## 2 Proposed Coupled Structure-Electromagnetic Statistical Model
The antenna array mean power pattern function is derived based on variance property \(\sigma_{E_{sr}}^{2}=\overline{E_{sr}\left(\theta,\,\phi\right)E_{sr}^{*}(\theta,\, \phi)}-E_{sr}(\theta,\,\phi)E_{sr}^{*}(\theta,\,\phi)\) as  
\[\overline{P_{sr}(\theta,\phi)}=\overline{E_{sr}\left(\theta,\,\phi\right)E_{sr }^{*}(\theta,\,\phi)}=\sigma_{E_{sr}}^{2}+\overline{E_{sr}(\theta,\,\phi)E_{sr }^{*}(\theta,\,\phi)} \tag{6}\]  
Putting \(E_{sr}(\theta,\,\phi)\) function into Equation (6), the pattern function mean power is given as  
\(\sigma_{E_{sr}}^{2}\) defines the variance of \(E_{sr}\) (\(\theta,\,\phi\)), and its equation can be deduced. According to [24], \(E_{sr}\left(\theta,\,\phi\right)\) can be expressed in real and imaginary parts. Let \(X=\sum_{m=0}^{M-1}\sum_{m=0}^{N-1}X_{mn}\) and \(Y=\sum_{m=0}^{M-1}\sum_{m=0}^{N-1}Y_{mn}\) represent the real and imaginary parts, respectively. The variance \(  
The amplitude and phase of the initial excitation current are equal. Initiating structure error random samples, a saddle-shaped distortion is introduced with minimum displacement \(\lambda/6\) in the \(z\) direction as systematic distortion. It is equally assumed that the random position errors along the \(x\), \(y\), and \(z\) planes satisfy normal distribution with \(\lambda/30\) variance and \(0\) mean. The performance of the array with the associated error sample of the structure is separately simulated using HFSS software with no coupling effect. The average performance of the array is computed by finding average total performances estimated from the total samples of error due to structure. One thousand (1000) samples (structure error) were considered because there is no further change in the results when the mean of the array performance is more than 1000 samples. The computed performance of the proposed model is compared with the result generated from HFSS. The parameters are listed in Table 1, and the compared results are as depicted in Figure 2:0 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Performance** & **Coupled Structure Model Result** & **HFSS-Based Result** \\ \hline \hline \
\begin{tabular}{c} Boresight pointing (\({}^{\circ}\)) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Beam width (\({}^{\circ}\)) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Gain (dB) \\ First SLL (dB) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Second SLL (dB) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Third SLL (dB) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Fourth SLL (dB) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ Fifth SLL (dB) \\ \(\phi=0^{\circ}\) \\ \(\phi=90^{\circ}\) \\ \hline \hline \end{tabular} &
\begin{tabular}{c} 0.14 \\ 0.12 \\ 0.12 \\ 8.70 \\ 8.69 \\ 61.32 \\ 51.17 \\ 50.87 \\ 44.88 \\ 41.82 \\ 41.95 \\ 38.71 \\ 35.09 \\ \hline \hline \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 1: Coupled structure model versus HFSS results parameters (SLL means sidelobe level: all SLL are right SLL). Figure 4: mmWave microstrip patch antenna array designed for verification purposes. The coupled structure model and HFSS-based results agree well, and they exhibit consistency in the main lobe and side lobe areas. The boresight pointing, beamwidth, and gain are all the same in both planes. This implies that the coupled structured model and HFSS-based results have the same main lobe areas (Figure 5). The first sidelobe shows some differences in the absolute figures of 0 dB and 0.09 dB in \(\phi=0^{\circ}\) and \(\phi=90^{\circ}\) planes, correspondingly. On the other hand, for the second, third, fourth, and fifth sidelobes, the peak variations in absolute figures are 0.43 dB and 0.48 dB in \(\phi=0^{\circ}\) and \(\phi=90^{\circ}\) planes, respectively. The above results obtained demonstrate the effectiveness and efficiency of the proposed model in the analysis of the element position randomness effect on the electromagnetic radiation performance of the antenna array.