Performance of RSMA-Based UOWC Systems Over Oceanic Turbulence Channel With Pointing Errors

Underwater optical wireless communication (UOWC) systems face significant challenges due to oceanic turbulence and pointing errors (PE), which can degrade system performance. Moreover, interference is another challenge in UOWC, considering the consistent increase in underwater optical devices. In this study, the performance of rate splitting multiple access (RSMA)-based UOWC systems is investigated over an exponential-generalized gamma (EGG)-distributed oceanic turbulence channel with generalized PE. The RSMA scheme is employed to facilitate communication between a source and multiple users in the UOWC system while accounting for the combined effects of oceanic turbulence and generalized PE. The statistical characterization of the UOWC system is analyzed, including the probability density function (PDF) of the signal-to-noise ratio (SNR), outage probability, throughput, and sum ergodic capacity. Additionally, closed-form expressions for the outage probability and throughput are derived, and asymptotic expressions for the outage probability in the high SNR regime are provided. Moreover, the diversity order of the system is evaluated, and the impact of different parameters on the system performance is discussed. Our results demonstrate the effectiveness of the RSMA-based UOWC system in mitigating the adverse effects of oceanic turbulence and PE while achieving improved performance in terms of capacity and outage probability. The findings of this study provide valuable insights for the design and optimization of UOWC systems in challenging underwater environments.


I. INTRODUCTION
U NDERWATER wireless communications are of paramount importance in the field of communications.The ability to establish reliable and efficient underwater communication links opens up a wide range of applications and opportunities for various industries and scientific research.Radio frequency (RF) signals can penetrate the water surface to a certain depth, allowing for limited underwater communication.Acoustic waves, on the other hand, can travel long distances through water, making them a practical choice for underwater communication over extended ranges.However, RF communications are energy intensive [1], [2], and acoustic communications are characterized by low speed and limited bandwidth [3], [4].Consequently, underwater optical wireless communication (UOWC) has been recently proposed as a promising alternative for marine communications.UOWCs are mainly based on light exchange between different underwater communication sensors.This technology is characterized by high data rates in the order of Gbps, low-cost implementation, low power consumption, and low latency [5].Nevertheless, UOWCs are highly affected by underwater turbulence and pointing errors (PE), where the transmitted beam is not perfectly facing the receiver.This significantly degrades the link quality and connectivity of such communication systems.Moreover, the upsurge of UOWC applications has led to a remarkable increase in optical devices underwater.This has resulted in denser underwater optical networks, emphasizing the impact of interference on UOWC systems.The interference between different optical links can further deteriorate the communication quality of the transmitted signals.Hence, studying the performance of such links becomes crucial.
Several statistical models were proposed in the literature to model underwater turbulence, namely the lognormal, log-logistic, gamma, exponential Weibull (EW), and generalized gamma (GG) distributions [6], [7].The exponential-generalized gamma (EGG) is another distribution that was recently introduced as a perfect model for air bubbles and temperature-induced oceanic turbulence under different channel conditions [8].Considering its ability to various oceanic turbulence levels in a unified manner, we select the EGG distribution to model the oceanic turbulence in our system.
Another challenge faced by multi-user UOWC systems is the interference between the different optical links.The increase of devices deployed underwater led to denser networks, and consequently stronger interference effects.In this regard, different multiple access schemes have been proposed to mitigate the interference effect on the system's performance.Space division multiple access (SDMA) and non-orthogonal multiple access (NOMA) are two adopted multi-access schemes in the literature, to name a few.In SDMA, users are separated in the spatial domain using a linear precoding technique, and the residual interference between users is considered as noise.As for NOMA, users are superposed in the power domain using linearly precoded superposition coding with successive interference cancellation (SIC)-based decoding.This technique is based on user grouping and ordering and enforces some users to fully decode and omit the interference generated by other users.These multi-access schemes indeed enhance the performance of multi-user systems by enabling multiple users to simultaneously transmit over the same resource block.However, they are extreme techniques as they fully treat interference as noise (i.e., SDMA) or fully decode the interference (i.e., NOMA).This extreme behavior degrades the performance of SDMA with the increase in the number of users as more transmit antennas are needed to effectively manage the multi-user interference, and remarkably increases the complexity of NOMA with the increase in the number of users.Hence, the performance of these multiple access schemes is limited in dense networks [9].As an alternative, the rate splitting multiple access (RSMA) scheme was proposed to overcome the aforementioned complexity issue.This technique was shown to be promising in terms of robustness, low complexity, and high spectral efficiency.RSMA can be considered as a combination between of NOMA and SDMA.Relying on linearly precoded rate splitting (RS) at the transmitter and SIC at the receivers, the interference is partially decoded and partially treated as noise.It is, therefore, a softer interference management scheme compared to NOMA and SDMA.

A. RELATED WORK
Several papers have considered the EGG distribution for modeling underwater turbulence in the literature.In [10], a NOMA-based reflecting intelligent surface (RIS)-assisted hybrid RF-UOWC system is assessed considering the EGG turbulence model for the optical link without PE.The performance of multi-hop UOWC systems is analyzed in [11] using both amplify-and-forward (AF) and decodeand-forward (DF) relaying schemes, considering the EGG turbulence model without PE.Authors in [12] study a vertical multi-layer underwater wireless optical communication (UWOC) system using EGG distribution without PE.In [13], [14], hybrid terrestrial-underwater wireless optical communications are considered assuming the EGG underwater turbulence with zero boresight PE model.
In [15], the selection combining technique is used for a hybrid RF-UWOC system, where the RF link is a backup for the optical link, which is modeled as an EGG turbulence link without PE.AF and DF relaying schemes are used in [16], [17], [18], [19] for joint RF-UOWC systems considering EGG underwater turbulence for the optical link without PE.In [20], the AF relaying scheme is proposed for hybrid free space optical (FSO)-UWOC systems using the EGG turbulence with zero-boresight PE model for the UOWC link.
Recently, RSMA has emerged as a promising solution for enhancing the spectrum efficiency of wireless systems.It has been considered by numerous researchers in terrestrial RF communications.Authors in [21], [22], [23] investigate the RSMA for the downlink transmission in unmanned aerial vehicle (UAV)-based systems to increase the data rate, throughput, and ergodic capacity.The RSMA scheme is analyzed in [24] for RIS-assisted UAV-based multi-user communication networks in the presence of co-channel interference.In [25], RSMA is proposed for uplink between ultra-reliable low-latency communication (URLLC) users and the base station (BS) to improve the communication's sum rate and reliability.Researchers in [26] study the energy efficiency of RSMA-based and RIS-assisted wireless communication systems.The RSMA technique is also employed in satellite communications to increase the sum rate and optimize the power budget in [27].
In addition to RF communications, many researchers use RSMA in visible light communications (VLCs).In [28], [29], [30], the RSMA technique is used in VLC networks to improve the system's sum rate compared to NOMA or SDMA.It is also demonstrated in [31] that RSMA-based VLC broadcast systems achieve better energy efficiency than NOMA-based and SDMA-based VLC systems.Researchers in [32] introduce a single-layer RSMAbased VLC system that maximizes the spectral efficiency under the constraint of the non-linear effect of a lightemitting diode (LED), compared to NOMA-based VLC systems.Considering the benefits of RSMA on the system's performance, this technique was recently introduced in UOWC systems as well.In [33], the ergodic rate of an RSMA-based UOWC system is assessed, where lognormal weak oceanic turbulence is considered without PE.However, to the best of the authors' knowledge, there are no existing results for the performance of RSMA-based UOWC systems over EGG turbulence with a generalized PE model.In Table 1, we present a comprehensive state-of-the-art summary.This table provides an overview of the key considerations, system models, and techniques employed in various studies conducted in the literature.It serves as a valuable reference for understanding the advancements and current state of research in UOWC systems and RSMA, highlighting the different aspects researchers have focused on and the approaches they have taken to address the challenges faced.

B. NOVELTY AND CONTRIBUTIONS
In this paper, we use the RSMA for communication between a source and multiple users over an underwater wireless link.We consider the combined impact of oceanic turbulence and generalized PE in our analysis.The novelty and contributions of this work can be summarized as follows: • We assess the performance of RSMA-based multi-user downlink UOWC over EGG oceanic turbulence with generalized PE. • We derive a novel analytical expression for the probability density function (PDF) of the signal-to-noise ratio (SNR) over EGG-distributed oceanic turbulence and generalized PE. • We derive the exact expression of the outage probability and throughput of RSMA-based multi-user UOWC considering the joint effect of oceanic turbulence and PE.The asymptotic expressions of these two metrics are also derived in this work.• We develop an analytical approximation for the ergodic capacity along with its corresponding asymptotic expression.
• The diversity order of the studied system is derived from the asymptotic expressions of the outage probability, throughput, and ergodic capacity.• The performance of the proposed system in terms of outage probability, throughput, and ergodic capacity metrics is assessed using simulations considering different oceanic turbulence and PE values.
• The effect of the number of users and the resource distribution among the broadcast and private links on the performance of the proposed system is assessed through simulations.

C. NOTATIONS AND ORGANIZATION
Notations: (•) i denotes parameters for the i-th user, N denotes the total number of users.We represent the zero-order modified Bessel function of the first kind by I 0 (•) [34, eq.(8).Organization: The remainder of this paper is structured as follows.Section II provides a detailed explanation of the channel model, including EGG distributed oceanic turbulence and generalized PE, for RSMA-based multi-user UOWC systems.In Sections III and IV, we present the statistical characterization and performance evaluation of UOWC systems, focusing on the PDF of the SNR, outage probability, throughput, and sum ergodic capacity.Section V showcases simulation results and discussions.Finally, in Section VI, we conclude this paper.

II. SYSTEM MODEL
We consider a downlink multi-user UOWC system (as shown in Fig. 1) that uses RSMA signaling to transmit information data to N users (D 1 , D2, . . ., D N ), simultaneously.The quality of the transmitted signal is degraded by various multiplicative channel effects, including path loss, oceanic turbulence, and PE.Therefore, the channel coefficient between the source and D i is given as: where h l i = e −αd is the oceanic path loss coefficient, d is the link distance in meters, and α is the extinction attenuation coefficient.Moreover, h t i is the EGG-distributed oceanic turbulence, and h p i is the Rician-distributed PE with non-zero boresight and random jitter.The PDF of h t i is given as [8]: where λ i , a i , b i , c i , and ω i represent the parameters of the EGG distribution, satisfying the condition 0 < ω i < 1.The PDF of h p i is given as [35], [36]: where 0 ≤ x ≤ A i .The parameter w 2 z eq i is defined as , with represents the ratio of aperture radius r i to beamwidth w z i .Furthermore, ρ i is defined as ρ i = w zeq i 2σ i , where σ i denotes the standard deviation of the jitter and w z eq i represents the equivalent beamwidth.In this context, s i = μ 2 x i + μ 2 y i = 0 is used to model the non-zero boresight, where μ x i = 0 and μ y i = 0 indicate the horizontal and vertical displacements between the center of the beam and the center of the detector, respectively.It is important to note that the presented in ( 3) is a generalized distribution, which includes the special case of zero boresight for s i = 0 [37].
In the RSMA scheme, each user's message is divided into two parts: common message and private message.The common messages from all the users are combined and encoded into a unified common message (s c ) using a common codebook accessible to all the users.Additionally, the private message of user D i is encoded as private message s i .The unified common message (s c ) represents the portion of the message shared among all users and is decoded by each user.On the other hand, the private message s i is specifically intended for user D i and is only decoded by D i .Consequently, the superimposed symbol transmitted by the source can be expressed as follows: where P t denotes the optical transmit power at the source.The power coefficients allocated to the common and private messages are denoted as a c and a p i , respectively, satisfying the constraint a c + N i=1 a p i = 1.The signal received at the i-th user can be expressed as: Common message

+ a p i P t h i s i
Private message where n i represents the additive white Gaussian noise (AWGN) with zero mean and variance σ 2 n i .Based on (5), it is evident that in addition to the intended common and private messages, each user also receives private messages from other users, resulting in interference during the decoding process.As a result, each user employs a two-step decoding approach to extract the desired information from the received signal.In the first step, the common message is decoded, considering all other messages as interference.Using the non-coherent intensity modulation/direct detection (IM/DD) scheme, the corresponding signal-to-interference plus noise ratio (SINR) for decoding the common message at the i-th user can be expressed as follows: where being the instantaneous SNR of the on-off keying (OOK) signaling scheme.Upon successful decoding of the common message, each user proceeds to the second step, where they decode their desired private messages.This is achieved by subtracting the decoded common message from the received signal while considering the private messages of all other users as interference.The SINR for decoding the private message at the i-th user can be expressed as follows:

III. STATISTICAL CHARACTERIZATION
In this section, we derive a novel closed-form expression for the PDF of the SNR over combined EGG-distributed oceanic turbulence and generalized PE.Theorem 1: The SNR PDF over EGG distributed oceanic turbulence and generalized PE for UOWC is given as: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: Since it is intractable to find the closed form expression of combined SNR PDF with the modified Bessel function, we employ the series expansion of the modified (j !) 2 in (3) to get: It is important to note that the converging series expansion presented in (9) enables us to analyze the system performance effectively.The authors in [38] have presented a plot that demonstrates the convergence of the series expansion employed in (9).The plot shows that the series converges rapidly, where approximately 20 terms is sufficient for accurate approximation.
Applying the theory of product distribution, the PDF of h tp i = h p i h t i can be expressed as: Substituting ( 9) and ( 2) in (10), and taking the integral definition of Meijer's G-function, we get: where After substituting log A i h p i = t to solve the inner integrals, I 1 and I 2 , we obtain: and Now, we substitute ( 12) and ( 13) in (11) and apply Meijer's G-function.Using the transformation γ i = γ 0 |h tp | 2 , we get the SNR PDF for UOWC in (8).

IV. PERFORMANCE ANALYSIS
In this section, we derive closed-form expressions for the outage probability and throughput, as well as analytical expressions for the ergodic sum rate of RSMAbased multiuser UOWC.The analysis takes into account oceanic turbulence following the EGG distribution and generalized PE.

A. OUTAGE PROBABILITY AND THROUGHPUT
In RSMA, if the SINRs for decoding the common and private messages fall below certain threshold values, γ c,i th and γ p,i th respectively, an outage occurs in the link between the UAV-BS and i-th user.Specifically, γ c,i th is set' to 2 2R c,i −1 , representing the target rate for decoding the common message, and γ p,i th is set to 2 2R p,i −1 , representing the target rate for decoding the private message.
Theorem 2: The outage probability of the RSMA-based multi-user UOWC link between the source and the i-th user is given in (14), at the bottom of the next page, where γ c,i th = , and, .
Proof: The outage probability of the i-th user can be expressed as follows: Using the SINRs of the common and private messages from ( 6) and ( 7) in (15), we get: where B = N j=1,i =j a p j .After some algebraic manipulations ( 16) can be written as: where γth = max( γ c,i th , γ p,i th ).Using (8) in the above equation, we get: Substituting the integral form of the Meijer-G function and using the inner integral γth 0 γ th , we get: Substituting γth = max( γ c,i th , γ p,i th ) in ( 19), we get the closed-form expressions for the outage probability in (14).
To find the diversity order of the proposed system, we derive the asymptotic expansion of the outage probability in the following Lemma.
Lemma 1: To derive the asymptotic expression for the outage probability in the high SNR regime γ 0 → ∞ in (20), at the bottom of the page, we can utilize the series expansion of the Meijer G-function, where Proof: Taking the asymptotic expansion of the Meijer-G function, [39, eq.(07).34.06.0006.01]at high SNR (γ 0 → ∞) in ( 14), we get the asymptotic expansion of the outage probability in (20).
Remark 1: Taking the exponent of γ 0 in (20), the diversity order of the system can be expressed as It is crucial to emphasize that the diversity order remains unchanged and unaffected by the boresight parameters.
Remark 2: It is evident from ( 14) that the successful decoding of both messages is crucial for enhancing the user's outage performance.Furthermore, parameters, such as the transmit power, power coefficients, target rate, etc., have a direct impact on the user's outage performance.Hence, careful selection of these parameters is crucial to achieve the desired performance.By utilizing (14), the throughput at the i-th user, measured in bits per channel use (bpcu), can Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
be evaluated as follows: where R i = min{R c,i , R p,i } with R c,i and R p,i representing the target rates for decoding the common and private messages, respectively.
The sum throughput of the overall system can be given as:

B. SUM ERGODIC CAPACITY
For RSMA, the sum ergodic capacity of the system can be calculated by adding the minimum capacity of the common message among all users and the sum of the capacities of the private messages of all users.Therefore, the sum ergodic capacity can be expressed as follows [23]: where In the following lemma, we derive the analytical expressions for the sum ergodic capacity.
Lemma 2: The analytical expression of the sum ergodic capacity of RSMA-based multi-user UOWC over EGG distributed oceanic turbulence and generalized PE is given in (24), at the bottom of the page, where ξ c,a 1 = {0, { Proof: First, we evaluate the ergodic capacity for the common message using the following expression: Applying the approximation log(1 + x) ≈ 2x (2+x) [40] in (25), we get: To compute the above integral, we substitute ( 6) and ( 8) in (26), yielding: Applying the identity [39, eq.(07).34.21.0086.01],we get the analytical expression of the ergodic capacity for the common message: Similarly, we determine the ergodic capacity for the private message using the following expression: To solve the above integral, we substitute ( 7) and ( 8) into (29) as follows: Applying the identity [39, eq.(07).34.21.0086.01],we get an approximation of the ergodic capacity for the private message: Therefore, the sum ergodic capacity in ( 24) is obtained by substituting ( 28) and ( 31) in (23).
Finally, we derive an asymptotic expression for the sum ergodic capacity of RSMA-based UOWC over EGGdistributed oceanic turbulence and generalized PE in the following Lemma.
Remark 3: We can further determine the diversity order of the proposed system as G i = min{ 1  2 , ρ 2 i 2 , a i c i 2 } by evaluating the exponent of γ 0 in the asymptotic expression of the sum ergodic capacity in (32).It is important to highlight that both MATLAB and MATHEMATICA provide standard computational built-in functions for calculating Meijer-G functions [39].Additionally, the PDF of the SNR over oceanic turbulence and generalized PE enables precise statistical analysis and an accurate asymptotic representation using Gamma functions.This allows for a comprehensive evaluation of RSMA-based multi-user UOWC systems' performance and an understanding of the impact of various parameters on the system performance.In contrast, numerical integration may not provide straightforward insights into the system's performance.It is worth mentioning that Monte Carlo simulations, although feasible, require longer computational time to assess the performance of the considered system.

V. SIMULATION RESULTS AND DISCUSSIONS
In this section, we assess the performance of the proposed downlink RSMA-based multi-user UOWC system under EGG turbulence with zero boresight PE.We consider a distance d = 50m between the source and the users.Two different bubble levels (BLs) are considered to evaluate the system performance, namely BL= 2.4 and BL=16.5.Two different PE parameters are considered for simulations, namely (A i = 0.04, ρ 2 = 3.5377) and (A i = 0.054, ρ 2 = 37.0657).The SNR threshold is set to γ th = 6 dB.Utilizing Monte-Carlo (MC) simulation averaged across 10 7 channel realizations, we validate the derived analytical expressions.Moreover, the asymptotic expression of the outage probability aligns with both the analytical and simulation results in the high SNR range.For the computation of the Meijer-G function, we utilize the standard built-in MATLAB library "meijerG".The simulation parameters are summarized in Table 2.We consider the outage probability, the throughput, and the sum ergodic capacity metrics in evaluating the performance of the proposed system.

A. OUTAGE PROBABILITY
First, we analyze the outage probability performance of the proposed system as a function of the PE parameters and resource distribution between common and private channels.Fig. 2 depicts the effect of the change in turbulence and PE parameters on the outage probability of the second user P out 2 .We can observe that the simulation and derived analytical results provide the same results.It is also noticed that BL = 2.4 yields 88% less outage probability at 25 dBm than BL = 16.5, and the PE parameters (A i = 0.054, ρ 2 = 37.0657) yield 23% less outage probability at 25 dBm than (A i = 0.04, ρ 2 = 3.5377).This is because increasing these   variables decreases the PE effect, which decreases the outage probability of the system.Fig. 3 analyzes the second user's outage probability performance for different values of the parameter a c while keeping an equal distribution among a p i s.It is observed that increasing a c from 0.5 to 0.6 decreases the outage by 45% at 25 dBm.Hence, it can be deduced that the common channel power allocation significantly affects the system performance.Furthermore, the slope of the outage probability aligns with the diversity order of the system

B. THROUGHPUT
Second, we evaluate the throughput performance of the proposed RSMA-based multi-user UOWC system.We set demonstrates the performance of the system's throughput η sys for different oceanic turbulence and PE parameters.The throughput is an increasing function of the transmit power since increasing power improves the link quality.Also, BL = 2.4 yields 23% higher throughput than BL = 16.5 at 5 dBm, while decreasing the PE parameters from (A i = 0.054, ρ 2 = 37.0657) to (A i = 0.04, ρ 2 = 3.5377) decreases the throughput by 9% at 5 dBm.This aligns with the results found in Fig. 2 for outage probability.At high transmit power P t , the performance of the system is similar for different oceanic turbulence and PE parameters.Fig. 5 shows the effect of the number of users on the system throughput η sys .In fact, increasing the number of users from three to five increases the throughput by 67% at 5 dBm.This can be explained by the fact that having more users will allow more efficient spectrum use since the available bandwidth is shared by more users.

C. SUM ERGODIC CAPACITY
Finally, we assess the sum ergodic capacity performance of the considered system for BL = 16.5 and PE parameters A i = 0.04 and ρ 2 = 3.5377.We evaluate the effect of the number of users and the power distribution between the common channel a c and private channels a p s. Fig. 6 shows the sum ergodic capacity as a function of the transmit power for 4 and 6 users.The sum ergodic capacity is an increasing function of the transmit power, which aligns with the theory that increasing the power improves the link quality.The results also show that the simulations match the numerical results, which validates the accuracy of our analysis.Furthermore, the sum ergodic capacity decreases when increasing the number of users from four to six by 3.6% at 25 dBm transmit power, which can be explained by the fact that adding more users to the system increases interference.Fig. 7 presents the sum ergodic capacity as a function of the transmit power for four users.We consider two different values for a c while equally dividing the remaining power among the private channels.Moreover, we evaluate the performance of the proposed system by maintaining a fixed value for a c = 0.4 and dividing the rest of the power resource equally or unequally between the private links.We observe that changing the value of a c from 0.4 to 0.6 increased the sum ergodic capacity by 24% at 25 dBm while moving from an equal to unequal distribution of a p increases the sum ergodic capacity by only 3.6% at 25 dBm.Hence, we can say that common channel power allocation has a stronger effect on the system performance than private channel power allocation.
Finally, we can conclude that decreasing the effect of PE by increasing the values of its parameters enhances the performance of the system.Also, increasing the BL decreases the performance of the system as it strengthens the oceanic turbulence.Moreover, the common channel power allocation significantly controls the performance of the system.

VI. CONCLUSION
The presence of oceanic turbulence and PE poses significant challenges to UOWC systems, affecting their performance and reliability.To overcome these challenges, RSMA has been employed as a promising multiple-access technique that optimizes resource allocation in the presence of interference.Through statistical characterization, outage probability analysis, and evaluation of throughput and sum ergodic capacity, we have gained valuable insights into the performance of RSMA-based UOWC systems in oceanic turbulence channels with PE.Our results demonstrate that by appropriately allocating the power coefficients and setting the target rates, RSMA allows for efficient transmission and reception of both common and private messages, enhancing the overall system performance.
The analytical expressions and asymptotic analysis presented in this study provide valuable guidance for the design and optimization of RSMA-based UOWC systems.By carefully selecting parameters and considering the system constraints, such as transmit power, power coefficients, and target rates, the performance of UOWC systems can be significantly improved.Overall, the findings of this study emphasize the importance of RSMA in addressing the challenges posed by oceanic turbulence and PE in UOWC systems.Further research can focus on exploring advanced techniques and algorithms to enhance the performance and reliability of UOWC systems, ultimately enabling robust UOWCs in challenging underwater environments.