Improving Physical Layer Security for Reconfigurable Intelligent Surface aided NOMA 6G Networks
Zhe Zhang, Chensi Zhang, Chengjun Jiang, Fan Jia, Jianhua Ge, Fengkui Gong
aa r X i v : . [ c s . I T ] J a n Improving Physical Layer Security forReconfigurable Intelligent Surface aidedNOMA 6G Networks
Zhe Zhang, Chensi Zhang,
Member, IEEE , Chengjun Jiang, Fan Jia, Jianhua Ge,and Fengkui Gong,
Member, IEEE
Abstract —The intrinsic integration of the nonorthogonal mul-tiple access (NOMA) and reconfigurable intelligent surface (RIS)techniques is envisioned to be a promising approach to sig-nificantly improve both the spectrum efficiency and energyefficiency for future wireless communication networks. In thispaper, the physical layer security (PLS) for a RIS-aided NOMA6G networks is investigated, in which a RIS is deployed to assistthe two “dead zone” NOMA users and both internal and externaleavesdropping are considered. For the scenario with only internaleavesdropping, we consider the worst case that the near-enduser is untrusted and may try to intercept the information offar-end user. A joint beamforming and power allocation sub-optimal scheme is proposed to improve the system PLS. Thenwe extend our work to a scenario with both internal and externaleavesdropping. Two sub-scenarios are considered in this scenario:one is the sub-scenario without channel state information (CSI)of eavesdroppers, and another is the sub-scenario where theeavesdroppers’ CSI are available. For the both sub-scenarios,a noise beamforming scheme is introduced to be against theexternal eavesdroppers. An optimal power allocation scheme isproposed to further improve the system physical security forthe second sub-scenario. Simulation results show the superiorperformance of the proposed schemes. Moreover, it has alsobeen shown that increasing the number of reflecting elementscan bring more gain in secrecy performance than that of thetransmit antennas.
Index terms—
Reconfigurable intelligent surface (RIS),power allocation, beamforming, nonorthogonal multiple access(NOMA), physical layer security.I. I
NTRODUCTION W ITH the rapid development of the society, industryand economy, the demand for data and access areexploding. It is forecasted that the connections will growto 28.5 billion and the global mobile data traffic will reach2.5 exabytes per day by 2022 [1]. From 1G to 5G, everygeneration of mobile communications was driven by theexponential growth of the data and access demands. Currently,in most communities, 5G networks have begun to be gradually
This work was supported in part by the key R & D plan of Shaanxi Province(2019ZDLGY07-02), in part by the joint Fund of Ministry of Education ofChina (6141A02022338), the Fundamental Research Funds for the CentralUniversities, in part by the National Natural Science Foundation of China(61501347), in part by the Project Funded by China Postdoctoral ScienceFoundation (2015M580816), in part by the Postdoctoral Fund of Shaanxiprovince,in part by the “111” project (B08038).Z. Zhang, C. Zhang, C. Jiang, F. Jia, J. Ge and F. Gong are with the StateKey Lab. of Integrated Service Networks, Xidian University, Xi’an, China.(e-mail: [email protected], [email protected], {cjjiang,fjia}@stu.xidian.edu.cn, {jhge, fkgong}@xidian.edu.cn). deployed and various 5G mobile devices have appeared inthe market. As one of the key technologies proposed in 5G,nonorthogonal multiple access (NOMA) have been widelyinvestigated, owing to its ability to significantly improve thespectrum efficiency (SE) of wireless communication systems.Especially for power-domain (PD) multiplexing NOMA, mul-tiple users are allowed to share the same spectrum resourcesand uses serial interference cancellation (SIC) technology torealize multi-user detection. NOMA can meet the data trafficneeds of different users in the same resource block in time-domain, frequency-domain and code-domain [2].On the other hand, to further satisfy the traffic requirementsof the emerging data-intensive applications, the relevant re-searchers have shown great interest in 6G technologies. Theforward-looking vision of 6G had been present in [3]–[5]. As anew technology, reconfigurable intelligent surface (RIS) haveattracted much attention [4]–[8]. Specifically, RIS consistsof massive low-cost passive reflectors, which can reflect thesignal independently by controlling its amplitude or phaseto achieve passive beamforming for signal enhancement ornulling [8]. Its typical applications including wireless coverageand network throughput enhancement, interference cancela-tion, secure communication, wireless information and powertransfer and so on. Importantly, the coverage performancecan be significantly improved by properly adjusting the angleof reflection of each RIS element [9]. This is a promisingapproach to solve the “dead zone” problem in mmWave com-munications [8]. Particularly, compared to the conventionalrelays, RIS only reflects the received signals passively insteadof performing the amplify-and-forward (AF) process actively,and works in full-duplex without self-interference. Thus, RIShas no additional transmission power consumption, which caneffectively improve the system energy efficiency (EE) withhigh SE. Inspired by the merits of both NOMA and RIS, theintrinsic integration of the NOMA and RIS techniques wereapplied to improve the performance of the wireless system[10].In addition, due to the broadcast nature of the radio signals,wireless communications are more vulnerable to eavesdrop-ping than wired ones. Therefore, physical layer security (PLS)without relying on the higher layer encryption algorithms,is getting more attention in the designs of future wireless Also known as Software Controlled Metasurface, Intelligent Wall, PassiveIntelligent Mirror (PIM), Smart Reflect Array, Intelligent Reflecting Surface(IRS), Large Intelligent Surfaces (LIS) in the literature. networks [11], [12]. PLS takes advantage of the dynamicnature of wireless communication channels to ensure thatlegitimate users can successfully decode the data while pre-venting eavesdroppers from decoding the data. Compared withtraditional cryptography, PLS does not require secret keys andcomplicated encryption processing, showing wide prospects infuture wireless networks [13].Many relevant researchers are making outstanding contribu-tions in aforementioned areas. Inspired by these studies, weconsider a RIS-aided NOMA networks and explore the PLSperformance of the system.
A. Relate Work
In recent years, RIS has received considerable attentionowning to the potential and availability of this technology. Themodel of RIS applied to wireless networks was considered andthe theoretical performance limit of RIS-aided communicationsystem was discussed by a mathematical technique in [14]. Theauthors of [9] studied the free-space path loss of RIS-assistedwireless communications by developing the free-space pathloss models for RIS. At first glance, RIS and relay look alittle similar in application, but they are completely differentconcepts. The RIS-aided network and the decode-and-forward(DF) relaying aided network were compared in [15] and theresults shown that the RIS has better transmission performancethan DF relay when the number of RIS reflection elementsis large enough. The authors of [16] presented an Energy-efficient design for RIS which makes the EE of the RIS system higher than that of AF relay system. In a MIMO systemwith RIS-aided, beamforming technique was proposed in [17],[18] to enhance the link quality of wireless communication.Robust design is important in communication systems. Underthe RIS-aided communication system with imperfect CSI, theauthors of [19]–[21] made robust transmission design underthe different conditions. RIS was applied to a simultaneouswireless information and power transfer (SWIPT) aided systemand multicell MIMO communications by the authors of [22],[23], respectively.Recently, NOMA and RIS techniques were integrated toimprove both the EE and SE of the wireless networks. In[24], RIS technique was applied in NOMA transmission tomake the directions of users’ channel vectors align effectively.For continuous phase shifts and discrete phase shifts of RISelements, the authors of [25] presented different algorithms toimprove the performance of the system. The downlink trans-mit power minimization problem for IRS-empowered NOMAnetwork was considered in [26]. The Bit Error Rate (BER)performance of the RIS assisted NOMA system was analyzedin [27]. In [28], RIS-assisted NOMA system was comparedwith traditional orthogonal multiple access (OMA) systemwith/without RIS and traditional NOMA system without RISand simulation results shown that RIS-assisted NOMA systemhas better rate performance than others. In [29], an RIS-assisted uplink NOMA system was considered, where theauthors maximized the sum rate of all users under individualpower constraint. RIS based unmanned aerial vehicles (UAV)assisted MISO NOMA downlink network was investigated in[30]. Security has always been one of the key indicators toevaluate the quality of a communication system. To improvethe PLS of a wireless network with an eavesdropper, the RISwas deployed near the eavesdropper to cancel out the effectivesignal received by eavesdropper in [8], which can effectivelyreduce the information leakage to improve the PLS of thesystem. Under the constraint of secrecy rate, beamformingwas used to minimize the transmitted power of the systemin [31]. When the channel of eavesdropper is superior to theuser’s and both channels are highly correlated in space, jointbeamforming was used to improve the secrecy rate of the userin [32]. The secrecy outage probability (SOP) was derived inRIS-aided wireless communication system and the effect of thenumber of the reflectors in the RIS on the secrecy performancewas analyzed in [33]. Two algorithms were presented in [34] toenhance the PLS of RIS-aided MISO system. When the systemcontains multiple legitimate users and multiple eavesdroppers,a minimum-secrecy-rate maximization problem was solved toimprove the secrecy performance of the whole system in [35].In [36], artificial noise (AN) was added into the wirelessnetwork and simulation results shown that it is beneficial to thesecrecy performance of the system if AN is used appropriately.An RIS assisted MISO network with independent cooperativejamming was studied in [37] and the EE of the system wasbalanced while ensuring secrecy transmission.
B. Motivations and Contributions
Through the above analysis, we can observe that currentresearch interests of RIS are the general applications of RIS,the intrinsic integration of the NOMA and RIS, and PLS ofRIS-aided wireless networks. However, to our knowledge, fewpapers have studied the PLS for RIS-aided NOMA networks.PD multiplexing NOMA technology enables the system touse non-orthogonal transmission at the transmitting end. Then,the receiving end detects users according to the different powerof users’ signals, that is, eliminate multi-access interference(MAI) through SIC. Compared with the traditional orthogonaltransmission, complexity of the receiver is increased, but SEof the system can be greatly improved. In the context of 6G,RIS, as a revolutionary new technology, can improve the EEof the system by reflecting the received signals passively to theusers. RIS can also solve the “dead zone” problem in mmWavecommunications. Inspired by the merits of both NOMA andRIS, we consider a NOMA system with two NOMA userswhich are located in the “dead zone” of the communicationbecause of the obstruction between BS and users. Therefore,RIS is adopted for collaborative transmission to improve thecoverage performance of the system. The RIS-aided NOMAsystem inherits the advantages of NOMA and RIS and isexpected to be applied in 6G networks.As is known to all, security has always been one of thekey indicators during the designs of wireless communicationsystems. Therefore, this paper investigates a joint beamform-ing and power allocation scheme to improve the PLS ofa RIS-aided NOMA network with two NOMA users . We In the scenario with multi-users, the proposed beamforming method andpower allocation scheme can also be used to adjust the order of SIC to leteavesdroppers be the preferred demodulated users. first consider the scenario with internal eavesdropping andthen extend our work to the scenario with both internaland external eavesdropping. For the internal eavesdroppingscenario, we consider the worst case that the near-end user(NU) is untrusted and may try to intercept the information offar-end user (FU). The contributions of this paper are mainlyas follows: • For the scenario with internal eavesdropping, we firstestablish a joint beamforming and power allocation op-timization problem to improve the system PLS. A sub-optimal algorithm is proposed to solve this problem. Tobe specific, the problem is solved in two steps, wherestep 1 focuses on beamforming optimization to enhancethe channel of FU, and step 2 is responsible for powerallocation to optimize the secrecy rate of the system. Analternate iterative algorithm is presented to obtain thebeamforming vector of base station (BS) and the phaseshifts of RIS in step 1. After step 1, FU is enhanced andthe order of SIC could be switched, i.e., the NU wasfirst demodulated. The simulation results show that theproposed algorithm can significantly improve the secrecyperformance of the system. • For the scenario with both internal and external eaves-dropping, there are two sub-scenarios. One is the sub-scenario without channel state information (CSI) ofeavesdroppers, another is the sub-scenario where theeavesdroppers’ CSI are available. For the both sub-scenarios, a noise beamforming scheme is introduced tobe against the external eavesdroppers, i.e., AN is usedto prevent external eavesdroppers from eavesdroppingNU. Depending on whether the eavesdroppers’ CSI areavailable or not, two algorithms based on the Schmidtorthogonalization are presented respectively to obtain thenoise beamforming matrix which can allocate AN into thenull space of the channel for NOMA users. An optimalpower allocation scheme is proposed to further improvethe system physical security for the second sub-scenario. • For the both scenarios, simulation results show thatincreasing the number of reflecting elements of RIS ortransmit antennas of BS have a positive impact on thesystem secrecy performance where increasing the numberof reflecting elements have larger impact than that oftransmit antennas.
C. Organization and Notations
The remainder of this paper is organized as follows. InSection II, we analyze the model of RIS-aided NOMA net-works without external eavesdroppers. The model of RIS-aidedNOMA networks with external eavesdroppers is considered inSection III. Simulation of the model is shown in Section IV.Finally, conclusions are given in Section V.
Notations : Superscripts ( · ) T , ( · ) ∗ and ( · ) H represent trans-pose, conjugate and conjugate transpose, respectively. Diag ( · ) is diagonal matrix with main diagonal ( · ) . |·| and k·k rep-resent the absolute value of scalar and 2-norm of complexvector, respectively. C N × N denotes N × N complex matrices. x ∈ CN ( a, b ) represents that x is complex Gaussian variable with mean a and variance b . y ∈ U ( c, d ) represents that y isa ( c, d ) uniformly distributed random variable. Rank ( · ) is therank of the matrix ( · ) .II. S ECRECY D ESIGN A GAINST I NTERNAL E AVESDROPPING
In this section, we consider a RIS-aided downlink NOMAtransmission model with an untrusted near user. A suboptimaljoint beamforming and power allocation scheme is proposedto improve the system PLS.
A. System Model
As shown in Fig. 1(a), the model consists of a BS with
N s ( N s ≥ transmit antennas, a RIS with N r reflectingelements and two single-antenna users ( U , U ), where U is closer to RIS than U . There is no direct transmissionpath between BS and users because of the obstruction amongthemselves. The obstruction will block the signal from BSto users. In traditional transport schemes, relays are gener-ally used for collaborative transmission. But relay not onlyconsumes additional energy, but also adds additional com-putational complexity. In our scenario, RIS is adopted forcollaborative transmission. BS sends the signal to the RIS, andthe RIS reflects the signal to the users through the reflectingsurface. We assume that the CSI between transport nodescan be accurately estimated. The channel coefficients of theBS-RIS link, RIS- U link and RIS- U link are denoted as H RS ∈ C Nr × Ns , h RU ∈ C Nr × and h RU ∈ C Nr × ,respectively . The acquisition of CSI between the RIS andBS/users is discussed in [8].Each reflecting element of RIS will reflect the signal fromthe BS to the users. All of them can adjust the phase ofthe reflected signal by adjusting the angle of reflection. Sothe phase shifts applied by the RIS can be expressed as Φ = Diag (cid:0) e jϕ , e jϕ , · · · , e jϕ Nr (cid:1) [14], where ϕ n ∈ [0 , π ] is phase shift of reflecting element for n = 1 , · · · N r .The mixed signal from BS to RIS is denoted by x = w (cid:16)p (1 − α ) P x + √ αP x (cid:17) , (1)where x and x are the signal for U and U , respectively,and w ∈ C Ns × ( k w k = 1) is the transmit beamformer of theBS, P and α ∈ (0 , are transmit power and power sharingfactor at BS. Then, the signals received at U and U can beexpressed as y = h HRU ΦH RS w (cid:16)p (1 − α ) P x + √ αP x (cid:17) + n , (2) y = h HRU ΦH RS w (cid:16)p (1 − α ) P x + √ αP x (cid:17) + n , (3) Our system model is suitable for the signal with high frequency, such assome candidate frequency bands of mmWave include 24, 28, 39 and 60GHz.These signals with high frequency have poor diffraction performance. Whenthe base station sends signals to the user, these signals are easily blocked byobstruction, such as the huge building in the typical urban communicationsystem. The structure and element spacing of the transmit antennas have consider-able impact on the channel coefficients. We assume that the elements in eachchannel coefficient are independently and identically distributed (i.i.d) underthe design of the transmit antennas. (cid:258) BS U U control RIS RS H RU h RU h (a) (cid:258) BS U U control RIS (cid:258) Eav M Eav RS H RU h RU h i RE h (b)Fig. 1. System model: (a) RIS-aided NOMA model with an internaleavesdropper. (b) RIS-aided NOMA model with both internal eavesdropperand external eavesdroppers. where n i ∈ CN (0 , N ) is the additive white Gaussian noise(AWGN) at user i for i = 1 , .Because U is closer to RIS than U , the transmissionchannel quality between U and RIS is superior to thatbetween U and RIS. Between BS and RIS, two users sharea transmission channel. Therefore, the integrated transmissionchannel of U is better than U . In the traditional NOMAsystem, BS will assign more power to U (weak user). U directly decodes x by treating x as noise. U needs to decode x first, and then decode x after subtracts x by the SIC. Weconsider the worst case that U will maliciously eavesdrop onthe signal of U . In this scheme, the order of SIC will beswitched to protect U from eavesdropping by U . BS willallocate more power to U . In this case, x is the first signalto be decoded by users. As a result, the SNR at U and U can be respectively expressed as γ ,x = h (1 − α ) Ph αP + N , γ ,x = h (1 − α ) Ph αP + N , (4) γ ,x = h αPN , γ ,x = h αPN , (5)where h i = (cid:13)(cid:13) h HRU i ΦH RS w (cid:13)(cid:13) is the integrated transmissionchannel gain of U i and γ i,x j is the SNR of U i to decode x j with i, j = 1 , . The secrecy rate of the system is given by R S = max [0 , log (1 + γ ,x ) − log (1 + γ ,x )] . (6) B. Joint Beamforming and Power Allocation
In this subsection, we propose a joint beamforming andpower allocation scheme to improve the system PLS. Our goalis to maximize the secrecy rate by adjusting the value of Φ , w , α . In addition, to guarantee the communication quality ofthe users, data rate requirements are also taken into account.The joint beamforming and power allocation problem can begiven by max Φ , w ,α log (1 + γ ,x ) − log (1 + γ ,x ) s.t. ( log (1 + γ ,x ) > R ,th log (1 + γ ,x ) > R ,th , (7)where R i,th is the minimum transmission rate threshold of U i for i = 1 , . However, due to the complexity of theobjective function, it is hard to obtain the global optimallysolution. Therefore, a suboptimal algorithm is proposed tosolve this problem, which can effectively improve the secrecyperformance of the system.The problem is solved in two steps, where step 1 focuses onbeamforming optimization and step 2 is responsible for powerallocation:Step 1: We adopt a beamforming method to improvethe integrated channel condition of U . Specifically, analternate iterative algorithm is presented to determine Φ and w to maximize the h . The optimization problem canbe given by: max w , Φ (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) . (8)Step 2: Then, we design an optimization equation tocalculate the optimal power sharing factor α . The op-timization problem can be given by: max α (log (1 + γ ,x ) − log (1 + γ ,x )) s.t. h (1 − α ) Ph αP + N > γ ,th h αPN > γ ,th . (9)where γ i,th = 2 R i,th − for i = 1 , . Step 1 : Because U is NU, the integrated channel conditionof U is better than that of U , which leads to the secrecyrate to be zero. So, we maximize h to enhance the channelcondition of U by adjusting Φ and w . The optimizationproblem is given by (8). However, it is still too difficult toobtain the closed form solution to (8). To this end, an alternateiterative algorithm is presented to find the appropriate valueof Φ and w . Before applying the algorithm, we set h HRU def = (cid:0) a e jβ , a e jβ , · · · , a Nr e jβ Nr (cid:1) , H RS w k def = (cid:0) b k e jθ k , b k e jθ k , · · · , b kNr e jθ kNr (cid:1) T , where a i , b kj ∈ [0 , ∞ ) , β i , θ kj ∈ [0 , π ] are the amplitudeand phase of corresponding vectors respectively for i, j =1 , , · · · , N r ; k = 0 , , ... . The proposed alternate iterativealgorithm is shown in Algorithm 1. The detailed derivationprocess in Algorithm 1 and the theoretic proof of the stationaryconvergence of solution are given in Appendix A. Algorithm 1
Alternate Iterative Algorithm for w and Φ Initialization: ε = 0 . , h = 0 , k = 0 , ϕ i = − β i ,where i = 1 , · · · , N r ; w k = ( h HRU ΦH RS ) H (cid:13)(cid:13)(cid:13) h HRU ΦH RS (cid:13)(cid:13)(cid:13) ; h k = (cid:13)(cid:13) h HRU ΦH RS w k (cid:13)(cid:13) ; while (cid:12)(cid:12) h k − h (cid:12)(cid:12) ≥ ε do h = h k ; k = k + 1 ; ϕ i = − β i − θ ( k − i for i = 1 , · · · , N r ; w k = ( h HRU ΦH RS ) H (cid:13)(cid:13)(cid:13) h HRU ΦH RS (cid:13)(cid:13)(cid:13) h k = (cid:13)(cid:13) h HRU ΦH RS w k (cid:13)(cid:13) ; end while w = w k ; Obviously, the objective function is non-convex. The proofis given in Appendix B. As a result, the solution obtainedby Algorithm 1 is actually locally optimal solution. However,Algorithm 1 has high searching efficiency, simple structureand convenient use. Moreover, simulation shows that the gapbetween the proposed algorithm and the optimal one is nomore than . with a 94 percent probability and the gap isno more than with a 99 percent probability. Therefore,Algorithm 1 can be safely adopted in practical systems.Applying Algorithm 1, h is significantly enhanced. How-ever, the event h > h may still happen with a certainprobability when the channel condition of U is far betterthan U . When h > h , the secrecy rate is zero and thesystem is outage. In the sequel, we assume that h > h always holds and the outage probability will be presented insimulation section. Step 2 : Then, we need to optimize the power sharing factor α . An optimization equation could be constructed as (9). Bysimplifying (9), we have max α h αP + N h αP + N s.t. α h P − γ ,th N h P ( γ ,th + 1) α > N γ ,th h P . (10)Observing the two constraints, it can be found that if h P − γ ,th N h P ( γ ,th +1) < N γ ,th h P , α has no feasible region, i.e., theproblem (10) has no solution. Only when h P − γ ,th N h P ( γ ,th +1) ≥ N γ ,th h P holds, i.e., P ≥ ( γ ,th + 1) N γ ,th h + γ ,th h N , the problem has an optimal solution.By taking the derivative of the objective function of (10)with respect to α , it can be easily proved that the objectivefunction is an increasing function of α . So we can get theoptimal solution, which is α ∆ = h P − γ ,th N h P ( γ ,th + 1) . (11) III. S ECRECY D ESIGN A GAINST BOTH I NTERNAL AND E XTERNAL E AVESDROPPING
In this section, we consider a RIS-aided downlink NOMAtransmission model with both untrusted near user and M unknown external eavesdroppers. Two scenarios are consid-ered: a) the scenario without CSI of eavesdroppers; b) thescenario where the eavesdroppers’ CSI are available. For theboth scenarios, a noise beamforming scheme is introduced tobe against the external eavesdroppers. Moreover, an optimalpower allocation scheme is proposed to further improve thesystem physical security for the second scenario. A. System Model
As shown in Fig. 1(b), there exist both internal untrustednear user and M external eavesdroppers. There is also nodirect transmission path between the external eavesdroppersand the BS. The external eavesdroppers will eavesdrop on thesignal reflected from RIS. We assume that external eavesdrop-pers are eavesdropping independently and are only interestedin the x . In MIMO wireless communication system with RISaided, AN was used to enhance the system security perfor-mance [38]. In order to prevent the external eavesdroppersfrom eavesdropping, we also use AN to interfere with externaleavesdroppers. v = ( v , v , · · · , v Nv ) T is the AN vector where N v is thenumber of the noise v i ∈ CN (cid:16) , (1 − ψ ) PNv (cid:17) and all the v i are independent of each other for i = 1 , , · · · , N v . The noisebeamforming matrix is denoted as T = ( t , t · · · t Nv ) , where t i ∈ C Ns × is unit column vector for i = 1 , , · · · , N v .The mixed signal sent from BS to RIS is denoted by x = w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + Tv . (12) ψ ∈ (0 , is the power sharing factor between the effectivesignal and AN. RIS reflects x from BS to the users. The signalsreceived at U and U can be expressed as y = h HRU ΦH RS w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + h HRU ΦH RS Tv + n , (13) y = h HRU ΦH RS w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + h HRU ΦH RS Tv + n , (14)where h HRU i ΦH RS Tv is the AN interference received by U i ,and other symbols have the same meaning as the model ofSection II. In this model, U will still maliciously eavesdropon the signal of U . B. Joint Beamforming and Power Allocation without Eaves-droppers’ CSI
In this scenario, the CSI of the external eavesdroppersare not available. In this subsection, the integrated channelcondition of U is still need to be enhanced to make sure thatthe secrecy rate not be zero. So, Φ and w can be achieved byAlgorithm 1 in Section II. In this subsection, our main goalis to determine the noise beamforming matrix T . To this end,a Schmidt orthogonalization based method is developed. The noise beamforming matrix T is used to allocate ANinto the null space of the channel for NOMA users, i.e., h HRU i ΦH RS T = 0 . The problem can be converted intogetting N v linearly independent unit vectors t j to make h HRU i ΦH RS t j = 0 for j = 1 , , · · · , N v ; i = 1 , . The authorof [39] design a method to project AN into the null space ofthe target vector. Based on the Schmidt orthogonalization, amethod denoted as Algorithm 2 is presented to find T . Algorithm 2 : Set
N v = N s − . Randomly generate N v column vectors p i to make Rank ( c , c , p , p , · · · , p Nv ) = N s be true where c j = (cid:16) h HRU j ΦH RS (cid:17) T . Then, N v unitvectors t i orthogonal to c i will be obtained by Schmidtorthogonalization for i = 1 , , · · · , N v ; j = 1 , .Then, the signals received at U and U can be rewritten as y = h HRU ΦH RS w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + n , (15) y = h HRU ΦH RS w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + n , (16)The SNR at U and U can be respectively expressed as γ ,x = h (1 − α ) ψPh αψP + N , γ ,x = h αψPN , (17) γ ,x = h (1 − α ) ψPh αψP + N , γ ,x = h αψPN . (18)where γ i,x j is the SNR of U i to decode x j with i, j = 1 , .We assume that the value of ψ is prescribed since theeavesdroppers’ CSI are not available, and the optimizationproblem can be given by: max α (log (1 + γ ,x ) − log (1 + γ ,x )) s.t. h (1 − α ) ψPh αψP + N > γ ,th h αψPN > γ ,th . (19)Similar to the solution of (9), only when P ≥ ( γ ,th + 1) N γ ,th h ψ + γ ,th h ψ N holds, the problem has an optimal solution, which is α ∆ = h ψP − γ ,th N h ψP ( γ ,th + 1) . (20) C. Joint Beamforming and Power Allocation with Eavesdrop-pers’ CSI
In this scenario, if the external eavesdroppers are untrustedlegitimate users, not NOMA users, the CSI of the externaleavesdroppers are available. The channel coefficient betweenRIS and the i th external eavesdropper is denoted as h RE i ∈ C Nr × for i = 1 , , · · · , M . Similar to the previous subsec-tion, Φ and w can be obtained through Algorithm 1. Basedon the Schmidt orthogonalization, Algorithm 3 is presented tofind T which not only can allocate AN into the null spaceof the channel for NOMA users, but also can project AN onthe channel of the external eavesdroppers as much as possible. Before applying the algorithm, set c i = (cid:0) h HRU i ΦH RS (cid:1) T and d j = (cid:16) h HRE j ΦH RS (cid:17) T for j = 1 , , · · · , M ; i = 1 , . Algorithm 3 : According to the number of external eaves-droppers, the algorithm is divided into two cases.
Case I : Set
N v = M while M N s − is true. For each d i and t i , an optimization equation could be constructed as max t i d iT t i s.t. ( c T t i = 0 c T t i = 0 , According to the knowledge of the Schmidt orthogonalization,a vector ω i which is orthogonal to c and c can be obtainedby decomposing d i , i.e., d i = ω i + k i c + k i c , where k ij is easy to be obtained for i = 1 , , · · · , N v ; j = 1 , . Then,we can get t i = ω ∗ i | ω i | . Case II : Set
N v = N s − while M > N s − is true. Sort d i , i.e., if i < j , (cid:13)(cid:13) d Ti w (cid:13)(cid:13) > (cid:13)(cid:13) d Tj w (cid:13)(cid:13) is true for i = 1 , , · · · , M ; j = 1 , , · · · , M . Witha small sequence number as priority, select N v columnvectors d l i to make Rank ( c , c , d l , d l , · · · , d l Nv ) =Rank ( c , c , d , d , · · · , d M ) for i = 1 , , · · · , N v and l < l < · · · < l Nv M . By doing the sameprocess as Case I to these column vectors, i.e., construct aseries of optimization equations with each d l i and t i for i = 1 , , · · · , N v . T can be obtained by solving theseoptimization equations.Then, our main goal is to maximize the secrecy rate byadjusting the power sharing factors α and ψ . There willbe a high computational complexity if we consider multipleexternal eavesdroppers for power allocation at the same time .So, we only consider the eavesdropper with the best channelcondition, which is denoted as E .The signals received at E can be expressed as y E = h HRE ΦH RS w (cid:16)p (1 − α ) ψP x + p αψP x (cid:17) + h HRE ΦH RS Tv + n E , (21)where n E ∈ CN (0 , N ) is the AWGN at E and h RE ∈ C Nr × is the channel coefficients of the RIS- E link. Weconsider the worst case that x has being decoded by E . TheSNR at E can be expressed as γ E,x = h E αψP (1 − ψ ) PNv h E + N , (22) Similar to the Problem (24), a optimization problem can be constructed ifwe consider multiple external eavesdroppers for power allocation at the sametime. The method to solve the problem is given as follows: according to theeavesdropper’s SNR, ψ can be divided into different segments in the feasibledomain. Each segment has a corresponding eavesdropper playing a major rolein eavesdropping. The optimal value of the secrecy rate in each segment canbe obtained. The maximum of these optimal values is the solution of theproblem. The E is the eavesdropper with the best channel condition, whichmeans the channel coefficient between RIS and E is h RE k where k =arg max i =1 ··· M (cid:13)(cid:13)(cid:13) h HRE i ΦH RS w (cid:13)(cid:13)(cid:13) . In the power allocation, we only consider E for external eavesdropping, but in the Section IV-B, the effect of all externaleavesdroppers on the system is considered. where h E = (cid:13)(cid:13) h HRE ΦH RS w (cid:13)(cid:13) , h E = (cid:13)(cid:13) h HRE ΦH RS T (cid:13)(cid:13) .The secrecy rate of the system is given by R S = max [0 , log (1 + γ ,x ) − log (1 + max ( γ ,x , γ E,x ))] . (23)The power sharing factors α and ψ can be jointly optimized.The optimization problem can be given by: max ψ,α (log (1 + γ ,x ) − log (1 + max ( γ ,x , γ E,x ))) s.t. h (1 − α ) ψPh αψP + N > γ ,th ( a ) h αψPN > γ ,th ( b ) < ψ . (24)The boundary constructed by (24.a) and (24.b) can be ex-pressed as α = h ψP − N γ ,th h ψP (1 + γ ,th ) , (25) α = N γ ,th h ψP . (26)The two-dimensional optimization region for ψ and α isshown in Fig. 2, in which the shaded region ( D R ) is thefeasible region of ( ψ, α ) . Points B ( B x , B y ) , C ( C x , C y ) and D ( D x , D y ) are intersection points between the constraintboundaries. The coordinates of points in Fig. 2 are shownin Appendix C. D A G CB
Fig. 2. Optimization Region for ψ and α . The target function f R ( ψ, α ) can be expressed as: (cid:26) log (1 + γ ,x ) − log (1 + max ( γ ,x , γ E,x )) , ( ψ,α ) ∈ D R , otherwise . (27) C y ≥ B y is required to ensure ( ψ, α ) has feasible region. Thatis, only when P ≥ ( γ ,th (1 + γ ,th ) h + h γ ,th ) N h h holds, the equation has an optimal solution. Compare the values of γ ,x and γ E,x , a bound respect to ψ can be found as: O bound def = ( h − h E ) N vN + P h h E P h h E . (28)Then, according to the value of O bound , the solution ofproblem (24) is divided into three cases case I 1 ≤ O bound case II O bound < D x case III otherwise . (29) Case I : When O bound ≥ , i.e., h ≥ h E , γ ,x ≥ γ E,x always holds. In this case, the internal eavesdropper U playsa major role in eavesdropping. The system model reduces tothe system model without external eavesdroppers in SectionII. Thus, (24) can be simplified as (9). The optimal solutionis point C in the feasible region, i.e., ψ ∆ = C x , α ∆ = C y . Case II : When O bound ≤ D x , γ ,x ≤ γ E,x is true. In thiscase, the external eavesdropper E dominates the system PLS.Then (24) can be simplified as: max ψ,α (cid:18) h αψPN (cid:19) h E αψP (1 − ψ ) P h E Nv + N ! − s.t. α h ψP − N γ ,th h ψP (1 + γ ,th ) α > N γ ,th h ψPD x < ψ . (30)By taking the partial derivative of the objective function of (30)with respect to α , it can be easily proved that the objectivefunction is a monotone function with respect to α . The optimalsolution of the equation must lie on the boundaries of thefeasible region. There are three boundaries on the feasibleregion, they are (25), (26) and ψ = 1 , respectively.For each boundary, the problem (30) can be solved bysubstituting the boundary into the objective function of (30).Specifically, D ( D x , D y ) and C ( C x , C y ) are the optimal solu-tion on the boundary in (26) and ψ = 1 , respectively. For theboundary in (25), G ( G x , G y ) is a stagnation point (maximumpoint), and D ( D x , D y ) and C ( C x , C y ) are also boundarypoints.To sum up, the optimal solution (cid:0) ψ ∆ , α ∆ (cid:1) of the problem(30) in this case can be given by ( ( G x , G y ) G x ∈ [ D x , ψ,α ( f R ( D x , D y ) , f R ( C x , C y )) otherwise . (31) Case III : When O bound ∈ ( D x , , ψ = O bound , a verticaldotted line in Fig. 2, divides the feasible domain into two parts: γ ,x ≥ γ E,x and γ ,x < γ E,x . Point A ( A x , A y ) is theintersection between ψ = O bound and (25). When ψ = O bound holds, γ ,x = γ E,x is true. In the part of γ ,x ≥ γ E,x . (24) can be simplified as: max ψ,α (log (1 + γ ,x ) − log (1 + γ ,x )) s.t. h (1 − α ) ψPh αψP + N > γ ,th h αψPN > γ ,th D x < ψ O bound . (32)It is easy to find that A ( A x , A y ) is the optimal solution inthis part.In the part of γ ,x < γ E,x , (24) can be simplified as: max ψ,α (cid:18) h αψPN (cid:19) h E αψP (1 − ψ ) P h E Nv + N ! − s.t. α h ψP − N γ ,th h ψP (1 + γ ,th ) α > N γ ,th h ψPO bound < ψ . (33)Similar to the Case II, the optimal solution (cid:0) ψ ∆ , α ∆ (cid:1) ofthe problem (33) in this case is ( ( G x , G y ) G x ∈ [ O bound , ψ,α ( f R ( A x , A y ) , f R ( C x , C y )) otherwise . (34)IV. S IMULATION R ESULTS
In this section, numerical results are presented to verify theproposed schemes. As shown in Fig. 3, cartesian coordinates isestablished with BS as the origin where
Eav i , U and U aredistributed in the positive X-axis and RIS is above the X-axis . Eav i is the i th external eavesdropper for i = 1 , , · · · , M . RISBS U U Eav i U d U d i E d x R d y R d Fig. 3. Simulation model.
The channels from RIS to U and U are Los, and theusers will receive not only reflected signal from RIS butalso the multipath signals from the environment, so does thechannel from RIS to BS. The channels between nodes are For the convenience of simulation, we assume users and external eaves-droppers are stay in a line with BS. In fact, our proposed scheme can alsoapply to other scenarios, i.e., users and external eavesdroppers stay in arbitrarylocations of plane in the manner of our model.
Rician channels. We assume that the K factor of the Ricianchannel and the path loss exponent of channel are 10 and 2,respectively, e.g., h ijRS ∈ CN (cid:16)p K /( K + 1) , K + 1) (cid:17) isthe element of the H RS of BS-RIS link for i = 1 , , · · · , N r ; j = 1 , , · · · , N s . Then, the channel coefficients are weightedaccording to the path loss. We normalize the distance betweennodes, i.e., set d R y = 0 . and then adjust other distanceparameters proportionally. We set P = 25 dBm, N = 0 dBmand R ,th = R ,th = 1 bps/Hz. Unless otherwise specified,these parameters are used in the following simulations. Sincewe care more about the relative values between the differentalgorithms rather than the absolute values, thus all the averagesecrecy rates are normalized in the following numerical results. A. Scenario with Internal Eavesdropping
In this subsection, we consider the scenario without externaleavesdropping. In this scenario, alternate iterative algorithm(see Algorithm 1) is proposed for beamforming (BF) toenhance the channel condition of U . In Fig. 4, we plot theSOP of the model against d U with d R x = 0 . and d U = 2 .Additionally, the results without BF design, i.e., Φ = I Nr and w = e Ns N s − , are also presented for the comparisonpurpose, which is denoted as Algorithm 4. As shown in Fig.4, the SOP increased with the increase of d U because thechannel conditions for U will deteriorate as U moves awayfrom RIS. The SOP obtained by Algorithm 4 is generally muchhigher than that of Algorithm 1, which show that the channelcondition of U is better than that of U , i.e., h > h with ahigh probability. The result can demonstrate that the proposedalgorithm could significantly enhance the channel condition of U , i.e., h < h with a high probability. Moreover, from Fig.4 we can see that SOP decreases to a large extent by increasing N r from 16 to 32, while the performance gain is limitedby increasing
N s . This observation shows that increasing thenumber of reflecting elements can bring more gains in secrecyperformance than that of BS antennas. Finally, since Algorithm4 does not perform beamforming operation, we can find thatthe values of
N r and
N s have little effect on the SOP of thesystem.The average secrecy rate under the proposed joint powerallocation and beamforming scheme (Algorithm 1, Problem(9)) are shown in Fig. 5 against d R x with d U = 1 and d U = 3 . For comparison, two contrasting schemes are alsoconsidered. For the downlink of multiuser MIMO systems, theauthors of [40], [41] use the technology of space-division mul-tiple access (SDMA) to optimize the transmission performanceof the system. A transmit preprocessing technique is usedto decompose the multiuser channel into multiple single-userchannels without inter-user interference in [41]. Accordingto the algorithm of [41], transmit preprocessing vectors areused to avoid mutual eavesdropping between two users inScheme I. In this scheme, the rest of the power is allocated to U after meeting the minimum transmission rate threshold of U . For Scheme II, only beamforming design is adopted with I Nr ∈ C Nr × Nr and e Ns ∈ C Ns × are the identity matrix and acolumn vector with all elements equal to one, respectively, i.e., I Nr =diag (1 , , · · · , and e Ns = (1 , , · · · , T . Fig. 4. SOP against d U with d R x = 0 . and d U = 2 . fixed power allocation α = 0 . (Algorithm 1). Fig. 5 showsthat our proposed scheme is better than Scheme I. Comparedwith Scheme II, the result shows that our power allocationscheme can further improve the secrecy rate of the system.Generally speaking, our scheme can effectively improve thesecrecy performance compared with Scheme I and Scheme II.The impact of the value of N r and
N s on the systemis also provided in Fig. 5. Similarly, we can find that moreperformance gain can be achieved by increasing the number ofreflecting elements (i.e.,
N r ). Moreover, we can also observethat there are three extreme points in our proposed scheme,one of which is the local minimum point and the others arethe local maximum points. When RIS moves to the vicinityof U or BS, the average secrecy rate reaches local minimumor local maximum, respectively. This observation can providea useful guideline for the deployment of RIS. -1 -0.5 0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91 Fig. 5. Average secrecy rate against d R x with d U = 1 and d U = 3 . -1 -0.5 0 0.5 1 1.5 200.10.20.30.40.50.60.70.80.91 Fig. 6. Average value against d R x with d U = 1 and d U = 3 . In our model, the performance of the system is greatlyaffected by the values of h and h , especially the differencevalue between them. Therefore, relative mean values of h and h are simulated to analyze the trend of system performancein Fig. 5. Based on the Fig. 5, the average value of h and h against d R x are shown in Fig. 6. With the influence ofAlgorithm 1, the channel condition of U is optimal whenRIS is near the BS, that is, h reaches the maximum. h and h decrease when the RIS is away from the BS to the left,resulting in a decline in system performance. When the RISmoves from BS to U , h decreases while h remains flat.And when the RIS moves from U to U , the downward trendof h gradually flattens out and even increases, while h startsto decrease. But when RIS moves to the right by more than avalue, the value of h cannot meet the minimum transmissionrate threshold of U because of the limitation of power. Inthis case, the performance of system decreased. The resultshows that our analysis is consistent with the trend of systemperformance in Fig. 5.In this article, we assume that all the CSI of channelsbetween nodes are perfect. But it is a challenge to obtainthe perfect CSI due to the time-varying characteristics ofthe channel and the limited signal processing capability ofRIS. The authors of [19]–[21] studied the robust performanceoptimization method of RIS under the condition with imperfectCSI. Our proposed scheme is also suitable for the scenariowith imperfect CSI. But the imperfect CSI still has an impacton the results obtained by our proposed scheme. It is necessaryto analyze the secrecy performance of the system with im-perfect CSI. Therefore, we consider a scenario with imperfectCSI. Specifically, there are errors in the CSI between transportnodes. The estimated channel can be modeled as ∧ h = h + e ,where h is the real CSI of channel and e ∈ CN (cid:0) , σ e (cid:1) is theerror of estimation [42]. The variance of h is denoted as σ h .We set t = q σ e (cid:14) σ h to determine the value of e . Base on Fig.5, our proposed scheme is applied to the scenario with perfect CSI and imperfect CSI, respectively. The relative error against t with d R x = 0 . , d U = 2 and d U = 3 are shown in Fig. 7.We can see that the relative error of the system increased withthe increase of t . But, when the estimation error reaches . ,the relative error is only . , i.e., there is small relative errorin our system. The result shows that our proposed scheme issuitable for the scenario with imperfect CSI. Fig. 7. Relative error against t with d R x = 0 . , d U = 2 and d U = 3 . B. Scenario with both Internal and External Eavesdropping
In this subsection, the scenario with both internal andexternal eavesdropping is considered. AN is used to interferewith external eavesdroppers. We assume that there are 10eavesdroppers in the simulation model, i.e., M = 10 . Theeavesdropper is uniformly distributed in a certain range, i.e., d E i ∈ U (1 , . for i = 1 , , · · · , M . Fig. 8. Average secrecy rate against ψ with d R x = 0 . and d U = 2 , d U = 3 . In the scenario without eavesdroppers’ CSI, Algorithm 2is used to generate noise beamforming matrix T to assisttransmission for all the schemes in this scenario. The averagesecrecy rate against ψ with d R x = 0 . and d U = 2 , d U = 3 are shown in the Fig. 8. In Fig. 8, the proposed schemerefers to joint power allocation and beamforming scheme(Algorithm 1, Problem (19)), and Scheme III refers to thescheme with only BF (i.e., Algorithm 1) with fixed powerallocation α = 0 . . As shown in Fig. 8, power allocationdesigns can significantly increase the system performance,especially for the case where RIS equipped more reflectingelements. We can see from Fig. 8 that the point of ψ = 1 , i.e.,AN is not used in this point, is lower than most other points.Therefore, it is necessary to use AN to improve the secrecyperformance of the system. Fig. 9. Average secrecy rate against d U with d R x = 0 . , d U = 2 and Ns = Nr = 16 . In addition, from Fig. 8 we found that the performanceundergone great changes against ψ , illustrating that the valueof ψ has a big impact on secrecy performance. Therefore, itis necessary to optimize ψ when the CSI of eavesdroppers areknown. In the scenario with eavesdroppers’ CSI, the averagesecrecy rate against d U with d R x = 0 . , d U = 2 and N s = N r = 16 are shown in Fig. 9. In Fig. 9, the proposedscheme refers to joint power allocation and beamformingscheme (Algorithm 1, Problem (24)), and Algorithm 3 isused to generate noise beamforming matrix T . Base on theproposed scheme, Algorithm 2 is used to generate T inScheme IV to prove the better performance of Algorithm 3.AN is not used in Scheme V which refers to the schemewith power allocation and beamforming scheme in Section II(Algorithm 1, Problem (9)). Scheme VI refers to the schemewith BF (i.e., Algorithm 1) and Algorithm 3 with fixed powerallocation α = 0 . , ψ = 0 . . In Algorithm 3, according tothe number of external eavesdroppers, the algorithm is dividedinto two cases. In order to fully show the performance of thetwo cases in Algorithm 3, we set M = 10 and M = 20 respectively in the simulation, i.e. M N s − (Algorithm 3,case I) and M > N s − (Algorithm 3, case II). We can see from Fig. 9 that Algorithm 3 has better secrecy performancethan Algorithm 2 by comparing with the Scheme IV and theperformance of the system deteriorated with the increase ofeavesdroppers. By comparing Scheme V and Scheme VI, wecan find that AN and joint power allocation can improve thesecrecy performance to a great extent.Then, we consider a scenario with dynamic users, i.e., usersand eavesdroppers move randomly over a range, rather thanstanding still at a certain point. Instead of being confined toa line with BS, they move randomly in the plane. In thisscenario, eavesdroppers, U and U are uniformly distributedin the circle with ( , , ( , and ( , as the center and 0.5as the radius, respectively. The average secrecy rate against P with d R x = 0 . , d R y = 1 and N s = N r = 16 areshown in Fig. 10. We can see that the performance of thesystem increased with the increase of P . When the powerincreases to a certain value, the power gain of scheme V isobviously lower than that of other schemes. Because if wedo not use AN in the system, it is no longer the power thatlimits the performance of the system, but the impact of externaleavesdropping on the system. We can find from Fig. 10 thatour proposed scheme can apply to the scenario with dynamicusers and can significantly improve the secrecy performanceof the system.
10 12 14 16 18 20 22 24 26 28 3000.10.20.30.40.50.60.70.80.91
Fig. 10. Average secrecy rate against P with d R x = 0 . , d R y = 1 and Ns = Nr = 16 . V. C
ONCLUSIONS
In this paper, PLS for RIS-aided NOMA system wasinvestigated. We first consider the scenario with only in-ternal eavesdropping, i.e., NU is untrusted and may try tointercept the information of FU. A joint beamforming andpower allocation scheme was proposed to improve the PLSof the system. Due to the complexity of the problem, asuboptimal joint beamforming and power allocation schemewas proposed to solve the question. Then, our work wasextended to the scenario with both internal and external eavesdropping. There are two sub-scenarios: one is the sub-scenarios without CSI of eavesdroppers, another is the sub-scenarios where the eavesdroppers’ CSI are available. Forthe both sub-scenarios, AN is used to prevent the externaleavesdroppers from eavesdropping. Depending on whether theeavesdroppers’ CSI are available or not, two algorithms basedon the Schmidt orthogonalization are presented respectively toobtain the noise beamforming matrix which can allocate ANinto the null space of the channel for NOMA users. In addition,an optimal power allocation scheme is proposed to optimallyallocate the power for jamming and transmitting signals forthe second sub-scenario. It has been also shown that increasingthe number of reflecting elements of RIS or transmit antennasof BS will improve the secrecy performance of the system.Specifically, increasing the number of reflecting elements canbring more gain in secrecy performance than that of transmitantennas. The simulation result shows that there is small errorof our proposed scheme in the scenario with imperfect CSI,which prove that our scheme is suitable for the scenario.A
PPENDIX
AThe detailed derivation process in Algorithm 1 is given asfollows:We regard Φ and w as variables of two different dimen-sions, respectively. First, the initial value is assigned to the Φ , i.e., ϕ i = − β i for i = 1 , · · · , N r . Then, take w as thevariable to obtain the maximum value of objective function (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) , i.e., w = arg max w (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) = (cid:16) h RU H ΦH RS (cid:17) H (cid:13)(cid:13)(cid:13) h RU H ΦH RS (cid:13)(cid:13)(cid:13) . Then alternate iteration loop is performed for Φ and w . In the k th loop, a new value of Φ can be obtained by max Φ (cid:13)(cid:13) h HRU ΦH RS w k − (cid:13)(cid:13) , i.e., ϕ i = − β i − θ ( k − i for i = 1 , · · · , N r and then we canget w k = arg max w (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) = (cid:16) h RU H ΦH RS (cid:17) H (cid:13)(cid:13)(cid:13) h RU H ΦH RS (cid:13)(cid:13)(cid:13) . Execute the loop until (cid:12)(cid:12) h k − h k − (cid:12)(cid:12) < ε is true where h k = (cid:13)(cid:13) h HRU ΦH RS w k (cid:13)(cid:13) for k = 1 , , · · · . The accuracy of erroris denoted as ε . When the loop is completed, the resulting Φ and w k are the phase shifts matrix and beamforming vectorof the system.It is necessary to prove the convergence of the algorithmresults, i.e., prove that the sequence h k is convergent. It is wellknown that if a sequence is increasing and bounded above,then it must be convergent. From the algorithm derivation, itis easy to find that the sequence h k is increasing. Accordingto the nature of the norm, (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) (cid:13)(cid:13) h HRU Φ (cid:13)(cid:13) k H RS w k is true. It is easy to find that (cid:13)(cid:13) h HRU Φ (cid:13)(cid:13) and k H RS w k are bounded in their feasible region, respectively. So (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) is bounded, i.e., h k has an upper bound.So, the sequence h k is convergent.A PPENDIX
BIt is hard to directly prove that the objective functionis non-convex because of massive variables. So, we treatall variables as constants except ϕ i , where ϕ i is the phaseshift of the i th element of RIS. Then, the objective function (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) can be simplified as (cid:13)(cid:13) q e jϕ i + q (cid:13)(cid:13) where q k is a constant for k = 1 , . We can find a ϕ i to make q = 0 be true. Further, function can be simplified as: (cid:13)(cid:13) q e jϕ i + q (cid:13)(cid:13) = (cid:0) q e jϕ i + q (cid:1) H (cid:0) q e jϕ i + q (cid:1) = k q k + k q k + Re (cid:0) q q H e jϕ i (cid:1) = k q k + k q k + τ Re (cid:16) e j ( µ + ϕ i ) (cid:17) = k q k + k q k + τ cos ( µ + ϕ i ) where q q H = τ e jµ . Obviously, the function is non-convex in range of ϕ i ∈ (0 , π ) . So, the objective function (cid:13)(cid:13) h HRU ΦH RS w (cid:13)(cid:13) is non-convex.A PPENDIX
CThe coordinates of points in Fig. 2 are shown as follow: A x = ( h − h E ) N vN + P h h E P h h E .A y = (( N vh − N vh E − h E γ ,th ) N + P h h E ) (( h − h E ) N vN + P h h E ) − × (1 + γ ,th ) − .B ( B x , B y ) = (cid:18) , N γ ,th h P (cid:19) .C ( C x , C y ) = (cid:18) , P h − N γ ,th h P (1 + γ ,th ) (cid:19) .D x = ( γ ,th (1 + γ ,th ) h + h γ ,th ) N h h P .D y = γ ,th h γ ,th (1 + γ ,th ) h + h γ ,th .G x = (cid:0) − P h h (1 + γ ,th ) h E − h ((1 + γ ,th ) h − h E γ ,th ) N vN h E + ( h h E h E (1+ γ ,th ) N v (( − N γ ,th + h P ) h E + N h × N v ) ( h E (( − N γ ,th + h P ) h + N h × (1 + γ ,th )) + N h ( h − h E ) N v )) (cid:17) × ( h P h E h (( − γ ,th − h E + h E N v )) − . G y = (cid:0) − h (1 + γ ,th ) ( − N γ ,th + h P ) h E − N h h × h E (1 + γ ,th ) N v + ((1 + γ ,th ) h E h h E × N v (( N vN + P h E ) h − N h E γ ,th ) × ( h (( h E γ ,th + h E + N vh − h E N v ) × N + P h h E ) − N γ ,th h h E )) (cid:17) × (1 + γ ,th ) − (cid:0) − P h h (1 + γ ,th ) h E − h N h E × N v (( h − h E ) γ ,th + h ) + ((1 + γ ,th ) h E × h N vh E (( N vN + P h E ) h − N × h E γ ,th ) ( h (( h E γ ,th + h E + N vh − h E N v ) × N + P h h E ) − N γ ,th h h E )) (cid:17) − . R Proc. of the IEEE
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