Robust Secure Transmission in MISO Simultaneous Wireless ...

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 1, JANUARY 2015

Correspondence Robust Secure Transmission in MISO Simultaneous Wireless Information and Power Transfer System Renhai Feng, Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin

Abstract—The secure transmission in the multiple-input–single-output simultaneous wireless information and power transfer (SWIPT) system is an important issue. Considering channel uncertainties, we investigate the robust secure transmission scheme, which maximizes the worst-case secrecy rate under transmit power constraint and energy-harvesting (EH) constraint. The optimization problem is a nonconvex problem. Omitting the rank-one constraint on transmit covariance, it is transformed into a solvable semidefinite program (SDP) where the rank relaxation performance upper bound is obtained. Since the obtained rank relaxation transmit covariance may not be rank one, we propose a lower bound-based rank-one suboptimal (LB-Sub) solution by employing Charnes–Cooper transformation. We also propose the suboptimal Gaussian randomized (GR) solutions based on the rank relaxation upper bound and the lower bound, respectively. Simulation results have shown that our proposed LB-Sub solution and suboptimal GR solution based on the rank relaxation upper bound outperform the nonrobust scheme. Index Terms—Energy harvesting (EH), multiple-input–single-output (MISO), secrecy rate, simultaneous wireless information and power transfer (SWIPT), worst-case model.

I. I NTRODUCTION Energy-constrained wireless systems are typically powered by batteries, which makes the energy supply a bottleneck. Fortunately, the bottleneck can be unblocked by the technique of simultaneous wireless information and power transfer (SWIPT). The SWIPT schemes for multiple-input–multiple-output (MIMO) and multiple-input–singleoutput (MISO) broadcast systems have been investigated in [1]–[3]. The works in [1]–[3] do not consider the security issue in wireless communications. Because of the broadcast nature of radio propagation and the inherent randomness of wireless channel, radio transmission is vulnerable to attacks from unexpected eavesdroppers [4]–[7]. The secure communications in MISO SWIPT systems were studied in [8]– [11] where the perfect channel state information (CSI) was considered. In practice, it is difficult to obtain the perfect CSI because of channel estimation and quantization errors. Without the SWIPT scheme, the robust secrecy transmission in the conventional MISO broadcast system was widely studied in the literature [5]–[7] where the channel uncertainties are modeled by the worst-case model. The robust secrecy transmission schemes without and with artificial noise were considered in [5] and [6], respectively. Considering the SWIPT scheme, Ng et al. proposed novel robust secure beamforming in the multiuser MISO SWIPT system, which consists of a transmitter, multiple colocated Manuscript received July 30, 2013; revised March 8, 2014 and April 9, 2014; accepted May 2, 2014. Date of publication May 6, 2014; date of current version January 13, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61173148 and Grant 61202498 and in part by the Scientific and Technological Project of Guangzhou City under Grant 12C42051578. The review of this paper was coordinated by Dr. M. Elkashlan. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, China (e-mail: fengrenhai@ gmail.com; [email protected]; [email protected]; issqjy@ mail.sysu.edu.cn). Digital Object Identifier 10.1109/TVT.2014.2322076

Fig. 1. System model for the secure transmission for SWIPT in the MISO broadcast system.

information-decoding (ID) and energy-harvesting (EH) receivers, and multiple passive eavesdroppers [12]. In this paper, considering the separated ID and EH receivers, we investigate the robust secure transmission problem in the MISO SWIPT system where a transmitter with multiple antennas transmits signals to a single-antenna ID receiver, an EH receiver, and an eavesdropper. In addition to the difference of system models of colocated and separated ID and EH receivers, the differences between our work and that in [12] are explained as follows. In [12], the normalized upper bound channel gain vectors of the eavesdropping channel are modeled as independent and identical distributed (i.i.d.) Rayleigh random variables, whereas in this paper, the channel uncertainties from the transmitter to the ID receiver to the EH receiver and to the eavesdropper are modeled by the worst-case model as in [13] and [14]. Furthermore, the transmit power minimization problem is considered in [12], whereas the secrecy rate maximization problem is considered here. In this paper, our objective is to maximize the worst-case secrecy rate under the transmit power constraint and the EH constraint. The optimization problem is a nonconvex problem. Without the rankone constraint on transmit covariance matrix, we will prove that the problem can be transformed into a solvable semidefinite program (SDP) by employing the S-procedure where the rank relaxation performance upper bound is obtained. Since the obtained rank relaxation transmit covariance may not be rank one, we propose a lower boundbased rank-one suboptimal solution by employing Charnes–Cooper transformation. We also propose the suboptimal Gaussian randomized (GR) solutions based on the rank relaxation upper and lower bounds, respectively. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. A† , tr(A), and rank(A) denote the conjugate transpose, trace, and rank of matrix A, respectively. By A 0 or A 0, we mean that matrix A is positive semidefinite or positive definite, respectively. a denotes the Euclidean norm of a. II. S YSTEM M ODEL Consider a wireless MISO system that consists of a transmitter, an ID receiver, an EH receiver, and an eavesdropper, as shown in Fig. 1. The transmitter is equipped with N antennas. Each of the other nodes is equipped with a single antenna. Denote the channel responses from the transmitter to the ID receiver, from the transmitter to the EH receiver, and from the transmitter to the eavesdropper as hd , hg , and

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he , respectively. We assume that the transmitter knows the imperfect CSI on hd , hg , and he . In practice, the CSI is imperfect because of the channel estimation and quantization errors. In this paper, we model the channel uncertainties by the worst-case model as in [13] and [14] ˆ i + Δhi , hi = h

i ∈ {d, g, e}

(1)

ˆ d, h ˆ g , and h ˆ e denote the estimates of the channels from the where h transmitter to the ID receiver, the EH receiver, and the eavesdropper, respectively; Δhd , Δhg , and Δhe denote the channel uncertainties of hd , hg , and he , respectively. In (1), Δhd , Δhg , and Δhe are bounded by the elliptical regions, denoted as Hd , Hg , and He , respectively, [13], [14]

Hi = Δhi |Δh†i Wi Δhi ≤ 1 ,

i ∈ {d, g, e}

and

ye = h†e x + ne

(3)

respectively, where x ∈ CN ×1 is the transmitted signal vector from the transmitter; nd and ne are the circularly symmetric complex Gaussian noises at the ID receiver and the eavesdropper, respectively, with zero mean and variance σ 2 = 1. Transmitted signal vector x is assumed to follow a complex Gaussian distribution with mean vector 0 and transmit covariance matrix S = E[xx† ], which is to be designed [6]. The transmitted signal is subject to the following constraints: tr(S) ≤ P and h†g Shg ≥ Γ

∀ Δhg ∈ Hg

constraint since Δhg ∈ Hg has infinite values. The optimization problem (5) is difficult to solve directly. Let t=

rank(S) = 1, h†g Shg ≥ Γ

ˆ = tS and S

(6)

ˆ d max min t + h†d Sh ˆ g − Γt ≥ 0 s.t. h†g Sh ˆ e≤1 t + h†e Sh

∀ Δhg ∈ Hg ∀ Δhe ∈ He

ˆ − P t ≤ 0. tr(S)

(7)

The equality of problems (5) and (7) is proved following the method employed in [4]. Since the constraint set in (7) is nonconvex, the proving method employed in [4] is not applicable. Denoting the optimal solution to problem (5) as So and the optimal solution to ˆ o , to ), we have the following proposition to prove problem (7) as (S the equality of problems (5) and (7). ˆ o , to ) to problem (7) should Proposition 1: The optimal solution (S satisfy the following equation: ˆ o he = 1. to + max h†e S

(8)

Δhe

Proof: See Appendix A. ˆ o = to So . Since problems (5) and (7) are equal, we have S By using the epigraph form of it, (7) is equivalently rewritten as max

ˆ S0,t≥0,τ ≥0

τ

(9a)

ˆ d≥τ s.t. t + h†d Sh

where P is the transmit power constraint at the transmitter, and Γ is the EH constraint at the EH receiver. We formulate the secrecy rate optimization problem considering the worst-case channel uncertainties as [4]–[6]

s.t.

1 + h†e She

ˆ Δhd S0,t≥0

(4)

1 + h†d Shd max min log2 S0 Δhd ,Δhe 1 + h†e She

1

where t ≥ 0 is a complimentary slackness parameter. Using Charnes–Cooper transformation [15], we rewrite the optimization problem (5) as follows:

(2)

where the matrices Wd 0, Wg 0, and We 0, assumed to be known, determine the qualities of CSI. The received signals at the ID receiver and the eavesdropper are expressed as yd = h†d x + nd

401

∀ Δhd ∈ Hd

ˆ g − Γt ≥ 0 h†g Sh ˆ e≤1 t + h†e Sh

∀ Δhg ∈ Hg ∀ Δhe ∈ He

∀ Δhg ∈ Hg

(5)

where function log2 (·) is omitted in the following derivations since the logarithm is a monotonically increasing function that has no effect on the optimization problem. Remark 1: The eavesdropper equipped with a single antenna is considered here. If the eavesdropper is equipped with multiple antennas, the optimization problem can be efficiently solved using the similar method as in [6] where the EH constraint was not considered. Remark 2: Problem (5) may be unfeasible because the transmitter may not simultaneously provide secure information transmission and energy transferring to both the ID and EH receivers. The unfeasible condition for problem (5) will be discussed in Section IV.

(9e)

by introducing a variable τ ≥ 0. Problem (9) has semi-infinite constraints (9b)–(9d), which are intractable. To make the problem tractable, we employ the S-procedure [16] to convert the constraints (9b)–(9d) into linear matrix inequalities (LMIs) [17]. For (9b), we substitute (1) into (9b) and rewrite it as follows:

∀ Δhd : Δh†d Wd Δhd − 1 ≤ 0 (10) †ˆˆ †ˆ ˆ† ˆ ˆ† S ˆˆ τ −h d hd − hd SΔhd − Δhd Shd − Δhd SΔhd − t ≤ 0.

Applying the S-procedure, we convert (10) into an LMI as follows:

ˆd ˆh S ˆ† S ˆ d + t − τ − λ1 ˆh h d

ˆ λ1 W d + S ˆ† S ˆ h d

0

(11)

where λ1 ≥ 0 is a slack variable. Using the similar method, we transform problem (9) into max

ˆ S0,t,τ,λ 1 ,λ2 ,λ3 ≥0

III. R ANK R ELAXATION P ERFORMANCE U PPER B OUND Here, we omit the rank-one constraint on S in (5) and obtain the rank relaxation upper bound for the robust secrecy transmission in the MISO broadcast system with the SWIPT scheme. Without the rankone constraint on S, the optimization problem (5) is still nonconvex because of the nonconvex objective function and the nonconvex EH

(9c) (9d)

ˆ − Pt ≤ 0 tr(S)

tr(S) ≤ P

(9b)

τ

s.t. B1 0, B2 0, B3 0, where

B1 =

ˆ λ1 W d + S ˆ† S ˆ h d

ˆ − Pt ≤ 0 tr(S)

ˆd ˆh S ˆ† S ˆ d + t − τ − λ1 ˆh h d

(12)

(13)

402


B2 =

B3 =

ˆ λ2 W g + S †ˆ ˆ hg S

ˆg ˆh S †ˆˆ ˆ hg Shg − Γt − λ2

ˆ λ3 W e − S †ˆ ˆ − he S

ê ˆh −S †ˆˆ ˆ − h e S h e − t + 1 − λ3

(14)

Thus, we obtain the lower bound of ξd , ξ d ≥ v † Hd v

(15)

in which λ2 ≥ 0 and λ3 ≥ 0 are slack variables. SDP problem (12) is convex and can be effectively solved using the interior point method [18]. We have the following theorem. Theorem 1: The rank of the obtained rank relaxation transmit covariance So of problem (12) is less than or equal to 2. Proof: See Appendix B.

where

ξe ≤ v† He v and ξg ≥ v† Hg v

†

s.t. tr(vv ) ≤ P,

ξg ≥ Γ

(26)

1

1

(28) .

(29)

Thus, the unfeasible condition for problem (5) is P ψg < Γ where ψg is the largest eigenvalue of Hg [1]. Using (25) and (27), we propose the lower bound-based suboptimal solution. The suboptimal solution optimizes the lower bound of problem (16), which is expressed as max v

1 + v † Hd v 1 + v † He v

s.t. tr(vv† ) ≤ P, v† Hg v ≥ Γ.

(30)

We transform problem (30) into

ξd = ξe = ξg =

min |v† hd |

2

Δhd ∈Hd

max |v† he |

2

1 + tr(Hd S) 1 + tr(He S)

s.t. rank(S) = 1, tr(S) ≤ P, tr(Hg S) ≥ Γ.

(31)

Let 2

min |v† hg | .

(19)

Δhg ∈Hg

ˆ d + Δhd )(h ˆ d + Δhd )† v + ζtr Δh† Wd Δhd −1 L = v † (h d

(20)

where ζ ≥ 0 denotes the Lagrangian variable. Because L is convex in Δhd , the Karush–Kuhn–Tucker (KKT) conditions are sufficient to obtain the exact minimum of ξd . Thus, we have

ˆdh ˆ† v min |v† hd | = − v† Wd−1 vv† h d

Δhd ∈Hd

− 12

ˆd Wd−1 vv† h

2 ˆ dh ˆ † + W−1 − 2α 12 W−1 v min |v† hd | = v† h d d d

Δhd ∈Hd

(21)

ˆ dh ˆ† v v† h d . −1 v † Wd v

(23)

1 tr(He S) + 1

and

Z = νS.

(32)

Omitting the rank-one constraint on S and using Charnes–Cooper transformation [15], problem (31) can be rewritten as max

Z0, ν≥0

ν + tr(Hd Z)

s.t. tr(He Z) + ν = 1, tr(Z) ≤ νP, tr(Hg Z) ≥ νΓ. (33) Denote the optimal solution to problem (33) as (Zo , νo ). If rank(Zo ) = 1, the optimal solution to problem (31) is given by √ v = (1/ νo )zo where Zo = zo z†o . If rank(Zo ) ≥ 2, from rank-one ˆz ˆ† in decomposition theorem [19], we can find a rank-one matrix z polynomial time such that ˆz ˆ† } = tr{Hi Zo } tr{Hi z

(34)

where i ∈ {d, g, e}. The optimal solution to the problem (31) can be √ z. computed as v = (1/ νo )ˆ B. Suboptimal GR Solution Based on the Rank Relaxation Upper Bound

It is noted that (23) is a generalized Rayleigh quotient. We have ˆdh ˆ † , W−1 α ≥ βmax h d d

ν=

(22)

where

S0

(18)

Δhe ∈He

α=

max

(17)

The equivalence of (5) and (16) is because the minimization of a fraction is equivalent to the simultaneous maximization of its denominator and minimization of its numerator. We obtain the lower boundbased suboptimal solution by neglecting the correlation between ξd , ξe , and ξg . To obtain the exact minimum of ξd , we write its Lagrangian function as follows:

2

.

(27)

2 ˆgh ˆ † + W−1 1 − 2βmax ˆgh ˆ † , W−1 h Hg = h g g g g

(16)

where

arg

2 ˆ eh ˆ † + W−1 1 + 2βmax êh ˆ † , W−1 h He = h e e e e

Since the rank relaxation transmit covariance So in Section III may not be rank one, we propose three suboptimal solutions to problem (5), which obtain the rank-one transmit covariance.

1 + ξd max v 1 + ξe

It is noted that the derived lower bound on ξd is tight in the sense that there exists v, which satisfies the equality in (23). Similarly, we have

IV. S UBOPTIMAL ROBUST S ECURE T RANSMISSION S CHEMES

Decompose the rank-one transmit covariance S as vv† . We rewrite problem (5) as follows:

1

2 ˆ dh ˆ † + W−1 1 − 2βmax ˆdh ˆ † , W−1 h Hd = h d d d d

where

A. Lower Bound-Based Suboptimal Solution

(25)

(24)

where βmax (A, B) denotes the largest generalized eigenvalue of the matrix pair (A, B).

The Gaussian randomization method is widely employed to obtain the rank-one solution for the optimization problem after rankone relaxation [20]. In this paper, with the obtained rank relaxation transmit covariance in Section III, we employ the Gaussian randomization method to generate the suboptimal rank-one solution. When the transmit covariance So is known, we generate L possible GR solutions as follows: √ 1 (35) v = P So u− 2 So u


403

where u is a randomly generated Gaussian vector. In all the L possible GR solutions, the best solution should satisfy the EH constraint and maximize the worst-case secrecy rate simultaneously. From derivations (20)–(23), we know that the EH constraint is equivalent to

⎛

ˆgh ˆ † + W−1 − 2 v† ⎝h g g

ˆgh ˆ† v v† h g

12

v† Wg−1 v

⎞ Wg−1 ⎠ v ≥ Γ.

(36)

The worst-case secrecy rate is obtained by solving min

Δhd ,Δhe

log2

1 + h†d vv† hd 1 + h†e vv† he

.

(37)

After the similar derivations as in Section III, problem (37) is equivalently transformed into the following SDP: max

t,τ,λ1 ,λ3 ≥0

s.t.

τ

λ1 Wd + tvv ˆ † vv† th d

†

λ3 We − tvv† ˆ † vv† −th e

†ˆ

tvv hd ˆ † vv† h ˆ d + t − τ − λ1 th d

Fig. 2. Average worst-case secrecy rates versus P/σ 2 ; comparison of our proposed robust secure transmission schemes with the nonrobust secure transmission scheme where Γ/σ 2 = 5 dB.

0

ê −tvv† h ˆ † vv† h ˆ e − t + 1 − λ3 −th e

0. (38)

Denoting the optimal objective value of (38) as τo , the worst-case secrecy rate is log2 τo . C. Suboptimal GR Solution Based on the Lower Bound From the previous section, we know that complexity to obtain the suboptimal GR solution based on the rank relaxation upper bound is high. The number of possible GR solutions determines the number of SDPs to be solved. To reduce the computational complexity, we propose the suboptimal GR solution based on the lower bound. After the possible GR solution generation by using (35), we check whether the EH constraint is satisfied by juging of whether the inequality v† Hg v ≥ Γ is valid. If the EH constraint is satisfied, we select the GR solution as follows: arg max v

1 + v † Hd v . 1 + v † He v

(39)

V. S IMULATION R ESULTS Here, we evaluate the performance of the proposed scheme via computer simulations. In the simulations, the transmitter is equipped ˆg, ˆ d, h with N = 4 antennas. All the entries of the channel estimates, h ˆ e , are i.i.d. complex Gaussian random variables with zero mean and h and unit variance. All the channel uncertainties are assumed to be norm −1 −1 bounded, i.e., Wd = −1 d I, Wg = g I, and We = e I, where −3 −2 −1 (d , g , e ) = (10 , 10 , 10 ) specifies the sizes of uncertainty regions. We produce 1000 randomly generated channel realizations and compute the average worst-case secrecy rate. In Fig. 2, we compare the rank relaxation upper bound obtained in Section III (denoted as “RRUB” in the legend) and the average worstcase secrecy rates achieved by our proposed lower bound-based suboptimal solution (denoted as “LB-Sub”), our proposed suboptimal GR solution based on the rank relaxation upper bound (denoted as “GRRRUB”), our proposed suboptimal GR solution based on the lower bound (denoted as “GR-LB”), and the nonrobust secure transmission ˆ g , and h ˆ e as the ˆ d, h scheme that directly used the channel estimates h

actual channel responses without considering the channel uncertainties (denoted as “Non-Robust”). The nonrobust secure transmission scheme is obtained by solving (9), where Δhd = 0, Δhg = 0, and Δhe = 0. Without considering the eavesdropper, the average worstcase rates achieved by employing the robust SWIPT scheme proposed in [3] (denoted as “No Eve”) are also presented as a benchmark. At the EH receiver, the EH constraint Γ is Γ/σ 2 = 5 dB. For the GR solutions, we select the best solution from L = 50 generated possible GR solutions each time. In Fig. 2, it is observed that the performance of our proposed GR-RRUB solution approaches the rank relaxation upper bound. Our proposed GR-RRUB and LB-Sub solutions outperform the nonrobust scheme. When 0 dB ≤ P/σ 2 ≤ 9 dB, our proposed GR-LB solution also outperforms the nonrobust scheme. It is noted that in Fig. 2, when P < Γ, the average worst-case secrecy rate does not decrease to zero. This is because the transmitter is equipped with N = 4 antennas, which accounts for about 6-dB array gain. Furthermore, when the randomly generated channels may or may not be suitable for secure transmission and EH, the obtained average worst-case secrecy rate is the averaging over the randomly generated channels. In Fig. 3, we present the percentage of feasible problems for different values of the EH constraint Γ where P/σ 2 = 10 dB. In Fig. 3, it is observed that the percentage of feasible problems is reduced as the increase in the EH constraint Γ. It is also found that the performance of our proposed GR-RRUB solution approaches the rank relaxation upper bound. Our proposed GR-RRUB and LB-Sub solutions outperform the nonrobust scheme. VI. C ONCLUSION In this paper, we have obtained the rank relaxation upper bound for the MISO SWIPT system when the transmitter has the imperfect CSI of the channels. Since the obtained rank relaxation transmit covariance may not be rank one, we propose the lower bound-based rank-one suboptimal (LB-Sub) solution by employing Charnes–Cooper transformation. We also propose the suboptimal GR solutions based on the rank relaxation upper bound and the lower bound, respectively. Simulation results have shown that our proposed LB-Sub solution and suboptimal GR solution based on the rank relaxation upper bound outperform the nonrobust scheme.

404


ˆ and applying Taking the partial derivative of (40) with respect to S KKT conditions, we have −D1 C1 D†1 − D2 C2 D†2 + D3 C3 D†3 + μI − C4 = 0.

Fig. 3. Percentage of feasible problems versus Γ/σ 2 ; comparison of our proposed robust secure transmission schemes with the nonrobust secure transmission scheme where P/σ 2 = 10 dB.

We prove Proposition 1 by reductio ad absurdum. Consider the folˆ o he < 1 and h† S ˆ lowing two situations. If to + maxΔhe h†e S g o hg − Γto > 0, we can always find t = to + Δt, i.e., Δt > 0, which satisfies the constraints of problem (7). ˆ o he < 1 and h† S ˆ If to + maxΔhe h†e S g o hg − Γto = 0, the optimal ˆ o) − solution is found when the power constraint is activated, i.e., tr(S ˆ o + ΔS, ˆ to + Δt), i.e., ΔS ˆ P to = 0. We can find a solution (S ˆ o he < 1 0, Δt > 0, which makes the constraint to + maxΔhe h†e S ˆ o ) − P to = 0. ˆ o hg − Γto = 0 and tr(S tighter while satisfying h†g S ˆ These contradict that (So , to ) is the optimal solution of problem (7). A PPENDIX B P ROOF OF T HEOREM 1 Assume that the dual variables C1 ∈ H+ , C2 ∈ H+ , C3 ∈ H+ , C4 ∈ H+ , and μ ≥ 0 are corresponding to constraints B1 0, B2 ˆ 0, and tr(S) ˆ − P t ≤ 0 in (12), respectively. The 0, B3 0, S Lagrangian dual function of primal problem (12) is given by L¯ = τ − tr(C1 B1 ) − tr(C2 B2 ) − tr(C3 B3 )

ˆ − P t − tr(C4 S). ˆ +μ tr(S)

(40)

Since C1 , C2 , and C3 are Hermitian, we have

ˆ i + tr(Ci Fi ), i ∈ {1, 2} tr(Ci Bi ) = tr Ci D†i SD tr(C3 B3 ) = − tr

ˆ 3 C3 D†3 SD

+ tr(C3 F3 )

(41) (42)

ˆ d ], D2 = [I h ˆ g ], D3 = [I h ˆ e ], and where D1 = [I h

F1 =

F2 =

F3 =

It is noted that C1 B1 = 0 from KKT conditions. Since the size of C1 and B1 is (N + 1) × (N + 1), we have rank(C1 ) + rank(B1 ) ≤ N + 1. ˆ 0 has full rank. We In matrix B1 , λ1 ≥ 0. If λ1 > 0, λ1 Wd + S will prove that λ1 = 0 by reductio ad absurdum. If λ1 = 0, constraint Δh†d Wd Δhd ≤ 1 is not active because from (10) and (11), λ1 is its dual variable. It is noted that constraint Δh†d Wd Δhd ≤ 1 is the only constraint on Δhd . If Δh∗d , where Δh∗d † Wd Δh∗d < 1, is the worst channel uncertainty which minimizes the achievable secrecy rate log2 (1 + h†d Shd )/(1 + h†e She ), we can always find a scalar ρ > 1 that satisfies ρ2 Δh∗d † Wd Δh∗d = 1. Substituting the channel uncertainty ρΔh∗d into log2 (1 + h†d Shd )/(1 + h†e She ), we obtain the lower achievable secrecy rate than that obtained by Δh∗d . It is contradictory to the assumption that Δh∗d minimizes the achievable secrecy rate. Thus, λ1 = 0. ˆ 0 has full rank, we have rank(B1 ) ≥ N . FurSince λ1 Wd + S thermore, rank(C1 ) = 0. Thus, rank(C1 ) = 1. Similarly, rank(C2 ) = ˆ we have rank(C3 ) = 1. Multiplying both sides of (46) with S,

A PPENDIX A P ROOF OF P ROPOSITION 1

λ1 W d 0

0 t − τ − λ1

λ1 W g 0

0 −Γt − λ2

λ1 W e 0

0 . 1 − t − λ3

(43)

(44)

(45)

(46)

ˆ = D 1 C1 D † + D 2 C2 D † S ˆ μI + D3 C3 D†3 S 1 2

(47)

ˆ = 0. Since μI+D3 C3 D† has full rank, and where it is noted that C4 S 3 † rank(D1 C1 D1 +D2 C2 D†2 ) ≤rank(D1 C1 D†1 )+rank(D2 C2 D†2 ) = 2, we have

ˆ = rank(S) ˆ ≤ 2. rank μI + D3 C3 D†3 S

(48)

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Resource Allocation for Power Minimization in the Downlink of THP-Based Spatial Multiplexing MIMO-OFDMA Systems Marco Moretti, Member, IEEE, Luca Sanguinetti, Member, IEEE, and Xiaodong Wang, Fellow, IEEE

Abstract—In this paper, we deal with resource allocation in the downlink of spatial multiplexing multiple-input–multiple-output (MIMO)orthogonal frequency-division multiple-access (OFDMA) systems. In particular, we concentrate on the problem of jointly optimizing the transmit and receive processing matrices, the channel assignment, and the power allocation with the objective of minimizing the total power consumption while satisfying different quality-of-service (QoS) requirements. A layered architecture is used in which users are first partitioned in different groups on the basis of their channel quality, and then channel assignment and transceiver design are sequentially addressed starting from the group of users with most adverse channel conditions. The multiuser interference among users belonging to different groups is removed at the base station (BS) using a Tomlinson–Harashima precoder operating at user level. Numerical results are used to highlight the effectiveness of the proposed solution and to make comparisons with existing alternatives. Index Terms—Linear programming (LP), multiple-input–multipleoutput (MIMO), resource allocation, Tomlinson–Harashima precoding (THP).

Manuscript received August 1, 2013; revised February 10, 2014 and April 16, 2014; accepted April 22, 2014. Date of publication April 29, 2014; date of current version January 13, 2015. The work of L. Sanguinetti was supported by the People Program (Marie Curie Actions) under FP7 PIEF-GA2012-330731 Dense4Green. This work was also supported by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications (NEWCOM) under Grant 318306. The review of this paper was coordinated by Prof. N. Arumugam. M. Moretti is with the Dipartimento di Ingegneria dell’Informazione, University of Pisa, 56122 Pisa, Italy (e-mail: [email protected]). L. Sanguinetti is with the Dipartimento di Ingegneria dell’Informazione, University of Pisa, 56122 Pisa, Italy, and also with Alcatel-Lucent Chair on Flexible Radio, Supélec, 91192 Gif-sur-Yvette, France (e-mail: luca. [email protected]). X. Wang is with the Department of Electrical Engineering, Columbia University, New York, NY 10027, USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TVT.2014.2320587

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I. I NTRODUCTION Dynamic resource allocation in multiple-input multiple-output (MIMO) systems based on orthogonal frequency-division multipleaccess (OFDMA) technologies has gained considerable research interest [1]. In most cases, subcarriers are assigned to the active users in an exclusive manner without taking advantage of the multiuser diversity offered by the spatial domain. A possible solution to exploit the spatial dimension is to make use of space-division multiple-access (SDMA) schemes, which allow the simultaneous transmission of different users over the same frequency band. The main impairment of SDMA is represented by multiple-access interference (MAI). In downlink transmissions, MAI mitigation can only be accomplished at the base station (BS) using prefiltering techniques. The most common approach for interference mitigation is zero-forcing (ZF) linear beamforming, which relies on the idea of preinverting the channel matrix at the transmitter. Another approach is represented by the block-diagonalization ZF (BD-ZF) scheme originally proposed in [2]. Particular attention has been also devoted to dirty-paper coding (DPC) techniques [3] although their implementation is still much open. A possible solution in this direction is represented by Tomlinson–Harashima precoding (THP), which can be seen as a 1-D DPC technique [4] and has been widely used in the downlink of single- and multi-user MIMO systems [5]–[8]. In combination with prefiltering, another way to deal with interference in SDMA-OFDMA systems is user partitioning, which basically consists in properly selecting the set of users transmitting on the same subcarriers. As illustrated in [9], a common approach is to first group together users whose channels have low spatial cross correlation and then to assign the subcarriers to the various groups. In [10], the authors follow a completely different approach in which the users are first divided into groups such that the spatial cross correlations among users in different groups are low as much as possible, and then subcarriers are sequentially assigned within each group. From the discussion, it follows that the use of SDMA schemes in MIMO-OFDMA systems makes the problem of resource allocation more challenging as it requires the joint optimization of: 1) channel assignment and user partitioning; 2) power allocation over all active links; and 3) transmit and receive filters. To the best of our knowledge, there exists only a few works dealing with all these problems together. In [11], the authors employ BD-ZF and Lagrange dual decomposition to derive a resource allocation scheme to minimize the power consumption when individual user rate constraints are imposed. The limiting factor of this approach is that an exhaustive search is required to find the best user allocation on each subchannel. Reduced complexity solution is illustrated in [12], in which a two-step procedure is adopted to decouple BD-ZF beamforming from subcarrier and power allocation. Although simpler than [11], it still requires an exhaustive search over a subset of users. In [13], the authors exploit a layered architecture in which a user partitioning technique (resembling that discussed in [10]) is first used in conjunction with BD-ZF to partially remove multiuser interference, and then carrier assignment is performed jointly with transceiver design using a linear programming (LP) formulation of the allocation problem [14]. In this paper, we return to the layered architecture investigated in [13] and extend it in several directions. First, we reformulate the power minimization problem assuming that the quality-of-service (QoS) constraint of each user is given as a sum of the mean square errors (MSEs) over all subcarriers rather than on the sum of the achievable rates. Second, transceiver design is carried out employing a nonlinear THP precoder operating at user level at the transmitter. Third, the choice of the user partitioning strategy is motivated by its combination with the THP

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