Full Duplex Beamforming - IEEE Xplore

IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 6, JUNE 2014

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A High Data Rate Wireless Communication System With Improved Secrecy: Full Duplex Beamforming Özge Cepheli, Semiha Tedik, and Güne¸s Karabulut Kurt, Member, IEEE

Abstract—We formulate a joint beamforming vector and power optimization problem for multi-antenna full duplex transmission systems and show that simultaneous transmissions of information bearing signals of legitimate nodes can be optimized to act as artificial noise against eavesdroppers. The proposed system improves the overall throughput, while maintaining secrecy and QoS levels within desired signal-to-interference-plus-noise ratio bounds, without additional power, as verified via simulations. Index Terms—Artificial noise, beamforming, full duplex communication, PHY security.

I. I NTRODUCTION

S

EVERAL physical layer (PHY) security approaches have been proposed to improve the secrecy levels of communication systems. These approaches mainly rely on beamforming and artificial noise (AN) techniques to address secrecy problems [1]–[4]. Inspired by full duplex (FD) systems that allow transmission and reception of data at the same time and frequency slots [5], we use beamforming and AN as tools to design a high throughput and guaranteed secrecy multi-antenna wireless communication system. Our system makes use of full duplex beamforming (FDB), that is proposed in [6] to increase the total ergodic capacity. Our main motivation stems from the fact that FD transmission between a pair of legitimate nodes with a predetermined quality of service (QoS) constraint can be designed to act as AN against eavesdroppers. Hence, a target secrecy level can be maintained due to the ambient interference. Following this motivation, we propose a novel guaranteed-secrecy FDB system without a dedicated AN signal generation block. Although the FD security systems have been proposed [4], [7], [8], here we introduce a guaranteed secrecy and QoS for legitimate users for the first time. We formulate a joint beamforming vector and power optimization problem for the proposed system, satisfying secrecy and QoS constraints. We obtain nonconvex quadratically constrained quadratic problems for both the maximal ratio combining (MRC) and optimum combining (OC) schemes in the presence of self-interference due to FD transmission. We derive convex approximations by deploying semidefinite relaxation (SDR) for both MRC and OC, and prove

Manuscript received June 23, 2013; revised December 11, 2013, February 10, 2014, and April 14, 2014; accepted April 14, 2014. Date of publication April 30, 2014; date of current version June 6, 2014. The associate editor coordinating the review of this paper and approving it for publication was M. C. Gursoy. The authors are with the Department of Electronics and Telecommunications Engineering, Istanbul Technical University, Istanbul 34469, Turkey (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LCOMM.2014.2321152

Fig. 1. System Model: (a) Our proposed secure FDB design. (b) AN aided design with AN generated at transmitter (AN@Tx) only. (c) AN aided design with AN generated at receiver (AN@Rx) only.

that solutions of our approximations yield the exact solution of the original problems. Our main contribution is the joint optimization of the transmit beamforming vectors of FD nodes for secrecy and QoS constraints. Operational significance of this system stems from the use of FDB, which almost doubles the overall throughput and enables us to use the information bearing signals as AN (according to the desired secrecy and QoS constraints), hence improving the power efficiency, as will be demonstrated via simulation results. II. S YSTEM M ODEL AND O PTIMALITY C ONDITIONS A wireless network with two legitimate FD nodes (A, B) and one eavesdropper node (E) is considered. All legitimate nodes use the dedicated bandwidth at the same time to communicate with each other. The minimum required signal-to-interferenceplus-noise ratio (SINR) of A and B define the QoS level of the system, while SINR constraints on E induces the desired secrecy level. A and B are equipped with Na and Nb transmit and receive antennas which enable them to deploy FDB and AN techniques. In the FDB technique, the transmit and receive antennas are separated for each node. E has Ne receive antennas. To maximize the received SINR, OC scheme is used at receivers, as it is known to be the combiner that maximizes output SINR in the presence of interferers [9]. We also extend the results to MRC scheme as it is a frequently utilized signal combination scheme. The proposed system is given in Fig. 1(a), the interference signals associated with FDB are used to provide the desired secrecy level instead of AN. In order to compare the proposed model with the state of the art techniques, we consider two of the most efficient transmission methods in the related literature, which we refer to as AN@Tx [2] and AN@Rx [4]. We also extend these to multi-antenna scenarios in order to provide a fair performance comparison. In AN@Tx method, illustrated in Fig. 1(b), all nodes are operating in half duplex mode, and optimum beamforming vectors and AN parameters are determined for the required QoS and secrecy [2]. In AN@Rx

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method, shown in Fig. 1(c), the AN signal is only generated by the receiving node with FD transmission [4]. Notation: Boldface lowercase (uppercase) letters donate vectors (matrices). The set of n-dimensional complex vectors is denoted by Cn and n × n Hermitian matrix is defined by Hn . In is an n × n identity matrix. x ∼ CN (μ, Σ) refers to a random vector x which follows a complex circular Gaussian distribution with mean μ and covariance Σ. The trace operator and Euclidean norm are denoted by Tr(·) and · , respectively. E{·} is the expectation, r(·) is the rank operator. AH is the conjugate transpose of A. B 0 and C 0 state that B and C are positive semi-definite (PSD) and positive definite (PD) matrices, respectively.

α > 0 is a scaling factor and Ri is the interference-plus-noise correlation matrix at node i. Assuming that noise and interferH ence are independent, we calculate Ri = κ2 HH ii wi wi Hii + σn2 INi for nodes A and B. After OC, the output SINR of j − i link can be expressed as

A. System Model

Maximal Ratio Combiner Scheme: Although MRC is suboptimal in the presence of interferers, it is a frequently used reception method that needs to be investigated. When MRC is used instead of OC, SINR expressions become

In our model, legitimate nodes use beamforming in FD transmission mode. Nodes deploy both transmit and receive beamforming to maintain secrecy with improved resistance to spatial correlation [10]. Although FD transmission has increased spectral efficiency, it has not been frequently used in practice due to the generated self-interference [11]. However, recent studies target to alleviate this performance degradation problem by using self-interference cancellation (SIC) techniques [12], [13], showing the potential of FD transmission. Suppose A sends the signal sa to B, and B sends to A the signal sb concurrently in FD mode. sa and sb are modeled as unit energy signals. Transmitted signals of A and B are xa = wa sa and xb = wb sb ; where wa ∈ CNa and wb ∈ CNb are the transmit beamforming vectors of sa and sb , respectively. Adopting the notation of [2] for our system, received signals at A, B and E can be modeled as (1a) xb + κa HH x a + na ra = wrHa HH ba aa H H H rb = wrb Hab xa + κb Hbb xb + nb (1b) H re,i = wrHe,i HH (1c) x + H x + n a b e ae be where wra , wrb and wre,i represent the receive beamforming vectors of the corresponding nodes for i ∈ {a, b}. Signals ra and rb are obtained at the output of the SIC block and κa and κb represent the residual self-interference coefficient in case of imperfect self-interference channel estimation. In order to provide a fair comparison framework we assume that E has two receiver chains dedicated to detect signals from both A and B as given with the relation in (1c) to obtain re,a and re,b . na , nb and ne are independent and identically distributed additive noise vectors where ni ∼ CN (0, σi2 INi ), i ∈ {a, b, e}. For simplicity of notation, it is assumed that σa = σb = σn and κa = κb = κ. Hab ∈ CNa ×Nb and Hba ∈ CNb ×Na define the channel matrices of links a − b and b − a, respectively. Hae ∈ CNa ×Ne and Hbe ∈ CNb ×Ne are the channel matrices of a − e and b − e links. The self-interference channels’ matrices are denoted by Haa ∈ CNa ×Na and Hbb ∈ CNb ×Nb . All required channel estimates are assumed to be available at all nodes. Optimum Combiner Scheme: At all receivers the OC technique, which maximizes the SINR by optimally combining received signals according to their channel noise level and interference [9], can be used. Here, we initially obtain the SINR of information bearing links. The maximum SINR is achieved for j − i link, i, j ∈ {a, b}, j = i, when the receive beamformH ing vector is selected according to wri = αR−1 i Hji wj , where

H Γij (OC) = wjH Hji R−1 i Hji wj .

(2)

E can also use OC and needs to calculate two distinct beamforming vectors for A and B to maximize their SINR values. H H H For i ∈ {a, b}, wre,i = αR−1 ei Hie wi Rei = Hje wj wj Hje + σe2 INe and the corresponding SINRs become H Γei (OC) = wiH Hei R−1 ei Hei wi .

wjH RHji wj κ2 wiH RHii wi + σn2 wjH RHje wj Γej (M RC) = H wi RHie wi + σe2

(3)

Γij (M RC) =

(4)

where RHji = Hji HH ji . Note that when the residual interference terms are zero (κ = 0), OC is also simplified to MRC. FDB Problem: Finally, our proposed FDB power optimization problem is defined as min

wa ,wb

wa 2 + wb 2

s.t. Γab (wa , wb ) ≥ γa Γba (wa , wb ) ≥ γb Γea (wa , wb ) ≤ γe,a Γeb (wa , wb ) ≤ γe,b

(5a) (5b) (5c) (5d) (5e)

where γa are γb are the SINR constraints for A and B, and γe,a and γe,b are the secrecy constraints against E for A and B, respectively. The total power wa 2 + wb 2 = P will lie on the global minima of the feasible set. If the system has a sum power constraint of Pm , the solution is applicable when P ≤ Pm , whereas P > Pm leads to infeasibility. Note that use of different modulation schemes, coding techniques, antenna arrays or joint detection performance of eavesdropper do not change our problem statements. However, these variations solely affect the thresholds of the constraints. B. Optimality Conditions of the FDB Problem Although the secrecy conditions in (5d) and (5e) are (can be transformed into) quadratical constraints that preserve convexity for OC (MRC), due to the SINR constraints in (5b) and (5c), the FDB problem is a non-convex quadratically constrained quadratic problem both for OC and MRC schemes. Here, we initially show that the FDB problem can be approximated as a semi-definite program (SDP) and later prove that the SDP problem’s solution is identical to that of (5). SDP for OC Scheme: In order to achieve the optimal solution we use the coordinate descent method [14], where wa and wb are optimized step-by-step in an iterative fashion, by setting an initial value to wa . Here, using the definition Qi = H Hji R−1 i Hji , SINR constraint in (5b) becomes a quadratical

CEPHELI et al.: WIRELESS COMMUNICATION SYSTEM: FULL DUPLEX BEAMFORMING

constraint for the steps that wa is known and (5c) in the steps that wb is known and can be expressed as H H Γij (OC) : wjH Hji R−1 i Hji wj = wj Qi wj ≥ γi

(6)

where i, j ∈ {a, b}, j = i. The constraints with optimization variable in the covariance matrix which include inverse operation, we use the matrix inversion lemma [15] and obtain H 1 κ2 HH ii wi wi Hii H ≥ γi (7) fij Γij (OC) : 2 fij Ini − 2 σn σn +κ2 wiH HH ii Hii wi where fij = wjH Hji becomes a known vector. After this point we can make use of SDR techniques to obtain a problem that can be solved efficiently by interior point method algorithms. A crucial first step in deriving an SDP is to observe that wjH Qi wj = Tr wjHQi wj = Tr Qi wj wjH = Tr(Qi Wj ) (8) where Wj = wj wjH , Qi 0, ∀ i. Notice that defining the new variable Wj is a rank-1 symmetric PSD matrix, we need to include two new constraints; Wj 0, r(Wj ) = 1. The problem is still hard to solve, due to the non-convex r(Wj ) = 1 constraint. Applying SDR to make the problem convex [16], we relax it by neglecting the rank constraint. For the case of initialized Wa , the approximate SDP for OC for becomes min

Wb ∈HNb

Tr(Wa ) + Tr(Wb )

s.t. Tr(Qa Wb ) ≥ γa H 2 gb (Wb )fba fba − κ2 Sba Wb SH ba ≥ σn gb (Wb )γb H H 2 ge (Wb )fea fea − Sea Wb Sea ≤ σe ge (Wb )γe,a Tr(Qe,b Wb ) ≤ γe,b Wa , Wb 0 (9)

−1 H H where Qe,i = HH ei Rei Hei , Sea = fea Hbe , Sba = fba Hbb , and 2 gi (Wb ) = σi + Tr(Wb Rhbi ), for i ∈ {b, e}. As the next iteration, we perform the identical procedure to obtain Wa , until convergence. Readers are referred to the Sections 7.7 and 8.9 of [14] for the background behind the global convergence of cyclic coordinate descent algorithms. SDP for MRC Scheme: Considering Γij (M RC) for i, j ∈ {A, B}, j = i, SINR constraints in (5b) and (5c) can be expressed as wjH RHji wj /γi − wiH RHii wi κ2 ≥ σn2 . By applying the same technique in (8) for MRC scheme as by using Wi = wi wiH , and by relaxing r(Wi ) = 1 one can easily show that the SDP for MRC can be obtained as

min

Wa ∈HNa ,Wb ∈HNb

Tr(Wa ) + Tr(Wb )

s.t. Tr(RHba Wb ) /γa −Tr(RHaa Wa )κ2 ≥ σn2 Tr(RHab Wa )/γb −Tr (RHbb Wb ) κ2 ≥ σn2 Tr(RHae Wa ) /γe,a −Tr (RHbe Wb ) ≤ σe2 Tr(RHbe Wb ) /γe,b −Tr (RHae Wa ) ≤ σe2 Wa , Wb 0. (10) In the SDPs given in (9) and (10) we omitted the non-convex rank-1 constraints on WA and WB . In Proposition 1 we show that although these SDPs are approximations of the original FDB problem in (5), and they yield our original problem. Proposition 1: The solution of the SDR problems in (9) and (10) yields the exact solution of the non-relaxed problems.

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Proof-OC Scheme: The relation can be driven by examining the Karush-Kuhn-Tucker (KKT) conditions for every iteration step. Let α , β , ψ ≥ 0 and μ ≥ 0 be the optimum dual variables of the SINR constraints in the problem given in (9). Moreover, let Ωa 0 and Ωb 0 be the optimum dual variables of the constraints Wa 0 and Wb 0 respectively. The KKT conditions of Wa and Wb include Ωa Wa 0 and Ωb Wb 0, implying that Wa and Wb should lie in the nullspaces of Ωa and Ωb . The KKT condition of Wa in OC scheme is H 2 −κ2 SH Qb Ωa = INa −α RHaa fab fab ab Sab −σn RHaa γa −β H H 2 + μ Qe,a + ψ RHae feb feb − Seb Seb − σe RHea γe 0, (11) and it can be expressed as Ωa = Θ − Φ, where Θ 0. Hence, Θ1/2 is invertible. By observing the fact that H Ωa = Θ1/2 INa − Θ−1/2 Φ1/2 Θ−1/2 Φ1/2 Θ1/2 (12) it can be shown that r(INa − (Θ−1/2 Φ1/2 )(Θ−1/2 Φ1/2 )H ) ≥ Na − 1. The same also holds for Wb . Hence, r(Ωi ) ≥ Ni − 1 and r(Wi ) ≤ 1 for i ∈ {a, b}. Since Wi = 0 is not a feasible solution, the rank of feasible solutions of r(Wi ) can only be 1, proving the that solution of (9) yields (5). Proof-MRC Scheme: The KKT condition of Wa in MRC scheme can be written as Ωa = INt +α RHaa −β RHab +μ RHae −ψ RHae 0 (13) can be expressed as Ωa = Θ − (β RHab + ψ RHae ) where Θ is again PSD. Hence, Θ1/2 is invertible. Following the same approach in OC case, one can easily show that r(Wa ) = 1 also holds for MRC scheme. This can be extended to Wb . Using the problems in (9) and (10) and Proposition 1, we can compute the optimal beamforming vectors wa and wb [16]. III. S IMULATION R ESULTS In order to explore the performance of the proposed system, we developed a simulation environment, where we used MATLAB as the simulation platform and SeDuMi [17] as the convex optimization engine. The number of Monte-Carlo runs is set to 2000. Constraints are set as γa = γb = 20 dB, γe,a = γe,b = 0 dB, σn2 = 0 dB. In the FDB system both OC and MRC schemes are considered, where we assume that all nodes use the same type of combiner. All wireless channels are modeled as unit energy circularly symmetric complex Gaussian channels. The original AN@Tx [2] and AN@Rx [4] methods are extended to multi-antenna case and power optimization is applied to these scenarios with same parameters in order to maintain a fair performance comparison with our proposed method. To obtain the same secrecy level in two way communications between A and B, we assume that AN@Tx and AN@Rx use frequency division duplexing. Performance metrics we analyze are the total power consumption and ergodic channel capacity of the system. The simulation results are given in Figs. 2 and 3, where we consider improving channel conditions for E (i.e., decreasing σe2 ). These results are presented for the case where E tries to detect A. Total power consumption of the system consists of signal and

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Fig. 2. Reciprocals of eavesdropper’s noise power are shown on the x axis: (a) Total powers with κ = 10−1/2 , 10−3 for Na = Nb = 4, Ne = 1. (b) Total powers with Na = Nb = 4, Ne = 1, 2 for κ = 10−3 . (c) SINRs of eavesdropper for Na = Nb = Ne = 4 and κ = 10−3 .

ACKNOWLEDGMENT The authors would like to thank ˙Ilker Bayram and the anonymous reviewers for their valuable comments and suggestions. R EFERENCES

Fig. 3. Channel and secrecy capacities versus the reciprocals of eavesdropper’s noise power with Na = Nb = Ne = 4 and κ = 10−3 .

AN in AN-aided methods, whereas in our system it implies the total power invested only on the information bearing signals. As can be observed from Fig. 2(a), our proposed system model with κ = 10−3 can satisfy the defined constraints with a power level of 19.59 dB, when all the nodes deploys MRC (or OC), while AN@Tx and AN@Rx need 20.36 dB and 20.02 dB respectively, for 1/σe2 = 20 dB considering κ = 10−3 . As one can expect, as the SIC performance deteriorates (i.e., κ increases) performances of the proposed system and AN@Rx degrade, while AN@Tx system is not affected. The results of varying number of antennas at E is given in Fig. 2(b). Here, we can see that it gets more (less) costly to satisfy the secrecy constraint on E while Ne increases, for OC (MRC) scheme. In Fig. 2(c) SINRs of E capturing the signal of A(Γea ) is shown, where for all systems eavesdropper’s secrecy constraint is maintained. The main advantage of our proposed system is shown in Fig. 3, where we plot the ergodic channel capacity (Ci,j = E{log(1 + Γij )} for i, j ∈ {a, b, e}, i = j) and the secrecy capacity of a − b link (Ca,b (s) = Ca,b − Ca,e ) [4]. Note that both the channel capacity and the secrecy capacity of a-b link are doubled in the proposed FDB system. IV. C ONCLUSION In this paper, we proposed a joint beamforming vector and power optimization technique for FDB systems where the generated interference is optimized to act as AN in order to improve secrecy, while preserving the QoS levels between legitimate nodes. Our method is more advantageous than the existing ANbased techniques due to the increased throughput and power efficiency for the same secrecy level.

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