## 1 Introduction  
Information security has always been an important problem for wireless communications, which provide a wide coverage at the cost of exposing information to undesired users with eavesdroppers (Eves) amongst them [1]. Traditionally, the higher-layer encryption techniques have been widely adopted to handle the security problem. However, with the explosive development of mobile Internet, the key-based methods need a secure channel to exchange the private key, which may be insufficient or even not suitable [2, 3]. Fortunately, physical layer security (PLS) exploiting the characteristics of wireless channels, e.g., interferences, noise, and fading, enables the secure communications without extra resources and signaling overhead, which can be considered as a significant complement to traditional cryptographic techniques [4, 5]. Directional modulation (DM), as an emerging promising physical layer wireless security technique, has attracted widespread concern and attention from many researchers globally over the past decade [6]. The initial concept of DM can be traced back to the near-field direct antenna modulation (NFDAM) techniques in [7, 8], where direction-dependent signals are formed by a transmit beam and a reflected beam at the radio frequency (RF) frontend. An angle-dependent DM technique for phased arrays was proposed in [9]. By changing the phase shift of each element, a symbol with desired amplitude and phase for generating a standard constellation can be projected along a pre-specified direction, while purposely scrambling the constellation to be very hard to demodulate in other unintended directions. In [10], the authors further demonstrated a DM transmitter to verify the feasibility of the proposed DM technique. A dual-beam antenna array excited by the I and Q signals was put forward to perform secure wireless DM transmission in [11]. Pattern synthesis approach was applied to DM systems in [12; 13]. This technique has been further extended to antenna subset modulation (ASM) technique in [14] for PLS communications. Bit error rate (BER) driven DM synthesis approaches were demonstrated in [9; 10; 15] to secure wireless communications. A switched antenna array [16], a linear sparse array [17], a sparse linear array [18], and a 4-D antenna array [19] were proposed to perform wireless DM transmission at the RF frontend. Afterwards, the artificial noise (AN) and DM concepts were first linked in [20], and further extended via the orthogonal vector approach in [21], which transferred the DM synthesis method from the RF frontend to the baseband. Furthermore, the orthogonal vector approach for an angle-dependent multi-beam DM system was implemented in [22]. Then, robust synthesis methods based on AN for secure single-beam and multi-beam DM were demonstrated by [PERSON] et al. [23] and [PERSON] et al. [24], respectively. The AN was also employed in [25; 26; 27] to secure DM transmission using the zero-forcing, precoding vector, and iteration convex optimization methods, respectively. However, the above-mentioned single- and multi-beam DM schemes can only achieve angle-dependent secure transmission, the security of which cannot be guaranteed once an Eve is aligned with the desired direction or very close to the desired user. To solve this neighbor security problem, one solution is to adopt a frequency diverse array (FDA) to synthesize both angle and range dependent DM [28; 29; 30; 31]; another solution is to utilize a multi-path model, where signals via both line of sight (LOS) and reflected paths are combined at the receiver side [32; 33; 34; 35; 36; 37; 38; 39]. The above solutions are all based on a single antenna array model. There are few studies regarding secure DM transmission for a multiple antenna arrays model. In [40], the authors presented a concept of secure and precise wireless transmission by using DM, random subcarrier selection with randomization procedure, and phase alignment at the DM transmitter for PLS.