Linear Precoding for Fading Cognitive Multiple Access Wiretap Channel with Finite-Alphabet Inputs

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Secondary-user nels, they are hardly implemented in practice. It is well known Receiver (SR) that practical inputs are drawn from finite constellation sets w1 such as phase-shift keying (PSK), pulse-amplitude modulation zE yR Primary-user F1,1 (PAM), or quadrature amplitude modulation (QAM). More Receiver (PR)-1 H1 H2 Eavesdropper G2 importantly, the common approach that designs linear precoder (ED) F2,1 ... G1 in a MIMO system under Gaussian inputs and then apply it to wJ the practical system may lead to significant performance loss F1,J [15], [16]. Therefore, the precoding design with finite-alphabet Primary-user F2,J Receiver (PR)-J inputs has drawn increasing research interest in recent years [17]–[28]. P1s1 P2s2 As illustrated in Fig. 1, we consider the underlay cog- Secondary-user Secondary-user nitive multiple access wiretap channel (CMAC-WT), where Transmitter (ST)-1 Transmitter (ST)-2 two secondary-user transmitters (STs) communicate with one Fig. 1: System model of the fading cognitive multiple access wiretap channel. secondary-user receiver (SR) in the presence of an eavesdropper (ED) and subject to interference threshold constraints results show that when considering finite-alphabet inputs, our at primary-user receivers (PRs). Each node in the system is proposed algorithm significantly outperforms the conventional equipped with multiple antennas. To the best of our knowl- Gaussian precoding method, which designs precoding matrices edge, this is a general model that has not been addressed yet. to maximize the average secrecy sum rate under Gaussian We design linear precoding matrices to achieve the maximum inputs, in the medium and high signal-to-noise ratio (SNR) average secrecy sum rate under finite-alphabet inputs and regimes. statistical CSI at STs. The problem setting is much closer The rest of this paper is organized as follows. Section to practical systems because it targets finite-alphabet inputs II introduces the system model and formulates the linear directly and exploits statistical CSI of fading channels. How- precoding problem, Section III develops a numerical algorithm ever, this problem is extremely difficult to solve due to two to maximize the average secrecy sum rate under finite-alphabet reasons: First, the computational complexity for evaluating inputs and statistical CSI, Section IV presents several numer- the average secrecy sum rate is prohibitively high. Second, ical results and Section V draws the conclusion. and more importantly, the optimization problem itself is a Notations: Boldface lowercase letters, boldface uppercase non-convex and non-deterministic polynomial time (NP)-hard letters, and calligraphic letters are used to denote vectors, problem. matrices and sets, respectively. The superscripts ( )T and ( )H A subset of non-convex optimization, which is called the represent transpose and Hermitian operations,· respective· ly. difference of convex functions (DC) optimization, has been [ ]+ denotes max( , 0); diag( ) represents a block diagonal studied extensively by exploiting its underlying structure [29]– matrix· whose diagonal· elements· are matrices. tr( ) is the trace [31]. DC optimization aims to maximize a DC function under of a matrix; vec( ) is a column vector formed by· stacking the some DC constraints. In [29], a basic outer approximation columns of a matrix;· denotes the Euclidean norm of a framework was proposed for solving DC problems. In [30], a vector; A B is the Kroneckerk·k product of two matrices A and new DC algorithm was introduced by exploiting the duality B; E( ) represents⊗ the statistical expectation; ( ) and ( ) theory of DC optimization. In [31], the authors presented denote· the real and image parts of a complex vectorℜ · or matrix;ℑ · the convex-concave procedure, which can be regarded as and are defined component-wise. I and 0 denote an iden- a special case of the algorithm in [30]. Since any twice ≥tity matrix≤ and a zero matrix, respectively, with appropriate continuously differentiable function is a DC function [29], dimensions; A 0 denotes the positive semidefiniteness of A. our linear precoding problem is a DC optimization problem. The symbol ( ) represents the mutual information; log( ) and However, no practical algorithm is known to construct a DC ln( ) are usedI for· the base two logarithm and natural logarithm,· decomposition for arbitrary twice continuously differentiable respectively.· function. Moreover, if we do not carefully design the DC representation of the average secrecy sum rate, the algorithms II. SYSTEM MODEL AND PROBLEM FORMULATION in [29]–[31] suffer from very slow convergence [32]. There- We consider the fading CMAC-WT depicted in Fig. 1. The fore, the DC representation is a main factor that affect the i-th ST has NT antennas, i =1, 2, the SR has NR antennas, performance of DC algorithms. i the ED has NE antennas, and the j-th PR has N antennas, We solve our problem efficiently by combining the convex- j j =1, 2, ..., J. The channel output at the SR, the ED and the concave procedure with an outer approximation framework. j-th PR are, respectively, given by We first exploit an accurate approximation of the average secrecy sum rate to reduce the complexity, and then reformulate yR = H1P1s1 + H2P2s2 + nR the approximated average secrecy sum rate as a DC function. zE = G1P1s1 + G2P2s2 + nE Subsequently, we generate a sequence of relaxed sets, which w = F P s + F P s + n , j =1, 2, ..., J (1) can be expressed explicitly as the union of convex sets, to j 1,j 1 1 2,j 2 2 j approach the non-convex feasible set. In this way, near opti- where Hi, Gi and Fi,j are complex channel matrices from the mal precoders are obtained by maximizing the approximated i-th ST to the SR, the ED, and the j-th PR, respectively; Pi average secrecy sum rate over these convex sets. Numerical is the linear precoding matrix at the i-th ST, i =1, 2; si is the 3 input data vector at the i-th ST with zero-mean and covariance III. LINEAR PRECODING UNDER FINITE-ALPHABET H Esi [sisi ]= I, i = 1, 2; nR, nE and nj are independent and INPUTS identically distributed (i.i.d.) zero-mean circularly symmetric In this section, we solve problem (6) under finite-alphabet complex Gaussian noises with covariance matrix 2I, 2I and σR σE inputs. We assume that each symbol of the input data vec- σ2I, respectively. j tor si is taken independently from an equiprobable discrete The channel matrices considered in this paper are modeled constellation with cardinality Mi, i = 1, 2. The average as [33] constellation-constrained mutual informations EH (s1, s2; yR) 1 1 2 2 I Hi = Φ H˜ iΨ , i =1, 2 and EG (s1, s2; zE ) can then be expressed respectively as [19] h hi I 1 1 2 ˜ 2 N N Gi = Φg GiΨgi , i =1, 2 1 H H n 1 1 E (s; yR) = log N E , R log 2 2 I − N F Φ F˜ Ψ (2) m=1 k=1 i,j = fj i,j fi,j , (i, j) ∀ X 2 X 2 HPe + nR + nR where H˜ , G˜ and F˜ are random matrices with i.i.d. zero- mk (7) i i i,j exp −k 2 k k k σR mean unit variance complex Gaussian entries; Φh, Φg and Φfj N N 1 are positive semidefinite receive correlation matrices of Hi, EG (s; zE ) = log N EG,n log G and F , respectively; Ψ , Ψ and Ψ are positive I − N E i i,j hi gi fi,j m=1 k=1 semidefinite transmit correlation matrices of Hi, Gi and Fi,j , X 2 X 2 GPe + nE + nE mk (8) respectively. exp −k 2 k k k σE We assume that the SR has instantaneous channel realiza- NT1 NT2 tions of H1, H2 , the ED has instantaneous channel realiza- where s sT sT T ; is a constant, equals to ; { } =[ 1 , 2 ] N M1 M2 tions of G1, G2 , and STs only know the transmit and re- P = diag(P , P ); e is the difference between d and { } 1 2 mk m ceive correlation matrices of H1, H2, G1, G2, Fi,j , (i, j) dk, with dm and dk representing two possible distinct signal {˜ ˜ ˜ ˜ ˜ ∀ } as well as the distributions of H1, H2, G1, G2, Fi,j , (i, j) . vectors from s. { ∀ } Under these assumptions, the following secrecy sum rate is Obviously, the evaluation and optimization of the above av- achievable [9]: erage mutual informations is a difficult task. In order to obtain + EH (s; yR) and EG (s; zE ), we need to calculate expectations (s1, s2; yR H) (s1, s2; zE G) I | − I | overI H and G asI well as n and n . Unfortunately, these ¯ ¯ + R E = EH (s1, s2; yR H=H) EG (s1, s2; zE G=G) I | − I | expectations have no closed-form expressions. Although we ¯ ¯ where H = [H1, H2], G = [G1, G2]; H and G represent the can use Monte Carlo method to estimate these expectations, instantaneous channel realizations of H and G, respectively. the computational complexity is prohibitively high especially For notational simplicity, we omit the given channel realization when the dimensions of H and G are large. condition in mutual information expressions and then the This difficulty can be mitigated by employing accurate average secrecy sum rate can be expressed as approximations of (7) and (8). Based on [21], EH (s; yR) and I + EG (s; zE ) can be approximated respectively as Ravg(P1, P2)= EH (s1, s2; yR) EG (s1, s2; zE ) . (3) I N N I − I 1 We maximize Ravg(P1, P2) subject to power constraints at A(s; yR) = log N log I − N STs and interference threshold constraints at PRs. The average m=1 k=1 X X −1 transmit power conforms to the power constraint βi: hq H H 1+ e P ΨhPemk (9) H H H 2σ2 mk Es tr Pisis P = tr P Pi βi, i =1, 2 (4) q R i i i i ≤ Y N N and the average interference power at the j-th PR is limited 1 A(s; zE ) = log N log by γj : I − N m=1 k=1 2 X X −1 H H H gq H H Es ,F tr F P s s P F 1+ e P Ψ Pe (10) i i,j i,j i i i i i,j 2σ2 mk g mk i=1 q E X h i Y 2 1 1 H 2 H H 2 where Ψh = diag(Ψh1 , Ψh2 ) and Ψg = diag(Ψg1 , Ψg2 ); F˜ ˜ ˜ = E i,j tr Pi (Ψf ) Fi,j Φfj Fi,jΨf Pi i,j i,j hq and gq represent the q-th eigenvalue of Φh and Φg, i=1 h i X 2 respectively. Approximations (9) and (10) are very accurate for H arbitrary correlation matrices and precoders, and the computa- = tr(Φf ) tr P Ψf Pi γj , j. (5) j · i i,j ≤ ∀ i=1 tional complexity of (9) and (10) is several orders of magnitude X The second equality in (5) holds because each element of lower than that of the original average mutual informations F˜ is i.i.d. complex Gaussian variable with zero-mean and [21]. i,j + ˜ By replacing Ravg(P1, P2) with [ A(s; yR) A(s; zE )] , unit variance, and Fi,j is independent to si. Then the average I − I secrecy sum rate maximization problem is formulated as problem (6) can be approximated as + maximize Ravg(P1, P2) maximize A(s; yR) A(s; zE ) P1,P2 (6) P1,P2 I − I (11) subject to (4) and (5). subject to (4) and (5). 4

A. Precoder vectorization Then problem (11) is converted into a vectorized form We reformulate problem (11) into a vectorized form by + maximize f(p) g(p) (25) employing the precoder vectorization technique [28], [34]. p∈P − This reformulation can better exploit the inherent structure where f(p) and g(p) are given below of (11). For convenience, we first reformulate A(s; yR) by N N I −1 precoder vectorization, and then the same procedure can be 1 gq T f(p)= log 1+ p Bmkp (26) applied for A(s; zE ) and the constraints of problem (11). 2 N σE · I H H m=1 k=1 q We start by rewriting e P ΨhPemk as X X Y mk N N −1 1 hq T 2 g(p)= log 1+ p Amkp (27) H H H H N σ2 · e P ΨhPemk = e P Ψh Piemk,i (12) m=1 k q R mk mk,i i i X X=1 Y i=1 X and is the feasible set T T T P where emk = [emk,1, emk,2] . Using the following matrix T T = p p Cip βi,i =1, 2, p Dj p γj , j . (28) equation [35]: P ≤ ≤ ∀ T T T n o tr(A BAC) = vec(A) (C B) vec(A) (13) The feasible set is convex and compact because it can be · ⊗ · interpreted geometricallyP as the intersection of multiple ellip- H H + emk,iPi Ψhi Piemk,i can be rewritten as soids. The objective function f(p) g(p) is continuous over because both f(p) and g(p) are− continuous functions. eH PH Ψ P e = tr PH Ψ P ET mk,i i hi i mk,i i hi i mk,i Therefore,P the existence of a globally optimal solution is H = vec(Pi) (Emk,i Ψhi ) vec(Pi) (14) guaranteed by the Weierstrass extreme value theorem [36]. In · ⊗ · addition, the operator [ ]+ has no effect on the optimal value where E = (e eH )T . By letting mk,i mk,i mk,i of problem (25) and thus· can be removed from the objective vec(P ) pˆ function because p = 0 always belongs to . However, it pˆ = 1 , p = ℜ{ } (15) P vec(P2) pˆ is extremely difficult to solve problem (25) due to the fol- ℑ{ } lowing reasons: First, both f(p) and g(p) are neither convex and nor concave, thus (25) is a purely non-convex optimization ˆ 1 problem. Second, problem (25) is a NP-hard problem because Amk = diag Emk,1 Ψh1 , Emk,2 Ψh2 (16) 2 · ⊗ ⊗ a specialized problem with particular parameters Amk and ˆ ˆ Amk Amk Bmk is NP-hard [37]. Amk = ℜ{ } −ℑ{ } (17) Aˆ Aˆ Although f(p) g(p) is non-concave, it can be expressed mk mk ℑ{ } ℜ{ } as a DC function− by adding a convex term A(s; yR) can be expressed alternatively as I T N N σ(p)= k p p, k> 0. (29) 1 · A(s; yR) = log N log I − N We can prove that both f(p)+ σ(p) and g(p)+ σ(p) are m=1 k=1 convex functions if X X −1 hq T 1+ p Amkp . (18) σ2 · k α max tr(Φh) λmax(Ψh), tr(Φg) λmax(Ψg) (30) q R ≥ · · · Y 2 1 where α = m,k emk , λmax( ) represents the maximum T 2 2 k k · Here Amk 0 because p Amkp is equal to Ψh Pemk , eigenvalue of a matrix. Then p p p p k k [f( )+ σ( )] [g( )+ σ( )] which is non-negative. is an explicitP DC function, and problem (25)− can be solved ˆ Similarly, we define Bmk and Bmk as by DC algorithms. However, this DC representation is not efficient because k is too large [32]. Through extensive ˆ 1 Bmk = diag Emk, Ψg1 , Emk, Ψg2 (19) 2 · 1 ⊗ 2 ⊗ simulations, we observe that even when each node in the system is only equipped with two antennas, the DC algorithm Bˆ mk Bˆ mk Bmk = ℜ{ } −ℑ{ } 0 (20) with this representation cannot converge within hundreds of Bˆ mk Bˆ mk ℑ{ } ℜ{ } thousands of iterations. Therefore, a computationally efficient Cˆ i and Ci as DC representation of the approximated average secrecy sum rate is crucial for designing our algorithm. Cˆ i = diag I (2 i)I, I (i 1)I (21) ⊗ − ⊗ − ˆ ˆ C i Ci B. Outer Approximation of the Feasible Set Ci = ℜ{ } −ℑ{ } 0 (22) Cˆ i Cˆ i ℑ{ } ℜ{ } We first rewrite (25) with an additional hyperrectangle init B Dˆ and D as j j maximize f(p) g(p) (31) p∈P∩Binit − ˆ Dj = tr(Φfj ) diag I Ψf1,j , I Ψf2,j (23) · ⊗ ⊗ in which the hyperrectangle init is given by ˆ ˆ B Dj Dj Dj = ℜ{ } −ℑ{ } 0. (24) p l p u (32) Dˆ j Dˆ j init = ( init) ( init) . ℑ{ } ℜ{ } B B ≤ ≤ B n o

To ensure that problems (31) and (25) are equivalent, the to compute ϕ( k) , that is, { F } hyperrectangle init should contain the feasible set , i.e., B P ϕ( k)= max ϕ( ( i)). (39) . Let ui and li denote the i-th component of u( ) P⊆Binit Binit F 1≤i≤k C B and l( init), respectively. init can be obtained via solving the followingB concave maximizationB problem Based on (38), achieving ϕ( init) may need a sufﬁciently large number of iterations, whichF is not practical when the

ui = maximize pi (33) computational time is concerned. In order to address this issue, p∈P we also generate a lower bound of ϕ( ) in each iteration. Finit where pi is the i-th component of p. Due to the symmetry of Denote the optimal solution for ϕ( k) at the k-th iteration opt opt F problem (33), li can be set as ui. by (Qk , pk ). We extract a feasible solution of problem − T opt By introducing a new variable Q = pp , we define a (31) from Qk , and the corresponding approximated average set function ϕ( ) as the optimal value of the following secrecy sum rate is denoted by ϕL( k), which serves as a F F optimization problem lower bound of ϕ( init). In the remainingF part of this subsection, we construct , ϕ( ) maximize F (Q) G(Q) (34) k explicitly as the union of convex sets ( i) . The F (Q,p)∈F − approximated{F } average secrecy sum rate maximization{C B proble} m where F (Q) and G(Q) are given below over ( i) and an efficient method to generate the lower bound C B ϕL( k) are investigated in the next subsection. N N F 1 gq −1 For ease of exposition, we first define a convex set ( ) as F (Q)= log 1+ tr(BmkQ) (35) N σ2 · C B m=1 k=1 q E T X X Y Q pp , tr(CiQ) βi,i =1, 2, N N ( ), (Q, p) ≤ (40) 1 hq −1 C B Q p D Q Q A Q (36) ( ( , ) ( ), tr( j ) γj , j ) G( )= log 1+ 2 tr( mk ) . ∈ S B ≤ ∀ N σR · m=1 k=1 q where ( ) is another convex set given by X X Y S B Note that F (Q) and G(Q) are convex functions because Q L LT l l T 0 −1 p p + ( ) ( ) , 1) log k qfq,k can be written as log kexp( q lnfq,k); − − T B · B ≥T −  Q Up Up +u( ) u( ) 0, 2) log k exp(gk) is convex whenever gk are convex [38]. ( ), (Q, p) − − B · B ≥ (41) P Q P P S B  T T  Therefore, F (Q) G(Q) is a DC function. Furthermore, when  Q Lp Up +l( ) u( ) 0,  P −  − − B · B ≥  init, given by l p u F ( ) ( )  B ≤ ≤ B  T  T T  Q = pp , tr(Dj Q) γj , j, with L =l( ) p and U = u( ) p . The following two = (Q, p) ≤ ∀ (37) p  p  init propositions areB · the foundation forB constructing· . F ( p init, tr(CiQ) βi,i =1, 2) k ∈B ≤ Proposition 1: If we split the initial hyperrectangle{F } Binit is equivalent to the feasible set , ϕ( init) serves as the into K smaller hyperrectangles such that = ... K , P F Binit B1 ∪ ∪B optimal value of problem (31). However, it is very difficult then ( ) ... ( K). Finit ⊆C B1 ∪ ∪C B to obtain ϕ( init) directly because init is a non-convex Proof: See Appendix A. F F set. Although we can use semidefinite relaxation (SDR) to Proposition 2: If we split a hyperrectangle into two B relax init into a convex set by relaxing the non-convex part smaller hyperrectangles 1 and 2 such that = 1 2 F T B B B B ∪B Q = pp , the solution obtained by SDR is not optimal and 1 2 = , then ( 1) ( 2) ( ). and cannot be improved iteratively. Hence we need tighter BProof:∩B See∅ AppendixC B A. ∪C B ⊆C B relaxations to overcome the shortcomings of SDR. With the help of Proposition 1, the first relaxed set 1 is The key idea of our proposed precoding algorithm is to obtained F generate a sequence of asymptotically tight sets k to ap- {F } = ( ). (42) proach init, and then ϕ( init) can be approached iteratively F1 C Binit from aboveF by solving aF sequence of optimization problems Similarly, in the second iteration, we generate 2 by partition- ϕ( k) . The sequence k should satisfy the following F { F } {F } ing the initial hyperrectangle init into two non-intersection three properties: hyperrectangles and B B1 B2 1 2 ... init = ( ) ( ) . (43) F ⊇F ⊇ ⊇F F2 C B1 ∪C B2 ⊆F1 lim ϕ( k)= ϕ( init) k→∞ F F We continue this process to generate a sequence of relaxed k sets k satisfying (38). At the k-th iteration, is split {F } Binit k = ( i), k (38) into non-intersection hyperrectangles such that F C B ∀ k 1, 2, ..., k i=1 B B B [ k = ( 1) ... ( k). (44) where ( i) is a convex set to be defined in (40). The first F C B ∪ ∪C B C B property implies that ϕ( k) is a monotonically decreasing The outer approximation algorithm is summarized in Algo- { F } sequence bounded below by ϕ( init). The second property rithm 1. guarantees that ϕ( ) can beF readily obtained by the se- The convergence of Algorithm 1 is presented by the follow- Finit quence ϕ( k) . The last property provides a trackable way ing proposition. { F } 6

Algorithm 1 : The outer approximation algorithm Therefore, by replacing F (Q) G(Q) with a concave lower − 1) Initialization: Set the maximum number of iterations bound B Kmax, k =1, = init , 1 = ( init), U1 = ϕ( 1) ˆ T {B } F C B F F (Q; Qc)=F (Qc)+tr F (Qc) (Q Qc) G(Q) (49) and L = ϕL( ). ∇ − − 1 F1 2) Stopping criterion: if k Kmax go to the next step, oth- we obtain the following concave maximization problem ≤ erwise STOP. ˆ maximize F (Q; Qc). (50) 3) Partition criterion: (Q,p)∈C(B) a) select g = argmax B ϕ( ( )) . B B∈ C B The convex-concave procedure obtains a locally optimal b) split g along any of its longest edge into two small B solution of problem (45) by solving a sequence of concave hyperrectangles, I and II , with equal volume. B B maximization problems (50) with different Qc. Once the c) remove g from B, and add I and II into B. B B B optimal solution of (50) in the ﬁrst iteration is found at d) compute the upper and lower bounds of ϕ( init) ∗ ∗ F initial Qc, denoted as Q1, the algorithm replaces Qc with Q1 and then solve (50) again. At the n-th iteration, the optimal k+1 = ( ) F C B solution of (50) is obtained by replacing Q with Q∗ , which B∈B c n−1 [ is the optimal solution at the (n 1)-th iteration. The convex- Uk+1 = ϕ( k+1) = max ϕ( ( )) − F B∈B C B concave procedure for solving problem (45) is summarized in

Lk = ϕL( k ). Algorithm 2. +1 F +1 Algorithm 2 : The convex-concave procedure 4) Set k := k +1 and go to step 2). 1) Initialization: Given tolerance ǫ> 0, choose a random initial point Q 0, set N =1, s = F (Q ) G(Q ), 0 0 0 − 0

Proposition 3: The sequence ϕ( k) converges to ∗ ∗ ∗ s1 = F (Q1) G(Q1). Let Qn represent the optimal ϕ( ), i.e., ε > 0, K > 0, such{ F that} k >K implies − Finit ∀ ∃ solution of (50) at the n-th iteration. ϕ( ) < ϕ( k) < ϕ( )+ ε. init init 2) Stopping criterion: if sn sn−1 >ǫ go to the next step, F Proof: SeeF AppendixF B. | − | It is worth remarking that each relaxed set k is tighter F otherwise STOP. than the set relaxed by SDR. We denote sdr = Q Q 3) Convex approximation: F | T 0, tr(CiQ) βi,i =1, 2, tr(Dj Q) γj , j . Since pp Q Q∗ Q∗ ≤ ≤ ∀ a) set c = n and solve problem (50) to obtain n+1. 0, we have Q (Q, p) k for any k. Thus the | ∈ F ⊆ Fsdr solution obtained by Algorithm 1 is better than that of the ∗ ∗ ∗ b) set sn+1 = F (Qn+1) G(Qn+1) and Qopt = Qn+1. SDR method. − 4) Set n := n +1 and go to step 2). C. DC Optimization Over the Convex Set 5) Output: Qopt and sn. In this subsection, we maximize the approximated average sum rate over the convex set ( ) by employing the convex- The stopping criterion in Algorithm 2 is guaranteed to be concave procedure [31]. TheC convex-concaveB procedure is a satisﬁed due to the following proposition. Proposition 4: The sequence sn generated by Algorithm general polynomial time algorithm for solving DC problems, { } and it works quite well in practice [39]–[41]. We ﬁrst rewrite 2 is monotonically increasing, i.e., sn+1 sn. Proof: Since the feasible set of (50)≥ does not change in the optimization problem as follows ∗ each iteration, the optimal solution in the n-th iteration Qn is ϕ( ( )) = maximize F (Q) G(Q). (45) a feasible point in the (n + 1)-th iteration. Thus we have C B (Q,p)∈C(B) − ∗ ∗ ∗ ∗ Fˆ(Q ; Q ) Fˆ(Q ; Q )= sn. (51) The objective function of problem (45) is a DC function, and n+1 n ≥ n n the convex part F (Q) can be lower bounded by its tangent at According to (46), it follows that any point Qc 0 ∗ ∗ ∗ ∗ sn+1 = Fˆ(Q ; Q ) Fˆ(Q ; Q ). (52) n+1 n+1 ≥ n+1 n F (Q) F (Q )+tr F (Q )T (Q Q ) (46) ≥ c ∇ c − c Therefore sn is monotonically increasing. Since problem{ } (45) is non-convex, Algorithm 2 is not where F (Q ) is the gradient of F (Q) at Q ∇ c c guaranteed to converge to the globally optimal value ϕ( ( )). T C B 1 gq Bmk Therefore, by embedding Algorithm 2 into Algorithm 1, we Q opt F ( c)= wmk 2 · (47) ∇ −N σ + gq tr(BmkQ ) obtain a near optimal solution Qk and the corresponding m,k q E c X X · approximated upper bound of ϕ( ) at the k-th iteration of Finit with Algorithm 1. Simulation results show that the gap between 2 the approximated upper bound and the actual upper bound is 1 exp σ + gq tr(BmkQc) w = E · . (48) usually very small because Algorithm 2 is insensitive to the mk 2 ln(2) · exp σ + gq tr(BmkQ ) k E · c initial point Q0. P 7

opt After obtaining Qk at the k-th iteration, we need to get a A. Convergence and Complexity Analysis feasible precoder pair (P1, P2) and the corresponding lower The convergence behavior of the proposed algorithm is bound ϕL( k). The feasible precoders can be obtained by demonstrated by considering a two-user fading CMAC-WT F opt extracting a feasible solution of (31) from Qk . There are with two STs, one SR, one ED, and one PR. Each node in the several rank one approximation methods to do this, and we system has two antennas. The correlation matrices are given adopt the Gaussian randomization procedure [42], which is by summarized in Algorithm 3. Φh = C(0.3), Ψh1 = C(0.95), Ψh2 = C(0.85)

Algorithm 3 : Gaussian randomization procedure Φg = C(0.6), Ψg1 = C(0.4), Ψg2 = C(0.95)

1) Given a number of randomizations L, and set l =1. Φf = C(0.5), Ψf1 = C(0.3), Ψf2 = C(0.5). (55) 2) If l L go to the next step, otherwise STOP. ≤ opt The maximum transmit power is constrained by β1 = β2 = 3) Generate ξl N(0, Q ), and construct a feasible point ∼ k 2. The interference threshold is given as γ = 0.2. The input data vectors s1 and s2 are drawn independently from BPSK p˜l 2 2 constellation, and the noise power is set as σR = σE =0.1. ξl p˜ 1 l = T T . ξl Ciξl ξl Dj ξl max i=1,2, ∀j 0.9 { βi } { γj } q 0.8 4) Set l := l +1 and go to step 2). 0.7 5) Choose p˜ = argmax1≤l≤L f(p˜l) g(p˜l). − 0.6 6) Set ϕL( k)= f(p˜) g(p˜). F − 0.5 7) Recover (P1, P2) from p˜. 0.4 Cumulative Distribution 0.3

0.2 D. Complexity Analysis 0.1

The computational complexity of Algorithm 1 is analyzed 0 0.86 0.865 0.87 0.875 0.88 0.885 0.89 0.895 0.9 as follows. In each iteration, Algorithm 1 invokes Algorithms The Output of Algorithm 2 2 and 3 twice to calculate the approximated upper bound Fig. 2: Empirical cumulative distribution of the output sn of Algorithm 2 from 3000 and the lower bound. Since the complexity of Algorithm 3 random initial points. is negligible, the complexity order for Algorithm 1 is given The empirical cumulative distribution of the output sn of by Algorithm 2 from 3000 random initial points is shown in Fig. 2. The tolerance ε in Algorithm 2 is set as 0.002. l( ) 2K C (53) B max · and u( ) are given as l( ) = √2 1, u( ) = √2 1. The empiricalB cumulative distributionB − illustrates· B that Algor· ithm 2 where Kmax is the maximum number of iterations, and C is the complexity order for Algorithm 2. Algorithm 2 obtains is insensitive to the initial point. Therefore, although problem a local maxima of problem (45) by solving a sequence of (45) is non-convex, the approximated upper bound obtained concave maximization problems (50). Each concave maxi- by Algorithm 2 is accurate. mization problem (50) can be solved by the interior point 0.9 3 method, and the complexity order is about (N ) [38], where 0.89 N = 4(N 2 + N 2 )2 + 2(N 2 + N 2 ) isO the total number T1 T2 T1 T2 0.88 of optimization variables in problem (50). Assuming that Algorithm 2 solves problems (50) T times, the complexity 0.87 order for Algorithm 2 is given by (T N 3). Based on 0.86 O · (53), the overall complexity order for Algorithm 1 is then 0.85 (2K T N 3). O max · 0.84

0.83 Evolution of the Objective Function IV. NUMERICAL RESULTS 0.82

In this section, we provide numerical results to demonstrate 0.81 Approximated Upper Bound Lower Bound the efficacy of our proposed algorithm for the fading CMAC- 0.8 0 5 10 15 20 25 30 35 40 45 50 WT under finite-alphabet inputs. For illustration purpose, we Number of Iterations adopt the exponential correlation model: Fig. 3: Evolution of the objective function in (31) with BPSK inputs. |i−j| [C(ρ)]i,j = ρ , (i, j) (54) ∀ Fig. 3 illustrates the evolution of the approximated upper where the scalar ρ [0, 1) depicts the interference coupling bound and the lower bound of ϕ( init). In order to guarantee between different antennas.∈ that the approximated upper boundF is accurate enough, the 8 tolerance ε in Algorithm 2 is set as 0.001. In each iteration 1) In the low SNR regime, our proposed precoding al- of Algorithm 1, we invoke Algorithm 2 to generate the gorithm and the Gaussian precoding method have the same approximated upper bound, which can be seen as the actual performance. According to [17], the low-SNR expansion of upper bound of ϕ( init) according to the result in Fig. 2. the mutual information is irrelevant to the input distribution, We also invoke AlgorithmF 3 to generate feasible precoding thus the optimal precoders designed under Gaussian inputs are matrices and the corresponding lower bound ϕL( init). Note also optimal for finite-alphabet inputs case. that when all hyperrectangles in B shrink down toF a point, we 2) In the medium and high SNR regime, our proposed can ensure that the approximated upper bound serves exactly precoding algorithm offers much higher average secrecy sum as the actual upper bound. From the figure, we can see that rate than the Gaussian precoding method. In Fig. 4, the after 10 iterations, the gap between the approximated upper normalized optimal precoders designed by our proposed pre- bound and the lower bound is less than 0.005. Moreover, near- coding algorithm in the high SNR regime are given by optimal precoders within 0.002 tolerance is obtained through 1 opt 0.663+0.008i 1.188+0.277i Algorithm 1 after 30 iterations. P = − (61) σ 1 0.663+0.008i 1.188+0.277i − 1 opt 0.578+0.399i 1.209 0.459i B. Comparison with Other Possible Methods P = − − . (62) σ 2 0.578+0.399i 1.209 0.459i In this subsection, we consider a secure cognitive radio − − system that has two STs, one SR, one ED, and one PR. Each Equations (61) and (62) imply that when the noise power σ2 node in the system has two antennas. The correlation matrices is decreased, we should reduce the optimal transmit power opt H opt opt H opt are given by tr (P1 ) P1 and tr (P2 ) P2 such that the average secrecy sum rate is kept at the maximum value 1.0265 bpz/Hz Φh = C(0.25), Ψh1 = C(0.95), Ψh2 = C(0.9) in the high SNR regime. Furthermore, the performance of

Φg = C(0.75), Ψg1 = C(0.5), Ψg2 = C(0.3) the Gaussian precoding method degrades severely with the

Φf = C(0.5), Ψf1 = C(0.8), Ψf2 = C(0.5). (56) increasing SNR in the high SNR regime. The reason is that both EH (s; yR) and EG (s; zE ) in (7) and (8) will saturate The transmit power constraint is set as β1 = β2 = β =2. The at log NIin the high SNRI regime. Therefore, if we do not interference thresholds γ1 =0.2 and γ1 =0.02 are considered. carefully control the transmit power, the average secrecy 2 2 The modulation is QPSK and the noise variance σR = σE = sum rate with finite-alphabet inputs EH (s; yR) EG (s; zE ) 2 2 σ . Then the SNR can be defined as SNR = β/σ . approaches zero in the high SNR regime.I Since− the GaussianI Figs. 4 and 5 depict the comparison results among the Gaus- precoding method ignores the saturation property of finite- sian precoding method and no precoding case. The Gaussian alphabet inputs systems, the corresponding average secrecy precoding method is to design transmit covariance matrices sum rate with finite-alphabet inputs degrades severely in the that maximize the average secrecy sum rate under Gaussian high SNR regime. signaling, i.e., 3) Since the average secrecy sum rate for the Gaussian

maximize EH1,H2 (R1) EG1,G2 (R2) precoding method decreases with the increasing SNR in the Q1,Q2 − high SNR regime, we can use a portion of the available subject to tr(Qi) βi, i =1, 2 (57) ≤ transmit power to make sure that the SNR is maintained at a tr(Φfj ) tr(Q1Ψf1,j +Q2Ψf2,j ) γj , j certain level. The average secrecy sum rate is then kept at its · ≤ ∀ maximum value. This simple power control method has been where Qi is the transmit covariance matrix of the i-th ST, i =1, 2; R and R are given by used in [11], [12] to improve the secrecy sum-rate performance 1 2 at the high SNR regime. 1 1 R = log det(I + H Q HH + H Q HH ) (58) 4) The interference threshold constraints have a huge impact 1 σ2 1 1 1 σ2 2 2 2 R R on the system performance. For example, when SNR is 20 dB, 1 1 R = log det(I + G Q GH + G Q GH ). (59) the average secrecy sum rate is 0.90 bps/Hz and 0.31 bps/Hz 2 σ2 1 1 1 σ2 2 2 2 E E for γ1 = 0.2 and γ1 = 0.02, respectively. More specifically, Problem (57) is a DC optimization problem, thus it can be given the set of all feasible precoding matrices solved efficiently by DC algorithms proposed in [41]. After H tr Pi Pi βi,i =1, 2, obtaining the optimal transmit covariance matrices (Q¯ 1, Q¯ 2), ≤ = (P , P ) 2 (63) we can evaluate the finite-alphabet based average secrecy sum  1 2 H  1 1 P tr(Φfj ) tr Pi Ψfi,j Pi γj , j 2 2  · ≤ ∀  rate under the corresponding optimal precoders (Q¯ , Q¯ ). In  i=1  1 2 X the no precoding case, we set precoding matrices as P = i we define the following parameters  βi I, i =1, 2, and then scale them down to meet interference NTi γj threshold constraints: β¯i = min min ,βi (64) 1 1≤j≤J tr(Φf ) λmin(Ψf ) 2 − 2 j i,j tr(Φfj ) n · o P¯ PH Ψ P P i = max tr i fi,j i i. (60) where λ (A) represents the smallest eigenvalue of A. Then 1≤j≤J γj · · min i=1 X for all (P1, P2) , we can easily prove that Based on the results in Figs. 4 and 5, we have the following ∈P H tr P Pi β¯i, i =1, 2. (65) remarks: i ≤ 9

1.2 1.2 QPSK, Proposed Algorithm QPSK, Proposed Algorithm QPSK, Gaussian Precoding QPSK, Gaussian Precoding 1 QPSK, No Precoding 1 QPSK, No Precoding

0.8 0.8

0.6 0.6

0.4 0.4 Average Secrecy Sum Rate (bps/Hz) Average Secrecy Sum Rate (bps/Hz) 0.2 0.2

0 0 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 SNR (dB) SNR (dB)

Fig. 4: The interference threshold at the PR is 10 dB less than the transmit Fig. 5: The interference threshold at the PR is 20 dB less than the transmit power (γ1 = 0.2). power (γ1 = 0.02).

¯ Equation (65) implies that when βi < βi, i = 1, 2, the 1 power constraints in are inactive, i.e., only a portion of BPSK P 0.9 QPSK the available transmit power can be used in order to meet all 8PSK interference threshold constraints. In the case of Figs. 4 and 0.8 5, (β¯1, β¯2) is calculated as 0.7 0.6 (0.5, 0.2) γ1 =0.2 (β¯1, β¯2)= (66) 0.5 (0.05, 0.02) γ1 =0.02 ( 0.4

Since (β1,β2) = (2, 2), the sum rate performance in Figs. 4 0.3

and 5 is only constrained by interference threshold constraints. Average Secrecy Sum Rate (bps/Hz) 0.2

5) The performance of no precoding case is very poor 0.1 because we do not exploit any statistical CSI from STs to 0 the SR and the ED. 0 5 10 15 20 25 30 35 SNR (dB)

Fig. 6: Average secrecy sum rate for the fading CMAC-WT with different modulations. C. Comparison of Different Modulations Finally, we investigate the average secrecy sum rate with presented a two-layer precoding algorithm, which exploits sta- different modulations. We consider a secure cognitive radio tistical CSI of fading channels, to maximize the approximated system with two STs, one SR, one ED and two PRs. Each average secrecy sum rate. The key idea of our algorithm node is equipped with two antennas. The correlation matrices is to ﬁnd a computationally efﬁcient DC representation of are given by the approximated average secrecy sum rate. By introducing a new matrix variable, we have reformulated the approximated Φh = C(0.3), Ψh1 = C(0.9), Ψh2 = C(0.95) average secrecy sum rate as a DC function, and then generated

Φg = C(0.6), Ψg1 = C(0.7), Ψg2 = C(0.2) a sequence of relaxed sets to approach the non-convex feasible set. Each relaxed set can be expressed as the union of con- Φf1 = C(0.4), Ψf1 1 = C(0.6), Ψf2 1 = C(0.4) , , vex sets. Finally, near optimal precoding matrices have been Φ 2 = C(0.5), Ψ 1 2 = C(0.3), Ψ 2 2 = C(0.5). (67) f f , f , obtained iteratively by maximizing the approximated average The maximum transmission power at the i-th ST is given as secrecy sum rate over a sequence of relaxed sets. Several numerical results have been provided to demonstrate β1 = β2 =2. The interference threshold at the j-th PR is set 2 2 2 the efﬁcacy of our proposed precoding algorithm. They have as γ1 = γ2 =0.2. The noise variance is σR = σE = σ . Fig. 6 plots the average secrecy sum rate with BPSK, QPSK also shown that the proposed precoding algorithm is superior and 8PSK modulations. Results in Fig. 6 show that the average to the conventional Gaussian precoding method and no pre- secrecy sum rate is an increasing function with respect to coding case in the medium and high SNR regimes. the order of modulation. They also indicate that our proposed APPENDIX A precoding design can achieve robust performances for a large PROOFS OF PROPOSITIONS 1-2 range of SNR with different modulations. Proof of Proposition 1: We rewrite init as the union of K subsets F V. CONCLUSION K In this paper, we have considered the precoding design for = ¯i (68) Finit F the fading CMAC-WT with ﬁnite-alphabet inputs. We have i=1 [ 10

where ¯i is given by APPENDIX B F PROOF OF PROPOSITION 3 ¯i = (Q, p) (Q, p) init, p i . (69) F | ∈F ∈B Proof: Since ϕ( k) is a monotonically decreasing ¯ { F } For any (Q, p) i, the following inequalities hold sequence lower bounded by ϕ( ), the limit of ϕ( k) ∈ F Finit { F } T exists [36]. Suppose that (p l( i)) (p l( i)) 0 (70) − B · − B ≥ p u p u T 0 (71) lim ϕ( k)= v > ϕ( init) (81) ( ( i)) ( ( i)) k→∞ F F − B · − B T ≥ (p l( i)) (p u( i)) 0 (72) then for any ε > 0, there exists K > 0 such that for any − BT · − B ≤ Q = pp , l( i) p u( i). (73) k>K, B ≤ ≤ B Thus, ¯i can be rewritten as v < ϕ( ( g)) < v + ε. (82) F C B ¯ Let r( ) denote the length of the longest edge of a hyperrect- i = (Q, p) (Q, p) init, p i ( i). (74) B F | ∈F ∈B ∩ S B angle satisfying init. In each iteration of Algorithm 1, T ¯ T B B⊆B By relaxing Q = pp in i into Q pp , one can easily we divide g along r( g) into two hyperrectangles. Therefore, F B B obtain the following r( g) should satisfy the following condition: B ¯ i ( i), i. (75) lim r( g)=0. (83) F ⊆C B ∀ k→∞ B Therefore, ( ) ... ( K). This completes the init 1 We further denote the center of g by pg, i.e., pg = (l( g)+ proof. F ⊆ C B ∪ ∪C B B B u( g))/2. When r( g) 0, we have Proof of Proposition 2: We divide ( ) into two subsets B B → S B T ( g) (Q, p) Q = pgpg , p = pg . (84) ( )= 1( ) 2( ) (76) S B →{ | } S B S B ∪ S B Therefore, ( g) converges to a point when pg belongs to the C B where 1( ) and 2( ) are given by feasible set , otherwise ( g) is an empty set. Thus we have S B S B P C B 1( )= (Q, p) (Q, p) ( ), p 1 lim ϕ( ( g)) = f(pg) g(pg), pg . (85) S B | ∈ S B ∈B r(Bg)→0 C B − ∈P 2( )= (Q, p) (Q, p) ( ), p 2 . (77) S B | ∈ S B ∈B Combining (83) and (85), we conclude that It is obvious that if we can prove lim ϕ( ( g)) = f(pg) g(pg) < v (86) ( ) ( ) k→∞ C B − S B1 ⊆ S1 B which is contradictory to (82). Therefore, ϕ( k) converges ( 2) 2( ) (78) { F } S B ⊆ S B to ϕ( init). This completes the proof. then ( ) ( ) ( ). We will restrict our attention to F C B1 ∪C B2 ⊆C B show ( ) ( ), and ( ) ( ) can be proved in REFERENCES S B1 ⊆ S1 B S B2 ⊆ S2 B the same way. [1] A. Goldsmith, S. A. Jafar, I. Marić, and S. Srinivasa, “Breaking spectrum Since 1 , we have gridlock with cognitive radios: An information theoretic perspective,” B ⊆B Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2009. l( ) l( 1) u( 1) u( ). (79) [2] L. Zhang, Y.-C. Liang, and Y. Xin, “Joint beamforming and power B ≤ B ≤ B ≤ B allocation for multiple access channels in cognitive radio networks,” Therefore, the following inequalities hold for any l( 1) p IEEE J. Sel. Areas Commun., vol. 26, no. 1, pp. 38–51, Jan. 2008. u( ): B ≤ ≤ [3] L. Zhang, Y. Xin, and Y.-C. Liang, “Weighted sum rate optimization B1 for cognitive radio MIMO broadcast channels,” IEEE Trans. Wireless Commun., vol. 8, no. 6, pp. 2950–2959, Jun. 2009. [l( ) l( )][p l( )]T +[p l( )][l( ) l( )]T 0 B1 − B − B1 − B B1 − B ≥ [4] H. Chan, A. Perrig, and D. Song, “Random key predistribution schemes [u( ) u( )][p u( )]T+[p u( )][u( ) u( )]T 0 for sensor networks,” in Proc. IEEE Symp. SP, 2003, pp. 197–213. 1 1 1 [5] C. E. Shannon, “Communication theory of secrecy systems,” Bell Syst. B − B − BT − B B − BT ≥ [l( 1) l( )][p u( )] +[p l( 1)][u( 1) u( )] 0. Tech. J., vol. 28, no. 4, pp. 656–715, Oct. 1949. B − B − B − B B − B ≤ [6] A. D. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, The above inequalities can be rewritten respectively as pp. 1355–1387, Oct. 1975. [7] I. Csiszár and J. Korner, “Broadcast channels with confidential mes- T T sages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May 1978. Q Lp( ) Lp( ) +l( ) l( ) [8] E. Tekin and A. Yener, “The Gaussian multiple access wire-tap channel,” − B − B B · TB ≥ T Q Lp( 1) Lp( 1) +l( 1) l( 1) IEEE Trans. Inf. Theory, vol. 54, no. 12, pp. 5747–5755, Dec. 2008. − B T− B BT · B [9] ——, “The general Gaussian multiple-access and two-way wiretap Q Up( ) Up( ) +u( ) u( ) − B − B B · B ≥ channels: Achievable rates and cooperative jamming,” IEEE Trans. Inf. Q U ( ) U ( )T +u( ) u( )T Theory, vol. 54, no. 6, pp. 2735–2751, Jun. 2008. p 1 p 1 1 1 [10] G. Geraci, M. Egan, J. Yuan, A. Razi, and I. B. Collings, “Secrecy sum- − B T− B T B · B Q Lp( ) Up( ) +l( ) u( ) rates for multi-user MIMO regularized channel inversion precoding,” IEEE Trans. Commun., vol. 60, no. 11, pp. 3472–3482, Nov. 2012. − B − B B · T B ≤ T Q Lp( 1) Up( 1) +l( 1) u( 1) (80) [11] G. Geraci, R. Couillet, J. Yuan, M. Debbah, and I. B. Collings, “Large − B − B B · B system analysis of linear precoding in MISO broadcast channels with T T where Lp( )= l( ) p , and Up( )= u( ) p . Inequal- confidential messages,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. B B · B B · 1660–1671, 2013. ities (80) provide a sufficient condition for ( 1) 1( ). [12] N. Yang, G. Geraci, J. Yuan, and R. Malaney, “Confidential broadcasting Therefore, ( ) ( ) ( ). This completesS B the⊆ S proof.B C B1 ∪C B2 ⊆C B via linear precoding in non-homogeneous MIMO multiuser networks,” IEEE Trans. Commun., vol. 62, no. 7, pp. 2515–2530, 2014. 11

[13] J. Yang, I.-M. Kim, and D. I. Kim, “Joint Design of Optimal Cooperative [39] A. Beck, A. Ben-Tal, and L. Tetruashvili, “A sequential parametric Jamming and Power Allocation for Linear Precoding,” IEEE Trans. convex approximation method with applications to nonconvex truss Commun., vol. 62, no. 9, pp. 3285–3298, Sep. 2014. topology design problems,” J. Global Optim., vol. 47, no. 1, pp. 29– [14] M. F. Hanif, L.-N. Tran, M. Juntti, and S. Glisic, “On linear precod- 51, 2010. ing strategies for secrecy rate maximization in multiuser multiantenna [40] O. Mehanna, K. Huang, B. Gopalakrishnan, A. Konar, and N. Sidiropou- wireless networks,” IEEE Trans. Signal Process., vol. 62, no. 14, pp. los, “Feasible point pursuit and successive approximation of non-convex 3536–3551, Jul. 2014. QCQPs,” IEEE Signal Process. Lett., vol. 22, no. 7, pp. 804–808, Jul. [15] A. Lozano, A. M. Tulino, and S. Verdú, “Optimum power allocation 2015. for parallel Gaussian channels with arbitrary input distributions,” IEEE [41] A. Khabbazibasmenj, F. Roemer, S. A. Vorobyov, and M. Haardt, “Sum- Trans. Inf. Theory, vol. 52, no. 7, pp. 3033–3051, Jul. 2006. rate maximization in two-way AF MIMO relaying: Polynomial time [16] C. Xiao and Y. R. Zheng, “On the mutual information and power solutions to a class of DC programming problems,” IEEE Trans. Signal allocation for vector Gaussian channels with finite discrete inputs,” in Process., vol. 60, no. 10, pp. 5478–5493, Oct. 2012. Proc. IEEE Globecom, 2008, pp. 1–5. [42] Z.-Q. Luo, W.-K. Ma, A.-C. So, Y. Ye, and S. Zhang, “Semidefinite [17] F. Pérez-Cruz, M. R. Rodrigues, and S. Verdú, “MIMO Gaussian relaxation of quadratic optimization problems,” IEEE Signal Process. channels with arbitrary inputs: Optimal precoding and power allocation,” Mag., vol. 27, no. 3, pp. 20–34, May 2010. IEEE Trans. Inf. Theory, vol. 56, no. 3, pp. 1070–1084, Mar. 2010. [18] C. Xiao, Y. R. Zheng, and Z. Ding, “Globally optimal linear precoders for finite alphabet signals over complex vector Gaussian channels,” IEEE Trans. Signal Process., vol. 59, no. 7, pp. 3301–3314, Jul. 2011. [19] M. Wang, W. Zeng, and C. Xiao, “Linear precoding for MIMO multiple access channels with finite discrete inputs,” IEEE Trans. Wireless Commun., vol. 10, no. 11, pp. 3934–3942, Nov. 2011. [20] W. Zeng, C. Xiao, J. Lu, and K. B. Letaief, “Globally optimal precoder design with finite-alphabet inputs for cognitive radio networks,” IEEE J. Sel. Areas Commun., vol. 30, no. 10, pp. 1861–1874, Nov. 2012. [21] W. Zeng, C. Xiao, M. Wang, and J. Lu, “Linear precoding for finite- alphabet inputs over MIMO fading channels with statistical CSI,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 3134–3148, Jun. 2012. [22] S. Bashar, Z. Ding, and C. Xiao, “On secrecy rate analysis of MIMO wiretap channels driven by finite-alphabet input,” IEEE Trans. Commun., vol. 60, no. 12, pp. 3816–3825, Dec. 2012. [23] Y. Wu, C. Xiao, Z. Ding, X. Gao, and S. Jin, “Linear precoding for finite- alphabet signaling over MIMOME wiretap channels,” IEEE Trans. Veh. Technol., vol. 61, no. 6, pp. 2599–2612, Jul. 2012. [24] J. Harshan and B. S. Rajan, “A novel power allocation scheme for two-user GMAC with finite input constellations,” IEEE Trans. Wireless Commun., vol. 12, no. 2, pp. 818–827, Feb. 2013. [25] S. Vishwakarma and A. Chockalingam, “Decode-and-forward relay beamforming for secrecy with finite-alphabet input,” IEEE Commun. Lett., vol. 17, no. 5, pp. 912–915, May 2013. [26] ——, “Power allocation in MIMO wiretap channel with statistical CSI and finite-alphabet input,” in Proc. National Conf. Commun, 2014, pp. 1–6. [27] M. Girnyk, M. Vehkapera, and L. K. Rasmussen, “Large-system analysis of correlated MIMO multiple access channels with arbitrary signaling in the presence of interference,” IEEE Trans. Wireless Commun., vol. 13, no. 4, pp. 2060–2073, Apr. 2014. [28] W. Zeng, Y. Zheng, and C. Xiao, “Multi-antenna secure cognitive radio networks with finite-alphabet inputs: A global optimization approach for precoder design,” IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 3044–3057, 2016. [29] R. Horst and H. Tuy, Global optimization: Deterministic approaches. Springer Science & Business Media, 1996. [30] P. D. Tao et al., “Duality in DC (difference of convex functions) optimization. Subgradient methods,” in Trends in math. optim. Springer, 1988, pp. 277–293. [31] A. L. Yuille and A. Rangarajan, “The concave-convex procedure,” Neur. Comput., vol. 15, no. 4, pp. 915–936, 2003. [32] A. Ferrer and J. E. Mart´ınez-Legaz, “Improving the efficiency of DC global optimization methods by improving the DC representation of the objective function,” J. Global Optim., vol. 43, no. 4, pp. 513–531, Aug. 2009. [33] C. Xiao, J. Wu, S.-Y. Leong, Y. R. Zheng, and K. Letaief, “A discrete- time model for triply selective MIMO Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1678–1688, Sep. 2004. [34] W. Xu, X. Dong, and W.-S. Lu, “Joint precoding optimization for multiuser multi-antenna relaying downlinks using quadratic programming,” IEEE Trans. Commun., vol. 59, no. 5, pp. 1228–1235, 2011. [35] G. A. Seber, A matrix handbook for statisticians. John Wiley & Sons, 2008, vol. 15. [36] W. Rudin, Principles of mathematical analysis. McGraw-Hill New York, 1964, vol. 3. [37] A. Nemirovski, C. Roos, and T. Terlaky, “On maximization of quadratic form over intersection of ellipsoids with common center,” Math. Pro- gram., vol. 86, no. 3, pp. 463–473, 1999. [38] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press, 2004.