# Robust Synthesis Algorithm for Directional Modulation Signal with Array Manifold Vectors Uncertainty

Authors: Tao Hong
Journal: International Journal of Antennas and Propagation
 (2020)

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## Abstract

Directional modulation (DM) has become a new research hotspot of physical layer security (PLS) communication at the transmitter side. In this paper, we propose a robust synthesis algorithm for DM signal under the condition of the array manifold vectors perturbation. This algorithm optimizes the constraints of sidelobe level and Euclidean distance of constellation points by considering the worst case performance of array manifold vectors. Furthermore, we also design an active constellation extension (ACE) method to relax the equality constraint of desired modulation symbols into a robust inequality constraint at the desired direction. These constraints can be reformulated in a convex form withl2 and l∞ regularization, which are computationally tractable. Simulation results show better performance of the proposed robust algorithm compared with the benchmark synthesis algorithms in the presence of array manifold vectors uncertainty.


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## Body

## 1. Introduction

With the development of wireless communication technology, an increasing amount of data is transmitted over wireless networks. However, the broadcasting nature of the wireless channel makes the information vulnerable to intercept. Currently, encryption techniques are used to guarantee transmit information confidential without considering the physical properties of the wireless communication channel. Due to the development of computing techniques (e.g., cloud computing), these classics security solutions are becoming ever more challenging. Considering the physical properties of the wireless communication channel, physical layer security (PLS) is a recently developed technology to guarantee secure transmission at the physical layer regardless of the eavesdropper’s computational capability. Thus, PLS techniques have drawn a lot of attention from information theory to security engineering by the research community [1].In recent years, directional modulation (DM) is a popular transmitter design technology in the field of PLS communication. The fundamental concept of DM technology was first introduced in [2], which implies that the modulation function can be achieved in the RF device and antenna level of the transmitter compared with the traditional wireless communication transmitter at the baseband. In this way, the constellation points of the DM signal hold their location in relation to each other as traditional baseband modulation symbols in the desired direction, while the constellation points change into disorganized in the undesired directions. As a result, eavesdroppers’ communication performance is degraded owing to the undesired directional information even if the similar signal power is received by eavesdroppers. Following this idea, many papers presented different DM transmitter structures and synthesis algorithms to meet the requirement of DM signal. Daly proposed a typical DM transmitter structure using a phased array in [3]. The modulation symbols were synthesized via a genetic algorithm (GA) by setting phase shift values at the desired direction. In [4], a dual-beam DM synthesis method was presented by employing a corner reflector. In [5], Zhu investigated a switching transmit DM signal by a time-modulated-array. Ding proposed a phase-conjugation DM signal by a retrodirective array in [6]. With the development of DM signal, DM transmitter employing a phased array turns into a hotspot in this research filed for the reasons of the synthesis principle [3, 7], the mature hardware conditions [8, 9], and the general metrics [10]. On this basis, Ding proposed two DM signal synthesis algorithms based on Fourier transform by considering the radiation patterns of DM signal: one is used to improve the power efficiency (PE) of DM system by optimizing the sidelobe of radiation patterns in [11] and the other is used to reduce interference with other system by generating a null in the sidelobe of radiation patterns in [12]. For the DM system with the practical digital phase shifters, Li derived a closed-form expression of the SINR performance loss associated with the number of quantization bits [13]. To improve the secrecy rate performance of DM technique, Wan designed a null-space projection beamforming scheme and derived its closed-form expression of optimal PA strategy between the confidential messages and the artificial noise projection matrix in [14]. Meanwhile, Yu et al. [15] proposed a general power iterative- (GPI-) based beamforming scheme to maximize the secrecy rate. For the DM signal with high peak-to-average-power ratio (PAPR) characteristic, He proposed an iterative projection method to guarantee secrecy performance and low PAPRs simultaneously in [16]. In [17], Chen tested a two-step peak clipping scheme to specifically decrease the DM signal’s PAPRs and presented a highly efficient DM transmitter with digital predistortion. Shu et al. designed a practical wireless transmission scheme to transmit confidential messages to the desired user securely and precisely by the joint use of multiple techniques, including artificial noise projection, phase alignment, and random subcarrier selection (RSCS) based on orthogonal frequency division multiplexing (OFDM) and DM in [18]. Zhang et al. presented a multicarrier DM signal and also pointed out the PAPR problem of proposed DM signal in [19]. To improve the efficient DM transmitter, the author also proposed a widebeand beam and phase pattern formation by Newton’s method (WBPFN) to reduce the PAPR value of proposed DM signal. To address the problem of DM signal with multiuser, Shi investigated a multiuser DM signal via a phased array with 2-bit phase shifters in [20]. Ding and Fusco [21] proposed an orthogonal vector method for synthesis multiuser DM signal. In [22], Xie et al. presented a dynamic multiuser DM signal based on an artificial-noise-aided zero-forcing synthesis approach. Hafez proposed a discrete Fourier-transform- (DFT-) based multiuser DM signal synthesis algorithm that divided the spatial dimension into orthogonal narrow subbeams in [23]. Christopher tested a multiuser directional modulation with artificial noise aided based on an iterative convex optimization in [24]. In [25], Shu et al. presented a multigroup DM signal by two different secure schemes in a multicast scenario. To simplify RF structure of DM transmitter, Zhang et al. proposed a compressive sensing-based sparse array synthesis algorithm for DM signal in [26]. In [27], Hong et al. proposed a sparse array synthesis algorithm for DM signal not only synthesizing a sparse array but also improving multiple metrics of DM system. In the aforementioned works about DM signal synthesis, algorithms by a phased array were mainly supposed that DM transmitter has the information of desired direction accurately. However, this assumption does not always hold in the practical scenario. In [28], a robust synthesis method for DM signal with imperfect information of desired direction was proposed to minimize the distortion of the constellation points along the desired direction based on conditional minimum mean square error. This robust synthesis method was also extended to the multiuser scenario in [29], MU-MIMO system in [30], and DM-based relay networks in [31]. For the performance of DM in multipath channel, the combination of DM technology and MIMO system has developed application scenarios from free space environment to multipath environment [32, 33]. In addition, DM signal was also employed in the secure wireless information and power transfer (SWIPT) system [34] and frequency diverse arrays to enlarge the application fields [35].The robust algorithm for synthesis DM signal was first proposed in [28] to solve the problem of the uncertain information of the desired direction compared with the transmitter with the precise information of the desired direction in [2–27]. Following this original work, the robust DM signal synthesis algorithm was extended to the multiuser scenario in [29], MU-MIMO system in [30], and DM-based relay networks in [31]. However, these following works still focused on the same uncertain factor in DM transmitter. For the DM via a phased array, the information of array manifold vectors is another uncertainty factor to degrade the performance of DM signal. The array manifold vectors, which include the spatial characteristics of the phased array, are assumed to be known exactly in the previous DM synthesis methods via a phased array [2–31]. However, the actual value of the array manifold may differ from the assumed value; i.e., the knowledge of the array manifold vectors can be imprecise, which often happen in practice. This manifold vectors mismatch or uncertainties can be caused by elements sensitivity mismatch, element position perturbations, channel and phase mismatch, mutual coupling between the elements, etc., and may cause undesirable far-field pattern degradation [36]. Therefore, we focus on the problem of the phased array manifold vectors with uncertain information for DM signal synthesis in this paper. To solve this problem, a robust synthesis algorithm via a phased array for DM signal is proposed based on convex optimization. This algorithm optimizes the constraints of sidelobe level and Euclidean distance of constellation points by considering the worst case performance of array manifold vectors. Furthermore, we also design an active constellation extension (ACE) method to relax the equality constraint of desired modulation symbols into a robust inequality constraint at the desired direction. These constraints of the proposed robust algorithm can be reformulated by l2 and l∞ regularization with a convex form, which can find the solution efficiently with a second-order cone programming (SOCP) solver. Simulation results show better performance of this robust algorithm in the presence of array manifold vectors uncertainty.The remainder of this paper is organized as follows.Section 2 introduces the robust synthesis algorithm for the DM signal. Section 3 describes the synthesis results compared with the benchmark problem. Section 4 draws concluding remarks.


### 1.1. Notations

Scalar variables are denoted by italic symbols. Vectors are denoted by boldface italic symbols. Given a complex numberx, x, Rex, and Imx denote the modulus, real part, and imaginary part of x. Given a complex vector y, y and y∞ denote the Euclidean norm and infinite norm of y, respectively. Given a complex vector y, y∗, yT, and yH denote the conjugate, transpose, and conjugate transpose of y.


## 1.1. Notations

Scalar variables are denoted by italic symbols. Vectors are denoted by boldface italic symbols. Given a complex numberx, x, Rex, and Imx denote the modulus, real part, and imaginary part of x. Given a complex vector y, y and y∞ denote the Euclidean norm and infinite norm of y, respectively. Given a complex vector y, y∗, yT, and yH denote the conjugate, transpose, and conjugate transpose of y.


## 2. Robust Synthesis Algorithm for Directional Modulation Signal

### 2.1. Problem Formulation for Directional Modulation Signal

Without loss of generality, we suppose a DM transmitter equipped with a linear phased array. The structure of DM transmitter is shown in Figure1 which consists of N isotropic elements along the x-axis. The distance between two elements is denoted by dx. Therefore, the array factor of this DM transmitter is presented as follows [3, 11]:(1)fθ=wHAθ=∑n=1n=Nwn∗anθwithanθ=ej2π/λxncosθ,where θ denotes the azimuth angle, Aθ=a1θ,a2θ,…,aNθT denotes the N-dimensional ideal (presumed) manifold vector, w=w1,w2,…,wNT denotes the weighted vector, wn is the complex weighted of the nth element, and xn denotes the position coordinate of the nth element.Figure 1
Structure of a DM transmitter.The problem of DM signal synthesis by a phased array can be expressed as one of the types of constrained optimization, in which the constraints are designed to guarantee the performance of DM signal in the desired direction and the undesired directions, while the objective function is chosen to obtain optimal metrics of DM transmitter. To compare with benchmark problems, we assume that QPSK signal with setΓi∈ej1/4π,ej3/4π,ej5/4π,ej7/4π,i=1,2,3,4 is synthesized at the desired direction. Thus, the constraint of radiation patterns satisfies as(2)g1:fiθdwi=Γi,i=1,2,…,4,where wi is the ith weighted vector for ith QPSK symbol in the symbol set Γ and θd is the desired direction. For the constraint of radiation patterns in the sidelobe regions as shown in Figure 2(a), we define three nonoverlapping regions ΘSL1, ΘSL2, and ΘSB, where ΘSL1, ΘSL2, and ΘSB denote the left sidelobe region, right sidelobe region, and mainlobe region, respectively. The constraint of sidelobe level is upper-bounded by an envelope u1, which is written as(3)g2:maxp=1,…,PwiHAθp=wiHAθp∞≤u1,θp∈ΘSL1,maxq=1,…,QwiHAθq=wiHAθq∞≤u1,θq∈ΘSL2,where θ1,…,θP∈ΘSL1 and θ1,…,θQ∈ΘSL2 are the angular grid points to approach the sidelobe region ΘSL1 and ΘSL2. Furthermore, this constraint is same as the conception of power-mask in [11, 12], which is useful for restraining the interference with the other system in the sidelobe region and improving the PE of DM system. The security performance of static DM system depends on the performance of bit-error-ratio (BER) in the sidelobe region. In [3, 10], two types of eavesdropping receiver were proposed to evaluate the security performance of DM signal: one is a standard QPSK receiver that decodes received QPSK symbols based on which quadrant the constellation points belong to; the other is a more advanced DM receiver that can carry out the minimum Euclidean distance demodulation for the eavesdropper. We consider that the eavesdropper equips an advanced receiver in our system model. Thus, we design a constraint of Euclidean distance of synthesized constellation points to ensure the performance of BER in the sidelobe region, which is denoted by the infinite norm form with the upper-bounded u2 as follows (Figure 2(b)):(4)g3:maxp=1,…,Pf1θp−f2θp=w1HAθp−w2HAθp∞≤u2,θp∈ΘSL1,maxq=1,…,Qf3θq−f4θq=w3HAθq−w4HAθq∞≤u2,θq∈ΘSL2.Figure 2
The constraint of (a) sidelobe and (b) Euclidean distance for DM synthesis algorithm.
(a)
(b)The optimization problem can be formulated to improve different metrics of DM system according to different application scenarios such as, PE of DM system [11], radiation power in the sidelobe region [12], and sparse array [22]. In this paper, we consider the PE of DM system as the objective function to compare the performance with the benchmark synthesis algorithm in [11]. Thus, the optimization problem for DM signal is written as follows:(5)minwi∈CN∑i=14wis.t.g1:fiθdwi=Γi,i=1,2,…,4g2:wiHAθp∞≤u1,i=1,2,3,4,θp∈ΘSL1wiHAθq∞≤u1,i=1,2,3,4,θq∈ΘSL2g3:w1HAθp−w2HAθp∞≤u2,θp∈ΘSL1w3HAθq−w4HAθq∞≤u2,θq∈ΘSL2,where the constraint g1 means that the desired modulation symbols are synthesized at the desired direction, the constraint g2 is used to restrain the interference with other system in the sidelobe region and improve the PE of DM system by bounding the peak value of the sidelobes, and the constraint g3 is used to improve the security performance by limiting Euclidean distance of synthesized constellation points in the sidelobes. In (5), the objective function and constraints denoted by l2 and l∞ are convex optimization form, which can find the solution with a SOCP solver such as SeDuMi. A solution of the proposed optimization problem is regarded as a nominal optimal beamformer with the metrics of PE of the DM system.


### 2.2. Robustness Design

According to aforementioned presentation, the system only has an imprecise information of the actual manifold vector in practical scenarios. The actual manifold vectorA˜θ=a˜1θ,a˜2θ,…,a˜NθT with uncertainties can be expressed as follows:(6)A˜θ=Aθ+Δθ,where Δθ is a random complex vector that describes the uncertainty of array manifold vector. The only available information of A˜θ for transmitter design is that it belongs to an uncertainty set Uθ because of Δθ with the random characteristics. Without the loss of generality, we assume that array manifold vector distortion Δθ is upper-bounded by a preknown constant ε>0 as Δθ≤ε. Therefore, the uncertainty set of A˜θ can be written as follows:(7)Uθ≜uθ:uθ=Aθ+Δθ,Δθ≤ε,where Uθ is an ellipsoid because of the imprecise information of the array manifold vector.To guarantee the robustness of the constraintg1, we propose an ACE method to transform g1 into a robust inequality constraint. For traditional DM signal, the synthesized constellation points hold their location in relation to each other as the digital baseband modulation symbols. If we increase the Euclidean distance of constellation points, it is useful for the BER performance of desired receiver. The method to increase the Euclidean distance of constellation points is shown in Figure 3. Following this idea, taking QPSK symbol Γ1 as example, the constraint g1 is transformed as follows:(8)g˜1forΓ1:Rew1HA˜θd≥22Imw1HA˜θd≥22.Figure 3
The principle of the proposed ACE method, where the black dots denote the standard QPSK symbols, the red dots denote an example of synthesis symbols based on ACE method, the gray area is the extension area of the proposed ACE method, a black line with arrows at both ends denotes the minimum Euclidean distance of standard QPSK symbols at the desired direction, and a red line with arrows at both ends denotes an example of the minimum Euclidean distance of the proposed ACE method.Substituting (6) into (8), we have(9)g˜1forΓ1:Rew1HA˜θd+Rew1HΔθd≥Rew1HAθd−εw1H≥22Imw1HAθd+Imw1HΔθd≥Imw1HAθd−εw1H≥22.In (9), we take the worst case performance of the array manifold vector into consideration to relax the inequality constraint (8) as a convex form. In a similar way for other QPSK symbols, we obtain the robust constraint o as follows:(10)g˜1:Rew1HAθd−εw1H≥22,i=1,4Rew1HAθd+εw1H≤−22,i=2,3Imw1HAθd−εw1H≥22,i=1,2Imw1HAθd+εw1H≤−22,i=3,4.It is important that the designed ACE method increase the Euclidean distance of the synthesized constellation points at the cost of improving transmit power of the DM system. Therefore, the far-field patterns via the proposed synthesis algorithm must be normalized to compare BER performance with benchmark synthesis algorithms.The sidelobe level constraint ofg2 for the left sidelobe region can be transformed as follows:(11)g˜2forΘSL1:maxp=1,…,PwiHA˜θp=maxp=1,…,PwiHA˜θp+wiHΔθp≤maxp=1,…,PwiHA˜θp+εwiH=wiHA˜θp∞+εwiH≤u1.In a similar way for the right sidelobe region, the robust constraintg˜2 is written as follows:(12)g˜2:wiHA˜θp∞+εwiH≤u1,i=1,2,3,4,θp∈ΘSL1wiHA˜θq∞+εwiH≤u1,i=1,2,3,4,θq∈ΘSL2.The BER constraint ofg3 for the left sidelobe region can be transformed as follows:(13)g˜3forΘSL1:maxp=1,…,Pf1θp−f2θp=maxp=1,…,Pw1HA˜θp−w2HA˜θp≤w1HAθp−w2HAθp∞+εw1H+εw2H≤u2.In a similar way, the robust constraintg˜3 is shown as follows:(14)g˜3:w1HAθp−w2HAθp∞+εw1H+εw2H≤u2,θp∈ΘSL1w3HAθq−w4HAθq∞+εw3H+εw4H≤u2,θq∈ΘSL2.Based on the aforementioned analysis, the proposed robust optimization problem for DM signal can be written as follows:(15)minwi∈CN∑i=14wis.t.g˜1,g˜2,g˜3.The optimization problem (15) is regarded as the l2 and l∞ regularization with convex form. We can find the solution with an SOCP solver. In this section, we consider the uncertainty information of array manifold vectors in the DM signal synthesis algorithm. The worst case performance of the array manifold vectors is considered to relax the constraints. In this way, the performance of sidelobe level and security will be degraded to tolerate the uncertain information of array manifold vectors compared with the nominal optimal beamformer for the DM system.


## 2.1. Problem Formulation for Directional Modulation Signal

Without loss of generality, we suppose a DM transmitter equipped with a linear phased array. The structure of DM transmitter is shown in Figure1 which consists of N isotropic elements along the x-axis. The distance between two elements is denoted by dx. Therefore, the array factor of this DM transmitter is presented as follows [3, 11]:(1)fθ=wHAθ=∑n=1n=Nwn∗anθwithanθ=ej2π/λxncosθ,where θ denotes the azimuth angle, Aθ=a1θ,a2θ,…,aNθT denotes the N-dimensional ideal (presumed) manifold vector, w=w1,w2,…,wNT denotes the weighted vector, wn is the complex weighted of the nth element, and xn denotes the position coordinate of the nth element.Figure 1
Structure of a DM transmitter.The problem of DM signal synthesis by a phased array can be expressed as one of the types of constrained optimization, in which the constraints are designed to guarantee the performance of DM signal in the desired direction and the undesired directions, while the objective function is chosen to obtain optimal metrics of DM transmitter. To compare with benchmark problems, we assume that QPSK signal with setΓi∈ej1/4π,ej3/4π,ej5/4π,ej7/4π,i=1,2,3,4 is synthesized at the desired direction. Thus, the constraint of radiation patterns satisfies as(2)g1:fiθdwi=Γi,i=1,2,…,4,where wi is the ith weighted vector for ith QPSK symbol in the symbol set Γ and θd is the desired direction. For the constraint of radiation patterns in the sidelobe regions as shown in Figure 2(a), we define three nonoverlapping regions ΘSL1, ΘSL2, and ΘSB, where ΘSL1, ΘSL2, and ΘSB denote the left sidelobe region, right sidelobe region, and mainlobe region, respectively. The constraint of sidelobe level is upper-bounded by an envelope u1, which is written as(3)g2:maxp=1,…,PwiHAθp=wiHAθp∞≤u1,θp∈ΘSL1,maxq=1,…,QwiHAθq=wiHAθq∞≤u1,θq∈ΘSL2,where θ1,…,θP∈ΘSL1 and θ1,…,θQ∈ΘSL2 are the angular grid points to approach the sidelobe region ΘSL1 and ΘSL2. Furthermore, this constraint is same as the conception of power-mask in [11, 12], which is useful for restraining the interference with the other system in the sidelobe region and improving the PE of DM system. The security performance of static DM system depends on the performance of bit-error-ratio (BER) in the sidelobe region. In [3, 10], two types of eavesdropping receiver were proposed to evaluate the security performance of DM signal: one is a standard QPSK receiver that decodes received QPSK symbols based on which quadrant the constellation points belong to; the other is a more advanced DM receiver that can carry out the minimum Euclidean distance demodulation for the eavesdropper. We consider that the eavesdropper equips an advanced receiver in our system model. Thus, we design a constraint of Euclidean distance of synthesized constellation points to ensure the performance of BER in the sidelobe region, which is denoted by the infinite norm form with the upper-bounded u2 as follows (Figure 2(b)):(4)g3:maxp=1,…,Pf1θp−f2θp=w1HAθp−w2HAθp∞≤u2,θp∈ΘSL1,maxq=1,…,Qf3θq−f4θq=w3HAθq−w4HAθq∞≤u2,θq∈ΘSL2.Figure 2
The constraint of (a) sidelobe and (b) Euclidean distance for DM synthesis algorithm.
(a)
(b)The optimization problem can be formulated to improve different metrics of DM system according to different application scenarios such as, PE of DM system [11], radiation power in the sidelobe region [12], and sparse array [22]. In this paper, we consider the PE of DM system as the objective function to compare the performance with the benchmark synthesis algorithm in [11]. Thus, the optimization problem for DM signal is written as follows:(5)minwi∈CN∑i=14wis.t.g1:fiθdwi=Γi,i=1,2,…,4g2:wiHAθp∞≤u1,i=1,2,3,4,θp∈ΘSL1wiHAθq∞≤u1,i=1,2,3,4,θq∈ΘSL2g3:w1HAθp−w2HAθp∞≤u2,θp∈ΘSL1w3HAθq−w4HAθq∞≤u2,θq∈ΘSL2,where the constraint g1 means that the desired modulation symbols are synthesized at the desired direction, the constraint g2 is used to restrain the interference with other system in the sidelobe region and improve the PE of DM system by bounding the peak value of the sidelobes, and the constraint g3 is used to improve the security performance by limiting Euclidean distance of synthesized constellation points in the sidelobes. In (5), the objective function and constraints denoted by l2 and l∞ are convex optimization form, which can find the solution with a SOCP solver such as SeDuMi. A solution of the proposed optimization problem is regarded as a nominal optimal beamformer with the metrics of PE of the DM system.


## 2.2. Robustness Design

According to aforementioned presentation, the system only has an imprecise information of the actual manifold vector in practical scenarios. The actual manifold vectorA˜θ=a˜1θ,a˜2θ,…,a˜NθT with uncertainties can be expressed as follows:(6)A˜θ=Aθ+Δθ,where Δθ is a random complex vector that describes the uncertainty of array manifold vector. The only available information of A˜θ for transmitter design is that it belongs to an uncertainty set Uθ because of Δθ with the random characteristics. Without the loss of generality, we assume that array manifold vector distortion Δθ is upper-bounded by a preknown constant ε>0 as Δθ≤ε. Therefore, the uncertainty set of A˜θ can be written as follows:(7)Uθ≜uθ:uθ=Aθ+Δθ,Δθ≤ε,where Uθ is an ellipsoid because of the imprecise information of the array manifold vector.To guarantee the robustness of the constraintg1, we propose an ACE method to transform g1 into a robust inequality constraint. For traditional DM signal, the synthesized constellation points hold their location in relation to each other as the digital baseband modulation symbols. If we increase the Euclidean distance of constellation points, it is useful for the BER performance of desired receiver. The method to increase the Euclidean distance of constellation points is shown in Figure 3. Following this idea, taking QPSK symbol Γ1 as example, the constraint g1 is transformed as follows:(8)g˜1forΓ1:Rew1HA˜θd≥22Imw1HA˜θd≥22.Figure 3
The principle of the proposed ACE method, where the black dots denote the standard QPSK symbols, the red dots denote an example of synthesis symbols based on ACE method, the gray area is the extension area of the proposed ACE method, a black line with arrows at both ends denotes the minimum Euclidean distance of standard QPSK symbols at the desired direction, and a red line with arrows at both ends denotes an example of the minimum Euclidean distance of the proposed ACE method.Substituting (6) into (8), we have(9)g˜1forΓ1:Rew1HA˜θd+Rew1HΔθd≥Rew1HAθd−εw1H≥22Imw1HAθd+Imw1HΔθd≥Imw1HAθd−εw1H≥22.In (9), we take the worst case performance of the array manifold vector into consideration to relax the inequality constraint (8) as a convex form. In a similar way for other QPSK symbols, we obtain the robust constraint o as follows:(10)g˜1:Rew1HAθd−εw1H≥22,i=1,4Rew1HAθd+εw1H≤−22,i=2,3Imw1HAθd−εw1H≥22,i=1,2Imw1HAθd+εw1H≤−22,i=3,4.It is important that the designed ACE method increase the Euclidean distance of the synthesized constellation points at the cost of improving transmit power of the DM system. Therefore, the far-field patterns via the proposed synthesis algorithm must be normalized to compare BER performance with benchmark synthesis algorithms.The sidelobe level constraint ofg2 for the left sidelobe region can be transformed as follows:(11)g˜2forΘSL1:maxp=1,…,PwiHA˜θp=maxp=1,…,PwiHA˜θp+wiHΔθp≤maxp=1,…,PwiHA˜θp+εwiH=wiHA˜θp∞+εwiH≤u1.In a similar way for the right sidelobe region, the robust constraintg˜2 is written as follows:(12)g˜2:wiHA˜θp∞+εwiH≤u1,i=1,2,3,4,θp∈ΘSL1wiHA˜θq∞+εwiH≤u1,i=1,2,3,4,θq∈ΘSL2.The BER constraint ofg3 for the left sidelobe region can be transformed as follows:(13)g˜3forΘSL1:maxp=1,…,Pf1θp−f2θp=maxp=1,…,Pw1HA˜θp−w2HA˜θp≤w1HAθp−w2HAθp∞+εw1H+εw2H≤u2.In a similar way, the robust constraintg˜3 is shown as follows:(14)g˜3:w1HAθp−w2HAθp∞+εw1H+εw2H≤u2,θp∈ΘSL1w3HAθq−w4HAθq∞+εw3H+εw4H≤u2,θq∈ΘSL2.Based on the aforementioned analysis, the proposed robust optimization problem for DM signal can be written as follows:(15)minwi∈CN∑i=14wis.t.g˜1,g˜2,g˜3.The optimization problem (15) is regarded as the l2 and l∞ regularization with convex form. We can find the solution with an SOCP solver. In this section, we consider the uncertainty information of array manifold vectors in the DM signal synthesis algorithm. The worst case performance of the array manifold vectors is considered to relax the constraints. In this way, the performance of sidelobe level and security will be degraded to tolerate the uncertain information of array manifold vectors compared with the nominal optimal beamformer for the DM system.


## 3. Simulation Results

To evaluate the proposed robust algorithm for DM signal, we consider the optimization problem (5) as a nominal optimal beamformer and employ the benchmark synthesis algorithm in [11] to compare the performance with the array manifold vectors uncertainty. The BER performance of DM transmitter is computed with a stream of random QPSK symbols in an additive white Gaussian noise (AWGN) channel. In addition, the signal-to-noise ratio (SNR) is calculated based on the desired direction. An advanced receiver that can carry out minimum Euclidean distance demodulation for the eavesdropper ([7]) is employed. The parameters of the phased array are used as follows: dx=λ/2, N=11, and θd=60°. The parameters of the optimization problem (5) u1=−15dB, u2=−25dB, ΘSL1=0°,45°, ΘSL2=75°,180°, and ΘSB=45°,75°. To tolerance the manifold vectors perturbation, the constraint values are relaxed to u1=−13dB and u2=−18dB, respectively, for the optimization problem (15). We also suppose that each element of the manifold vector is distorted with a zero-mean circularly symmetric complex Gaussian random variable normalized with ε=0.2. We implement 100 Monte Carlo simulations to obtain the average far-field pattern, minimum Euclidean distances, and BER performance as follows.Figure4 shows the far-field patterns of different synthesis algorithms for the DM signal. It is found that the maximum sidelobe level performance degrades to −10.3 dB and −13.6 dB, respectively, for the synthesis algorithm in [11] and the proposed robust synthesis algorithm in (16) compared with −15 dB of the nominal optimal beamformer for DM transmitter in (5). According to the PE equation of the DM system [10, 27], the PE of the DM system reaches to 35.13% and 51.9%, respectively, for the synthesis algorithm in [11] and the proposed robust synthesis algorithm in (16) compared with 64.77% of the nominal optimal beamformer for the DM system with the metrics of PE. It means that the proposed robust synthesis algorithm tolerates the array manifold vectors perturbation at the expense of the sidelobe level performance, which reduces the PE of the DM system.Figure 4
Far-field patterns of different synthesis algorithms for the DM signal.Figure5 shows the minimum Euclidean distances of different synthesis algorithms for the DM signal. It is found that proposed ACE method increases the minimum Euclidean distances of synthesized constellation points compared with the nominal optimal beamformer in (5); meanwhile, mitigate the degradation of the minimum Euclidean distances at the sidelobe region compared with the synthesis algorithm in [11]. It is to note that the far-field patterns via the proposed synthesis algorithm must be normalized to compare BER performance with the synthesis algorithm in [11] as follows.Figure 5
Minimum Euclidean distances of different synthesis algorithms for the DM signal.Figure6 presents the BER performances of different DM transmitters when the SNR equals to 30 dB for demonstrating the sidelobe region clearly and the SNR equals to 10 dB for demonstrating the mainlobe region clearly. We found that (1) the BER performance of the proposed robust synthesis algorithm achieves 8.321×10−4 (almost coincide with the nominal optimal beamformer 7.827×10−4) compared with the 1.024×10−3 of the synthesis algorithm in [11] at the desired direction; (2) at the maximum sidelobe direction, the BER performance of the proposed robust synthesis algorithm achieves 0.05 compared with the 0.01 of the synthesis algorithm in [11]. It means the proposed robust synthesis algorithm has better robustness for the array manifold vectors perturbation both in the desired direction and undesired directions.Figure 6
BER performances of different synthesis algorithms for the DM signal.


## 4. Conclusion

In this paper, we presented a robust synthesis algorithm to solve the problem of the array manifold vector uncertainty. In the desired direction, we design an ACE method to relax the equality constraint into a robust inequality constraint, while optimizing the constraints of sidelobe level and Euclidean distance of constellation points by considering the worst case performance of array manifold vectors at the undesired directions. Simulation results show better performance of the proposed synthesis algorithm compared with the benchmark synthesis algorithms. In the future work, we will focus on other influence factors in a practical DM system such as the digital phase shifter with quantization error and mutual coupling between the elements.



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*Source: 9060732-2020-01-08.xml*