Directional modulation (DM) is a physical layer security technique which keeps known constellation mappings in a desired direction or directions, while scrambling them for the remaining ones [1, 2]. Both reconfigurable arrays and phased array structures can be used for its design. For reconfigurable arrays, their elements are switched for each symbol to make their constellation points not scrambled in desired directions, but distorted in other directions . For phased arrays [4, 5], DM can be implemented by phase shifting the transmitted antenna signals properly. In , a method named dual beam DM was introduced, where the I and Q signals are transmitted by different antennas. The bit error rate (BER) performance of a system based on a two-antenna array was studied using the DM technique for eight phase shift keying modulation in . A more systematic pattern synthesis approach was presented in 
, followed by an orthogonal vector approach which produces an artificial orthogonal noise vector from information to achieve DM in, and energy-constrained design in . Recently, the time modulation technique was introduced to DM to form a four-dimensional (4-D) antenna array in .
However, the polarisation information is not exploited in all aforementioned DM designs, and a single signal is transmitted through the propagation channel. To increase system capacity, we can exploit the polarisation information of the electromagnetic signal and send two different signals simultaneously at the same direction, same frequency, but different polarisations. This can be achieved by employing polarisation-sensitive arrays, such as tripole arrays and crossed-dipole arrays [12, 13, 14, 15, 16, 17, 18]. These two signals can also be considered as one composite signal using the four dimensional (4-D) modulation scheme across the two polarisation diversity channels [16, 19, 20]. In this paper, based on this idea, we proposed a DM design using crossed-dipole arrays, which can send two separate signals ( and ) with orthogonal polarisation states to the same direction simultaneously. To receive and separate the two orthogonally polarised signals, at the receiver side, a crossed-dipole antenna or array is needed, similar to the transmitter side . However, the polarisation directions of the antennas at the receiver side do not need to match those of the transmitted signals, as cross-interference due to a mismatch can be solved easily using some standard signal processing techniques .
The remaining part of this paper is structured as follows. A review of polarised beamforming based on crossed-dipole arrays is given in Sec. II. Directional modulation designs for two signals transmitted to the same direction, but with orthogonal polarisation states are presented in Sec. III. In Sec. IV, design examples are provided, with conclusions drawn in Sec. V.
Ii Review of Polarised Beamforming
A narrowband linear array for transmit beamforming based on crossed-dipole antennas is shown in Fig. 1. For each antenna, there are two orthogonally orientated dipoles, and the one parallel to the x axis is connected to a complex valued weight coefficient represented by , and the one parallel to the y axis is connected to for . The spacing from the first antenna to its subsequent antennas is represented by for . The elevation angle is denoted by and azimuth angle is represented by . The spatial steering vector of the array, a function of elevation angle and azimuth angle , is given by
where is the transpose operation, and is the speed of propagation. For a uniform linear array (ULA) with a half-wavelength spacing (), the spatial steering vector can be simplified to
where is the auxiliary polarisation angle and represents the polarisation phase difference.
For convenience, we split the array structure into two sub-arrays: one is parallel to the x-axis and the other to the y-axis. Then, the steering vectors of the two sub-arrays are given by
The beam response of the array can be given by 
where represents the Hermitian transpose, , a vector representing the overall steering vector of the array, is given by
and w is the complex valued weight vector
Iii Directional Modulation Design
In traditional crossed-dipole array design, the weight coefficients are designed for one single beam response and therefore can only cater for one transmitted DM signal. To increase the transmission capacity, in this section, we design a set of common weight coefficients for two signals ( and ) with orthogonal polarisation states transmitted to the same direction simultaneously. Based on the parameter representations in Sec. II, we have , , , and to represent steering vectors for and , and beam responses for and , respectively.
For -ary signaling, such as multiple phase shift keying (MPSK), each of and can create constellation points ( symbols), leading to desired responses. Then, for and transmitted simultaneously, there are different symbols and sets of response pairs, represented by a vector for , and each response corresponds to a weight vector . Here, we assume points are sampled in the mainlobe and points in the sidelobe range; then the responses for two signals in the mainlobe direction and the sidelobe range are given by
Here we have implicitly assumed that all the responses are considered with a fixed . Moreover, we put all the steering vectors of both and at the sidelobe region into an matrix , and the steering vectors at the mainlobe direction into an matrix, denoted by , where
Then, for the -th symbol, its corresponding weight coefficients can be solved by
where denotes the norm. The objective function in (10) ensures a minimum difference between the desired and designed responses for two signals in the sidelobe, and the constraint keeps a desired constellation value of two signals to the mainlobe. To ensure that the constellation is scrambled in the sidelobe regions, the phase of the desired response for two signals at different sidelobe directions can be randomly generated.
As these two signals and are modulated to generate a 4-D modulated signal in the far-field, to separate the corresponding components from the received composite signal, at the receiver side, we need a crossed-dipole antenna array in the same way as given in .
Iv Design examples
In this section, we provide design examples based on a aperture ULA with a half wavelength spacing between adjacent antennas to show the performance of the proposed formulations. The polarisation characteristics are defined by for for a horizontal polarisation, and for for a vertical polarisation. For each of and , the desired response is a value of one (magnitude) with phase shift at the mainlobe (QPSK), i.e., symbols ‘00’, ‘01’, ‘11’, ‘10’ correspond to , , and , respectively, and a value of (magnitude) with random phase shifts over the sidelobe regions. Therefore, for signals and transmitted simultaneously, we can construct different symbols. Moreover, without loss of generality, we assume the mainlobe direction is for and the sidelobe regions are for , sampled every , representing two signals are transmitted through the plane.
With the above parameters and design formulations in (10), the composite beam patterns for symbols ‘00,00’, ‘00,01’, ‘00,11’ and ‘00,10’ are shown in Figs. 2, 2, 2 and 2, where all main beams are exactly pointed to (3.01dB or magnitude) with a reasonable sidelobe level, and its corresponding components for and coincide in the mainlobe direction with dB mainlobe level, representing that a value of one for magnitude of desired signals and has been satisfied. The phase patterns for these symbols are displayed in Figs. 3, 3, 3, 3. We can see that the phases of and in the desired direction match the standard QPSK constellation, while for the rest of the angles, phases are random, demonstrating that directional modulation has been achieved effectively. The beam and phase patterns for other symbols are not shown as they have the same features as the aforementioned figures.
A crossed-dipole antenna array for directional modulation has been proposed for the first time, and a set of common weight coefficients has been designed for two signals and with orthogonal polarisation states transmitted to the same direction, based on a fixed given array geometry. As these two signals are modulated to generate a 4-D modulated signal in the far-field, to receive and separate the corresponding components from the resultant , at the receiver side, a crossed-dipole antenna array is needed. As shown in the provided design examples, for both and , a mainlobe and standard phase pattern have been achieved in the desired direction with low sidelobe level and scrambled phases in other directions.
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