#### 6.1. SER

A. Single beam with fixed polarization information

First, we will consider broadside and off-broadside design examples for single beam to transmit two independent data streams with different PSs to verify that the introduction of PS can increase the spectral efficiency. That can also be viewed as the polarization multiplexing. The capacity performance of our method is equal to the polarization multiplexing that two separate signals are transmitted by a PSA. We assume that the signals are Gray-coded QPSK modulated. The PSs are defined by ${\mathsf{\Theta}}_{1}=({45}^{\circ},{90}^{\circ})$ for data stream 1 for a left-hand circular polarization, and ${\mathsf{\Theta}}_{2}=({45}^{\circ},-{90}^{\circ})$ for data stream 2 for a right-hand circular polarization at arbitrary directions. For each data stream, the desired beam pattern is a value of one with 90${}^{\circ}$ phase shift at the desired mainlobe, i.e., symbols “00”, “01”, “11”, “10” correspond to 45${}^{\circ}$, 135${}^{\circ}$, −135${}^{\circ}$, −45${}^{\circ}$. Meanwhile, the desired beam patterns over the sidelobe region are random complex numbers with the amplitudes being approximate zero.

For broadside design example, the desired direction is set ${\theta}_{ML}={0}^{\circ}$; for off-broadside design example, the desired direction is set ${\theta}_{ML}={30}^{\circ}$. The sidelobe regions are sampled every 1${}^{\circ}$ except the mainlobe direction.

The beam patterns for broadside for symbol pairs “00,00”, “00,01”, “00,11” and “00,10” are depicted in

Figure 6, where all main beams are exactly pointed to 0

${}^{\circ}$ with normalized magnitude 0 dB level, denoting that the amplitude of the desired data streams as expected. The phase patterns for broadside for those symbol pairs are shown in

Figure 7. It is not hard to see that the phases of the two data streams in the desired direction are in line with the standard QPSK constellation, while in the sidelobe regions, phases are random enough. The beam and phase patterns for other twelve symbol pairs are not displayed here on account of the similar features as the four symbol pairs mentioned above. The resulting SER curves are demonstrated in

Figure 8. It is indicated that the low SER is achieved in the desired direction, while in other undesired directions, the SER approximates the upper bound of a QPSK transmission system (0.75), representing that the directional modulation has been realized effectively.

The beam and phase patterns for off-broadside

${\theta}_{ML}={30}^{\circ}$ for symbol pairs “00,00”, “00,01”, “00,11” and “00,10” are depicted in

Figure 9 and

Figure 10, respectively. The resulting SER curves are displayed in

Figure 11. It is easy to see that the designed responses are slightly less desirable because the wider mainlobe. Even so, the PLS can still be enhanced.

B. Multiple beams with fixed polarization information

Next, we will consider design examples for multiple beams to transmit different or the same modulation information. Take two beams for example, one data stream transmits the QPSK modulation symbols in two desired directions with a horizontal polarization

${\mathsf{\Theta}}_{1}=({0}^{\circ},{0}^{\circ})$, and another data stream transmits the BPSK modulation symbols in the same two directions with a vertical polarization

${\mathsf{\Theta}}_{1}=({90}^{\circ},{0}^{\circ})$. The simulated far-field (a) magnitude patterns and (b) phase patterns for 100 random symbols are shown in

Figure 12.

Thus, from

Figure 12, it can be observed that standard QPSK and BPSK constellation patterns are only along the prescribed directions,

${0}^{\circ}$ and

${30}^{\circ}$ as expected, with the signal IQ formats along all other directions being distorted, in such a manner to lower the possibility of interception by eavesdroppers located in these regions.

Figure 13 shows the SER performance versus elevation angle for the two data streams transmitted when SNR equals 12 dB. It is obvious to find that the SER performance of the two data streams is the same as the traditional QPSK or BPSK signal at desired directions (

${0}^{\circ}$ and

${30}^{\circ}$), while the SER performance is deteriorated seriously when the elevation angle is off the desired directions. Therefore, the channel capacity is double increased, and this characteristic of the designed signals is also beneficial for PLS enhancement.

C. Multiple beams with variable polarization information

In the following design example, it is assumed that a signal stream modulated with QPSK are projected along broadside

${0}^{\circ}$, while another independent data stream modulated with BPSK, is transmitted along off-broadside

${30}^{\circ}$ by a 21-element uniform linear PSA. Meanwhile, we designate the polarization

${\mathsf{\Theta}}_{1}=({45}^{\circ},{45}^{\circ})$ at the direction

${0}^{\circ}$, the polarization

${\mathsf{\Theta}}_{2}=({45}^{\circ},-{45}^{\circ})$ at the direction 30

${}^{\circ}$, and the PSs are generated randomly at other undesired directions by the polarization control unit in

Figure 3. The simulated far-field (a) magnitude patterns and (b) phase patterns for 100 random symbols are shown in

Figure 14.

In

Figure 15, the SER simulation results versus elevation angle obtained for receivers using a polarization sensitive antenna and a single-polarized antenna are depicted. Two observations can be generalized from

Figure 15: (1) the SERs of the LUs using a polarized sensitive antenna at the desired directions (

${0}^{\circ}$ and

${30}^{\circ}$) are as good as the ideal case; (2) No matter where the Eve is located, it cannot intercept the exact confidential information using a single-polarized antenna due to the high SER. Further, the SER performance versus SNR for receivers at the desired direction

${0}^{\circ}$ utilizing a polarization sensitive antenna and a single-polarized antenna is displayed in

Figure 16. From

Figure 16, we can see that even if the eavesdroppers locate in the desired directions with high SNR enough, the confidential information still cannot be demodulated exactly, when the eavesdroppers use the antennas different from the LUs, i.e.,

${\mathsf{\Gamma}}^{\mathrm{EVE}}\ne {\mathsf{\Gamma}}_{{}^{k}}^{\mathrm{LU}}$,

$k\in \{1,2,\cdots ,K\}$. Therefore, the DM technique based on a PSA is an effective approach to enhance PLS.