s.t. Tr(Q) ⤠Ps, Q ⽠0,. (4) where Ps > 0 is the average transmit power limit of Alice. By introducing a slack variable t, we can rewrite problem (4) as max. Q,t.
Iterative Precoding for MIMO Wiretap Channels Using Successive Convex Approximation Bing Fang, Zuping Qian, Wei Zhong and Wei Shao College of Communications Engineering, PLA University of Science and Technology, Nanjing, China Email: bingfang ch@163.com, qzp811@sina.com, weizhong@ieee.org, swlxssz@126.com n0 and n1 are independent identically distributed (i.i.d.) zeromean circularly symmetric complex Gaussian (ZMCSCG) with zero mean and unit variance. Further assuming x ∼ CN (0, Q), the following secrecy capacity can be achieved [3]
Abstract—In this paper, we study the precoding problem for physical layer security in a general multiple-input multiple-output (MIMO) wiretap channel. Since the resultant secrecy capacity maximization (SCM) problem is nonconvex in general, we solve it by employing a successive convex approximation method, where the nonconvex part of the formulated problem is approximated by its first-order Taylor expansion. Thus, the SCM problem can be iteratively solved through convex programming of its convexified version. Finally, an iterative precoding algorithm with provable convergence is presented. Numerical simulations are also provided to verify the proposed algorithm.
Cs (Q) = C0 (Q) − C1 (Q), where C0 (Q) = log |I + HQHH |, C1 (Q) = log |I + GQGH |.
max Cs (Q) Q
I NTRODUCTION
s.t. Tr(Q) ≤ Ps , Q 0,
In recent years, there are growing interests in exploiting the available spatial dimensions provided by MIMO to enhance the secrecy capabilities of the wireless channels, see [1] and references therein. In this paper, we focus on a general MIMO wiretap channel, which consists of a common transmitter, a legitimate receiver and an eavesdropper, all mounted with multiple antennas. Although the secrecy capacity of such a MIMO wiretap channel has already been known [2], there is still no effective way for the optimal transmit covariance matrix (i.e., the precoding matrix). More recently, a promising way, named as “alternating optimization (AO) algorithm”, was presented in [3]. However, such an AO method is still not so compute efficient as stated. The main contribution of this paper is to propose an iterative precoding algorithm, which is not only compte efficient but also easy to implement. Simulation results further show that our algorithm can converge fast to a “near” optimal solution. II.
max C0 (Q) − t Q,t
s.t. Tr(Q) ≤ Ps , C1 (Q) ≤ t, Q 0, t > 0.
(5)
However, problem (5) is still nonconvex. The nonconvexity of the problem (5) lies in the constraint C1 (Q) ≤ t. Fortunately, such kind of nonconvexity can be dealt with by employing a successive convex approximation method. III.
S UCCESSIVE C ONVEX A PPROXIMATION A LGORITHM
According to [4], the differential of C1 (Q) can be calculated as dC1 (Q) = Tr[GH (I + GQGH )−1 GdQ].
(6)
can be written Thus, its first-order Taylor expansion around Q as
In this paper, we consider a general MIMO wiretap channel, which consists of a transmitter (Alice), a legitimate receiver (Bob), and a eavesdropper (Eve). Assuming that quasistatic frequency-flat fading environment for all communication links, the signal received by Bob and Eve can be given as
+ Tr[D(Q − Q)], C1 (Q) ∼ = C1 (Q)
(7)
H )−1 G. where D is calculated as D = GH (I + GQG obtained in the nth iteration, the followTherefore, with Q ing convex optimization problem [5]
(1)
max C0 (Q) − t Q,t
where H ∈ CNd ×Ns represents the channel from Alice intended to Bob, G ∈ CNe ×Ns is the channel from Alice to Eve, x ∈ CNs is the confidential signal transmitted by Alice to Bob. We denote by Ns , Nd and Ne the number of antennas employed by Alice, Bob and Eve, respectively, and assume the elements of
c 978-1-4799-8897-6/15/$31.00 2015 IEEE
(4)
where Ps > 0 is the average transmit power limit of Alice. By introducing a slack variable t, we can rewrite problem (4) as
S YSTEM M ODEL AND P ROBLEM F ORMULATION
y0 = Hx + n0 , y1 = Gx + n1 ,
(3)
Therefore, the secrecy capacity maximization (SCM) problem for such a wiretap channel can be formulated as
Keywords—MIMO wiretap channel, physical layer security, successive convex approximation, convex optimization.
I.
(2)
− Tr(DQ) + Tr(DQ) ≤ t, s.t. C1 (Q) Tr(Q) ≤ Ps , Q 0, t > 0,
(8)
can be solved in the (n + 1)th iteration. Thus, the yielded iterative precoding algorithm for SCM in MIMO wiretap
67
Algorithm 1: Iterative Precoding Algorithm for SCM. and i = 0. 1: initially set H, G, Ps , Q,
Secrecy capacity (nps/channel use)
2: 3: 4: 5: 6: 7:
5.94
repeat update i = i + 1, and compute D with Q. compute Q by solving the problem (8) with a CVX solver [6]. with Q. compute Cs (Q), and update Q until the termination criteria is satisfied. return Q and Cs .
channel can be summarized as Algorithm 1. In addition, the convergence property of Algorithm 1 can be analyzed in the following Lemma. Lemma 1: The iterative precoding algorithm presented as Algorithm 1 is convergent.
This work was supported by the Natural Science Foundation of China under Grant No. 61201218 and No. 61201241.
68
2
3
4
5
6
7
8
no Eve
Secrecy capacity (nps/channel use)
8
ZF Alg. 1
7 6 5 4 3 2 1
−5
0 Power (dB)
5
10
Fig. 2. The secrecy capacity versus the transmit power (with Ns = 6, and Nd = Ne = 3). 0
10
CDF of secrecy capacity
−1
10
no Eve ZF Alg. 1
−2
10
2
3
4
5
6
7
8
9
Secrecy capacity (nps/channel use)
Fig. 3. The CDFs of secrecy capacity with Ps = 7dB averaged over 1000 channel realizations (with Ns = 6, and Nd = Ne = 3).
R EFERENCES [1]
[3]
[4] [5]
ACKNOWLEDGMENT
1
0 −10
C ONCLUSION
In this paper, we have provided a novel iterative precoding algorithm based on successive convex approximation for secrecy capacity maximization in a general MIMO wiretap channel. Results show that our algorithm converges very fast to a “near” optimal solution.
5.89
9
[2]
V.
5.9
Fig. 1. The secrecy capacity versus the number of iterations (with Ns = 6, Nd = Ne = 3, and Ps = 10dB).
The convergence behaviour of the proposed algorithm is demonstrated in Fig. 1, which is obtained with Ps = 10dB. From this figure, it can be seen that the proposed algorithm converges in monotone increasing behaviour as shown by Lemma 1. Moreover, it has been shown in many simulations that the proposed algorithm always converge to a single point with different initial points. The secrecy capacity performance of Algorithm 1 is shown in Fig. 2 and Fig. 3, which are presented with a comparison to the zero-forcing (ZF) method and the no Eve present case. The secrecy capacity versus the transmit power of Alice is presented in Fig. 2, and the empirical cumulative distribution functions (CDFs) of the achieved secrecy capacity is presented in Fig. 3, which is obtained with Ps = 7dB over 1000 channel realizations. As the two figures have shown, the proposed algorithm always outperform the ZF method during many simulations. However, the existence of Eve has a dramatic effect on the achievable secrecy capacity of the whole system.
5.91
Number of iterations
P ERFORMANCE E VALUATION
In this section, numerical simulations are proposed to verify the performance of the proposed algorithm. During the simulations, all the channel matrices are modelled as ZMCSCG random matrices with unit variance, and the number of antennas employed by Alice, Bob, and Eve are set to be Ns = 6 and Nd = Ne = 3.
5.92
5.88
Proof: With Algorithm 1, the value of Cs (Q) is increased in each iteration, and also the value of Cs (Q) is upper bounded with a given transmit power constraint. Because an increasing sequence that is upper bounded always converges, the convergence of the proposed iterative precoding algorithm is thus guaranteed. Moreover, the limit point of the iterations is a stationary point of the problem (4). IV.
5.93
[6]
A. Mukherjee, S. A. A. Fakoorian, J. Huang, and A. L. Swindlehurst, “Principles of physical layer security in multiuser wireless networks: a survey,” Communications Surveys & Tutorials, IEEE, vol. 16, no. 3, pp. 1550-1573, Third Quarter 2014 F. Oggier, B.Hassibi, “The secrecy capacity of the MIMO wiretap channel,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 4961-4972. Aug. 2011. Q. Li, M. Hong, H.-T. Wai, Y.-F. Liu, W.-K. Ma, and Z.-Q. Luo, ”Transmit solutions for MIMO wiretap channels using alternating optimization,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 2704-2717, Sep. 2013 X. Zhang, Matrix Analysis And Applications, Beijing: Tsinghua Univ. Press, Sep. 2004. S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming [Online]. Available: http://stanford.edu/ boyd/cvx
2015 IEEE 4th Asia-Pacific Conference on Antennas and Propagation (APCAP)
