## 3 Numerical Analysis
### Benefits of an Octagonal Ground Cavity  
To verify the superiority of an octagonal ground cavity (Figure 1a) over a square one (Figure 1b), we compared the performance of an open prism antenna backed by these two types of ground cavities. All other antenna dimensions remained constant, with \(L_{g}=165\) mm, \(H_{sw}=80\) mm, \(g=11\) mm, \(s=3\) mm, \(l=53\) mm, \(w=30\) mm, \(l^{\prime}=1\) mm, and \(w^{\prime}=3.6\) mm. Figure 3a shows the differential reflection coefficient of the two antennas. The ground corner truncation leads to a small increase in the absolute impedance bandwidth (i.e., from 774 MHz to 813 MHz). With operating frequencies slightly shifting upward, the fractional impedance bandwidth does not change much. More specifically, both antennas offer a wide \(\mathbb{S}_{\text{dd11}}\leq-10\) dB impedance bandwidth: the one with the square ground cavity has an impedance bandwidth of 54.0%, and the one with the octagonal ground cavity has an impedance bandwidth of 54.3%. The gain and diagonal-plane cross-polarization of the two antennas are compared in Figure 3b. The average gains across the corresponding impedance bandwidths of the square and the octagonal ground cavity-backed antennas are almost the same, with the former being 9.1 dB and the latter being 9.0 dB. By contrast, the effect of the ground corner truncation on cross-polarization is more evident. It is worth clarifying that the cross-polarization values in negative dB refer to peak cross-polarization (ratio), i.e., the difference between the peak cross-polarization (in dBi) and the peak co-polarization (in dBi) in the specified \(\phi\)-plane (in this case, \(\phi=45^{\circ}\)) within the entire \(\theta\) range of interest (i.e., \(-90^{\circ}\leq\theta\leq 90^{\circ}\)), not cross-polarization on the axis (i.e., \(\theta=0^{\circ}\)). With the square ground cavity, across the operating frequency band, the average peak diagonal-plane cross-polarization is \(-21.2\) dB, and the cross-polarization requirement of \(\leq-25\) dB is not met at any frequencies; with the octagonal ground cavity, the average peak diagonal-plane cross-polarization is \(-27.3\) dB across the operating frequency band, and \(65.4\%\) of the operating frequencies meet the cross-polarization requirement. This verifies that the use of a corner truncated cavity can effectively suppress the cross-polarization in the diagonal planes. In addition to the above-mentioned advantage, the antenna with the octagonal ground cavity also enjoys a smaller (by 17.2%) footprint and a higher (by 9.3%) average aperture efficiency, as shown in Figure 3c. More precisely, the average aperture efficiencies across the corresponding operating frequency bands of the square and the octagonal cavity-backed antennas are 106.4% and 115.7%, respectively. Figure 3d shows the cross-polarization ratio of the octagonal cavity-backed antenna, at 1.1 GHz, in all elevation and azimuth planes. It verifies that: the maximum cross-polarization of the proposed antenna occurs in the diagonal planes (i.e., \(\phi=45^{\circ}\) and \(135^{\circ}\)), and there is zero cross-polarization in the principal planes (i.e., \(\phi=0^{\circ}\) and \(90^{\circ}\)). Therefore, we can use the peak diagonal plane cross-polarization to represent the maximum cross-polarization in all \(\phi\) planes (\(0^{\circ}\leq\phi\leq 360^{\circ}\)). The frequency of 1.1 GHz was chosen because it has the worst cross-polarization performance across the operating frequency band. Figure 3e,f shows the 3D far-field co- and cross-polarized radiation of the octagonal cavity-backed antenna at 1.1 GHz. They were computed using the predefined HFSS functions, i.e., GainL3Y and GainL3X, that are based on [PERSON]'s 3 rd definition [26]. The overlay co-polarized field shows that the antenna radiates in its broadside. The overlay cross-polarized field verifies the same information as Figure 3d, i.e., cross-polarization does not exist in the principal planes and reaches its maximum in the diagonal planes.