Performance Analysis of Dual-Hop Hybrid RF-UOWC NOMA Systems

The hybrid combination between underwater optical wireless communication (UOWC) and radio frequency (RF) is a vital demand for enabling communication through the air–water boundary. On the other hand, non-orthogonal multiple access (NOMA) is a key technology for enhancing system performance in terms of spectral efficiency. In this paper, we propose a downlink NOMA-based dual-hop hybrid RF-UOWC with decode and forward (DF) relaying. The UOWC channels are characterized by exponential-generalized Gamma (EGG) fading, while the RF channel is characterized by Rayleigh fading. Exact closed-form expressions of outage probabilities and approximated closed-form expressions of ergodic capacities are derived, for each NOMA individual user and the overall system as well, under the practical assumption of imperfect successive interference cancellation (SIC). These expressions are then verified via Monte-Carlo simulation for various underwater scenarios. To gain more insight into the system performance, we analyzed the asymptotic outage probabilities and the diversity order. Moreover, we formulated and solved a power allocation optimization problem to obtain an outage-optimal performance. For the sake of comparison and to highlight the achievable gain, the system performance is compared against a benchmark orthogonal multiple access (OMA)-based system.


Introduction
Underwater optical wireless communication (UOWC) has received substantial research interest as an efficient transmission technology for a wide range of underwater applications such as surveillance and oceanic monitoring. Many wireless data transmission techniques faced limitations while communicating underwater, including acoustic waves and radio-frequency (RF) signals. An acoustic-based underwater communication has many drawbacks such as high latency, low data rates, and high attenuation. The situation was not much different when using RF in underwater communication scenarios [1,2]. An acoustic-based underwater communication has many drawbacks such as high latency, low data rates, high bit error rates, and high attenuation. In addition, it severely suffers from malicious attacks. This is due to the fact that acoustic communication channels are uniquely designed for networks used on land; they require more sophisticated security mechanisms [3]. The situation was not much different when using RF in underwater communication scenarios [1]. The underwater RF communications suffers from high power wireless channel is characterized by Rayleigh fading with an additive white Gaussian noise (AWGN) and the UOWC links are characterized by EGG fading with AWGN. (2) We analyzed the diversity order of the OPs. (3) We proposed and solved a power allocation optimization problem to obtain an outage-optimal power allocation factor. (4) We validated the analytical derivations through Monte-Carlo simulations for varying underwater scenarios of air bubbles level (BL) under thermally uniform and temperature gradient UOWC channels, then we analyzed the impact of system parameters on the system performance. (5) Finally, we carried out a comparison between the proposed system with an OMA-based benchmark system.
The rest of the paper is organized as follows, the system model is introduced in Section 2. The performance of the considered system is analytically evaluated by deriving the OPs and ECs in Sections 3 and 4, respectively. The proposed power allocation algorithm is provided in Section 5. Analytical and simulation results are discussed and compared with a benchmark system in Section 6. Finally, conclusions are provided in Section 7.

System Model
In this paper, we propose a downlink NOMA-based dual-hop hybrid RF-UOWC system depicted in Figure 1, where the source (S) is equipped with an RF interface that aims to communicate with two destinations (D 1 and D 2 ) equipped with UOWC interface via an intermediate decode and forward relay (R). The relay has an RF interface to receive from S and then transmit to D 1 and D 2 through the UOWC interface, where D 1 is the far or weak user and D 2 is the near or strong user. Such a scenario can find applications in many areas in the UIoT [19] (e.g., offshore oil field exploration, oceanic monitoring, and data collection). The S-R channel (h w ) is assumed to be a RF channel characterized by Rayleigh fading with AWGN and the R-D i channels (h i ) are assumed to be UOWC channels characterized by EGG fading with AWGN, where i ∈ {1, 2}. For the sake of improving the spectral efficiency, we assume that S and R adopt PD-NOMA for multiplexing their messages. The communication is initiated at S by multiplexing the two messages x 1 and x 2 intended for D 1 and D 2 , respectively. The Sto-R message is x S = √ a 1 P S x 1 + √ a 2 P S x 2 , where P S is the total transmitted power at S and a i is the NOMA power allocation factor for D i at S. Without loss of generality, we assume that a 1 > a 2 and a 1 + a 2 = 1. The received message at R through the RF link is S-to-R link distance, v is the RF channel path-loss exponent, and n ω represents AWGN with n ω ∼ CN (0, σ 2 ω ). Utilizing NOMA concept, R decodes x 1 first, then applies the SIC operation, which is assumed to be imperfect, to decode x 2 . So, the signal-to-interferenceplus noise ratios (SINRs) for decoding x 1 and x 2 are expressed as , respectively, where ρ s = P S σ 2 ω , and 0 ≤ η ≤ 1 is the residual power factor of the imperfect SIC.
In the second phase, R retransmits the received messages over the UOWC channels that are characterized by independent but not necessarily identical mixture EGG distribution [5]. The relay multiplexes the detected messages using PD-NOMA again, such that where P R is the total transmitted power at R and b i is the NOMA power allocation factor for D i at R. Without loss of generality, b 1 > b 2 and b 1 + b 2 = 1. The received message at D 1 through the UOWC link h 1 is y D1 = εh 1 x R + n u , where h 1 is the EEG fading of UOWC channel from R-to-D 1 with expectation E[|h 1 | 2 ] = 1, ε is responsivity that is considered to be unity, and n u represents AWGN with n u ∼ CN (0, σ 2 u ). Utilizing NOMA concept, D 1 decodes x 1 first. So, the SINR for decoding x 1 at D 1 is expressed as The received message at D 2 through the UOWC link h 2 is y D2 = εh 2 x R + n u , where h 2 is the EEG fading of UOWC channel from R-to-D 2 with expectation E[|h 2 | 2 ] = 1. Following the NOMA principle, D 2 decodes x 1 first and then applies the SIC operation, which is assumed to be imperfect, to decode x 2 . So, the SINRs for decoding x 1 and x 2 are expressed

Channels Distributions:
We assume that the UOWC links h 1 and h 2 are characterized by the EGG distribution [5], which models the underwater turbulence fading resulting from air bubbles and gradient of temperature in an effective manner. EGG is a weighted combination of the exponential and generalized Gamma distributions, it effectively matches the experimental results obtained under different scenarios of channel impairments of UOWC. A closed-form expression for the cumulative distribution function (CDF) of EGG distribution is given as [5] where 0 < w < 1 represents the mixture ratio between exponential and generalized Gamma distributions, λ is the exponential distribution scale parameter of the exponential distribution, (a, b, c) are the parameters associated with generalized Gamma distribution, and G p,q m,n (.) is the Mejier-G function [20]. According to the receiver detection method, heterodyne detection (r = 1) or intensity modulation/direct detection (IM/DD) (r = 2), the electrical signal to noise ratio (SNR) is where Ω xi is the average SNR of the UOWC links. We assume that Ω x1 = Ω x2 = Ω x , thus µ r1 = µ r2 = µ r . The values of (w, λ, a, b, c) for different scenarios of air bubbles under thermally uniform and gradient-based UOWC channels are experimentally obtained in [5] ( Tables 1 and 2). Finally, the RF-links h w undergo a Rayleigh fading with AWGN noise, therefore |h i | 2 follows an exponential distribution whose CDF is given as (3) Table 1. EGG parameters for temperature gradient water [5].

Outage Probability Analysis
In this section, the system performance analysis in terms of OPs is presented. The OPs are defined as the probability that the received SINR falls below a certain threshold limit. We derived closed-form expressions for the outage at each destination as well as the overall system outage. Then, we derive an asymptotic expression for each of them at a high SNR regime. To gain more insight into the system performance, the outage diversity order is further derived.

Outage Probability OP 1
The outage event of D 1 , OP 1 , occurs if R or D 1 fails to decode x 1 , which can be formulated as

Outage Probability OP 2
The outage OP 2 occurs if R or D 2 fails to decode x 1 or x 2 ; this is due to NOMA SIC concept that involves receiving x 1 and cancels it before receiving x 2 . It is formulated as where (b) stems from the independence between h w and h 2 , . With the aid of CDFs in (1) and (3), we obtain a closed-form expression of OP 2 as in (7).
3.3. System Outage Probability OP sys The total system outage OP sys occurs if R or D 2 fails to decode any of the two messages or D 1 fails to decode x 1 . It is formulated as where (c) stems from the independence between h w , h 1 , and h 2 . With the aid of CDFs in (1) and (3), we obtain a closed-form expression of OP 2 as in (9).

Asymptotic Outage Probability
A deep insight on the system performance under high SNRs regime is obtained through the derivation of the asymptotic outage probabilities. A tight asymptotic expression for the CDF of the exponential and EGG distributions at high SNR are [5] Based on (10) and (11), we derive asymptotic expressions for OP 1 , OP 2 , and OP sys as

Diversity Order
To gain more insight, we study the achievable diversity order (DO) of the obtained OPs. DO is the slope of OP l where l ∈ {1, 2, sys}. According to [21], we can calculate diversity order as DO l = − lim ρ→∞ (log(OP l ) log(ρ)). It is clear from (12)-(14) that DO l = min(1, 1 r ). As ac r >> 1 in all scenarios, this result is consistent with the plots in Figure 2.

Ergodic Capacity Analysis
In this section, we derive an approximated closed-form expression for the ergodic capacity (EC) of the proposed system under the condition a i = b i . The instantaneous channel capacities for the two messages, C x l , C x 2 , are given by [13,22] The EC, defined as the expectation of the channel capacity, can be mathematically expressed as [21] where j ∈ {a, b}. The ergodic sum capacity (ESC) can be expressed as In the following subsections, we derive the individual ECs.

Ergodic Capacity EC x 2
The CDF F γ b (γ) is given as where (e) stems from the independence of the channels gain and 0 < γ < a 2 a 1 η . Then then applying variable transformation of τ b = γ (a 2 − ηa 1 γ) and using the Rayleigh CDF (3) and the tight approximated EGG CDF at high SNR (11), we can write where By Substituting (28) into (26), a closed-form expression of EC x 2 is obtained.

Proposed Power Allocation Algorithm
In this section, we propose a power allocation algorithm for optimizing the system OP under the condition a i = b i , where i ∈ {1, 2} or equivalently τ i = β i and τ = β. The proposed optimization problem is expressed as We provide the following Theorem to solve Problem (29). (29) is a convex problem, and the optimal power allocation factor value is

Theorem 1. Problem
Proof. See Appendix A. Figure 6 graphically verifies that the obtained result in Theorem 1 is correct. We set R 1 = 0.5 and R 2 = 0.75 as a test values, which implies that a * 1 ≈ 0.58 mathematically, which is consistent with the optimal value in Figure 6.

Results and Discussion
In this section, we provide a detailed discussion on the derived metrics of the proposed system under varying conditions of air bubbles for both fresh/salty and thermally uniform waters under heterodyne or IM/DD detection techniques to gain more insight and highlight some conclusions. The correctness of the obtained analysis is verified via a Monte-Carlo simulation with 10 6 samples. Throughout this section, we used the distribution parameters provided in Tables 1 and 2. Unless otherwise mentioned, the system parameters are set to a 1 = b 1 = 0.7, η = 0.1, R 1 = 0.5 bits/sec/Hz, and R 2 = 0.75 bits/sec/Hz; d = 0.8 is the normalized distance with respect to the cell radius, and v = 2, ρ s = ρ R = ρ, and Ω x = 1. In the following, we denote "Ana" as the analytical result, "Asym" as an asymptotic result, and "Sim" as Monte-Carlo simulation results. Figure 2 presents the outage probability for the proposed system under uniform temperature salty water for both IM/DD and heterodyne techniques. As expected, it can be deduced that the OPs significantly improve when heterodyne detection is implemented compared to IM/DD. This result is due to the ability of the heterodyne receiver to overcome the UOWC link's turbulence effects, while this leads to a more complex receiver compared to IM/DD receiver. For example, the OP sys of 10 −2 is achieved at ρ = 37 dB under the heterodyne receiver and ρ = 46 dB using the IM/DD receiver. It is remarkable that the analytical and the simulation results are a match, which validates our analytical derivations. Additionally, they match the asymptotic curves at high SNR regime. In addition, to validate the DO derived in Section 3.5, we can observe that for heterodyne detection r = 1, the OP sys = 0.0004747 at ρ = 50 dB and OP sys = 0.00004747 at ρ = 60 dB; therefore, the OP sys falls with a slope of log(0.0004747) − log(0.00004747) = 1. Following the same procedure for IM/DD, we can observe that the OP sys = 0.006119 at ρ = 50 dB while OP sys = 0.001835 at ρ = 60 dB, so the OP sys falls with a slope of log(0.006119)−log(0.001835) ≈ 0.5. These results are consistent with the diversity order DO l . Figure 3 depicts the OPs for the proposed system under uniform temperature salty water with varying air bubbles levels BL = 2.4 and BL = 4.7 L/min. It is clear that the increase in the level of air bubbles leads to a degradation in the OPs performance. This is due to the rise of the water turbulence. To evaluate the performance of the proposed system in this work, we compared its performance with a benchmark scheme: the OMAbased dual-hop hybrid RF-UOWC system. Figure 3 provides the comparison between the proposed NOMA-based system versus the OMA-based system under the same system settings. According to the figure, the proposed system outperforms the benchmark in terms of OPs performance. This is due to the fact that the NOMA technique is more spectral efficient than the OMA technique.  Figure 4 illustrates the influence of the residual power factor of imperfect SIC on OPs performance of the proposed system under uniform thermally salty water at BL = 2.4 L/min utilizing three varying levels of η = 0, 0.1, 0.2. We can see that the OPs performance degrades by increasing η while the best performance is achieved with the perfect SIC scenario (η = 0). This is due to the fact that an increase in η leads to a higher interference level, hence the SINRs γ 2 R and γ 2 D2 decrease while decoding the near user message. However, the SINRs γ 1 R , γ 1 D1 , and γ 1 D2 are not affected by changing η.  Figure 6 demonstrates the influence of the power allocation factor a 1 = b 1 , which varies from 0.5 to 0.99, on the OPs performance with ρ = 40 dB in two varying air bubble levels of BL = 2.4 and BL = 4.7 L/min. We can observe that the OP 1 enhances with the increase in a 1 due to the increase of its own message power. On the other hand, the OP 2 witnesses an improvement at first with a 1 increase as D 2 needs to decode x 1 first before decoding its own message x 2 . However, with the continuous increase in a 1 , an inflection point is reached since increasing a 1 means decreasing the allocated power for D 2 message (a 2 = 1 − a 1 ) that degrades the OP 2 . Finally, the OP sys follows the same trend as OP 2 with a bit increase. Additionally, this figure graphically proves the convexity of the optimization problem in (29).  Figure 7 illustrates the influence of the residual power factor of imperfect SIC on ECs performance of the proposed system under uniform thermally salty water at BL = 2.4 L/min where η = 0.01, and 0.05. We can see that the EC x 2 and ESC performance degrades by increasing η. This is due to the fact that an increase in η leads to a higher interference level at the decoding process of x 2 . On the other hand, the EC x 1 performance is not affected by changing η. The figure also shows a perfect agreement between the simulation and the obtained analytical results at high SNR with a small deviation at low SNR. This deviation is due to the usage of the tight approximated expression for the CDF of the EGG distributions at high SNR.  Moreover, Figure 9 shows the effect of TG on the ECs performance in salty water under the air bubbles level BL = 2.4 L/min. The figure investigated two different values of TG = 0.05, 0.15. It is obvious that the higher the level of the temperature gradient, the stronger the turbulence, leading to a ECs performance degradation. From Figures 8 and 9, we can conclude that the effect of the variation in water turbulence (BL, TG) is negligible at the high SNR regime.  Figure 10 demonstrates the influence of the power allocation factor a 1 , which varies from 0.5 to 0.99, on the ECs performance to gain insight into the effectiveness and the fairness with ρ = 50 dB, under uniform temperature salty water with BL = 2.4. We can see that EC x 1 increases as a 1 increases because the higher power allocation factor means a higher SINRs γ 1 R , γ 1 D1 , and γ 1 D2 , but EC x 2 drops as power allocation factor increases because the SINRs γ 2 R and γ 2 D2 degrade. Furthermore, we can see that ESC is approximately constant over the entire range of the power allocation factor, which is owing to the fact that the rate of increase in EC x 1 is approximately the same as the rate of decline in EC x 2 .

Conclusions
In this paper, we analyzed the system performance in terms of OP and EC and optimized the OP of a downlink NOMA-based dual-hop hybrid RF-UOWC system with DF relaying under the practical assumption of imperfect SIC, where the UOWC channels are characterized by EGG distribution. We derived new analytical closed-form expressions for OPs and ECs and asymptotic expressions for the OPs and the DO. To gain more insight, we investigated the influence of system parameters on performance. Consequently, we deduced that the increase in the level of air bubbles and/or temperature gradient leads to a degradation in the OPs and ECs performances, and the outage performance improves when implementing heterodyne detection compared to IM/DD. Moreover, we investigated the feasibility of obtaining an outage-optimal power allocation factor. Finally, we carried out a comparison with a benchmark system, from which we realize that our proposed system is suitable for UIoT applications. As a future work, we may study the a multi-underwater destination system with amplify and forward relay assuming imperfect channel state information.  For τ 1 > τ 2 or γ 1 1+γ 1 < a 1 < γ 1 (1+γ 2 ) γ 1 +γ 2 +γ 1 γ 2 (1+η) , we rewrite (A1) as To differentiate OP sys with respect to a 1 , we used the chain rule, which is always negative valued, this result indicates a monotonically decreasing function in this interval.