Robust Masked Beamforming for MISO Cognitive Radio ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 65, NO. 2, FEBRUARY 2016

Robust Masked Beamforming for MISO Cognitive Radio Networks With Unknown Eavesdroppers Jun Xiong, Dongtang Ma, Senior Member, IEEE, Kai-Kit Wong, Senior Member, IEEE, and Jibo Wei, Member, IEEE

Abstract—This paper studies a cognitive radio network (CRN), in which a multiple-input single-output (MISO) secondary transmitter (SU-Tx) aims to send confidential messages to its receiver (SU-Rx) in the presence of unknown eavesdroppers, while having to control its generated interference to the primary users (PUs) under a given threshold. Because the eavesdroppers are unknown, this paper considers the artificial noise (AN) approach or masked beamforming to provide the information secrecy. The objective of this paper is to maximize the power of AN in order to degrade the eavesdroppers, subject to the signal-to-interference-plus-noise ratio (SINR) constraint at the SU-Rx, as well as the interference temperature limits (ITLs) for the PUs. In the case of perfect channel state information (CSI), we reveal that the optimal strategy for the information-bearing signal is beamforming. Imperfect CSI cases of bounded and stochastic uncertainties are investigated. In the case of ellipsoid-bounded errors, we derive the equivalent forms for the SINR and ITL constraints and then transform the optimization problem into a form of semidefinite programming (SDP). For the case of probabilistic CSI uncertainties, we propose an outage-constrained robust formulation where the CSI errors are Gaussian distributed. With the aid of two kinds of Bernsteintype inequalities, we reexpress the probabilistic constraints into deterministic forms, which results in a safe approximate solution. Index Terms—Artificial noise (AN), channel errors, cognitive radio, masked beamforming, robust design, secrecy.

I. I NTRODUCTION

I

N recent years, there is increasing interest in providing physical-layer security using multiple antenna technologies. It has been illustrated that the secrecy performance of multiantenna systems is highly dependent on the amount of channel state information (CSI) at the transmitter. In the ideal case of precise knowledge of the main channel and the eavesdropper channel at the transmitter, the secrecy capacity of the Gaussian multiple-input multiple-output wiretap channel has been derived in [1] and [2]. However, in practice, the eavesdropper channel is usually not known, or the eavesdropper is not even known presently. In this case, a more sophisticated approach is Manuscript received June 15, 2014; revised November 2, 2014; accepted January 29, 2015. Date of publication February 5, 2015; date of current version February 9, 2016. This paper was supported in part by the China Natural Science Foundation under Grant 61101096 and Grant 61372099. The review of this paper was coordinated by Dr. S. K. Jayaweera. J. Xiong, D. Ma, and J. Wei are with the School of Electronic Science and Engineering, National University of Defense Technology, Changsha 410073, China (e-mail: [email protected]; [email protected]; wjbhw@nudt. edu.cn). K.-K. Wong is with the Department of Electronic and Electrical Engineering, University College London, London WC1E 6BT, U.K. (e-mail: kai-kit.wong@ ucl.ac.uk). 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/TVT.2015.2400452

to send artificial noise (AN) to confuse the eavesdroppers. The notion of using AN was first introduced by Goel and Negi in [3] and has drawn much attention in recent studies [4], [5]. On the other hand, cognitive radios have risen as a viable solution to the problem of rigid radio resource management by providing the opportunity for unlicensed cognitive users to transmit in the available licensed bands [6]. These unlicensed users are often referred to as secondary users (SUs) and the SUs should control their interference to the licensed users or primary users (PUs). Given that the interference power to each PU’s receiver is below an interference temperature limit (ITL), the concurrent transmission of cognitive radio networks (CRNs) and the PUs can achieve higher spectral efficiency, i.e., the underlay paradigm [7], [8]. In this respect, multiantenna technique is widely used to exploit the spatial diversity to steer the interference signals away from the PUs [9], [10]. However, CRN is believed to be more susceptible to eavesdropping because of the openness of the spectrum. In [11], the authors established the relationship between the CRN problem and the secrecy rate maximization (SRM) problem. Moreover, [12] addressed the optimal transmit design for the SRM problem, where a multiantenna SU transmitter sends confidential message to a single-antenna SU receiver in the presence of a single-antenna PU and a single-antenna eavesdropper (Eve). It was revealed that beamforming is optimal. Most recently, [13] has considered the worst-case robust transmit design for the SRM problem for the same scenario, in which the SU channel, PU channel, and Eve’s channel are all known imperfectly but with errors modeled within ellipsoid-bounded regions. In principle, to calculate the secrecy rate, one must know the full or partial eavesdropper’s CSI [1]–[5]. Such information is unlikely to be available in many scenarios, e.g., purely passive eavesdroppers. Unfortunately, to maximize the achievable secrecy rate, some prior knowledge of the eavesdropper channel was assumed in [12] and [13]. In this paper, we consider a downlink multiple-input singleoutput (MISO) CRN in the presence of multiple noncolluding eavesdroppers, where a multiantenna SU transmitter is communicating with a single-antenna SU receiver and shares the same spectrum with several PUs. No CSI about the eavesdroppers is assumed available, and we use AN to mask the desired signal for physical-layer secrecy. Our aim is to maximize the transmit power of AN for confusing any Eves, while maintaining a prescribed signal-to-interference-plus-noise ratio (SINR) at the SU receiver and satisfying the ITLs of the PUs. In the ANaided scheme, it is noted that imperfect CSI would cause interference leakage to the desired receiver, to result in significant performance degradation [14], [15]. Consequently, we develop

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XIONG et al.: BEAMFORMING FOR MISO COGNITIVE RADIO NETWORKS WITH UNKNOWN EAVESDROPPERS

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robust approaches that are insensitive to CSI errors. Our main contributions are summarized as follows. • For the perfect CSI case, the proposed transmit power optimization problem is a semidefinite programming (SDP). Furthermore, the optimal transmission strategy in the CRN for the information-bearing signal is beamforming. • When both the CSIs for the SU receiver and the PUs at the SU transmitter are imperfect, we address both the ellipsoid-bounded uncertainty model [16] and the stochastic uncertainty model [17], [18] for the CSI errors. • With bounded CSI errors, we derive equivalent forms to the SINR and ITL constraints and reformulate the worst case robust optimization problem into an SDP. • When the CSI errors are Gaussian with known distributions, we propose an outage-constrained robust formulation. In this case, we minimize the transmit power for the information-bearing signal, while guaranteeing the outage probability constraint of the SU and certain probabilities that the interference to the PUs exceeds the ITLs. With the aid of Bernstein-type inequalities, we obtain an approximation solution, which converts the complex probabilistic constraints into solvable deterministic forms. • The proposed MISO CRN network model embraces the conventional MISO wiretap channel in [3] and [4]. Thus, the proposed robust approaches in this paper can be directly extended to the corresponding transmit design for MISO wiretap channels with imperfect transmitter CSIs. The remainder of this paper is organized as follows. Section II describes the system model of a MISO CRN and formulates the optimization problem. In Section III, we study the worst case robust design with ellipsoid-bounded CSI errors for channel vectors and channel covariances, respectively. In Section IV, we investigate the robust transmit design with stochastic CSI errors. Simulation results are presented in Section V, and finally, we conclude this paper in Section VI. Notations: Throughput this paper, conjugate, transpose, inverse, and conjugate transpose are expressed as (·)∗ , (·)T , (·)−1 and (·)† , respectively. · returns the Frobenious norm of a vector or matrix. X 0(X 0) means that X is a Hermitian positive semidefinite (definite) matrix. In addition, λmax (A), Tr(A), and rank(A) present the maximal eigenvalue, trace, and rank of A, respectively. Furthermore, vec(A) is the vectorization of A by stacking its columns and mat(x) gives an N × N square matrix of the elements of an N 2 -vector x. Pr{·} denotes the probability of an input event, whereas ⊗ denotes the Kronecker product. In addition, we use IN to denote an N × N identity matrix and CN to represent the set of all N -dimensional complex vectors. Moreover, the notation x ∼ CN (0, σ 2 IN ) means that x is a random vector following a complex circular Gaussian distribution with zero mean and covariance σ 2 IN . II. S YSTEM M ODEL AND P ROBLEM F ORMULATION As shown in Fig. 1, we consider a communication scenario in a downlink CRN where the SU transmitter (SU-Tx) sends a confidential message to an SU receiver (SU-Rx) on the same

Fig. 1.

MISO CRN in the presence of multiple eavesdroppers.

band as K(≥ 1) PUs (PU-Rx). At the same time, M (≥ 1) noncolluding eavesdroppers (Eve-Rx) are trying to interpret the message sent by the SU-Tx. The SU-Tx has Nt antennas and all other terminals are equipped with only one antenna. We denote the channels between the SU-Tx and the SU-Rx, the kth PU-Rx, and the mth eavesdropper by hs , hp,k , and he,m , respectively, where hs , hp,k , he,m ∈ CNt . These channels are independent, and the elements are independent and identically distributed (i.i.d.) complex Gaussian random variables. The AN-aided transmit signal at the SU-Tx can be expressed as x = u + z [3], where u is the information-bearing signal with the covariance matrix of Su = E{uu† }, and z is the AN with the covariance matrix of Sz = E{zz† }. The received signals at SU-Rx, the kth PU-Rx, and the mth Eve-Rx are, respectively, expressed as ys = h†s u + h†s z + ns yp,k =

h†p,k u

+

h†p,k z

+ np,k ,

ye,m = h†e,m u + h†e,m z + ne,m ,

(1a) for k = 1, . . . , K

(1b)

for m = 1, . . . , M (1c)

where ns , np,k , and ne,m are additive white complex Gaussian noises. We assume ns , np,k , ne,m ∼ CN (0, σ 2 ). Thus, the instantaneous SINR at SU-Rx and the mth Eve-Rx can be, respectively, given by SINRs = SINRe,m =

h†s Su hs † hs Sz hs + σ 2 h†e,m Su he,m , h†e,m Sz he,m +σ 2

(2a) for m = 1, 2, . . . , M (2b)

and the interference temperature to the kth PU-Rx is ITp,k = h†p,k (Su + Sz )hp,k ,

for k = 1, 2, . . . , K.

(3)

Since no information of the eavesdropper channel is available, the best approach for the SU-Tx is to allocate as much power as possible to transmit the AN for secrecy. Obviously, the best secrecy performance for such an approach cannot be guaranteed; a fortuitous eavesdropper in the right location could

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end up with a better quality signal. Here, the goal is to reduce the likelihood of such an event [14]. Assuming that the total transmit power at the SU-Tx is P , our aim is to minimize the transmit power of the information signal with the SINR requirement of the SU-Rx γs and the ITL {Γk } by jointly optimizing the signal and AN covariance matrices Su , Sz , which is equivalent to maximizing transmit power of AN [14], [15]. Mathematically, it is expressed as

which means that rank

(1+η)I+

K

Su ,Sz 0

s.t.

Tr(Su )

Since (1 + η)I +

h†s Sz hs h†p,k (Su

+

σ2

≥ γs

+ Sz )hp,k ≤ Γk

(4b) ∀k

Tr(Su + Sz ) = P.

(4c) (4d)

It is worth mentioning that, when the constraint (4d) is replaced with Tr(Su + Sz ) ≤ P , the optimal solution is Sz = 0, i.e., AN should not be used. It can be found from the later simulation results that the usage of AN can enhance secure communications compared with “NO-AN.” Furthermore, without the PU ITL constraints (i.e., K = 0), the power minimization problem (4) is reduced to the transmit design in the conventional MISO wiretap channel [3], [4], where the optimal signal covariance matrix is Su = qq† with q = (hs /hs ), and the AN will be generated in the form of an isotropically distributed spatial noise on the orthogonal subspace of hs , i.e., h†s z = 0. Finally, the optimal objective value can be obtained in closed form, i.e., Tr(Su ) = min{P, (γs σ 2 /hs 2 )}. However, the AN may take any spatial pattern in the presence of the PU ITL constraints. With perfect CSI at the SU-Tx, (4) is known to be an SDP [19] and can be solved using standard optimization software packages such as SeDuMi [20]. Furthermore, the following lemma always holds true. Lemma 1: The optimal S∗u is rank-one, i.e., rank(S∗u ) = 1. Proof: We write the Lagrangian function of (4) as L (X, Y, λ, {μk }, η) = Tr(Su ) − λ Tr (Su − γs Sz )hs h†s − γs σ 2 K

+ μk Tr (Su + Sz )hp,k h†p,k − Γk k=1

+ η [Tr(Su + Sz ) − P ] − Tr(Su X) − Tr(Sz Y)

(5)

where λ, μk , η ≥ 0, X, Y 0 are the dual variables. Furthermore, the Karush–Kuhn–Tucker optimality conditions that are relevant to the proof are given by I − λhs h†s +

K

μk hp,k h†p,k + ηI − X = 0

(6a)

XSu = 0 λ, μk , η ≥ 0, X, Su 0.

(6b) (6c)

k=1

Postmultilying (6a) by Su and using (6b), we obtain K † (1 + η)I + μk hp,k hp,k Su = λhs h†s Su k=1

(8)

K

† k=1 μk hp,k hp,k

(1 + η)I +

(4a)

h†s Su hs

0

K

μk hp,k h†p,k

Su . (9)

k=1

Combining (8) and (9), we can obtain rank(Su ) ≤ 1. Since the case of Su = 0 is trivial, rank(Su ) = 1 always holds true. This completes the proof of Lemma 1. Lemma 1 indicates that for the perfect CSI case, beamforming is optimal for the information-bearing signal. III. ROBUST D ESIGN W ITH E LLIPSOIDAL C HANNEL S TATE I NFORMATION E RRORS Here, we assume that both the channels for the SU-Rx and PUs are imperfect at the SU-Tx and are estimated in the forms of channel vector and channel covariance, where the error is modeled by an ellipsoid-bounded uncertainty region. The original problem (4) becomes a worst case based design problem, which aims to ensure the signal and AN covariance matrices Su , Sz that satisfy the constraints for all channel realizations within the ellipsoidal regions. These problems are generally very hard to solve. After some heuristic manipulations, we reformulate them into tractable SDP problems. A. Ellipsoidal Channel Vector Uncertainty If the CSIs are estimated in the form of channel vectors, then the channels at the SU-Tx can be modeled as [16] ˆ s + hs hs = h ˆ p,k + hp,k hp,k = h

(10a) ∀k

(10b)

ˆ p,k } are the CSI estimates at the SU-Tx, ˆ s and {h in which h while hs and { hp,k } are the CSI errors. Depending on the estimation methods or feedback schemes, the channel errors follow specific random distributions. Here, we consider the general case where hs , { hp,k } are subject to colored noise and bounded by ellipsoids, i.e., Hs = hs : h†s Cs hs ≤ 1 (11a) † (11b) Hp,k = hp,k : hp,k Cp,k hp,k ≤ 1 where Cs , Cp,k 0 determine the quality of CSI and are assumed known at the SU-Tx. Moreover, it is found that when Cs and Cp,k approach infinity, the CSI becomes perfect, whereas the CSI is the worst if these vectors are zeros. Taking the CSI errors into account, the original optimization problem (4) can be reformulated as min Tr(Su )

Su ,Sz 0

(7)

Su = rank λhs h†s Su ≤ 1.

k=1

rank(Su ) = rank min

μk hp,k h†p,k

s.t.

Tr (Su − γs Sz )hs h†s ≥ γs σ 2

(12a) ∀ hs ∈ Hs

(12b)


Tr (Su +Sz )hp,k h†p,k ≤ Γk

∀ hp,k ∈ Hp,k , ∀ k (12c)

Tr(Su + Sz ) = P.

(12d)

It can be easily seen that (12) is a minimum convex semiinfinite programming because the objective function is linear and the constraints are defined by an infinite number of convex sets [19]. To handle the problem (12), we need to seek an efficient way to manage the infinitely many inequalities. By using S-Procedure [19], [21], the constraints (12b) and (12c) can be turned into linear matrix inequalities (LMIs). Lemma 2 (S-Procedure): Let fk (x) = x† Ak x + 2 Re{b†k x} + ck

(13)

where k = {1, 2}, and Ak is an N × N Hermitian matrix. The implication f1 (x) ⇒ f2 (x) holds if and only if there exists ρ ≥ 0 such that A1 b1 A2 b 2 ρ (14) b†1 c1 b†2 c2 ˜ such that f1 (˜ x) < 0. provided that there exists a point x ˆ s + hs into the SINR constraint We substitute hs = h (12b) and the following implication will hold:

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vectors. In [22]–[24], the worst-case SRM problems for the deterministic uncertainty model were investigated, where [22], [23] considered the case that the eavesdropper channel mismatch is norm-bounded and subsequently, [24] considered the robust cooperative jamming problem with aid of an external helper where the channel mismatches from the source and the helper to the eavesdropper are modeled as norm-bounded uncertainty regions. A more recent work in [25] studied the SRM problem where both the main and eavesdropper channel vectors are imperfect at the transmitter and the channel errors are modeled as norm-bounded uncertainty regions. However, it is noted that the robust AN-aided transmit design presented in (19) with K = 0 is very different from [22]–[25] in the following aspects: i) We assume that no information about the eavesdroppers is available at the transmitter, and ii) the channel mismatches are modeled as general ellipsoidal regions. B. Ellipsoidal Channel Covariance Uncertainty Compared with the channel vector, the second-order statistics of the channel changes more slowly. Thus, the estimated CSIs in the form of channel covariances are more practical. In this case, we model channel covariance uncertainties as [26] ˆ s + Hs Hs hs h†s = H

h†s Cs hs ≤ 1

ˆ † (Su − γs Sz ) hs ⇒ h†s (Su − γs Sz ) hs + 2 Re h s ˆ s − γs σ 2 ≥ 0. ˆ † (Su − γs Sz ) h +h s

(15)

According to S-Procedure, (12b) holds if and only if there exists a ρs ≥ 0 such that ˆs ρs Cs + Su − γs Sz (Su − γs Sz )h ˆ † (Su − γs Sz )h ˆ s − γs σ 2 − ρs 0. ˆ † (Su − γs Sz ) h h s s (16) Similarly, the ITL constraint (12c) can be expressed as h†p,k Cp,k hp,k ≤ 1

ˆ † (Su + Sz ) hp,k ⇒ h†p,k (Su + Sz ) hp,k + 2 Re h p,k ˆ † (Su + Sz ) h ˆ p,k − Γk ≤ 0. +h p,k

(17)

By using S-Procedure, (17) can be replaced by ˆ p,k ρk Cp,k − Su − Sz −(Su + Sz )h ˆ † (Su + Sz ) −h ˆ † (Su + Sz )h ˆ p,k + Γk − ρk 0 −h p,k

p,k

k ∈ {1, 2, . . . , K},

∃ρk ≥ 0. (18)

Su ,Sz 0 ρs ≥0,{ρk ≥0}

s.t.

Tr(Su )

(16) and (18) Tr(Su + Sz ) = P.

(19a)

When K = 0, (19) is reduced to the robust AN design for conventional MISO wiretap channels under imperfect channel

(20b)

where the parameters Cs , Cp,k 0 are known a prior. Taking the channel covariance uncertainties into account, the original optimization problem (4) can be recast into min Tr(Su )

s.t.

Tr(Su Hs ) ≥ γs Tr(Sz Hs ) + σ 2 Tr ((Su + Sz )Hp,k ) ≤ Γk

(19b)

∀k

ˆ s and {H ˆ p,k } represent the CSI estimates at the SUwhere H Tx while Hs and { Hp,k } correspond to the CSI errors. Similarly, we consider the more practical scenario where the channel covariance matrices are estimated in the presence of colored noises, and Hs and { Hp,k } are bounded by ellipsoidal regions. In addition, Hs and { Hp,k } should be set to guarantee the positive semidefiniteness properties of the ˆ p,k + Hp,k , i.e., ˆ s + Hs and H matrices H ˆ s + Hs 0 ˜ s = Hs : Hs = H† , H H s Tr( H†s Cs Hs ) ≤ 1 (21) ˆ p,k + Hp,k 0 ˜ p,k = Hp,k : Hp,k = H† , H H p,k Tr( H†p,k Cp,k Hp,k ) ≤ 1 (22)

Su ,Sz 0

Thus, (12) can be converted into a convex SDP min

ˆ p,k + Hp,k Hp,k hp,k h†p,k = H

(20a)

Tr(Su + Sz ) = P.

(23a) ˜s ∀ Hs ∈ H ˜ p,k ∀ k ∀ Hp,k ∈ H

(23b) (23c) (23d)

Note that (23) is more challenging to solve than (12), due to the complex constraints under channel covariance errors. In

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CRN beamforming design without secrecy consideration [27], the robust solution involves several additional approximations and the SINR constraint (23b) is usually replaced by min Hs ∈H˜s Tr(Su Hs ) max Hs ∈H˜s Tr(Sz Hs ) + σ 2

≥ γs

(24)

max

Tr(Su Hp,k ) +

max

˜p,k Hp,k ∈H

Tr(Sz Hp,k ) ≤ Γk .

(25) The approximation is a conservative way to find the minimum of the SINR and the maximum of the ITL. For example, if (24) is satisfied, then the SINR constraint (23b) will also be satisfied. The reverse statement however does not hold. Both approximations may cause performance loss. To circumvent that, the authors of [28] have considered the transmit power minimization problem in a multiuser MISO CRN under normbounded channel covariance uncertainties and directly obtained the exact minimum of SINR, i.e., min Tr ((Su − γs Sz )Hs ) ≥ γs σ 2

Hs

ˆ s − εs Su − γs Sz ≥ γs σ 2 ⇔ Tr (Su − γs Sz )H

(26)

where the channel covariance error is modeled as a normbounded region Hs ≤ εs . However, the approach in [28] ignores the positive semidefiniteness property of the mismatched covariance matrix. Specifiˆ s + Hs 0 may be not satisfied because Su − γs Sz cally, H cannot be positive semidefinite at the optimality. An improved robust beamforming approach for conventional CRN without secrecy consideration has been presented in [16], which takes into account the positive semidefiniteness constraints on the covariance matrices. Next, we will extend the concepts of [16], [28], [29] to the AN-aided transmit design. Our approach avoids the aforementioned approximation and derives the exact equivalent forms for the SINR constraint and the ITL constraint through the use of Lagrange duality. First, we use Lagrange duality to solve the minimization problem in (23b) and obtain the equivalence form for the SINR constraint, which is summarized in the following lemma. Lemma 3: Let Hs = H†s , Tr( H†s Cs Hs ) ≤ 1, and ˆ s + Hs 0. The following equivalence holds: H

s.t. Hs = H†s Tr H†s Cs Hs ≤ 1 (29)

The corresponding Lagrangian function is written as L( Hs , Φ, λ) ˆ s + Tr ((Su − γs Sz ) Hs ) = Tr (Su − γs Sz )H ˆ s + Hs ) + λ Tr H†s Cs Hs − 1 − Tr Φ(H (30) where λ ≥ 0 is the Lagrange multipler, and Φ is used to ˆ s + Hs 0. guarantee that H By differentiating this function and equating it to zero, we have ∂L = (Su − γs Sz ) + λCs Hs + λ Hs Cs − Φ = 0 ∂ H∗s (31) and applying the Kronecker operation, we then obtain ˜ −1 vec(Φ − Su + γs Sz ) . (32) λ Hs = mat C s Furthermore, we obtain the Lagrangian dual function for (29) as (a)

G(Φ, λ) =

(b)

=

inf

L( Hs , Φ, λ)

inf

ˆ s) ˆ s − λ − Tr(ΦH Tr (Su − γs Sz )H

˜s Hs ∈H

˜s Hs ∈H

+ Tr ( Hs ((Su − γs Sz ) + λCs Hs − Φ)) (c) ˆ s − Tr(ΦH ˆ s) = inf Tr (Su − γs Sz )H ˜s Hs ∈H

− λ − λTr H†s Cs Hs (d) ˆ s − Tr(ΦH ˆ s ) − 2λ = Tr (Su − γs Sz )H

(33)

(27)

where (c) follows from (31) and (d) is due to the fact that Tr( H†s Cs Hs ) = 1 holds at the optimality. It is clear that the original problem (29) is convex in Hs . Furthermore, there always exists a nonempty solution, e.g., √ −(1/2) , to make it feasible. Thus, we can Hs = (1/ Nt )Cs conclude that the strong duality holds [19], i.e., the maximal value of the dual function (33) is equal to the minimal value of (29). Accordingly, min Hs ∈H˜s Tr ((Su − γs Sz )Hs ) ≥ γs σ 2 holds if there exists some Φ 0 such that ˆ s ) − 2λ ≥ γs σ 2 . ˆ s − Tr(ΦH (34) Tr (Su − γs Sz )H

(28)

Moreover, using (32) and Tr( H†s Cs Hs ) ≤ 1, we obtain 1 2 ˜ −1 vec(Φ − Su + γs Sz ) (35) ≤ λ. Cs mat C s

min Tr ((Su − γs Sz )Hs ) ≥ γs σ 2 ⎧ ˆ s ) − 2λ ≥ γs σ 2 ˆ s − Tr(ΦH ⎪ Tr (S − γ S ) H ⎪ u s z ⎪ ⎪ ⎨ 1 2 ˜ −1 vec(Φ − Su + γs Sz ) ⇔ Cs mat C ≤λ s ⎪ ⎪ ⎪ ⎪ ⎩ λH ˆ s + mat C ˜ −1 vec(Φ − Su + γs Sz ) 0 s

˜s Hs ∈H

where λ ≥ 0, Φ 0 are slack variables, and ˜ s = IN ⊗ Cs + CT ⊗ I N . C s t t

Hs

ˆ s + Hs 0. H

and the ITL constraint (23c) is approximated as ˜p,k Hp,k ∈H

Proof: We consider the optimization problem ˆ s + Tr ((Su − γs Sz ) Hs ) min Tr (Su − γs Sz )H


ˆ s + Hs 0 In addition, according to (32) and H ˆ s + mat C ˜ −1 vec(Φ − Su + γs Sz ) 0. λH s

(36)

Combining (34)–(36), we conclude the equivalence in (27) is true. This completes the proof of Lemma 3. Following the same steps as in Lemma 3, we can obtain the equivalent form for the ITL constraint (23c). Furthermore, because Su + Sz 0 always holds, the positive semidefiniteness ˆ p,k + Hp,k constraint for the channel covariance matrix H can be guaranteed in general, and hence, (23c) can be replaced by the following deterministic form:

ˆ p,k Tr (Su + Sz )H

the SU’s SINR below a predefined level and maintaining the probabilities of the interferences to the PUs exceeding the ITL below some preset thresholds. The outage-constrained robust optimization problem can be expressed as min

Su ,Sz 0

Tr(Su )

s.t.

Pr

(40a) h†s Su hs

h†s Sz hs + σ 2

≤ γs

≤ Ps

Pr h†p,k (Su + Sz )hp,k ≥ Γk ≤ Pp,k

(40b) ∀k (40c)

Tr(Su + Sz ) = P

1 2 ˜ −1 vec(Su + Sz ) + 2 Cp,k mat C ≤ Γk p,k

∀k

(37)

˜ p,k = IN ⊗ Cp,k + CT ⊗ IN . where C t t p,k Finally, based on the aforementioned results, the optimization problem (23) can be also transformed into an SDP: min

Su ,Sz 0, λ≥0,Φ0

Tr(Su )

⎧ ⎨ (34)−(37) s.t.

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⎩ Tr (S + S ) = P. u z

(38)

Without the ITL constraints at the PUs (i.e., K = 0), (38) is reduced to the robust AN design for conventional MISO wiretap channels under imperfect channel covariances. To the best of our knowledge, the robust AN-aided transmit design under imperfect channel covariances has not been investigated in the literature. The equivalent form for the SINR constraint to obtain the optimal AN covariance design is entirely new. IV. ROBUST D ESIGN W ITH S TOCHASTIC C HANNEL S TATE I NFORMATION E RRORS Here, our endeavor is to extend the robust design to the scenario where the CSI errors are modeled as Gaussian random variables with known statistical distributions, i.e., [30] ˆ s + hs hs = h ˆ p,k + hp,k , hp,k = h

(39a) for k = 1, 2, . . . , K

(39b)

where hs and { hp,k } are the errors, which are zero-mean Gaussian random variables with covariances Cs and Cp,k , i.e., hs ∼ CN (0, Cs ) and hp,k ∼ CN (0, Cp,k ). As hs and hp,k are unbounded, meeting the SINR and ITL constraints in all time would be impossible. Despite this, the worst case approach in Section III often leads to overly conservative solutions against some extremely rare conditions that occur only with very low probability. Here, we propose a more flexible approach that minimizes the transmit power for the information signal, while ensuring the outage probability of

(40d)

where 0 < Ps ≤ 1 is the maximum outage probability of the SU-Rx SINR, and 0 < Pp,k ≤ 1 is the maximum tolerable probability of the kth PU exceeding the ITL. Apart from the outage-constrained formulation, one can also adopt the average SINR and average ITL constraints by taking expectation over all possible channel error realizations. In contrast to the average-based formulation, the outage-constrained formulation caters for delay-critical applications and provides a performance tradeoff between the SU and the PUs through probabilistic constraints [30]–[32]. It can be illustrated that the optimization problem (40) is more challenging to solve than the aforementioned worst-case-based problems (12) and (23), because of the fact that the probabilistic constraints have no closed-form expressions. To tackle these challenges, we will propose an approximation to obtain a solution for (40), which does not require explicitly calculating these probabilities. 1/2 To this end, let hs = Cs gs with gs ∼ CN (0, INt ) and through some derivations, (40b) can be rewritten as Pr

h†s Su hs

≤ γs

≤ Ps h†s Sz hs + σ 2 ⎞ ⎛ 1 1 gs† Cs2 (Su − γs Sz )Cs2 gs ⎟ ⎜ 1 † 2 ˆ ⎟ ⇔ Pr ⎜ ⎝ +2Re gs Cs (Su − γs Sz )hs ⎠ ≤ Ps . ˆ † (Su − γs Sz )h ˆs ≤ γs σ 2 − h s

(41)

The right-hand side of (41) is a chance constraint and is still hard to deal with. Here, we use the recently proposed Bernsteintype inequalities [33] to transform the chance constraint into a deterministic form, which can effectively control the quadratic forms of Gaussian variables. Lemma 4 (Bernstein-type Inequalities): Let ϑ = g† Ag + 2Re{g† a} where A is an N × N Herimitian matrix, a ∈ CN , and g ∼ CN (0, IN ). Then, for any σ ≥ 0, we can obtain √ $ Pr ϑ ≤ Tr(A) − 2σ vec(A)2 + 2a2 − σs− (A) ≤ exp (−σ) (42) √ $ Pr ϑ ≥ Tr(A) + 2σ vec(A)2 + 2a2 + σs+ (A) ≤ exp(−σ)

(43)

750


where s− (A) = max {λmax (−A), 0} s+ (A) = max {λmax (A), 0} .

(44) (45)

Proof: See [33, Lemma 0.2]. Using the Bernstein-type inequality (42), (41) holds true if the inequality (46), shown at the bottom of the page, is satisfied where σs = − lnPs . In other words, (46) results in an upper bound on the probability that the SINR is smaller than the threshold γs . As a result, (46) serves as an approximation for (41). Furthermore, through introducing auxiliary variables λs and μs , (46) is equivalently rewritten as 1 $ 1 Tr Cs2 (Su − γs Sz )Cs2 − 2 ln Ps−1 μs + λs ln Ps ˆ † (S − γs Sz )h ˆs ≥ γ σ2 − h ⎡ s 1 s u ⎤ 1 vec Cs2 (Su − γs Sz )Cs2 ⎣ √ 1 ⎦ ≤ μs ˆs 2Cs2 (Su − γs Sz )h 1

(47a) (47b)

1

λs INt + Cs2 (Su − γs Sz )Cs2 0.

(47c)

Similarly, (40c) can be rewritten as Pr h†p,k (Su + Sz )hp,k ≥ Γk ≤ Pp,k ⎞ ⎛ 1 1 † 2 2 Cp,k (Su + Sz )Cp,k gp,k gp,k 1 ⎟ ⎜ ˆ p,k } ⎠ ≤ Pp,k ⇔ Pr ⎝ +2Re{g† C 2 (Su + Sz )h p,k p,k ˆ † (Su + Sz )h ˆ p,k ≥ Γk − h

V. N UMERICAL R ESULTS (48)

p,k

1/2

where gp,k ∼ CN (0, INt ) and hp,k = Cp,k gp,k . Furthermore, (48) can be replaced by the following deterministic forms using the Bernstein-type inequality (43) such that ) 1 1 −1 2 2 Tr Cp,k (Su + Sz )Cp,k μk − λk ln Pp,k + 2 ln Pp,k ˆ † (Su + Sz )h ˆ p,k ≤ Γk − h p,k ⎡ ⎤ 1 1 2 2 vec Cp,k (Su + Sz )Cp,k ⎣ √ 1 ⎦ ≤ μk 2 ˆ p,k 2Cp,k (Su + Sz )h 1

(49a) (49b)

1

2 2 (Su + Sz )Cp,k 0 λk INt − Cp,k

(49c)

where {μk }, {λk } are auxiliary variables. Finally, the original nonconvex problem (40) can be transformed into the following problem: min

Su ,Sz 0 μs ,λs ≥0,{μk },{λk ≥0}

s.t.

1 2

Tr(Su )

(47) and (49) Tr(Su + Sz ) = P.

1 2

Tr Cs (Su − γs Sz )Cs

−

$

Problem (50) has a linear objective and constraints that are in the form of LMIs or affine functions, which can be efficiently solved by existing convex optimization solvers such as SeDuMi [20]. As a final remark, we will give the computational complexities of the proposed algorithms. It is known that the number of variables, the number of SDP constraints and the size of SDP dominate the overall complexity [16]. For the perfect CSI case, the optimization problem (4) has 2Nt2 design variables with K + 2 scalar constraints. For the case of deterministic channel vector uncertainty, the optimization problem (19) has 2Nt2 design variables and K + 1 slack variables, with K + 1 SDP constraints of size Nt + 1 and one scalar constraint. For the case of deterministic channel covariance uncertainty, the optimization problem (38) has 2Nt2 design variables and Nt2 + 1 slack variables, with one SDP constraint of size Nt and K + 3 scalar constraints. Finally, for the case of stochastic channel uncertainty, (50) has 2Nt2 design variables and 2K + 2 slack variables, with K + 1 SDP constraints of size Nt and 2K + 3 scalar constraints. In addition, all methods should add two SDP constraints of size Nt , i.e., Su , Sz 0, and there are also some computations required to perform matrix decomposition and multiplication in the proposed methods.

(50)

Here, we present simulation results to evaluate the secrecy performance of our proposed AN-aided designs. In the simulations, we considered that Nt = 4 and each entry of channel vectors hs , hp,k , and he,m is randomly generated from an i.i.d. complex Gaussian distribution with zero mean and unit variance. Simulations with 3000 randomly generated channels (hs , hp,k , he,m ) were averaged and plotted. In addition, we assumed that all PUs have the same ITL, i.e., Γk = Γ, ∀ k, and the maximal transmit power P is 5 dB. In the simulations, we considered Cs = Cp,k = (1/ξchn )INt for channel matrix uncertainty, Cs = Cp,k = (1/ξcov )INt for channel covariance uncertainty, and hs , hp,k ∼ CN (0, ξran INt ) for stochastic channel uncertainty. The probabilistic constraints in the outage problem (50) are set to 0.1, i.e., Ps = Pp,k = 0.1 ∀ k. As shown in [30] and [34], there exists a close relationship between channel matrix uncertainty and random error, i.e., Pr hs 2 ≤ ξchn = 1 − Ps . (51) From (51), the CSI error bound of ξchn can be calculated −1 as ξchn = CDF−1 2Nt (1 − Ps )(ξran /2), where CDF2Nt (·) is the inverse cumulative distribution function (CDF) of Chi-square random variable with 2Nt degrees of freedom [35]. In other

* 1 2 1 1 2 ˆ s 2σs vec Cs2 (Su − γs Sz )Cs2 + 2 Cs2 (Su − γs Sz )h 1 1 ˆ † (Su − γs Sz )h ˆs − σs s− Cs2 (Su − γs Sz )Cs2 ≥ γs σ 2 − h s

(46)


751

Fig. 2. Maximal SINR among Eves versus the SINR requirement γs with K = 2, M = 3, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty.

Fig. 3. Maximal SINR among Eves versus the SINR requirement γs with K = 2, M = 3, Γ = −2 dB, and ξcov = 5 × 10−3 for the case of deterministic channel covariance uncertainty.

words, the error bound of ξchn is chosen so that it covers 90% of the uncertainty region. Furthermore, the following benchmarks are compared with our proposed algorithms:

CSI” but at the cost of no control of the SINR for SU-Rx and interference to PUs. Fig. 3 presents the maximal SINR among Eves versus the SINR requirement γs with K = 2, M = 3, Γ = −2 dB, and ξcov = 5 × 10−3 for the case of deterministic channel covariance uncertainty. As we can see, for the case of imperfect channel covariances, the maximal SINR of Eves for all approaches increases with an increase in γs . However, the proposed robust approach “Robust, Channel Covariance” has a significant performance gain over “NO-AN.” Similarly, since the SU transmitter has to sacrifice secrecy performance to satisfy the ITL constraints to the PUs, the SINR obtained by “Robust, Channel Covariance” is lower that that of “NO-ITL, Channel Covariance.” For further insight, we show the histogram of the achieved ITL to the PUs for “Naive” and the proposed robust approaches in Figs. 4 and 5, respectively for K = 2, M = 3, γs = 2 dB, and Γ = −2 dB. The simulations are performed for ξran = 10−2 and ξcov = 5 × 10−3 , and ξchn is obtained according to (51). As we can see, “Naive” approaches cannot satisfy the requirement due to the ignorance of CSI errors. However, as shown in Fig. 5, for the cases of imperfect channel vectors and channel covariances, using proposed “Robust” approaches, the ITL to PUs is always below the required level, whereas the required probability of the ITL to PUs above −2 dB is met for “Robust, Random Channel.” The corresponding histogram of the achieved SINR of SURx for “Naive” and the proposed robust methods is provided in Figs. 6 and 7, respectively. As expected, in the cases of “Naive” approaches, a number of SINR constraints is not met. However, from Fig. 7, both the proposed robust approaches “Robust, Channel Vector” and “Robust, Channel Covariance” have an absolute control over the SINR constraint, while in the “Robust, Random Channel” approach, the outage probability of the SINR of SU-Rx below 2 dB is much less than 0.1. Fig. 8 gives the maximal SINR among Eves versus different M values with M ∈ [2, 30], K = 2, γs = 0 dB, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty. As expected,

• perfect CSI: solving problem (4) based on the perfect CSIs for {hs , hp,k , he,m } at the SU-Tx; • naive: solving problem (4) based on the estimated CSIs ˆ p,k , h ˆ e,m } at the SU-Tx; ˆ s, h {h • NO-AN: solving problem (4) with Sz = 0 based on the perfect CSIs for {hs , hp,k , he,m } at the SU-Tx;1 • NO-ITL: solving problems (19), (38), and (50) without the PU interference constraints (i.e., K = 0), respectively. These optimization problems correspond to the robust AN designs for conventional MISO wiretap channels. Fig. 2 shows the maximal SINR among Eves versus the SINR requirement γs with K = 2, M = 3, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty. We observe that the maximal SINR of Eves for all approaches increases as the SINR requirement γs increases and our proposed AN-aided methods outperform “NO-AN” in all SINR regions, even for imperfect CSI cases. Results also show that the proposed outageconstrained approach “Robust, Random Channel” is more efficient than the worst case approach “Robust, Channel Vector.” Recall that the worst case method obtains absolute control of the SINR requirement for SU and ITL to PUs on every possible CSI error realization, whereas the outage-constrained method can provide an efficient balance between the SINR requirement and ITL and secrecy performance using probabilistic constraints. In addition, the ITL constraints would cause a secrecy performance loss and the performance of the robust designs “Robust, Random Channel” and “Robust, Channel Vector” is worse than that of “NO-ITL, Random Channel” and “NO-ITL, Channel Vector,” respectively. Finally, it can be found that “Naive” results in a lower SINR than proposed robust approaches and obtains nearly the same performance as “Perfect 1 The

constraint (4d) in problem (4) is replaced with Tr(Su + Sz ) ≤ P .

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Fig. 4. Histogram of the achieved ITL to PUs with K = 2, M = 3, γs = 2 dB, and Γ = −2 dB for “Naive” approaches.

Fig. 5. Histogram of the achieved ITL to PUs with K = 2, M = 3, γs = 2 dB, and Γ = −2 dB for proposed robust approaches.

Fig. 6. Histogram of the achieved SINR of SU-Rx with K = 2, M = 3, γs = 2 dB, and Γ = −2 dB for “Naive” approaches.

Fig. 7. Histogram of the achieved SINR of SU-Rx with K = 2, M = 3, γs = 2 dB, and Γ = −2 dB for proposed robust approaches.

Fig. 8. Maximal SINR among Eves versus different M values with K = 2, γs = 0 dB, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty.

the maximal SINR among Eves for all the approaches increases with an increase in M . It can be found that when M is sufficiently large, the maximal SINR among Eves would be higher than the SINR constraint of SU-Rx, which means that the achievable secrecy rate is zero. When M > 10, the resulting maximal SINR among Eves for “NO-AN” is higher than the SINR constraint 0 dB. However, for “Robust, Channel Vector” and “Robust, Random Channel,” in order to drive the secrecy rate to zero, the number of the eavesdroppers should be larger than 18 and 27, respectively. This further shows that our proposed AN-aided schemes provide substantial performance gains over “NO-AN” design for both perfect and imperfect CSI cases. The similar results can be found in Fig. 9 for the case of deterministic channel covariance uncertainty, where M ∈ [2, 30], K = 3, γs = 0 dB, Γ = −2 dB, and ξcov = 5 × 10−3 . Fig. 10 presents the maximal SINR among Eves versus different K values with M = 3, γs = 0 dB, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty. We can see that the maximal SINR among Eves for all approaches increases


Fig. 9. Maximal SINR among Eves versus different M values with K = 3, γs = 0 dB, Γ = −2 dB, and ξcov = 5 × 10−3 for the case of deterministic channel covariance uncertainty.

Fig. 10. Maximal SINR among Eves versus different K values with M = 3, γs = 0 dB, Γ = −2 dB, and ξran = 10−2 for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty.

as the number of PUs K increases. This is because we should sacrifice more transmit power to guarantee the ITL constraints. Furthermore, as shown in Fig. 2, it can also be found that the outage-constrained approach “Robust, Random Channel” outperforms the worst case approach “Robust, Channel Vector.” Obviously, given hs and hp,k may not support the desired SINR and ITL constraints with a total transmit power P , which would make the optimization problem infeasible. Fig. 11 shows the feasible rates of the proposed approaches versus CSI error variance ξran with γs = 0 dB and Γ = −2 dB for the cases of deterministic channel vector uncertainty and stochastic channel uncertainty. For “perfect CSI“, “NO-AN,” “NO-ITL” and “Naive,” the problem has more lenient SINR and ITL constraints, and as a consequence the feasible rates are always above 95%. As expected, the larger the CSI error variance, the smaller the feasible rate, and vice versa. In addition, the proposed outage-constrained approach “Robust, Random Channel” has much higher feasible rate than that of the worst-case approach “Robust, Channel Vector” over the whole CSI error regions. Combined this result with Fig. 2, we can conclude that

753

Fig. 11. Feasible rates versus error variance ξran with K = 2, M = 3, γs = 0 dB, and Γ = −2 dB.

Fig. 12. Feasible rates versus error bound ξcov with K = 2, M = 3, γs = 0 dB, and Γ = −2 dB.

proposed outage-constrained approach “Robust, Random Channel” provides much better performance than “Robust, Channel Vector” at the cost of slight degradation for the SINR of SU-Rx and the ITL to PUs. In other words, the proposed outage-constrained approach obtains a performance tradeoff between the SINR of SU-Rx and the ITL to PUs and the secrecy performance via some probabilistic constraints. Fig. 12 shows the feasible rates of the proposed approaches versus the CSI error bound ξcov when K = 2, M = 3, γs = 0 dB, and Γ = −2 dB for the case of deterministic channel vector uncertainty. Similarly, it can be observed that the feasible rate is almost 100% for “perfect CSI,” “NO-AN,” “NO-ITL,” and “Naive” approaches, which serve as the upper bound. We also see that the larger the CSI error bound, the smaller the feasible region for the proposed approach “Robust, Channel Covariance.” VI. C ONCLUSION This paper has studied the AN-aided transmit optimization for a MISO CRN in the presence of multiple passive eavesdroppers. The aim was to maximize the power of AN available while

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Jun Xiong received the B.S. and Ph.D. degrees from the School of Electronic Science and Engineering, National University of Defense Technology (NUDT), Changsha, China, in 2009 and 2014, respectively. He is currently a Lecturer with the School of Electronic Science and Engineering, NUDT. His research interests include cooperative communications, physical-layer security, and resource allocation.


Dongtang Ma (SM’14) received the B.S. degree in applied physics and the M.S. and Ph.D. degrees in information and communication engineering from the National University of Defense Technology (NUDT), Changsha, China, in 1990, 1997, and 2004, respectively. From 2004 to 2009, he was an Associate Professor with the School of Electronic Science and Engineering, NUDT. Since 2009, he has been a Professor with the Department of Communication Engineering, School of Electronic Science and Engineering, NUDT. From August 2012 to February 2013, he was a Visiting Professor with the Centre for Communication Systems Research, University of Surrey, Surrey, U.K. His research interests include physical-layer security, cooperative communications and networks, multiple-input–multiple-output, and space communications. Prof. Ma is a member of the Executive Directors of Hunan Electronic Institute.

Kai-Kit Wong (SM’08) received the B.Eng., M.Phil., and Ph.D. degrees from the Hong Kong University of Science and Technology, Hong Kong, in 1996, 1998, and 2001, respectively, all in electrical and electronic engineering. He is a Reader in Wireless Communications with the Department of Electronic and Electrical Engineering, University College London, London, U.K. He held faculty and visiting positions with the University of Hong Kong, Hong Kong; Lucent Technologies, Inc., Murray Hill, NJ, USA; Bell Labs, Holmdel, NJ, USA; the Smart Antennas Research Group, Stanford University, Stanford, CA, USA; and the Department of Engineering, University of Hull, Kingston upon Hull, U.K. He is a Fellow of the Institution of Engineering and Technology (IET). He is also on the Editorial Board of the IEEE W IRELESS C OMMUNICATIONS L ETTERS, the IEEE C OMMUNICATIONS L ETTERS, the IEEE Communications Society/KICS Journal of Communications and Networks, and IET Communications. He is a Senior Editor of the IEEE C OMMUNICATIONS L ETTERS. He also served as an Editor of the IEEE T RANSACTIONS O N W IRELESS C OMMUNICATIONS from 2005 to 2011 and the IEEE S IGNAL P ROCESSING L ETTERS from 2009 to 2012.

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Jibo Wei (M’03) received the B.S. and M.S. degrees in electronic engineering from the National University of Defense Technology (NUDT), Changsha, China, in 1989 and 1992, and the Ph.D. degree in electronic engineering from Southeast University, Nanjing, China, in 1998. He is currently the Director and a Professor of the Department of Communication Engineering with NUDT. His research interests include wireless network protocol and signal processing in communications, more particularly, the areas of multipleinput-multiple–output, multicarrier transmission, cooperative communications, and cognitive networks. Prof. Wei is the member of the IEEE Communication and Vehicular Technology Societies and a Senior Member of the China Institute of Communications and Electronics. He is also on the Editorial Board of the Journal on Communications and China Communications.