## 2 The Extraction of Manufacturing Process Uncertainty for the Case of Patch Antennas  
One of the main tasks on the way to derive the stochastic ANN model of a manufactured ordinary patch antenna was to estimate the geometrical uncertainty of the manufacturing process. Hence, two patch antenna arrays were designed and manufactured in a series--the first one for the C-band, and the second for the X-band and Ka-band. They are presented in Figures 1-3.  
Figure 1: Patch antenna array design for the frequency of 3.4 GHz—Top view.  
Figure 3: A cross-section of the patch antenna arrays.  
Figure 2: Patch antenna array design for the frequencies of 11.5 and 34.5 GHz—Top view.  
The designs contain copper ground and antenna layers and an FR4 substrate layer between the copper layers. The substrate has a relative permittivity equal to 4.4 and a nominal height equal to 1.5 mm. The main geometrical design parameters are \(g_{0}\), \(w_{0}\), \(y_{0}\), \(d\), and \(q\). The number of the manufactured antenna arrays for each design is 85. The antenna arrays were made by more than 10 different manufacturers.  
The nominal values of design parameters for the case of the antenna arrays from Figures 1 and 2 are \(y_{0}=2.5\) mm, \(g_{0}=20.0\) mm, \(w_{0}=21.6\) mm, \(d=30.0\) mm, and q = 12.0 mm, and \(y_{0}=1.4\) mm, \(g_{0}=5.1\) mm, \(w_{0}=6.7\) mm, \(d=5.5\) mm, and q = 3.6 mm, respectively. The author performed a comprehensive sensitivity analysis of the variation of the design parameters on the reflection coefficient for the above-presented antenna arrays. In particular, the Sobol indices analysis [16] was conducted. The analysis concluded that the only parameter whose uncertainty had a meaningful impact on the variance of the reflection coefficient of the designed antenna arrays was \(y_{0}\). Hence, the procedure for the extraction of geometrical uncertainty of the manufacturing process can be illustrated, as in Figure 4.  
In this procedure, a histogram of measurement results is compared with histograms of the \(\mid\)\(S_{11}\)\(\mid\) simulation results obtained for different PDFs of \(\mathbf{y}_{0}\). The histogram-matching analysis was performed for \(\mathbf{y}_{0}\) having Gaussian, Beta, as well as Uniform distribution. For the case of the Gaussian distribution, the value of \(\sigma_{y0}\) --the standard deviation of \(\mathbf{y}_{0}\) was the subject of the analysis. To make a reliable comparison of histograms for the three aforementioned PDFs, the support of Beta and Uniform distributions were \(-3.5\sigma_{y0}\leq y_{0}\leq 3.5\sigma_{y0}\). Three shapes of Beta PDF were considered, one symmetrical and two asymmetrical. The symmetrical one had shape parameters \(\alpha=6\), \(\beta=6\). This PDF is close in shape to the Gaussian PDF. The shape parameters \(\alpha=4\), \(\beta=6\) and \(\alpha=6\), \(\beta=4\) were applied to the asymmetrical Beta PDFs. Unknown \(\sigma_{y0}\) was searched with a step equal to 0.001 mm in the range 0.005 mm \(\leq\sigma_{y0}\leq\) 0.1 mm. This procedure was repeated for frequencies 3.4, 11.5, and 34.5 GHz. The comparison was made using an \"eye-view\" analysis. The author compared the shape and spread of the histograms. It turned out that the best match of the measurement with simulation results for three considered frequencies occurs when PDF of \(\mathbf{y}_{0}\) is assumed to be Gaussian, while 0.023 mm \(\leq\sigma_{y0}\leq\) 0.025 mm. A comparison of the histograms for the best match case for frequencies 3.4, 11.5, and 34.5 GHz is presented in Figures 5-7. The exemplary simulation results for the case of \(\mathbf{y}_{0}\) having Uniform and Beta  
Figure 4: The illustration of the procedure of the extraction of the geometrical uncertainty of the PCB antenna manufacturing process.  
The estimated geometrical uncertainty of the manufacturing process is used in the next section of the paper.