Semi-Adaptive Transmission in Slow Fading MISOME Wiretap Channels Zongmian Li, Pengcheng Mu, Zongze Li, Xiaoyan Hu and Hui-Ming Wang MOE Key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, China 710049 Email:
[email protected],
[email protected],
[email protected]
Abstract—The design of artificial-noise-aided secure transmission over multiple-input, single-output, and multi-antenna eavesdropper (MISOME) slow fading channels is addressed in this paper. We consider the scenario where the messages sent by an N -antenna transmitter are overheard by a K-antenna eavesdropper. The instantaneous channel state information of the eavesdropper’s channel is assumed to be unknown to the transmitter, and we use the secrecy outage probability as the security metric. Differing from the well-known adaptive and non-adaptive schemes, we consider the newly proposed semiadaptive (S-ADP) scheme, in which the communication rate is fixed and the secrecy rate is adaptively adjusted. By fixing the communication rate, the codebook can be fixed, thus the encoder and decoder designs are both simplified. The communication rate is assumed to be predefined as in many practical scenarios, and we maximize the secrecy rate under reliability and secrecy constraints. Simulation results show that the S-ADP scheme can achieve satisfactory performance.
I. I NTRODUCTION Secret-key-based cryptographic techniques are traditionally used to guarantee the secrecy of wireless data transmission. However, the security provided by encryption algorithms depends heavily on the eavesdropper’s computational capability, and the management of secrecy key is also risky. In this context, physical-layer security, founded by the seminal works [1]–[3] in the 1970’s, emerged as a new approach which provides secrecy from the angle of information theory. Motivated by these pioneering contributions, various techniques were studied considering different systems models and assumptions. In particular, a lot of research efforts have been devoted to secure multi-antenna transmissions. A basic approach to enhance secrecy using multiple antennas is transmit beamforming, by which the transmit signal is concentrated toward the legitimate receiver, and the signal leaked to the eavesdropper is reduced accordingly [4]–[7]. Another approach is transmitting the information-bearing signal in conjunction with artificial noise (AN) to interfere with the eavesdropper [8]. The AN is proved to be an effective measure to enhance secrecy and has received much attention in recent years [9]–[15]. Since fading is an important phenomenon in wireless communications, the design of secure transmissions in fading channels has attracted much attention [16]–[23]. Specifically, Zhang et al. investigated the design of AN-aided multiantenna transmission in slow fading multiple-input, singleoutput, and single-antenna eavesdropper (MISOSE) wiretap channels in [17]. They studied two transmission schemes: 1) the non-adaptive (N-ADP) scheme, in which the secrecy
rate, the communication rate, and the power allocation parameter remain fixed for all transmissions; 2) the adaptive (ADP) scheme, in which all these parameters are adaptively adjusted according to the legitimate channel’s instantaneous channel state information (CSI). The N-ADP scheme is easy to apply, but its performance is quite limited. In contrast, the ADP scheme can achieve good performances but is more complicated to implement, since it requires adaptive encoding and decoding with multiple codebooks to meet the need of adaptive communication rate. In fact, the codebook can be designed for several discrete communication rates in practice. In this case, adaptively adjusting the communication rate is impossible and fixed-rate transmission is preferable. To this end, a new secure transmission scheme with fixed communication rate and adaptive secrecy rate for single-antenna wiretap channels has been proposed in our recent work [24] and is named semi-adaptive (S-ADP) scheme. It is observed that the S-ADP scheme performs well when the secrecy requirement is strict. Since high secrecy is usually required in practice and the S-ADP scheme simplifies the encoder and decoder design, the S-ADP transmission scheme can achieve a good trade-off between complexity and performance in the single-antenna case. In this paper, we design the S-ADP scheme for multipleinput, single-output, and multi-antenna eavesdropper (MISOME) wiretap channels. The MISOME wiretap channel corresponds to the practical scenario of downlink communication in cellular systems where a multi-antenna base station communicates with a single-antenna mobile user in the presence of a multi-antenna eavesdropper. Our contribution consists in maximizing the secrecy rate under constraints on reliable transmission and secrecy outage probability (SOP) and deriving the optimal secrecy rate and power allocation ratio. Numerical results are presented to verify the satisfactory performance of the S-ADP scheme. The rest of this paper is organized as follows. In Section II, we present the system model and the principles of S-ADP transmission. Then we maximize the secrecy rate subject to reliability and secrecy constraints in Section III. Although the general solution to the secrecy rate maximization problem cannot be expressed explicitly, it is shown in Section IV that closed-form expressions for the optimal solution actually exist in some special cases. In Section V, we present numerical results to show the the S-ADP scheme’s performance. Finally, we conclude our work in Section VI.
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Notations: Boldfaced lowercase (resp. uppercase) letters represent vectors (resp. matrices); the symbols (·)H , {·}+ and · denote respectively conjugate transpose, max{0, ·} and the Euclidean norm; CN denotes the set of N × 1 complex vectors. IN represents an N × N identity matrix. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider a wireless communication system in which an N -antenna transmitter (Alice) transmits confidential messages to a single-antenna legitimate receiver (Bob), in the presence of a K-antenna passive eavesdropper (Eve). It is assumed that 1 ≤ K ≤ N − 1, otherwise Eve can always remove the AN signal [9], [22]. The channels from Alice to Bob and to Eve are assumed to be independent and are denoted by an N × 1 vector h and an N × K matrix G, respectively. We consider delay-intolerant communications, thereby model all channels as quasi-static Rayleigh fading channels. Consequently, the entries of h and G are i.i.d. complex Gaussian random variables with mean zero and variance ΓB and ΓE , respectively. We assume that Bob can estimate h accurately and uses a perfect link to feed back h to Alice. However, Alice does not know the instantaneous CSI of the eavesdropper’s channel since Eve is a passive eavesdropper. When Alice transmits a signal vector x ∈ CN , the received signals at Bob and Eve are given by, respectively, yB = hH x + nB , H
yE = G x + nE ,
(1) (2)
where nB is the additive white Gaussian noise (AWGN) at Bob with unit variance, and nE is the AWGN vector with covariance matrix σ 2 IK . A. Artificial-Noise-Aided Beamforming We adopt the conventional AN-aided beamforming structure for passive eavesdroppers in [8]. Namely, Alice chooses h as = h as the beamforming direction. Based on as , Alice generates an orthonormal basis of CN as A = [as , Ac ] to describe the AN-aided transmit signal. The transmitted signal vector x is thus expressed as (3) x = P φas s + P (1 − φ)Ac z, where s is the information-bearing signal and z is the AN vector. s and z are statistically independent, Gaussiandistributed, with mean zero, variance one and covariance matrix N 1−1 IN −1 , respectively. P and φ denote respectively the total transmit power and the fraction of power allocated into the space of as . With (1) and (3), the received signal at Bob becomes yB = P φ h s + nB , (4) and the signal-to-noise ratio at Bob is ρB = P φγ, 2
(5)
where γ = h represents the effective channel gain. γ is a Gamma-distributed random variable with shape N and scale ΓB , i.e., γ ∼ Gamma(N, ΓB ).
With (2) and (3), the received signal at Eve becomes yE = P φGH as s + P (1 − φ)GH Ac z + nE .
(6)
Since the noise power at Eve is unknown to Alice, a robust approach, as done in [17], is to design for the worst case in which Eve’s noise power equals zero (i.e., σ 2 = 0). Therefore, the signal-to-interference ratio at Eve is given by [22], [25] (N − 1)φ H H H −1 as G GAc AH Gas . (7) ρE = c G 1−φ Since the entries of G are i.i.d. zero-mean complex Gaussian variables with variance ΓE and A is unitary, the entries of GH A are also i.i.d. zero-mean complex Gaussian variables with variance ΓE . Applying the system model in [25] and using [25, eq.(18)], we obtain the complementary cumulative distribution function (CCDF) of ρE as −1 −1 k k K−1 1 + k=1 Nk−1 φN −1 ρ QE (ρ; φ) = , (8) N −1 φ−1 −1 1 + N −1 ρ where ρ ≥ 0. We should note that, with the assumption of zero noise power at Eve, P and ΓE do not appear in the expression of QE (ρ; φ). Also note that QE (ρ; φ) may change with h because φ can be adjusted according to h. B. Secrecy Outage Probability According to [16], perfect secrecy is achieved if the following inequality holds: +
Rs ≤ {CB − CE } ,
(9)
where Rs , CB = log2 (1 + ρB ) and CE = log2 (1 + ρE ) denote respectively secrecy rate, capacities of the legitimate channel and of the eavesdropper’s channel. The case in which (9) is not satisfied is described as the secrecy outage. Since Alice does not know the exact value of G in the passive eavesdropping scenario, perfect secrecy cannot always be achieved. In this case, CE becomes a random variable and the concept of SOP is used to measure the security of transmission. By adopting the SOP formulation used in [17] and [24], we can express the SOP as pso (RB , Rs ; h) = Pr { Rs > RB − CE | h} ,
(10)
where RB is the communication rate with Rs ≤ RB ≤ CB . C. Semi-Adaptive Transmission In the S-ADP scheme, the communication rate RB takes a predefined positive value and remains invariant for all transmissions, while the secrecy rate Rs and the power allocation parameter φ are adaptively adjusted according to h. Therefore, we use Rs (h), φ(h) and ρB (h) instead of Rs , φ and ρB to emphasize their dependance on h. The primary design requirement is to ensure reliable transmission, i.e., RB ≤ log2 (1 + ρB (h)). The system’s secrecy requirement is quantified as a maximum tolerable SOP, i.e., pso (RB , Rs (h); h) ≤ ε with 0 < ε ≤ 1. Moreover, the secrecy rate Rs (h) should be
nonnegative and cannot exceed the communication rate, i.e., 0 ≤ Rs (h) ≤ RB . As in [17] and [24], we implement the on-off transmission protocol to prevent undesirable transmissions that incur reliability or secrecy outages. Namely, Alice transmits only when the effective channel gain γ exceeds an on-off threshold μ. The threshold μ is nonnegative and needs to be optimized. Our objective is to maximize the secrecy rate for any given positive communication rates, subject to the constraints on reliability, security, and nonnegative secrecy rate. Finally, the optimal S-ADP design is the solution to the following secrecy rate maximization problem: max
Rs (h),φ(h),μ
s.t.
Rs (h),
(11a)
RB ≤ log2 (1 + ρB (h)) , pso (RB , Rs (h); h) ≤ ε, 0 ≤ Rs (h) ≤ RB ,
(11b) (11c) (11d)
0 < φ(h) ≤ 1, μ ≥ 0.
(11e) (11f)
This problem will be solved in the next section. III. S ECRECY R ATE M AXIMIZATION WITH S EMI -A DAPTIVE S CHEME The objective of this section is to solve the secrecy rate maximization problem (11). The problem is solved in the following order. First, we derive lower bounds for φ(h) and γ from the reliability constraint (11b). Then, the inequality constraint (11c) is transformed into an equality constraint, from which the optimal Rs (h) and φ(h) are derived. Finally, we determine the optimal μ to ensure the existence of Rs (h) and φ(h) while achieving the maximum secrecy rate. In the subsequent parts, we omit the parameter h after the optimization variables for brevity. However, it should be noted that Rs and φ are dependent on h. A. Reliability Constraint Since ρB (h) = P φγ, the reliability constraint (11b) can be expressed as RB ≤ log2 (1 + P φγ), which is equivalent to φ≥
2RB − 1 . Pγ
(12)
Namely, to ensure reliable transmission, φ should be greater than the lower bound in (12). Since the range of φ is also restricted by the constraint (11e), the above lower bound RB should not exceed 1, i.e. 2 P γ−1 ≤ 1, or equivalently, γ≥
2RB − 1 . P
(13)
Eq. (13) implies that the optimal on-off threshold μ must lie RB in the interval [ 2 P−1 , +∞), since transmission occurs once γ is greater than μ .
B. Secrecy Outage Probability Constraint The expression of SOP can be obtained based on the definition of SOP in (10) and the CCDF of ρE given in (8). The calculation is detailed as follows: pso (RB , Rs ; h) =Pr { Rs > RB − log2 (1 + ρE )| h}
=Pr ρE > 2RB −Rs − 1 h =QE (2RB −Rs − 1; φ) −1 −1 k R −R k K−1 1 + k=1 Nk−1 φN −1 2 B s −1 = . N −1 −1 −1 1 + φN −1 (2RB −Rs − 1)
(14)
From (14) we can deduce that pso is non-decreasing with ∂pso so respect to Rs and φ, i.e., ∂p ∂φ ≥ 0 and ∂Rs ≥ 0. Using the differentiation rule for implicit functions, we have ∂pso
∂Rs ∂φ = − ∂pso ≤ 0, ∂φ ∂R
(15)
s
which means that Rs is non-increasing with respect to φ for any value of pso . Therefore, to maximize the secrecy rate, we should choose φ to be as small as possible, which implies that the lower bound in (12) is optimal for φ, i.e., φ =
2RB − 1 . Pγ
(16)
Substituting φ into the expression of SOP given in (14), we can rewrite the SOP constraint (11c) as F (Rs , γ) ≤ ε,
(17)
where the function F (Rs , γ) is defined as F (Rs , γ) K−1 N −1
(P γ−2RB +1)(2RB −Rs −1) 1 + k=1 k (N −1)(2RB −1) = N −1 (P γ−2RB +1)(2RB −Rs −1) 1+ (N −1)(2RB −1)
k .
(18)
From (18) we know that F (Rs , γ) is monotonically increasing with respect to Rs , and F (Rs = RB , γ) = 1. Therefore, to maximize the secrecy rate, the equal sign in the inequality (17) should always hold. Namely, the optimal secrecy rate Rs is the unique solution to the following equation: F (Rs , γ) = ε.
(19)
C. Secrecy Rate Constraint Please note that the solution to equation (19) does not always exist due to the secrecy rate constraint (11d). The reason is detailed as follows. With the secrecy rate constraint (11d), the domain of the newly defined function F (Rs , γ) is restricted to Rs ∈ [0, RB ]. On the one hand, we can verify that F (0, γ) > 0 and F (Rs = RB , γ) = 1; on the other hand, the function F (Rs , γ) increases with respect to Rs . Therefore, the value range of F (Rs , γ) is limited to [F (0, γ) , 1]. As a
consequence, in the case where F (0, γ) is greater than ε, the equation (19) is no longer solvable. The above discussion implies that reliability and secrecy cannot be simultaneously achieved for some values of γ, no matter how Rs and φ are adjusted. Therefore, it is important to find the set of γ corresponding to the existence of the solution to (19). To determine this set, we examine the left endpoint of the interval [F (0, γ) , 1], i.e., k K−1 RB +1 1 + k=1 Nk−1 P γ−2 N −1 . (20) F (0, γ) = N −1 R B +1 1 + P γ−2 N −1
channel’s quality is, the less power is needed to guarantee reliability, and the more power is allocated to the AN. In fact, the power allocated to the information-bearing signal is the minimum amount that is needed to support the communication rate RB , and the rest of the power is all allocated to the AN to degrade the eavesdropper’s reception.
It can be proved that the value of F (0, γ) as γ in decreases 2RB −1 = 1 and creases. In addition, we can verify that F 0, P
A. Single-Antenna Eavesdropper
2RB −1
limγ→+∞ F (0, γ) = 0 (please note that γ ≥ P should always be satisfied to ensure reliability, cf. eq. (13)). Consequently, there is one, and only one γ0 satisfying F (0, γ0 ) = ε: for all γ > γ0 , we have F (0, γ) < ε; for all γ such that 2RB −1 ≤ γ < γ0 , we have F (0, γ) > ε. To maximize the P secrecy rate, the on-off threshold should be chosen as small as possible on the premise of reliable and secure transmission. Therefore, μ = γ0 is the optimal on-off threshold. In summary, for any given communication rate, the optimal S-ADP scheme can be implemented in the following way: Alice uses the on-off transmission protocol to determine whether transmit or not. The optimal on-off threshold μ lies RB in the interval [ 2 P−1 , +∞), and is the unique solution to the equation −2RB +1 k K−1 1 + k=1 Nk−1 P μ N −1 = ε. (21) N −1 R −2 B +1 1 + Pμ N −1 Owing to the monotonicity of the left side of eq. (21), μ can 2 be determined by numerical methods. When h < μ , Alice 2 turns off the transmission. When h ≥ μ , she resumes the transmission, and adjusts the power allocation ratio to its optimal value 2RB − 1 φ (h) = (22) 2 . P h In this case, the maximum secrecy rate Rs (h) lies in [0, RB ], and is the unique solution to the following equation: k K−1 N −1 (P h2 −2RB +1) 2RB −Rs (h) −1 1 + k=1 k (N −1)(2RB −1) = ε, N −1 (P h2 −2RB +1)(2RB −Rs (h) −1) 1+ (N −1)(2RB −1) (23) Since the left side of eq. (23) is also monotone with respect to Rs (h), Rs (h) can be easily determined numerically. Finally, we make some comments on the optimal power allocation ratio φ (h). According to (22), φ (h) decreases with 2 respect to h , which means that the better the legitimate
IV. S PECIAL C ASES We should highlight that, although Rs (h) and μ do not have explicit expressions in general, their value can be readily obtained using numerical approaches. Moreover, as shown in the following, the closed-form expressions for Rs (h) and μ do exist in some special cases. Consider the case where Eve is equipped with a single antenna, i.e. K = 1. We can thus simplify eq. (21) as 1−N P μ − 2RB + 1 1+ = ε, (24) N −1 from which μ can be solved as 1 (N − 1) ε 1−N − 1 + 2RB − 1 μ = . (25) P Similarly, by substituting K = 1 into (23) and transforming the resultant equation, we can derived the closed-form expression of Rs (h) as Rs (h) = RB − 1 ⎛ ⎞ (N − 1) ε 1−N − 1 2RB − 1 ⎠. log2 ⎝1 + 2 P h − 2RB + 1
(26)
B. Eavesdropper Equipped With N − 1 Antennas The most pessimistic estimate is to suppose that Eve has K = N − 1 antennas. By using K = N − 1 and applying the binomial theorem, eq. (21) can be rewritten as N −1 P μ − 2RB + 1 1− = ε, (27) N − 1 + P μ − 2RB + 1 which is equivalent to 1 N − 1 + 2RB − 1 (1 − ε) 1−N − 1 μ = . 1 P (1 − ε) 1−N − 1
(28)
The closed-form expression of the maximum secrecy rate in this case can be obtained accordingly: Rs (h) = RB − ⎛
⎞ (N − 1) 2RB − 1 ⎠ . log2 ⎝1 + 1 2 (1 − ε) 1−N − 1 P h − 2RB + 1 (29)
From (26) and (29) we clearly observe that Rs (h) increases 2 and approaches RB as h increases for both the cases K = 1 and K = N − 1. We should note that the results in (26)
10
14
Secrecy throughput (bits/s/Hz)
9
R =8 bits/s/Hz B
7 6
R =6 bits/s/Hz
5
B
*
RB or Rs (bits/s/Hz)
8
4
R =4 bits/s/Hz
3
B
2
R
1
R*s
0
B
0
2
4
6
γ
8
10
12
14
Secrecy throughput (bits/s/Hz)
7 6 5 4
K=N−1
2
K=1
6
4
K=N−1
0 −10
0
10
20
30
40
Fig. 3. Performances of the three schemes versus total transmit power, with N = 4, K = 1, 3, ε = 0.1 and ΓB = 0dB.
μ
1 0
8
increase the communication rate RB . Secondly, we evaluate the performance of the S-ADP scheme under different SOP constraints. The ADP and the N-ADP schemes for MISOSE wiretap channels proposed in [17] are also simulated for comparison. We chose the secrecy throughput (also referred to as the average secrecy rate) as the performance metric. The secrecy throughput is computed by averaging the secrecy rate over all realizations of the legitimate channel. For the S-ADP scheme, the secrecy throughput can be calculated by the following equation: +∞ ¯s = R Rs (r)fγ (r)dr, (30)
K=1
3
10
P (dB)
9 8
S−ADP ADP NADP
2
Fig. 1. Variation of Rs of the S-ADP scheme with respect to γ, with N = 6, K = 2, ε = 0.1, ΓB = 0dB, P = 20dB and RB = 4, 6, 8bits/s/Hz.
ADP N−ADP S−ADP
12
0
0.2
0.4
ε
0.6
0.8
1
Fig. 2. Performances of the three schemes versus SOP constraint, with N = 4, K = 1, 3, ΓB = 0dB and P = 20dB.
and (29) are particularly meaningful when the exact number of antenna at Eve is unknown to Alice, since they can provide estimations for the best and worst cases. V. N UMERICAL R ESULTS In this section, we present numerical results to evaluate the performance of the S-ADP scheme with different numbers of eavesdropper antennas. For better understanding of the working process of the SADP scheme, we firstly simulate the variation of the optimal secrecy rate Rs with respect to γ. The simulation parameters are: N = 6, K = 2, ε = 0.1, ΓB = 0dB, P = 20dB and RB = 4, 6, 8bits/s/Hz. As verified in Fig. 1, the optimal secrecy rate Rs increases with respect to γ and approaches the communication rate RB as γ increases. We can also observe that the optimal on-off threshold μ becomes higher if we
where fγ (r) is the probability density function of γ. It should ¯ s varies with RB , since be noted that the secrecy throughput R μ and Rs both depend on RB . Although the value of RB is predefined in most practical scenarios, it is still meaningful to ¯ s with respect to RB . However, due to page further maximize R limit, this problem is not addressed here. In Fig. 2 and Fig. 3, the values of RB for the S-ADP scheme are deliberately chosen as the optimal communication rates for the N-ADP scheme. In the following parts, we only simulate for the best (K = 1) and the worst (K = N − 1) cases to estimate the performance limits of the S-ADP scheme. Please note that the performance curves for the general cases 2 ≤ K ≤ N − 2 must lie between the best and the worst performance curves in the figures. 1) The secrecy throughputs of the three schemes with respect to the SOP constraint are plotted in Fig. 2. The simulation settings are as follows: N = 4, K = 1, 3, ΓB = 0dB and P = 20dB. We can observe that, when K = 1, the S-ADP scheme always outperforms the NADP scheme, but is not as good as the ADP scheme. Specifically, it is interesting that the S-ADP scheme’s performance is quite close to the ADP scheme’s when the SOP constraint is strict (i.e., when ε is small).
2) Fig. 3 compares the performances of the three schemes with respect to the total transmit power P , with N = 4, K = 1, 3, ε = 0.1 and ΓB = 0dB. In general, we can see that the secrecy throughputs of all the three schemes increase as P increases. When K = 1, although the secrecy throughput of the S-ADP scheme is smaller than that of the ADP scheme, it is always greater than the NADP scheme. In particular, we observe that the S-ADP curve is close to the ADP curve and is much higher than the N-ADP curve. This is because we chose a relatively strict SOP constraint. VI. C ONCLUSION In this paper, we extended the S-ADP secure transmission scheme to the MISOME wiretap channel model with ANaided beamforming. The S-ADP scheme fixes the communication rate while adaptively adjusts the secrecy rate and the power allocation ratio according to the legitimate channel’s instantaneous CSI. By using a fixed communication rate, the S-ADP scheme can fix the codebook for encoding and decoding, thus simplifies the encoder and decoder design. With a predefined communication rate, the optimal parameters that maximize the secrecy rate were derived. Specifically, closedform expressions for the maximum secrecy rate were obtained for the extreme cases K = 1 and K = N − 1. We used numerical simulations to compare the S-ADP scheme with the N-ADP and the ADP schemes proposed in previous works. It is shown that the S-ADP scheme always outperforms the N-ADP scheme, and its performance is close to the ADP scheme’s when the SOP constraint is strict. Since strict secrecy is usually required in practice and fixed communication rate simplifies the coding process, the S-ADP transmission can be a practical approach to provide secure transmissions. Before ending this paper, we highlight that the communication rate that is used to simulate the S-ADP scheme in Section V is suboptimal in the sense of secrecy throughput maximization. A meaningful extension of this work is to maximize the secrecy throughput with respect to the communication rate to further improve the performance of the S-ADP scheme. ACKNOWLEDGEMENT This work was funded by the NSFC (Nos. 61172092, 61302069), the Foundation for Innovative Research Groups of the NSFC (No. 61221063), the National High-Tech Research and Development Program of China (No. 2015AA011306), and the Research Fund for the Doctoral Program of Higher Education of China (No. 20130201130003). R EFERENCES [1] A. D. Wyner, “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, pp. 1355–1387, Oct. 1975. [2] I. Csisz´ar and J. K¨orner, “Broadcast channels with confidential messages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May 1978. [3] S. Leung-Yan-Cheong and M. E. Hellman, “The Gaussian wire-tap channel,” IEEE Trans. Inf. Theory, vol. 24, no. 4, pp. 451–456, Jul. 1978. [4] Z. Li, W. Trappe, and R. Yates, “Secret communication via multi-antenna transmission,” in Proc. Ann. Conf. Inf. Sci. Syst., Baltimore, MD, USA, Mar. 2007, pp. 905–910.
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