## I Introduction

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, 4, 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 as a significant complement to cryptographic techniques [6, 7, 8, 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, 11, 12, 13, 14, 15, 16, 17, 18, 19, 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 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 non-convex 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. , , , and denote the transpose, the conjugate transpose, the trace, the rank and the real part of the matrix A, respectively. denotes that A is a positive semi-definite matrix. and denote the module of the scalars and the Frobenius norm of the matrices. , , and denote the statistical expectation, the logarithm to base 2, the logarithm to base 10 and the natural logarithm, respectively. denotes the inner product of the vector x and y. denotes the set of all complex matrices. denotes an identity matrix.

denotes a circularly symmetric complex Gaussian random variable

with zero mean and variance

.## Ii system model and problem formulation

### Ii-a Channel Model

In this paper, the DM transmitter is assumed to be equipped with uniform isotropic array. The array is composed of elements with an adjacent distance of , while the direction angle between the transmit antenna array and the receiver is denoted by . The normalized steering vector for the transmit antenna array is expressed as

(1) |

where is

(2) |

where 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

(3) |

where is the path loss in dB at a given distance , is the frequency of the transmit signal radio and 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

(4) | ||||

where is the distance between the transmit antenna array and the receiver, and denotes the path loss between the transmit antenna array and the receiver.

### Ii-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 transmit antennas, () single-antenna destination users (), and single-antenna eavesdroppers (), wishing to wiretap the confidential information sent from the BS to the destination users. The total transmit power of the BS is , and the direction angle between the BS and each destination user is (). The transmit beamforming technique is employed by the BS to steer the information signal beams toward the destination users. The direction angle between the BS and each eavesdropper is (). 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 () 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 denote the signal vector formed by the independent data streams sent from the BS to the destination users with . Let and denote the information beamforming matrix and the AN beamforming matrix, respectively. The total transmit signals of the BS can be expressed as

(5) |

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

(6) | ||||

where the first term of the right hand side of (6) is the useful signal received by , 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 (AWGN) with .

Similarly, the signal received at the -th eavesdropper is

(7) | ||||

where the right hand side of (7) is composed of the -th confidential message intercepted by , the messages sent to the other (-1) destination users that are intercepted by , the AN to disturb and the AWGN satisfying .

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

(8) | ||||

where denotes the system sum secrecy rate, and denote the signal-to-interference-plus-noise ratio (SINR) at the destination user and the eavesdropper , respectively.

Then, according to (6), the SINR at the -th destination user is

(9) |

And according to (7), the SINR at the -th eavesdropper intending to intercept the -th destination user is

(10) |

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 (II-B) on the following page.

(11) |

### Ii-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 direction angle between the BS and the -th eavesdropper is expressed as

(12) |

where is the estimated direction angle between the BS and the -th eavesdropper, denotes the estimation error which is assumed to follow the Von Mises distribution [27] over the interval . Note that 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

(13) |

where represents the estimation error of the direction angle, denotes the mean of , denotes the concentration parameter and controls the width of , and with is the modified Bessel function of the first kind with order computed by

(14) |

## 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 (II-B), the item can be expressed as

(15) | ||||

To simplify the expression, we define

(16) |

where . The necessity of the expectation in (16) is explained as follows. If we remove the expectation, since is a random variable, then would be random as well as the objective function of (II-B), and it would be very difficult to find a solution to (II-B).

Let denote the -th row and the -th column entry of , then can be written as

(17) |

The detailed procedure for deriving and based on the Von Mises distribution is shown in Appendix A. And and are expressed as (Appendix A) and (Appendix A), respectively.

In order to make (II-B) more tractable, we substitute the new matrix as well as the positive semi-definite matrix variables , into (II-B). Then the optimization problem (II-B) can be rewritten in a simple form as shown in (III) at the top of the next page.

(18) |

By solving the problem (III), we can obtain the optimal information beamforming matrices and the AN beamforming matrix Q. However, the objective function in (III) 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 (III) 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 (III), i.e.,

(19) |

(20) |

(21) |

(22) |

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

(23) |

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

(24a) | |||

(24b) | |||

(24c) | |||

(24d) | |||

(24e) | |||

(24f) | |||

(24g) |

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, and can be linearized based on the first-order Taylor approximation as

(25) |

(26) |

where

(27) |

(28) |

are the points around which the approximation are made. So the problem (II-B) eventually becomes

(29a) | |||

(29b) | |||

(29c) | |||

(29d) | |||

(29e) | |||

(29f) | |||

(29g) |

As a convex problem, (29) can be solved iteratively by the Algorithm 1 using the CVX optimization software. According to the definition, has to satisfy , so the optimal 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) |

1: Given and randomly that |

are feasible to (29); |

2: Set , and set =0; |

3: Repeat |

4: Substituting and into (27) and (28) yields |

and ; |

5: Increment ; |

6: Substituting and into (29) yields the optimal solution |

and ; |

7: Until convergence; |

8: Obtain and by decomposition of |

for all in the case of ; |

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

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 -th eavesdropper can be expressed as

(30) |

where is the estimated channel between the BS and the -th eavesdropper, is the channel estimation error, and is the bound of the norm of the channel estimation error of the -th eavesdropper.

Then we derive the lower bound of the objective function in (II-B). For that purpose, we first derive the upper and lower bounds of and in (II-B). The upper bound of is

(31) | ||||

Similarly, the lower bound of is

(32) | ||||

It should be noted that in (31) and (32) the second-order error term is omitted because it is quite small compared to the other two terms. And note that the step () in (31) results from the following derivation process. According to the Cauchy-Schwarz inequality theorem, we have

(33) |

Since the inequality

(34) |

is obviously true, then we can obtain the following inequality

(35) |

The inequality in (35) holds with equality when and only when x and y are linearly dependent. The derivation of the step () in (31) is completed.

Similarly, the upper bound of is

(36) | ||||

and the lower bounds of is

(37) | ||||