Outage Performance of Multiple Hybrid Active Relays and RISs-Assisted NOMA Networks

As the demand for high-performance wireless communication continues to grow, it becomes essential to explore efficient network designs that can improve system efficiency, coverage, and capacity. In this letter, we investigate the performance of hybrid active relays and reconfigurable intelligent surfaces (HARRIS)-aided non-orthogonal multiple access (NOMA) networks. We derive the outage probability of the proposed HARRIS architecture in closed-form expression over Rician fading channels and develop a framework for selecting the best HARRIS node considering different quality of service (QoS) requirements of the users. Simulation results include comparisons with pure RIS and pure relay in both NOMA and orthogonal multiple access (OMA) networks. Under favorable path loss, HARRIS approaches pure RIS performance, while in severe path loss, it closely aligns with purely active relay. Moreover, OMA outperforms NOMA for low QoS requirements, but NOMA excels in high QoS scenarios.

Abstract-As the demand for high-performance wireless communication continues to grow, it becomes essential to explore efficient network designs that can improve system efficiency, coverage, and capacity.In this letter, we investigate the performance of hybrid active relays and reconfigurable intelligent surfaces (HARRIS)-aided non-orthogonal multiple access (NOMA) networks.We derive the outage probability of the proposed HARRIS architecture in closed-form expression over Rician fading channels and develop a framework for selecting the best HARRIS node considering different quality of service (QoS) requirements of the users.Simulation results include comparisons with pure RIS and pure relay in both NOMA and orthogonal multiple access (OMA) networks.Under favorable path loss, HARRIS approaches pure RIS performance, while in severe path loss, it closely aligns with purely active relay.Moreover, OMA outperforms NOMA for low QoS requirements, but NOMA excels in high QoS scenarios.

I. INTRODUCTION
R ECONFIGURABLE intelligent surfaces (RISs) are increasingly recognized as a highly promising technology for next-generation wireless communications.RISs consist of numerous low-cost, reconfigurable, and energy-efficient passive elements, where each element has the capability to precisely control the phase shift and magnitude of the incident electromagnetic waves with high energy efficiency [1].This transformative technology can create programmable wireless propagation environments [2].Additionally, the expected costeffectiveness of these passive meta-surfaces makes them readily integratable into current wireless communication networks.Moreover, the concept of simultaneously transmitting and reflecting RIS (STAR-RIS) has recently gained growing attention within the research community owing to its enhanced capabilities comparing to traditional reflecting-only RIS.With STAR-RIS, the incoming signal is divided into transmitted and reflected components, providing more advanced signal manipulation and control in wireless communication environments [3].
Evidently, the working principle of RISs is similar to classical relaying technologies [4], and it is useful to scrutinize these analogies to gain a holistic understanding of the possibilities and obstacles associated with their coexistence in future communication systems.Unlike active relays, RISs are unable to amplify incoming signal magnitudes due to their natural passive design, thereby limiting the achievable performance improvements.This suggests that future wireless networks will incorporate both RISs and active relays, resulting in transmissions being aided not only by pure RISs or active relays but also by a combination of the two technologies.Further, the hybrid deployment of active relays and RISs enhances network coverage and connectivity in metropolitan areas by addressing signal attenuation and interference challenges through signal amplification and controllable reflections [5].Hence, instead of perceiving the coexistence of both technologies as a challenge, it is imperative to foster a symbiotic convergence that enhances communication reliability by their joint utilization.
On the other hand, non-orthogonal multiple access (NOMA) is a promising concept in wireless communications that aims to accommodate multiple users via the same resource blocks, which significantly enhances system spectral efficiency [6].In NOMA networks, users are grouped according to their channel conditions [6] or quality of service (QoS) requirements [7], [8], and assigned to different power levels, where the signals of all users are superimposed into a single message transmitted from the access point (AP), while the receiver decodes the signals using successive interference cancellation (SIC) technique.
In this context, few important contributions have investigated he possibilities of integrating both RIS and relay technologies in NOMA networks.For instance, a downlink transmission framework for coexisting passive RIS and active relay is proposed in [4], wherein both the RIS and relay contribute to assist communication between the AP and a pair of NOMA users.In [9], a RIS-aided cooperative multiple-input single-output (MISO)-NOMA system involving two users is investigated.Reference [10] has proposed a strategy for task offloading and computing in a mobile edge computing system using a RISaided NOMA relaying network.Nevertheless, existing works in this letter area have commonly focused on the coexistence of a single relay and a single RIS, not yet tackling practical scenarios where multiple distributed relays and RISs collaborate simultaneously.This observation motivates this letter to study the performance metrics of such system setups.
This letter delves into the outage analysis of employing multiple hybrid active relays and RISs (HARRIS1 ) distributed in NOMA networks.To this end, the system outage performance is statistically characterized, and its closed-form expression is derived.Additionally, simulations are provided to validate the obtained analytical expressions.Moreover, we provide a comparative analysis under various data-rate targets per user for orthogonal multiple access (OMA) and NOMA to identify the optimal usage scenarios.
Notation: A ∼ CN (μ A , σ2 A ) denotes that A is a circularly symmetric complex Gaussian distributed with μ A mean and σ 2 A variance.The notation FX (.) is the complementary cumulative density function of the random variable X. E[.] and Var[.] are the expectation and variance operators, respectively.

II. SYSTEM MODEL
As illustrated in Fig. 1, we consider coexisting K passive STAR-RISs and Z active DF relays to assist NOMA transmissions from a single-antenna AP to a pair of singleantenna users (U i ) distributed at the opposite sides of the STAR-RISs. 2Each STAR-RIS consists of M elements, for m ∈ {1, 2, . . ., M }.Due to obstacles, direct links from the AP to users do not exist, i.e., transmission is only feasible through a selected STAR-RIS/relay node.The channel vectors between the AP and the k th STAR-RIS, and the k th STAR-RIS and U i are denoted by h k ∈ C M ×1 and g k ,i ∈ C 1×M , respectively, where The channel between the AP and the z th relay, and the z th relay and U i are denoted by h z and g z ,i , respectively.Rician fading is assumed for all channels, and thus, the small-scale coefficients to and from the , respectively, where ξ k ,m and δ k ,m,i are the amplitude coefficients, while ϑ k ,m and υ k ,m,i are the phase shifts.Similarly, the channels to and from the z th relay are h z = ξ z e −j ϑz and g z ,i = δ z ,i e −j υ z ,i , respectively, where ξ z and δ z ,i are the amplitude coefficients, while ϑ z and υ z ,i are the phase shifts.The Rician factors of the channels to and from the STAR-RIS/relays are denoted by κ h and κ g , respectively.Further, we denote λ k , λ k ,i , λ z , and λ z ,i as the large-scale fading coefficients, including path-loss from the AP to k th STAR-RIS, k th STAR-RIS to U i , AP to z th relay, and z th relay to U i , respectively.These fading coefficients depend on their corresponding distances, such that z ,i , and η is the path-loss exponent.We assume that the STAR-RISs and relays are closely clustered such that they have equivalent distances to the same node, and hence, all channel gains are independent identically distributed (i.i.d.), consistent with common assumptions in related works [11].
Moreover, we assume that the AP can acquire perfect channel state information (CSI). 3Unlike conventional NOMA, users are categorized by their QoS requirements [7], [8], enabling more flexible and personalized resource allocation.Particularly, assume that U 1 performs low data rate communications, meanwhile U 2 requires high data rates.For instance, U 1 can be ultra reliable and low latency communications (URLLC) that receive information containing a few bytes per second, whilst U 2 can be an enhanced Mobile Broadband (eMBB) user with high data rate requirements.The AP transmits the two signals as a superimposed combination, , where s i denotes the signal of U i , and α i denotes the power allocation coefficient for U i .We adopt α 1 > α 2 since R 1 < R 2 , and α 1 + α 2 = 1.According to the selected node for transmission (STAR-RIS or DF relay), two signal models are described.

A. STAR-RIS Signal Model
The STAR-RIS model differs from conventional RIS, where each element can operate simultaneously in two modes; transmission and reflection, i.e., it allows portion of the incoming signal to pass through while reflecting the other, and reconfigures the propagation to both users.To minimize information exchange overhead between the AP and STAR-RISs, we assume all elements have the same amplitude coefficients [12].Let (t, r) denote transmission or reflection, and thus, the k th STAR-RIS coefficient matrices for transmission or reflection region can be given as [12] where β (t,r )   k and e j φ (t,r )   k ,m are the amplitude and phase shift adjustments imposed on the incident signal by the k th STAR-RIS, respectively, i.e., β (t,r )   k ∈ [0, 1], and φ (t,r )  k ,m ∈ [0, 2π) for m ∈ {1, . . ., M }.Further, β (t)  k + β (r ) k = 1 should be satisfied. 4ence, if a STAR-RIS is selected, the received signal at U i is where w i ∼ CN (0, σ 2 w ) is the additive white Gaussian noise (AWGN) at U i .Thereby, U 1 decodes its signal with signalto-interference-plus-noise ratio (SINR): is the transmit signal-to-noise ratio (SNR) of the AP.Meanwhile, U 2 perform interference cancellation, hence the received SNR after applying SIC is (4) Note that, by considering smart phase shifting elimination capability of the STAR-RIS for both users, the cascaded channel gain is maximized, and thus, can be expressed as [1]

B. DF-Relay Signal Model
On the other hand, if a DF relay node is selected, the received signal at the z th relay is given by where w z ∼ CN(0, σ 2 w ) is the AWGN at relay z.The conditions for a relay to decode U 1 's and U 2 's signals, respectively: During the second time slot, relay z superimposes the two signals ( and transmit to both users.Thereby, The SINR to decode s 1 at U 1 is where ρ z = pz σ 2 w .Conditioned on successfully decoding s 1 , U 2 apply SIC to cancel s 1 .Accordingly, the SNR to detect s 2 is III. HARRIS NODE SELECTION SCHEME In this section, we develop analytical framework for HARRIS node selection. 5Firstly, we build a subset of STAR-RISs and relays, which can guarantee successful decoding of s 1 at U 1 and U 2 [13]: where , (rel ) 1 = 2 2R 1 − 1.Therefore, among the group of STAR-RISs and relays in A, only the optimal single node, denoted as x * , that 5 Node selection can provide maximum diversity gain for the outage performance [13].Furthermore, NOMA can be implemented within each twouser group, where the analysis of each group is mutually independent [14], and hence, selecting the best node for each group is guaranteed.maximizes the rate for U 2 , is selected either from the K STAR-RISs or the Z active DF relays: where IV. OUTAGE PROBABILITY Based on the described selection scheme, the overall outage probability can be statistically founded as follows: P o = P (1)  out + P (2)  out , ( where P (1)  out denotes the outage event that s 1 cannot be decoded by U 1 and U 2 through any STAR-RIS or relay.In (17), P (2)   out denotes the outage event that s 2 cannot be decoded by U 2 , while s 1 is successfully decoded at both users.Therefore, P (1)   out can be calculated as follows: where |A RIS | and |A rel | denotes cardinality of set A RIS , and A rel , respectively.Since the channels are assumed to be i.i.d Rician distributed random variables, P (1)  out can be given as where Ω h,ψ (rel ) I 0 (ax ) dx , is the generalized Marcum Q-function [15], I 0 (•) is the modified zero-order Bessel function of the first kind [16, eq.(8.431)], ) [17], Γ(.) is the Gamma function, and Γ(., .) is the upper incomplete Gamma function, c and θ are given as [17] where: α 2 , otherwise the system is in outage.Assuming |A| > 0, P (2)  out can be given as [11]: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where: , and ψ (rel ) 2 = 2 2R 2 −1 ρz λ z ,2 α 2 .Further, n and l are the number of STAR-RISs and relays satisfied (12), and (13), respectively.We first evaluate P(ζ k ) and P(ζ z ) as follows: which is equivalently written as The above equation can be evaluated as follows On the other hand, P(ζ z ) can be expressed as Thereby, P(ζ z ) can be evaluated as Proof: Please refer to the Appendix.Meanwhile, P(|A RIS | = n) and P(|A rel | = l ) can be given as By substituting (28), (30), (31), and (32) into (23), the closedform expression for the overall outage probability of HARRIS can be written as follows V. NUMERICAL RESULT In this section, we conduct simulations to validate the performance of HARRIS.For a fair comparison, we assume p AP = p z , same number of active and passive nodes in HARRIS design (K = Z), and β (t,r ) k = 0.5 for all STAR-RISs.The distances of the AP-RIS/relay, and RIS/relay-U i links, are set as d f = d f ,i = d for f ∈ {k , z }.The power allocation coefficients α 1 and α 2 are chosen to be 3  4 and 1 4 , respectively.The targeted data rates for both users are R 1 and R 2 in bit per channel use (BPCU).The noise variance is set to σ 2 w = In Fig. 2, we investigate the outage performance of HARRIS for NOMA and compare it to purely passive and active networks.As observed, the performance gains of purely RIS are considerably degraded by severe path loss.However, HARRIS not only achieves the minimal outage performance gains of a purely RIS system but also reaches the performance levels of purely relay networks.Specifically, under favorable path loss conditions, HARRIS performs similarly to a purely RIS setup.In severe path loss scenarios, HARRIS significantly outperforms purely RIS and approaches the performance of purely active relay network.Moreover, the analytical curves in (33) exactly match with the simulation results.
Fig. 3 investigates the performance of HARRIS for NOMA and OMA compared to purely passive and active networks.For OMA, two and four time slots are needed for STAR-RISs and DF relays networks, respectively.As illustrated, HARRIS with OMA outperforms NOMA for low-targeted data rate (R 2 = 0.5 BPCU).However, due to the prelog penalty associated with OMA, the required power grows rapidly with higher rates (R 2 = 4 BPCU), and hence, HARRIS NOMA regains significant superiority for high-data rates.