## 4 Simulation Example and Analysis  
The structural deformation of the antenna array in practice usually causes an error due to random element positioning and systematic deformation. The randomness in the element positioning error is quantified as a random variable. Also, the distortion exhibited in practice is generally saddle-shaped. According to the surface mathematical properties, the phase center of a saddle-shaped element in position (m, n) is expressed as  
\[\Delta Z_{mn}=\Delta Z_{max}\left(\frac{x_{mn}^{2}}{x_{max}^{2}}-\frac{y_{mn}^{ 2}}{y_{max}^{2}}\right) \tag{14}\]  
where \(\Delta Z_{\max}\) is the maximum displacement of antenna array elements (systematic error) in z coordinate, and \(y_{\max}\) and \(x_{\max}\) are the array aperture width and half-length, respectively. In this section, antenna array with both element random position error and saddle shape distortion is addressed. Consider a rectangular 10 \(\times\) 10 microstrip patch array operating at 28.9 GHz frequency (Figure 6). The element spacing along \(x\) and \(y\) coordinates is \(\lambda/2\). Random error (s) in the \(x\), \(y\), and \(z\) coordinates is (are) assumed normal distribution of zero average and \(\lambda/20\) variance. \(\lambda/5\) is selected as the saddle-shaped maximum displacement in the z plane. Rogers R5880 substrate is used. It has a relative permittivity of 2.2, thickness height of 0.254 mm, and tangential loss tan \(\delta\) of 0.001. We used the corporate feeding method for the antenna. The corporate feeding technique is most popular in microstrip antenna, and its corporate feeding system is flexible. The systematic distortion is considered saddle-shaped, while three random samples (random samples 1-3) are emulated based on similar normal distribution. The antenna gain used is \(P_{sr}\) (\(\theta\), \(\phi\)) \(E_{sr}^{*}\)(\(\theta\), \(\phi\)), where \(E_{sr}\) (\(\theta\), \(\phi\)) is the field intensity in Equation (3). The proposed statistical model is then used to determine the performance of the deformed antenna array, and the results obtained are presented in Figure 7 with the parameters listed in Table 2. The results here can be discussed in terms of beamwidth, sidelobe level, gain, and boresight. (a) There are small variations in the beamwidth per random sample, and the values are nearly equal to the statistical random error. The maximum changes exhibited are 0.05\({}^{\circ}\) and 0.06\({}^{\circ}\) in \(\phi\) = 0\({}^{\circ}\) and \(\phi\) = 90\({}^{\circ}\), respectively. Therefore, the beamwidth is mainly affected by systematic distortion. (b) When there are both saddle shape and random errors, the gain is greatly reduced. The magnitude of the highest loss in the gain is \(-\)1.98 dB. The gain error is computed from statistical random error by substituting random error variances with systematic distortion into the proposed model. It is also computed by adding systematic distortion with the three random samples as total structural error. There is a negligible difference between the two results, and the peak difference is 0.41 dB. Hence, the change in the gain is a function of systematic distortion. (c) When the saddle shape is symmetric with no effect on target direction, the boresight varies majorly from various random error distributions. Therefore, calculation from statistical performance is preferable to the method of adding random samples to the systematic distortion. (d) The sidelobe level varies differently with the three random samples in the two planes. The sidelobe levels from the proposed model are the values of the evaluation because they are from the mean antenna gain pattern. For all the samples, 1000 rounds of computations were conducted. The mean gain loss is \(-\)3.44 dB, which is the same as the one computed from the proposed statistical model.