# Polarization Sensitive Array Based Physical-Layer Security

## Abstract

We propose a framework exploiting the polarization sensitive array (PSA) to improve the physical layer security of wireless communications. Specifically, the polarization difference among signals is utilized to improve the secrecy rate of wireless communications, especially when these signals are spatially indistinguishable. We firstly investigate the PSA based secure communications for point-to-point wireless systems from the perspectives of both total power minimization and secrecy rate maximization. We then apply the PSA based secure beamforming designs to relaying networks. The secrecy rate maximization for relaying networks is discussed in detail under both the perfect channel state information and the polarization sensitive array pointing error. In the later case, a robust scheme to achieve secure communications for relaying networks is proposed. Simulation results show that the proposed PSA based algorithms achieve lower total power consumption and better security performance compared to the conventional scalar array designs, especially under challenging environments where all received signals at destination are difficult to distinguish in the spatial domain.

Physical layer security, polarization sensitive arrays, point-to-point wireless systems, relaying networks

## 1 Introduction

The issue of information security in wireless networks has attracted extensive attention in recent years considering the openness of wireless links [1, 2]. Traditionally, encryption techniques are utilized to ensure secure communications, which are generally applied in the upper layer of network and have a high design complexity [3]. Therefore, an intrinsic approach exploring the characteristics of wireless fading channels to improve information security emerges as a prominent technique, which is referred to as the physical layer security [4]. The fundamental theory for physical layer security was firstly established by Shannon [5]. Following Shannon’s work, Wyner [6] introduced the famous wiretap channel model and further defined the channel secrecy capacity. The work [7] proposed a Gaussian degraded wiretap channel which is widely used to model the wireless propagation environment.

Based on these pioneering theoretical concepts, a large amount of literature focusing on various design aspects of secure communications have sprung up. By applying multiple antennas at communication nodes to exploit spatial freedom, these researches aimed to significantly improve the physical layer security of wireless networks [8, 9, 10, 11]. For example, an artificial noise scheme was proposed for wiretap channels in [8] to study the impact of antenna selection on security performance of multi-input multi-output (MIMO) two-way relaying networks. The work [9] introduced an effective method called cooperative jamming to confuse the eavesdropper deliberately. With the aid of the game theory, a collaborative physical-layer security transmission scheme was designed in [10] to effectively balance the security performance among different links. All these works however assume that the wireless channels are ideally Rayleigh distributed, which ignores the influence of array directivity and correlation. A technique known as the directional modulation was also investigated to realize secure communications. In the work [12], the directional modulation technique was applied to the phased array to offer security. Specifically, by shifting each array element’s phase appropriately, the desired symbol phase and amplitude in a given direction is generated. The study [13] on the other hand adopted the directional modulation technique to enhance the security of multi-user MIMO systems. Different from the standard secrecy rate optimization, the secure communications of multi-user MIMO systems are achieved by increasing the symbol error rate at the eavesdropper. It can be seen that the directional modulation technique designs the weighting coefficients of the phased array. As will be shown, our polarization sensitive array (PSA) based technique designs the spatial pointing of each antenna to effectively extract the signals’ polarization information for realizing secure communications.

Generally, the polarization status, similar to the amplitude and phase, is a feature of the signal. Many researches have indicated that the direction-finding performance and short-wave communication quality can be improved by means of the polarization difference among signals [14]. However, in many practical communication scenarios, such as radar and electronic reconnaissance, the conventional scalar array (CSA) is widely deployed. In essence, the CSA is the uniformly spaced linear array with the same spatial properties in all its array elements. Generally, CSA is blind to the polarization status of signal and sensitive to the array aperture and signal wavelength [15]. Worse still, in some specific array alignment, a CSA may present the morbid response to the polarization status of signal. Different from the CSA, the PSA consists of a certain number of antennas with different spatial pointings, which can be utilized to extract the signal information more meticulously and comprehensively in a vector way [16, 17]. The spatial pointings of the PSA offer extra design degrees of freedom for physical layer security of wireless networks. In most practical wireless networks, jammer is typically introduced to effectively interfere with the eavesdropper, but it simultaneously causes the interference to the destination. When the jammer signal has approximately the same spatial properties as the source, the CSA based destination beamforming optimization is unable to suppress the interference, as it can only rely on the signals’ spatial characteristics. By contrast, since different polarization information can be extracted by the spatial pointings of PSA, the PSA based destination beamforming optimization is capable of suppressing the interference effectively, even when the signals are indistinguishable in the spatial domain. Therefore, utilizing the PSA to realize secure communications for wireless networks can achieve superior performance over the CSA design.

However, most existing PSA related works focus on the problem of estimating the signal’s direction of arrival (DOA). In [18], a two-step maximum-likelihood signal estimation procedure was developed under the PSA. Based on the sparse polarization sensor measurements, the DOA estimation of the transmitted signal was conducted in [19]. There also exist some works specifically related to the optimization of dual-polarization array to enhance the system capacity. Compared to the single polarization array, the orthogonal dual polarization antenna can enhance MIMO spatial multiplexing gain remarkably by means of the eigenvalue ratio decomposition [20]. The study [21] designed a linear-polarized dual-polarization frequency reuse system to increase spectrum utilization and further improve the system capacity, while the work [22] compared three different transmission schemes for MIMO networks to achieve the maximum diversity under a dual-polarization channel model. All these works do not consider utilizing PSA to enhance secure communications.

Against the above background, this paper investigates the PSA based secure transmission strategy for wireless networks. Specifically, we first consider the PSA based secure communications for the point-to-point single-input multi-output (SIMO) network with the aid of jammer. In this case, the secure beamforming is firstly designed aiming at minimizing the total transmit power subject to the secrecy rate requirements. Then the secrecy rate maximization scheme is proposed to improve the secrecy capacity of SIMO network as much as possible. Further extending our research into the more complicated scenario where the relay is employed to enlarge the communication coverage of source nodes, we consider the secrecy rate maximization under both perfect channel state information (CSI) and imperfect PSA pointing, respectively. It is worth noting that convex optimization techniques [23] can be utilized to solve the optimization problems formulated in this paper effectively.

The rest of the paper is organized as follows. In Section 2, the system model of PSA is briefly introduced. In Section 3, the point-to-point secrecy beamforming is designed for SIMO networks, while the one-way relaying network is considered in Section 4, where the corresponding secrecy rate optimization problems are formulated. Section 5 presents the simulation results, and our conclusions are given in Section 6.

The normal-faced lower-case letters denote scalars, while bold-faced lower-case and upper-case letters stand for vectors and matrices, respectively. denotes the absolute value and denotes the Euclidean norm, while , , and represent the conjugate, transpose, conjugate transpose and inverse operators, respectively. An optimal solution is marked by , while and denote the trace and rank of matrix, respectively. The th row of matrix is given by , and the th-row and th-column element of is . means that is a positive semidefinite matrix. The vector stacking operator stacks the columns of a matrix on top of one another, and is the diagonal matrix with the diagonal elements . is the identity matrix, and is the matrix with all zero elements. means that is a complex Gaussian distributed random vector with the zero mean vector and the covariance matrix , while is the expectation operator. The determinant operation is denoted by , and denotes the Kronecker product. Finally, , and .

## 2 Polarization Sensitive Array System Model

Without loss of generality, we assume that a total of antennas are located in the -axis and the distance between the adjacent antennas is half wavelength, as illustrated in Fig. 1 (a). Here two plane electromagnetic (EM) signals are considered, i.e., the desired EM signal and the jamming EM signal . They arrive at the antennas of the PSA from different incident angles. As is well known, the EM wave is traveling in a single direction, where the electric component and the magnetic component are perpendicular to each other as well as perpendicular to this propagation direction. Taking the electric component as an example, we define the transverse electric field vectors of the EM signal as

(1) |

where and are the electric field projections on the and directions, respectively. As a result, the magnetic field biases of the EM signal are and , respectively, for keeping the orthogonality [24]. Furthermore, it is assumed that the EM signals are completely polarized signals which means that the time varying and can be formulated as an ellipse. As described in Fig. 1 (b), and are the polarization orientation and ellipse angle, respectively, which represent the track of the EM signal’s electric vector and are thereafter called the POA for short. According to the EM theory [24, 19, 20, 21, 22, 25, 26], we can express the EM signal in a vector form with its DOA and POA as follows

(2) |

where and are the azimuth and elevation angles of the EM signal , respectively, while

(3) |

(4) |

is the steering matrix of , which is composed of the electric and magnetic field bases of the EM signal, while and are the corresponding rotation and ellipticity matrix of , respectively, [26].

In addition to the polarization of EM signals, the antenna polarization should also be considered. It is noted that only short dipole antennas are adopted in our work, and thus the array magnetic response can be neglected. Besides, the polarization sensitive matrix which represents the polarization characteristics of the array is defined by the spatial pointing angles of the antennas of the PSA, i.e., for , where and are the azimuth and elevation pointing angle of the th antenna of the PSA, respectively. Mathematically, we have

(5) |

with

(6) |

where (generally taking the value of 1) is the antenna gain when the polarization status of the EM signal perfectly matches the antenna. Note that the matrix included in indicates that the array magnetic response is ignored. In addition, for the matrix , we have

(7) |

It is worth emphasizing that different from [19], where the PSA consists of the aligned short dipole antennas, each antenna of the PSA in our paper is deployed with a different spatial pointing angle, which becomes an optimization variable for secure communications.

Furthermore, the space phase matrix of the EM signal impinging on the PSA is given by

(8) | ||||

(9) |

where denotes the propagation vector of the EM signal , is the position vector of the th polarization antenna, and is the wavelength of . Based on (2), (5) and (8), the spatio-polarized manifold for the EM signal is defined as

(10) |

for . For notational convenience, we will simplify as in the sequel.

For the sake of maximizing secrecy rate, the PSA’s spatial pointings need to be optimized. In order to perform this optimization conveniently, the formulation (10) is rewritten as

(11) |

for , where

(12) | ||||

(13) |

Clearly, the new vector denotes the PSA’s spatial pointings and thus becomes our optimization variables.

## 3 Point-to-Point Secrecy Beamforming Design

We consider the simplest but most representative wiretap channel as a source, a
destination and an eavesdropper. In most cases, the capacity of the wiretap channel is
higher than the main channel owing to the concealment and intention of the eavesdropper.
In order to realize secure communications, we introduce a jammer to disturb the
eavesdropper sufficiently. In this four-terminal network as depicted in Fig. 2, the source and jammer are equipped with single antenna, while the eavesdropper
and the destination are equipped with and antennas, respectively. More importantly, in our work, the -antenna PSA at destination is assumed instead of the conventional CSA to fully show the advantage of PSA for secure communications ^{1}

(18) |

Let be the channel gain vector from node to node , where and . Furthermore, we assume far field communications related to destination , and we denote and as the channel gains from source and jammer to the reference antenna (the first antenna) of the PSA at destination , respectively. It is worth pointing out that the eavesdropper in our work is a legitimate, active but non-intended receiver, which means that can simultaneously transmit signals to other nodes and intercept the confidential signal from source. Based on this assumption, the CSI of eavesdropper is available through a training-based channel estimation technique. For the sake of improving security performance of the SIMO network, a beamforming vector satisfying is applied to the antennas of the PSA to maximize the received confidential signal to interference plus noise ratio (SINR). Based on this setting, source and jammer simultaneously transmit the confidential signal and the jammer signal to the destination and eavesdropper , respectively. Here, is assumed. Since and are far field signals relative to the PSA, the signals and impinging on the reference antenna of the PSA from source and jammer are represented as and , respectively, where and denote the maximum transmit powers of source and jammer , respectively.

(27) |

Because both source and jammer are far-field narrowband synchronized
transmitters ^{2}

(14) | ||||

(15) |

where with is the receive beamforming vector of eavesdropper , while and are the received Gaussian noise vectors at destination and eavesdropper , respectively. From the perspective of eavesdropper , the optimal is designed to achieve the maximum amount of wiretapped information, i.e., to maximize its desired SINR, which is obtained by solving the following problem

(16) |

Clearly, the above problem is a standard generalized Rayleigh quotient problem, whose optimal solution is the generalized eigenvector corresponding to the largest generalized eigenvalue of the matrix pencil [28]. Owing to the fact that the matrix is nonsingular, the optimal eavesdropper’s receive beamforming vector is equivalent to the normalized eigenvector associated with the maximum eigenvalue of the matrix , that is,

(17) |

where is a normalized factor to satisfy and denotes the eigenvector corresponding to the maximum eigenvalue of the matrix . Considering the rank-1 property of the matrix , the matrix is also rank-1 and only has one nonzero eigenvalue. Specifically, we have the formulation (3) given at the top of this page. Thus the unique nonzero eigenvalue and the corresponding eigenvector are and , respectively. Thus, the optimal eavesdropper’s receive beamforming vector (17) can be written as

(19) |

Based on (14) as well as (15) and (19), we formulate the received SINRs at destination and eavesdropper as

(20) | ||||

(21) |

respectively, where applies because is a real vector. To realize secure communication of the SIMO network, the security metric called the maximum achievable secrecy rate [6] is considered, which is defined as follows

(22) |

where denotes the achievable secrecy rate, is the mutual information between source and destination, and is the mutual information between source and eavesdropper. With the assumption of Gaussian wireless channels, and can readily be calculated as and , respectively. Thus the maximum achievable secrecy rate of the SIMO network is formulated as

(23) |

For the point-to-point SIMO network, we consider two optimization problems, which are the total power minimization under secrecy rate constraint and the secrecy rate maximization under transmit power constraints, respectively.

### 3.1 Total Power Minimization

The optimization problem is defined as the one that minimizes the total transmit power of the SIMO network subject to the minimum secrecy rate constraint , that is,

(24) |

where and , in which the sparse matrix is defined as

(25) |

Note that the constraint in (24) is equivalent to the property of PSA spatial pointings given in (7). When and are given, the optimal for the problem (24) is obtained, similar to the derivation of , as

(26) |

Next we substitute (26) into (24) to reformulate the total power minimization problem as (27), which is given at the top of this page. Unfortunately, because of the nonlinear and coupled term , which is given by

(28) |

the optimization problem (27) is generally nonconvex and difficult to solve directly. Hence we propose a suboptimal algorithm for the optimization problem (27), i.e., (24). With this method, the optimization of is performed independently from and . Specifically, since the received desired signal strength at destination in the SIMO network satisfies

(29) |

we can consider the term as the optimization objective for the PSA spatial pointings by introducing . Thus, the secrecy optimization problem with respect to can be formulated as

(30) |

where the constraint indicates that the interference introduced by jammer to destination can be canceled completely. However, the problem (30) is nonconvex and NP-hard due to the rank-1 constraint.

In order to find an efficient way of solving the optimization (30), we firstly relax it to a standard semidefinite programming (SDP) problem by neglecting the rank-1 constraint temporarily. Then the penalty based method [29] is utilized to obtain the finally rank-1 satisfied solution for the problem (30). To be specific, let be the optimal solution of (30) without considering the rank-1 constraint. Then is actually an upper bound of in the objective function of the problem (30). With the penalty based method, this is adopted as the initial point for the iterative optimization given in (31):

(31) |

where the auxiliary variable satisfying , and the superscript denotes the iteration number, while is the maximum eigenvalue of and denotes the corresponding eigenvector. For a fixed , we can obtain the optimal rank-1 satisfied solution by solving the optimization problem (31) iteratively, and the corresponding optimal is calculated through the eigenvalue decomposition of . We utilize the bisection method [30] to perform one-dimensional search for obtaining the optimal auxiliary variable , so as to obtain the optimal solution . The convergence of utilizing this penalty based method to solve the problem (31) is proved in Appendix.

Once the optimal is given, the optimal and the SINR at destination are derived respectively from (26) and (20) as

(32) | ||||

(33) |

By substituting and into the original problem (24), the reformulated total power minimization problem is given by

(34) |

After performing some mathematical transformations, we have

(35) |

where . For effectively solving the optimization problem (35), we consider different cases of the required secrecy rate threshold , which corresponds to different optimal solutions of . Firstly, two bounds of are defined as

(36) | ||||

(37) |

Based on (36) and (37), the following three cases of are discussed.

#### Case 1.

In this case, the optimization problem (35) is actually a standard geometric programming (GP) problem, which is

(38) |

where

(39) | ||||

(40) | ||||

(41) |

Obviously, this optimization can be efficiently solved using the convex optimization technique to yield corresponding optimal total transmit power .

#### Case 2.

In fact, the expression denotes the maximum secrecy rate of the SIMO network without introducing jammer under a high SINR condition. If is required, it makes no sense to introduce jammer and thus is designed. Therefore, the optimization problem (35) is transformed into