Low-Complexity Linear Precoding for Secure Spatial Modulation

1 Lo w-Comple xity Linear Precoding for Secure Spatial Modulation Feng Shu, Member , IEEE , Z hengwang W ang, Shihao Y an, Member , IEEE , Xiaobo Z h ou, Jun Li, Senior Member , IEEE , Xiangyu n Zhou, Senior Memb er , IEEE Abstract —In this w ork, we inv estigate lin ear pr ecodin g for secure spatial modulation. With secure spatial modulation, the achiev abl e secrecy rate does not hav e an easy-to -compute math- ematical expression, and hence, has to be eva luated numerically , which leads to high complexity in th e opt i mal precoder design. T o address this issu e, an accurate and analytical approximation of the secrecy rate i s d eriv ed in this work. Using this approximation as the objective fun ction, two low-complexity l inear precoding methods based on gradient descend (GD) and successive con vex approximation (S CA) are proposed. T h e GD-based method has much lower complexity but usually con verges to a local optimum. On the other hand, the S CA-based method uses semi-definite relaxa tion to deal with th e n on-con vexity in the precoder opti- mization problem and achieves near -opti mal solution. Compared with the existin g GD-based precoder design in the li terature that directly u ses the exact and numerically evaluated secrecy capacity as the objective function, the two proposed designs hav e significantl y lower complexity . Our SCA- b ased design ev en achiev es a h igher secrec y rate than the existing GD-based design. Index T erms —MIMO, secure spatial modulation, l i near pre- coding, gradient d escend, successive con vex approxima tion. I . I N T R O D U C T I O N S P A T IAL MODULA TION (SM), propo sed by Raed Y . Mesleh in [1], constitutes a pr o mising multiple-input- multiple-ou tput (MIMO) co mmunicatio n technology for futur e wireless commun ication systems. The b asic idea of SM is to use b oth th e indices of tr ansmit anten nas and constellation signals to carry a block of in formatio n bits [ 2]–[4]. Compared to single-inpu t-single-ou tput (SISO) systems, SM systems improve the overall spectral efficiency (SE) b y the lo garithm with base 2 of the num b er of tran smit an tennas [1]. Due to one antenna bein g active at any time instant in SM system s, the impacts of in ter-channel inter ference (I CI) a n d inter-antenna synchro n ization (IAS) can be mitigated [5], an d thus the practical im p lementation co mplexity at the tran smitter and This work was in part supported by the National Natura l Science Founda- tion of China (Grant Nos. 61771244 , 6147219 0, 61702258, 61501238, and 61602245), the China Postdoctoral Scienc e Foun dation (2016M591852), the Postdoctor al research funding program of Jiangsu Province(1 601257C), the Natural Science Foundation of J iangsu Province (Grants No. BK20150791) , and the open resea rch fund of National Mobile Communicatio ns Research Laboratory , Southeast Uni versity , China (No. 2013D02). F . Shu, Z. W ang, X. Zhou, and J. Li are with School of Elect ronic and Optical Enginee ring, Nanjing Univ ersity of Science and T echnol ogy , Nanjing, 210094, China. F . Shu is also with the College of Computer and Information Science s, Fujian Agriculture and Forestry Uni versity , Fuzhou 350002, China. S. Y an is with the School of Engineering, Macquari e Uni versity , Sydney , NSW 2109, Australia (e-mail : shihao.yan@mq.edu.au .) X. Z hou is with Researc h School of Engineeri ng, T he Australia n Nationa l Uni versity , A CT 2601, Australi a (e-mail: xiangyu n.zhou@anu.e du.au.) receiver is significantly reduce d [6]. Compar ed to space-time block cod ing (STBC) and bell labora tories layered space-time (BLAST) arch itectures, th e SM techn ique strikes an attractive tradeoff b etween the SE and energy efficiency (EE), thu s it is app licable to som e futur e energy-efficient scenarios such as internet o f thin g s, 5G and beyond mobile n etworks. Howev er, it is very likely that the con fidential messages ar e intercepted by un in tended receivers, due to the br oadcast nature and openn e ss of wir eless chann el. Therefore, it is very important and necessary to addr ess security issues for such SM systems. Recently , physical layer security on co n ventio nal MIMO systems has be en widely investigated in [7]– [ 11], which exploits th e u niquen e ss and tim e - varying cha r acteristics of wireless ch annel to obtain secur e transmissions against eaves- dropp ers. Particularly , the autho rs in [ 12], [13] exploited the direction mo dulation (DM) technique and random frequ e ncy div erse arrays to ob tain secure and pr ecise transmissions, where only desired users with prede fined specific directio ns and d istances ca n receiv e confid ential messages. Ho wever , some works on physical layer security of co n ven tio nal MIMO systems assum e d that the inp ut signa ls follow Gau ssian dis- tribution [14], wh ic h is not practical in SM systems d ue to the numb e r of transmit antennas being finite. In fact, fin ite alphabet inp uts should b e con sidered in SM systems. In [15]– [26], the au thors inv e stig a te d secure tra nsmissions for SM systems from different aspects. The authors in [15] derived the secr ecy mutual info rmation with finite alph a b et input for a SM m ultiple-inp ut-single- output (MISO) system, and propo sed a pr ecoding sch eme to degrad e the dete c tion perfo rmance at an eavesdropper by decr easing the Euclidean distance at the eavesdropper . Howe ver, the precod ing scheme is n ot applicable fo r SM- MIMO systems. Both [16] and [ 17] add ressed the security of SM systems with the aid of artificial no ise (AN), which was proje c ted into the null space of legitimate ch annels. In [16], the ran d om AN signals were ra diated at the tran sm itter and th e secr e cy rate ( SR) was analyzed . In [17], a f ull-dup lex receiver with self-interferen ce c ancelation ca p ability was em- ployed to transmit AN signals. Mean while, the preco ding- aided spatial modulation (PSM) was generalized to provide secure tran smissions f or o n e legitimate u ser in [1 8], [ 19] o r multiple legitimate users in [20]. The secur ity of PSM was enhanced by con structing time-varying precod er [1 8], [2 0] or op timizing the pr e c oder throug h join tly minimizing th e receive power at eavesdropper and maximizing the receive power at desired user [19]. Unlike traditional SM systems, the PSM uses the index of receiv e an tenna to carry informatio n 2 bits an d activ ates a ll tran smit anten nas, which red u ces the complexity at receiver but r esults in the issues of I AS and ICI at th e transmitter . In [22], [ 23], the au thors explored the schemes of transmit antenna selec tio n to enhance the security of SM-MIMO systems. A n ew idea of redefinin g the mapping rules between the informatio n bits and the indice s of transmit antennas was pro posed to achieve secu re transmission s in [24], and th e n the au thors in [25], [26] further improved the security by exploiting th e knowledge of the legitimate channel state inform ation (CSI) to ro ta te bo th the indices of the tra nsmit antennas and th e constellation symb ols. In [24]–[26], the legitimate CSI actually ca n b e regarded as an encryp tion key to en crypt the confidential messages. In other words, the scheme of rotating bo th the in dices of the transmit anten n as an d the constellation symb ols actu ally does not destroy the receive signals at eav e sd ropper s. Ther e fore, when the legitimate channel is slo w fading chan nel, th e key is in variant dur ing a long cohe r ence in terval, an d thus the eav esdropp e r s may decod e th e key by using th e cryptanalysis or m achine learning meth ods. On the other hand, linear p recodin g for SM sy stem s has received some r esearchers’ attentio n in [27]–[3 2 ] . The au- thors in [ 2 7] pro posed two design schemes of g eneralized precod e r, maximu m m inimum distance (MM D) and gu aran- teed Euclidean distance (GED), and a n iterative algorithm to acquire effecti ve so lutions. Then a low-complexity method with sm a ller n umber s of iterations was pro posed to solve the MMD and GED prob le m s in [31]. Meanwhile, the a u - thors in [28], [29] con stru cted a similar MMD problem for space shift keying (SSK) systems, where the optim ization problem was efficiently solved using semi-defin ite r elaxation (SDR) techniques. In [30], the authors intro d uced two design criterions, maximizin g the min imum Euclidean distance and minimizing the bit erro r r ate (BER), to op timize the diag onal linear preco ding m atrix. In [32], a new precoding scheme was propo sed to improve the mutua l infor mation for gen eralized spatial m odulation (GSM) systems by conv erting the GSM systems into a vir tual MI MO system and emp loying the extended ellipsoid algorith m. Howe ver, all the aforementio ned works optimized the lin ear preco ding vector to improve th e BER per forman ce regardless o f security . In [21], the auth o rs in vestigated the design of linear preco d ing to maximize the actual SR (Max-SR) f or secure SSK systems and the M ax- SR based on gr adient d escend (GD) method (Max -SR-GD) was also proposed to solve the correspo n ding optimizatio n problem , but the comp utational co mplexity is extremely high due to the require m ent of a high computatio nal amount to ev aluate the actual SR. Compared to [21], we will focus on the d esign of linear precod in g with lower com plexity or highe r perfor mance in this work. Our main contributions are summ arized as fo llows: 1) D u e to th e absence of a closed- form expression for SR in secure SM systems with finite alphab et input, it is har d to design a practica l feasible linear p recoder of Max- SR with lower-complexity . T o red uce the compu tational complexity of Max-SR pre coder, an approx imated SR (ASR) is deriv e d and presen ted, which is used as an optimization o bjective function . Numerical Mon te-Carlo simulations show that the obtained ASR is very close to the actual SR fo r different n umber s of transmit anten nas and dif f erent typ es of modulation . 2) By m aking use of the closed-fo r m expression of ASR, the op timization pro blem of maximizin g ASR (Max- ASR) is casted and pro posed. T o solve the optimiz a tion problem , a low-complexity precoder, called Max -ASR- GD, is propo sed to iterativ e ly solve the pr oblem by GD algorithm. It is well-known that the GD m ethod can usually co n verge to a locally optimal solution. Our simulation results also show that the pr oposed Max - ASR-GD ac h iev es a slightly lo wer SR than th e Max- SR-GD pro p osed in [21] but with a sign ificantly lower complexity . 3) T o furth er improve the SR perfor mance in the o pti- mization prob lem of Max- ASR, whic h is a typical non- conv ex quadratically constrained quadr atic programm ing (QCQP) problem a nd NP-h ard in general, its o bjective function is first relaxed an d repr esented as the d if- ference betwe en two convex functions by using SDR technique s. T hen, the succ e ssi ve conv ex approxim ation (SCA) method is e m ployed to iteratively solve the con- vex su b prob lems a n d obtain an approxim ately op timal solution. Finally , the prop osed Max-ASR-SCA is proved to be convergent. From simu lation results, it f ollows that the prop osed Max -ASR-SCA outperf orms the Max -SR- GD method an d M a x-ASR-GD in terms of achieving a higher SR. Mor e importan tly , the numb er of itera tio ns of the prop osed Max -ASR-SCA is smaller than those of Max-SR-GD and Max- ASR-GD. The remain d er of this paper is organ ized as follows. In Section I I , the system model of the co nsidered secur e SM- MIMO is presented, an d the optimiz a tion problem o f Max- ASR is fo rmulated in Section I II. In Section IV , two p r ecoding methods, Max -ASR-GD and Ma x-ASR-SCA, are pro posed to solve the optimiz a tion pr oblem, and their co mputatio n al complexities are analyzed. Numerical r esults are presented in Section V . Finally , we make ou r conclu sions in Section VI. Notation: Matrices, vectors, a nd scalars ar e deno te d b y letters of bold upper case, bold lower case, and lower case, respectively . Signs ( · ) − 1 and ( · ) H denote in verse and con- jugate tra nspose, r espectively . Notation E {·} stand s fo r the expectation oper ation. I N denotes the N × N iden tity matrix , and tr( · ) deno tes matrix trace. I I . S Y S T E M M O D E L A. Secure S patial Mod ulation A typical secure SM-MIMO system model is shown in Fig. 1. In th is system, there are a transmitter (Alice) with N t transmit antenn a s, a legitimate receiv er (Bob) with N b receive antennas, and an eavesdropper (Eve) with N e receive antennas. Alice activates one of her N t transmit antennas to emit M -ary amplitude phase mod ulation (APM) sy mbol, and the index of activ ated antenna is also used to carry inform ation. As a result, the SE is log 2 M N t bits per chan n el use (bpcu). T o enhance th e security of such a SM system, a linear precod e r and AN pr ojection at tr ansmitter is adopted to 3 RF Ă 1 N t Eve Ă N e 1 Bob Ă 1 N b A N projection Bit stream blocking Bit stream Mo dul a t e d b i t s Spa tial b i t s Symbol m a pping Linea r precoding G H A ct i vate one antenna to c ar ry m e ss age Ă Fig. 1. System model of linea r precoding schemes for secure SM system guaran tee the secure transmission against Eve’ s eav esdroppin g. Similar to [16], the transm itted baseband signal at Alice may be co nstructed as follows x = p P 1 Vs n,m + p P 2 T AN n , (1) where V ∈ C N t × N t denotes the linear precodin g matrix, and T AN ∈ C N t × N t is the AN pro jec ting matrix. P 1 and P 2 are the power of confiden tial signal and AN, respectively , with power con straint P 1 + P 2 ≤ P t holding , wh ere P t is the total transmit power . n ∈ C N t × 1 is the ran d om AN vector following standard com plex G a ussian distribution C N (0 , I N t ) . s n,m = e n s m , n ∈ [1 , N t ] , m ∈ [1 , M ] , is the input symbo l vector . s m is the norm alized inp ut sy mbol, with E ( | s m | 2 ) = 1 , which is drawn equipro bably from discrete M -ary constellation, an d e n is th e n th column of iden tity matrix I N t . It is worth n oting that the linear precodin g for SM-MIMO systems is d ifferent fro m conventional MIMO systems. In SM-MIMO system s, only o ne an tenna is activated to tran smit modulated symb ol, so the linear precod ing matrix is a d iag- onal m a trix, i.e. , V = diag ( v ) , where v ∈ C N t × 1 stands for th e linear preco ding vector . Moreover , the eleme n ts of linear p recodin g vector v canno t be a c ti ve simultaneo usly . I n other words, on ly the elem ent corresp onding to the ac tivated transmit an tenna is active during each symbo l interval [27]. Let H ∈ C N b × N t and G ∈ C N e × N t denote the comp lex channel m atrices cor respond ing to the legitimate channe l and eav esdropp in g chann el, respectiv ely . This paper assumes th a t Alice has the perfect CSIs of both ch annels, which may b e tru e for active Eve [ 2 1]. Meanwh ile, Bob and Eve attain their own CSIs thro ugh pilot-assisted channel estimation , respectively . It is assumed that chan nel matrices H a nd G both are flat Rayleig h fading with e ach elem e n t being the stand a r d complex-Gau ssian distribution C N (0 , 1) and keep constant during each co herence time. Accordin gly , th e receive signals at Bob and Eve are respectively stated as fo llows y b = Hx + n b = p P 1 HVs n,m + p P 2 HT AN n + n b , (2) y e = Gx + n e = p P 1 GVs n,m + p P 2 GT AN n + n e , (3) where n b and n e are the ad ditive white Gau ssian noise (A WGN) vectors at Bob and E ve with obeying C N (0 , σ 2 b I N b ) and C N (0 , σ 2 e I N e ) , respecti vely . Consequently , with the kn owledge of H and V , Bob can employ the maximu m likelihood detector (MLD) as follows [ ˆ n, ˆ m ] = arg min n ∈ [1 ,N t ] ,m ∈ [1 ,M ]    y b − p P 1 HVs n,m    2 . (4) Here, we consider the worst case when Eve knows the lin e ar precod in g matrix V , and th us Eve may also carr y ou t MLD as f ollows [ ˆ n, ˆ m ] = arg min n ∈ [1 ,N t ] ,m ∈ [1 ,M ]    y e − p P 1 GVs n,m    2 . (5) Howe ver, the terms of √ P 2 HT AN n an d √ P 2 GT AN n in (2) and (3) is time-varying interference, which will seriou sly degrade their detecting perf ormanc e. I n what follows, we will derive an expression of SR with respect to T AN and V , then we pro ject the AN signals into the null spa ce of legitimate channel b y designin g the P AN . Finally , o ur main aim of this paper is to design a linear p r ecoding matrix to imp rove SR perfor mance. B. Secr ecy Ra te for Secure S patial mod ulation Similar to [21], we c h aracterize the security b y evaluating av e r age SR defined a s ¯ R s = E H , G ( R s ) , (6) where R s is the instantaneo us SR. Accord in g to [7], the SR for secure SM system is defined as follows R s = [ I ( s ; y b ) − I ( s ; y e )] + , (7) where [ a ] + = max { a, 0 } . I ( s ; y b ) and I ( s ; y e ) denote the mutua l info rmation over legitimate and eaves dropp ing channels, respectively . Howe ver , du e to the interf erence plus noise not being Gaussian distributed in (2) and ( 3), it is no t straightfor ward to derive the expression of SR. Similar to [1 6], the interferen ce plus n oise at Bob is de noted as w b = p P 2 HT AN n + n b , (8) 4 then we u tilize a linear whitening tr ansforma tio n matrix Q b to con vert w b into a white noise vecto r n ′ b = Q − 1 / 2 b w b , (9) where Q b is th e covariance m atrix of w b giv en by Q b = P 2 HT AN T H AN H H + σ 2 b I N b . (10) Substituting (10) into (9) and considerin g that n and n b are indepen d ent, we have E ( n ′ b ) = 0 and E ( n ′ b n ′ H b ) = I N b , thus n ′ b ∼ C N (0 , I N b ) ca n be c onsidered as an A WGN vector . By pre-mu ltiplying the linear m atrix Q b , th e receiver signal in ( 2) can b e simplified as y ′ b = p P 1 Q − 1 / 2 b HVs n,m + n ′ b , (11) which yields, fo r a specifical H , the conditio n al p robab ility density function (PDF) of p ( y ′ b | s n,m ) p ( y ′ b | s n,m ) = 1 π N b exp  −    y ′ b − p P 1 Q − 1 / 2 b HVs n,m    2  . (12) W e assume that each a n tenna is activ ated eq uiprob ably to transmit co nfidential message an d each symbo l s m is uni- formly selected fr om M -ary constellation . Thus, the complex receive signal vecto r y b at Bob obeys the distribution as follows p ( y ′ b ) = 1 M N t N t X n =1 M X m =1 p ( y ′ b | s n,m ) . (13) As per (12) and (13), following the method given in [33], the mutual information I ( s ; y ′ b ) is written as I ( s ; y ′ b ) = log M N t 2 − 1 M N t N t X n =1 M X m =1 E n ′ b " log 2 N t X n ′ =1 M X m ′ =1 exp  k n ′ b k 2 −    α n ′ ,m ′ n,m + n ′ b    2  !# , (14) where α n ′ ,m ′ n,m = p P 1 Q − 1 / 2 b HV ( s n,m − s n ′ ,m ′ ) . (15) It shou ld be noted th at the I ( s ; y ′ b ) is equivalent to I ( s ; y b ) , since the tran sf o rmation is lin ear [16]. Similar to (14), w e can derive the mutual in formatio n I ( s ; y e ) as fo llows I ( s ; y e ) = log M N t 2 − 1 M N t N t X n =1 M X m =1 E n ′ e " log 2 N t X n ′ =1 M X m ′ =1 exp  k n ′ e k 2 −    δ n ′ ,m ′ n,m + n ′ e    2  !# , (16) with δ n ′ ,m ′ n,m = p P 1 Q − 1 / 2 e GV ( s n,m − s n ′ ,m ′ ) , (17) n ′ e = Q − 1 / 2 e ( p P 2 GT AN n + n e ) , (18) where Q e = P 2 GT AN T H AN G H + σ 2 e I N e is the linear whiten- ing tr ansforma tio n matrix a t Eve. C. Design the AN Pr ojecting Matrix In this paper , we consid e r the scenarios with N t ≥ N b , thus the AN signals can be p rojected into the null sp ace of legitimate ch a nnel. Here, we present a closed-fo r m expression of AN projection matrix , which is different from th e singular value d ecompo sition (SVD) method in [16]. Accordingly , the T AN can b e directly calcu lated as follows [ 34] T AN = 1 µ h I N t − H H  HH H  − 1 H i , (19) where µ = k I N t − H H  HH H  − 1 H k F is the normalized factor f or satisfying tr( T AN T H AN ) = 1 with the k A k F representin g Frob enius nor m of a matrix A . It can be seen from (19) that T AN is the projecto r of the null space o f legitimate chan nel, thus we have the following e quality HT AN = 0 , Q b = σ 2 b I N b . (20) I I I . O P T I M I Z A T I O N P RO B L E M O N T H E L I N E A R P R E C O D I N G M AT R I X In this sectio n , we first p resent a o ptimization p roblem of Max-SR to o ptimize the lin ear precod ing matrix . T hen, a simple expression of ASR is developed to be the co n ven ie n ce of dealing with the original optimization problem o f Max- SR. Further more, due to the linear preco ding matrix V being diagona l, we turn to optimize the lin ear precod in g vector v = Diag ( V ) for sake o f convenience. A. Maximizing the Secrecy Rate After designing the AN projecting matrix , we turn to optimize th e lin ear precodin g matrix by solvin g the following problem of Max- SR maximize V R s ( V ) (21a) sub ject to : P 1 tr( VV H ) N t ≤ P t − P 2 . (21b) It is worth no tin g that the ab ove optimiz a tion pro blem of d e- signing the linear preco der matrix is intrac table in g e n eral, and we need m a ny ev alu ations for calcu la tin g the actual SR du e to the absence of the closed- f orm expression of SR. Actu ally , the above optimizatio n prob lem can be so lved by using GD method as illustrated in [2 1], but it has a high com putation a l complexity . As we will see later , to reduce the c o mputatio n al complexity , we present a closed -form appro x imated e x pression of SR, which can be used as an effective metr ic to optimize the linea r pr ecoding matr ix . B. Appr ox imated Expression fo r S ecr ecy Rate In this subsection, we pr esent the tight lower b o unds on mutual in formatio n over legitimate and eavesdropping chan- nels, respectively , and p rovide an a c curate approx imation to SR. Similar to [33] and [35], by using Jensen ’ s inequality and 5 the integrals of exponential function , we have the tight lower bound of I ( s ; y b ) I ( s ; y b ) LB = log M N t 2 − 1 M N t N t X n =1 M X m =1    log 2    N t X n ′ =1 M X m ′ =1 exp    −    α n ′ ,m ′ n,m    2 2          . (22) Similarly , the mutual inf ormation of eavesdropping c hannel can b e lo wer boun ded by I ( s ; y e ) LB = log M N t 2 − 1 M N t N t X n =1 M X m =1    log 2    N t X n ′ =1 M X m ′ =1 exp    −    δ n ′ ,m ′ n,m    2 2          . (23) Then an accur a te app roximatio n of SR can be given by R ′ s = [ I ( s ; y b ) LB − I ( s ; y e ) LB ] + . (24) T o further make the optimization variable clear, the term    α n ′ ,m ′ n,m    2 in (22) can be represented as [31]    α n ′ ,m ′ n,m    2 = P 1 ·    Q − 1 / 2 b HV ( s n,m − s n ′ ,m ′ )    2 (25a) = P 1 · tr  Q − H b HV∆ n ′ ,m ′ n,m V H H H  (25b) = P 1 · v H  H H Q − H b H ⊙ ∆ n ′ ,m ′ n,m  v (25c) = P 1 · v H B n ′ ,m ′ n,m v , (25d) where ∆ n ′ ,m ′ n,m = ( s n,m − s n ′ ,m ′ )( s n,m − s n ′ ,m ′ ) H , (26) B n ′ ,m ′ n,m = h ( H H Q − H b H ) ⊙ ∆ n ′ ,m ′ n,m i . (27) Similarly , the term    δ n ′ ,m ′ n,m    2 in (23) can be represented as    δ n ′ ,m ′ n,m    2 = P 1 · v H h  G H Q − H e G  ⊙ ∆ n ′ ,m ′ n,m i v (28a) = P 1 · v H E n ′ ,m ′ n,m v , (28b) with E n ′ ,m ′ n,m = h  G H Q − H e G  ⊙ ∆ n ′ ,m ′ n,m i , (29) where ⊙ denotes th e Hadam ard product of two matrices. W e note that (25) and (28) are achieved by utilizing the trace proper ty , i.e . , tr( AB ) = tr ( BA ) for m a tr ices A and B . C. Maximizing the App r oximated S R Making use of the above accu rate approxima tion o f SR, the Max-SR op timization problem in (21) can be converted into the following simplified pro b lem maximize v R ′ s ( v ) (30a) sub ject to : tr( vv H ) ≤ N t , (30b) where R ′ s ( v ) is given in (31) a t the top of n ext page, and the (30b) is obtained with P 1 + P 2 ≤ P t . The above Max- ASR can significantly reduce the co m plexity compared to Max-SR in (21) as sho wn in what f ollows. Howe ver , it can be seen from (31) that the ab ove optimization problem is a non -conve x QCQP pro blem, thus it is NP-hard in gen eral. In what follows, we present a lower-complexity GD method and propo se the SCA me th od to addre ss the ab ove op tim ization pro blem. I V . P R O P O S E L O W - C O M P L E X I T Y L I N E A R P R E C O D I N G S C H E M E S O F M A X I M I Z I N G A S R T o provide a rapid solution to the optimization problem in (30), two low-comp lexity m ethods, Max-ASR-GD and Max- ASR-SCA, are presented in this section . For the fo rmer, simi- lar to [2 1], using a gradie n t vector of the clo sed-form ASR, th e iterativ e GD algo rithm can be employed to yield an extremely low-complexity solu tion to the optimization prob lem as given in (30), which converges to a locally optimal solu tion. For the latter , the original pr oblem is first relaxed to a difference of conv ex (DC) pr o gramm in g b y using the SDR method, and the first-order T a ylor approximatio n expansion and SCA way are combined to achieve a n app roximate ly optimal solution. A. Pr oposed Max-ASR-GD Metho d Due to th e nonco n vexity of th e problem (30), it is in tractable to ob ta in a glob ally optimal solution in gen eral. Howe ver, it is possible to employ the GD method , wh ich iteratively searches for a locally op tim al solu tion. The Max-ASR-GD method firstly needs to calculate th e gradien t of R ′ s ( v ) with respect to v . This grad ient is directly shown in (32) at the top of th e next p age. In (32), κ b and κ e are d efined as follows κ b = N t X n ′ =1 M X m ′ =1 exp − P 1 · v H B n ′ ,m ′ n,m v 2 ! , (33) κ e = N t X n ′ =1 M X m ′ =1 exp − P 1 · v H E n ′ ,m ′ n,m v 2 ! . ( 34) Based on the grad ient of R ′ s ( v ) over v in (3 2) , we upda te the linea r pr ecoding vector in the following way v k +1 = v k + µ ∇ v R ′ s ( v k ) , (35) where µ and k are the step size and iterative index, respec- ti vely . It should be noted that the upd ated linea r precoding vector should satisfy the po wer constraint, the po wer normal- ization p rocedu re is written as v = q N t / tr( vv H ) v . (36) The de ta iled p rocedu re of th e pro posed Max-ASR-GD method is illustrated in Algo rithm 1 . The algorith m iterati vely searches for an optimal p . It is guara n teed that this algo rithm conv e rges to a locally optimal solution. It should be noted that the a u thors in [21] prop osed a Max-SR-GD method to solve an optim ization p r oblem similar to (2 1). Howev er , the me th od Max -SR-GD in [ 2 1] is to optimize the linear p recodin g vector in secu re SSK sy stem s, 6 R ′ s ( v ) = 1 M N t N t X n =1 M X m =1 " log 2 N t X n ′ =1 M X m ′ =1 exp − P 1 tr( v H E n ′ ,m ′ n,m v ) 2 !! − log 2 N t X n ′ =1 M X m ′ =1 exp − P 1 tr( v H B n ′ ,m ′ n,m v ) 2 !!# (31) ∇ v R ′ s ( v ) = P 1 M N t N t X n =1 M X m =1      N t P n ′ =1 M P m ′ =1  exp  − P 1 · v H B n ′ ,m ′ n,m v 2  B n ′ ,m ′ n,m v  2 κ b · ln2 − N t P n ′ =1 M P m ′ =1  exp  − P 1 · v H E n ′ ,m ′ n,m v 2  E n ′ ,m ′ n,m v  2 κ e · ln2      (32) Algorithm 1 The prop osed Max-ASR-GD method for solving problem (30) 1: Initialize v 0 with satisfying tr( v 0 v H 0 ) ≤ N t , set th e step size µ , the minimum tolerance µ min and k = 0 2: Calculate R ′ s ( v k ) b y u sing (3 1) 3: Calculate ∇ v R ′ s ( v k ) b y u sing (3 2) 4: If µ ≥ µ min , goto step 5, otherwise stop and r eturn v k 5: Calculate v k +1 by using (35) and no rmalize v k +1 by using (36) 6: Calculate R ′ s ( v k +1 ) b y u sing (3 1) 7: If R ′ s ( v k +1 ) ≥ R ′ s ( v k ) , th e n goto step 8, otherwise, let µ = µ/ 2 and g oto step 4 8: Do k = k + 1 goto step 3 9: Output the v k , then calculate the SR R s ( v k ) by using (14) and (16) and it can be steadily extended to secu re SM systems. There exists two serio u s prob lems of facing Max-SR-GD: no closed- form expression for SR and h ig h-com p lexity . No closed -form expression me a n s th a t it req uires com plex comp utation to ev aluate the actual value of SR and its g radient per iteratio n . B. Pr oposed Max-ASR -SCA Method The optimization pro blem (30) is a n on-co n vex QCQP problem , wh ich motiv ates us to explore the SDR method . T he SDR is a powerful and computation ally ef ficient app roxima- tion technique , wh ich is particularly ap p licable to non -conve x QCQP prob lems. Many pra ctical expe r iences have indicated that SDR is capable of providing near optimal appro ximation s [36]. Let us introduc e a new matrix variable W = vv H , then we obtain the equivalence o f (3 0) as follows maximize W R ′ s ( W ) (37a) sub ject to : tr( W ) ≤ N t (37b) W  0 (37c) rank( W ) = 1 , (37d) where W  0 deno tes that W is a semi- d efinite ma tr ix. Howe ver, the above o ptimization pro blem still is intr a ctable due to the existence of the non-conve x r ank-o ne constraint. Actually , af te r removin g the non-co n vex rank-one co nstraint in the above optimizatio n pro b lem, (37) is relaxed a s maximize W R ′ s ( W ) (38a) sub ject to : tr( W ) ≤ N t (38b) W  0 . (38c) By d roppin g the ran k-one constrain t, the above SDR achieves a upper bou nd on the optimal solution to the primal p roblem (38). If the o ptimal solution W ∗ to (38) is rank- o ne, the optimal V ∗ is the eig e n vector c orrespon ding to the largest eigenv alue of W ∗ . Oth erwise, the ra ndomiza tio n technique may b e used to ob tain ap proxim a ti ve optimu m. In what follows, we f ocus o n solvin g the optimization p roblem in ( 3 8). In (38), the two constraints are conv ex sets. But its objectiv e function is in form of subtraction of two conve x fun ctions [37]. In general, th is difference is non-co ncave. Obviously , the objectiv e function in (3 8) is also non- concave. It is well- known that SCA is an efficient way to solve this no n-conve x DC p rogra m ming pro blem. T h e algor ithm begins with an initial feasible point, the non-c o n vex objective function is approx imated b y a strictly conve x arou nd this point. The resulting conv ex pr o blem is solved to obtain the feasible p oint for next iteration. The iteration is repeated until the stopp ing criterion is satisfied [38]. For convenience of presen tation, we r ewrite the objective function as R ′ s ( W ) = 1 M N t N t X n =1 M X m =1 [ f 1 ( W ) − f 2 ( W )] , (39) where f 1 ( W ) and f 2 ( W ) are defined as f 1 ( W ) = lo g 2 N t X n ′ =1 M X m ′ =1 exp − tr( WE n ′ ,m ′ n,m ) 2 ! , (40) f 2 ( W ) = lo g 2 N t X n ′ =1 M X m ′ =1 exp − tr( WB n ′ ,m ′ n,m ) 2 ! . (4 1 ) In (39), f 1 ( W ) and f 2 ( W ) are log-su m-exp fun ctions, wh ich are proved to be conv ex [3 7], as a result, R ′ s ( W ) is non- concave. It is well-known that a linear fun ction is bo th conve x and concave. I n the following, we em ploy the first-order T ay lo r 7 series expan sion around a feasible po int W 0 on f 1 ( W ) to transform f 1 ( W ) into a linear fu n ction. Due to f 1 ( W ) being conv ex and differentiable, th e fir st-o rder T aylo r app r oximation actually is a glob al underestima to r of the fu n ction [37], i.e. , the following ineq uality h olds f 1 ( W ) ≥ f 1 ( W 0 ) + tr [ ∇ f 1 ( W 0 ) ( W − W 0 )] , (42) where ∇ f 1 ( W 0 ) is defined as the g radient of the functio n f 1 ( W ) at th e point W 0 , which is derived as ∇ f 1 ( W 0 ) = − N t P n ′ =1 M P m ′ =1  exp  − tr  W 0 E n ′ ,m ′ n,m  2  E n ′ ,m ′ n,m H  2ln 2 N t P n ′ =1 M P m ′ =1 exp  − tr  W 0 E n ′ ,m ′ n,m  2  . (43) Then, given a fixed W k − 1 , th e p roblem (38) can be iterati vely computed by solvin g the following conve x subp roblem maximize W k 1 M N t N t X n =1 M X m =1 f ( W k , W k − 1 ) (44a) sub ject to : tr( W k ) ≤ N t (44b) W k  0 , (44c) with f ( W k , W k − 1 ) (45) = f 1 ( W k − 1 ) + tr [ ∇ f 1 ( W k − 1 ) ( W k − W k − 1 )] − f 2 ( W k ) where f ( W k , W k − 1 ) is in f o rm of d ifference o f line ar func- tion an d conve x functio n, thus it is a co ncave fu nction with respect to W k . Ther e fore, the pr oblem (44) is a c o n vex semi- definite p rogram ming (SDP) prob le m , which can be h andled conv e n iently and e fficiently b y using the con vex optimizatio n toolbox CVX or interior-point metho d. Finally , the detailed proced u res of the proposed Max-ASR-SCA method ar e pre- sented in Algo r ithm 2. Algorithm 2 The proposed Max -ASR-SCA metho d for solv- ing p roblem (38) 1: Initialization: find a f e asible po int v 0 , W 0 = v 0 v 0 H , with satisfying the constraints in pro blem (38), set the tolerance ǫ , and k = 0 2: Rep eat 3: Solve the prob le m (44) with W k , and ob tain W ∗ k 4: Set k ← k + 1 5: Update W k = W ∗ k 6: Calculate R ′ s ( W k ) by u sin g (3 9) 7: Un til | R ′ s ( W k ) − R ′ s ( W k − 1 ) |≤ ǫ is met 8: Output the optim a l W ∗ = W k − 1 It is worth no ting th at as long as the each app roxima tio n of objective function satisfies the tightness and differentiation condition s, the proposed Max- ASR-SCA method is guaranteed to conv e rge to a p oint th at satisfies the Kar ush-Kuhn- Tucker (KKT) optim ality cond itio ns of problem (3 8) [38]. Mea nwhile, we present a simple pr oof for the conver gence of the Max- ASR-SCA method in Appendix A . By using the p r oposed Max -ASR-SCA me th od, we actually obtain a SDR solu tion W ∗ to problem (3 7). It is worth while noting that the solu tion is an a pprox imately optima l solu - tion. I f the solution W ∗ is ran k-one , th e optimal v ∗ is the eigenv e ctor corre sponding to the largest eigen value of W ∗ . In fact, the solu tio n may not be rank on e, we may emp loy the ran d omization metho d to search an approx imated solution. Such a rando mization method has be em pirically found to provide efficient approximatio ns fo r a variety of applicatio ns [36]. Until now , we complete our d esign of th e propo sed two methods. C. Complexity Analysis and Comp arison In this subsectio n , we condu ct the complexity analysis o n our pro posed two linear preco ding methods and c o mpare them with the complexity of th e GD me th od in [21]. Here, consider that a matrix -vector multiplication y = Ax , where A ∈ C m × n costs 2 mn floatin g-poin t opera tions (FLOPs), and a m atrix- matrix product C = AB , where A ∈ C m × n and B ∈ C n × p , costs 2 mnp FLOPs (See Appendix C in [37] f o r d etails). For the pr oposed GD method in subsection A, its co m puta- tional co mplexity dep ends m ainly o n the following three parts: (a) comp ute the app roximatio n o f SR, (b) compu te the gradie n t of ASR, an d (c) the num ber of iteration s. After the complexity of expo nential and logar ithm oper a tio ns in (2 2) and ( 2 3) a r e omitted, we have the computation al co mplexity of p art (a ) C a = 2 M 2 N 2 t (4 N 2 t + N b + N e ) . (46) In (32), co mputing the terms E n ′ ,m ′ n,m and B n ′ ,m ′ n,m both req uire about 3 N 2 t FLOPs, where we ignore the complexity of c om- puting the H H Q − H b H and G H Q − H e G , which are in variant matrices fo r a specifical channel. I n fact, ∆ n ′ ,m ′ n,m is a sparse matrix, which h as at most fo ur non-z e r o eleme n ts. Therefo re, the co mplexity of part (b ) ma y be a pprox imately given by C b = 2 M 2 N 2 t (3 N 2 t + 2 N 2 t + 2 N t + 2 N 2 t ) . ( 4 7) Let D 1 denotes the numb er of iterations in Algo rithm 1, then the final computational co m plexity of the Max-ASR-GD method is as follows C Alg . 1 = 2 D 1 M 2 N 2 t (11 N 2 t + 2 N t + N b + N e ) . (48) For the Ma x -SR-GD me thod in [21], its comp lexity analysis is similar to (48). Let N samp denote the nu mber of realization s of no ise f o r accu rately ev alu a ting the actual SR, then the computatio nal complexity of ev aluating SR is about C e = 4 M 2 N t 2 N samp  2 N t 2 + N b + N e  . (49 ) It is worthwhile notin g that the actual SR in (14) r equires av e r aging over at least N samp = 50 0 realizations of n oise for accurately ev alua tin g SR. Similar ly , the com putationa l complexity of e valuating th e curr ent grad ient vector is C g = 2 M 2 N 2 t N samp (3 N 2 t + 2 N 2 t + 2 N t + 2 N 2 t ) . (50) Let D 2 denote the numb er of iterations for Algo rithm.1 in [21], th en its complexity is written as C = 2 D 2 M 2 N 2 t N samp (11 N 2 t + 2 N t + 2 N b + 2 N e ) . (51) 8 Finally , fo r the pro p osed Max- ASR-SCA m e th od, its c o m- putational complexity also dep ends on the following three aspects: (a) compu te the (39), (b ) solve the conve x op- timization pr oblem in (44), and (c) th e nu mber of iter a- tions. The computationa l co mplexity of aspect (a) is abo ut 2 M 2 N t 2  2 N 3 t + 3 N t 2  FLOPs. It is noted that the exact com- plexity of aspec t (b) depen ds heavily on the specific con vex solver . Here, it is assumed that the interior point method [39] is used. The o bjective f unction o f problem (44) co nsists of M N t conv ex fu nctions, one lin ear inequality co nstraint, and one line ar matrix inequ ality constraint. Acc o rding to [ 3 6], [39], this p r oblem in (44) may be solved with a worst-case complexity C sdp = O ( N 4 . 5 t log(1 /ς )) + 2 M 2 N 2 t (3 N 3 t + 4 N 2 t ) , ( 5 2) where ς is a given solutio n accuracy , and the secon d term is the comp lexity of constru cting the objective fun ction. Let D 3 denotes the numb er of iter ations in Algor ithm 2. T herefor e, the com plexity of the propo sed Max-ASR-SCA meth od in Algorithm. 2 is approximated as C Alg . 2 = D 3 C sdp + 2 D 3 M 2 N t 2  2 N 3 t + 3 N 2 t  . (53) In su m mary , from th e above complexity analysis, it can b e seen that the th ree method s all have a poly n omial complexity . The pr oposed Max-ASR-GD in Algorith m. 1 ha s the lowest complexity a mong the three schemes, an d the propo sed Max- ASR-SCA metho d is in b etween Max -ASR-GD and Max- SR- GD in [21] in terms of complexity du e to N samp ≫ N t . This means that Max-SR-GD is o f th e highest comp lexity among the three m ethods. From (48) and (53), it fo llows that the complexities of Max-ASR-GD an d Max -ASR-SCA have the orders O ( N 4 t ) and O ( N 5 t ) provided that o ther parameter s are fixed, r espectively . Th erefore, w h en the number of transmit antennas tends to large-scale, the p roposed Max -ASR-GD method ha s resulted in a dramatic redu ction on co m putation a l complexity compa r ed to Max-ASR-SCA an d Max -SR-GD. V . N U M E R I C A L R E S U LT S A N D D I S C U S S I O N S In this section, we pr esent o ur n umerical simulation results to ev aluate the p roposed two linear precod in g method s. I n the fo llowing simulation s, the noise power o f Bob and Eve is assumed to be equal, i.e. , σ 2 b = σ 2 e , an d the achievable SR is evaluated by usin g (6) and calcu lated by averaging over 50 realizations of channels. M e anwhile, some n ecessary parameters are set as f ollows: N b = N e = 2 ; µ = 0 . 5 , and µ min = 0 . 01 fo r Algorith m 1; ǫ = 0 . 001 for Algor ithm 2. First, in or d er to verify th e v a lid iness of using ASR as the design m etric. Fig. 2 plots the cu r ves of the a p prox im ated and simu lated actu al SRs versus signal-to- noise ratio (SNR), where Sim-SR stand s for the simulated actual SR for the sake of simplicity . It is no ted that the approxim ated and actual SR are calculated by ( 24) and (7), respectively . It can be seen that all ASR cur ves ar e very close to these of simulated actual SR for almost all SNR regions. In other words, the rate difference between ASR and simulated actua l SR is tri vial. Therefor e, it is reasonab le to employ the closed -form expression of ASR in (24) as the design metric. -15 -10 -5 0 5 10 15 SNR(dB) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 SR (bps/Hz) ASR N t =8 M=4 Sim-SR N t =8 M=4 ASR N t =8 M=2 Sim-SR N t =8 M=2 ASR N t =4 M=2 Sim-SR N t =4 M=2 Fig. 2. Approximated and simulated SR -15 -10 -5 0 5 10 15 SNR (dB) 0 1 2 3 4 5 SR (bps/Hz) Proposed Max-ASR-GD Proposed Max-ASR-SCA Max-SR-GD in [21] No linear precoding Fig. 3. SR performa nce with N t = 8 and M = 4 . Fig. 3 demonstrates the cu rves of SR per forman ce versus SNR for three linear preco ding methods Max-ASR-GD, Max- ASR-SCA, and Max-SR-GD with no linear prec oding schem e as a per forman ce ben chmark . It c an be seen that the three methods can achieve different SR p erform ance gains over the no preco ding case. The SR perfor mance of p roposed Max- ASR-GD metho d is worse than Max -ASR-SCA but tend s to conv e n tional Max-SR-GD in [21]. It has a h igh pro bability that the two based-GD method s, Max -ASR-GD and M a x-SR-GD, conv e rge to their locally op timal solution s while th e p roposed Max-ASR-SCA meth od can o btain the appro ximately optimal solution. The refore, th e pro posed Max-ASR-SCA achieves the best SR perfo rmance amon g the three pr ecoding scheme s. Fig. 4 shows the cumu lativ e distribution func tio n (CDF) curves of SR fo r the thr ee linear p recoding meth ods with different ty pical values of SNRs: -5 dB, 0d B, and 5dB. Th e curves are gener ated by calcu lating SR over 500 rea lizations of all chann els. F o r all thr e e methods, as SNR increases, the CDF curves of SR moves towards the right-han d side. This 9 0 1 2 3 4 5 SR (bps/Hz) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 CDF Proposed Max-ASR-GD Proposed Max-ASR-SCA Max-SR-GD in [21] No linear precoding SNR=0dB SNR=-5dB SNR=5dB Fig. 4. CDF of SR with N t = 8 and M = 4 . -15 -10 -5 0 5 10 15 SNR (dB) 0 0.5 1 1.5 2 2.5 3 SR (bps/Hz) Proposed Max-ASR-GD Proposed Max-ASR-SCA Max-SR-GD in [21] No linear precoding Fig. 5. SR performa nce with N t = 4 and M = 2 . means that the value o f SR b ecomes large as SNR incr eases. Additionally , for all three given values of SNR, the fou r methods have a decreasing order in ter ms of SR perform a nce as follows: Max-ASR-SCA, Max- SR-GD in [2 1], Max-ASR- GD, and witho ut linear precod ing. In ord er to examine the im pact of N t and M on SR perfor mance, Fig. 5 p resents the curves o f SR per forman ce versus SNR for the three metho ds with N t = 4 , an d M = 2 . It can be seen th a t th e the SR p erform ance trend in Fig. 5 is similar to that in Fig. 3. Compared to no linear pr ecoding , the prop osed M ax-ASR-SCA and Max- ASR-GD, an d the Max-SR-GD method in [21] a c hieve 1 b ps/Hz, 0 . 7 bps/Hz, and 0.6 bps/Hz SR improvement at SNR=15 dB, respectiv e ly . Compared to conventional Max-SR-GD, the SR impr ovemen t gain ach iev able by the propo sed Max -ASR-SCA is abou t 13% at SNR=15dB, which is a nice result. Fig. 6 illustrates the h isto g ram of pr obability mass fun ction (PMF) of num ber of iterations for the prop osed Max-ASR- SCA, and Max -ASR GD with conventional Max - SR-GD in 0 10 20 30 40 50 60 0 0.1 0.2 0.3 0.4 PMF Max-SR-GD in [21] 0 10 20 30 40 50 60 0 0.1 0.2 0.3 0.4 PMF Proposed Max-ASR-GD 10 20 30 40 50 60 Number of iterations 0 0.1 0.2 0.3 0.4 PMF Proposed Max-ASR-SCA Fig. 6. PMF of numbers of iteration s with N t = 4 , M = 2 , and SNR = 5 dB. [21] as a performan ce referen ce. I t is noted that 500 channe l trials are condu c ted to show the distribution of number of iterations. It can be seen that nea r 90 perc e nt trials converge within 30, 25 and 8 iter a tio ns for c o n ventional Max-SR- GD in [21], the pr oposed Max -ASR-GD, and Max-ASR-SCA methods, respec ti vely . In oth er words, the prop osed Max - ASR-GD con verges slightly faster th an con ventio nal Max -SR- GD but slower th an pr o posed Max-ASR-SCA. Althou g h the propo sed Max-ASR-SCA meth od co n verges within a fewer number of iterations, its co mplexity of ea ch iteration is hig her than that of the p r oposed Max-ASR-GD method. Fig. 7 presents th e curves of the average number o f itera- tions versus SNR for the above thr e e linear pr e coding schemes. It is o bserved th at the average number of iter ations of f o r the two based-GD methods decrea se as SNR increases. Howe ver, for the Max-ASR-SCA sch eme, the a verag e nu mber of itera- tions slightly increases wh en SNR increases. When SNR tends to be larger tha n 15dB, the average nu mber of iterations nearly keep constant for thr ee linear preco d ing schemes. From Fig. 7, it also c a n be seen that the three sch emes hav e an increasing order in ter ms o f con vergence p erform ances as f ollows: Max- SR-GD, Max-ASR-GD, and Max-ASR-SCA. T o make a detailed comp lexity compar ison amon g the three lin ear precoding schemes, Fig. 8 shows the curves of their co m plexities versus N t . I n acco r dance with Fig. 6, th e number s of iterations are set as D 1 = 25 , D 2 = 30 , D 3 = 8 for in suring 90 % trials conv e rge. It can be seen that the Max- 10 -15 -10 -5 0 5 10 15 20 25 SNR (dB) 0 5 10 15 20 25 30 35 40 45 Average number of iterations Proposed Max-ASR-GD Proposed Max-ASR-SCA Max-SR-GD in [21] Fig. 7. A vera ge number of iteratio ns versu SNR with N t = 4 and M = 2 . 10 20 30 40 50 60 N t 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 FLOPs Proposed Max-ASR-GD Max-SR-GD in [21] Proposed Max-ASR-SCA Fig. 8. Numbers of FLOPs versu N t with M = 4 . SR-GD method in [21] has an extremely high comp lexity compare d to the two p roposed metho ds. The complexities of the p roposed Max-ASR-GD a n d Max -ASR-SCA are two- order-of-mag nitude, and one-o rder-of-mag n itude lower than that of conv entional Max- SR-GD as th e value of N t is larger than 30, resp ectiv e ly . T hus, the pro p osed two metho ds leads to a significan t complexity reduction over M a x -SR-GD. This makes them app licable to the fu ture secure spatial modulation networks. Furthermo re, compar ed to Max-SR-GD and Max- ASR-SCA, the main a d vantage of the proposed Max-ASR-GD is its extreme ly-low-complexity . Esp ecially , as the value o f N t tends to medium -scale and large-scale, its compu tational complexity is far lower tha n tho se o f Max-SR-GD and Max- ASR-SCA. V I . C O N C L U S I O N In this pap er , we designed two linear precoding schemes for secure SM system s. In order to simplify the or ig inal optimization p roblem of Max- SR-GD an d reduce its com- putational complexity , a simple an d analytical expression of ASR fo r SR was der iv ed as a d esign or optimization metric. Subsequen tly , two low-complexity linear prec o ding schemes, namely Max-ASR-GD and Max- ASR-SCA, were developed based on convex optimization techniqu es. Ou r examination first demo nstrated that the p r oposed Max-ASR-SCA ou tper- forms the con ventio nal Max-SR-GD in terms of achieving a higher SR with a o ne-ord er-of-magnitude lower complexity . In add ition, th e pro posed Max- A SR-GD achieves a two-order- of-mag nitude lower com plexity tha n Max-SR-GD at the cost of a negligib le SR p erform ance loss, wh ich shows tha t the propo sed Max- ASR-GD strikes a g ood balance between the complexity and SR perf o rmance. The pro posed two low- complexity linear precoder s are attractive for fu ture MIMO networks with secure sp a tial m odulation . A P P E N D I X A P R O O F O F T H E C O N V E R G E N C E F O R T H E M A X - A S R - S C A First, conside r that W k is a feasible solution of problem (38) an d f 1 ( W ) is a conve x fu n ction with respe c t to W , so we ha ve the following ineq uality f 1 ( W k +1 ) ≥ f 1 ( W k ) + tr( ∇ f 1 ( W k ) ( W k +1 − W k )) , (54) where W k +1 is a updated solu tion o f problem ( 38) by solving the p roblem (4 4) with a given feasible p oint W k . Then, accordin g to th e objec tive function in ( 4 4a), we have the following inequality N t X n =1 M X m =1 f ( W k +1 , W k ) ≥ N t X n =1 M X m =1 f ( W k , W k ) . (55) Based on the de fin ition of f ( W , W 0 ) in (45), f ( W k +1 , W k ) and f ( W k , W k ) is given by f ( W k +1 , W k ) (56) = f 1 ( W k ) + tr [ ∇ f 1 ( W k ) ( W k +1 − W k )] − f 2 ( W k +1 ) , f ( W k , W k ) = f 1 ( W k ) − f 2 ( W k ) . (57) Finally , by substituting (56) and (57) into ( 55) and using the inequality in (54), we ha ve th e following ineq uality N t X n =1 M X m =1 [ f 1 ( W k +1 ) − f 2 ( W k +1 )] ≥ N t X n =1 M X m =1 [ f 1 ( W k ) − f 2 ( W k )] . (58) It is o bserved the achieved ob jectiv e fu n ction value in (38) is mo notone n on-d e creasing a n d b o unded as the number of iterations increases, thus we may conclud e the SCA algorithm conv e rges. Until now , we com plete the p roof of the co n ver- gence fo r the Max-A SR-SCA. 11 R E F E R E N C E S [1] R. Y . Mesleh, H. Haas, S. Sinanovic, W . A. Chang, and S. Y un, “Spatial modulati on, ” IE EE T rans. V eh. T echnol . , vol . 57, no. 4, pp. 2228–2241, Jul. 2008. [2] M. D. Renzo, H. Haas, and P . M. G rant, “Spatial modulation for multiple -antenna wireless systems: a surv ey , ” IEE E Commun. Mag . , vol. 49, no. 12, pp. 182–191, Dec. 2012. [3] J. Jegana than, A. Ghrayeb, and L . 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