Robust Directional Modulation Design for Secrecy

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Robust Directional Modulation Design for Secrecy Rate Maximization in Multi-User Networks

arXiv:1808.07607v1 [cs.IT] 23 Aug 2018

Linqing Gui, Mengxia Yang, Xiaobo Zhou, Feng Shu, Jun Li, Jiangzhou Wang, Fellow, IEEE, and Jinhui Lu

Abstract—In this paper, based on directional modulation (DM), robust beamforming matrix design for sum secrecy rate maximization is investigated in multi-user systems. The base station (BS) is assumed to have the imperfect knowledge of the direction angle toward each eavesdropper, with the estimation error following the Von Mises distribution. To this end, a Von Mises distribution-Sum Secrecy Rate Maximization (VMDSSRM) method is proposed to maximize the sum secrecy rate by employing semi-definite relaxation and first-order approximation based on Taylor expansion to solve the optimization problem. Then in order to optimize the sum secrecy rate in the case of the worst estimation error of direction angle toward each eavesdropper, we propose a maximum angle estimation errorSSRM (MAEE-SSRM) method. The optimization problem is constructed based on the upper and lower bounds of the estimated eavesdropping channel related coefficient and then solved by the change of the variable method. Simulation results show that our two proposed methods have better sum secrecy rate than zero-forcing (ZF) method and signal-to-leakage-andnoise ratio (SLNR) method. Furthermore, the sum secrecy rate performance of our VMD-SSRM method is better than that of our MAEE-SSRM method. Index Terms—Directional modulation, multi-user systems, direction angle estimation error, beamforming, artificial noise, sum secrecy rate.

I. I NTRODUCTION With the explosive growth of the mobile Internet, wireless communications have played an increasingly important role in daily life [1], [2]. However, due to the inherent broadcasting nature of the wireless communications, the transmit signals are easily wiretapped by the unauthorized receivers. Therefore, wireless communications are facing the serious privacy and security problems. Traditional communication methods based on the cryptographic techniques need an additional secure channel for private key exchanging, which may be insufficient with the development of the mobile Internet [3]–[5]. Recently, physical layer security exploiting the characteristics of the wireless channels can realize the secure wireless communications without the upper layer data encryption, being identified This work is supported in part by the National Natural Science Foundation of China (Nos. 61602245, 61771244, 61501238, 61702258, and 61472190), in part by the Natural Science Foundation of Jiangsu Province (No. BK20150791). L. Gui, M. Yang, X. Zhou, F. Shu, J. Li and J. Lu are with the Department of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing, 210094 China. E-mail: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]. J. Wang is with the School of Engineering and Digital Arts, University of Kent, Canterbury CT2 7NT, U.K. E-mail: [email protected].

as a significant complement to cryptographic techniques [6]– [9]. Directional modulation (DM), which is regarded as one of the promising physical layer wireless security techniques, has attracted wide attentions in recent years [10]–[20]. The main ideal of DM is to project the confidential information signals into the desired spatial directions while simultaneously distorting the constellation of the signals in the other directions [10]. In [11], the authors proposed a DM method which employed the near-field direct antenna modulation technique to overcome the security challenges. A similar simplified method of DM synthesis was proposed in [12] for a far-field scenario. With the development of the artificial noise (AN) aided technique, DM has been further developed. The AN is added to the transmit signals expecting that the AN would interfere with the eavesdroppers and not affect the legitimate receivers. In [13], a novel baseband signal synthesis method was proposed to construct the AN signal based on the null space of the channel vector in the desired direction. An orthogonal projection (OP) method in [10] was designed for the multi-beam DM systems. And in [14], the authors proposed a multiple-direction DM transmission scheme which employed the space domain to transmit the multiple data streams independently and increased the capacity of the system. All the methods mentioned above assumed that the transmitter knows the precise direction angles toward all users (including the eavesdroppers). However, in practical scenarios, the perfect direction angles are hard to obtain because of the estimation errors induced by the widely-used estimation algorithms such as multiple signal classification (MUSIC) and Capon. So in [21] the authors considered a single-user DM system where the estimation error of the direction angle toward the desired receiver was assumed to follow the uniform distribution. [22] considered the imperfect desired direction angles in a multi-beam DM scenario, and assumed the desired direction angle estimation errors follow the truncated Gaussian distribution. Compared to the direction angles of the desired receivers, the precise direction angles toward the eavesdroppers are more difficult to be obtained as they rarely expose their accurate locations. In this paper, we consider the imperfect direction angles toward eavesdroppers. Moreover, since the Von Mises distribution is regarded as the best statistical model for the direction angles, we assume that the angle estimation error follows the Von Mises distribution [23]. Meanwhile, the transmit beamforming technique and the AN-aided technique are also employed to realize the secure wireless communications. The

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main contributions of this paper are summarized as follows: 1) We propose a Von Mises distribution-Sum Secrecy Rate Maximization (VMD-SSRM) robust DM scheme which designs the robust signal beamforming matrix and AN beamforming matrix on the assumption that the estimation errors of direction angles toward the eavesdroppers follow the Von Mises distribution. First, the expectation of the estimated eavesdropping channel related coefficient is derived. Then the system sum secrecy rate maximization problem subject to the total transmit power of the base station (BS) is formulated. Comprising the logarithms of the product of fractional quadratic functions, the sum secrecy rate expression is nonconvex and difficult to tackle. In order to solve this problem, we employ semi-definite relaxation and first-order approximation based on Taylor expansion to transform the original problem into a convex problem. 2) In order to optimize the system sum secrecy rate in the case of the worst estimation error of direction angle toward each eavesdropper, we propose a maximum angle estimation error-SSRM (MAEE-SSRM) method. To design the robust signal beamforming matrix and AN beamforming matrix, first the upper and lower bounds of the estimated eavesdropping channel related coefficient are derived, then the system sum secrecy rate maximization problem based on the derived upper and lower bounds is constructed. Since the optimization problem is still non-convex, we use the change of the variable method to convert it into a convex problem and then solve it by convex optimization tools. 3) Through simulation and evaluation, we investigate the performance of our two proposed methods. We compare our methods with some typical DM methods such as ZF method [24] and SLNR method [25]. Simulation results show that our two proposed robust methods have much better sum secrecy rate than ZF method and SLNR method. Meanwhile the performance of our VMD-SSRM method is better than that of MAEE-SSRM method. The rest of this paper is organized as follows. Section II introduces the channel model and the system model. In Section III, we design the information beamforming matrix and AN beamforming matrix to maximize the worst-case system sum secrecy rate with the VMD-SSRM method. In Section IV, we consider the maximum angle estimation error of the eavesdropper and design the information beamforming matrix and AN beamforming matrix to maximize the worst-case system sum secrecy rate with the MAEE-SSRM method. Section V provides the simulation results to validate the effectiveness of the two proposed methods as well as the complexity of our two proposed methods, and followed by our conclusions in Section VI. Notations: In this paper, we use boldface lowercase and uppercase letters to denote the vectors and matrices, respectively. AT , A H , Tr(A), rank(A) and Re{A} denote the transpose, the conjugate transpose, the trace, the rank and the real part of the matrix A, respectively. A  0 denotes that A is a positive semidefinite matrix. | · | and k·k denote the module of the scalars and the Frobenius norm of the matrices. E{·}, log(·), log10 (·) and In(·) denote the statistical expectation, the logarithm to base 2, the logarithm to base 10 and the natural logarithm,

respectively. hx, yi denotes the inner product of the vector x and y. Cm×n denotes the set of all complex m × n matrices. I M denotes an M × M identity matrix. x ∼ CN (0, σ 2 ) denotes a circularly symmetric complex Gaussian random variable x with zero mean and variance σ 2 . II. SYSTEM MODEL AND PROBLEM FORMULATION A. Channel Model In this paper, the DM transmitter is assumed to be equipped with uniform isotropic array. The array is composed of N elements with an adjacent distance of l, while the direction angle between the transmit antenna array and the receiver is denoted by θTR . The normalized steering vector for the transmit antenna array is expressed as ˆ TR ) = √1 [e j2πΨθTR (1), e j2πΨθTR (2),· · ·, e j2πΨθTR (N ) ]T , (1) h(θ N where ΨθTR (n) is   n − (N + 1)/2 l cos (θTR ) ΨθTR (n) = − , n = 1, 2,· · ·, N, (2) λ where λ is the wavelength of the transmit signal radio. In the line of sight (LOS) communication scenarios, e.g., unmanned aerial vehicles (UAV) communication and suburban mobile communication, it is widely accepted to use free-space path loss model [26], given by  4πdf  2 PL(d) = 10log10 , (3) c where PL(d) is the path loss in dB at a given distance d, f is the frequency of the transmit signal radio and c is the radio velocity. Then according to (1) and (3), the channel model between the transmit antenna array and the receiver can be expressed as s  2 c ˆ TR ) h(θTR ) = × h(θ 4πdTR f (4) 1 j2πΨθ (1) j2πΨθ (2) √ j2πΨ (N ) T θTR TR TR = gTR √ [e ,e ,··· ,e ] , N where dTR is the distance between the transmit antenna array  2

and the receiver, and gTR = 4πdcTR f denotes the path loss between the transmit antenna array and the receiver. B. Multi-User System Model with Multiple Eavesdroppers

In this paper, we consider a flat fading multiuser multi-input single-output (MU-MISO) downlink system as shown in Fig. 1. The system comprises a BS with N transmit antennas, M (M < N) single-antenna destination users Di (i = 1, 2, · · · , M), and K single-antenna eavesdroppers Ek (k = 1, 2, · · · , K), wishing to wiretap the confidential information sent from the BS to the destination users. The total transmit power of the BS is Pt , and the direction angle between the BS and each destination user is θ BDi (i = 1, 2, · · · , M). The transmit beamforming technique is employed by the BS to steer the M information signal beams toward the destination users. The direction angle between the BS and each eavesdropper is θ BEk

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Similarly, the signal received at the k-th eavesdropper Ek is Main channel

yEk = h H (θ BEk )x + nEk

Eavesdropping channel

= h H (θ BEk )wi xi +

E1

M Õ

h H (θ BEk )wm xm (7)

m=1 m,i

D1

+

E2

L Õ

h H (θ BEk )ql zl + nEk ,

l=1

D2

EK

Base Station

DM

Fig. 1. Multi-User Downlink System Model with Multiple Eavesdroppers

(k = 1, 2, · · · , K). In order to decode the confidential messages sent from the BS to all destination users, the eavesdroppers are assumed to locate closer to the BS than the destination users. And in order to interfere with the eavesdroppers, the BS employs the beamforming technique to steer L (L ≤ N) AN beams toward the eavesdroppers. These AN beams are transmitted along with the information signal beams, thus the received information signal qualities of the eavesdroppers are degraded. Let xˆ = [x1, · · · , x M ]T denote the signal vector formed by the M independent data streams sent from the ˆ = BS to the destination users with E{xˆ xˆ H } = I M . Let W ˆ = [q1, · · · , q L ] ∈ C N ×L denote [w1, · · · , w M ] ∈ C N ×M and Q the information beamforming matrix and the AN beamforming matrix, respectively. The total transmit signals of the BS can be expressed as

x=

M Õ

wm xm +

m=1

L Õ

ql zl,

(5)

i=1 M Õ

min Rik k

  min log(1 + SINRDi )−log(1 + SINREk )

M Õ

k

i=1

= log

M Ö

M  Ö  (1 + SINRDi ) −log max(1 + SINREk ) ,

i=1

i=1

SINRDi =

(6)

h H (θ BDi )ql zl + nDi ,

l=1

where the first term of the right hand side of (6) is the useful signal received by Di , the second one is the multiuser interference from other destination users, the third one is the AN, and the last one is the additive white Gaussian noise 2 ). (AWGN) with nDi ∼ CN (0, σD

k

|h H (θ BDi )wi | 2 . (9) M L Í Í H H 2 2 2 |h (θ BDi )wm | + |h (θ BDi )ql | +σD l=1

And according to (7), the SINR at the k-th eavesdropper Ek intending to intercept the i-th destination user is |h H (θ BEk )wi | 2 . (10) M L Í Í |h H (θ BEk )wm | 2 + |h H (θ BEk )ql | 2 +σE2

m=1 m,i

h H (θ BDi )wm xm

(8)

where Rs denotes the system sum secrecy rate, SINRDi and SINREk denote the signal-to-interference-plus-noise ratio (SINR) at the destination user Di and the eavesdropper Ek , respectively. Then, according to (6), the SINR at the i-th destination user Di is

SINREk =

m=1 m,i

+

=

M Õ

m=1 m,i

yDi = h H (θ BDi )x + nDi

L Õ

Rs =

l=1

where zl is the AN signal sent by the BS. Then the received signal at the i-th destination user Di is given by

= h H (θ BDi )wi xi +

where the right hand side of (7) is composed of the i-th confidential message intercepted by Ek , the messages sent to the other (M-1) destination users that are intercepted by Ek , the AN to disturb Ek and the AWGN satisfying nEk ∼ CN (0, σE2 ). It should be noted that we consider the non-colluding eavesdropping scenario where each eavesdropper relies on itself to overhear and decode the messages and there is no information exchange between the eavesdroppers. In this paper, the worst-case system sum secrecy rate is used to evaluate the system performance in the multiuser scenario, which can be expressed as

l=1

Now, according to (8), (9) and (10), the optimization problem that maximizes the worst-case system sum secrecy rate under the constraint of the total transmit power of the BS is constructed as shown in (11) on the following page. C. Direction Angle Estimation Error Following Von Mises Distribution Since the BS is assumed to have the imperfect knowledge of the direction angles toward the eavesdroppers, the actual

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maximize {wi }, {ql }

log

M Ö i=1

M Í m=1 M Í m=1 m,i

s.t.

L Õ l=1

|h H (θ BDi )wm | 2 + |h H (θ BDi )wm | 2 +

kql k 2 +

M Õ

L Í l=1 L Í l=1

2 |h H (θ BDi )ql | 2 +σD  2 |h H (θ BDi )ql | 2 +σD

M L Í Í |h H (θ BEk )wm | 2 + |h H (θ BEk )ql | 2 +σE2  M Ö m=1 l=1 − log max M L k Í Í i=1 |h H (θ BEk )wm | 2 + |h H (θ BEk )ql | 2 +σE2 m=1 m,i

l=1

kwi k 2 ≤ Pt .

(11)

i=1

direction angle between the BS and the k-th eavesdropper is expressed as θ BEk = θˆBEk + ∆θ BEk , k = 1, · · · , K,

(12)

where θˆBEk is the estimated direction angle between the BS and the k-th eavesdropper, ∆θ BEk denotes the estimation error which is assumed to follow the Von Mises distribution [27] over the interval [−∆θ max, ∆θ max ]. Note that ∆θ max represents the maximum angle error and is a positive value up to the beamwidth between the first nulls [21]. The Von Mises distribution, also known as the circular normal distribution, is a continuous probability distribution on the circle and mainly describes the information related to the directional angles. In [23], the Von Mises distribution was considered to be the best statistical model for circular parameters such as the direction angle. Subsequently, according to (12), the estimation error of the direction angle should also follow the Von Mises distribution. The probability density function (PDF) of the Von Mises distribution is f (θ| µ, κ) =

eκ cos(θ−µ) , θ ∈ [−π, π), 2πI0 (κ)

(13)

where θ represents the estimation error of the direction angle, µ denotes the mean of θ, κ ≥ 0 denotes the concentration parameter and controls the width of θ, and In with n = 0 is the modified Bessel function of the first kind with order n computed by ∫ 1 π x cos(t) In (x) = e cos(nt)dt, x ∈ (−∞, ∞). (14) π 0 III. ROBUST BEAMFORMING MATRIX DESIGN WITH VON-MISES DISTRIBUTED DIRECTION ANGLE ERROR AT EAVESDROPPERS In the previous section, the estimation error of the direction angle toward each eavesdropper has been modeled to follow the Von Mises distribution. Then in order to maximize the worst-case system sum secrecy rate, in this section, we propose a Von Mises distribution-SSRM (VMD-SSRM) robust DM scheme. The procedure of the proposed scheme, i.e., the design of the optimal beamforming matrices, will be illustrated as follows.

In (11), the item |h H (θ BEk )wm | 2 can be expressed as |h H (θ BEk )wm | 2 =   H Tr h(θ BEk )h H (θ BEk )wm wm   H = Tr h(θˆBEk + ∆θ BEk )h H (θˆBEk + ∆θ BEk )wm wm .

(15)

To simplify the expression, we define n o HBEk = E h(θˆBEk + ∆θ BEk )h H (θˆBEk + ∆θ BEk ) , k ∈ K, (16) where HBEk ∈ C N ×N . The necessity of the expectation in (16) is explained as follows. If we remove the expectation, since ∆θ BEk is a random variable, then HBEk would be random as well as the objective function of (11), and it would be very difficult to find a solution to (11). Let HBEk (u, v) denote the u-th row and the v-th column entry of HBEk , then HBEk (u, v) can be written as HBEk (u, v) = Υ1k (u, v) − jΥ2k (u, v).

(17)

The detailed procedure for deriving Υ1k (u, v) and Υ2k (u, v) based on the Von Mises distribution is shown in Appendix A. And Υ1k (u, v) and Υ2k (u, v) are expressed as (A17) and (A24), respectively. In order to make (11) more tractable, we substitute the new matrix H(θ BDi ) = h(θ BDi )h H (θ BDi ) as well as the positive L Í semi-definite matrix variables Wi = wi wiH , Q = ql qlH into l=1

(11). Then the optimization problem (11) can be rewritten in a simple form as shown in (18) at the top of the next page. By solving the problem (18), we can obtain the optimal information beamforming matrices {Wi } and the AN beamforming matrix Q. However, the objective function in (18) is non-convex and difficult to tackle, as it comprises the logarithms of the product of fractional quadratic functions. In order to solve this problem, we employ the semi-definite relaxation and the first-order approximation technique based on the Taylor expansion to transform the original problem into a convex problem [28]. Then this convex problem can be solved by using the convex optimization toolbox (CVX). The specific process for converting (18) into a convex one is illustrated as follows. First, the exponential variables are used to substitute the numerators and denominators of the fractions in the objective function in (18), i.e., M     Õ 2 e pi = Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD , (19) m=1

5

maximize {Wi },Q

log

M Ö i=1

M Í

    2 Tr H(θ BDi )Wm +Tr H(θ BDi )Q +σD 

m=1 M Í

− log     2 Tr H(θ BDi )Wm +Tr H(θ BDi )Q +σD

M Ö

M Í

max k

i=1

M Õ

m=1 M Í

    Tr HBEk Wm +Tr HBEk Q +σE2

m=1 m,i

s.t. Tr(Q) +

    Tr HBEk Wm +Tr HBEk Q +σE2 

m=1 m,i

Tr(Wi ) ≤ Pt ,

i=1

Wi, Q  0, ∀i.

e qi =

M Õ

(18)

    2 , Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD

(20)

m=1 m,i

e ck =

M Õ

    Tr HBEk Wm + Tr HBEk Q + σE2 ,

(21)

m=1

e

=

di, k

M Õ









Tr HBEk Wm + Tr HBEk Q +

σE2 .

(22)

m=1 m,i

M Õ

(pi − qi ) − max (ck − di,k ) .


k=1, ··· ,K

i=1

(23)

S

s.t. M Õ m=1 M Õ m=1 m,i M Õ m=1 M Õ

(25)

= e (ck − c¯k + 1),

(26)

e

ck

c¯ k

(pi − qi ) − max (ck − di,k )




k=1, ··· ,K

i=1

q¯i = ln

M Õ

     2 Tr H(θ BDi )Wm +Tr H(θ BDi )Q +σD ,

(27)

m=1 m,i

c¯k = ln

M Õ

     Tr HBEk Wm + Tr HBEk Q + σE2 ,

(28)

m=1

Meanwhile pi , qi , ck , di,k are constrained by the right hand sides of each equation in (19), (20), (21) and (22). So the problem (18) can be transformed to a semi-definite programming (SDP) problem as

maximize

eqi = eq¯i (qi − q¯i + 1), where

Then according to the properties of exponential and logarithmic functions, the objective function in (18) can be rewritten as

M Õ

The objective function (24a) is a concave function because it comprises a sum of the affine functions minus a sum of the maxima of the affine functions. However, the constraints (24c) and (24d) are non-convex. In order to transform these two constraints into convex ones, eqi and eck can be linearized based on the first-order Taylor approximation as

(24a)

are the points around which the approximation are made. So the problem (11) eventually becomes M Õ
(pi − qi ) − max (ck − di,k ) (29a) maximize S

s.t. M Õ

k=1, ··· ,K

i=1

    2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≥ e pi , ∀i,

m=1

(29b) 















2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≥ e pi , ∀i, (24b)

(24d)

    Tr HBEk Wm + Tr HBEk Q + σE2 ≥ e di, k , ∀i, ∀k, (24e)

m=1 m,i

Tr(Q) +









2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD

m=1 m,i

2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≤ eqi , ∀i, (24c)

    Tr HBEk Wm + Tr HBEk Q + σE2 ≤ eck , ∀k,

M Õ

≤ eq¯i (qi − q¯i + 1), ∀i, M     Õ Tr HBEk Wm + Tr HBEk Q + σE2

(29c)

m=1

≤ ec¯k (ck − c¯k + 1), ∀k, (29d) M     Õ Tr HBEk Wm + Tr HBEk Q + σE2 ≥ e di, k , ∀i, ∀k, (29e) m=1 m,i

M Õ

Tr(Wi ) ≤ Pt ,

(24f)

i=1

Wi, Q  0, ∀i.

Tr(Q) +

M Õ

Tr(Wi ) ≤ Pt ,

(29f)

i=1

(24g)

Wi, Q  0, ∀i.

(29g)

6

As a convex problem, (29) can be solved iteratively by the Algorithm 1 using the CVX optimization software. According to the definition, Wm has to satisfy rank(Wm ) = 1, so the optimal Wm of (29) should satisfy this rank-one constraint. Otherwise the randomization technology [29] would be utilized to get a rank-one approximation. Algorithm 1 Algorithm for solving the problem (29) ˜ i , i = 1, · · · , M } and {q˜ l , l = 1, · · · , L } randomly that 1: Given {w are feasible to (29); L Í ˜ ˜ i [0] = w ˜ iH , i = 1, · · · , M, Q[0] ˜ iw = q˜ l q˜ lH and set n=0; 2: Set W l=1

3: Repeat ˜ i [n] and Q[n] ˜ 4: Substituting W into (27) and (28) yields q¯ i [n + 1] and c¯ k [n + 1]; 5: Increment n = n + 1; 6: Substituting q¯ i [n] and c¯ k [n] into (29) yields the optimal solution ˜ i [n] and Q[n]; ˜ W 7: Until convergence; ˜ and {w∗, i = 1, · · · , M } by decomposition of 8: Obtain Q i ˜ i [n]) = 1; ˜ i [n] = (w∗ )(w∗ ) H for all i in the case of rank(W W i i otherwise the randomization technology [29] would be utilized to get a rank-one approximation.

IV. ROBUST BEAMFORMING MATRIX DESIGN WITH MAXIMUM DIRECTION ANGLE ERROR AT EAVESDROPPERS

upper bound of |h H (θ BEk )wm | 2 is |h H (θ BEk )wm | 2 H = h H (θ BEk )wm wm h(θ BEk )     H ˆ H H = h (θ BEk ) + ∆hBEk wm wm h(θˆBEk ) + ∆hBEk o n H H H ˆBEk ) w w h( θ = h H (θˆBEk )wm wm h(θˆBEk )+2Re ∆hBE m m k (a)

H H ≤ h H (θˆBEk )wm wm h(θˆBEk ) + 2εk kwm wm h(θˆBEk )k.

(31)

Similarly, the lower bound of |h H (θ BEk )wm | 2 is |h H (θ BEk )wm | 2 n o H H H ˆ = h H (θˆBEk )wm wm h(θˆBEk )+2Re ∆hBE w w h( θ ) m m BEk k H H ≥ h H (θˆBEk )wm wm h(θˆBEk ) − 2εk kwm wm h(θˆBEk )k.

(32) It should be noted that in (31) and (32) the second-order H H ∆h error term ∆hBE wm wm BEk is omitted because it is quite k small compared to the other two terms. And note that the step (a) in (31) results from the following derivation process. According to the Cauchy-Schwarz inequality theorem, we have |x H y| ≤ kxk kyk.

(33)

− |x H y| ≤ Re{x H y} ≤ |x H y|,

(34)

Since the inequality

is obviously true, then we can obtain the following inequality In the previous section, the system was designed based on the Von Mises distribution of the eavesdropper direction angle error. Thus the system sum secrecy rate was maximized for all possible values of the direction angle error. Nevertheless, it is also important to exclusively investigate the case of the worst direction angle error. Thus in this section, the system will be designed when the eavesdropper direction angle error reaches its maximum value and the robust beamforming matrices are obtained by the maximum angle estimation error-SSRM (MAEE-SSRM) method. If the direction angle estimation error toward each eavesdropper is bounded, we can prove that the channel estimation error between the BS and each eavesdropper is norm-bounded (the detailed procedure is illustrated in Appendix B). Then the channel between the BS and the k-th eavesdropper can be expressed as

− kxk kyk ≤ Re{x H y} ≤ kxk kyk.

(35)

The inequality in (35) holds with equality when and only when x and y are linearly dependent. The derivation of the step (a) in (31) is completed. Similarly, the upper bound of |h H (θ BEk )ql | 2 is |h H (θ BEk )ql | 2 ≤ h H (θˆBEk )ql q H h(θˆBEk ) + 2εk kql q H h(θˆBEk )k, l

(36)

l

and the lower bounds of |h H (θ BEk )ql | 2 is |h H (θ BEk )ql | 2 ≥ h H (θˆBEk )ql q H h(θˆBEk ) − 2εk kql q H h(θˆBEk )k. l

(37)

l

With the upper and lower bounds of |h H (θ BEk )wm | 2 and |h (θ BEk )ql | 2 substituted into (11), the maximization of the lower bound of the objective function in (11) can be written as (38) at the top of the next page. So we have converted (11) into a problem to maximize the lower bound of the sum secrecy rate. But (38) is non-convex and difficult to tackle. To solve this problem, we use the change of variable method in [28]. First, the new matrices H(θ BDi ) = h(θ BDi )h H (θ BDi ), H(θˆBEk ) = h(θˆBEk )h H (θˆBEk ) and the positive semi-definite L Í matrix variables Wi = wi wiH , Q = ql qlH are introduced H

h(θ BEk ) = h(θˆBEk ) + ∆hBEk , k∆hBEk k ≤ εk , k = 1,· · ·, K, (30) where h(θˆBEk ) is the estimated channel between the BS and the k-th eavesdropper, ∆hBEk is the channel estimation error, and εk is the bound of the norm of the channel estimation error of the k-th eavesdropper. Then we derive the lower bound of the objective function in (11). For that purpose, we first derive the upper and lower bounds of |h H (θ BEk )wm | 2 and |h H (θ BEk )ql | 2 in (11). The

l=1

to make (38) more tractable. And (38) can be rewritten in a simple form as shown in (39) at the top of this page.

7

maximize {wi }, {ql }

log

M Ö i=1

M Í m=1 M Í m=1 m,i

|h H (θ BDi )wm | 2 + |h H (θ BDi )wm | 2 +

L Í l=1 L Í l=1

2 |h H (θ BDi )ql | 2 + σD 

−

2 |h H (θ BDi )ql | 2 + σD

M L Í Í H H H 2 H h(θˆ H (h H (θˆBEk )wm wm BEk )+2εk kwm wm h(θˆBEk )k)+ (h (θˆBEk )ql ql h(θˆBEk )+2εk kql ql h(θˆBEk )k)+σE  M Ö m=1 l=1 . log max M L k Í Í H ˆ H ˆ H H 2 H H i=1 ˆ ˆ ˆ ˆ (h (θ BEk )wm wm h(θ BEk )−2εk kwm wm h(θ BEk )k)+ (h (θ BEk )ql ql h(θ BEk )−2εk kql ql h(θ BEk )k)+σE m=1 m,i

l=1

(38) maximize {Wi },Q

log

M Ö i=1

M Í m=1 M Í m=1 m,i

log

M Ö i=1

    2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD  

m=1 M  Í

max

s.t. Tr(Q) +





Tr H(θ BDi )Wm + Tr H(θ BDi )Q + M  Í

k



m=1 m,i M Õ

−

2 σD

   Tr(H(θˆBEk )Wm )+2εk kWm h(θˆBEk )k + Tr(H(θˆBEk )Q)+2εk kQh(θˆBEk )k +σE2     Tr(H(θˆBEk )Wm )−2εk kWm h(θˆBEk )k + Tr(H(θˆBEk )Q)−2εk kQh(θˆBEk )k +σE2

Tr(Wi ) ≤ Pt ,

i=1

Wi, Q  0, ∀i.

(39)

Then, the numerators and denominators of the fractions in the objective function in (39) are substituted by the exponential variables as follows M     Õ 2 e si = Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD , (40)

At the same time, si , ti , ak , bi,k are constrained by each expression at the right hand sides of (40), (41), (42) and (43). Then (39) can be rewritten as

m=1

e ti =

M Õ









2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ,

(41)

m=1 m,i

e ak =

M  Õ

Tr(H(θˆBEk )Wm ) + 2εk kWm h(θˆBEk )k





+ Tr(H(θˆBEk )Q) + 2εk kQh(θˆBEk )k + ebi, k =

M  Õ

(42)

σE2 ,

Tr(H(θˆBEk )Wm ) − 2εk kWm h(θˆBEk )k



m=1 m,i

(43)

  + Tr(H(θˆBEk )Q) − 2εk kQh(θˆBEk )k + σE2 .

i=1

m=1 M Õ m=1 m,i M Õ

(si − ti ) − max (ak − bi,k ) . k=1, ··· ,K

k=1, ··· ,K




(45a)

    2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≥ esi , ∀i, (45b)     2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≤ eti , ∀i, (45c) 

Tr(H(θˆBEk )Wm ) + 2εk kWm h(θˆBEk )k



  + Tr(H(θˆBEk )Q) + 2εk kQh(θˆBEk )k + σE2 ≤ e ak , ∀k, (45d) M  Õ




i=1

(si − ti ) − max (ak − bi,k )

m=1

Thus we can rewrite the objective function in (39) as M Õ

S1

s.t. M Õ



m=1

maximize

M Õ

(44)

m=1 m,i

Tr(H(θˆBEk )Wm ) − 2εk kWm h(θˆBEk )k



8

  + Tr(H(θˆBEk )Q) − 2εk kQh(θˆBEk )k + σE2 ≥ ebi, k ,∀i,∀k, (45e) M Õ Tr(Q) + Tr(Wi ) ≤ Pt , (45f) i=1

Wi, Q  0, ∀i.

(45g)

Now, the objective function (45a) is concave, but the constraints (45c) and (45d) are non-convex. In order to convert these two constraints into convex ones, we linearize the exponential terms eti and e ak based on the first-order Taylor approximation as eti = et¯i (ti − t¯i + 1),

(46)

e ak = e a¯ k (ak − a¯k + 1),

(47)

where t¯i = ln

M Õ

     2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD , (48)

m=1 m,i

a¯k = ln

M  Õ

Tr(H(θˆBEk )Wm ) + 2εk kWm h(θˆBEk )k

m=1





 (49)



+ Tr(H(θˆBEk )Q) + 2εk kQh(θˆBEk )k + σE2 , are the points around which the linearization are made. Thus the problem (11) finally becomes maximize S1

s.t. M Õ m=1 M Õ

M Õ i=1

(si − ti ) − max (ak − bi,k )




k=1, ··· ,K

(50a)

    2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD ≥ esi , ∀i, (50b)     2 Tr H(θ BDi )Wm + Tr H(θ BDi )Q + σD

m=1 m,i

≤ et¯i (ti − t¯i + 1), ∀i, M   Õ Tr(H(θˆBEk )Wm ) + 2εk kWm h(θˆBEk )k

(50c)

m=1

  + Tr(H(θˆBEk )Q) + 2εk kQh(θˆBEk )k + σE2 ≤ e a¯ k (ak − a¯k + 1), ∀k, M   Õ Tr(H(θˆBEk )Wm ) − 2εk kWm h(θˆBEk )k

l=1

3: Repeat ˜ i [n] and Q[n] ˜ 4: Substituting W into (48) and (49) yields t¯i [n + 1] and a¯ k [n + 1]; 5: Increment n = n + 1; 6: Substituting t¯i [n] and a¯ k [n] into (50) yields the optimal solution ˜ i [n] and Q[n]; ˜ W 7: Until convergence; ˜ and {w∗, i = 1, · · · , M } by decomposition of 8: Obtain Q i ˜ i [n]) = 1; ˜ i [n] = (w∗ )(w∗ ) H for all i in the case of rank(W W i i otherwise the randomization technology [29] would be utilized to get a rank-one approximation.

Next, we derive the complexity of our two proposed methods. According to [30], the per-iteration complexity of the proposed VMD-SSRM method and MAEE-SSRM method can be both approximately calculated as Comp per−iter = p O (K M + K + 2M + 1)(2N + 2) + K M + (M + 1)N no [no2 + no ((K M + K + 2M + 1)(2N + 2)2 + K M 2 +

(51)

(M + 1)N 2 ) + (K M + K + 2M + 1)(2N + 2)3 + K M 3 + 1  (M + 1)N 3 ]ln( ) ,  where N, M and K are the number of transmit antennas at the BS, the number of destination users and the number of eavesdroppers, respectively. no = (K + 3)M + K + (M + 1)N 2 is the dimension of real design variables, and  is a computational accuracy. Then the complexity of VMD-SSRM method is Comp per−iter · IVMDma x while the complexity of MAEESSRM method is Comp per−iter · I MAEEma x , where IVMDma x and I MAEEma x are the iteration times of VMD-SSRM method and MAEE-SSRM method, respectively. According to (51), the per-iteration complexity of VMDSSRM method and MAEE-SSRM method increases with the increase of N, M and K. When N, M and K are small, for example, N = 4, M = 2, K = 1, the complexity of proposed VMD-SSRM method and MAEE-SSRM method is slightly higher than that of ZF method and SLNR method. V. SIMULATION AND ANALYSIS

(50d)

m=1 m,i

  + Tr(H(θˆBEk )Q) − 2εk kQh(θˆBEk )k + σE2 ≥ ebi, k ,∀i,∀k, (50e) M Õ Tr(Q) + Tr(Wi ) ≤ Pt , (50f) i=1

Wi, Q  0, ∀i.

Algorithm 2 Algorithm for solving the problem (50) ˜ i , i = 1, · · · , M } and {q˜ l , l = 1, · · · , L } randomly that 1: Given {w are feasible to (50); L Í ˜ ˜ i [0] = w ˜ iw ˜ iH , i = 1, · · · , M, Q[0] = q˜ l q˜ lH and set n=0; 2: Set W

(50g)

Now, (50) is a convex problem and can be solved iteratively by Algorithm 2 using CVX.

In this section, we present the simulation results to evaluate the performance of our proposed VMD-SSRM method and MAEE-SSRM method. Our methods are compared to ZF method and SLNR method. The two methods are based on the schemes in [24] and [25], respectively, but the eavesdropping channels in these two methods are estimated in the same way as our VMD-SSRM method. The simulation parameters are set as shown in Table I, unless otherwise stated. The distance between the BS and each destination user is set to 80m, while the distance between the BS and each eavesdropper is set to 50m. Fig. 2 shows the locations of the BS, destination users and eavesdroppers in the Cartesian coordinate system.

9

TABLE I S IMULATION PARAMETERS S ETTING Parameters The number of antennas at the BS (N )

Values 6

The number of destination users (M)

2 4

The number of eavesdroppers (K) The maximum angle estimation error toward each eavesdropper (∆θmax ) The mean of the angle estimation error (µ) The concentration parameter of the angle estimation error (κ) 2 , σ2 ) The power of receiver noise (σD E The transmit power of the BS (Pt ) The direction angles toward the destination users ({θ BD1 , θ BD2 }) The direction angles toward the eavesdroppers ({θ BE1 , θ BE2 , θ BE3 , θ BE4 })

6◦ 0 100 −30dBm 40dBm { π6 ,

π 3

}

π π π 5π , 12 , 4 , 12 {− 12

}

y D2 E4

D1

E3 E2

BS

x

Fig. 3 shows how the sum secrecy rate changes with the transmit power of the BS. It can be observed that the sum secrecy rate grows with the increase of the transmit power at the BS for all four methods. The reason is explained as follows. When the transmit power of the BS increases, the destination user’s SINR increases more than the eavesdropper’s SINR, because these four methods all try to provide the eavesdroppers less confidential signal and more AN by designing the best signal beamforming matrix and AN beamforming matrix, thus limiting the eavesdropper’s SINR. It can also be observed that our VMD-SSRM method achieves better sum secrecy rate than our MAEE-SSRM method. The reason is that MAEE-SSRM method aims to maximize the system performance in the worst case when the estimation error of direction angle toward each eavesdropper is the largest, while VMD-SSRM method can deal with more cases because most angle estimation errors are taken into consideration through expectation computation. Moreover, the sum secrecy rate of SLNR method increases slowly. The reason is explained as follows. As in [25], SLNR method fixes the power allocation factors for both confidential signal and AN, and the power allocation factor for confidential signal is much higher than that of AN. So when transmit power of the BS increases, the power allocated to confidential signal is much more than that of AN, providing the eavesdroppers more chance to decipher confidential signals. So the sum secrecy rate of SLNR method increases very slowly.

E1 4

Fig. 2. The locations of the BS, destination users and eavesdroppers.

Sum Secrecy Rate (bits/s/Hz)

6 5

Sum Secrecy Rate (bits/s/Hz)

3.5 3 2.5 2 1.5

proposed VMD−SSRM method proposed MAEE−SSRM method ZF method SLNR method

1

proposed VMD−SSRM method proposed MAEE−SSRM method ZF method SLNR method

0.5 6

8

10

12

14

16

18

The Number of Transmit Antennas at the BS

4 3

Fig. 4. Sum secrecy rate versus the number of transmit antennas at the BS for M = 2, K = 4, Pt = 40dBm, ∆θmax = 6◦ .

2

Fig. 4 compares the sum secrecy rate against different number of transmit antennas at the BS. It can be seen that when the number of transmit antennas increases, the sum secrecy rates of four methods also increase. The reason is explained as follows. These four methods all try to obtain the optimal sum secrecy rate by designing the optimal signal beamforming vectors and AN beamforming vectors. On the other hand, when the BS has more antennas, the beams generated by the BS can be narrower. So the signal beams will be more precisely directed toward destination users and the AN beams will also be more precisely directed toward eavesdroppers. It can also be observed that when the number of transmit antennas exceeds

1 0 36

38

40

42

44

46

48

Transmit power (dBm)

Fig. 3. Sum secrecy rate versus the transmit power of the BS for N = 6, M = 2, K = 4, ∆θmax = 6◦ .

10

3

3.5 3 Sum Secrecy Rate (bits/s/Hz)

some value (such as 16), the sum secrecy rates of four methods increase slowly. The reason can be explained as follows. When the number of transmit antennas exceeds 16, the dimension of the beamforming vectors is already high enough to concentrate the confidential signal on the corresponding desired user as well as concentrate the AN on eavesdroppers. Since the system secrecy performance is already optimized, merely increasing the number of transmit antennas cannot effectively improve the system secrecy performance any more.

2.5 2 1.5 1

Sum Secrecy Rate (bits/s/Hz)

0.5 2.5

0 1

proposed VMD−SSRM method proposed MAEE−SSRM method ZF method SLNR method

1.5

2

2.5

3

3.5

4

4.5

5

The Number of Eavesdroppers (K)

2

Fig. 6. Sum secrecy rate versus the number of the eavesdroppers for N = 6, π π π 5π 7π M = 2, {θ BE1 , θ BE2 , θ BE3 , θ BE4 , θ BE5 } = {− 12 , 12 , 4 , 12 , 12 }, Pt = 40dBm, ∆θmax = 6◦ .

1.5 1 0.5 0 0

proposed VMD−SSRM method proposed MAEE−SSRM method ZF method SLNR method

1

2

3

4

5

6

The Maximum Angle Estimation Error (degree)

Fig. 5. Sum secrecy rate versus the maximum angle estimation error for N = 6, M = 2, K = 4, Pt = 40dBm.

In Fig. 5, fixing the transmit power of the BS, we investigate the impact of the maximum angle estimation error ∆θ max on the sum secrecy rate. With the increase of ∆θ max , the sum secrecy rates of the four methods degrade slowly. The reason is that these four methods all take the estimation error of direction angle toward each eavesdropper into consideration when constructing the optimization problems. Similar to our VMD-SSRM method, the ZF method and the SLNR method both obtain the expectation of eavesdropping channel related coefficient on the condition of Von-Mises-distributed direction angle estimation error. As such, they are robust against the effects of the direction angle estimation errors. Moreover, our proposed VMD-SSRM method and MAEE-SSRM method obviously outperform ZF method and SLNR method. The main reason is that both VMD-SSRM method and MAEESSRM method are aiming to directly maximize the sum secrecy rate, while ZF method and SLNR method do not take the sum secrecy rate as the direct optimization objective. The optimization objective of ZF method is to maximize the confidential signal for each destination user whist nullify signal interference at each destination user. On the other hand, SLNR method aims to minimize the confidential signal leakage to other users including all eavesdroppers and meanwhile to maximize AN power leakage to all eavesdroppers [25]. Fig. 6 shows how the sum secrecy rate changes with the number of the eavesdroppers. It can be observed that the sum secrecy rates of four methods decrease when the number of the eavesdroppers increases. The reason is explained as follows. When there are more eavesdroppers, the confidential signal

will be more likely to be wiretapped by the eavesdroppers. Although the AN is used to interfere the eavesdroppers, since the BS has only 6 transmit antennas, the BS cannot generate enough AN beams to interfere the additional eavesdroppers. It can also be observed that when the number of the eavesdroppers exceeds 4, the sum secrecy rates of four methods decrease slowly. The reason is that the fifth eavesdropper E5 (as well as subsequent eavesdroppers) are set to be far away from the destination users. Thus even if the number of the eavesdroppers increases, the additional eavesdroppers can hardly capture confidential signals, resulting in small effect on the sum secrecy rate. VI. C ONCLUSION In this paper, we have investigated the robust beamforming matrix design for sum secrecy rate maximization based on directional modulation in multi-user systems. It was assumed that the BS has the imperfect knowledge of the direction angle toward each eavesdropper and the direction angle estimation errors follow the Von Mises distribution. Following this assumption, we first proposed a VMD-SSRM method to maximize the sum secrecy rate. In addition, to optimize the sum secrecy rate in the case of the worst estimation error of direction angle toward each eavesdropper, we proposed a MAEE-SSRM method. From the analysis and the simulations, our VMD-SSRM method and MAEE-SSRM method achieve better sum secrecy rate than ZF method and SLNR method. Moreover, the sum secrecy rate performance of the proposed VMD-SSRM method is better than that of the proposed MAEE-SSRM method. A PPENDIX A HBEk (u, v) can be expressed as n o HBEk (u, v) = E hu (θˆBEk + ∆θ BEk )hvH (θˆBEk + ∆θ BEk ) . (A1)

11

According to (4), (A1) can be written as ∫ ∆θmax 2π(v−u)l gBEk ˆ HBEk (u, v) = e j λ cos(θ BE k +∆θ BE k ) N −∆θmax f (∆θ BEk)d(∆θ BEk ).

(A2)

The exponential function in (A2) can be expanded by using Euler’s formula as HBEk (u, v) = ∫ ∆θmax n  gBEk exp jαuv cos(θˆBEk ) cos(∆θ BEk ) N −∆θmax o ˆ − sin(θ BEk ) sin(∆θ BEk ) f (∆θ BEk)d(∆θ BEk ),

(A3)

. It is assumed that ∆θ max is so small where αuv = 2π(v−u)l λ that it approaches zero. The trigonometric functions of the direction angle estimation error in (A3) can be expanded by using second-order Taylor expansion. Thus we can get the following approximate expressions sin(∆θ BEk ) ≈ ∆θ BEk ,

(A4)

∆θ 2BEk

. (A5) 2 Substituting (A4) and (A5) into (A3), we have the expression of HBEk (u, v) as cos(∆θ BEk ) ≈ 1 −

Now the task is to derive the analytic expressions of Υ1k (u, v) and Υ2k (u, v). For this purpose, we first expand the Cosine function in (A11) into the following form k cos(ξu,v )=    ∆θ 2BEk  cos αuv cos(θˆBEk ) cos αuv sin(θˆBEk )∆θ BEk 2    ∆θ 2BEk  sin αuv sin(θˆBEk )∆θ BEk . − sin αuv cos(θˆBEk ) 2 (A13) Then the second-order Taylor series is used to approximate each term in (A13), and (A13) can be rewritten as k cos(ξu,v )= 2 sin2 (θˆ 2 2  αuv αuv cos2 (θˆBEk )∆θ 4BEk   BEk )∆θ BEk  1− 1− 8 2   ∆θ 2BEk   − αuv cos(θˆBEk ) αuv sin(θˆBEk )∆θ BEk . 2 (A14) Expand (A14) as a sum of several terms and omit the higher order terms about ∆θ BEk , then (A14) can be further written as

k cos(ξu,v )≈1−

∫  ∆θ 2BEk gBEk jαuv cos(θˆBE ) ∆θmax n k exp − jαuv cos(θˆBEk ) e N 2  o−∆θmax + sin(θˆBEk )∆θ BEk f (∆θ BEk)d(∆θ BEk ). (A6) According to Euler’s formula, (A6) can be further written as ∫ ∆θmax   k k HBEk (u, v) =Hˆ BEk(u, v) cos(ξu,v )− j sin(ξu,v ) (A7) −∆θmax f (∆θ BEk)d(∆θ BEk ), where

gBEk jαu v cos(θˆBE ) k , e (A8) N denotes the u-th row and the v-th column entry of h(θˆBEk )h H (θˆBEk ), and Hˆ BEk(u, v) =

  ∆θ 2BEk k = αuv cos(θˆBEk ) + sin(θˆBEk )∆θ BEk . ξu,v 2 For simplify, (A7) can be written as HBEk (u, v) = Υ1k (u, v) − jΥ2k (u, v),

Υ2k (u, v) = Hˆ BEk(u, v)

∆θmax

−∆θmax

∫

∆θmax

−∆θmax

k cos(ξu,v )f (∆θ BEk)d(∆θ BEk ),

(A11) k sin(ξu,v )f (∆θ BEk)d(∆θ BEk ).

(A12)

2

(A15) .

(A16)

Combining (A11), (A15) and (A16), we can obtain the expression of Υ1k (u, v) at the top of the next page. Note that in (A17), ∆θ 1 = ∆θ max + µ, ∆θ 2 = ∆θ max − µ and the step (b) results from the following equations [31] ∫

x

√ e−q

2t2

x

∫

te−q

x

∫

2 −q 2 t 2

t e

∫ 0

x

t 3 e−q

2t2

π erf(qx), 2q

(A18)

2 2 1 (1 − e−q x ), 2 2q

(A19)

dt =

0

0

∫

∆θ 3BEk

1 cos(∆θ BEk − µ) ≈ 1 − (∆θ BEk − µ)2 . 2

0

(A10)

sin(θˆBEk ) cos(θˆBEk )

Since the direction angle estimation error ∆θ BEk is assumed to be very small, µ is also a small value near zero. Therefore, we can expand the trigonometric function in the Von Mises PDF as

(A9)

where Υ1k (u, v) = Hˆ BEk(u, v)

2

2 αuv

−

HBEk (u, v) =

2 sin2 (θˆ 2 αuv BEk )∆θ BEk

dt =

√  2 2 1  π dt = 3 erf(qx) − qxe−q x , 2 2q

2t2

dt =

 1  2 2 −q 2 x 2 1 − (1 + q x )e , 2q4

where erf(x) represents the error function defined as ∫ x 2 2 erf(x) = √ e−t dt. π 0

(A20)

(A21)

(A22)

12

2 sin2 (θˆ αuv α2 sin(θˆBEk ) cos(θˆBEk ) 3  BEk ) ∆θ 2BEk − uv ∆θ BEk f (∆θ BEk )d(∆θ BEk ) 2 2 −∆θmax r r r  ˆ BEk (u, v)eκ n π  (b) H κ κ ≈ ∆θ 1 ) + erf( ∆θ 2 ) erf( 2πI0 (κ) 2κ 2 2 r r √     2 2 2 ˆ α sin (θ BEk ) (1 + κ µ ) π κ κ 2 2 κ κ − uv erf( ∆θ ) + erf( ∆θ 2 ) − ∆θ 2 e− 2 ∆θ1 − ∆θ 1 e− 2 ∆θ2 √ 1 2κ 2 2 2κ r r √    2 2 ˆ ˆ αuv sin(θ BEk ) cos(θ BEk ) (3 + κ µ )µ π κ κ erf( ∆θ 1 ) + erf( ∆θ 2 ) − √ 2κ 2 2 2κ  o κ κ 2 2 2 2 + ( + ∆θ 12 − 3µ)e− 2 ∆θ1 − ( + ∆θ 22 + 3µ)e− 2 ∆θ2 . κ κ

Υ1k (u, v) ≈ Hˆ BEk (u, v)

∫

∆θmax



1−

(A17)

 cos(θˆBEk ) 2 ∆θ BEK + αuv sin(θˆBEk )∆θ BEK f (∆θ BEk )d(∆θ BEk ) 2 −∆θmax r r √   Hˆ BEk (u, v)eκ n αuv cos(θˆBEk )  (1 + κ µ2 ) π  κ κ 2 2 κ κ ≈ ∆θ 1 ) + erf( ∆θ 2 ) − ∆θ 2 e− 2 ∆θ1 − ∆θ 1 e− 2 ∆θ2 erf( √ 2πI0 (κ) 2κ 2 2 2κ r r o   rπ  κ κ 2 2 κ κ 1 erf( (A24) ∆θ 1 ) + erf( ∆θ 2 ) + (e− 2 ∆θ1 − e− 2 ∆θ2 ) . + αuv sin(θˆBEk ) µ 2κ 2 2 κ

Υ2k (u, v) ≈ Hˆ BEk (u, v)

∫

∆θmax

α

uv

k ) in (A12) can be approximately expressed Similarly, sin(ξu,v

as k sin(ξu,v )≈  α cos(θˆ ) uv BEk



+ 1−



2 sin2 (θˆ 2 αuv BEk )∆θ BEk 

∆θ 2BEk 1 − 2 2 cos2 (θˆ 4 αuv BEk )∆θ BEk  

2 αuv sin(θˆBEk )∆θ BEk

 (A23)

8 αuv cos(θˆBEk ) 2 ≈ ∆θ BEk + αuv sin(θˆBEk )∆θ BEk . 2

Then, combining (A12), (A16) and (A23), we can obtain the expression of Υ2k (u, v) at the top of this page. Now, combining (A10), (A17) and (A24), we can finally obtain the analytic expression of HBEk (u, v) and the proof is completed.

direction angle estimation error in (B1) by using second-order Taylor expansion, and (B1) is transformed as follows h p (θ BEk ) =   r ∆θ 2 k )−sin(θˆ gBEk −jαp cos(θˆBE k )(1− BE BE k )∆θ BE k 2 e N   r ∆θ 2 k +sin(θˆ gBEk −jαp cos(θˆBE ) jαp cos(θˆBE k ) BE BE k )∆θ BE k 2 ke = e N   = h p (θˆBEk ) cos(ξ pk ) + j sin(ξ pk ) = Γ1k (p) + jΓ2k (p). (B2) where h p (θˆBEk ) =

r

gBEk −jαp cos(θˆBE ) k e N

(B3)

denotes the p-th entry of h(θˆBEk ), and A PPENDIX B The p-th entry of h(θ BEk ) can be expressed as h p (θ BEk ) = r gBEk −j2π (p−( N +1)/2)l cos(θˆBE +∆θ BE ) λ k k e (B1) N   r gBEk −jαp cos(θˆBE k ) cos(∆θ BE k )−sin(θˆBE k ) sin(∆θ BE k ) = e , N where αp = 2π(p−(Nλ +1)/2)l . Since ∆θ max is a minimum value nears zero, we can expand the trigonometric functions of the

  ∆θ 2BEk + sin(θˆBEk )∆θ BEk , ξ pk = αp cos(θˆBEk ) 2

(B4)

Γ1k (p) = h p (θˆBEk ) cos(ξ pk ),

(B5)

Γ2k (p) = h p (θˆBEk ) sin(ξ pk ).

(B6)

Then performing the same operation on the trigonometric functions in (B5) and (B6) as shown in APPENDIX A, we

13

finally obtain the expression of h p (θ BEk ), h p (θ BEk ) =   h p (θˆBEk ) cos(ξ pk ) + j sin(ξ pk ) = h p (θˆBEk ) + jαp sin(θˆBEk )h p (θˆBEk )∆θ BEk  1  (B7) + αp j cos(θˆBEk ) − αp sin2 (θˆBEk ) h p (θˆBEk )∆θ 2BEk 2 1 − αp2 sin(2θˆBEk )h p (θˆBEk )∆θ 3BEk 4 = h p (θˆBEk ) + ∆hkp, where ∆hkp = jαp sin(θˆBEk )h p (θˆBEk )∆θ BEk  1  + αp j cos(θˆBEk ) − αp sin2 (θˆBEk ) h p (θˆBEk )∆θ 2BEk (B8) 2 1 2 − αp sin(2θˆBEk )h p (θˆBEk )∆θ 3BEk 4 is the p-th entry of the channel estimation error of the k-th eavesdropper. Since the direction angle estimation error satisfies ∆θ BEk ∈ [−∆θ max, ∆θ max ], and ∆θ max is a minimum value nears zero, then we can omit the terms of the second-order and the thirdorder of ∆θ BEk in (B8). Therefore, ∆hkp can be approximately rewritten as ∆hkp ≈ jαp sin(θˆBEk )h p (θˆBEk )∆θ BEk ,

(B9)

and the modulus of ∆hkp is |∆hkp | = | jαp sin(θˆBEk )h p (θˆBEk )||∆θ BEk |.

(B10)

Apparently, |∆hkp | is a monotonically increasing function with respect to |∆θ BEk |. Therefore, |∆hkp | can take the maximum value when and only when ∆θ BEk equals to the maximum value ∆θ max or the minimum value −∆θ max , i.e., |∆hkp | ≤ | jαp sin(θˆBEk )h p (θˆBEk )||∆θ max |.

(B11)

|∆hkp |

It shows that is bounded, then each entry in |∆hBEk | is also bounded, thus |∆hBEk | is norm-bounded, which means that the channel estimation error of the eavesdroppers are norm-bounded and the proof is completed. We use εk to represent the maximum value of the bound of the norm of the channel estimation error of the k-th eavesdropper, then v u tÕ N |αp sin(θˆBEk )h p (θˆBEk )∆θ max | 2 . (B12) εk = p=1

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