## 2 Proposed Coupled Structure-Electromagnetic Statistical Model  
Figure 2 shows the configuration of antenna array elements whose numbers are \(M\times N\), with equal spacing. \(dy\) and \(dx\) are the distances between the radiators in both \(x\) and \(y\) directions. (\(\theta,\phi\)) is the pointing far-field direction based on the Oxyz coordinate in Figure 3 with cosine direction as (_cos ax_, _cos ax_, _cos ax_) [29]. Figure 1: Antenna array deformation mode. (**a**) Sag. (**b**) potato chip, (**c**) saddle, (**d**) Bessel [9]. Figure 2: Configuration of planar antenna array element. Antenna fabrication and operating process result in structure errors, which include systematic distortion and random error. Assuming that the radiator (\(m\), \(n\)), (\(0\leq m\leq M-1\), \(0\leq n\leq N-1\)) has a random error of (\(\Delta x^{\prime}_{mn}\), \(\Delta y^{r}_{mn}\), \(\Delta z^{r}_{mn}\)), phase difference \(\Delta\Phi^{r}_{mn}\) with respect to Figure 1's coordinate origin is expressed as  
\[\Delta\Phi^{r}_{mn}=k(\Delta x^{r}_{mn}sin\theta cos\phi+\Delta y^{r}_{mn}sin \theta sin\phi+\Delta z^{r}_{mn}cos\theta) \tag{1}\]  
Also, by assumption that the radiator (\(m\), \(n\)) systematic distortion is given as (\(\Delta x^{s}_{mn}\), \(\Delta y^{s}_{mn}\), \(\Delta z^{s}_{mn}\)), then the phase difference \(\Delta\Phi^{s}_{mn}\) with respect to Figure 1's coordinate origin is given as  
\[\Delta\Phi^{s}_{mn}=k(\Delta x^{s}_{mn}sin\theta cos\phi+\Delta y^{s}_{mn}sin \theta sin\phi+\Delta z^{s}_{mn}cos\theta) \tag{2}\]  
Based on the antenna array superposition principle without coupling, the planar rectangular active antenna array field pattern density function with random position error and systematic distortion is presented as  
\[E_{sr}(\theta,\phi)=f_{e}(\theta,\phi)\sum\limits_{m=0}^{M-1}\sum\limits_{n=0 }^{N-1}I_{mn}e^{j(\Delta\Phi_{mn}+\Delta\Phi^{r}_{mn}}+\Delta\Phi^{s}_{mn}+ \varphi_{mnB} \tag{3}\]  
where \(f_{e}(\theta,\phi)\) represents the free space pattern function of the element, \(I_{mn}\) represents the amplitude excitation current of the element (\(m\), \(n\)), \(\varphi_{mnB}\) is the inherent phase difference in the array decided by the phase shifter, and \(\Delta\Phi_{mn}=kmdx~{}sin~{}\theta~{}cos~{}\phi+kndy~{}sin~{}\theta~{}sin~{}\phi\) denotes the initial spatial phase distribution. Practically, the element random position error (s) is probabilistic in nature; it then becomes important to statistically evaluate the characteristic of EM for antenna array, including random error. Let random position errors in \(x\), \(y\), and \(z\) directions be \(\Delta x^{r}_{mn}\), \(\Delta y^{r}_{mm}\), and \(\Delta z^{r}_{mn}\) subjected to normal distribution, with zero means and \(\sigma^{2}_{x},\sigma^{2}_{y}\), and \(\sigma^{2}_{z}\) variances, respectively. Hence, the random phase difference \(\Delta\Phi^{r}_{mn}\) in Equation (1) is a normal distribution function, and the variance is expressed as  
\[E_{sr}(\theta,\phi)\sigma^{2}_{\Phi_{r}}=k^{2}[\sigma^{2}_{x}(sin\theta cos \phi)^{2}+\sigma^{2}_{y}(sin\theta sin\phi)^{2}+\sigma^{2}_{z}(cos\phi)^{2}] \tag{4}\]  
For all random parameters that are normally distributed as \(x\sim N\) (\(0\), \(\alpha_{\rm x}\)), the associated relationship \(\overline{cos~{}x}\) and \(\overline{sin~{}x}\) is presented in reference [30; 31; 32; 33; 34]. Employing the average of exponential function, \(e^{jx}=e^{-\sigma^{2}~{}x/2}\), the average field density pattern function \(E_{sr}\) (\(\theta\), \(\phi\)) in Equation (3) is expressed as  
\[\overline{E_{sr}(\theta,\phi)} =f_{e}(\theta,\phi)\sum\limits_{m=0}^{M-1}\sum\limits_{n=0}^{N-1} I_{mn}e^{j(\Delta\Phi_{mn}+\Delta\Phi^{s}_{mn}+\varphi_{mnB})}e^{\overline{ \Delta\Phi^{r}_{mn}}} \tag{5}\] \[=E_{s}(\theta,\phi)e^{-\left(\frac{1}{2}\right)}\sigma^{2}_{\Phi_ {r}}.\]  
Figure 3: Geometrical spacing relationship of the target far-field. \(E_{s}\left(\theta,\phi\right)\) represents the field density of the pattern function together with the associated systematic distortion.