Low-complexity and High-performance Receive Beamforming for Secure Directional Modulation Networks against an Eavesdropping-enabled Full-duplex Attacker

In the past decade, physical layer security (PLS) has attracted wide attention from academia [1, 2, 3, 4, 5, 6, 7, 8]. Directional modulation (DM), as an advanced PLS transmission technique, is suitable for line-of-sight (LoS) propagation channel. Beamforming technology with artificial noise (AN) has a capability of creating a secure transmission in the LoS propagation channel [9, 10, 11, 4].

In [12], the authors presented a DM technique that uses the phased arrays to generate modulation and provides security by deliberately distorting signals in other directions and making the signal independent of direction. An orthogonal vector approach was proposed in [9] for the synthesis of multi-beam DM transmitters. In [10, 13], the authors developed robust synthesis schemes, in which DM is able to enhance the security performance of desired directions and distort the constellation points of undesired directions. These systems have the capability of concurrently projecting independent data streams along different specified spatial directions while simultaneously distorting signal constellations in all other directions. A directional modulation technique using frequency diverse array (FDA) for secure communications was proposed in [14, 15, 16, 17, 18, 19, 20]. In [14, 15], the authors developed a novel DM scheme based on random frequency diverse arrays with artificial noise (RFDA-DM-AN) to enhance physical layer security of wireless communications. In order to address the limitations of the previous works and further enhance physical layer (PHY) security, an AN-aided index modulation (IM) scheme with cooperative LUs based on FDA beamforming was proposed in [16]. Multi-beam directional modulation schemes based on FDA were investigated in [16, 17, 18]. Different from the traditional directional modulation that only implements angle dependent directional modulation, the authors proposed some precise secure transmission schemes of achieving two-dimensional dependence on angle and range dimensions in [14, 19, 20]. In the multi-carrier based DM antenna array system design, to solve the problems of high peak-to-average-power ratio (PAPR) and phase pattern formation simultaneously, a method, called wideband beam and phase pattern formation by Newton’s (WBPFN), was proposed in [21]. In [22], the authors developed a multi-carrier based DM framework using antenna arrays, which can realize data transmission on multiple frequencies simultaneously, so as to obtain a higher data rate. A practical wireless transmission scheme was proposed in [11] to transmit the confidential message (CM) to the desired user securely and precisely by jointly exploiting multiple techniques, including artificial noise (AN) projection, phase alignment/beamforming, and random subcarrier selection (RSCS) based on orthogonal frequency division multiplexing (OFDM)[23, 24], and DM. Particularly, if a DM transmitter intends to transmit CM to Bob and to interfere eavesdropper (Eve) with AN, she needs to know the directions of Bob and Eve in advance. In [25], two phase alignment (PA) methods, hybrid analog and digital PA and hybrid digital and analog PA, were proposed to estimate direction of arrival (DOA) based on the parametric method. In [26], the authors proposed an improved hybrid analog-digital (HAD) estimation of signal parameters via rotational invariance techniques (ESPRIT) at HAD transceiver to measure the DOA of a desired user.

However, the aforementioned existing works focused on the scenarios with only a passive eavesdropper Eve having no active malicious attacking capacity. In such a situation, it is hard for Alice to obtain the channel state information (CSI) from Alice to Eve. This is an embarrassing issue usually raised by secure expert. How to address this challenging issue? If Eve behaves like Mallory, with an active malicious attacking ability, this problem naturally disappears. By channel estimating, Alice may obtain the CSI from Alice to Mallory and Bob can obtain the CSI from Mallory to Bob. In [27], the authors proposed a jamming detection method for non-coherent single input multiple output (SIMO) systems without CSI. A hybrid wiretapping wireless system with a half-duplex adversary was proposed in [28], where the adversary can decide to either jam or eavesdrop the transmitter-to-receiver channel. In the presence of a full-duplex eavesdropper having both eavesdropping and jamming capabilities, the authors in [29] proposed a novel secure transmission strategy against eavesdropping by the sophisticated adversary. The authors in [30] studied the secrecy outage probability and mean SR of the system to evaluate the security performance of the system. In this paper, our focus is on how to suppress the malicious jamming and improve the secure performance by designing receive beamforming (RBF) at Bob in the presence of the malicious jamming.

In this paper, we present a novel DM scenario with a full-duplex (FD) malicious attacker Mallory. Moreover, Mallory also eavesdrops the CM conveyed from Alice to Bob. To reduce the impact of jamming from Mallory on Bob and improve the secure performance, four RBF methods are proposed. Our main contributions are summarized as follows:

1. 1.

   To alleviate the jamming from Mallory on Bob and improve the secure performance, the conventional MRC RBF at Bob is presented to strengthen the confidential signal. To enhance its performance, a WFMRC RBF is proposed by converting the colored interference plus noise at Bob into a white one with its covariance matrix being a multiple of identity matrix. Then, two RBFs Max-SR and conventional MMSE are derived. From simulation results, the proposed two RBFs WFMRC and Max-SR obviously outperform the conventional MRC in the medium and high SNR regions and achieve the same performance as conventional MMSE in terms of SR and BER.

2. 2.

   To completely remove the jamming from Mallory, a new RBF of maximizing the WF-based receive power at Bob is proposed to force the malicious jamming onto the null-space (NSP) of the channel spanning from Alice to Bob. By exploiting the rank-one property of channel steering vector, a low-complexity MMSE is proposed to reduce the complexity of conventional MMSE without performance loss. According to simulation, we find that the proposed NSP-based Max-WFRP is slightly worse than Max-SR, WFMRC and low-complexity MMSE in terms of the SR performance. Its main advantage is that its SR performance is independent of the change in malicious jamming power. Moreover, the proposed low-complexity MMSE has achieved the same performance as three proposed high-performance RBFs WFMRC, conventional MMSE, and Max-SR with an extremely low-complexity.

The remainder of this paper is organized as follows. Section II presents the DM system model. In Section III, four schemes of design RBF vector $\mathbf{v}_{BR}$ is proposed and its closed-form expression is given. Simulation and numerical results are shown in Section IV. Finally, we draw our conclusions in Section V.

Notations: Throughout the paper, matrices, vectors, and scalars are denoted by letters of bold upper case, bold lower case, and lower case, respectively. Signs $(\cdot)^{T}$, $(\cdot)^{-1}$, $(\cdot)^{\dagger}$, $(\cdot)^{H}$, $\parallel\cdot\parallel$ and $\rm tr(\cdot)$ represent transpose, inverse, Moore-Penrose, conjugate transpose, norm and trace, respectively. $\textbf{I}_{N}$ denotes the $N\times N$ identity matrix. The notation $\mathbb{E}\{\cdot\}$ represents the expectation operation.

The proposed DM system is illustrated in Fig. 1. It consists of an $N_{A}$-antennas base station (Alice), a legitimate $N_{B}$-antennas user (Bob) and an illegal $N_{M}$-antennas malicious attacker (Mallory). Here, the Alice sends CM to the Bob. In addition, there is an illegal malicious attacker Mallory trying to intercept the CM. Mallory operates on the FD model. That is, he/she can eavesdrop on CM, and simultaneously initiate an active attack on Bob. The baseband transmit signal can be expressed as

where $P_{A}$ is the total transmit power, $\beta_{1}$ stands for the power allocation (PA) factor, $\mathbf{v}_{A}\in\mathbb{C}^{N_{A}\times 1}$ denotes the transmit beamforming vector of the CM, and $\mathbf{T}_{A,AN}\in\mathbb{C}^{N_{A}\times N_{A}}$ is the projection matrix for controlling AN to the undesired direction, where $\mathbf{v}^{H}_{A}\mathbf{v}_{A}=1$ and $\rm tr[\mathbf{T}_{A,AN}\mathbf{T}^{H}_{A,AN}]=1$. In addition, $\mathbf{v}_{A}$ is defined as the transmit beamforming vector for the CM. In (1), $d_{A}$ is the CM with average power $\mathbb{E}[\|d_{A}\|^{2}]=1$ and $\mathbf{z}_{A,AN}\in\mathbb{C}^{N_{A}\times 1}$ denotes the AN vector with complex Gaussian distribution, i.e., $\mathbf{z}_{A,AN}\sim\mathcal{C}\mathcal{N}(0,\mathbf{I}_{N_{A}})$. The malicious attacking signal at Mallory can be expressed as

where $\mathbf{T}_{M,AN}\in\mathbb{C}^{N_{M}\times N_{J}}$ is the projection matrix for forcing the jamming to Bob, $N_{J}\in\left\{1,2,....,N_{M}-1\right\}$. Herein, $P_{M}$ is the transmit power at Mallory, $\mathbf{z}_{M,AN}\in\mathbb{C}^{N_{J}\times 1}$ denotes the AN vector with complex Gaussian distribution, i.e., $\mathbf{z}_{M,AN}\sim\mathcal{C}\mathcal{N}(0,\mathbf{I}_{N_{J}})$.

The corresponding received signal at Bob can be written as

where $\mathbf{H}^{H}(\theta_{AB})=\mathbf{h}(\theta_{r,AB})\mathbf{h}^{H}(\theta_{t,AB})$ and $\mathbf{H}^{H}(\theta_{MB})=\mathbf{h}(\theta_{r,MB})\mathbf{h}^{H}(\theta_{t,MB})$. $\mathbf{H}^{H}(\theta_{AB})\in\mathbb{C}^{N_{B}\times N_{A}}$ denotes the channel matrix from Alice to Bob. $\mathbf{H}^{H}(\theta_{MB})\in\mathbb{C}^{N_{B}\times N_{M}}$ denotes the channel matrix from Mallory to Bob. The normalized steering vector $\mathbf{h}(\theta)$ as

where the phase function $\Psi_{\theta}(n)$ is defined by [31]

where $\theta$ is the direction of arrival or departure, $n$ is the index of antenna, $d$ represents the element spacing in the transmit antenna array, and $\lambda$ is the wavelength. In (II), $\mathbf{n}_{B}\in\mathbb{C}^{N_{B}\times 1}$ is the complex additive white Gaussian noise (AWGN) vector, distributed as $\mathbf{n}_{B}\sim\mathcal{C}\mathcal{N}(0,\sigma_{B}^{2}\mathbf{I}_{N_{B}})$. Notably, $g_{AB}=\frac{\alpha}{d_{AB}^{c}}$ represents the path loss from Alice to Bob. Here, $d_{AB}$ is the distance between Alice and Bob, $c$ denotes the path loss exponent and $\alpha$ means the path loss at reference distance $d_{0}$. Moreover, $\mathbf{v}_{BR}\in\mathbb{C}^{N_{B}\times 1}$ represents the receive beamforming vector of Bob. Using the concept of the null-space projection, we can design

Similarly, the receive signal at Mallory is given by

where $\mathbf{H}^{H}(\theta_{AM})=\mathbf{h}(\theta_{r,AM})\mathbf{h}^{H}(\theta_{t,AM})$. $\mathbf{H}^{H}(\theta_{AM})\in\mathbb{C}^{N_{M}\times N_{A}}$ denotes the channel matrix from Alice to Mallory. $g_{AM}=\frac{\alpha}{d_{AM}^{c}}$ represents the corresponding path loss, and $d_{AM}$is the distance between Alice and Mallory. Additionally, $\mathbf{v}_{MR}$ represents the receive beamforming vector at Mallory. The complex AWGN at Mallory is denoted by $\mathbf{n}_{M}\sim\mathcal{C}\mathcal{N}(0,\sigma_{M}^{2}\mathbf{I}_{N_{M}})$. $\sqrt{\rho}\mathbf{H}^{H}_{M}\in\mathbb{C}^{N_{M}\times N_{M}}$ is the residual self-interference (RSI) channel matrix of Mallory, $\rho\in[0,1]$ denotes the residual self-interference parameter of the Mallory after self-interference cancelation [29].

As per (II) and (II), we can derive the achievable rate along Bob and Mallory as

and

respectively, where $\mathbf{R}_{M}$ is the covariance matrix of RSI channel,

Then we arrive the achievable SR, $R_{s}$, as follows:

In this section, to improve the security rate performance and reduce the effect of jamming from Mallory on Bob, four RBF schemes are proposed. Additionally, the conventional MRC is also presented as a performance benchmark for the future comparison.