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On secrecy performance analysis of multiantenna STARRISassisted downlink NOMA systems
EURASIP Journal on Advances in Signal Processing volume 2022, Article number: 118 (2022)
Abstract
To accommodate the stringent requirements of enhanced coverage quality and improved spectral efficiency, simultaneous transmitting and reflecting reconfigurable intelligent surface (STARRIS)aided communication has been perceived as an interesting research topic. This paper investigates a downlink STARRISaided nonorthogonal multiple access (NOMA) system, where a STARRIS is deployed to enhance the transmission qualities between users and a multipleantenna base station (BS). The considered STARRIS utilizes the energy splitting (ES) protocol to serve a pair of NOMA users located at both sides of STARRIS. Based on the ES protocol, each reconfigurable element can operate in the modes of transmission and reflection simultaneously. In an effort to characterize the secrecy performance, we first derive the closedform expressions of secrecy outage probability (SOP) for STARRISaided NOMA system. Then, the asymptotic performance of the derived SOP is analyzed. For gleaning further insights, secrecy diversity order (SDO) is derived according to the asymptotic approximation in the high signaltonoise ratio and maintoeavesdropper ratio regimes. As a further advance, the system parameters are optimized to minimize the SOP of the system. Our analytical results demonstrate that the multipleantenna BS has almost no impact on SDO for STARRISaided NOMA system. In simulations, it is demonstrated that the theoretical results match with the simulation results very well and the SOP of STARRISaided NOMA is less than that of conventional orthogonal multiple access (OMA) system obviously.
1 Introduction
The upcoming sixthgeneration (6G) wireless networks have advantages of supplying enhanced spectral efficiency (SE), global coverage and more intelligence as well as meeting the requirements of dynamic businesses [1]. Reconfigurable intelligent surface (RIS) has been emerged as a promising technique for 6G systems [2]. Specifically, by utilizing a large number of passive reconfigurable elements, a RIS can intelligently control the phase shifts of incident signals to satisfy the requirements of various wireless applications. Its typical applications include wireless coverage and network throughput enhancement, interference cancelation, and so on [3,4,5]. Different from the conventional fullduplex relay, RIS is capable of manipulating the phase shifts of the incident signal without introducing selfinterference.
As a promising technology, nonorthogonal multiple access (NOMA) has advantages in supplying high SE and supporting massive connectivity [6]. Moreover, different from the conventional orthogonal multiple access (OMA), NOMA transmission strategies can achieve better user fairness, in which superposition coding (SC), successive interference cancellation (SIC), and message passing (MP) are employed at the transmitters or receivers. As a result, NOMA transmission schemes are widely considered to be promising solutions to handle the emerging heterogeneous services and applications, and thus have drawn significant research interests in the past few years [7, 8]. By extending NOMA to cooperative communication, cooperative NOMA scheme was proposed in [9]. To further enhance SE, the authors of [10] analyzed the outage performance of cooperative NOMA systems. For an indepth performance evaluation, the closedform expressions of the outage probability and ergodic capacity for NOMA systems with multiple users were surveyed in [11, 12]. Apart from the above treatises, NOMA technique has been widely applied to a lot of communication scenarios. Due to the fact that both RIS and NOMA techniques can be leveraged to enhance SE, the intrinsic integration of RIS and NOMA techniques was applied to improve the performance of wireless networks [13].
Up to now, from the perspective of performance analysis, a RISaided NOMA network has been discussed in [14,15,16]. Particularly, the influence of coherent/random phase shifts on the outage behavior was studied for RISNOMA networks in [14]. Furthermore, a simple design of RISNOMA transmission scheme was proposed, where an increasing number of reflection elements can effectively reduce the outage probability [15]. Inspired by the above works, the authors of [16] investigated the outage probability, ergodic rate and energy efficiency (EE) of RISaided NOMA systems. As a further development, by collaboratively optimizing the transmit power and the phase shifts, the transmit power efficiency for RISenabled multigroup NOMA networks were examined in [17]. Given the user’s rate requirement, the authors of [18] investigated the transmit power minimization problem with discrete phase shifts for RISaided NOMA comparing to OMA. According to whether there is a direct link between base station (BS) and users, the outage performance of multiple RISsassisted NOMA networks with discrete phase shifting was analyzed in [19]. Overall, the findings in [14,15,16, 19] showed a significant performance improvement achieved by either RISassisted uplink or downlink NOMA. Furthermore, the ergodic rate of RISassisted uplink and downlink NOMA networks was examined in [20], which revealed the superiority of RIS over fullduplex decode and forward (DF) relaying.
Despite the aforementioned advantages, the conventional RIS can only serve the users located on the same side. That is to say, the reflectingonly RIS offers only halfspace coverage, which limits the flexibility of deployments. To overcome this limitation, the novel concept of simultaneous transmitting and reflecting RIS (STARRIS) [21] or intelligent omnisurface (IOS) [22] has been proposed. More specifically, in contrast to conventional reflectingonly RIS, the incident signal on STARRIS is divided into the transmitted and reflected signals propagating into both sides of the surface, and thus STARRIS is capable of achieving a fullspace reconfigurable wireless environment. Currently, depending upon usage scenarios and foreseeable requirements, a STARRIS can operate in three different modes, namely, energy splitting (ES), mode switching (MS), and time switching (TS). Particularly, for the ES mode, each element may have a different reflectiontransmission amplitude coefficient ratio and all elements can operate in transmission and reflection mode simultaneously. For the MS mode, each element can operate in full transmission or reflection mode. In addition, in the TS mode, all elements periodically switch between the transmission and reflection modes in the different time slots. As a result, by deploying STARRIS, the transmitters and receivers do not have to be located on the same side of the surface comparing to the conventional RIS. Motivated by this characteristic, growing research efforts have been devoted to examining the benefits of deploying STARRIS in wireless networks. For instance, the authors proposed three practical operating protocols for STARRIS and studied the corresponding joint beamforming design problems in both unicast and multicast scenarios in [23]. Moreover, the fundamental coverage limits of STARRIS were characterized by the authors of [24], where both NOMA and OMA schemes were considered.
The deployment of STARRIS is beneficial to NOMA systems. On one hand, for NOMA users with weak channel conditions, STARRIS is able to create stronger transmission links. On the other hand, since STARRIS has the ability to adjust channel gains of different NOMA users, it can offer a flexible decoding order according to the priority of users. Nevertheless, the research on STARRISaided NOMA networks is still in its infancy. For example, three STARRIS operating protocols in a singlecell NOMA network were firstly evaluated in a recent work [25]. As a further advance, the authors in [26] investigated the performance of a pair of NOMA users for STARRIS networks in terms of outage probability and ergodic rate. However, secure communication is still an intractable problem when there exist several eavesdroppers in the considered system model.
Physical layer security (PLS) has gained widely attentions and several techniques have been proposed to improve the secrecy performance of wireless systems, such as cooperative diversity and spatial diversity. From the perspective of information theory, PLS has been widely used to enhance the security performance of wireless communication systems. The key idea behind the PLS is to use the different characteristics between the legitimate channel and the eavesdropper’s channel. The perfect security can be achieved when the quality of the eavesdropper’s channel is inferior to that of the legitimate channel. Since PLS depends on the physical characteristics of the propagation environment, it is interesting to research the PLS performance with the assistance of RIS [27, 28]. More specifically, with the goal of maximizing the secrecy rate, the authors of [27] introduced some low complexity approaches by jointly designing the access point’s transmit beamforming and the RIS’s reflect beamforming. Moreover, from the perspectives of hardware and system design, the authors of [28] proposed a practical design for active RIS, and the secrecy performance of RISaided networks was also investigated. However, few works have studied the PLS of RISassisted NOMA systems. For example, the authors of [29] investigated the PLS of the downlink in RISaided NOMA networks in the presence of an eavesdropper. Besides, the authors in [30] proposed a novel scheme to enhance the PLS in RISaided NOMA network.
Since the aforementioned significant literatures have laid a solid foundation for understanding of NOMA, STARRIS, and PLS techniques in communication networks, it is an effective approach to enhance the SE and EE by integrating these three promising technologies. In [26], the performance of a pair of users, i.e., the celledge user and cellcenter user, was investigated for STARRISaided NOMA networks in terms of outage probability and ergodic rate. However, the PLS of such a system has never been reported. Additionally, the authors of [31] studied the secrecy performance of a RISaided wireless communication system in the presence of an eavesdropping user with Rayleigh fading channels. Furthermore, the average required transmit power, the outage probability, and the diversity order were investigated of multiantenna RISassisted NOMA network in [32]. To the best of our knowledge, the combination of PLS with STARRIS has not been reported in multiantenna NOMA system. The STARRIS can not only improve the channel gain of legitimate users, but also reduce the received signal strength at the eavesdropper. Thus, by carefully designing the phase shifts of the STARRIS, the secrecy performance of NOMA system can be boosted in a costeffective manner. In addition, the STARRISassisted NOMA communication system has greater advantages than the reflectingonly RISaided NOMA system because it can provide the service for the users located within 360degree coverage area. Motivated by the aforementioned observations, in this work, we propose a novel secure communication scheme for multiantenna STARRISassisted NOMA system. More specifically, we investigate the secrecy performance of two NOMA users with different requirements of quality of services (QoSs) in the presence of an eavesdropping user located at the same side of each NOMA user. The direct links from the BS to both the cellcenter user and the eavesdropper are taken into account. According to the aforementioned explanations, the key contributions of this work are summarized as follows:

1.
We propose a STARRISassisted scheme to enhance PLS with a multiantenna BS, a STARRIS, a pair of users and two eavesdroppers in NOMA system, where the channel correlation between reconfiguration elements of each column on STARRIS is taken into consideration.

2.
We analyze the secrecy performance of the proposed scheme. More specially, we derive the exact SOP with closedform expressions. For the sake of comparison, we also derive the expression of the exact SOP for the multiantenna STARRISassisted OMA scheme as a benchmark. To provide further useful insights, we derive the asymptotic expressions of the SOP for multiantenna STARRISassisted NOMA as well as OMA networks. Besides, we also derive the expression of secrecy diversity order (SDO) for multiantenna STARRISaided NOMA and OMA systems.

3.
Since the power allocation (PA) coefficient and the ES coefficient have significant impact on the secrecy performance for multiantenna STARRISaided NOMA communication schemes, for the sake of giving scope to the positive influence of the parameters on secrecy performance, the PA and the ES parameters are studied and optimized to reduce the system SOP in our proposed scheme.
The remainder of this paper is organized as follows. In Sect. 2, the system model of multiantenna STARRISaided NOMA system is introduced. The secrecy outage behaviors of multiantenna STARRISassisted NOMA networks are examined in Sect. 3. More specially, the exact expressions of SOP for celledge user and cellcenter user are provided. The asymptotic SOP for multiantenna STARRISaided NOMA system is evaluated in Sect. 4. In Sect. 5, the SDO of multiantenna STARRISaided NOMA communication system is derived. The parameters of the multiantenna STARRISNOMA scheme are optimized in Sect. 6. Simulation results and their discussions are presented in Sect. 7. Conclusion is drawn in Sect. 8.
Notations: The main notations in this paper used are shown as follows. The probability density function (PDF) and cumulative distribution function (CDF) of a random variable X are denoted by \(f_{X}(\cdot )\) and \(F_{X}(\cdot )\), \(\Gamma (\cdot )\) denotes the Gamma function, respectively. \(\Gamma (\cdot , \cdot )\) denotes the upper incomplete Gamma function, \(\gamma (\cdot ,\cdot )\) denotes the lower incomplete Gamma function, respectively. The superscript \((\cdot )^{H}\) denotes conjugatetranspose operation and \( \cdot \) means the 2norm of vectors, respectively. \({\textbf{B}}(\cdot ,\cdot )\) denotes Beta functions and \({}_p \!F_q\) denotes Generalized hypergeometric series, respectively.
The aim of this paper is to analyze the secrecy performance of downlink NOMA based on a multiantenna BS and a STARRIS assistance, specifically, analyzing the secrecy outage probability and secrecy diversity order of this STARRISaided NOMA system. For multiantenna STARRISNOMA systems, we propose a transmit antenna beamforming and ES protocol for STARRIS scheme, which minimizes the secrecy outage probability. There exists a direct link between the BS and the cellcenter user \(\textrm{U}_2\), while the BS can only communicate with the celledge users \(\textrm{U}_1\) through STARRIS. Specifically, the user \(\textrm{U}_2\) and the eavesdropper \(\textrm{E}_2\) can receive the superposition of the transmitted signal from the BS and the reflected signal from STARRIS, while the user \(\textrm{U}_1\) and the eavesdropper \(\textrm{E}_1\) can only receive the refraction signal from STARRIS. A brief introduction to the design idea: The BS employs multiantenna technique to process transmit data in order to satisfy the rate requirements of the cellcenter user. The STARRIS utilizes ES protocol and adjustable phase shifts to meet the rate requirements of both NOMA users. Since the PA and the ES coefficients have significant impact on the secrecy performance for multiantenna STARRISaided NOMA communication schemes. We propose an effective method to optimize the PA and the ES parameters.
2 System model and secrecy capacity
2.1 Network description
We consider a downlink NOMA system as shown in Fig. 1, where the superposed signals from BS are reflected and transmitted to a pair of NOMA users,^{Footnote 1} i.e., the celledge user \(\textrm{U}_1\) and the cellcenter user \(\textrm{U}_2\), simultaneously. The NOMA communication system is assisted by a STARRIS in the presence of two eavesdroppers^{Footnote 2} (i.e., \(\textrm{E}_1\), \(\textrm{E}_2\)), which are located on both sides of the STARRIS. Due to the serious blockage and complicated wireless environment, we assume that the links from the BS to both user \(\mathrm{U_1}\) and eavesdropper \(\mathrm{E_1}\) are negligible or even fall into complete outage status. Due to the long propagation distance between the BS and the celledge user, there will have severe obstacle blockage (buildings or trees) between the celledge user and the BS. Thus, it is reasonable to assume that the direct link is blocked for the celledge user. In the case that both the direct link and the reflecting link are considered for the celledge user, it will bring new challenges in the derivation of the security outage performance. Specifically, the distribution characteristics of the sum of the cascaded channel and the direct channel are reshaped. More specifically, the user \(\mathrm{U_2}\) and eavesdropper \(\mathrm{E_2}\) are located on the same side of the BS in comparison to STARRIS, which have ability to receive both the signal from the BS and the signal reflected by the STARRIS. However, the user \(\textrm{U}_1\) and eavesdropper \(\textrm{E}_1\) can receive only the signals refracted by the STARRIS. In the considered STARRISaided NOMA system, the direct links between the BS and the users located in the reflecting region are considered, whereas the direct links between the BS and the users located in the refracting area are neglected. This assumption is reasonable because the direct transmission links of the users in the refraction region will be blocked by the STARRIS.
It is assumed that the BS has T \(\ge\) 2 antennas, while both the NOMA users and the eavesdroppers are equipped with a single antenna. Besides, the STARRIS consists of \(M\times N\) (i.e., M rows and N columns) reconfigurable elements. In this paper, we employ the ES operating protocol for the STARRIS, in which the incident signal is split into two portions (refract and reflect signals) by each of the elements on the STARRIS. ES is a widely used protocol for STARRIS, which is considered in this paper. For the energy split mode, the incident signal can be divided into reflection and refraction signals through each reconfigurable unit. Each reconfigurable unit can adjust the energy ratio of reflection and refraction, i.e., the power split coefficient. Generally, the secrecy outage probability of the two NOMA users will be influenced by power split coefficients. We denote the channel vectors (or matrices) from BS to STARRIS, from STARRIS to user \(\mathrm{U_1}\), from STARRIS to eavesdropper \(\mathrm{E_1}\), from STARRIS to user \(\mathrm{U_2}\), from STARRIS to eavesdropper \(\mathrm{E_2}\), from BS to user \(\mathrm{U_2}\), and from BS to eavesdropper \(\mathrm{E_2}\) by \({\textbf{G}}\in {\mathbb {C}}^{M N\times T}\), \({\textbf{g}}_{1}\in {\mathbb {C}}^{1\times M N}\), \({\textbf{p}}_{1}\in {\mathbb {C}}^{1\times M N}\), \({\textbf{g}}_{2}\in {\mathbb {C}}^{1\times M N}\), \({\textbf{p}}_{2}\in {\mathbb {C}}^{1\times M N}\), \({\textbf{g}}_{3}\in {\mathbb {C}}^{1\times T}\), and \({\textbf{p}}_{3}\in {\mathbb {C}}^{1\times T}\), respectively. Moreover, the channel vectors (matrices) can be modeled as \({{\textbf{G}}}=\sqrt{\alpha _{SR} d_{SR}^{\eta _{0}}}{\textbf{H}}\), \({{\textbf{g}}_k}=\sqrt{\alpha _{\Pi _1 \Pi _2} d_{\Pi _1 \Pi _2}^{\eta _{0}}}{\textbf{h}}_k\) with \(k \in \{1, 2, 3 \}\), and \({{\textbf{p}}_k}=\sqrt{\alpha _{\Pi _1 \Pi _3} d_{\Pi _1 \Pi _3}^{\eta _{0}}}{\textbf{e}}_k\), where \(\alpha _{SR}\), \(\alpha _{\Pi _1 \Pi _2}\), and \(\alpha _{\Pi _1 \Pi _3}\) are the amplitude coefficients,^{Footnote 3}\(d_{SR}\), \(d_{\Pi _1 \Pi _2}\), and \(d_{\Pi _1 \Pi _3}\) are the distances from the \(\textrm{BS}\) to the STARRIS, from the node \(\Pi _1\) to \(\Pi _2\), and from the node \(\Pi _1\) to \(\Pi _3\), with \(\Pi _1 \in \{\textrm{BS}, \mathrm{STARRIS} \}\), \(\Pi _2 \in \{\textrm{U}_1, \textrm{U}_2 \}\), and \(\Pi _3 \in \{\textrm{E}_1, \textrm{E}_2 \}\), \({\textbf{H}}\), \({\textbf{h}}_k\) and \({\textbf{e}}_k\) represent the small scale fading of the channels and \(\eta _0\) denotes the path loss exponent. Moreover, we adopt the independent Rayleigh fading for all the wireless communication links, i.e., the elements of \({\textbf{H}}\), \({\textbf{h}}_k\) and \({\textbf{e}}_k\) are independent and identically distributed (i.i.d.) following the distribution of \({{\mathcal {C}}}{{\mathcal {N}}}(0,1)\). Let \({\textbf{H}}=[{\textbf{h}}_{sr}^{1t},...,{\textbf{h}}_{sr}^{it},...,{\textbf{h}}_{sr}^{M N t}]^{H}\), where \({\textbf{h}}_{sr}^{it}=[{h}_{sr}^{i1},..., {h}_{sr}^{it} ,...,{h}_{sr}^{iT}]\), \(t \in \{ 1,...,t,...,T\}\), \({\textbf{h}}_{1}=[h_{rU_{1}}^{1},...,h_{rU_{1}}^{i},...,h_{rU_{1}}^{M N}]\), \({\textbf{h}}_{2}=[h_{rU_{2}}^{1},...,h_{rU_{2}}^{i},...,h_{rU_{2}}^{M N}]\), and \({\textbf{h}}_{3}=[h_{sU_{2}}^{1},...,h_{sU_{2}}^{t},...,h_{sU_{2}}^{T}]\) denote the channel coefficients from the tth transmit antenna on BS to the ith reconfigurable element on STARRIS, from the ith reconfigurable element to user \(\mathrm{U_{1}}\), from the ith reconfigurable element to user \(\mathrm{U_{2}}\), and from the tth transmit antenna to user \(\mathrm{U_{2}}\), respectively. In addition, the fading gains \(h_{sr}^{it}\), \(h_{rU_{\nu }}^{i}\), and \(h_{sU_{2}}^{t}\) are complex Gaussian distributed with zero mean and unit variance, i.e., \(h_{sr}^{it}\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,1 )\), \(h_{rU_{\nu }}^{i}\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,1 )\), and \(h_{sU_{2}}^{t}\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,1 )\).
The effective cascaded channel coefficient from the BS to the STARRIS and from the BS to node \(\Pi _2\cup \Pi _3\) can be written as \(\mathbf {\Xi }_{k} \mathbf {\Phi } {\textbf{H}}\), with \(\Xi _{k}\in \{ {\textbf{h}}_{k},{\textbf{e}}_{k} \}\), where \(\mathbf {\Phi } = \text {diag} \{\sqrt{\beta _{1}^{\chi _{\nu }}}e^{j \theta _{1}^{\chi _{\nu }}},... ,\sqrt{\beta _{i}^{\chi _{\nu }}}e^{j \theta _{i}^{\chi _{\nu }}},... ,\sqrt{\beta _{M N}^{\chi _{\nu }}}e^{j \theta _{M N}^{\chi _{\nu }}}\}\) denotes the refractive/reflective coefficient matrix of the STARRIS. Here, the notation \(\chi _{\nu }\) represents the transmission mode for the signal with \(\nu \in \{ 1, 2\}\), i.e., \(\chi _{1} = tra.\) and \(\chi _{2} = ref.\) represent the refracting and reflecting modes for the signal, respectively. \(\sqrt{\beta _{i}^{\chi _{\nu }}}\in [0,1]\) and \(\theta _{i}^{\chi _{\nu }}\in [0,2\pi )\) denote the ES coefficient and phase shift of the ith element for reflecting or refracting responses, respectively. On account that the STARRIS is passive and the energy consumption of circuits is assumed to be ignorable, we obtain \(\beta _{i}^{tra.}+\beta _{i}^{ref.}=1\). Since the channel estimation and feedback processes are not the focus of this manuscript, the STARRIS is assumed to apply adjustable phase shifts that are controlled through a communicationoriented software. Similar to [33], it is assumed that the perfect channel status information (CSI) from the BS to STARRIS links, from the STARRIS to two NOMA users links and from the BS to cellcenter user links are available, which corresponds to the best scenario in terms of system operation and constitutes a performance benchmark for practical implementations.^{Footnote 4}
2.2 Secrecy capacity for STARRISaided NOMA
According to the downlink NOMA transmissions, the users’ messages are distinguished in the power domain. At the BS, the transmit signals are superposition coded and beamformed, which can be expressed as
where \({\textbf{x}}\in {\mathbb {C}}^{T\times 1}\) is the transmit vector, \({\textbf{w}}\in {\mathbb {C}}^{T\times 1}\) is the beamforming vector, \(s_{\nu }\) is the signal for \(\textrm{U}_{\nu }\) with unit power, \(P_s\) is the total transmit power, the PA coefficient of \(s_1\) is \(\alpha _o\) with \(0.5< \alpha _o < 1\), and the rest transmit power is allocated to the signal \(s_2\) [9].
It is clear that the optimal beamforming design should consider both users in NOMA systems. However, to the best our knowledge, the optimal beamformer cannot be achieved in the closedform for twouser multiantenna NOMA systems currently. As a result, the secrecy performance cannot be analyzed for the optimal beamformer in the considered multiantenna NOMA downlinks. In order to obtain an analytical solution of the secrecy performance, a suboptimal beamformer is adopted in this paper, i.e., the beamformer based on the channel of the cellcenter user is used. According to the benefits of NOMA over OMA, the system performance can also be improved when the channel difference between two NOMA users is exaggerated. In this work, since the direct link of user \(\mathrm{U_2}\) is more helpful to the capacity of \(\mathrm{U_2}\) than the reflected link created by the STARRIS when N is not so large, we adopt the matched filter (MF) beamforming with respect to the direct link of \(\mathrm{U_2}\).^{Footnote 5} Thus, the beamforming vector with normalized power, \({\textbf{w}}\), can be written as
The BS broadcasts the superposed signals \({\textbf{x}}\) to a pair of NOMA users, the signals received by both NOMA users, denoted by \(y_{U_{1}}\) and \(y_{U_{2}}\), respectively, can be given by
and
where \(n_{U_{\nu }}\) is the additive white Gaussian noise (AWGN) with mean power \(N_{U_{\nu }}\) at \(\textrm{U}_{\nu }\).
From (3), the signalplusinterferencetonoise ratio (SINR) of user \(\textrm{U}_{1}\) to decode the information \(s_{1}\) can be given by
where \(\rho _{U_{1}}=P_s/N_{U_{1}}\) denotes the signaltonoise ratio (SNR) of user \(\textrm{U}_{1}\).
In addition, the cellcenter user \(\textrm{U}_2\) performs SIC to first detect the signal \(s_1\) of user \(\textrm{U}_1\), then proceeds to subtract \(s_1\) and decode its own signal \(s_2\). Hence, the SINR to decode the information \(s_1\) and the SNR to decode the information \(s_2\) of user \(\textrm{U}_2\) can be, respectively, obtained by
and
where \(\rho _{U_{2}}=P_s/N_{U_{2}}\) denotes the SNR of user \(\textrm{U}_{2}\).
We also assume that the eavesdroppers can utilize parallel interference cancellation (PIC), i.e., eavesdroppers can detect \(s_1\) (or \(s_2\)) without being interfered by \(s_2\) (or \(s_1\)). This assumption may overestimate the eavesdropper’s detection ability, which refers to the worstcase scenario of the considered system model.
The received signals at the eavesdropper \(\textrm{E}_{\nu }\) can be, respectively, given by
and
where \(n_{E_{\nu }}\) is the AWGN with mean power \(N_{E_{\nu }}\) at eavesdropper \(\textrm{E}_{\nu }\).
Furthermore, the SNR at the eavesdropper \(\textrm{E}_{\nu }\) to detect \(s_{\nu }\) can be formulated as
and
where \(\rho _{E_{\nu }}=P_s/N_{E_{\nu }}\) denotes the ratio of the transmitting power of the signal source to the noise power at the eavesdropper \(\textrm{E}_{\nu }\). Let \({\textbf{e}}_{1}=[h_{rE_{1}}^{1},...,h_{rE_{1}}^{i},...,h_{rE_{1}}^{M N}]\), \({\textbf{e}}_{2}=[h_{rE_{2}}^{1},...,h_{rE_{2}}^{i},...,h_{rE_{2}}^{M N}]\), and \({\textbf{e}}_{3}=[h_{sE_{2}}^{1},...,h_{sE_{2}}^{t},...,h_{sE_{2}}^{T}]\) denote the channel coefficients from the ith reconfigurable element to eavesdropper \(\mathrm{E_{1}}\), from the ith reconfigurable element to eavesdropper \(\mathrm{E_{2}}\), and from the tth transmit antenna at the BS to eavesdropper \(\mathrm{E_{2}}\). The fading gains \(h_{rE_{\nu }}^{i}\) and \(h_{sE_{2}}^{t}\) are complex Gaussian distributed with zero mean and unit variance, i.e., \(h_{rE_{\nu }}^{i}\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,1 )\), and \(h_{sE_{2}}^{t}\sim {{\mathcal {C}}}{{\mathcal {N}}}(0,1 )\).
Based on the SINR (or SNR) derived above, the capacity of the in node \(\Pi _2\) to decode signal \(s_{\nu }\) is expressed as
Furthermore, the secrecy capacities obtained at user \(\textrm{U}_\nu\) are, respectively, given by
and
where \([x]^+=\max \left\{ {x,0}\right\}\).
2.3 STARRISOMA
In this subsection, the STARRISaided OMA scheme is selected as a baseline for comparison, in which the STARRIS is deployed to assist the BS to send the information to both user \(\textrm{U}_1\) and user \(\textrm{U}_2\). Thus, the detecting SNR of node \(\Pi _2\cup \Pi _3\) for STARRISOMA scheme can be, respectively, given by
and
where \(\Pi _4 \in \{\mathrm{U_1}, \mathrm{E_1} \}\), \(\Pi _5 \in \{\mathrm{U_2}, \mathrm{E_2} \}\), and \(\omega \in \{{\textbf{g}},{\textbf{p}} \}\).
The entire wireless communication process includes two time slots for STARRISOMA networks. More specifically, in the first time slot, the BS sends the information \(s_1\) through STARRIS to refract to user \(\textrm{U}_1\), and the BS sends \(s_2\) to reflect to user \(\textrm{U}_2\) by the aid of STARRIS in the second slot. As a result, the capacity of the node \(\Pi _3\) to decode signal \(s_{\nu }\) can be expressed as
Thus, the secrecy capacity obtained at user \(\textrm{U}_\nu\) is given by
2.4 Channel statistical properties
In this subsection, the channel statistical properties of cascaded Rayleigh channels are provided, which will be employed to evaluate secrecy outage behaviors for STARRISaided NOMA systems in the following sections.
In view of the above discussions, \(h_{sr}^{it}\) and \(h_{rU_{\nu }}^{i}\) are the channel coefficients from the BS to STARRIS and from STARRIS to user \(\mathrm{U_{\nu }}\), respectively. Unfortunately, the exact distribution of cascaded Rayleigh channels \(X_{\nu ,t}=\sum _{t=1}^{ T}\sum _{j=1}^{ N}h_{rU_{\nu }}^{i}{{\textbf{h}}}_{sr}^{it}{\textbf{w}}\) is difficult to derive, we propose to approximate the PDF of \(X_{\nu ,t}\) as a Gamma distribution by exploiting the Laguerre series approximation [34], which can be given by
where \(\Gamma (\cdot )\) denotes the Gamma function [35], \(a=\frac{c^2}{d}1\), \(b =\frac{d}{c}\) with \(c=\frac{ N\pi }{2}\), \(d=4N\left( 1\frac{\pi ^2}{16} \right)\).
Moreover, the exact distribution of \(Y={\textbf{h}}_{3}{\textbf{w}}^2={\textbf{h}}_{3}^2\) follows the Gamma distribution with shape parameter T and scale parameter 1. Thus, the PDF of Y can be written as
Additionally, we assume that the distance of each row on STARRIS is less than half wavelength, and the distance of each column on STARRIS is larger than half wavelength. Thus, the received signals between rows are correlated, whereas the received signals between any two columns are independent. As a result, the correlation matrix of the nth column on STARRIS is defined as^{Footnote 6}
where \(\Upsilon \in \{ W, V, Q, L, G \}\), \(\Lambda \in \{ \eta , \xi , \zeta , \phi , \varsigma \}\), \(W_n\) represent the receiving correlation matrix of the nth column on the STARRIS, \(V_n\) and \(Q_n\) represent the refracting correlation matrix of the nth column on the STARRIS for \(\textrm{U}_1\) and \(\textrm{E}_1\), and \(L_n\) and \(G_n\) represent the reflecting correlation matrix of the nth column on the STARRIS for \(\textrm{U}_2\) and \(\textrm{E}_2\), respectively. In addition, \(\Lambda ^{n} _{m,m^{'}}\) \((1\le m \le M, 1\le m^{'} \le M, m\ne m^{'} )\) denotes the correlation coefficient between the mth and the \(m^{'}\)th element of the nth column on the STARRIS.
Referring to (5), the SINR of user \(\textrm{U}_{1}\) to decode \(s_1\) can be formulated as
where \(A_1=\frac{ \beta _{j}^{\chi _{1}}}{({d_{SR}d_{RU_{1}}})^{\eta _{0}}}\left( 1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\xi _{mm^{'}}^{n}\right)\), \(X_{U_1,t}=\sum _{t=1}^{T}\sum _{j=1}^{N} h_{rU_{1}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}}\), and \(\beta _{j}^{\chi _{1}}\) is the transmitting coefficient.
In order to obtain better system security performance, we propose a new scheme, that is, each of the reconfigurable elements serves user \(\textrm{U}_1\) and user \(\textrm{U}_2\) at the same time. Specifically, the reflective links serve user \(\textrm{U}_2\) and the refractive links serve user \(\textrm{U}_1\). It is well known that, the STARRIS can adjust the phase shift of each reconfigurable element according to the legitimate user’s channel information, so that the sum of signal amplitudes obtained by the legitimate user can be maximized. Unfortunately, since it is hard to obtain the eavesdropping CSI, we only need to maximize the SINR or SNR received by the legitimate users. For the celledge user, the data transmission is through the STARRIS link only. According to the SINR expression in Eq. (5), we can get that if the cascaded channel gain is the largest (i.e., \(\left {\textbf{g}}_{\textbf{1}}\mathbf {\Phi Gw}\right\) is the maximum), the SINR is the maximum for the celledge user. By applying some mathematical operations, the suboptimal phase shifts for maximizing the SINR can be obtained by \(\theta _{j,opt.}^{\chi _{1}}=arg[[h_{rU_{1}}^{j}][{\textbf{h}}_{sr}^{jt}{\textbf{w}}]]\). It is shown that the design of phase shifts depends on the suboptimal beamforming vector. Thus, the phase shifts obtained here is also suboptimal.
Proposition 1
The CDF of \(\gamma _{{U_{1}}, s_1}\) in (22) can be written as
where \(B_{1}= \alpha _o \rho _{U_{1}} A_1\), and \(\gamma (x,y)\) denotes the lower incomplete Gamma function [35].
Proof
Please refer to “Appendix 1”.
For the cellcenter user, there exist a direct link and a cascaded channel. According to the SINR expressions in Eqs. (6)–(7), it is obvious that when the overall channel gain (the direct link and the cascaded channel) is the largest, the SINR/SNR achieves the largest. In such case, it requires that \(\left {\textbf{g}}_{\textbf{3}}{\textbf{w}}+{\textbf{g}}_{\textbf{2}}\mathbf {\Phi Gw}\right\) is the maximum. Thus, the suboptimal phase shifts for the cellcenter user are denoted by \(\theta _{j,opt.}^{\chi _{2}}=arg[h_{sU_{2}}^{t}]arg[[h_{rU_{2}}^{j}][{\textbf{h}}_{sr}^{j}{\textbf{w}}]]\). Let \(X_{2,t}=\sum _{t=1}^{T}\sum _{j=1}^{N}h_{rU_{2}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}}\), \(Y_1={\textbf{h}}_{3}\).
Based on NOMA, the SINR and SNR at cellcenter user \(\textrm{U}_2\) can be formulated as
and
where \(A_2=\frac{ \beta _{j}^{\chi _{2}}}{({d_{SR}d_{RU_{2}}})^{\eta _{0}}}(1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\phi _{mm^{'}}^{n})\), \(C_1= d_{SU_2}^{\eta _{0}}\).
In order to obtain the CDF of \(\gamma _{{U_{2}}, s_{\nu }}\), we first derive a closeform expression of \(Z_{1}^2\), with \(Z_{1}^{2}=\left( \sqrt{C_1} Y_1+\sqrt{A_2}X_{2,t} \right) ^2\), in the following proposition.
Proposition 2
The CDF of \(Z_1^2\) can be written as
where \({\textbf{B}}(x,y)\) is Beta function and \({}_1 \!F_1\) is degenerate hypergeometric function [35].
Proof
Please refer to “Appendix 2”.
Based on (26), the CDF of \(\gamma _{U_{2}, s_1}\) is given by
In order to avoid redundancy, the detailed derivations are no longer presented.
Referring to the solution process of \(F_{\gamma _{U_{2}, s_{1}}}(x)\), the CDF of \(\gamma _{{U_2}, s_{2}}\) can be written as
Furthermore, referring to (10), we rewrite \(\gamma _{{E_{1}}, s_{1}}\) as
where \(A_3=\frac{ \beta _{j}^{\chi _{1}} }{({d_{SR}d_{RE_{1}}})^{\eta _{0} }}\left( 1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\zeta _{mm^{'}}^{n}\right)\), \(Z_{1,t}= \sum _{t=1}^{T}\sum _{j=1}^{N} h_{rE_{1}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}} e^{j\theta _{j}^{\chi _{1}}}\).
According to the central limit theorem (CLT), \(Z_{1,t}\) can be approximated with a complex Gaussiandistributed random variable. As a result, the square of \(Z_{1,t}^2\) can be approximated with an exponential random variable with parameter N, respectively.
Based on the above operations, the CDF of \(\gamma _{{E_{1}}, s_{1}}\) can be expressed as
where \(\lambda _{E_{1}}= \alpha _{o}N\rho _{E_{1}}A_3\).
Referring to (11), we rewrite \(\gamma _{{E_{2}}, s_{2}}\) as
where \(A_4=\frac{ \beta _{j}^{\chi _{2}}}{d_{SR}^{\eta _{o}}d_{RE_2}^{\eta _{1} }}\left( 1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\varsigma _{mm^{'}}^{n}\right)\), \(C_2=d_{SE_{2}}^{\eta _{1}}\), \(Y_{2}= {\textbf{e}}_3 {\textbf{w}} e^{j\theta _{s}}\), \(Z_{2,t}=\sum _{t=1}^{T} \sum _{j=1}^{N} h_{rE_{2}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}} e^{j\theta _{j}^{\chi _{2}}}\).
According to the CLT, \(\left( \sqrt{A_4} Z_{2,t}+\sqrt{C_2}Y_2\right)\) can be approximated with a complex Gaussiandistributed random variable. As a result, the \(\left( \sqrt{A_4} Z_{2,t}+\sqrt{C_2}Y_2\right) ^2\) can be approximated with an exponential random variable with parameter \(NA_4+C_2\), respectively.
After the above operations, the CDF of \(\gamma _{{E_{2}}, s_{2}}\) can be expressed as
where \(\lambda _{E_{2}}= (1\alpha _{o})\rho _{E_{2}}\left( NA_4+C_2\right)\).
3 Secrecy outage probability analysis
In this section, the performance of STARRISassisted NOMA networks is investigated in terms of secrecy outage probability, where the exact expressions of SOP for both user \(\textrm{U}_1\) and user \(\textrm{U}_2\) are derived.
3.1 The SOP of the celledge user
According to the NOMA protocol, if the celledge user \(\textrm{U}_1\) cannot detect safely the refracting signal \(s_1\), the secrecy outage event will happen. The corresponding SOP can be written as
According to (23) and (33), we present a closedform expression of user \(\textrm{U}_1\) in the following lemma.
Lemma 1
The exact SOP expression of user \(\textrm{U}_1\) in our proposed STARRISaided NOMA scheme is given by
where \(\gamma _{s,1}=2^{R_1^s}\), \(I_1=\cos \left( \frac{2l_{0}1}{2N_0}\pi \right)\), and \(N_0\) denotes the number of terms for the Gauss–Chebyshev quadrature approximation.
Proof
Please refer to “Appendix 3”.
3.2 The SOP of the cellcenter user
The SOP of user \(\textrm{U}_2\) to detect \(s_2\) is given by
Since \(C_{S,{U_{2}, s_1}} \ge R_{1}^{s}\) and \(C_{S,U_{2}, s_2}<R_{2}^{s}\) are independent of each other, referring to [36], we can rewrite (35) as
Referring to (26) and “Appendix 3”, we can easily obtain the approximate closedform expression of \(\Psi _1\) in the following lemma.
Lemma 2
The approximate closedform expression of \(\Psi _1\), which can be written as
where \(\gamma _{s,2}=2^{R_2^s}\).
Then, by substituting (37) into (36), we obtain the approximate SOP of user \(\textrm{U}_2\). Thus, the total approximate SOP can be expressed as
3.3 The SOP of STARRISOMA
For STARRISOMA systems, the SOP of user \(\textrm{U}_{\nu }\) can be expressed as
Referring to (39), we can derive the approximate SOP of user \(\textrm{U}_{\nu }\) for OMA systems in the following lemma.
Lemma 3
The approximate closedform expression of \(P_{U_{\nu }}^{OMA}\), which can be expressed as
where \(\gamma _{s,1}^{OMA}=2^{2R_{1}^s}\), \(\gamma _{s,2}^{OMA}=2^{2R_{2}^s}\), \(\lambda _{E_3}=\rho _{E_1} A_3N\), and \(\lambda _{E_4}=\rho _{E_2}A_4\).
Thus, the total approximate SOP for STARRISaided OMA can be expressed as
From Lemmas 1, 2 and 3, we can observe that the closed expression of exact SOP has a high computation complexity, which is difficult to draw valuable insights. Motivated by this, in the following section, we will focus on the asymptotic SOP achieved by our proposed multiantenna STARRISaided NOMA scheme. Based on the asymptotic SOP, we will provide an explicit evaluation on the secrecy performance of the proposed scheme.
4 The asymptotic secrecy outage probability analysis
In order to provide pivotal insights into the secrecy performance achieved by our proposed STARRISaided NOMA scheme, in this section, we will derive a closedform expression for the asymptotic SOP in the high SNR regime. When \(\rho _{s}\rightarrow \infty\) and \((1\alpha _o)\rho _{U_{1}} A_1\left( \sum _{j=1}^{N} h_{rU_{1}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}} \right) ^2\ll 1\), we can obtain \(\gamma _{U_{1}\leftarrow s_{1}}\approx \alpha _{o}\rho _{U_{1}} A_1\left( \sum _{j=1}^{N} h_{rU_{1}}^{j}{\textbf{h}}_{sr}^{jt}{\textbf{w}} \right) ^2\).
4.1 The asymptotic SOP of celledge user
Referring to (34), we first derive a closedform expression to approximate \(P_{U_1}\) in the following theorem.
Theorem 1
When \(\rho _{s}\rightarrow \infty\), the approximate SOP of user \(\textrm{U}_1\) to detect \(s_1\) is given by
4.2 The asymptotic SOP of cellcenter user
Recalling to (34) and (42), we present a closedform expression to approximate \(P_{U_2}\) in the following theorem.
Theorem 2
The approximate SOP of user \(\textrm{U}_2\) with closedform expression is expressed as
where \(\varphi _1=\frac{\rho _{E_{2}}\gamma _{s,2}(NA_4+C_2)}{\rho _{U_{2}}}\).
As a result, the total approximate SOP can be written as
Based on Theorems 1 and 2, it indicates that the asymptotic SOP of the multiantenna STARRISaided NOMA scheme approaches a certain positive constant.
4.3 The asymptotic SOP of STARRISOMA
Referring to (40), we derive the SOP of the multiantenna STARRISassisted NOMA scheme in the following theorem.
Theorem 3
The closedform expression of SOP for user \(\textrm{U}_{\nu }\), which is approximated as
and
where \(\varphi _2=\frac{\rho _{E_{2}}\gamma _{s,2}^{OMA}(NA_4+C_2)}{\rho _{U_{2}}}\).
As a result, the total approximate SOP for STARRISaided OMA network can be written as
Based on Theorem 3, we also clarify that the asymptotic SOP of the multiantenna STARRISaided OMA scheme approaches a certain positive value.
Comparing our proposed multiantenna STARRISaided NOMA scheme with OMA scheme, we can obtain the following two remarks from the above analyses.
Remark 1
When \(\rho _s \rightarrow \infty\), the average asymptotic SOP for our proposed multiantenna STARRISaided NOMA system approaches a floor. This observation is also applicable to multiantenna STARRISaided OMA system. It means that when the transmission power is large, blindly increasing the transmission power cannot improve the system security performance.
Remark 2
For multiantenna STARRISaided NOMA system, the entire communication procedure occupies a total time slot. Thus, a factor of 1 is used in computing the channel capacities for the multiantenna STARRISaided NOMA scheme. However, for multiantenna STARRISaided OMA system, the entire communication procedure is completed by using two orthogonal time slots. A factor of \(\frac{1}{2}\) is used for the multiantenna STARRISaided OMA scheme. Besides, the value of Eq.(47) is always greater than that of Eq.(44) when the transmit power is in the high regime. It indicates that the advantage of the multiantenna STARRISaided NOMA scheme over the multiantenna STARRISaided OMA scheme.
5 Secrecy diversity order analysis
In order to gain more deep insights, the SDO is usually selected to evaluate the secrecy outage behaviors for communication networks, which is able to describe how fast the SOP decreases when both the transmitting SNR and maintoeavesdropper ratio (MER) are sufficiently high [37,38,39]. Moreover, let \(\lambda _{m,e}=\frac{\Omega _{main}}{\Omega _{eave}}\) denote the ratio of average channel gain between the source and the destination to that between the source and the eavesdropper, which is referred to the MER. Hence, the SDO can be defined as
where \(P_{out}^{\infty }\) denotes the total asymptotic SOP.
Furthermore, we define \(\Omega _{RU_1}=\lambda _{m,e} \Omega _{RE_1}\), \(\Omega _{SR}=\lambda _{1} \lambda _{m,e} \Omega _{RE_1}\), \(\Omega _{RU_{2}}=\lambda _{2} \lambda _{m,e} \Omega _{RE_1}\), \(\Omega _{SU_{2}}=\lambda _{3} \lambda _{m,e} \Omega _{RE_1}\), \(\Omega _{RE_2}=\lambda _{4} \Omega _{RE_1}\), and \(\Omega _{SE_2}=\lambda _{5} \Omega _{RE_1}\), where \(\lambda _{1}\), \(\lambda _{2}\), \(\lambda _{3}\), \(\lambda _{4}\), and \(\lambda _{5}\) are positive constants, \(\Omega _{RE_1}\), \(\Omega _{RE_2}\), \(\Omega _{RU_1}\), \(\Omega _{RU_2}\), \(\Omega _{SR}\), \(\Omega _{SU_2}\), and \(\Omega _{SE_2}\) stand for the channel gain of the link from the STARRIS to the eavesdropper \(\textrm{E}_1\), the channel gain of the link from the STARRIS to the eavesdropper \(\textrm{E}_2\), the channel gain of the link from the STARRIS to the user \(\textrm{U}_1\), the channel gain of the link from the STARRIS to the user \(\textrm{U}_2\), the channel gain of the link from the BS to the STARRIS, the channel gain of the link from the BS to the user \(\textrm{U}_2\), and the channel gain of the link from the BS to the eavesdropper \(\textrm{E}_2\), respectively.
5.1 The SDO of celledge user
Recalling (42), as \(\rho _s \rightarrow \infty\), we derive the approximate SOP of user \(\textrm{U}_1\) in the following corollary.
Corollary 1
The approximate SOP of celledge user \(\textrm{U}_1\) can be given by
where \(D_1=1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\xi _{mm^{'}}^{n}\), \(D_2=1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\zeta _{mm^{'}}^{n}\).
By substituting (49) into (48), we obtain the SDO of user \(\textrm{U}_1\) as \(\frac{a + 1}{2}\).
5.2 The SDO of cellcenter user
Recalling (43), as \(\rho _s \rightarrow \infty\), we present the approximate SOP of user \(\textrm{U}_2\) in the following corollary.
Corollary 2
The approximate SOP of cellcenter user \(\textrm{U}_2\) can be given by
where \(\varphi _3=N\beta _{i}^{\chi _{2}}\varphi _4\), \(\varphi _4=\lambda _1\lambda _4\Omega _{RE_1}\), \(\varphi _5=\beta _{i}^{\chi _{2}}\lambda _1\lambda _2 \Omega _{RE_1}\), \(\varphi _6=\lambda _5\), \(\varphi _7=\lambda _3\), \(D_3=1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\phi _{mm^{'}}^{n}\), \(D_4=1+\sum _{m=1}^{M}\eta _{mm^{'}}^{n}\varsigma _{mm^{'}}^{n}\).
By substituting (50) into (48), we obtain the SDO of user \(\textrm{U}_2\) as \(\frac{a + 1}{2}\).
Based on Corollaries 1 and 2, it is obvious that the SDO of the multiantenna STARRISaided NOMA scheme is a constant, i.e., \(\frac{a+1}{2}=\frac{N}{2}\frac{\pi ^2}{\left( 16\pi ^2 \right) }\), which depends on the number of columns on the STARRIS.
5.3 The SDO of STARRISOMA
Recalling (45) and (46), as \(\rho _s \rightarrow \infty\), we also present the approximate SOP of user \(\textrm{U}_2\) in the following corollary.
Corollary 3
The approximate SOP of user \(\textrm{U}_{\nu }\) for STARRISOMA scheme can be expressed as
and
By substituting (51) and (52) into (48), we obtain the SDO of user \(\textrm{U}_{\nu }\) as \(\frac{a + 1}{2}\).
Based on Corollary 3, we also clarify that the SDO of the multiantenna STARRISaided OMA scheme is a constant, i.e., \(\frac{a+1}{2}=\frac{N}{2}\frac{\pi ^2}{\left( 16\pi ^2 \right) }\), which is also dependent on the number of columns on STARRIS.
From Corollaries 1, 2 and 3, we have two remarks given as follows:
Remark 3
The SDO of our proposed system can be obtained, which is \(\frac{N}{2} \frac{\pi ^2}{\left( 16\pi ^2 \right) }\). This conclusion also applies to the STARRISaided OMA system. As expected, increasing the number of reconfigurable columns on the STARRIS will bring an increasing of the SDO and thus a decreasing of the SOP.
Remark 4
The SDO of the celledge user for STARRISaided NOMA and OMA systems is the same constant, namely \(\frac{N}{2} \frac{\pi ^2}{\left( 16\pi ^2 \right) }\). Interestingly, differing from the traditional SDO, the SDO of the cellcenter user for STARRISaided NOMA and OMA systems is not impacted by the number of transmit antennas on the BS. This is because that the STARRISaided links of the eavesdropper \(\textrm{E}_2\) (next to the cellcenter user) include the links from the BS to the STARRIS by comparing with conventional cooperative communication links.
6 Parameter optimization
Unlike most articles only on performance analysis, this paper analyzes the secrecy performance and optimizes the parameters those affect system performance. As far as we are concerned, both the PA and ES coefficients affect the secrecy performance of the considered network. As a result, selecting appropriate parameters can improve effectively the possibility of secure communication.
The PA and ES coefficients can be jointly optimized. Then, the optimization problem can be formulated as^{Footnote 7}
where \(\beta ^{tra.}\) represents the ES coefficient of each element and \(\alpha _o\) represents PA coefficient, respectively. Due to the relationships between the cellcenter and the celledge users for the PA and the ES coefficients, we only consider the PA and the ES coefficients of the celledge user as the optimization variables in the formulated optimization problem. After the optimal value of the celledge user is obtained, the optimal PA and ES coefficients of the cellcenter user can be, respectively, obtained by \(1 \alpha _o\) and \(\beta ^{ref.}=1\beta ^{tra.}\).
In this section, the optimization design of the two parameters is mainly carried out. It is clear that the optimal solution of the problem in (53) can be obtained by using twodimension search or alternating optimization method with the onedimension search. Due to higher computational complexity of twodimension search, therefore, an alternating optimization method with the onedimension search is invoked to find the optimal solution in this paper. The optimal parameters are obtained by alternating iteration method. More specifically, the detailed procedure is expressed as follows.
6.1 The PA coefficient optimization of STARRISNOMA
When the ES coefficient \(\beta ^{tra.}\) is fixed, according to the expression of \(P_{out}^{\infty }\), the derivative of SOP with respect to the PA coefficient \(\alpha _o\) is obtained, and its firstorder derivative is given as
where
The optimal PA coefficient can be obtained by setting (54) to zero, which is represented by \({\alpha _{o}^{*}}=\frac{\sqrt{\Delta _1\Delta _2}}{\sqrt{\Delta _1\Delta _2}+\sqrt{\Delta _3\Delta _4}}\).
6.2 The ES coefficient optimization of STARRISNOMA
When the PA coefficient \(\alpha _o\) is fixed, according to the expression of \(P_{out}^{\infty }\), the derivative of SOP with respect to the ES coefficient \(\beta ^{tra.}\) is obtained. However, it is difficult to determine the relationship between the derivative and zero, which cannot be determined as a unimodal function in the definition domain. In addition, the value of \(\beta ^{tra.}\) is a consecutive positive number. To sum up, onedimensional exhaustive search algorithm is adopted to find the optimal solution. The detailed steps of onedimensional exhaustive search algorithm is omitted here. In such case, the two parameters \(\alpha _o\), \(\beta ^{tra.}\) of this system have been optimized.
Remark 5
It is clear that the PA and the ES coefficients will affect the system secrecy outage probabilities; thus selecting appropriate the PA and the ES coefficients is beneficial to the improvement of system security performance. It has certain guiding significance in practical engineering application.
7 Results and discussion
In this section, simulation results are presented to validate the theoretical expressions. Monte Carlo (MC) simulation parameters used in this section are presented in Table 1. For the other propagation distances between the STARRIS and the users, the behaviors of performance curves are similar. BPCU is the abbreviation for bit per channel use. All the noise powers are assumed to the same, which is \(\sigma _{o}^{2}\).
We also assume that the received correlation matrix of the nth column \(W_{n}\) is equal,^{Footnote 8} and it is given by
where the coefficients \(\eta _1\), \(\eta _2\) and \(\eta _3\) can be obtained in \(W_n\). Moreover, the refractive correlation matrix of the nth column on the STARRIS of user \(\textrm{U}_1\) and eavesdropper \(\textrm{E}_1\) are written as
where the coefficients \(\xi _1\), \(\xi _2\) and \(\xi _3\) can be obtained in \(V_n\).
where the coefficients \(\zeta _1\), \(\zeta _2\) and \(\zeta _3\) can be obtained in \(Q_n\).
Additionally, the reflective correlation matrix of the nth column on STARRIS for user \(\textrm{U}_2\), and eavesdropper \(\textrm{E}_2\) can be given by
where the coefficients \(\phi _1\), \(\phi _2\) and \(\phi _3\) can be obtained in \(L_n\). Besides, we have
where the coefficients \(\varsigma _1\), \(\varsigma _2\) and \(\varsigma _3\) can be obtained in \(G_n\).
Figure 2 plots the SOP of our proposed scheme versus transmit power \(P_s\) for different transmit antenna selection schemes, i.e., beamforming (BF), optimal antenna selection (OAS) and random antenna selection (RAS) schemes. It can be observed that the secrecy outage behavior of STARRISaided NOMA system with BF scheme is superior to that of OAS and RAS schemes. This is due to the fact that multiantenna beamforming can increase the system capacity. It also confirms that the three transmit antenna selection schemes obtain the same SOP when \(P_s\) is in the high regime, which is consistent with Remark 1. The reason is that the SOP of the system is mainly determined by the rate requirements of the celledge user \(\textrm{U}_1\). Furthermore, we can also observe a close agreement between the simulation results and the derived theoretical analysis expressions.
Figure 3 depicts the SOP of multiantenna STARRISNOMA networks versus transmit power \(P_s\) with different benchmarks. It is observed that the SOP of the proposed scheme is less than various benchmark schemes. The reasons are given as follows. Compared with STARRISaided OMA, the benefit of our proposed scheme originates from the fact that our proposed scheme works in fullduplex (FD) mode and is not affected by selfinterference. Compared with the RISNOMA, the SOP is decreased because of the advantage of STARRIS over RIS. The reason for this phenomenon that the proposed scheme is superior to the RISNOMA is that the signal incident on all STARRIS elements is divided into the transmitted and reflected signals, while a composite smart surface consist of reflectingonly and transmittingonly RIS. Moreover, we observe that the scheme with the aid of STARRIS for cooperative NOMA has a better secrecy performance compared to [32]. The reason is that the STARRIS design aims at maximizing the SINRs of both user \(\textrm{U}_1\) and user \(\textrm{U}_2\) simultaneously. What’s more, our proposed scheme outperforms the STARRISNOMA with random transmit antenna selection and random shift phase modulation (RR) scheme because the multiple transmit antennas and phase alignment can be used to increase the capacity of the main channel. Furthermore, when the transmit power is in the middle and high regions, the SOPs of STARRISNOMA as well as STARRISOMA networks tend to be flat, as discussed in Remarks 1 and 2.
Figure 4 illustrates the SOP of STARRISaided cooperative NOMA network versus different number of columns on STARRIS. It can be seen from this figure that the SOP of all schemes except for the RR scheme of STARRISNOMA system decrease with the number of columns on STARRIS because the secrecy capacity of these schemes is enhanced by increasing the number of columns. However, with the number of columns increases slowly, the SOP of the STARRISNOMA with RR scheme is unchanged almost. This is because increasing the number of columns does not effectively increase the security capacity of two NOMA users.
Figures 5 and 6 demonstrate the SOP of STARRISaided cooperative NOMA network versus different eavesdropping distances. Specifically, Fig. 5 examines the SOP of STARRISaided cooperative NOMA scheme versus the distance between celledge user \(\textrm{U}_1\) and eavesdropper \(\textrm{E}_1\). One can observe that the SOPs of all schemes are decreased with the distance between the STARRIS and the eavesdropper \(\textrm{E}_1\) increases. The reason is that the capacity of the wiretap link is decreased when the propagation distance is increased. In addition, another observation is that the SOPs of all schemes are decreased when the distance between user \(\textrm{U}_2\) and the eavesdropper \(\textrm{E}_2\) increases in Fig. 6. The reason is similar to that in Fig. 5. Furthermore, compared with the impact of \(d_{U_1 E_1}\) on SOP, we also observe that the SOP changes slowly when \(d_{U_2 E_2}\) is in the medium and high regimes. This is due to the fact that the SOP of our proposed scheme is mainly determined by \(d_{U_1 E_1}\), when \(d_{U_2 E_2}\) tends to be larger.
Figures 7 and 8 illustrate the SOP of multiantenna STARRISaided cooperative NOMA system for different secrecy rate requirements. Specifically, on one hand, Fig. 7 examines the SOP of STARRISaided cooperative NOMA system versus the secrecy rate requirement of the celledge user \(\textrm{U}_1\). In this figure, it can be observed that the SOP of the proposed STARRISaided NOMA scheme and other schemes become degraded as the secrecy rate increases. This is because the successful transmission occurs when the requirement of secrecy rate increases. On the other hand, Fig. 8 examines the SOP of STARRISaided cooperative NOMA system versus the secrecy rate requirement for the cellcenter user \(\textrm{U}_2\). We can observe that, as expected, the SOPs increase as the secrecy rate increases. The reason is present in the forgoing analysis. As a further development, our proposed scheme achieves a better secrecy performance than the scheme in STARRISOMA network. The reason is that our proposed scheme considers the secrecy rate requirements of two NOMA users at the same time, while the STARRISOMA scheme only meets the secrecy rate requirements of a single user.
Figure 9 depicts the SOP of the STARRISaided NOMA scheme versus \(\lambda _{m,e}\) for different transmission schemes. As it can be observed, the proposed STARRISaided scheme of cooperative NOMA and OMA system, the conventional RISNOMA, and the scheme in [32], have a better secrecy performance than the STARRISNOMA with RR scheme, when \(\lambda _{m,e}\) is in the medium and high regimes. The reason is due to the fact that the SDO of the STARRISNOMA with RR scheme is one. Besides, it is noted that, the SDO of the conventional RISNOMA is a half of STARRISNOMA scheme. Interestingly, the SDOs of the proposed scheme, STARRISOMA scheme, and the scheme in [32] remain the same as discussed in Remark 3.
Figure 10 illustrates the SOP of the cellcenter user \(\textrm{U}_2\) versus \(\lambda _{m,e}\) for three transmission schemes, i.e., with the direct link for eavesdropper \(\textrm{E}_2\), without the direct link for eavesdropper \(\textrm{E}_2\) and without the STARRISaided link for eavesdropper \(\textrm{E}_2\). In this figure, we observe that the SDO of user \(\textrm{U}_2\) with the direct link for eavesdropper \(\textrm{E}_2\) and without the direct link for eavesdropper \(\textrm{E}_2\) schemes are the same, which is irrelevant to the number of transmit antennas on the BS. Moreover, we also observe that the scheme without the STARRISaided link for eavesdropper \(\textrm{E}_2\) can obtain the full order of secrecy diversity, which is relevant to both the number of transmit antennas on the BS and the number of columns on the STARRIS as discussed in Remark 4. The reason is that the STARRISaided links of eavesdropper \(\textrm{E}_2\) include the links from the BS to the STARRIS by comparing with conventional cooperative communication links.
8 Conclusion
In this paper, the multiantenna STARRISassisted downlink NOMA schemes have been investigated thoroughly. More specifically, we have investigated the SOP of STARRISaided NOMA systems with a multiantenna BS. The closedform and asymptotic expressions of SOP for two NOMA users are derived, respectively. Based on the analytical results, the SDOs of celledge user and cellcenter user are obtained, respectively. It has been also shown that the SOP of our proposed scheme outperforms that of STARRISOMA and other benchmark schemes. In our future research, on one hand, we will investigate the secrecy behaviors of a more complicated scenario with multiple STARRISsaided NOMA communication networks. On the other hand, the channel estimation error was taken into consideration in our future work.
Notes
It is worth noting that the NOMA transmission scenario with multiple pairs of user for STARRISassisted communication networks can further enrich the contents of the considered model, which will be set aside in our future work.
The scenario of multiple eavesdroppers surrounding each NOMA user is not the focus of this paper. It is also beyond the scope of our work.
It is assumed that both the distance between transmit antennas on BS and between reconfigurable elements on STARRIS is much less than that between the BS and the STARRIS. In order to enhance the signal strength, the amplitude coefficient is set to 1.
In practice, it is a nontrivial task of achieving perfect channel state information (CSI). As a result, our analytical results may overestimate the attainable secrecy performance for the considered systems. However, the case with imperfect CSI is beyond the scope of this paper.
In the design of the optimal beamformer, both the direct channel and the cascaded channel should be considered. However, the cascaded channel condition depends on the phase shifts at the STARRIS. Therefore, it is quite difficult to obtain the channel state information of cascaded channel in practice. Actually, in the considered model, the STARRIS is deployed to close to the celledge user, which may be far away from the cellcenter user. In such case, for the cellcenter user, the channel gain is dominant by the direct channel due to the productdistance path loss of the cascaded channel. Based on the above observations, the beamformer is designed based on the direct channel of the cellcenter user in this paper.
Currently, in most of the existing literature, it is assumed that the signals between each reconfigurable element are independent of each other. However, in practice, there exists the correlation between reconfigurable elements in general. In order to make the derivation result more practical, we relax the above assumption in this paper, i.e., the correlation between the rows is considered. This is applicable to the scenario that each column of reconfigurable units is well separated in the space. It is worth mentioning that the derived results can also be extended to the scenario with column correlation straightforwardly. However, for the situation with both row correlation and column correlation, it is more complicated to derive secrecy performance for STARRIS NOMA system, which is still an open problem and is scheduled to be studied in our future work.
For convenience, it is assumed that the transmission coefficients of all elements are the same, i.e., \(\beta ^{\chi _{1}}=\beta _{j}^{\chi _{1}}\).
It can be easily extended to the case that each column has different receiving correlation matrices. Due to different propagation environments for each node, the angle of departure or the angle of arrival may be different. Inspired by this, the different correlation coefficients are adopted in the simulations. For the other correlation coefficients, the behaviors of the curves plotted in the figures are similar.
Abbreviations
 RIS:

Reconfigurable intelligent surface
 STARRIS:

Simultaneous transmitting and reflecting reconfigurable intelligent surface
 NOMA:

Nonorthogonal multiple access
 OMA:

Orthogonal multiple access
 BS:

Base station
 ES:

Energy splitting
 SOP:

Secrecy outage probability
 SDO:

Secrecy diversity order
 SINR:

Signaltointerferenceplusnoise ratio
 SNR:

Signaltonoise ratio
 CDF:

Cumulative distribution function
 PDF:

Probability density function
 MER:

Maintoeavesdropper ratio
 6G:

Sixgeneration
 SE:

Spectral efficiency
 SC:

Superposition coding
 MP:

Message passing
 EE:

Energy efficiency
 DF:

Decodeandforward
 IOS:

Intelligent omnisurface
 MS:

Mode switching
 TS:

Time switching
 PLS:

Physicallayer security
 CSI:

Channel state information
 QoSs:

Quality of services
 PA:

Power allocation
 BF:

Beamforming
 OAS:

Optimal antenna selection
 RAS:

Random antenna selection
 MF:

Matched filter
 SIC:

Successive interference cancellation
 PIC:

Parallel interference cancellation
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The authors would like to thank the anonymous reviewers for their valuable comments and suggestions that helped improve the quality of this manuscript.
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This work was supported by the Natural Science Foundation of China under Grants 62171237 and 61901232.
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Appendix
Appendix
1.1 Proof of Equation (23)
According to (22), we rewrite the expression of \(\gamma _{U_{1}, s_1}\) as
The CDF of \(\gamma _{U_{1}, s_1}\) can be expressed as
where \(B_{1}= \alpha _o \rho _{U_{1}} A_1\).
By employing (19), (62) can be rewritten as
Thus, the expression of \(F_{\gamma _{U_{1}, s_1}}(z)\) is obtained.
1.2 Proof of Equation (26)
Since \(X_{2,t}\) is approximated as a Gamma distribution, \(Y_{1}\) is a squared Gamma distribution. Then, the PDF of \(X_{2,t}\) and the CDF of \(Y_{1}\) can be, respectively, expressed as
and
Additionally, we can easily obtain that \(Y_1\) is independent of \(X_{2,t}\). It is because \(E[X_{2,t} Y_1] = E[X_{2,t}]E[Y_1]=0\), thus, they are independent approximately.
Firstly, we can obtain the CDF of the sum of \(\sqrt{C_1} Y_1+\sqrt{A_2}X_{2,t} =Z_1\), which can be written as
Then, by substituting (64) and (65) into (68), we obtain the CDF of \(Z_1\), which can be written as
where \({\textbf{B}}(x,y)\) is Beta functions and \({}_1 \!F_1\) is Degenerate hypergeometric function [34].
As a result, the CDF of \(Z_1^2\) can be expressed as
Here, the proof of (26) is completed.
1.3 Proof of Equation (34)
Referring to (33), we can rewrite the probability \(P_{U_1}\) as
Unfortunately, it is challenging to derive the exact closedform of the integral in (69). By further applying Gaussian–Chebyshev quadrature into the above integral expression, we can obtain (34).
Here, the proof is completed.
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Xu, S., Liu, C., Wang, H. et al. On secrecy performance analysis of multiantenna STARRISassisted downlink NOMA systems. EURASIP J. Adv. Signal Process. 2022, 118 (2022). https://doi.org/10.1186/s13634022009535
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DOI: https://doi.org/10.1186/s13634022009535
Keywords
 Reconfigurable intelligent surface
 Nonorthogonal multiple access
 Simultaneous transmitting and reflecting
 Physicallayer security
 Secrecy outage probability