## 3 Digital Beam-Forming
### Log-Periodic Dipole Antenna Array  
The transmitting antenna array of the WIOBSS-AA consists of five log-periodic dipole antennas. These antennas are arranged in a line and the distance between adjacent antennas is 15.3 m. The height of each antenna is 12 m. It is well known that, in phased linear arrays, the optimum antenna separation for beam forming is \(\lambda\)/2 [22]. For the WIOBSS-AA frequency range of 6-30 MHz, this results in an optimum spacing between 5 m (at 30 MHz) and 25 m (at 6 MHz). Due to the length of the longest dipole element and the limited space, the antenna separation is selected as 15.3 m. There are five transmitting channels and the channels are independent of each other. Each transmitting channel is mainly comprised of the DDS device and a 2 kW solid state power amplifier. The output of each power amplifier is connected to the antenna. Based on the high-speed direct digital frequency synthesizer, the phase, as well as the amplitude difference between each transmitting channel, can be adjusted by modifying the phase offset register and amplitude register. The accuracy of the phase is about 0.02 degree (the phase offset resolution of AD9911 is 14-bit) and the accuracy of the amplitude is about 0.5 mV (the amplitude-scaling resolution of AD9911 is 10-bit).  
The computed radiation pattern of the whole array is shown in Figure 5. The symmetric antenna radiation pattern of the full array pointing to an elevation of 32\({}^{\circ}\) has a half power beam width of 22.8\({}^{\circ}\), a directive gain of 18.7 dBi, and an almost symmetric first side-lobe with more than 15 dB suppression with respect to the main lobe. Although the log periodic antenna is a directional antenna, it has a large beam width of 71.4\({}^{\circ}\), as shown in Figure 6 (blue dashed line). Compared with a single antenna, the antenna array has a relatively narrow beam width of only 22.8\({}^{\circ}\) and higher gain, as shown in Figure 6 (red solid line).  
Figure 4: Computed radiation pattern of a single LPDA at 10 MHz (main lobe vector alignment), where dBi values are with reference to an isotropic gain pattern.  
How changing the frequency affects the beam pattern of the array is demonstrated in Figure 7. The antenna radiation pattern of the antenna array has a half power beam width of 32.3\({}^{\circ}\) at 8 MHz (red line), 22.8\({}^{\circ}\) at 10 MHz (green line), and 19\({}^{\circ}\) at 12 MHz (blue line). The width of the beam decreases with the increased frequency. Unfortunately, the sidelobe and backlobe rise with decreased frequency. The beam pointing of the array has different elevation for different frequency. The elevation angle is 42\({}^{\circ}\) at 8 MHz, 32\({}^{\circ}\) at 10 MHz, and 30\({}^{\circ}\) at 12 MHz. With an increase in frequency, the elevation of the beam decreases. The 15.3 m antenna separation decides the optimum operating frequency is 10 MHz. The antenna separation limits the beam steerability [23]. Large grating lobes start appearing when the beam is steered beyond \(\pm\)36\({}^{\circ}\) from the boresight, as shown in Figure 8.  
Figure 5: Computed radiation pattern of the antenna array at 10 MHz (main lobe vector alignment), where dBi values are with reference to an isotropic gain pattern.  
Figure 6: Comparison of the radiation patterns between a single antenna (blue line) and the antenna array (red line) at 10 MHz (main lobe vector alignment), where dBi values are with reference to an isotropic gain pattern.