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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 9, SEPTEMBER 2015
AN-Aided Secrecy Precoding for SWIPT in Cognitive MIMO Broadcast Channels Bing Fang, Zuping Qian, Wei Zhong, and Wei Shao
Abstract—In this letter, we study the secrecy precoding problem for simultaneous wireless information and power transfer (SWIPT) in a cognitive multiple-input multiple-output (MIMO) broadcast channel. We adopt an artificial noise (AN)-aided precoding scheme and formulate the problem as a secrecy rate maximization (SRM) problem, which is subject to both an interference power constraint imposed to protect the primary user (PU) and an energy harvesting constraint required by the secondary energy receiver (SER). Since the formulated SRM problem constitutes a difference convex (DC)-type programming problem, we solve it by employing a successive convex approximation (SCA) method. With the SCA method, the nonconvex part of the SRM problem can be locally linearized to its first-order Taylor expansion. Then, relying on solving a series of convexified optimization problems, an iterative precoding algorithm is developed. Numerical simulations are also provided to demonstrate the proposed algorithm. Results show that our algorithm can achieve a near-optimal performance with guaranteed convergence. Index Terms—MIMO precoding, cognitive radio, energy harvesting, physical layer security, successive convex approximation.
I. I NTRODUCTION
C
OGNITIVE RADIO (CR) is a novel approach for enhancing utilization of the precious spectrum resources. One popular way for CR designing is to allow the secondary users (SUs) to transmit concurrently on the same frequency bands with the licensed primary users (PUs) as long as the resulting interference power is kept below the maximum interference power threshold that can be tolerated by the licensed PUs. Along with multiple-input multiple-output (MIMO) becoming a dominating technology for the next-generation cellular networks, the researches with regard to cognitive MIMO radio have gained more and more interests [1]. On the other hand, simultaneous wireless information and power transfer (SWIPT) in MIMO broadcasting systems has recently drawn significant interests for its dual use of the precious spectrum [2]. In practice, the energy receivers (ERs) are usually deployed closer to the base station (BS) than the information receivers (IRs). So that, ERs can easily eavesdrop the confidential message intended for IRs when they are not harvesting the RF energy as scheduled, and thus gives rise to a secrecy precoding problem in the MIMO SWIPT systems,
Manuscript received March 24, 2015; revised May 31, 2015; accepted June 10, 2015. Date of publication June 24, 2015; date of current version September 4, 2015. This work was supported by the Natural Science Foundation of China under Grants 61201218 and 61201241, and by the Natural Science Foundation of Jiangsu Province under Grant BK2012056. The associate editor coordinating the review of this paper and approving it for publication was Z. Ding. The authors are with the College of Communications Engineering, PLA University of Science and Technology, Nanjing 210007, China (e-mail: bingfang_
[email protected];
[email protected];
[email protected];
[email protected]). Digital Object Identifier 10.1109/LCOMM.2015.2449315
which is so important and has recently been considered in [3]. However, such an important secrecy precoding problem for MIMO SWIPT system is still not considered under a cognitive scenario, which has potential to be a major paradigm in the future wireless networks. In this letter, we first address the secrecy precoding problem for SWIPT in a cognitive MIMO broadcast channel, where a secondary transmitter (ST) opportunistically sends confidential message to a secondary information receiver (SIR) while guaranteeing the energy harvesting requirement of a secondary energy receiver (SER) on the licensed spectrum band of a PU. In such a cognitive MIMO SWIPT system with confidential information requirement of SIR, there are two conflicting goals for the transmit precoding matrix design of ST: i.e., the power level of the receive signals at SER are expected to be sufficient high for efficient energy harvesting, but also need to be kept sufficient low to prevent information eavesdropping. In order to resolve these two conflict goals, an artificial noise (AN)aided precoding scheme is adopted here. Note that, the ANaided precoding scheme has already been widely used in the conventional MIMO wiretap channels to improve the achievable secrecy rate [4], [5]. Through adding AN to the transmit signals of the BS, the eavesdropper’s channel can be effectively deteriorated. Thus, the difference of the receive signal quality between the legitimate receiver and the eavesdropper can be enlarged, which in turn results in higher achievable secrecy rate of the legitimate receiver. However, the AN employed in the cognitive MIMO SWIPT system also has another function, i.e., energy-carrying. With the AN-aided precoding scheme, we formulate the secrecy precoding problem as a secrecy rate maximization (SRM) problem, which subjects to both an interference power constraint imposed to protect the PU and an energy harvesting requirement to efficiently charge the SER. Since the formulated SRM problem naturally constitutes a difference convex (DC)type programming problem, a successive convex approximation (SCA) method [1], [6] is employed to solve it. With the SCA method, the nonconvex part of the SRM problem can locally linearized to its first-order Taylor expansion. Thus, the nonconvex SRM problem can be iteratively solved through successive convex programming of its convexified version. Finally, an ANaided iterative precoding algorithm with guaranteed convergence is developed. Simulation examples are also provided to demonstrate the proposed algorithm. Notations: Bold uppercase letters denote matrices and bold lowercase letters denote vectors; Cm×n defines the space of all m × n complex matrices; A 0 means that matrix A is positive semidefinite; Hermitian transpose of matrix A is represented as AH ; |A|, AF and Tr(A) means the determinant, the Frobenius norm, and the trace of matrix A, respectively; and log(·) denotes the natural logarithm.
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FANG et al.: AN-AIDED SECRECY PRECODING FOR SWIPT IN COGNITIVE MIMO BROADCAST CHANNELS
Fig. 1. System model of a cognitive MIMO broadcast channel intended for simultaneous wireless information and power transfer (SWIPT).
II. S YSTEM M ODEL AND P ROBLEM F ORMULATION In this letter, we study the secrecy precoding problem for SWIPT in a cognitive MIMO broadcast channel. As shown in Fig. 1, the system model considered here consists of a primary user (PU), a secondary transmitter (ST), a secondary information receiver (SIR), and a secondary energy receiver (SER), all mounted with multiple antennas. We denote by H0 ∈ CNp ×Ns the complex channel matrix from ST to PU, by H1 ∈ CNr ×Ns the complex channel matrix from ST to SIR, and by H2 ∈ CNe ×Ns the complex channel matrix from ST to SER, where Np , Ns , Nr and Ne are the number of antennas employed by PU, ST, SIR and SER, respectively. In order to provide enough degree of freedom for an AN-aided precoding scheme, Ns > Nr is required in this work. It is further assumed that the channel state information (CSIs) of all transmit links are perfectly known by ST. Assuming that a quasi-static frequency-flat fading environment for all transmit links, the vector-valued signals received by SIR and SER can be denoted as y1 = H1 x + n1 , y2 = H2 x + n2 ,
(1)
respectively, where x ∈ CNs is the transmitted signal of ST, n1 ∈ CNd is the additive noise received by SIR, and n2 ∈ CNe is the additive noise received by SER. It is assumed that the elements of n1 and n2 are independent identically distributed (i.i.d.) zero-mean circularly symmetric complex Gaussian (ZMCSCG) random noises with unit variance. With the AN-aided secrecy precoding scheme, the signal transmitted by ST can take the following form x = s + a,
(2)
where s ∈ CNs is the precoded confidential information intended for SIR, and a ∈ CNs is the AN created by ST for energy transfer to SER. Obviously, s and a should be independent from each other and precoded by different precoding matrices. Further assuming that s ∼ CN (0, Q) and a ∼ CN (0, Z), the following secrecy rate of SIR can be achieved [5] Cs (Q, Z) = C1 (Q, Z) − C2 (Q, Z), where C1 (Q, Z) and C2 (Q, Z) are defined as follows H C1 = log I + H1 (Q + Z)HH 1 − log I + H1 ZH1 , H C2 = log I + H2 (Q + Z)HH 2 − log I + H2 ZH2 .
(3)
(4)
According to [2], the energy harvested by SER is given by E = ηTr H2 (Q + Z)HH (5) 2 ,
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where 0 < η ≤ 1 is a constant that accounts for the loss in the energy transduction. Without loss of generality, it is assumed that η = 1 for convenience in the rest of this letter. In this work, we aim to maximize the achievable secrecy rate of SIR while guaranteeing the energy harvested by SER above a given threshold and the interference power inflicts on PU below a given threshold. Mathematically, such a SRM problem can be formulated as (P1) : max Cs (Q, Z) Q,Z s.t. Tr H2 (Q + Z)HH 2 ≥ E, Tr H0 (Q + Z)HH 0 ≤ , Tr(Q + Z) ≤ P, Q 0, Z 0,
(6)
where P is the maximum transmit power available for ST, is the interference power constraint imposed to protect PU, and E denotes the energy harvesting requirement of SER. Note that, the problem (P1) is applicable for the scenario when SER has a strict energy harvesting constraint while SIR only requires an opportunistic transmission. III. I TERATIVE P RECODING A LGORITHM In this section, an iterative precoding algorithm, based on the SCA method, is proposed to solve the problem (P1). A. Algorithm Design The problem (P1) is nonconvex, because the objective function Cs (Q, Z) is nonconcave over (Q, Z). In order to deal with such kind of nonconvexity, we reformulate the objective function as Cs (Q, Z) = φ(Q, Z) − ϕ(Q, Z),
(7)
where the two auxiliary functions φ(Q, Z) and ϕ(Q, Z) are defined as follows H φ = log I + H1 (Q + Z)HH 1 + log I + H2 ZH2 , H ϕ = log I + H2 (Q + Z)HH (8) 2 + log I + H1 ZH1 . Thus, the problem (P1) can be reformulated as (P2) : max φ(Q, Z) − ϕ(Q, Z) Q,Z s.t. Tr H2 (Q + Z)HH 2 ≥ E, Tr H0 (Q + Z)HH 0 ≤ , Tr(Q + Z) ≤ P, Q 0, Z 0.
(9)
Since the auxiliary functions φ(Q, Z) and ϕ(Q, Z) are concave over (Q, Z), the problem (P1) naturally constitutes a DCtype programming problem, which can be iteratively solved by employing a SCA method as detailed follows. According to [7], the first-order differential of ϕ(Q, Z) can be calculated as H H −1 I + H dϕ = Tr HH QH + H ZH H dQ 2 2 2 2 2 2 H H −1 H2 dZ + Tr HH 2 I + H2 QH2 + H2 ZH2 H −1 H1 dZ . (10) + Tr HH 1 I + H1 ZH1
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IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 9, SEPTEMBER 2015
Then, around a given point (Q, Z), the first-order Taylor expansion of ϕ(Q, Z) can be written as + Tr R(Z − ϕ∼ Z) + Tr T(Q − Q) Z) , (11) = ϕ(Q, where T and R are computed as −1 H + H2 T = HH I + H2 QH ZHH H2 , 2
2
H
2
H + H2 ZHH R = H2 I + H2 QH 2 2 −1 H H ZH1 + H1 I + H1 H1 .
−1
H2 (12)
Thus, with the given point (Q, Z), the problem (P2) can be turned into the following convex optimization problem [8]
Tr H2 (Q + Z)HH 2 ≥ E, Tr H0 (Q + Z)HH 0 ≤ ,
s.t.
Tr(Q + Z) ≤ P, Q 0, Z 0,
Therefore, it can be easily concluded that Cs (Q, Z) ≤ Cs (Q∗ , Z∗ ), ∀ (Q, Z)
(17)
always holds. Hence, the convergence of the proposed iterative precoding algorithm is thus guaranteed, because a monotonically non-decreasing sequence that is upper bounded always converges.
(P3) : max φ(Q, Z) − Tr(TQ) − Tr(RZ) Q,Z
always holds during the update process. Therefore, it can be concluded that the sequence Cs (Q(n), Z(n) ) is monotonically non-decreasing. On the other hand, the value of Cs (Q, Z) is always upper bounded by Cs (Q∗ , Z∗ ), where (Q∗ , Z∗ ) is the maxima of the proposed system under a given transmit power constraint. At the same time, because ϕ(Q, Z) is concave over (Q, Z), then, it always holds that + R(Z − ϕ(Q, Z) ≤ ϕ(Q, Z) + Tr T(Q − Q) Z) . (16)
(13)
where the constant terms in the objective has been discarded. Then, relying on solving a series of convexified problems, an iterative precoding algorithm for solving the problem (P1) can be developed, which is formally summarized as Algorithm 1.
B. Algorithm Analysis Because the problem (P1) is nonconvex in general, the convergence behavior of Algorithm 1 has to be analytically established in the following Lemma. Lemma 1: Suppose that the problem (P3) is strict convex, then the iterative precoding algorithm presented as Algorithm 1 is convergent. Proof: Specifically, letting Cs (Q, Z) denote the objective function of the problem (P3), which is the concave surrogate of Cs (Q, Z). Then, consider the following update process
Q(n) , Z(n) = arg max Cs Q, Z|Q(n−1) , Z(n−1) , (14) Q,Z
where n stands for the iteration index. Because the problem (P3) is strict convex, then the following inequlity
Cs Q(n−1), Z(n−1) , (15) Cs Q(n), Z(n) ≥
C. Regularized Approximation According to Lemma 1, it can be concluded that the convergence of Algorithm 1 is established on the strict convexity of the problem (P3). However, such a condition is not always satisfied, especially when the MIMO channels are spatially correlated. Therefore, a proximal point-based regularization approach is pursued here to ensure the convergence, without requiring any special restrictions on the channel ranks [1]. The idea of the proximal point-based regularization consists in penalizing the objective of the problem (P3) using a quadratic regularization term. So that, we can have the problem (P4), shown at the bottom of the page. Note that, in the regularization term, τ > 0 is a small value to force (Q, Z) to stay “close” to (Q, Z). So that, the strict convexity of the problem (P4) is ensured by the convexity of Frobenius norm. The regularized precoding algorithm can also be given as Algorithm 1, just with the problem (P4) replacing the problem (P3), and the regularized algorithm can achieve the same solution under different τ , because the Frobenius norm is equal to zero at the limit point. IV. S IMULATION E XAMPLES In this section, numerical simulations are provided to demonstrate the proposed algorithm(s). During the simulations, the elements of all channel matrices are modelled as ZMCSCG random variables with unit variance, and the number of antennas employed by ST, SIR, SER, and PU are set to be Ns = 6 and Nr = Ne = Np = 2. The convergence behavior of Algorithm 1 is demonstrated in Figs. 2 and 3. From these two figures, it can be seen that Algorithm 1 converges very fast and the achievable secrecy rate obtained by Algorithm 1 increases with the number of iterations in monotone way as shown by Lemma 1. In addition, Fig. 2
2 2 + ||Z − (P4) : max φ(Q, Z) − Tr(TQ) − Tr(RZ) − τ ||Q − Q|| Z|| F F Q,Z
s.t.
H Tr H2 (Q + Z)HH 2 ≥ E, Tr H0 (Q + Z)H0 ≤ , Tr(Q + Z) ≤ P, Q 0, Z 0.
(18)
FANG et al.: AN-AIDED SECRECY PRECODING FOR SWIPT IN COGNITIVE MIMO BROADCAST CHANNELS
Fig. 2. Convergence behavior of Algorithm 1 under different τ (P = 1, = 0.5, E = 6).
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Fig. 4. Achievable secrecy rate performance of Algorithm 1 (τ = 0.1).
It can also be seen from this figure that the existence of PU have a dramatic effect on the achievable secrecy rate, which means that the precoding matrices of ST must be carefully designed under the cognitive scenario. V. C ONCLUSION
Fig. 3. Convergence behavior of Algorithm 1 under different (P = 1, τ = 0.1, E = 6).
is demonstrated under different τ , while Fig. 3 is presented under different . From Fig. 2, it can be seen that Algorithm 1 always converges to a single point under different τ , and the convergence rate is lower when τ gets larger, which can be viewed as the price to pay for the guaranteed convergence. From Fig. 3, it can be seen that Algorithm 1 converges to different points under different , when gets larger, i.e., the interference power constraint is slightly relaxed, the achievable secrecy rate will be accordingly increased. The achievable secrecy rate performance of Algorithm 1 is demonstrated in Fig. 4, where 4 different cases are simultaneously presented for a comparison: case 1, no PU present and no energy harvesting constraint, the secrecy rate is obtained by Algorithm 1; case 2, no PU present, and the energy harvesting constraint is stated as E = 6, the secrecy rate is obtained by the hybrid zero-forcing (ZF) method, where the transmit signal is designed in the null space of the wiretap channel, while AN is designed in the null space of the main channel; case 3, no PU present, and the energy harvesting constraint is also stated as E = 6, the secrecy rate is obtained by Algorithm 1; case 4, the interference power constraint is = 0.5 and the energy harvesting constraint is E = 6, the secrecy rate is obtained by Algorithm 1. As shown by this figure, Algorithm 1 always outperforms the hybrid ZF method under the same conditions.
In this letter, we have studied the secrecy precoding problem for SWIPT in a cognitive MIMO broadcast channel. With an AN-aided precoding scheme, the problem is formulated as a SRM problem. Using the SCA method as a corner stone, an iterative precoding algorithm is developed to solve it. Moreover, a proximal point-based regularization approach is also pursued to ensure the convergence of the proposed algorithm without requiring any special restrictions on the channel ranks. Results show that our algorithm can achieve a satisfactory solution with guaranteed convergence. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. R EFERENCES [1] Y. Zhang, E. DallAnese, and G. B. Giannakis, “Distributed optimal beamformers for cognitive radios robust to channel uncertainties,” IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6495–6508, Dec. 2012. [2] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013. [3] Q. Shi, W. Xu, J. Wu, E. Song, and Y. Wang, “Secure beamforming for MIMO broadcasting with wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 14, no. 5, pp. 2841–2853, May 2015. [4] Q. Li and W.-K. Ma, “Spatially selective artificial-noise aided transmit optimization for MISO multi-eves secrecy rate maximization,” IEEE Trans. Signal Process., vol. 61, no. 10, pp. 2704–2717, May 2013. [5] S.-H. Tsai and H. V. Poor, “Power allocation for artificial-noise secure MIMO precoding systems,” IEEE Trans. Signal Process., vol. 62, no. 13, pp. 3479–3493, Jul. 2014. [6] A. Alvarado, G. Scutari, and J.-S. Pang “A new decomposition method for multiuser DC-programming and its applications,” IEEE Trans. Signal Process., vol. 62, no. 11, pp. 2984–2998, Jun. 2014. [7] X. Zhang, Matrix Analysis and Applications. Beijing, China: Tsinghua Univ. Press, Sep. 2004. [8] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [9] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming. [Online]. Available: http://stanford.edu/boyd/cvx
