Space Shift Keying with Reconfigurable Intelligent Surfaces: Phase Configuration Designs and Performance Analysis
Qiang Li, Miaowen Wen, Shuai Wang, George C. Alexandropoulos, Yik-Chung Wu
aa r X i v : . [ c s . I T ] F e b Space Shift Keying with Reconfigurable IntelligentSurfaces: Phase Configuration Designs andPerformance Analysis
Qiang Li,
Member, IEEE , Miaowen Wen,
Senior Member, IEEE , Shuai Wang,
Member, IEEE ,George C. Alexandropoulos,
Senior Member, IEEE , and Yik-Chung Wu,
Senior Member, IEEE
Abstract —Reconfigurable intelligent surface (RIS)-assistedtransmission and space shift keying (SSK) appear as promisingcandidates for future energy-efficient wireless systems. In thispaper, two RIS-based SSK schemes are proposed to efficiently im-prove the error and throughput performance of conventional SSKsystems, respectively. The first one, termed RIS-SSK with passivebeamforming (RIS-SSK-PB), employs an RIS for beamformingand targets the maximization of the minimum squared Euclideandistance between any two decision points. The second one,termed RIS-SSK with Alamouti space-time block coding (RIS-SSK-ASTBC), employs an RIS for ASTBC and enables the RISto transmit its own Alamouti-coded information while reflectingthe incident SSK signals to the destination. A low-complexitybeamformer and an efficient maximum-likelihood (ML) detectorare designed for RIS-SSK-PB and RIS-SSK-ASTBC, respectively.Approximate expressions for the average bit error probabilitiesof the source and/or the RIS are derived in closed-form assumingML detection. Extensive computer simulations are conducted toverify the performance analysis. Results show that RIS-SSK-PB significantly outperforms the existing RIS-free and RIS-based SSK schemes, and RIS-SSK-ASTBC enables highly reliabletransmission with throughput improvement.
Index Terms —Alamouti code, passive beamforming, perfor-mance analysis, reconfigurable intelligent surface, space shiftkeying.
I. I
NTRODUCTION T HE deployment of the fifth generation (5G) cellularnetworks is accelerating across the world. This wireless
The work was supported in part by the National Natural Science Foundationof China under Grant 61871190 and Grant 62001203, in part by the NaturalScience Foundation of Guangdong Province under Grant 2018B030306005and Grant 2020A1515110470, in part by the Pearl River Nova Program ofGuangzhou under Grant 201806010171, and in part by the FundamentalResearch Funds for the Central Universities under Grant 2019SJ02. (Cor-responding author: Miaowen Wen.)
Qiang Li is with the Department of Electronic Engineering, College ofInformation Science and Technology, Jinan University, Guangzhou 510632,China (e-mail: [email protected]).Miaowen Wen is with the School of Electronic and Information Engineer-ing, South China University of Technology, Guangzhou 510640, China (e-mail: [email protected]).Shuai Wang is with the Department of Electrical and Electronic Engineer-ing, and the Department of Computer Science and Engineering, SouthernUniversity of Science and Technology, Shenzhen 518055, China, and also withthe Sifakis Research Institute of Trustworthy Autonomous Systems, Shenzhen518055, China (e-mail: [email protected]).George C. Alexandropoulos is with the Department of Informaticsand Telecommunications, National and Kapodistrian University of Athens,Panepistimiopolis Ilissia, 15784 Athens, Greece (e-mail: [email protected]).Yik-Chung Wu is with the Department of Electrical and Elec-tronic Engineering, The University of Hong Kong, Hong Kong (e-mail:[email protected]). communication standard is expected to support lots of newapplications and services, which require various enablingtechnologies. Among these, energy-efficient transmission is akey enabler for energy-constrained networks, such as Internetof Things. However, the traditional digital modulation schemesthat alter the amplitude, phase, and/or the frequency of asinusoidal carrier signal often involve complicated operations,such as mixing and filtering.Recently, the concept of index modulation (IM) that lever-ages upon the ON/OFF state of the transmission entities toconvey information has created completely new dimensions forenergy-efficient transmission [1]–[4]. As a prominent memberof IM, spatial modulation (SM) uses the transmit antennasof a multiple-input multiple-output (MIMO) system in aninnovative fashion [5]–[7]. In SM, only a single transmitantenna is activated for each transmission of a constellationsymbol, and the index of the active antenna is used to conveyextra information bits. By limiting the active antenna to simplytransmit an unmodulated sinusoidal carrier signal, SM evolvesinto space shift keying (SSK) that employs the index ofthe active antenna only for information transfer [8]. Sincethe traditional signal modulation is avoided, SSK achieveshigh energy efficiency and enjoys low transceiver complexity.Therefore, SSK is a promising candidate for future energy-efficient wireless communications. However, SSK still faceschallenges. On one hand, SSK-based solutions are basedon the fact that different active antennas lead to differentchannel realizations, whereas they cannot configure the wire-less propagation environment itself. The system performanceof SSK highly depends on the distinctness of the channelsignatures associated with different active antennas. Hence,rich scattering in the propagation environment and/or a largenumber of receive antennas are required for SSK to avoid poorerror performance. Unfortunately, these requirements may notbe satisfied in general. On the other hand, the sinusoidal carriersignal itself transmitted from the active antenna does not carryany information, resulting in low system throughput. For SSK,embedding information in the carrier signal while avoidingtraditional digital modulation is still an open issue.The emerging technology of reconfigurable intelligent sur-faces (RISs) has also received a lot of research interests dueto the distinctive capability of turning an uncontrollable andunfavorable environment into a controllable and benign entity[9]–[11]. An RIS is a man-made planar surface consisting ofa large number of reflecting elements. Each of those elements is able to reflect the incident signals by applying an adjustablephase shift. This dynamically adjustable reflection, that goesbeyond Snell’s law, is accomplished without any form ofthe conventional power amplifiers, since no new signal isgenerated at the RIS side, and with no contamination of theimpinging signal with reception thermal noise [12]. In the liter-ature, an RIS is often deployed to modify channel links and/orto transmit its own information [13]. Specifically, to modifychannel links, the RIS-based dual-hop scheme was proposedin [14] for single-input single-output systems, where the RISis adjusted to maximize the instantaneous receive signal-to-noise ratio (SNR). In [15], an RIS was utilized to minimizethe symbol error rate of precoded spatial multiplexing MIMOsystems via the joint reflecting and precoding optimization. Acosine similarity-based low-complexity reflecting optimizationwas proposed in [16] for RIS-assisted spatial multiplexingMIMO systems by maximizing the overall channel gain. Theauthors in [17] employed an RIS to enhance the secrecy rateof MIMO transmission in the presence of an eavesdropperby jointly optimizing the transmit covariance and reflectioncoefficient matrices. References [18]–[21] applied the RIStechnology to multiuser downlink multiple-input single-outputsystems. In [18] and [19], the transmit power was minimizedvia the joint optimization of the active beamforming at thebase station and the passive beamforming (PB) at the RIS,by allowing continuous and discrete reflection coefficients,respectively. Aiming to maximize the energy efficiency, thetransmit power for each user and the reflection coefficientswith both infinite [20] and realistic low-resolution [21] phaseconfiguration codebooks were jointly optimized. Due to theattractive advantages, researchers are also utilizing RISs toenhance other existing techniques, such as non-orthogonalmultiple access [22] and orthogonal frequency division mul-tiplexing [23]. As the authors in [24] applied reconfigurableantennas to SSK, the authors in [25] proposed the intelligentRIS-SSK, which makes real-time adjustments to the reflectioncoefficients for maximizing the instantaneous receive SNRprovided that the active antenna index is known to the RISper transmission.Besides solely acting as a signal reflector, an RIS can beemployed for information transfer by adjusting the reflectioncoefficients [26]. For an RIS-aided channel, the capacity-achieving scheme was demonstrated to jointly encode infor-mation in the RIS configuration as well as in the transmittedsignal [27]. Both phase shift keying (PSK) and quadratureamplitude modulation can be achieved at each reflectingelement, thus providing a new design of MIMO transmission[28]. In [29], the reflecting modulation that uses the indices ofreflecting patterns for information embedding was introducedinto traditional MIMO frameworks. In [30] and [31], the SMprinciple was applied to an RIS, i.e., the information is carriedby the ON/OFF status of reflecting elements. In [32], the PBwas performed at the RIS to steer the signal towards a certainreceive antenna, thus using the index of the selected receiveantenna to convey information. Based on [32], [33] furtherapplied SM to the RF side, leading to even higher systemthroughput. Similar to the antenna-based MIMO, Alamoutispace-time block coding (ASTBC) and Vertical Bell Labs layered space-time (VBLAST) schemes was implemented withan RIS in [34].Motivated by the challenges faced by SSK and the re-cently identified potential of the amalgamation of RISs andSSK, we study RIS-based SSK in this paper towards fu-ture energy-efficient wireless communications. Two RIS-basedSSK schemes are proposed to improve the error and through-put performance of conventional SSK systems, respectively.The contributions of this paper are summarized as follows. • The first proposed scheme, which is called RIS-SSK withPB (RIS-SSK-PB), employs an RIS for PB and targetsthe maximization of the minimum squared Euclideandistance between any two decision points. Compared withthe intelligent RIS-SSK scheme in [25], RIS-SSK-PBachieves the following advantages: 1) the knowledge ofthe active antenna index is not required any more at theRIS; 2) the reflection coefficients are not adjusted onlineper transmission any more; and 3) higher diversity gainscan be obtained with the same detection complexity. Wedevise a semi-definite relaxation (SDR) method for theconsidered RIS reflection optimization problem as well asa low-complexity algorithm. We analyze the bit error rate(BER) performance of RIS-SSK-PB over Rayleigh fadingchannels assuming two transmit antennas and maximum-likelihood (ML) detection. An approximate expression forthe average bit error probability (ABEP) of RIS-SSK-PBis also derived in closed-form. • The second proposed scheme, which is called RIS-SSK with ASTBC (RIS-SSK-ASTBC), employs an RISfor ASTBC and enables the RIS to transmit its ownAlamouti-coded information, while reflecting the inci-dent SSK signals to the destination. We design a low-complexity ML detector for RIS-SSK-ASTBC. The BERperformance of RIS-SSK-ASTBC with ML detectionover Rayleigh fading channels is analyzed. Approximateand asymptotic ABEP expressions are derived in closed-form for the source and the RIS, which reveal that theirinformation bits have two-diversity-order protection withincreasing SNR or the number of reflecting elements.The rest of this paper is organized as follows. Thetransceiver structure and the performance analysis of RIS-SSK-PB are presented in Section II. Section III describes thescheme of RIS-SSK-ASTBC, including the system model andperformance analysis. Section IV presents computer simula-tion results, followed by the conclusion in Section V.
Notation:
Column vectors and matrices are in the form oflowercase and capital bold letters, respectively. Superscripts ∗ , T , and H stand for conjugate, transpose, and Hermitiantranspose, respectively. j = √− denotes the imaginary unit. tr( · ) and rank( · ) return the trace and rank of a matrix, re-spectively. diag( · ) transforms a vector into a diagonal matrix. ( C ) N ( µ, σ ) represents the (complex) Gaussian distributionwith mean µ and variance σ . The probability of an event andthe probability density function (PDF) of a random variableare denoted by Pr( · ) and f ( · ) , respectively. Q ( · ) representsthe Gaussian Q -function. E {·} and V ar {·} denote expectationand variance, respectively. |·| , ∠ , and ℜ{·} denote the absolute, (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:7)(cid:8)(cid:7) (cid:1) (cid:2)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:13)(cid:6)(cid:12)(cid:11)(cid:10)(cid:14)(cid:15)(cid:11)(cid:10)(cid:2)(cid:14) (cid:1) (cid:1) (cid:16)(cid:17)(cid:1) (cid:16)(cid:17)(cid:1)(cid:8)(cid:5)(cid:2)(cid:14)(cid:11)(cid:4)(cid:2)(cid:18)(cid:18)(cid:6)(cid:4) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:7)(cid:8)(cid:7) (cid:2)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:13)(cid:6)(cid:12)(cid:11)(cid:10)(cid:14)(cid:15)(cid:11)(cid:10)(cid:2)(cid:14) (cid:16)(cid:17)(cid:1) (cid:4)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:12)(cid:3)(cid:9)(cid:19)(cid:12)(cid:3)(cid:4)(cid:20)(cid:15)(cid:5)(cid:6)(cid:8)(cid:21)(cid:22)(cid:12)(cid:3)(cid:9)(cid:19)(cid:12)(cid:3)(cid:4)(cid:20)(cid:15)(cid:5)(cid:6)(cid:8)(cid:21)(cid:23)(cid:16)(cid:17)(cid:1)(cid:8)(cid:5)(cid:2)(cid:14)(cid:11)(cid:4)(cid:2)(cid:18)(cid:18)(cid:6)(cid:4)
Fig. 1. System model of RIS-SSK-PB. phase, and real part of a complex number, respectively. f i denotes the i -th entry of f , g l represents the l -th column of G , and g il stands for the ( i, l ) -th element of G . k · k denotesthe Frobenius norm. V (cid:23) means that V is positive semi-definite. II. T HE RIS-SSK-PB S
CHEME
In this section, we study the proposed RIS-SSK-PB scheme,in which the RIS merely reflects the incident SSK signalswithout transmitting its own information.
A. System Model
Fig. 1 depicts the system model of RIS-SSK-PB, whichconsists of a source, a destination, and an RIS. Due to anobstacle, the source and the destination communicate witheach other through the RIS. The source is equipped with N t transmit antennas, and the destination adopts single-antennareception for a simple receiver structure. The RIS is connectedto a controller and made up of N reflecting elements that aredeployed on a two-dimensional rectangular grid. The channelmatrices of the source-to-RIS and RIS-to-destination linksare denoted by G ∈ C N × N t and f ∈ C N × , respectively.Specifically, g il represents the channel coefficient between the l -th transmit antenna and the i -th reflecting element, while f i isthe channel coefficient between the i -th reflecting element andthe destination, where i ∈ { , . . . , N } and l ∈ { , . . . , N t } .The reflecting elements are assumed placed with horizontaland vertical inter-element distances equal or greater thanhalf the signal wavelength. In this special case, the fadingchannels can be approximately considered as independent andidentically distributed (i.i.d.) fading, i.e., g il and f i are i.i.d.complex Gaussian random variables following the distribution CN (0 , for i = 1 , . . . , N and l = 1 , . . . , N t [20], [25].We note that the practical case of correlated fading [35] willbe considered in future work. The reflection coefficient forthe i -th reflecting element of the RIS can be expressed as φ i = β i exp( jθ i ) , where β i ∈ [0 , and θ i ∈ [0 , π ) arethe amplitude coefficient and phase shift, respectively. Tocharacterize the performance limit of RIS-SSK-PB, we assumethat β i = 1 , θ i can be continuously varied in [0 , π ) for i = 1 , . . . , N , and the channel state information (CSI) of thesource-to-RIS and RIS-to-destination links is perfectly knownto the RIS and the destination. Note that the channel estimationin RIS-empowered wireless communications is a challengingproblem, however, with encouraging approaches already [23],[36].For each transmission, the source encodes b S bits into anindex l ∈ { , . . . , N t } , and then emits an unmodulated carriersignal from the antenna with the index l towards the RIS,resulting in the SSK signal x = (cid:2) , . . . , , | {z } l − , , . . . , | {z } N t − l (cid:3) T , (1)where b S = log ( N t ) , and N t is assumed to be an arbitraryinteger power of two, such that l can be easily obtained byconverting the b S bits into the decimal representation. TheRIS reflects the incident signal with reflection vector φ =[exp( jθ ) , . . . , exp( jθ N )] T ∈ C N × , leading to the receivedbaseband signal at the destination as follows: y = f T ΦGx + w = f T Φg l + w, (2)where Φ = diag( φ ) and w is the additive white Gaussiannoise (AWGN) at the destination, which follows the distribu-tion CN (0 , N ) . We define the transmit signal-to-noise ratio(SNR) as ρ = 1 /N . With (2), the destination adopts MLdetection to decode l , namely ˆ l = arg min l (cid:12)(cid:12) y − f T Φg l (cid:12)(cid:12) , (3)where ˆ l is the estimate of l . The destination then recovers b S information bits from ˆ l . We observe from (3) that the value of φ undoubtedly influences the error performance. In the nextsubsection, we will focus on the optimization of φ . B. Reflecting Optimization
In the intelligent RIS-SSK scheme of [25], φ is optimizedby maximizing the instantaneous receive SNR at the desti-nation. More specifically, the received signal in (2) can berewritten as y = N X i =1 f i g il exp ( jθ i ) + w. (4)Obviously, the instantaneous receive SNR is given by ρ | P Ni =1 f i g il exp( jθ i ) | , which is maximized when θ i = − ∠ f i − ∠ g il . This optimization criterion, however, requiresthat the knowledge of the active antenna index be availableat the RIS in real time, and that φ be adjusted online pertransmission even though the CSI remains unchanged, whichis unrealistic. Moreover, since the BER performance of SSKhighly depends on the channel difference associated with dif-ferent active transmit antennas [5], [6], maximizing the receiveSNR may not lead to good performance. These observationsmotivate us to propose other optimization algorithms. Let us first examine the conditional pairwise error proba-bility (PEP), which is defined as the probability of detecting l incorrectly as ˆ l conditioned on G , f , and φ , namely Pr (cid:16) l → ˆ l | G , f , φ (cid:17) = Pr (cid:16)(cid:12)(cid:12) y − f T Φg l (cid:12)(cid:12) > (cid:12)(cid:12) y − f T Φg ˆ l (cid:12)(cid:12) (cid:17) = Pr (cid:16) − (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) − ℜ (cid:8) w ∗ (cid:2) f T Φ ( g l − g ˆ l ) (cid:3)(cid:9) > (cid:17) = Pr ( D > , (5)where D is a Gaussian random variable with E { D } = − (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) , (6) V ar { D } = 2 N (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) . (7)Hence, we have Pr (cid:16) l → ˆ l | G , f , φ (cid:17) = Q s ρ (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) . (8)Then, according to the union bounding technique [37], anupper bound on the instantaneous error probability of activeantenna index detection can be expressed as P e ≤ N t N t X l =1 N t X ˆ l =1 l< ˆ l Pr (cid:16) l → ˆ l | G , f , φ (cid:17) ≤ ( N t − · Q (cid:18)r ρ d min (cid:19) , (9)where d min is defined as d min = min l, ˆ ll =ˆ l (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) . (10)Therefore, from (9), we propose to optimize φ by maximizing d min , namely (P1) : max φ d min (11) s.t. θ i ∈ [0 , π ) , i = 1 , . . . , N. (12)For N t = 2 , the optimal solution to problem (P1) can be easilyderived in closed-form as φ opt = arg max φ (cid:12)(cid:12) f T Φ ( g − g ) (cid:12)(cid:12) = arg max φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 f i ( g i − g i ) exp ( jθ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = [ − ∠ f − ∠ ( g − g ) ,. . . , − ∠ f N − ∠ ( g N − g N )] T . (13)Unfortunately, for N t > , it is not an easy task to solveproblem (P1) optimally due to the max-min criterion of (11).Therefore, we first apply the SDR method and then propose alow-complexity algorithm to solve problem (P1) sub-optimallyin the following.
1) SDR Method:
For SDR, by introducing an auxiliaryvariable t , problem (P1) can be equivalently reformulated as (P2) : max φ t (14) s.t. (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) ≥ t, ∀ l, ˆ l = 1 , . . . , N t , l = ˆ l, (15) θ i ∈ [0 , π ) , ∀ i = 1 , . . . , N. (16)However, problem (P2) is still non-convex due to the constraintin (15). Actually, the term on the left-hand side of (15) canbe expressed as (cid:12)(cid:12) f T Φ (cid:0) g l − g ˆ l (cid:1)(cid:12)(cid:12) = (cid:12)(cid:12) f T diag (cid:0) g l − g ˆ l (cid:1) φ (cid:12)(cid:12) = φ H R φ = tr (cid:0) R φφ H (cid:1) , (17)where R = ( f T diag( g l − g ˆ l )) H ( f T diag( g l − g ˆ l )) . Define V = φφ H with V (cid:23) and rank( V ) = 1 . Since the rank-oneconstraint is non-convex, we use SDR to relax this constraint.As a result, problem (P2) is reduced to (P3) : max V t (18) s.t. tr ( RV ) ≥ t, ∀ l, ˆ l = 1 , . . . , N t , l = ˆ l, (19) V ii = 1 , ∀ i = 1 , . . . , N, (20) V (cid:23) . (21)Obviously, problem (P3) is a convex semi-definite pro-gram and can be solved by existing convex optimizationsolvers, such as CVX [38]. However, the rank-one constraintis relaxed in problem (P3), such that the solution may notsatisfy rank( V ) = 1 . This implies that the optimal objectivevalue of problem (P3) is an upper bound on that of problem(P1). Therefore, extra steps are required to obtain a rank-onesolution from the solution to problem (P3). Specifically, aftertaking the eigenvalue decomposition of V as V = UΣU H ,where U ∈ C N × N and Σ ∈ C N × N are a unitary matrixand a diagonal matrix, respectively, we can obtain a sub-optimal solution to problem (P3) as φ = UΣ / r , where r ∈ C N × is a random vector with each element drawnfrom CN (0 , . Further, by generating a sufficiently largenumber of realizations of r and selecting the one, denoted by ˜ r , that maximizes d min , the sub-optimal solution to problem(P1) can be derived as φ = [ ˜ φ / | ˜ φ | , . . . , ˜ φ N / | ˜ φ N | ] T , where [ ˜ φ , . . . , ˜ φ N ] T = UΣ / ˜ r .
2) Low-Complexity Algorithm:
To achieve lower complex-ity than the SDR method, we propose a low-complexityalgorithm here. The core idea is to derive a set of φ , denotedby A , from which the φ capable of maximize d min in (10)is selected as the solution to problem (P1). Obviously, A isthe key. Inspired by the solution to the case of N t = 2 , aheuristic A with N t ( N t − / entities can be constructed by φ ij = [ − ∠ f − ∠ ( g i − g j ) , . . . , − ∠ f N − ∠ ( g Ni − g Nj )] T ,where i, j ∈ { , . . . , N t } , i < j . Note that since the optimal φ may not be included in A , the low-complexity algorithm issuboptimal. However, as will be shown in Section V, comparedwith the SDR method, this algorithm can achieve satisfyingperformance with much reduced complexity. C. Performance Analysis for N t = 2 Since the reflection matrix Φ is directly correlated with thechannel matrices f and G , as seen from Section II.B, it is verydifficult to derive the distribution of the composite channelmatrix f T ΦG and analyze the BER performance of RIS-SSK-PB for a general case of N t . Therefore, we focus only on theperformance analysis for N t = 2 to obtain insights.For N t = 2 , the optimal φ is given in (13). From (8), theinstantaneous BEP of RIS-SSK-PB can be expressed as P bi = Q s ρ | v | , (22)where v = P Ni =1 | f i | | g i − g i | . Under the assumption of N ≫ , and according to the central limit theorem (CLT) [39], v can be regarded as a Gaussian random variable followingthe distribution N ( µ v , σ v ) , where µ v and σ v can be derivedas follows: µ v = √ πN, σ v = (cid:18) − π (cid:19) N. (23)Hence, | v | is a noncentral chi-square random variable withone degree of freedom and noncentrality parameter a = π N / , whose moment generating function is given by Ψ | v | ( s ) = (cid:18) − σ v s (cid:19) / exp (cid:18) a s − σ v s (cid:19) . (24)Then, the ABEP of RIS-SSK-PB with N t = 2 can be obtainedby averaging P bi over | v | , namely P b = E | v | { P bi } ≈
112 Ψ | v | (cid:16) − ρ (cid:17) + 14 Ψ | v | (cid:16) − ρ (cid:17) , = 112 (cid:18)
22 + σ v ρ (cid:19) / exp (cid:18) − a ρ σ v ρ (cid:19) + 14 (cid:18)
33 + 2 σ v ρ (cid:19) / exp (cid:18) − a ρ σ v ρ (cid:19) , (25)where the approximation results from [40] Q ( x ) ≈
112 exp (cid:18) − x (cid:19) + 14 exp (cid:18) − x (cid:19) . (26)Since σ v and a grows linearly and quadratically with increas-ing N , respectively, we observe from P b that the performanceimprovement achieved by doubling N is greater than that bydoubling ρ , i.e., doubling N results in SNR gains greater than3 dB. Moreover, as exp( − x ) is a concave function, the SNRgains achieved by doubling N are increasingly small. Remark 1:
Since the proposed reflecting optimization isaimed to maximize the minimum squared Euclidean distancebetween any two decision points, RIS-SSK-PB is expectedto outperform the intelligent RIS-SSK in terms of BER.Moreover, the instantaneous knowledge of the active antennaindex is not required at the RIS, and the reflection coefficientsare not adjusted online per transmission, but only whenany of the channel matrices change, for RIS-SSK-PB. Thedrawback of RIS-SSK-PB is that the reflecting optimizationinvolves high computational complexity in comparison withthe intelligent RIS-SSK. Fortunately, the operation can be (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:7)(cid:8)(cid:7) (cid:2)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:13)(cid:6)(cid:12)(cid:11)(cid:10)(cid:14)(cid:15)(cid:11)(cid:10)(cid:2)(cid:14) (cid:16)(cid:17)(cid:1) (cid:16)(cid:17)(cid:1)(cid:8)(cid:5)(cid:2)(cid:14)(cid:11)(cid:4)(cid:2)(cid:18)(cid:18)(cid:6)(cid:4) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:7)(cid:8)(cid:7)(cid:8)(cid:7) (cid:1) (cid:2)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:13)(cid:6)(cid:12)(cid:11)(cid:10)(cid:14)(cid:15)(cid:11)(cid:10)(cid:2)(cid:14) (cid:1) (cid:1) (cid:16)(cid:17)(cid:1) (cid:1) (cid:4)(cid:3) (cid:9)(cid:10)(cid:11)(cid:12) (cid:12)(cid:3)(cid:9)(cid:19)(cid:12)(cid:3)(cid:4)(cid:20)(cid:15)(cid:5)(cid:6)(cid:8)(cid:21)(cid:22)(cid:12)(cid:3)(cid:9)(cid:19)(cid:12)(cid:3)(cid:4)(cid:20)(cid:15)(cid:5)(cid:6)(cid:8)(cid:21)(cid:23)(cid:16)(cid:17)(cid:1)(cid:8)(cid:5)(cid:2)(cid:14)(cid:11)(cid:4)(cid:2)(cid:18)(cid:18)(cid:6)(cid:4)
Fig. 2. System model of RIS-SSK-ASTBC. implemented offline, and the results remain unchanged untilthe CSI varies. In addition, the detection complexity and therequirement of CSI of RIS-SSK-PB are the same as those ofthe intelligent RIS-SSK.III. T HE RIS-SSK-ASTBC S
CHEME
In this section, we present the RIS-SSK-ASTBC scheme, inwhich the RIS not only reflects the incident SSK signals butalso transmits its own information via ASTBC.
A. System Model
The system model for RIS-SSK-ASTBC is depicted inFig. 2, where the RIS with an even number of reflecting ele-ments is divided into two sub-surfaces, namely sub-surface N/ reflecting elements.In RIS-SSK-ASTBC, the source transmits SSK symbols to thedestination via the RIS symbol-by-symbol with guard intervalsin between, while the RIS conveys its own information tothe destination through the Alamouti transmission scheme. Weassume that channel links experience small-scale flat Rayleighfading as in RIS-SSK-PB. It should be noted that, differentfrom the case in RIS-SSK-PB, the CSI of both links is requiredat the destination only, not at the RIS any more.Each complete transmission is comprised of two time slots.In the first time slot, the RIS’s own information of b R =2 log ( M ) bits is loaded into the RIS controller. Specifically,the first log ( M ) bits are used to adjust the phase shifts ofthe sub-surface θ i = α with i = 1 , . . . , N/ , and theremaining log ( M ) bits are used to adjust the phase shiftsof the sub-surface θ i = α with i = N/ , . . . , N ,where α , α ∈ M (= { , π/M, . . . , π ( M − /M } ) . Inthis manner, two virtual M -PSK symbols are generated at theRIS. Meanwhile, the source transmits b S = log ( N t ) bits viaan SSK signal towards the RIS. Therefore, the received signalat the destination in the first time slot can be expressed as y = exp( jα ) N/ X i =1 f i g il + exp( jα ) N X i = N/ f i g il + w , (27) where l denotes the index of the active transmit antenna and w represents the AWGN at the destination in the first time slot.The first and second summations of terms are from sub-surface θ i = π − α with i = 1 , . . . , N/ , and the phase shifts of the sub-surface θ i = − α with i = N/ , . . . , N . Thesource transmits the same SSK signal as that in the first timeslot. Therefore, the received signal at the destination in thesecond time slot can be given by y = − exp( − jα ) N/ X i =1 f i g il + exp( − jα ) N X i = N/ f i g il + w , (28)where w denotes the AWGN at the destination in the secondtime slot. The spectral efficiency of RIS-SSK-ASTBC is log ( M ) + log ( N t ) / bits per channel use (bpcu), while thatof traditional SSK is log ( N t ) bpcu. Obviously, RIS-SSK-ASTBC achieves a higher spectral efficiency than traditionalSSK in the case of M > √ N t .After re-organizing (27) and (28) into a matrix form, wehave y = [ y , y ] T = (cid:20) exp( jα ) exp( jα ) − exp( − jα ) exp( − jα ) (cid:21)| {z } C N/ P i =1 f i g ilN P i = N/ f i g il | {z } h l + w , (29)where w = [ w , w ] T . From (29), the optimal ML detectorcan be formulated as (cid:16) ˆ l, ˆ α , ˆ α (cid:17) = arg min l,α ,α k y − Ch l k , (30)where ˆ l , ˆ α , and ˆ α are the estimates of l , α , and α ,respectively. Unfortunately, the computational complexity ofthe optimal ML detector in (30) in terms of complex multi-plications is O ( N t M ) , which becomes impractical for largevalues of N t and M . The design of low-complexity MLdetector will be investigated as follows.Given a realization of l ∈ { , . . . , N t } , according to theclassical Alamouti principle, we can obtain the combinedsignals as follows: ( r ) l = y h ∗ l + y ∗ h l = (cid:16) | h l | + | h l | (cid:17) exp( jα ) + h ∗ l w + h l w ∗ , (31) ( r ) l = y h ∗ l − y ∗ h l = (cid:16) | h l | + | h l | (cid:17) exp( jα ) − h l w ∗ + h ∗ l w . (32)Then, in the case of the l -th antenna being activated, the MLmetric can be calculated by D ( l ) = min α ∈M (cid:12)(cid:12)(cid:12) ( r ) l − (cid:16) | h l | + | h l | (cid:17) exp( jα ) (cid:12)(cid:12)(cid:12) + min α ∈M (cid:12)(cid:12)(cid:12) ( r ) l − (cid:16) | h l | + | h l | (cid:17) exp( jα ) (cid:12)(cid:12)(cid:12) . (33) Finally, the ML receiver makes a decision onto the activeantenna index from ˆ l = arg min l D ( l ) , (34)and recovers α and α from ˆ α = arg min α ∈M (cid:12)(cid:12)(cid:12) ( r ) ˆ l − (cid:16)(cid:12)(cid:12) h l (cid:12)(cid:12) + (cid:12)(cid:12) h l (cid:12)(cid:12) (cid:17) exp( jα ) (cid:12)(cid:12)(cid:12) , (35) ˆ α = arg min α ∈M (cid:12)(cid:12)(cid:12) ( r ) ˆ l − (cid:16)(cid:12)(cid:12) h l (cid:12)(cid:12) + (cid:12)(cid:12) h l (cid:12)(cid:12) (cid:17) exp( jα ) (cid:12)(cid:12)(cid:12) . (36)As seen from (33), the computational complexity of thisdetector in terms of complex multiplications is reduced to O (2 N t M ) . B. Performance Analysis
In this subsection, approximate and asymptotic ABEP ex-pressions are derived in closed-form for the source and theRIS of RIS-SSK-ASTBC utilizing the optimal ML detector.
1) ABEP of the Source:
The source information is com-pletely conveyed through the active antenna index. Let us firststudy the conditional PEP from (30), which is the probabilityof detecting ( l, C ) as (ˆ l, ˆ C ) conditioned on G and f , namely Pr (cid:16) ( l, C ) → (ˆ l, ˆ C ) | G , f (cid:17) = Pr (cid:18) k y − Ch l k > (cid:13)(cid:13)(cid:13) y − ˆ Ch ˆ l (cid:13)(cid:13)(cid:13) (cid:19) = Q vuut ρ (cid:13)(cid:13)(cid:13) Ch l − ˆ Ch ˆ l (cid:13)(cid:13)(cid:13) . (37)Here, we resort to the CLT under the assumption of N ≫ for the calculation of this PEP. Under the CLT, each entity of Ch l and ˆ Ch ˆ l can be treated as a random variable following thedistribution CN (0 , N ) . Hence, when l = ˆ l , it can be shownthat k Ch l − ˆ Ch ˆ l k is a central chi-square random variablewith four degrees of freedom, whose PDF is given by f ( x ) = xAN exp (cid:16) − xBN (cid:17) , (38)where A = 4 and B = 2 . Averaging Pr(( l, C ) → (ˆ l, ˆ C ) | G , f ) over k Ch l − ˆ Ch ˆ l k results in the following unconditional PEP[41, Eq. (64)]: Pr (cid:16) ( l, C ) → (ˆ l, ˆ C ) (cid:17) = Z + ∞ Q (cid:18)r ρx (cid:19) f ( x ) dx = 3 p − p , (39)where p = 12 − s N ρ
N ρ ! . (40) According to the union bounding technique, an upper boundon the error probability of active antenna index detection isgiven by P e ≤ N t M X l, ˆ ll =ˆ l X C , ˆC Pr (cid:16) ( l, C ) → (ˆ l, ˆC ) (cid:17) = M ( N t − (cid:0) p − p (cid:1) . (41)Finally, the ABEP of the source can be expressed as [42, Eq.(13)] P bS ≈ P e · N t N t − . (42)
2) ABEP of the RIS:
It is obvious that the error eventsfor phase shift detection can be categorized into two comple-mentary types, depending on whether the index of the activeantenna is detected correctly or not. Thus, the overall ABEPof the RIS can be given by P bRIS ≈ P e + (1 − P e ) P A , (43)where P e is given in (41), P A is the ABEP of M -PSKdemodulation in the case of correct detection for the activeantenna index, and the factor / accounts for the ABEP inthe case of incorrect detection for the active antenna index.For the calculation of P A , the instantaneous receive SNRper virtual PSK symbol can be derived from (35) or (36) as ρ k h l k . Therefore, the conditional ABEP of M -PSK demod-ulation can be expressed as [43, Eq. (8.23)] P A | k h l k ∼ = 2max (log ( M ) , × max( M/ , X i =1 Q (cid:18)q ρg PSK ( i ) k h l k (cid:19) , (44)where g PSK ( i ) = sin ((2 i − π/M ) . By resorting to the CLT, || h l || can be approximated as a central chi-square randomvariable with four degrees of freedom, whose PDF is givenby (38) with A = 1 / and B = 1 / . Then, by averaging P A | k h l k over || h l || , we have P A ∼ = 2max (log ( M ) , × max( M/ , X i =1 q ( i )] − q ( i )] , (45)where q ( i ) = 12 − s ρg PSK ( i ) N ρg PSK ( i ) N ! . (46)Finally, by substituting (41) and (45) into (43), we obtain theABEP of the RIS.
3) Asymptotic Analysis:
By taking the Taylor series of theexponential function and ignoring higher order terms, (38)reduces to f ( x ) ≈ x N . (47) With the simplified PDF in (47), the unconditional PEP in (39)can be re-calculated asymptotically as Pr (cid:16) ( l, C ) → (ˆ l, ˆ C ) (cid:17) → · ( ρN ) − . (48)Putting (48) into (41) yields P e ≤ M ( N t −
1) ( ρN ) − . (49)Then, by substituting (49) into (42), we have the asymptoticABEP of the source as follows: P bS ≈ M N t ( ρN ) − , (50)which achieves a diversity order of two. Interestingly, we ob-serve from (50) that the achievable diversity order of the sourceis irrelevant to N , namely the number of reflecting elements.However, increasing N is equivalent to increasing ρ , both ofwhich decrease the PEP quadratically. These observations alsoapply to the RIS. Similar to (48), we have P A → · max (log ( M ) , × max( M/ , X i =1 ( ρg PSK ( i ) N ) − . (51)Further, at high SNR, (43) can be simplified to P bRIS ≈ P e + P A . (52)Substituting (49) and (51) into (52) results in the asymptoticABEP of the source. Obviously, we observe from P bS and P bRIS that the information bits from both the source and theRIS have two-diversity-order protection with increasing SNRor the number of reflecting elements. Moreover, doubling N is equivalent to doubling ρ , such that a consistent SNR gainof about 3 dB gain can be achieved every time N is doubledfor both the source and the RIS. Remark 2:
In RIS-SSK-ASTBC, the reflection coefficientscompletely depend on the information to be transmitted fromthe RIS, and no beamforming is performed at the RIS. Hence,the CSI is not needed at the RIS. Actually, if the CSI is avail-able at the RIS, we can implement the Alamouti transmissionand beamforming simultaneously at the RIS, further enhancingthe performance of RIS-SSK-ASTBC.IV. S
IMULATION R ESULTS AND C OMPARISONS
In this section, we perform Monte Carlo simulations toassess the uncoded BER performance of RIS-SSK-PB andRIS-SSK-ASTBC. In all simulations, we plot the BER versus ρ = 1 /N . The channels are assumed to be Rayleigh flatfading channels, the CSI is perfectly known to the destinationand/or the RIS, and any path loss effect is neglected. ForRIS-SSK-PB, the traditional SSK [8] and the intelligent RIS-SSK [25] are taken as reference schemes. The amplify-and-forward aided SSK and blind RIS-SSK are excluded from theperformance comparison, since they have been shown to beinferior to the intelligent RIS-SSK in [25]. For RIS-SSK-ASTBC, RIS-SSK-VBLAST that employs the two-antennabased VBLAST scheme to transmit RIS’s private informationis chosen as a benchmark. All considered schemes employsingle-antenna ML detection for fair comparisons. Each BERpoint is obtained by averaging over at least transmission. -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 SNR (dB) -5 -4 -3 -2 -1 B E R Traditional SSK [8]Intelligent RIS-SSK [25]RIS-SSK-PB, Sim.RIS-SSK-PB, Approx. N =4, 8, 16, 32, 64, 128 Fig. 3. Performance comparison among traditional SSK, intelligent RIS-SSK,and RIS-SSK-PB, where N t = 2 and N = 4 , , , , , . A. RIS-SSK-PB
Fig. 3 depicts the comparison results among traditionalSSK, intelligent RIS-SSK, and RIS-SSK-PB, where N t = 2 and N = 4 , , , , , . The approximate ABEP curvesderived in Section II.C are also presented for RIS-SSK-PB.As shown in Fig. 3, due to the nature of the CLT, theanalytical results become more accurate with increasing N .When N ≥ , the analytical ABEP curves can predict thesimulated counterparts very well. With increasing N , the BERperformance of both intelligent RIS-SSK and RIS-SSK-PBimprove, and both perform much better than traditional SSKfor all N in the overall SNR region. Moreover, RIS-SSK-PB significantly outperforms the intelligent RIS-SSK for all N and SNR values. In particular, RIS-SSK-PB achieves anadditional diversity gain over the intelligent RIS-SSK, and thegain becomes more prominent when N is a smaller value.Fortunately, even for N = 128 , approximately 3 dB SNR gainis obtained by RIS-SSK-PB over the intelligent RIS-SSK, at aBER value of − . Note that these advantages are achievedalong with the benefits of avoiding the requirements of activeantenna index information and online phase adjustments.In Fig. 4, we make comparisons among traditional SSK,intelligent RIS-SSK, and RIS-SSK-PB, where N t = 4 and N = 4 . For RIS-SSK-PB, both the SDR and low-complexitybeamforming algorithms are taken into account. The randomsimulation times in the SDR method for solving problem (P3)is set as 100. As seen from Fig. 4, with the aid of a 4-element RIS, both the intelligent RIS-SSK and RIS-SSK-PBperform much better than traditional SSK throughout the SNRregion. Moreover, no matter which beamforming algorithm isemployed, RIS-SSK-PB achieves a higher diversity gain than -15 -12 -9 -6 -3 0 3 6 9 12 15 18 21 24 27 30 SNR (dB) -5 -4 -3 -2 -1 B E R Traditional SSK [8]Intelligent RIS-SSK [25], N =4RIS-SSK-PB, N =4, Low-ComplexityRIS-SSK-PB, N =4, SDR Fig. 4. Performance comparison among traditional SSK, intelligent RIS-SSK,and RIS-SSK-PB, where N t = 4 and N = 4 . -20 -15 -10 -5 0 5 10 15 SNR (dB)(a) -5 -4 -3 -2 -1 B E R o f t h e S ou r ce -20 -15 -10 -5 0 5 10 15 SNR (dB)(b) -5 -4 -3 -2 -1 B E R o f t h e R I S Sim. Approx. N =4, 8, 16,32, 64, 128 N =4, 8, 16,32, 64, 128 Fig. 5. Performance of RIS-SSK-ASTBC for: (a) the source and (b) the RIS,where N t = 2 , M = 2 , and N = 4 , , , , , . the intelligent RIS-SSK and traditional SSK. In particular,RIS-SSK-PB with the SDR method outperforms the intelli-gent RIS-SSK when SNR is greater than -1 dB, achievingapproximately 19 dB SNR gain at a BER value of − .Notably, the low-complexity beamformer suffers from about3 dB performance loss as compared with the SDR method ata BER value of − . However, RIS-SSK-PB with the low-complexity beamforming is still superior to the intelligent RIS-SSK and traditional SSK with much reduced complexity. -15 -10 -5 0 5 10 15 20 25 SNR (dB)(a) -5 -4 -3 -2 -1 B E R o f t h e S ou r ce -15 -10 -5 0 5 10 15 20 25 SNR (dB)(b) -5 -4 -3 -2 -1 B E R o f t h e R I S RIS-SSK-VBLASTRIS-SSK-ASTBC, Sim.RIS-SSK-ASTBC, Approx.RIS-SSK-ASTBC, Asympt. N =32, 64, 128 N =32, 64, 128 N =32, 64, 128 N =32, 64, 128 Fig. 6. Performance comparison between RIS-SSK-ASTBC with M = 8 andRIS-SSK-VBLAST with M = 2 for: (a) the source and (b) the RIS, where N t = 4 and N = 32 , , . B. RIS-SSK-ASTBC
Fig. 5 presents the BER performance of RIS-SSK-ASTBC,where N t = 2 , M = 2 , and N = 4 , , , , , . Toverify the analysis given in Section III.B, we also plot theapproximate ABEP curves for the source, namely (42), andfor the RIS, namely (43). As seen from Fig. 5, since we resortto the CLT in the performance analysis, the analytical resultsbecome accurate with increasing N for both the source andthe RIS. In the cases of N ≥ , the theoretical curves agreewith the simulated counterparts very well in the SNR region ofinterest. It can be seen that increasing N yields performanceimprovement for both the source and the RIS. For example,about 3 dB SNR gain is observed at a BER value of − for the source and the RIS, when N increases from 64 to128. However, there is no diversity improvement, since thesource and the RIS achieve a diversity order of two, which isirrelevant to the value of N .Fig. 6 shows the performance comparison between RIS-SSK-ASTBC with M = 8 , and RIS-SSK-VBLAST with M =2 , where N t = 4 and N = 32 , , . All considered schemesachieve the same spectral efficiency of 4 bpcu. Both theapproximate and asymptotic results are also provided. FromFig. 6, we observe that the approximate and asymptotic ABEPcurves match the simulated counterparts very well. For boththe source and the RIS, RIS-SSK-ASTBC achieves a diversityorder of two, while RIS-SSK-VBALST achieves a diversityorder of unity. Due to the higher diversity order, RIS-SSK-ASTBC significantly outperforms RIS-SSK-VBLAST, thoughRIS-SSK-ASTBC employs a higher constellation size forachieving the same spectral efficiency as RIS-SSK-VBLAST.For instance, the source and the RIS of RIS-SSK-ASTBCachieve about 15 dB SNR gain over those counterparts of RIS-SSK-VBLAST, at a BER value of − . V. C ONCLUSION
In this paper, we have proposed RIS-SSK-PB and RIS-SSK-ASTBC schemes to improve the error performance andthroughput of traditional SSK systems, respectively. In RIS-SSK-PB, the RIS is employed for beamforming, which max-imizes the minimum squared Euclidean distance between anytwo decision points. The SDR and low-complexity algorithmshave been developed for the reflecting optimization. In RIS-SSK-ASTBC, the RIS is employed for ASTBC, which enablesthe RIS to transmits its own Alamouti-coded information whilereflecting the incident signals. A low-complexity ML detectorhas been studied for RIS-SSK-ASTBC. The approximateABEP expressions have been derived for the source and/orthe RIS in closed-form assuming ML detection. Computersimulations have corroborated the performance analysis andthe performance improvement achieved by RIS-SSK-PB andRIS-SSK-ASTBC. It could be concluded that the two pro-posed schemes are viable candidates for energy-efficient andultra-reliable communications. Also, RIS-SSK-ASTBC can beextended by employing general STBC with more than two sub-surfaces at the RIS. For future work, we intend to extend theproposed RIS-SSK schemes to more practical RIS models,including the RIS unit cell model of [44] and the mutual-impedances-based end-to-end model of [45].R
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