Covert Communication in Intelligent Reflecting Surface-Assisted NOMA Systems: Design, Analysis, and Optimization
Lu Lv, Qingqing Wu, Zan Li, Zhiguo Ding, Naofal Al-Dhahir, Jian Chen
aa r X i v : . [ c s . I T ] D ec Covert Communication in Intelligent ReflectingSurface-Assisted NOMA Systems: Design,Analysis, and Optimization
Lu Lv,
Member, IEEE , Qingqing Wu,
Member, IEEE ,Zan Li,
Senior Member, IEEE , Zhiguo Ding,
Fellow, IEEE ,Naofal Al-Dhahir,
Fellow, IEEE , and Jian Chen,
Member, IEEE
Abstract
In this paper, we investigate covert communication in an intelligent reflecting surface (IRS)-assistednon-orthogonal multiple access (NOMA) system, where a legitimate transmitter (Alice) applies NOMAfor downlink and uplink transmissions with a covert user (Bob) and a public user (Roy) aided by an IRS.Specifically, we propose new IRS-assisted downlink and uplink NOMA schemes to hide the existenceof Bob’s covert transmission from a warden (Willie), which cost-effectively exploit the phase-shiftuncertainty of the IRS and the non-orthogonal signal transmission of Roy as the cover medium withoutrequiring additional uncertainty sources. Assuming the worst-case covert communication scenario whereWillie can optimally adjust the detection threshold for his detector, we derive an analytical expression forthe minimum average detection error probability of Willie achieved by each of the proposed schemes. Tofurther enhance the covert communication performance, we propose to maximize the covert rates of Bobby jointly optimizing the transmit power and the IRS reflect beamforming, subject to given requirementson the covertness against Willie and the quality-of-service (QoS) at Roy. Simulation results demonstratethe covertness advantage of the proposed schemes and confirm the accuracy of the derived analyticalresults. Interestingly, it is found that covert communication is impossible without using IRS or NOMAfor the considered setup while the proposed schemes can always guarantee positive covert rates.
Index Terms
Intelligent reflecting surface, covert communication, low probability of detection, non-orthogonalmultiple access, power allocation, passive beamforming.
Lu Lv, Zan Li, and Jian Chen are with the State Key Laboratory of Integrated Services Networks, Xidian University,Xi’an 710071, China (e-mail: { lulv, zanli } @xidian.edu.cn; [email protected]). Qingqing Wu is with the State KeyLaboratory of Internet of Things for Smart City, University of Macau, Macau 999078, China (e-mail: [email protected]).Zhiguo Ding is with the School of Electrical and Electronic Engineering, The University of Manchester, Manchester M13 9PL,U.K. (e-mail: [email protected]). Naofal Al-Dhahir is with the Department of Electrical and Computer Engineering,The University of Texas at Dallas, Richardson, TX 75080, USA (e-mail: [email protected]). I. I
NTRODUCTION
Driven by the unprecedented demands for high-data rate applications and ubiquitous wire-less services, various advanced wireless technologies including massive multiple-input multiple-output (MIMO) and millimeter wave (mmWave) have been proposed to improve the networkperformance [1]. Nevertheless, the benefit generally comes at the expense of high energy con-sumption and/or hardware cost as well as complexity, due to the use of a huge number ofpower-hungry active components (i.e., radio-frequency (RF) chains). Against this background,intelligent reflecting surface (IRS), also referred to as reconfigurable intelligent surface, wasrecently proposed as a cutting-edge technology to achieve spectral- and energy-efficient wirelesscommunications [2]–[4]. Specifically, IRS is a two-dimensional (2D) surface of electromagnetic(EM) material (i.e., metasurface) that consists of a large number of passive and reconfigurablereflecting elements. Each reflecting element can be controlled by a smart controller to adjust theEM properties (e.g., phase and amplitude) of the incoming signals. By smartly controlling allthe reflecting elements of the IRS, a desirable radio propagation environment can be establishedto enhance the data rate or reception reliability [5]–[7], reduce energy consumption [8]–[10],extend coverage [11], and achieve massive connectivity [12]–[15].On the other hand, secrecy and privacy provisioning has become a critical task in developingthe sixth-generation (6G) wireless, since the soaring volume of confidential and sensitive data(e.g., financial details, e-health records, and identity authentication) is transmitted over the openwireless medium. This calls for physical layer security, that exploits the intrinsic randomness ofnoise and fading channels to prevent information leakage [16]–[18]. Motivated by its capabilityof reconfiguring wireless channels in a cost-effective manner, IRS is recently integrated withphysical layer security to safeguard information transmission. To be specific, with appropriatephase shifting, the IRS reflected signals can be added with the non-reflected signal coherently atthe legitimate user but destructively at the eavesdropper, thus significantly improving the secrecyrate. Assuming the knowledge of perfect channel state information (CSI) of the eavesdropper, thesecrecy rate maximization and transmit power minimization problems for IRS-assisted multiple-input single-output (MISO) transmission were investigated in [19], [20], respectively. After that,attention was shifted to an IRS-assisted MIMO scenario to maximize the secrecy rate in [21]. Inthe cases with and without eavesdropper’s CSI, the authors of [22], [23] showed that incorporatingartificial noise in transmit beamforming along with IRS reflect beamforming is helpful to increase the secrecy rate. In [24], a new information jamming scheme via IRS was proposed to guaranteesecrecy for two-way communications.However, in certain circumstances, protecting the content of communications using existingphysical layer security techniques is far from sufficient, and the communication itself is oftenrequired to hide from being detected [25]. For example, in tactical intelligent networks, militaryoperations desire to shield themselves from the adversary. Alternatively, in financial institutionnetworks, an entity hopes to protect its own secret activities from being monitored by anauthoritarian government. Thus, secure communication systems should also provide stealth orlow probability of detection, and catering to such security concern motivates the recent advancesin covert communication. Theoretically, covert communication aims to explore the fundamentallimits of hiding the amount of wireless information that can be covertly transmitted from alegitimate transmitter to its legitimate receiver, subject to a negligible probability of beingdetected by a warden [25], [26]. The work of [27] pointed out that a positive covert rate canbe achieved when the warden does not exactly know its received noise power. For the scenariowhere the warden has uncertainty about the aggregate received interference, the studies in [28],[29] investigated the maximum covert throughput and the covert outage probability in randomwireless networks. Covert communication with delay constraints was studied in [30], where theauthors showed that the incorporation of finite block length and the use of random transmitpower can effectively induce confusion to the warden. In [31], both conventional and truncatedchannel inversion power control strategies were proposed to prevent the warden from knowing theexistence of the legitimate transmitter and thus guarantee covertness. Furthermore, it was foundin [32]–[34] that covert communication with a positive covert rate can be achieved by embeddingthe covert signal into a superimposed signal and making use of the non-covert transmissions ofa greedy relay [32], a cellular user [33], and a non-orthogonal multiple access (NOMA) weakuser [34], as the shield.It is worth noting that the reviewed covert strategies in [27]–[34] may degrade the communi-cation performance at legitimate users to a certain extent, due to the resource-consuming issueand the stringent covertness constraint [35]. To address this dilemma, several recent studieshave attempted to exploit the IRS to facilitate covert wireless communication via simultaneouslyimproving the received signal quality at the legitimate user and weakening the signal strengthat the warden. In [35], by considering the noise uncertainty at the warden as the cover medium,the authors showed that a much higher covert rate can be achieved by using an IRS compared to that without an IRS. The work of [35] was then extended to a more general system setupwith both single antenna and multiple antennas at the legitimate transmitter in [36], where theimpact of different CSI availabilities of the warden on the covert rate performance is evaluated.More lately, the authors in [37] investigated the joint design of the transmit power and the IRSreflection coefficients to maximize the received signal power of the covert receiver, under theassumption of a finite number of channel uses.As for the aforementioned research efforts, we make two important observations below. • In many existing works (see, e.g., [32]–[34]) on using public actions as the cover mediumfor the covert action, it is assumed that different codebooks [32] or random transmit power[33], [34] is applied at the legitimate transmitter. However, we note that: i) the assumptionof different codebooks at the legitimate transmitter may not be feasible for all possiblescenarios; and ii) as indicated in [30], with random transmit power, both the number oftransmit power levels and codebooks for transmission rate should approach infinity, whichis often difficult to achieve in practical wireless networks. • Initial studies (see, e.g., [35]–[37]) proposed to use the IRS for covert communication, butextra uncertainty sources, e.g., noise uncertainty at the warden [35], [36], or finite blocklength requirement [37] (that is limited to some specified application scenarios) is needed.In fact, they all neglect a useful source of uncertainty that exists inherently in the consideredIRS-assisted covert communication system. Specifically, the phase shifts of the IRS can bedesigned to induce a deliberate confusion and degrade the signal detection performance atthe warden, while at the same time enhancing the signal reception quality of the legitimatereceiver to benefit covertness. However, such a novel design has not yet been consideredfor covert communication, to our best knowledge.Motivated by the above observations, as the first work, we study both downlink and uplinkcovert communications in an IRS-assisted NOMA system, where a legitimate transmitter appliesNOMA to communicate with a covert user and a public user against a warden. Our goal is tohide the communication between the legitimate transmitter and the covert user by making fulluse of the uncertainty inherent in the wireless system environment. The main contributions ofthis paper are summarized as follows.1) We propose novel IRS-assisted downlink and uplink NOMA schemes to achieve covertwireless communications. The phase-shift uncertainty of the IRS and the non-orthogonal signal transmission of the public user are jointly exploited as the new cover medium toshield the signal transmission of the covert user. As such, the proposed schemes do notrequire any other uncertainty sources such as the transmitter’s random transmit power or thewarden’s noise uncertainty, thus rendering them generally simpler and more cost-effectivethan existing covert communication approaches.2) Under the worst-case scenario from the perspective of the covert communication where thewarden can optimally choose its detection threshold, we derive an analytical closed-formexpression for the minimum average detection error probability of the warden achieved byeach of the proposed schemes.3) For each of the proposed schemes, a joint optimization framework of the transmit powerallocation and the IRS reflect beamforming is formulated to enhance the performance of thecovert communication. To deal with the formulated non-convex optimization problems, wefurther develop efficient algorithms based on the alternating optimization to optimize thepower allocation and reflection coefficients alternatively. In particular, at each iteration, theoptimal power allocation solutions for given IRS phase shifts are derived in closed form,and the optimal reflection coefficients for given power allocation are obtained relying onthe semidefinite relaxation (SDR) technique.4) Through analytical and numerical results, we obtain various useful insights: i) our schemescan always guarantee positive covert rates with non-zero transmit power; ii) increasing thetransmit power for the public user and the number of reflecting elements at the IRS areboth helpful to degrade warden’s detection performance. Particularly, as the transmit powerfor the public user grows large, the minimum average detection error probability of thewarden approaches one, implying that the warden’s detection seems like a random guess;and iii) it is impossible to achieve covert communications without using an IRS or NOMAin the considered system, thus validating the effectiveness of the proposed schemes.The rest of the paper is organized as follows. Section II introduces the system model underinvestigation. Sections III and IV provide the transmission design, performance analysis, andpower/beamforming optimization framework for the IRS-assisted downlink and uplink NOMAschemes, respectively, to achieve covert wireless communications. Simulation results are pre-sented in Section V. Finally, we conclude the paper in Section VI.
Notations:
In this paper, scalars are denoted by italic letters, vectors and matrices are denotedby bold-face lower-case and upper-case letters, respectively.
Pr( · ) denotes probability, f X ( · ) IRSIRS BobBob RoyAlice Willie
Is Alice transmitting to Bob?Is Alice transmitting to Bob?
Willie
Is Alice transmitting to Bob?
Direct channelsDirect channels Reflecting channelsReflecting channels IRSIRS
BobBob Roy
Alice
Willie
Is Bob transmitting to Alice?Is Bob transmitting to Alice?
Willie
Is Bob transmitting to Alice?
Direct channelsDirect channels Reflecting channelsReflecting channels (a) (b)
IRS
Controller
IRS
Controller
Fig. 1. Covert communication in an IRS-assisted NOMA system. (a) Downlink scenario. (b) Uplink scenario. denotes the probability density function (PDF), and arg( x ) denote the phase of the complex-valued random variable (RV) x . The rank and the trace of matrix S are denoted by rank( S ) and tr( S ) . Furthermore, S (cid:23) implies that S is a positive semidefinite matrix.II. S YSTEM M ODEL
We consider covert communication in an IRS-assisted NOMA system, which consists of alegitimate transmitter (called Alice), a covert user (called Bob), a public user (called Roy),an IRS, and a warden (called Willie), as shown in Fig. 1. The IRS is deployed to assist inthe downlink and uplink public/covert signal transmissions between Alice and Roy/Bob, whileguaranteeing a low probability of being detected by Willie. Each node is equipped with a singleantenna and operates in a half-duplex mode, which means that signal transmission and receptioncannot be carried out on the same time/frequency resource block. The IRS has N low-costreconfigurable reflecting elements. Each of the elements can reflect a phase shifted version ofthe incident signal independently, aiming at improving the signal reception qualities at Roy andBob (i.e., in downlink) or Alice (i.e., in uplink) as well as creating uncertainty at Willie forcovertness enhancement. Due to the severe path loss, we assume that the powers of the signalsthat are reflected by the IRS two or more times are sufficiently weak, so that they can be ignored[2], [9]. Furthermore, the IRS is connected to a smart controller, such as field-programmablegate array (FPGA), that is in charge of the reflection reconfiguration and assists in the channelestimation of the IRS-involved links [2], [3], [15]. All the wireless channels are assumed to experience a quasi-static block fading along with adistance-based path loss. The fading coefficient vectors between Alice/Bob/Roy/Willie and IRSare denoted by h a , h b , h r , and h w , with the corresponding distances being d a , d b , d r , and d w .The fading coefficients between Alice and Bob/Roy/Willie are denoted by h ab , h ar , and h aw , withthe corresponding distances being d ab , d ar , and d aw . The fading coefficients between Bob/Royand Willie are denoted by h bw and h rw , with the corresponding distances being d bw and d rw .We assume a time-division duplexing (TDD) protocol for downlink and uplink transmissions,such that channel reciprocity holds. Each entry of h a , h b , h r , and h w , as well as h ab , h ar , h aw , h bw , and h rw , are independent and identically distributed with zero mean and unit variance.Furthermore, each receiving node is corrupted by additive white Gaussian noise (AWGN) withzero mean and variance of σ .The CSI availability is discussed next. For the downlink scenario, we assume that: 1) Aliceknows the instantaneous CSI of the direct Alice-user links and the reflected Alice-IRS-user links,but only knows the statistical CSI of the direct Alice-Willie link and the IRS-Willie link. This isreasonable because Willie is usually an external warden and tries to hide its existence from thelegitimate system, and it is difficult for Alice to know its instantaneous CSI. 2) Willie possessesthe instantaneous CSI of the direct Alice-Willie link and the reflected Alice-IRS-Willie link,which is the worst-case scenario from the perspective of covert communication design. Similarly,for the uplink scenario, we assume that: 1) Roy knows the instantaneous CSI of the direct Roy-Alice link and the reflected Roy-IRS-Alice link but only the statistical CSI of the Roy-Willielink and the IRS-Willie link. In addition, Bob knows instantaneous CSI of the direct Bob-Alicelink and the reflected Bob-IRS-Alice link but only the statistical CSI of the Bob-Willie link andthe IRS-Willie link. 2) Willie possesses the instantaneous CSI of the direct Roy/Bob-Willie linksand the reflected Roy/Bob-IRS-Willie links.III. C OVERT C OMMUNICATION IN
IRS-A
SSISTED D OWNLINK
NOMAThis section first proposes a novel IRS-assisted downlink NOMA scheme to achieve covertwireless communication. Then, an analytical closed-form expression for the minimum averagedetection error probability is derived by assuming that Willie can optimally choose his detectionthreshold. Finally, an efficient alternating optimization algorithm is developed to solve the jointtransmit power and passive beamforming optimization problem to maximize the covert rate.
A. IRS-Assisted Downlink NOMA Scheme
As shown in Fig. 1(a), by applying the downlink NOMA principle, Alice transmits a publicsignal s r ( k ) and a covert signal s b ( k ) to Roy and Bob, respectively, where k = 1 , . . . , K denotesthe index of each signal sample, K is the total number of signal samples in a communicationslot, and s r ( k ) and s b ( k ) are Gaussian input signals with zero mean and unit variance. Here,Roy’s public signal s r ( k ) is exploited as a cover for Bob’s covert signal s b ( k ) . Thus, the receivedsignals at Bob and Roy are given, respectively, by y b ( k ) = (cid:18) h ab p L ( d ab ) + h Ha Θh b p L ( d a ) L ( d b ) (cid:19)(cid:16)p P r s r ( k ) + p P b s b ( k ) (cid:17) + z b ( k ) , (1) y r ( k ) = (cid:18) h ar p L ( d ar ) + h Ha Θh r p L ( d a ) L ( d r ) (cid:19)(cid:16)p P r s r ( k ) + p P b s b ( k ) (cid:17) + z r ( k ) , (2)where Θ = diag ( e jθ , . . . , e jθ N ) ∈ C N × N denotes the IRS diagonal reflection-coefficients matrixwith θ n ∈ [0 , π ) being the phase shift of the n th IRS element. Since the instantaneous CSI ofWillie is unavailable to Alice, the optimal Θ should be selected based on the CSI of the Alice-Bob/Roy and Alice-IRS-Bob/Roy links, which is discussed in detail in Section III-C. P r and P b denote the average transmit power of s r ( k ) and s b ( k ) , respectively. z r ( k ) and z b ( k ) denotethe AWGNs at Roy and Bob, respectively, and L ( d ) denotes the effective path loss function[12]. To facilitate the IRS-assisted downlink NOMA scheme, the users are ordered according totheir composite channel (which includes the direct Alice-user link and reflected Alice-IRS-userlink) gains, which has been widely adopted in the literature on IRS-assisted NOMA [12]–[14].Without loss of generality, we assume that the composite channel gains of the users are orderedas | g ab | ≥ | g ar | , where g ab = h ab √ L ( d ab ) + h Ha Θh b √ L ( d a ) L ( d b ) is the composite channel of Bob and g ar = h ar √ L ( d ar ) + h Ha Θh r √ L ( d a ) L ( d r ) is the composite channel of Roy. It is important to point out thatwith the above ordered channels, the SIC ordering at Bob should start from s r ( k ) first andthen move towards s b ( k ) , and thus, more transmit power is allocated to Roy, i.e., P r ≥ P b ,to guarantee successful SIC [12]. This setting is beneficial to the covertness of Bob, since ahigher transmit power of Roy can better protect Bob’s covert communication. Furthermore, thisSIC order does not affect the performance of Willie, due to the fact that Willie only uses theenergy detection to detect the existence of Bob’s transmission rather than decoding the signals. Therefore, the achievable rates of Bob to sequentially decode s r ( k ) and s b ( k ) are given by R b,s r ( k ) = log (cid:18) P r | g ab | P b | g ab | + σ (cid:19) , (3) R b,s b ( k ) = log (cid:18) P b | g ab | σ (cid:19) . (4)Roy directly decodes s r ( k ) by treating s b ( k ) as noise, yielding the achievable rate as R r,s r ( k ) = log (cid:18) P r | g ar | P b | g ar | + σ (cid:19) . (5)On the other hand, Willie tries to judge whether Alice is transmitting a covert signal to Bobor not by carrying out the Neyman-Pearson test based on his received signal sequence y w ( k ) for k = 1 , . . . , K [29]. As a result, Willie faces a binary detection problem: 1) the null hypothesis H which indicates Alice is not transmitting to Bob, and 2) the alternative hypothesis H whichindicates an ongoing covert transmission from Alice to Bob. Then, the received signals at Willieunder these two hypotheses are given, respectively, by H : y w ( k ) = (cid:18) h aw p L ( d aw ) + h Ha Θh w p L ( d a ) L ( d w ) (cid:19)p P r s r ( k ) + z w ( k ) , (6) H : y w ( k ) = (cid:18) h aw p L ( d aw ) + h Ha Θh w p L ( d a ) L ( d w ) (cid:19)(cid:16)p P r s r ( k ) + p P b s b ( k ) (cid:17) + z w ( k ) , (7)where z w ( k ) denotes the AWGN at Willie. Based on (6) and (7), Willie adopts a radiometer forthe binary detection. Using the average received power at Willie (i.e., P w = K P Kk =1 | y w ( k ) | )as the test statistic, the decision rule is given by P w D ≷ D τ dl , (8)where τ dl > is the detection threshold of Willie’s detector, D and D are the binary decisionsin favor of H and H , respectively. Similar to [27]–[29], we assume that Willie uses an infinitenumber of signal samples for the binary detection, i.e., K → ∞ , such that the uncertainties ofthe transmitted signals and the received AWGNs will vanish. Accordingly, the average receivedpower at Willie of the proposed IRS-assisted downlink NOMA scheme is obtained as P w = (cid:16) | h aw | L ( d aw ) + | h Ha Θh w | L ( d a ) L ( d w ) (cid:17) P r + σ , H , (cid:16) | h aw | L ( d aw ) + | h Ha Θh w | L ( d a ) L ( d w ) (cid:17)(cid:0) P r + P b (cid:1) + σ , H . (9) The performance of Willie’s hypothesis test can be measured by the detection error probability,which is defined as ξ w, dl , P FA + P MD , (10)where P FA = P ( D |H ) denotes the false alarm probability, P MD = P ( D |H ) denotes the missdetection probability, and ≤ ξ w, dl ≤ . To be specific, ξ w, dl = 0 implies that Willie can perfectlydetect the covert signal without error, while ξ w, dl = 1 implies that Willie fails to detect the covertsignal and his behavior is like a random guess. Remark 1:
To guarantee Bob’s covert communication, the key idea is to artificially create thephase-shift uncertainty at the IRS. Since the optimal Θ is designed according to the CSI ofthe Alice-Bob/Roy and Alice-IRS-Bob/Roy links, this makes | h Ha Θh w | in (9) a RV at Willieand only the distribution of | h Ha Θh w | is known to him. In this way, when Willie measures hisreceived power, he cannot tell whether the received power change is due to the public signaltransmission or the covert signal transmission, thus ensuring a low probability of detection withrespect to Bob’s covert communication. As a result, we conclude that the proposed IRS-assisteddownlink NOMA scheme is cost-effective and easy to implement, in the sense that the uncertaintyexists inherently in Roy’s non-orthogonal transmission and the phase-shift uncertainty of the IRS,without requiring any extra uncertainty sources, e.g., random transmit power at the legitimateuser [33], [34] or uncertain noise power at the warden [35], [36], which are usually difficult toimplement in practice. Remark 2:
Consider an IRS-assisted orthogonal multiple access (OMA) scenario where thesignals for Roy and Bob are transmitted separately in two orthogonal time slots, the covertness ofBob cannot be achieved for the considered setup. Since without the instantaneous CSI of Willie,it is impossible to design the IRS to help neutralize the signals received by Willie completely.Thus, Willie can exactly measure his non-zero received power of Bob’s covert communication.Furthermore, instead of using IRS, we consider a NOMA full-duplex relay scenario where theIRS is replaced by a full-duplex relay with constant transmit power. In this case, Bob’s covertcommunication cannot be achieved, since Willie can easily raise an alarm if additional transmitpower for Bob’s covert signal is received. B. Detection Error Probability of Willie
For ease of notation, we denote δ N = h Ha Θh w , which can be re-expressed as δ N = N X n =1 | h an || h wn | e − jψ n , (11)where h an and h wn are the fading coefficients from Alice and Willie to the n th element of theIRS, and ψ n = θ ∗ n + arg( h an ) + arg( h wn ) . Specifically, θ ∗ n denotes the optimal phase shift ofthe IRS in the downlink, which is a function of arg( h an ) , arg( h bn ) , arg( h rn ) , arg( h ab ) , and arg( h ar ) . Before deriving the distribution of δ N , we first characterize the distribution of ψ n inthe following lemma. Lemma 1:
The phase value ψ n follows an independent and uniform distribution on [0 , π ) ,namely, f ψ n ( x ) = π with x ∈ [0 , π ) . Proof:
Please refer to Appendix A.In fact, it is challenging to obtain the exact distribution of δ N since δ N is a sum of complex-valued RVs with correlated imaginary and real coefficients. However, when N is sufficientlylarge, using Lemma 1 and [12, Lemma 2], the CDF of δ N can be approximated by a complexGaussian RV with zero mean and variance equal to N , i.e., δ N ∼ CN (0 , N ) . Moreover, asvalidated by the numerical results in [12], such an approximation is also very tight with relativelysmall N .Accordingly, the false alarm probability of the proposed IRS-assisted downlink NOMA schemecan be derived as P FA = Pr (cid:18) P r | h aw | L ( d aw ) + P r δ N L ( d a ) L ( d w ) + σ > τ dl (cid:19) = e Pr | haw | /L ( daw )+ σ − τ dl PrN/ ( L ( da ) L ( dw )) | {z } ν , if τ dl > σ + P r | h aw | L ( d aw ) , , otherwise . (12)Moreover, the miss detection probability of the proposed IRS-assisted downlink NOMA schemeis computed by P MD = Pr (cid:18) ( P r + P b ) | h aw | L ( d aw ) + ( P r + P b ) δ N L ( d a ) L ( d w ) + σ < τ dl (cid:19) = − e ( Pr + Pb ) | haw | /L ( daw )+ σ − τ dl ( Pr + Pb ) N/ ( L ( da ) L ( dw )) | {z } ν , if τ dl > σ + ( P r + P b ) | h aw | L ( d aw ) , , otherwise . (13)Substituting (12) and (13) into (10), the detection error probability of the proposed IRS-assisteddownlink NOMA scheme is obtained as ξ w, dl = , if τ dl < σ + P r | h aw | L ( d aw ) ,ν , if σ + P r | h aw | L ( d aw ) ≤ τ dl ≤ σ + ( P r + P b ) | h aw | L ( d aw ) , ν − ν , if τ dl > σ + ( P r + P b ) | h aw | L ( d aw ) . (14)It is important to point out that the detection error probability in (14) is derived by assuming anarbitrary detection threshold τ dl . From a worst-case perspective of covert communication, Williecan optimally choose τ dl to achieve the minimum detection error probability, i.e., min τ dl ξ w, dl .The optimal solution of τ dl is provided in the following theorem. Theorem 1:
The optimal detection threshold of Willie to minimize the detection error proba-bility of the IRS-assisted downlink NOMA scheme is obtained as τ ∗ dl = σ + ( P r + P b ) | h aw | L ( d aw ) , if | h aw | ≥ P r NP b φ ln (cid:0) P r + P b P r (cid:1) ,σ + P r ( P r + P b ) NP b L ( d a ) L ( d w ) ln (cid:0) P r + P b P r (cid:1) , otherwise , (15)where φ = L ( d a ) L ( d w ) L ( d aw ) . Proof:
Please refer to Appendix B.Then, by substituting (15) into (14) and applying some algebraic manipulations, the minimumdetection error probability of the IRS-assisted downlink NOMA scheme is expressed as ξ ∗ w, dl = e − Pbφ | haw | PrN , if | h aw | ≥ P r NP b φ ln (cid:0) P r + P b P r (cid:1) , − P b P r (cid:0) P r + P b P r (cid:1) − Pr + PbPb e φ | haw | N , otherwise . (16)Recall that Alice does not know the instantaneous CSI of h aw . Hence, we adopt the minimumaverage detection error probability over all channel realizations of | h aw | as the performancemetric of covertness. The following theorem provides a closed-form expression for the minimumaverage detection error probability. Theorem 2:
The minimum average detection error probability of Willie achieved by the IRS- assisted downlink NOMA scheme is given by ¯ ξ ∗ w, dl = (cid:0) P r + P b P r (cid:1) − (1+ PrNPbφ ) P b φ ( P r N ) − + 1 + 1 − (cid:16) P r + P b P r (cid:17) − PrNPbφ − P b P r (cid:0) P r + P b P r (cid:1) − Pr + PbPb φ N − − (cid:20)(cid:16) P r + P b P r (cid:17) PrPb (1 − Nφ ) − (cid:21) . (17) Proof:
According to the Total Probability Theorem, the minimum average detection errorprobability of the IRS-assisted downlink NOMA scheme can be expressed as ¯ ξ ∗ w, dl = Z ∞ PrNPbφ ln (cid:0) Pr + PbPr (cid:1) e − Pbφ xPrN f | h aw | ( x ) dx + Z PrNPbφ ln (cid:0) Pr + PbPr (cid:1) (cid:20) − P b P r (cid:16) P r + P b P r (cid:17) − Pr + PbPb e φ xN (cid:21) f | h aw | ( x ) dx. (18)By calculating the integral in (18), we can readily obtain (17), which completes the proof. Remark 3:
Based on (17), we reveal the following useful insights: 1) ¯ ξ ∗ w, dl → , when P r isfinite but P b → ∞ ; and 2) ¯ ξ ∗ w, dl → , when P b is finite but P r → ∞ . The above results can bereadily proved using the L’Hopital’s Rule in the large P b or P r regime, and thus omitted herefor simplicity. C. Joint Power and Beamforming Optimization
To improve the covert performance of the IRS-assisted downlink NOMA system, we aim tomaximize the covert rate of Bob by jointly optimizing the transmit power of Alice and the reflectbeamforming of the IRS, subject to the total transmit power constraint at Alice, the quality-of-service (QoS) constraint at Roy, and the minimum average detection error probability constraintat Willie. Accordingly, the optimization problem can be formulated as(P1): max P r ,P b , Θ R b,s b ( k ) (19a)s.t. P r + P b ≤ P max a , P r ≥ P b , (19b) R r,s r ( k ) ≤ R b,s r ( k ) , (19c) R r,s r ( k ) ≥ R min r,s r ( k ) , (19d) ¯ ξ ∗ w, dl ≥ − ε, ε ∈ [0 , , (19e) ≤ θ n ≤ π, n = 1 , . . . , N. (19f) Specifically, (19b) contains the transmit power constraint and the power order constraint forSIC, where P max a is the maximum transmit power of Alice; (19c) guarantees that the SIC atBob can be successfully carried out; (19d) denotes the QoS constraint with R min r,s r ( k ) being theminimum rate requirement of Roy in downlink NOMA; (19e) is the covert constraint with ε beingpredefined to specify a certain covertness; and (19f) describes the phase-shift constraints of theIRS elements. The optimization problem, given its original formulation in (P1), is challenging tosolve, due to the highly coupled optimization variables P r , P b , and Θ in the objective functionand constraint (19d), the non-convex constraint (19c), and the complicated expression ¯ ξ ∗ w, dl in(19e). To tackle this challenge, in the following, we propose to optimize the transmit power andpassive beamforming alternatively, until the convergence is achieved.Due to the fact that β ( t ) = log (cid:0) P r tP b t + σ (cid:1) is an increasing function with t , (19c) is equivalentto | g ar | ≤ | g ab | , which is independent of the transmit power P r and P b . Then, for any givenphase shift Θ , problem (P1) is reduced to a transmit power optimization problem as(P2): max P r ,P b R b,s b ( k ) (20a)s.t. P r + P b ≤ P max a , P r ≥ P b , (20b) P r | g ar | ≥ (cid:0) R min r,sr ( k ) − (cid:1)(cid:0) P b | g ar | + σ (cid:1) , (20c) ¯ ξ ∗ w, dl ≥ − ε, ε ∈ (0 , , (20d)where γ th = 2 R min r,sr ( k ) − . Proposition 1:
The maximum objective value of (P2) is achieved only when P r + P b ≤ P max a in (20b) holds with equality, i.e., P r + P b = P max a . Proposition 2:
With P r + P b = P max a , ¯ ξ ∗ w, dl is monotonically decreasing with respect to P b . Proof:
Please refer to Appendix C.With Propositions 1 and 2, problem (P2) can be simplified as(P3): max P b R b,s b ( k ) (21a)s.t. P b ≤ min n P max a , Ξ , ¯ ξ ∗ ( − w, dl ( ε ) o , ε ∈ (0 , , (21b)where Ξ = P max a | g ar | − γ th σ (1+ γ th ) | g ar | . It is not difficult to verify that R b,s b ( k ) is monotonically increasingwith respect to P b , and thus, the optimal power allocation coefficients to maximize R b,s b ( k ) areobtained in closed-form as P ∗ b = min (cid:8) P max a , Ξ , ¯ ξ ∗ ( − w, dl ( ε ) (cid:9) and P ∗ r = P max a − P ∗ b . In particular, ¯ ξ ∗ ( − w, dl ( ε ) can be computed numerically using MATLAB.On the other hand, for any given feasible power allocation coefficients P r and P b , problem(P1) is reduced to the following passive beamforming optimization problem(P4): max Θ R b,s b ( k ) (22a)s.t. | g ar | ≤ | g ab | , (22b) (cid:0) P r − γ th P b (cid:1) | g ar | ≥ γ th σ , (22c) ≤ θ n ≤ π, ∀ n. (22d)Let u = [ u , . . . , u N ] H with u n = e jθ n , ∀ n . Then, constraints in (22d) are equivalent to | u n | = 1 , ∀ n . By applying the change of variables Λ i = diag( h Ha ) h i √ L ( d a ) L ( d i ) and v i = h ai √ L ( d ai ) with i ∈ { r, b } , aswell as introducing H i = (cid:20) Λ i Λ Hi Λ i v i v i Λ Hi (cid:21) and ¯ u = (cid:20) u (cid:21) , (23)we obtain that | g ai | = | u H Λ i + v i | = ¯ u H H i ¯ u + | v i | = tr( H i U ) + | v i | , where U = ¯ u ¯ u H (cid:23) and rank( U ) = 1 . With this result, problem (P4) can be equivalently transformed into(P5): max U tr( H b U ) (24a)s.t. tr( H r U ) + | v r | ≤ tr( H b U ) + | v b | , (24b) (cid:0) P r − γ th P b (cid:1)(cid:0) tr( H r U ) + | v r | (cid:1) ≥ γ th σ , (24c) U n,n = 1 , n = 1 , . . . , N + 1 , (24d) U (cid:23) , (24e) rank( U ) = 1 . (24f)Specifically, we use the fact in (24a) that the function log (cid:0) xσ (cid:1) is monotonically increasingwith x , and thus, maximizing log (cid:0) xσ (cid:1) is equivalent to maximizing x . However, problem(P5) is still non-convex due to the rank-one constraint (24f). To address this issue, we applySDR to relax the rank-one constraint (24f). Consequently, problem (P5) is reduced to(P6): max U tr( H b U ) (25a)s.t. (24b) , (24c) , (24d) , (24e) . (25b) Algorithm 1
Alternating Optimization Algorithm Initialize Θ (1) , P b (1) , and P r (1) . Set iteration index t = 1 . repeat Given Θ ( t ) , solve problem (P3) to obtain P b ( t + 1) and P r ( t + 1) . Given P b ( t + 1) and P r ( t + 1) , solve problem (P6) to obtain U ( t + 1) . Update t = t + 1 . until The increase of the objective value is below a threshold ̺ > . if rank (cid:0) U ( t + 1) (cid:1) = 1 then Compute the eigenvalue decomposition of U ( t + 1) to obtain the nonzero eigenvalue λ eigen and the respective eigenvector v eigen . return Θ ( t + 1) = diag( p λ eigen v eigen ) . else for q = 1 , . . . , Q do Compute U ( t + 1) = VΣV H and generate e q = VΣ r q , where r q ∼ CN ( N +1 , I N +1 ) . Compute Θ q = diag (cid:0) e j arg( e q [1] e q [ N +1] ) , . . . , e j arg( e q [ N ] e q [ N +1] ) (cid:1) . Obtain the objective value of problem (P5), denoted by R q . end for end if return Θ ( t + 1) = arg max q =1 ,...,Q R q .It is clear that problem (P6) is a convex semidefinite program (SDP), which can be efficientlysolved using standard convex optimization solvers such as CVX [38]. Generally, the relaxedproblem (P6) may not always yield a rank-one solution, i.e., rank( U ) = 1 , which indicates thatthe optimal objective value of problem (P6) is an upper bound of problem (P5). Hence, weapply the Gaussian randomization method [9] to obtain an approximate rank-one solution fromthe higher-rank solution to problem (P6). The overall algorithm is given in Algorithm 1 and itsconvergence performance is characterized in the following proposition. Proposition 3:
The developed Algorithm 1 is guaranteed to converge.
Proof:
Please refer to Appendix D.IV. C
OVERT C OMMUNICATION IN
IRS-A
SSISTED U PLINK
NOMAThis section develops a new IRS-assisted uplink NOMA scheme to achieve covert wirelesscommunication. From the worst-case perspective that Willie can optimally select his detectionthreshold, the minimum average detection probability of the proposed scheme is derived in closedform. A joint transmit power and passive beamforming optimization problem is formulated andsolved by an effective alternating optimization algorithm. A. IRS-Assisted Uplink NOMA Scheme
As illustrated in Fig. 1(b), using the uplink NOMA principle, Roy transmits his public signal s ˜ r ( k ) and Bob transmits his covert signal s ˜ b ( k ) to Alice simultaneously, where the public signaltransmission of Roy is exploited as a cover for the covert signal transmission of Bob. Therefore,the received signal at Alice is given by y ˜ a ( k ) = (cid:18) h ar p L ( d ar ) + h Hr Φh a p L ( d a ) L ( d r ) (cid:19)p P ˜ r s ˜ r ( k )+ (cid:18) h ab p L ( d ab ) + h Hb Φh a p L ( d a ) L ( d b ) (cid:19)p P ˜ b s ˜ b ( k ) + n a ( k ) , (26)where Φ = diag ( e jϕ , . . . , e jϕ N ) ∈ C N × N denotes the IRS diagonal reflection-coefficients matrixof uplink NOMA with ϕ n ∈ [0 , π ) , P ˜ r and P ˜ b denote the average transmit powers of Roy andBob, and n a ( k ) denotes the AWGN at Alice. The optimal Φ is designed using the CSI of theRoy/Bob-Alice and Roy/Bob-IRS-Alice links, which is detailed in Section IV-C. As indicatedin [13], for uplink NOMA, the users with better channel conditions are often decoded earlier.To facilitate the SIC decoding at Alice, without loss of generality, we assume that the users arearranged in a descending order according to their composite channels, i.e., | g ra | ≥ | g ba | , where g ra = h ar √ L ( d ar ) + h Hr Φh a √ L ( d a ) L ( d r ) is the composite channel of Roy and g ba = h ab √ L ( d ab ) + h Hb Φh a √ L ( d a ) L ( d b ) isthe composite channel of Bob. With the above ordered channels, the covert signal is decoded atthe last stage of SIC without suffering from severe inter-user interference, which is beneficial toBob’s covertness (i.e., the covert rate of Bob increases as the average transmit power increases).Then, the achievable rates of Alice to sequentially decode s ˜ r ( k ) and s ˜ b ( k ) are given by R a,s ˜ r ( k ) = log (cid:18) P ˜ r | g ra | P ˜ b | g ba | + σ (cid:19) , (27) R a,s ˜ b ( k ) = log (cid:18) P ˜ b | g ba | σ (cid:19) . (28)Similar to the downlink scenario, Willie performs the Neyman-Pearson test to judge whetherBob is transmitting a covert signal to Alice or not. As a result, the received signals under thenull hypothesis H and the alternative hypothesis H are given, respectively, by H : y ˜ w ( k ) = (cid:18) h rw p L ( d rw ) + h Hr Φh w p L ( d r ) L ( d w ) (cid:19)p P ˜ r s ˜ r ( k ) + z w ( k ) , (29) H : y ˜ w ( k ) = (cid:18) h rw p L ( d rw ) + h Hr Φh w p L ( d r ) L ( d w ) (cid:19)p P ˜ r s ˜ r ( k ) + (cid:18) h bw p L ( d bw ) + h Hb Φh w p L ( d b ) L ( d w ) (cid:19)p P ˜ b s ˜ b ( k ) + z w ( k ) . (30)With the received signals in (29) and (30), the average received power at Willie of the proposedIRS-assisted uplink NOMA scheme can be obtained as P ˜ w = (cid:16) | h rw | L ( d rw ) + | h Hr Φh w | L ( d r ) L ( d w ) (cid:17) P ˜ r + σ , H , (cid:16) | h rw | L ( d rw ) + | h Hr Φh w | L ( d r ) L ( d w ) (cid:17) P ˜ r + (cid:16) | h bw | L ( d bw ) + | h Hb Φh w | L ( d b ) L ( d w ) (cid:17) P ˜ b + σ , H . (31) Remark 4:
The optimal phase shift Φ is selected based on the CSI of the Roy/Bob-Alice andRoy/Bob-IRS-Alice links, which helps create the IRS phase-shift uncertainty to Willie to achieveBob’s covert communication, e.g., both | h Hr Φh w | and | h Hb Φh w | serve as RVs at Willie. B. Detection Error Probability of Willie
Let ζ N, = h Hr Φh w = P Nn =1 | h rn || h wn | e − jχ n and ζ N, = h Hb Φh w = P Nn =1 | h bn || h wn | e − jχ ′ n ,where h rn and h bn are the fading coefficients from Roy and Bob to the n th element of the IRS, χ n = ϕ ∗ n + arg( h rn ) + arg( h wn ) , and χ ′ n = ϕ ∗ n + arg( h bn ) + arg( h wn ) . In particular, ϕ ∗ n is theoptimal phase shift of the IRS in the uplink, which is a function of arg( h an ) , arg( h bn ) , arg( h rn ) , arg( h ab ) , and arg( h ar ) . The following lemma provides the statistics of χ n and χ ′ n . Lemma 2:
The phase values χ n and χ ′ n follow the same independent and uniform distributionon [0 , π ) , namely, f χ n ( x ) = f χ ′ n ( x ) = π with x ∈ [0 , π ) . Proof:
The proof is similar to that in Appendix A, which is omitted for brevity.With Lemma 2, and again applying [12, Lemma 2], we can approximate the CDF of ζ N, and ζ N, as complex Gaussian RVs with zero mean and variance equal to N , i.e., ζ N, , ζ N, ∼CN (0 , N ) . Consequently, the false alarm probability of the IRS-assisted uplink NOMA schemecan be calculated as P ′ FA = Pr (cid:18) P ˜ r | h rw | L ( d rw ) + P ˜ r ζ N, L ( d r ) L ( d w ) + σ > τ ul (cid:19) = e P ˜ r | hrw | /L ( drw )+ σ − τ ul P ˜ rN/ ( L ( dr ) L ( dw )) | {z } ν , if τ ul > σ + P ˜ r | h rw | L ( d rw ) , , otherwise . (32)The miss detection probability of the IRS-assisted uplink NOMA scheme is derived as P ′ MD = Pr (cid:18) P ˜ r | h rw | L ( d rw ) + P ˜ b | h bw | L ( d bw ) + P ˜ r ζ N, L ( d r ) L ( d w ) + P ˜ b ζ N, L ( d b ) L ( d w ) + σ < τ ul (cid:19) = − e P ˜ r | hrw | /L ( drw )+ P ˜ b | hbw | /L ( dbw )+ σ − τ ul ( P ˜ r/L ( dr )+ P ˜ b/L ( db )) N/L ( dw ) | {z } ν , if τ ul > σ + P ˜ r | h rw | L ( d rw ) + P ˜ b | h bw | L ( d bw ) , , otherwise . (33)Summarizing (32) and (33), the detection error probability of the IRS-assisted uplink NOMAscheme is obtained as ξ w, ul = , if τ ul < σ + P ˜ r | h rw | L ( d rw ) ,ν , if σ + P ˜ r | h rw | L ( d rw ) ≤ τ ul ≤ σ + P ˜ r | h rw | L ( d rw ) + P ˜ b | h bw | L ( d bw ) , ν − ν , if τ ul > σ + P ˜ r | h rw | L ( d rw ) + P ˜ b | h bw | L ( d bw ) . (34)The following theorem gives the optimal detection threshold τ ul to minimize the detection errorprobability (34). Theorem 3:
The optimal detection threshold of Willie to minimize the detection error proba-bility of the IRS-assisted uplink NOMA scheme is obtained as τ ∗ ul = σ + P ˜ r | h rw | L ( d rw ) + P ˜ b | h bw | L ( d bw ) , if | h bw | ≥ P ˜ r NP ˜ b φ ln (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) , P ˜ r | h rw | L ( d rw ) − P ˜ r φ | h bw | L ( d bw ) + σ + P ˜ r ( P ˜ r φ + P ˜ b ) NP ˜ b L ( d r ) L ( d w ) ln (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) , otherwise , (35)where φ = L ( d b ) L ( d r ) and φ = L ( d r ) L ( d w ) L ( d bw ) . Proof:
The proof is similar to that in Appendix B and thus omitted for simplicity.Substituting (35) into (34), the minimum detection error probability of the IRS-assisted uplinkNOMA scheme can be derived as ξ ∗ w, ul = e − P ˜ bφ | hbw | P ˜ rN , if | h bw | ≥ P ˜ r NP ˜ b φ ln (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) , − P ˜ b P ˜ r φ (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) − P ˜ rφ P ˜ bP ˜ b e φ | hbw | N , otherwise , (36)where φ = L ( d b ) L ( d w ) L ( d bw ) . Based on this result, the minimum average detection error probability inclosed-form is provided in the following theorem. Theorem 4:
The minimum average detection error probability of Willie achieved by the IRS-assisted uplink NOMA scheme is expressed as ¯ ξ ∗ w, ul = (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) − (1+ P ˜ rNP ˜ bφ ) P ˜ b φ ( P ˜ r N ) − + 1 + 1 − (cid:16) P ˜ r φ + P ˜ b P ˜ r φ (cid:17) − P ˜ rNP ˜ bφ − P ˜ b P ˜ r φ (cid:0) P ˜ r φ + P ˜ b P ˜ r φ (cid:1) − P ˜ rφ P ˜ bP ˜ b φ N − − (cid:20)(cid:16) P ˜ r φ + P ˜ b P ˜ r φ (cid:17) P ˜ rP ˜ b ( φ − Nφ ) − (cid:21) . (37) Proof:
The proof is similar to that of Theorem 2, which is omitted for brevity.
Remark 5:
According to (37), we have the following observations: 1) ¯ ξ ∗ w, ul → , if P ˜ r is finitebut P ˜ b → ∞ ; and 2) ¯ ξ ∗ w, ul → , if P ˜ b is finite but P ˜ r → ∞ . C. Joint Power and Beamforming Optimization
Similar to the downlink scenario, our goal is to maximize the covert rate of Bob via jointtransmit power and passive beamforming optimization, subject to the individual transmit powerconstraints at Roy and Bob, the QoS constraint at Roy, and the minimum average detection errorprobability constraint at Willie. Accordingly, the optimization problem is formulated as(P7): max P ˜ r ,P ˜ b , Φ R a,s ˜ b ( k ) (38a)s.t. P ˜ r ≤ P max r , P ˜ b ≤ P max b , (38b) | g ra | ≥ | g ba | , (38c) R a,s ˜ r ( k ) ≥ R min a,s ˜ r ( k ) , (38d) ¯ ξ ∗ w, ul ≥ − ε, ε ∈ (0 , , (38e) ≤ ϕ n ≤ π, n = 1 , . . . , N. (38f)In (38b), P max r and P max b denote the maximum transmit powers of Roy and Bob, respectively. In(38d), R min a,s ˜ r ( k ) denotes the minimum rate requirement of Roy in uplink NOMA. To handle thehighly coupled optimization variables P ˜ r , P ˜ b , and Φ in the objective function, we decomposeproblem (P7) into a transmit power optimization problem and a passive beamforming optimiza-tion problem, and apply the alternating optimization similar to that of the downlink scenario.By fixing Φ , problem (P7) is reduced to a transmit power optimization problem as(P8): max P ˜ r ,P ˜ b R a,s ˜ b ( k ) (39a)s.t. P ˜ r ≤ P max r , P ˜ b ≤ P max b , (39b) P ˜ r | g ra | ≥ ˜ γ th P ˜ b | g ba | + ˜ γ th σ , (39c) ¯ ξ ∗ w, ul ≥ − ε, ε ∈ (0 , , (39d) where ˜ γ th = 2 R min a,s ˜ r ( k ) − . The optimal solution to problem (P8) is provided in the followingproposition. Proposition 4:
The optimal transmit power allocation coefficients to problem (P8) are givenby P ∗ ˜ r = P max r and P ∗ ˜ b = min (cid:8) ¯ ξ ∗ ( − w, ul ( ε ) , P max r | g ra | − ˜ γ th σ ˜ γ th | g ba | , P max b (cid:9) . Proof:
Similar to that in Appendix C.On the other hand, for given P ˜ r and P ˜ b , problem (P7) is reduced to a passive beamformingoptimization problem as (P9): max Φ R a,s ˜ b ( k ) (40a)s.t. | g ra | ≥ | g ba | , (40b) P ˜ r | g ra | ≥ ˜ γ th P ˜ b | g ba | + ˜ γ th σ , (40c) ≤ θ n ≤ π, ∀ n. (40d)Let w = [ w , . . . , w N ] H with w n = e jϕ n , constraint (40d) is transformed into | w n | = 1 , ∀ n .Then, we apply the change of variable ˜ Λ i = diag( h Hi ) h a √ L ( d a ) L ( d i ) , i ∈ { r, b } and define the followingauxiliary matrices: ˜ H i = (cid:20) ˜ Λ i ˜ Λ Hi ˜ Λ i v i v i ˜ Λ Hi (cid:21) and ¯ w = (cid:20) w (cid:21) . (41)We have | g ia | = | w H ˜ Λ i + v i | = ¯ w H ˜ H i ¯ w + | v i | . As a result, problem (P9) can be equivalentlyformulated as(P10): max W tr( ˜ H b W ) (42a)s.t. tr( ˜ H r W ) + | v r | ≥ tr( ˜ H b W ) + | v b | , (42b) P ˜ r (cid:0) tr( ˜ H r W ) + | v r | (cid:1) ≥ ˜ γ th P ˜ b (cid:0) tr( ˜ H b W ) + | v b | (cid:1) + ˜ γ th σ , (42c) W n,n = 1 , n = 1 , . . . , N + 1 , (42d) W = ¯ w ¯ w H (cid:23) , (42e) rank( W ) = 1 . (42f)By applying SDR to relax the rank-one constraint (42f), problem (P10) is reduced to a convexSDP, which can be efficiently solved by CVX. In addition, the Gaussian randomization is usedto extract an approximate rank-one solution. The overall algorithm for solving problem (P7) is -15 -10 -5 0 5 10 1500.250.50.751 (a) Exact, eq. (17)Simulation-15 -10 -5 0 5 10 1500.250.50.751 (b)
Exact, eq. (17)Simulation
Fig. 2. Minimum average detection probability versus theaverage transmit power in downlink, where N = 32 .
10 20 30 40 50 60 70 80 90 1000.10.20.30.40.50.60.70.80.9
Fig. 3. Minimum average detection probability versus thenumber of reflecting elements at the IRS in downlink. similar to Algorithm 1, which thus is omitted.V. N
UMERICAL S TUDIES
This section presents numerical results to evaluate the system performance achieved by theproposed IRS-assisted downlink and uplink NOMA schemes. For illustration, we assume thatAlice, Bob, Roy, Willie, and the IRS (which forms a uniform rectangular array) are located at (0 , , (100 , , (100 , , (90 , − , and (90 , in meter (m) in a two-dimensional plane, respec-tively. The effective path loss function is modeled as L ( d ) (in dB) = 35 . . d ) − G t − G r ,where G t and G r denote the transmitter and receiver antenna gains with G t = G r = 10 dBi.The variance of the AWGN is σ = − dBm. Moreover, the stopping threshold in Algorithm 1is ̺ = 10 − , and the Monte-Carlo simulations are averaged over independent trials. A. IRS-Assisted Downlink NOMA Scheme
Fig. 2 shows the downlink minimum average detection probability ¯ ξ ∗ w, dl as a function of theaverage transmit power P r for Roy’s public signal in Fig. 2(a), and the average transmit power P b for Bob’s covert signal in Fig. 2(b), respectively. It is found that ¯ ξ ∗ w, dl monotonically increaseswith P r , as seen in Fig. 2(a), but monotonically decreases with P b , as seen in Fig. 2(b), which isproved in Proposition 2. The fundamental reason is that, in the proposed scheme, Bob exploits thepublic transmission of Roy as a cover for its covert communication. An increase in P r enhancesthe strength of the cover, and in turn creates a larger uncertainty to Willie, as shown in (9).However, when P b increases, it becomes easier for Willie to detect Bob’s covert communication. Fig. 4. Covert rate versus the transmit power budget indownlink, where N = 32 .
20 30 40 50 60 70 80 90 100 110 120012345678910
Fig. 5. Covert rate versus the number of reflecting elementsat the IRS in downlink, where P max a = 25 dBm. It is also observed from the figures that by further increasing P r and P b , ¯ ξ ∗ w, dl gradually goes to1 and 0, respectively, which is consistent with our findings in Remark 3. Furthermore, in bothFigs. 2(a) and (b), the exact results in (17) match well with the simulated ones, thus verifyingthe accuracy of the derived analytical results.In Fig. 3, we plot the downlink minimum average detection probability ¯ ξ ∗ w, dl as a function ofthe number of reflecting elements N at the IRS. A general trend from the figure is that ¯ ξ ∗ w, dl increases by increasing N , which is due to the fact that an increase in N is helpful in creatinga large IRS phase-shift uncertainty to confuse Willie. It is also observed from Fig. 3 that byfixing P b , increasing P r helps in achieving a larger ¯ ξ ∗ w, dl , i.e., from Case-1 to Case-2. Under thesame total transmit power, a smaller P b yields a larger ¯ ξ ∗ w, dl , i.e., from Case-3 to Case-2.Figs. 4 and 5 plot the covert rate of Bob achieved by different schemes versus the transmitpower budget at Alice. We compare the following benchmark schemes: 1) Random phase shift:we randomly select the phase-shift values of the IRS in [0 , π ) and perform the optimal powerallocation at Alice; 2) Fixed power: we set the transmit power allocation at Alice as P b = α P max a and P r = (1 − α ) P max a with optimized phase shifts of the IRS; 3) IRS-assisted OMA; 4) NOMAwithout IRS. From both figures, it is observed that the proposed scheme significantly outperformsthe four other benchmark schemes in terms of achieving a higher covert rate. In Fig. 4, for thefixed power with an incorrect choice of P b (e.g., when α = 0 . ), the covert rate will alwaysbe zero, since the QoS constraint at Roy or the minimum average detection error probabilityconstraint at Willie is not satisfied in this case. In Fig. 5, the covert rate of the random phaseshift increases much more slowly than that of the proposed scheme as N increases, due to the -30 -25 -20 -15 -10 -5 0 5 1000.250.50.751 (b) Exact, eq. (37)Simulation-10 -5 0 5 10 15 20 25 3000.250.50.751 (a)
Exact, eq. (37)Simulation
Fig. 6. Minimum average detection probability versus theaverage transmit power in uplink, where N = 32 .
10 20 30 40 50 60 70 80 90 10000.050.10.150.20.250.30.350.40.450.5
Fig. 7. Minimum average detection probability versus thenumber of reflecting elements at the IRS in uplink. passive beamforming gain loss with random phase shifts. All these observations demonstrate theeffectiveness of the joint power and beamforming design, that can dynamically adjust Alice’stransmit power and the IRS’s phase shifts to meet the QoS requirement and the covertnessconstraint, to maximize the covert rate at Bob. It can be also seen from Fig. 4 that the covertrate of the proposed scheme decreases as ε decreases or R min r,s r ( k ) increases. The main reason isthat, the covertness constraint at Willie and the QoS constraint at Roy become more stringentwith the decreased ε and the increased R min r,s r ( k ) , which results in a smaller covert rate for Bob.Furthermore, the IRS-assisted OMA and NOMA without IRS schemes always achieve the zerocovert rate, which confirms our findings in Remark 2. B. IRS-Assisted Uplink NOMA Scheme
In Fig. 6, the uplink minimum average detection probability ¯ ξ ∗ w, ul versus the average transmitpower of Roy P ˜ r and Bob P ˜ b . From Fig. 6(a) and (b), it is seen that ¯ ξ ∗ w, ul monotonically increaseswith P ˜ r while decreasing with P ˜ b , for which the reason is similar to that of the downlink NOMAscenario. Furthermore, by increasing P ˜ r and P ˜ b , ¯ ξ ∗ w, ul approaches 1 and 0, respectively, whichconfirms our results in Remark 5. In addition, the accuracy of the derived analytical results inTheorem 4 is also evaluated in these two subfigures.Fig. 7 shows the uplink minimum average detection probability ¯ ξ ∗ w, ul versus the number ofreflecting elements N at the IRS. As expected, ¯ ξ ∗ w, ul increases with N due to the significantphase-shift uncertainty of the IRS with a large N . It is also seen from this figure that a higher P ˜ r or a lower P ˜ b is beneficial to confound Willie, which yields a larger ¯ ξ ∗ w, ul . Fig. 8. Covert rate versus the transmit power budget in uplink,where N = 32 and P max0 = P max r = P max b .
20 30 40 50 60 70 80 90 100 110 120012345678
Fig. 9. Covert rate versus the number of reflecting elementsat the IRS in uplink, where P max r = P max b = 20 dBm. Figs. 8 and 9 show the uplink covert rate of Bob achieved by different schemes versus thecommon transmit power budget P max0 , where the random phase shifts, the fixed power, IRS-assisted OMA, and NOMA without IRS are used as the benchmarks. As expected, the proposedscheme achieves a remarkable covert rate gain than that achieved by the benchmark schemesas P max0 and N increase. Specifically, with the same covertness constraint ε = 0 . , the covertrate of the proposed scheme with R min r,s ˜ r ( k ) = 2 approaches that with R min r,s ˜ r ( k ) = 1 in the high P max0 regime, as seen in Fig. 8. The main reason is that, as P max0 becomes sufficiently large, theoptimal transmit power of Bob mainly depends on ¯ ξ ∗ ( − w, ul ( ε ) , thus yielding the same covert rate. Inaddition, as shown in Fig. 9, the random phase shift outperforms the fixed power in terms of thecovert rate with a large N , e.g., N = 120 . The underlying reason is that when N is sufficientlylarge but P max r is finite, the transmit power of Bob is determined by P max r | g ra | − ˜ γ th σ ˜ γ th | g ba | , as indicatedby Proposition 4. In this case, the transmit power allocation at Roy and Bob dominates the rateperformance.Fig. 10 demonstrates how individual transmit power budgets affect the covert rate of Bob.As seen from Fig. 10(a), by fixing P max b , the covert rate initially increases with P max r and thenreaches a plateau by further increasing P max r . This is due to the fact that the transmit powerof Bob is determined only by P max b with a large P max r , as implied in Proposition 4. A similarphenomenon can be observed from Fig. 10(b) as we fix P max r but increase P max b . (a) -10 -5 0 5 10 15 200123 (b) Fig. 10. Covert rate versus the individual transmit power budgets at Roy and Bob, where N = 32 . VI. C
ONCLUSION
In this paper, we proposed novel IRS-assisted NOMA schemes for improving the performanceof covert wireless communications, where the phase-shift uncertainty of the IRS and the non-orthogonal transmission of Roy are exploited as the new cover medium. Considering the worst-case scenario where Willie can optimally choose his detection threshold, we analytically derivedthe minimum average detection error probability of Willie to facilitate the performance evaluation.Furthermore, to maximize the covert rates of Bob, the transmit power and the IRS passivebeamforming were jointly optimized, subject to the minimum covertness requirement of Willieand the QoS requirement of Roy. Simulation results revealed that the proposed schemes canguarantee covert communications with positive covert rates and significantly outperform otherbenchmark schemes. Furthermore, it was found that increasing the transmit power for Roy’spublic signal and the number of reflecting elements at the IRS are beneficial to the covertcommunication performance. A
PPENDIX
A: P
ROOF OF L EMMA ψ n = ψ n, + ψ n, , n = 1 , . . . , N , where ψ n, = θ ∗ n + arg( h an ) and ψ n, = arg( h wn ) .According to [39], we know that ψ n , ψ n, , and ψ n, have a uniform circular distribution. Bydefinition, a circular distribution is a probability distribution of a RV whose values are angles,usually taken to be in the range [0 , π ) . Furthermore, since ψ n, is uniformly distributed on [0 , π ) , let ψ n, = x , we obtain that ψ n = x + ψ n, is uniformly distributed on [ x , x + 2 π ) =[ x , π ) ∪ [2 π, x + 2 π ) ( i ) = [ x , π ) ∪ [0 , x ) = [0 , π ) , where step (i) follows from the fact that ψ n is circularly distributed. Therefore, we prove that ψ n is uniformly distributed on [0 , π ) . On the other hand, for n = 1 , . . . , N , we know that ψ n, is independently distributed while ψ n, is not, because for different n , θ ∗ n in ψ n, is a function that is related to the same arg( h ab ) and arg( h ar ) . Thus, to prove the independence of ψ n , it is necessary to show that ψ n is independentof ψ n, . To this end, we express the joint PDF of ψ n and ψ n, as f ψ n, ,ψ n ( x , x ) = f ψ n, ,ψ n, ( x , x ) | J ψ n, ,ψ n ( x , x ) | , (43)where J ψ n, ,ψ n ( x , x ) is the Jacobian matrix. Utilizing the fact that ψ n, is independent of ψ n, and is uniformly distributed on [0 , π ) , we obtain that f ψ n, ,ψ n, ( x , x ) = f ψ n, ( x ) f ψ n, ( x ) = π f ψ n, ( x ) . Furthermore, the Jacobian matrix J ψ n, ,ψ n ( x , x ) can be derived as J ψ n, ,ψ n ( x , x ) = " ∂x ∂x ∂x ∂x ∂x∂x ∂x∂x = (cid:20) (cid:21) , (44)such that | J ψ n, ,ψ n ( x , x ) | = 1 . Substituting this result into (43), we have f ψ n, ,ψ n ( x , x ) = π f ψ n, ( x ) = f ψ n ( x ) f ψ n, ( x ) , where the last equality is based on the fact that ψ n is uniformlydistributed on [0 , π ) . Hence, we conclude that ψ n is independent of ψ n, .Summarizing the aforementioned results, Lemma 1 is proved immediately.A PPENDIX
B: P
ROOF OF T HEOREM ξ w, dl in the following three regions of τ dl : • If τ dl < σ + P r | h aw | L ( d aw ) , we show that ξ w, dl is always equal to 1 and cannot be optimized,which is the worst-case detection performance of Willie. Therefore, the optimal τ dl doesnot exist in this case. • If σ + P r | h aw | L ( d aw ) ≤ τ dl ≤ σ + ( P r + P b ) | h aw | L ( d aw ) , it is not difficult to verify that ξ w, dl is amonotonically decreasing function with τ dl , and thus, the optimal solution that minimizes ξ w, dl is τ ∗ dl = σ + ( P r + P b ) | h aw | L ( d aw ) . • If τ dl > σ + ( P r + P b ) | h aw | L ( d aw ) , we derive the first derivative of ξ w, dl with respect to τ dl as ∂ξ w, dl ∂τ dl = e ( Pr + Pb ) | haw | /L ( daw )+ σ − τ dl ( Pr + Pb ) N/ ( L ( da ) L ( dw )) ( P r + P b ) N/ ( L ( d a ) L ( d w )) − e Pr | haw | /L ( daw )+ σ − τ dl PrN/ ( L ( da ) L ( dw )) P r N/ ( L ( d a ) L ( d w )) . (45)By setting ∂ξ w, dl ∂τ dl = 0 , the optimal solution is obtained as τ † dl = σ + P r ( P r + P b ) NP b L ( d a ) L ( d w ) ln (cid:16) P r + P b P r (cid:17) . (46) Using results in (45) and (46), the monotonicity of ξ w, dl is discussed as follows: 1) If τ † dl ≥ σ + ( P r + P b ) | h aw | L ( d aw ) , we can prove that ∂ξ w, dl ∂τ dl < for τ dl ∈ (cid:0) σ + ( P r + P b ) | h aw | L ( d aw ) , τ † dl (cid:1) while ∂ξ w, dl ∂τ dl > for τ dl ∈ (cid:0) τ † dl , + ∞ (cid:1) . This implies that ξ w, dl first decreases and then increaseswith respect to τ dl , and the optimal solution for minimizing ξ w, dl is given by τ ∗ dl = τ † dl . 2)If τ † dl < σ + ( P r + P b ) | h aw | L ( d aw ) , it shows that ξ w, dl is monotonically increasing with respect to τ dl since ∂ξ w, dl ∂τ dl > for τ dl ∈ (cid:0) σ + ( P r + P b ) | h aw | L ( d aw ) , + ∞ (cid:1) . In addition, we know that ξ w, dl is acontinuous function of τ dl . Thus, the optimal solution is τ ∗ dl = σ + ( P r + P b ) | h aw | L ( d aw ) .Summarizing the above results, we prove Theorem 1.A PPENDIX
C: P
ROOF OF P ROPOSITIONS AND P r + P b ≤ P max does not hold, i.e., P r + P b < P max . Then, by fixing P r , we scale up P b by a factor η ( η > ) to make P r + P b = P max hold. In this case, we can always obtain a larger objective value of (P2) since R b,s b ( k ) is amonotonically increasing function with respect to P b . This contradicts the original assumptionof P r + P b < P max .Next, our attention is shifted to the proof of Proposition 2. Let P r = P max − P b and takethe first derivative of ¯ ξ ∗ w, dl with respect to P b , we can verify that ∂ ¯ ξ ∗ w, dl /∂P b < . Finally, wecomplete the proof. A PPENDIX
D: P
ROOF OF P ROPOSITION P r ( t + 1) and P b ( t + 1) arederived for given Θ ( t ) , and thus, we have the following inequality R b,s b ( k ) (cid:0) Θ ( t ) , P b ( t ) , P r ( t ) (cid:1) ≤ R b,s b ( k ) (cid:0) Θ ( t ) , P b ( t + 1) , P r ( t + 1) (cid:1) . (47)Then, as shown in step 4 of Algorithm 1, the optimal solution Θ ( t + 1) is obtained for given P r ( t + 1) and P b ( t + 1) , and thus, the following inequality holds R b,s b ( k ) (cid:0) Θ ( t ) , P b ( t + 1) , P r ( t + 1) (cid:1) ≤ R b,s b ( k ) (cid:0) Θ ( t + 1) , P b ( t + 1) , P r ( t + 1) (cid:1) . (48)By summarizing (47) and (48), we can readily obtain that R b,s b ( k ) (cid:0) Θ ( t ) , P b ( t ) , P r ( t ) (cid:1) ≤ R b,s b ( k ) (cid:0) Θ ( t + 1) , P b ( t + 1) , P r ( t + 1) (cid:1) . (49) This indicates that the objective value of problem (P1) is non-decreasing over the iterations. Onthe other hand, the objective value of problem (P1) clearly has an upper bound for any choiceof feasible Θ and P max a . Hence, Algorithm 1 is proved to converge.R EFERENCES [1] J. Zhang, E. Bj ¨ ornson, M. Matthaiou, D. W. K. Ng, H. Yang, and D. J. Love, “Prospective multiple antenna technologiesfor beyond 5G,” IEEE J. Sel. Areas Commun. , vol. 38, no. 8, pp. 1637–1660, Aug. 2020.[2] Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network,”
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