Fig. 2(a) with Fig. 2(b), the strength of turbulence has great influence to the outage probability performance as expected. In both subfigures, the Monte Carlo results match quite well with the theoretical results. In Fig. 3, the outage probability performances of two users under different targeted data rate are shown when ${\tilde{R}}_{1} = {\tilde{R}}_{2}$ , and ζ = 3 dB. In both subfigures, higher targeted data rate results in worse outage probability as it is more difficult to be satisfied. Moreover, the BER is shown in Fig. 4 when $σ_{R}^{2}$ = 0.1. It is clearly shown in the Fig. 4(a) that the BER performance of user 1 is better when ζ = 5 dB than that when ζ = 2 dB, and 3 dB, while user 2 holds the opposite results. The properties of BER are similar to the outage probability shown in Fig. 2. In terms of BER, there are an appropriate power back-off in order to achieve the minimum average BER. Figure 4(b) shows the average BER and individual BER when {ρ, $σ_{R}^{2}$ } = {25 dB, 0.1}. It verifies the analysis in Fig. 4(a). For user 1, the BER performance is improved with the increase of power back-off. However, for user 2, there are a minimum value with the varies of power back-off. With the increase of power back-off, the BER first decreases and then increases. The reason why the BER first decrease is that the better performance of user 1 helps to decode the message of user 2. However, if the power back-off is too large, the SNR of user 2 is too small. That is why the BER increases. In Fig. 5, the ergodic sum rate between NOMA and OMA are presented when ζ = 5dB, where degree of freedom allocated to two users is 0.5 in OMA. ζ = 5dB is just one of the special cases to give the visualized results of Proposition 1. Figure 5(a) demonstrated that the NOMA can achieve a larger sum rate than OMA under different turbulence strength as demonstrated in Proposition 1. The performance loss of NOMA is quite small compared with OMA with the increase of turbulence strength, and it is another advantage of NOMA technique. In Fig. 5(b), the difference value of ergodic sum rate between NOMA and OMA ( $R_{s u m}^{N O M A} - R_{s u m}^{O M A}$ ) is presented under ρ is 30 dB and 40 dB.

Fig. 2 Outage probability of NOMA for different ζ. (a) $σ_{R}^{2}$ = 0.1. (b) $σ_{R}^{2}$ = 1.

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Fig. 3 Outage probability of NOMA for different targeted data rate. (a) $σ_{R}^{2}$ =0.1. (b) $σ_{R}^{2}$ =1.

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Fig. 4 BER performance of the proposed NOMA system $σ_{R}^{2}$ =0.1. (a) BER of NOMA versus ρ. (b) BER of NOMA versus ζ.

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Fig. 5 Ergodic sum rate performance of NOMA. (a) Ergodic sum rate results of NOMA as the function of ρ. (b) Difference value of ergodic sum rate.

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As can be seen from the figure, the performance gain of NOMA increases with the increase of intensity strength as demonstrated in Proposition 2. In order to reinforce the statements for the user pairing scheme, we provide the simulation results in Fig. 6. In the simulations, three-user system is considered, and the link distances between users and central node are 1 km, 2 km, and 3 km, respectively. $σ_{R}^{2}$ = 1 is considered. The power back-off step ζ = 5dB. In the figure, “U_i-U_j” denotes that the i-th user and j-th user are paired together. It is clearly shown in the figure that the optimal performance gain is achieved when user 1 and user 3 are paired together.

Fig. 6 Performance gain of different user pairing cases.

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5. Conclusions

In this paper, NOMA technique with power control scheme has been proposed for WOC system. The performance of the system is evaluated from the outage probability and BER aspects. The ergodic sum rate of the WOC link is analyzed, where we show that NOMA achieves a superior ergodic sum data rate than OMA. With the characters of turbulence channel in mind, we further prove that NOMA is more suitable for WOC link than OMA, especially for strong turbulence channel. Because SIC has to be considered, the proposed system introduces additional complexity. It is quite important to achieve a tradeoff between performance and complexity. As an alternative method, the hybrid system is considered, in which the criterion of user pairing scheme is theoretically given. Monte Carlo simulations are presented, which corroborate the theoretical results.

Appendix A

With the utilization of law of total probability, (15) can be easily obtained. Therefore, the proof of the theorem can be converted to prove (18) and (19). The signal model when detecting k-th user can be expressed as

y_{k} = {\begin{cases} a_{k} g_{k} P_{a i m} + F_{k} + n, x_{k} = 1, \\ F_{k} + n, x_{k} = 0, \end{cases}

where

F_{k} = P_{a i m} \sum_{i = 1}^{k - 1} g_{i} a_{i} e_{i} + P_{a i m} \sum_{i = k + 1}^{K} g_{i} a_{i} x_{i},

where the first term is the residual interference caused by the previous detection, and the second term is the interference of the j-th user (j>k). Under ML criterion, the threshold of k-th user can be expressed as

τ_{k} = \frac{a_{k} g_{k} P_{a i m} + P_{a i m} \sum_{i = k + 1}^{K} a_{i} g_{i}}{2} .

Denote

x_{k} = (x_{k}, x_{k + 1}, \dots, x_{K})

. The

P_{e} (D_{k} | e_{k - 1}, \dots, e_{1}, x_{k}, h)

is

P_{e} (D_{k} | e_{k - 1}, \dots, e_{1}, x_{k}, h) = \int_{- \infty}^{τ_{k}} \frac{1}{\sqrt{2 π σ^{2}}} \exp [- \frac{{(x - (a_{k} g_{k} P_{a i m} + F_{k}))}^{2}}{2 σ^{2}}] d x = \frac{1}{2} e r f c (γ_{k}^{1}) .

With the utilization of law of total probability, (18) is obtained.

Similarly, when the transmitted message of k-th user is ‘0’, the conditional error rate can be expressed as

P_{e} (E_{k} | e_{k - 1}, \dots, e_{1}, x_{k}, h) = \int_{τ_{k}}^{+ \infty} \frac{1}{\sqrt{2 π σ^{2}}} \exp [- \frac{{(x - F_{k})}^{2}}{2 σ^{2}}] d x = \frac{1}{2} e r f c (γ_{k}^{2}) .

With the utilization of law of total probability, (19) is obtained.

The proof is completed.

Appendix B

The performance gain can be expressed as $R_{g a i n} = R_{s u m}^{N O M A} - R_{s u m}^{O M A}$ . For proving the proposition, we need to prove the first derivation of R_gain with respect of $σ_{R}^{2}$ is bigger than zero. Therefore, the proof can be converted to prove that $\partial R_{s u m}^{N O M A} / \partial σ_{R}^{2}$ is bigger than $\partial R_{s u m}^{O M A} / \partial σ_{R}^{2}$ . The ergodic sum rate of OMA is

\begin{array}{l} R_{s u m}^{O M A} = E {\sum_{k = 1}^{K} η_{k} {[\frac{1}{2} \log (1 + \frac{ρ {(μ_{k} g_{k} a_{k})}^{2}}{η_{k} A}) - ε_{ϕ}]}^{+}} \overset{(d)}{\approx} \frac{1}{2} \sum_{k = 1}^{K} η_{k} E [\log (\frac{ρ {(μ_{k} g_{k} a_{k})}^{2}}{η_{k} A})] \\ = \frac{1}{2} \log ρ + \frac{1}{2} \sum_{k = 1}^{K} η_{k} E [\log ({(μ_{k} g_{k} a_{k})}^{2})] - \frac{1}{2} \sum_{k = 1}^{K} η_{k} \log (η_{k} A), \end{array}

where η_k is the degree of freedom of k-th user, and

\sum_{k = 1}^{K} η_{k} = 1

. Step (d) is obtained at large ρ. Then,

\begin{array}{l} \frac{\partial R_{s u m}^{O M A}}{\partial σ_{R}^{2}} = \frac{1}{2} \frac{\partial}{\partial σ_{R}^{2}} {\sum_{k = 1}^{K} η_{k} E [\log ({(μ_{k} g_{k} a_{k})}^{2})]} = \frac{1}{2} \int_{g} \sum_{k = 1}^{K} η_{k} \log ({(μ_{k} g_{k} a_{k})}^{2}) \frac{\partial}{\partial σ_{R}^{2}} f (g, σ_{R}^{2}) d g \\ \leq \frac{1}{2} \int_{g} \log (\sum_{k = 1}^{K} η_{k} {(μ_{k} g_{k} a_{k})}^{2}) \frac{\partial}{\partial σ_{R}^{2}} f (g, σ_{R}^{2}) d g \leq \frac{1}{2} \int_{g} \log (\sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2}) \frac{\partial}{\partial σ_{R}^{2}} f (g, σ_{R}^{2}) d g, \end{array}

where g is (g₁, g₂, …, g_K), and f(g,

σ_{R}^{2}

) is the joint PDF of (g₁, g₂, …, g_K). Utilizing (21),

\partial R_{s u m}^{N O M A} / \partial σ_{R}^{2}

is derived as

\begin{array}{l} \frac{\partial R_{s u m}^{N O M A}}{\partial σ_{R}^{2}} = \frac{1}{2} \frac{\partial}{\partial σ_{R}^{2}} E {\log (\sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2})} = \frac{1}{2} \frac{\partial}{\partial σ_{R}^{2}} \int_{g} \log (\sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2}) f (g, σ_{R}^{2}) d g \\ = \frac{1}{2} \int_{g} \log (\sum_{k = 1}^{K} {(μ_{k} g_{k} a_{k})}^{2}) \frac{\partial}{\partial σ_{R}^{2}} f (g, σ_{R}^{2}) d g . \end{array}

Comparing (31) with (32), $\partial R_{s u m}^{N O M A} / \partial σ_{R}^{2}$ is bigger than $\partial R_{s u m}^{O M A} / \partial σ_{R}^{2}$ , which completes the proof. We obtain the statement of the Proposition 2.

Appendix C

The proof of the theorem should be divided into two parts. Firstly, we prove that j-th user performs better with NOMA. Secondly, we prove that j-th user’s power has to meet a necessary condition, in order to guarantee that i-th user performs better with NOMA. For the first part, the data rate expressions of NOMA and OMA of j-th user are expressed in (4) and (22). Then,

\begin{array}{l} R_{j} - R_{j}^{O M A} \approx \frac{1}{2} \log (1 + \frac{ρ {(μ_{j} g_{j} a_{j})}^{2}}{A}) - \frac{1}{4} \log (1 + \frac{2 ρ {(μ_{j} g_{j} a_{j})}^{2}}{A}) \\ = \frac{1}{4} \log (\frac{A^{2} + 2 A ρ {(μ_{j} g_{j} a_{j})}^{2} + {[ρ {(μ_{j} g_{j} a_{j})}^{2}]}^{2}}{A (A + 2 ρ {(μ_{j} g_{j} a_{j})}^{2})}) = \frac{1}{4} \log (1 + \frac{{[ρ {(μ_{j} g_{j} a_{j})}^{2}]}^{2}}{A (A + 2 ρ {(μ_{j} g_{j} a_{j})}^{2})}) > 0, \end{array}

where

0.5 ε_{ϕ}

is neglected, because

ε_{ϕ} = 0.016

. Therefore,

R_{j} > R_{j}^{O M A}

.

For the second part, the condition is derived by utilizing $R_{i} > R_{i}^{O M A}$ , as

\frac{1}{2} \log (1 + \frac{ρ {(μ_{i} g_{i} a_{i})}^{2}}{ρ {(μ_{j} g_{j} a_{j})}^{2} + A}) - ε_{ϕ} > 0.5 {\frac{1}{2} \log (1 + \frac{ρ {(μ_{i} g_{i} a_{i})}^{2}}{0.5 \times A}) - ε_{ϕ}} .

Neglecting

ε_{ϕ}

, after some algebraic manipulations, the condition can be obtained. To this end, Theorem 2 is proved.

Appendix D

The expression in (23) is equivalent to

\lim_{ρ \to \infty} E {\underset{I_{1}}{\underset{︸}{R_{i + 1} - R_{i}}} - \underset{I_{2}}{\underset{︸}{(R_{i + 1}^{O M A} - R_{i}^{O M A})}}} < 0.

Firstly, we derive the expressions of ℐ1 and ℐ2. Based on the derived expressions, (35) is proved. Utilizing (4),

\begin{array}{l} I_{1} = \frac{1}{2} \log (1 + \frac{ρ {(μ_{i + 1} g_{i + 1} a_{i + 1})}^{2}}{ρ {(μ_{j} g_{j} a_{j})}^{2} + A}) - ε_{ϕ} - \frac{1}{2} \log (1 + \frac{ρ {(μ_{i} g_{i} a_{i})}^{2}}{ρ {(μ_{j} g_{j} a_{j})}^{2} + A}) + ε_{ϕ} \\ = \frac{1}{2} \log (\frac{ρ {(μ_{j} g_{j} a_{j})}^{2} + A + ρ {(μ_{i + 1} g_{i + 1} a_{i + 1})}^{2}}{ρ {(μ_{j} g_{j} a_{j})}^{2} + A + ρ {(μ_{i} g_{i} a_{i})}^{2}}) . \end{array}

With (22),

\begin{array}{l} I_{2} = 0.5 [\frac{1}{2} \log (1 + \frac{2 ρ {(μ_{i + 1} g_{i + 1} a_{i + 1})}^{2}}{A}) - ε_{ϕ}] - 0.5 [\frac{1}{2} \log (1 + \frac{2 ρ {(μ_{i} g_{i} a_{i})}^{2}}{A}) - ε_{ϕ}] \\ = \frac{1}{4} [\log (\frac{A + 2 ρ {(μ_{i + 1} g_{i + 1} a_{i + 1})}^{2}}{A + 2 ρ {(μ_{i} g_{i} a_{i})}^{2}})] . \end{array}

Substituting (36) and (37) into(35),

\lim_{ρ \to \infty} E {R_{i + 1} - R_{i} - R_{i + 1}^{O M A} + R_{i}^{O M A}} = \frac{1}{2} E {\log \underset{I_{3}}{\underset{︸}{(\frac{{(μ_{j} g_{j} a_{j})}^{2} + {(μ_{i + 1} g_{i + 1} a_{i + 1})}^{2}}{{(μ_{j} g_{j} a_{j})}^{2} + {(μ_{i} g_{i} a_{i})}^{2}} \frac{μ_{i} g_{i} a_{i}}{μ_{i + 1} g_{i + 1} a_{i + 1}})}}} .

With some algebraic manipulations, it is easily to obtain ℐ₃<1. Therefore, (35) is proved. Meanwhile, the expression of (24) is equivalent to

\lim_{ρ \to \infty} E {{R^{'}}_{i} - R_{i} + R_{j + 1} - R_{j} - R_{j + 1}^{O M A} + R_{j}^{O M A}} > 0.

The proof of (39) is conducted in a similar way as the proof of (35), which is shown as

\begin{array}{l} \lim_{ρ \to \infty} E {{R^{'}}_{i} - R_{i} + R_{j + 1} - R_{j} - R_{j + 1}^{O M A} + R_{j}^{O M A}} \\ = \frac{1}{2} E {\log \underset{I_{4}}{\underset{︸}{(\frac{{(μ_{j + 1} g_{j + 1} a_{j + 1})}^{2} + {(μ_{i} g_{i} a_{i})}^{2}}{μ_{j + 1} g_{j + 1} a_{j + 1}} \frac{μ_{j} g_{j} a_{j}}{{(μ_{j} g_{j} a_{j})}^{2} + {(μ_{i} g_{i} a_{i})}^{2}})}}} > 0. \end{array}

With some algebraic manipulations, it is easily to obtain ℐ₄>1. Therefore, (40) is proved.

Combining (35) and (39), Theorem 3 is proved.

Funding

National Key Research and Development Program of China (2016QY02D0304); National Key Basic Research Program of China (2013CB329205); National Natural Science Foundation of China (NSFC) (60572002).

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Multi-user access in wireless optical communication system

Abstract

1. Introduction

2. System and channel models

2.1 System model

2.2 Channel model

3. Performance analysis

3.1 Outage probability

3.2 BER performance

3.3 Ergodic sum rate

3.4 User pairing

4. Simulation results

5. Conclusions

Appendix A

Appendix B

Appendix C

Appendix D

Funding

References

Cited By

Figures (6)

Equations (40)

Optics Express