## 2 Proposed Pattern Distortion Coefficient  
Figure 1a shows a two-antenna system composed of a conventional IFA and a probe-fed patch antenna are mounted on a conducting ground with dimensions of 200 mm \(\times\) 200 mm. Their design parameters are tuned for resonance at 1.5 GHz and 6 GHz, respectively, and detailed values are listed in Table 1. Note that this system is an example that describes an SAA with two antenna types operating in different frequency bands, and the antenna can be any type operating at any frequency that the SAA system requires. The pattern distortion coefficient, \(\rho_{PDC}\), is then defined by:  
\[\rho_{PDC}=\frac{\left|\iint_{4\pi}\left[\overline{E_{un}}(\theta,\phi)\cdot \overline{E_{aff}}^{*}(\theta,\phi)\right]g(\theta,\phi)d\Omega\right|}{\sqrt {\iint_{4\pi}\left[\overline{E_{un}}(\theta,\phi)\cdot\overline{E_{um}}^{*}( \theta,\phi)\right]g(\theta,\phi)d\Omega}\sqrt{\iint_{4\pi}\left[\overline{E_{ aff}}(\theta,\phi)\cdot\overline{E_{aff}}^{*}(\theta,\phi)\right]g(\theta,\phi)d \Omega}} \tag{1}\]  
where \(\overline{E_{un}}(\theta,\phi)\) indicates an unaffected far-field pattern that is obtained without the cause of interest, as in Figure 1b, and \(\overline{E_{aff}}(\theta,\phi)\) represents a distorted pattern of the AUT. The cross correlation between the unaffected and affected pattern--i.e., \(\overline{E_{un}}(\theta,\phi)\cdot\overline{E_{aff}}^{*}(\theta,\phi)\)--is weighted by the Gaussian function, which is written as:  
\[g(\theta,\phi)=\frac{1}{2\pi\sigma_{\theta}\sigma_{\phi}}\exp\left(-\frac{( \theta-\mu_{\theta})^{2}}{2\sigma_{\theta}^{2}}-\frac{(\phi-\mu_{\phi})^{2}}{2 \sigma_{\phi}^{2}}\right), \tag{2}\]where \(\mu\) and \(\sigma\) indicate the mean and standard deviation. This equation can be reduced to:  
\[g(\theta)=\frac{1}{\sqrt{2\pi\sigma_{\theta}}}\exp\left(-\frac{(\theta-\mu_{ \theta})^{2}}{2{\sigma_{\theta}}^{2}}\right), \tag{3}\]  
when the Gaussian weighting is applied only in the \(\theta\)-direction. In the same manner, the Gaussian function is weighted to the auto-correlation of each pattern with a denominator of (1) for normalization purposes. Figure 2a shows a Gaussian function determined by \(\mu_{\theta}=0^{\circ}\) and \(\sigma_{\theta}=43.5^{\circ}\). Figure 2b presents a comparison of the radiation patterns before and after the Gaussian weighting for a microstrip patch antenna, whose half-power beamwidth is \(102^{\circ}\). As can be seen, by applying the Gaussian weighting the degree of pattern distortion can be anticipated mainly within the target angular range required for its operation. Note that the definition of the proposed FoM is distinguished with an envelope correlation coefficient (ECC) because of the input far-field patterns. More specifically, the ECC is widely used to evaluate the channel capacity for multi-input and multi-output (MIMO) antennas and uses the far-field patterns of two different antennas that have the same resonant frequency. Conversely, the proposed approach takes an input from the same antenna before and after distortion; thus, the proposed FoM can be used to evaluate the distortion by any cause of interest--e.g., mutual coupling, platform effects, or both--at any frequency. Furthermore, by the given definition, the proposed FoM is also capable of evaluating a single-antenna system for pattern distortion affected by platform effects, whereas none of the other FoMs are capable of such evaluation.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \(W_{g}\) & \(L_{g}\) & \(L_{\text{IFA}}\) & \(H_{\text{IFA}}\) & \(G_{\text{IFA}}\) & \(W_{p}\) & \(L_{p}\) \\ \hline Value (mm) & 200 & 200 & 51.4 & 25.7 & 25.7 & 11.2 & 11.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of the IFA and probe-fed patch antenna.  
Figure 1: Illustration of the antenna system where (**a**) IFA is situated besides a patch antenna (affected) and (**b**) the patch antenna is mounted alone (unaffected).