## 3 Demonstration of the Proposed Approach  
### Parametric Studies  
Figure 3a shows the variations in \(\rho_{PDC}\) according to the distance between two antennas. In this observation, the position of the patch antenna is adjusted, while the port of the IFA is located at a reference point, \(P_{ref}\). The mutual coupling is presented by \(\mid\)S\({}_{21}\) \(\mid\), given as a red dashed line, and is compared to \(\rho_{PDC}\), which is calculated using weighted patterns with different \(\sigma_{\theta}\) from \(10^{\circ}\) to \(180^{\circ}\) for the same \(\mu_{\theta}\) of \(0^{\circ}\). The reason we provided comparisons between \(\rho_{PDC}\) and \(\mid\)S\({}_{21}\) \(\mid\) is to demonstrate if the proposed coefficient can reflect the effect of mutual coupling as well as the pattern distortion. Indeed, the electrically large geometry of the lower-band antenna and its harmonic resonances can cause unexpected increases in mutual coupling and pattern distortion in higher-band spectrum. The results show that \(\rho_{PDC}\) tends to increase as \(\sigma_{\theta}\) decreases because the observation window of \(\theta\) becomes narrower. In addition, both \(\rho_{PDC}\) and \(\mid\)S\({}_{21}\) \(\mid\) show a convergence behavior in an inverse proportional manner, and \(\rho_{PDC}\) drops to 0.83 for \(\sigma_{\theta}\) = \(180^{\circ}\) as \(\mid\)S\({}_{21}\) \(\mid\) rises to \(-11\) dB at 10 mm. These results imply that pattern distortion is highly correlated to mutual coupling, which is well-represented by \(\rho_{PDC}\). Therefore, we can give a design guideline for the SAA that, for instance, the distance between the two antennas should be maintained at greater than 20 mm for \(\rho_{PDC}\geq 0.9\). Figure 3b presents \(\rho_{PDC}\) variations in terms of frequency when the distance is fixed at 20 mm. \(\mid\)S\({}_{21}\) \(\mid\) varies between \(-20\) dB and \(-42\) dB and shows two peaks of \(-32\) dB and \(-20\) dB at 4.6 GHz and 6.2 GHz, respectively. The purpose of measuring the variation in \(\rho_{PDC}\) according to frequency is to confirm if our approach is effective for the out-of-band emission as well as the pattern distortion, simultaneously. In general, \(\mid\)S\({}_{21}\) \(\mid\) provides better insight to the system interference from the power standpoint; however, pattern distortion can be another critical issue caused by mutual coupling, especially in the SAA. It is worth pointing out that the lower peak at 4.6 GHz exhibits a more distorted pattern with \(\rho_{PDC}\) = 0.86 at \(\sigma_{\theta}\) = \(180^{\circ}\) compared to the higher peak whose \(\rho_{PDC}\) is greater than 0.9. This is because \(\rho_{PDC}\) does not measure the pattern distortion caused by only the mutual coupling but also the other causes from the surrounding environment, such as nearby antennas and platforms. Although the conventional approach of calculating the maximum pattern deviation can be an easy interpretation to the pattern distortion, it provides only the amplitude deviation. In contrast, the proposed coefficient considers both the amplitude and phase distortion of the radiation patterns, which has potential benefits in beamforming operations. Figure 4a presents the distorted radiation patterns of the patch antenna at 6 GHz with respect to the distance. The pattern distortion of the antenna becomes more obvious as the distance to the IFA decreases. This corresponds to the tendency shown in Figure 3a--that is, \(\rho_{PDC}\) has a lower value when the patch is closer to the IFA. The distorted radiation patterns of the patch antenna according to the frequency are shown in Figure 4b. This also shows that the proposed coefficient represents the amount of distortion in the radiation pattern of an AUT well.