# A Software-Defined Radio Platform for Teaching Beamforming Principles
## 4 Beamforming Platform Validation and Testing
### Design and Characterization of an Array of Patch Antennas  
A customized 1 \(\times\) 4 array of patch antennas was designed in CST Studio Suite to operate in the 3.5 GHz frequency range, emulating a device operating in the 5G FR1. The dimensions of each patch were set at 28 mm wide and 19 mm high, with a distance of 4.42 mm between each patch. To ensure proper feeding, each patch element was connected through a feeder with a width of 2.8 mm and a height of 8.32 mm, values calculated to obtain a characteristic impedance of 50 Ohms on each antenna port. In addition, to facilitate practical handling, an additional copper-free substrate element has been added to the bottom of the antenna to create a solid handle. Through the optimization process, adjustments were made to the patch dimensions, increasing the width by 0.138 mm and the height by 0.469023 mm. These modifications were implemented to correct the deviations identified in the simulation results and achieve optimized antenna parameters. The antenna was then printed on an FR4 substrate. Figure 10 presents the configuration of the designed (a) and manufactured (b) antenna. Figure 11 presents the array of patch antennas' 3D radiation pattern(a) as well as the horizontal 2D radiation pattern (b), extracted from simulation for 0\({}^{\circ}\) phase offset between the patch elements. One can observe the forward direction of the main radiation lobe with the presence of two smaller sidelobes. Figure 8: (**a**) SA measurement of USRP TX signal amplitude; (**b**) USRP TX channel amplitude linearity. Figure 9: (**a**) Experimental setup for RX parametrization; (**b**) USRP/SA RX amplitude linearity. By modifying the phase of the signal fed to each patch element, the user can shift the beam towards the desired azimuthal direction. From phased array antenna theory [23], the azimuthal orientation or the steering angle of the beam (\(\theta\)), relative to the forward direction, can be calculated based on the wavelength (\(\lambda\)), the element spacing (d), and the phase shift per element (\(\Delta\Phi\)) based on Equation (3). A schematic representation of the beamforming process for the designed patch antenna is presented in Figure 12.  
\[\theta=\arcsin(\frac{\lambda\Delta\phi}{2\pi\text{d}}) \tag{3}\]  
To validate the physical antenna, the scattering (S) parameters of the patch antenna were measured using a two-port R&S ZNB Vector Network Analyzer (VNA) and compared to the simulation results. Measurements were performed indo  
Figure 11: Array of patch antennas’ radiation pattern in (a) 3D and (b) azimuth. Figure 12: Phased array beamformer principle. Figure 10: (a) Designed patch antenna; (b) physical patch antenna. The S\({}_{11}\) parameter in Figure 13 corresponds to a single patch element (the first), while S\({}_{12}\) was measured between patch element 1 and 2. In the simulation, the antenna resonant frequency was 3.6 GHz, and the associated S\({}_{11}\) value was \(-\)12 dB, while measurements indicated an actual resonant frequency at 3.53 GHz, with a measured S\({}_{11}\) parameter of \(-\)8.5 dB. The observed 70 MHz shift in the resonant frequency (from 3.6 GHz to 3.53 GHz) is relatively small (around 2%). However, the S\({}_{11}\) discrepancy indicates that the antenna reflects more power than expected, suggesting either increased losses or a mismatch between the antenna and its feed. The discrepancies between simulated and measured values can be attributed to several factors, including fabrication tolerances, material properties, environmental factors, and model assumptions. Variations in the manufacturing process, such as slight changes in the dimensions of the antenna or the dielectric properties of the substrate (real versus simulated) can lead to the observed differences. One must also consider the presence of interference and reflections in the laboratory environment during measurements due to the presence of walls, furniture, and objects, which were not modeled in the CST Studio Suite simulation.