Precoding for secrecy rate maximisation in cognitive MIMO wiretap channels
H −1 dC2 (Q) = Tr[HH 2 (I + H2 QH2 ) H2 dQ]
Bing Fang✉, Zuping Qian, Wei Zhong and Wei Shao The problem of physical layer security in a cognitive MIMO wiretap channel is studied. The problem is formulated as a secrecy rate maximisation (SRM) problem, and solved by a successive convex approximation (SCA) method. With the SCA method, the non-convex part of the SRM problem is approximated by its first-order Taylor expansion. Then, relying on solving a series of convexified optimisation problems, an iterative precoding algorithm is developed.
Introduction: Cognitive radio (CR) is a novel approach for enhancing utilisation of precious spectrum resources. With MIMO becoming a dominating technology for next-generation cellular networks, the researches with regard to cognitive MIMO radio have attracted increasing interest. On the other hand, physical layer security technology is becoming a promising complement to the traditional cryptographic technology and gaining considerable research interest [1]. Generally, there are two kinds of physical layer secrecy problems considered in the literature on CR networks: one is to enhance secrecy between the PUs and the other is to achieve secrecy among secondary users [2, 3]. In this Letter, attention is focused on the later. In this Letter, we study the precoder design problem for a cognitive MIMO wiretap channel. The problem is formulated as a secrecy rate maximisation (SRM) problem, which subjects to both an interference power constraint and a transmit power constraint. Since the SRM problem naturally constitutes a difference convex (DC)-type programming problem, we solve it by employing a successive convex approximation (SCA) method [4]. System model and problem formulation: In this Letter, we study the precoder design problem for a cognitive MIMO wiretap channel. The system considered consists of a PU, a secondary transmitter (Alice), a secondary receiver (Bob), and an eavesdropper (Eve), all mounted with multiple antennas. We denote by H0 [ CNp ×Ns the channel matrix from Alice to the PU, H1 [ CNd ×Ns the channel matrix from Alice to Bob, and H2 [ CNe ×Ns the channel matrix from Alice to Eve, where Np, Ns, Nd, and Ne are the number of antennas employed by the PU, Alice, Bob, and Eve, respectively. Assuming a quasi-static frequency-flat fading environment for all communication links, the signals received by Bob and Eve can be given as y1 = H1 x + n1 ,
(1)
y2 = H2 x + n2
where x ∈ ℂNs is the transmit signal of Alice, n1 ∈ ℂNd is the additive receive noise of Bob, and n2 ∈ ℂNe is the additive noise received by Eve. It is assumed that the elements of n1 and n2 are independent identically distributed zero-mean circularly symmetric complex Gaussian (ZMCSCG) random noises with unit variance. Further assuming that x CN (0, Q), the following secrecy rate of the secondary system can be achieved: Cs (Q) = C1 (Q) − C2 (Q)
C2 (Q) = log |I + H2 QHH 2|
(3)
Then, the SRM problem for such a cognitive MIMO wiretap system can be formulated as (P1): max Cs (Q) Q
s.t.
Tr(H0 QHH 0)
≤ G,
(4)
Tr(Q) ≤ P, Q X 0 where Γ > 0 is the maximum interference power constraint imposed by the PU and P > 0 is the transmit power constraint of Alice. Since both C1(Q) and C2(Q) are concave over Q, the problem (P1) naturally constitutes a DC-type programming problem, which can be iteratively solved by employing a SCA method.
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(5)
Then, its first-order Taylor expansion around a given point Q can be written as Q) + Tr[D(Q − Q)] C2 (Q) C2 (
(6)
where D is calculated as H −1 D = HH 2 (I + H2 QH 2 ) H2
(7)
Then, with a given point Q, the problem (P1) is turned into the following convex optimisation problem [6] (P2): max C1 (Q) − Tr(DQ) Q
s.t.
(8)
Tr(H0 QHH 0 ) ≤ G, Tr(Q) ≤ P, Q X 0
where the constant terms in the objective has been discarded. Then, an iterative precoding algorithm for solving the problem (P1) can be developed, which is formally summarised as Algorithm 1. Since the problem (P1) is non-convex, convergence of Algorithm 1 with local convex approximation has to be analytically established in the following Lemma. Lemma 1: Suppose that the problem (P2) is strictly convex, then, the iterative precoding algorithm presented as Algorithm 1 is convergent. Proof: Specifically, letting Cs (Q) denote the objective function of the problem (P2), which is the concave surrogate of Cs(Q). Then, consider Cs (Q|Q(n−1) ) Q(n) = arg max
(9)
Q
where n stands for the iteration index. Therefore, it holds that Cs (Q(n) ) ≥ Cs (Q(n−1) )
(10)
which comes from the convexity of the problem (P2). Then, it can be concluded that the sequence Cs (Q(n) ) is monotonically non-decreasing. On the other hand, the value of Cs(Q) is always upper bounded by Cs(Q*), where Q* is the maxima of the proposed system under a given transmit power constraint. At the same time, because C2(Q) is concave over Q, it always holds that Q) + Tr[D(Q − Q)], C2 (Q) ≤ C2 (
∀Q
(11)
Then, it can be easily concluded that Cs (Q) ≤ Cs (Q∗ ),
∀Q
(12)
always holds. Hence, the convergence of the proposed precoding algorithm is guaranteed, because a monotonically non-decreasing sequence that is upper bounded always converges. □
(2)
where C1 (Q) = log |I + H1 QHH 1 |,
Iterative precoding algorithm: According to [5], the first-order differential of C2(Q) can be calculated as
From Lemma 1, it can be concluded that the convergence of Algorithm 1 is established on the strict convexity of the problem (P2). However, such a condition is not always satisfied, especially when the MIMO channels are spatially correlated. Therefore, a proximal point-based regularisation approach is further pursued here to ensure the convergence, without requiring restrictions on the antenna configurations and the channel ranks. The idea consists in penalising the objective of the problem (P2) using a quadratic regularisation term, and thus we have Q||2F (P3): max C1 (Q) − Tr(DQ) − t||Q − Q
s.t.
Tr(H0 QHH 0 ) ≤ G,
(13)
Tr(Q) ≤ P, Q X 0 where τ > 0 is a small value to force Q to stay ‘close’ to Q. The iterative precoding scheme with the proximal point-based regularisation can also be given as Algorithm 1, with the problem (P3) replacing the problem (P2). However, the price to pay is that a possibly slower
Doc: {EL}Articles/Pagination/EL20151486.3d Wireless communications
Algorithm 1: Iterative precoding algorithm for solving problem (P1) 1: initially set H, H2, P, Γ, τ, Q, and i = 0. 2: repeat 3: update i = i + 1, and compute D with Q. 4: compute Q by solving the problem (P2) with a CVX solver [7]. Q with Q. 5: compute Cs(Q), and update 6: until the termination criteria is satisfied. 7: return Q and Cs. Performance evaluation: In this Section, numerical simulations are proposed to verify the proposed algorithm. 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 Alice, Bob, Eve, and the PU are set to be Ns = 6 and Nd = Ne = Np = 2.
Algorithm 1 with an interference power constraint Γ = 0.5. As this Figure show, the proposed algorithm always outperforms the ZF method and the existence of Eve and PU have a dramatic effect on the achievable secrecy rate of the whole cognitive MIMO wiretap system. 6.5 no PU, no Eve no PU, ZF method no PU, algorithm 1 algorithm 1, G = 0.5
6.0 achievable date rate, nps/Hz
convergence rate is common to the proximal point-based approximation algorithm.
5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5
1.620
achievable secrecy rate, nps/Hz
1 1.615
2
3 4 5 6 transmit power constraint of Alice, dB
7
8
Fig. 3 Achievable secrecy rate performance of Algorithm 1 (τ = 0.2)
Conclusion: In this Letter, we have studied the SRM problem in a cognitive MIMO wiretap channel. On the basis of the SCA method, an iterative precoding algorithm is developed to solve it. Results show that our algorithm can achieve a near-optimal solution with guaranteed convergence.
1.610
1.605 t = 0.1 t = 0.2 t = 0.3
1.600
Acknowledgment: This research is supported by the National Science Foundation of China under grant nos. 61201241 and 61201218.
1.595 1
2
3
4 5 iterations
6
7
8
Fig. 1 Convergence behaviour of Algorithm 1 under different τ (Γ = 0.5)
The convergence behaviour of Algorithm 1 is demonstrated in Figs. 1 and 2. From these two Figures, it can be seen that Algorithm 1 converges very fast, in no more than five times of iteration, and the achievable secrecy rate obtained by Algorithm 1 increases with the iteration number in a monotone way as shown by Lemma 1. From Fig. 1, it can be seen that Algorithm 1 always converges to a single point from different τ, and the convergence speed is lower when τ gets larger. From Fig. 2, it can be seen that Algorithm 1 converges to different points with different Γ, when Γ gets larger, i.e. the interference power constraint is slightly relaxed, the achievable secrecy rate will be increased.
achievable secrecy rate, nps/Hz
1.625
1.620
1.615
1.610
G = 0.49 G = 0.50 G = 0.51
1.605
1.600 1
2
3
4
5
6
7
8
iterations
Fig. 2 Convergence behaviour of Algorithm 1 under different Γ (τ = 0.2)
The achievable secrecy rate performance of Algorithm 1 is shown in Fig. 3, where four different cases are simultaneously presented for a comparison: case 1, no Eve and no PU present; case 2, no PU present obtained by the classic zero-forcing (ZF) method; case 3, no PU present by Algorithm 1; and case 4, both PU and Eve present by
© The Institution of Engineering and Technology 2015 Submitted: 2 June 2015 doi: 10.1049/el.2015.1486 One or more of the Figures in this Letter are available in colour online. Bing Fang, Zuping Qian, Wei Zhong and Wei Shao (College of Communications Engineering, PLAUST, Nanjing 210007, People’s Republic of China) ✉ E-mail:
[email protected] References 1 Mukherjee, A., Fakoorian, S.A.A., Huang, J., et al.: ‘Principles of physical layer security in multiuser wireless networks: a survey’, IEEE Commun. Surv. Tutor., 2014, 16, (3), pp. 1550–1573, Third Quarter 2014 2 Pei, Y., Liang, Y.-C., Zhang, L., et al.: ‘Secure communication over MISO cognitive radio channels’, IEEE Trans. Wirel. Commun., 2010, 9, (4), pp. 1494–1502 3 Elkashlan, M., Wang, L., Duong, T.Q., et al.: ‘On the security of cognitive radio networks’, IEEE Trans. Veh. Technol., 2015, 64, (8), pp. 3790–3795 4 Zhang, Y., DallAnese, E., and Giannakis, G.B.: ‘Distributed optimal beamformers for cognitive radios robust to channel uncertainties’, IEEE Trans. Signal Process., 2012, 60, (12), pp. 6495–6508 5 Zhang, X.: ‘Matrix analysis and applications’ (Tsinghua University Press, Beijing, 2004) 6 Boyd, S., and Vandenberghe, L.: ‘Convex optimization’ (Cambridge University Press, Cambridge, UK, 2004) 7 Grant, M., and Boyd, S.: ‘CVX: MATLAB Software for Disciplined Convex Programming [Online]. Available at http://www.stanford.edu/ boyd/cvx
