## 3 Planar FSS Transmission Coefficient Vector Fitting  
Before applying planar PM FSS to the curved radome, further data processing has to be accomplished. The PM FSS dielectric constant data of wide incident angles and broadband frequencies are too colossal and complicated, a direct computation based on the data may cause too much stagnation and cost too much time. A more effective way is to fit the transmission coefficients of the PM FSS as a function of frequency and incident angle. The vector fitting method is an efficient algorithm to adopt rational functions to approximate S parameters, the numerical results can be expressed as analytic functions using vector fitting. In high frequency analysis, the calculation of the radome transmission coefficients is very important but it is difficult to get the analytic expressions of the transmission coefficients in general. The numerical table lookup method requires a considerable amount of computation and a long calculation time. Vector fitting can approximate the analytic functions of the radome transmission coefficients, which will greatly simplify the complexity of analysis and calculation. To describe the relationship between the transmission coefficients and frequency, the rational fraction model, polynomial model, and partial fraction model can be used. In the vector fitting method, the partial fraction model and residues are utilized to approximate the variation relation of transmission coefficients and frequency [19]. Equation (17) shows a general expression of partial fractions:  
\[f(s)=\sum_{j=1}^{N}\frac{c_{j}}{s-a_{j}}+d+e\cdot s, \tag{17}\]  
where \(a_{j}\) is the complex pole, \(c_{j}\) is the complex residue at the pole, \(e\) and \(d\) are real constants, and \(s=j\omega\). If \(a_{j}\) is known, the \(1/(s-a_{j})\) could be seen as a basic function, thus, the least squares method can be used to fit the data. However, in actual situations the poles are unknown and fitting the poles is necessary in order to practically fit the transmission coefficients data. To find the poles and residues, we begin with a set of starting poles \(\left\{a_{j}\right\}\) and build the following functional expressions:  
\[\sigma(s)f(s) = \sum_{n=1}^{N}\frac{c_{n}}{s-\overline{a_{n}}}+d+s\cdot h, \tag{18}\] \[\sigma(s) = \sum_{j=1}^{N}\frac{\widetilde{c_{n}}}{s-\overline{a_{n}}}+1. \tag{19}\]  
Equation (19) multiplied by \(f(s)\) leads to the following equation:  
\[\sum_{n=1}^{N}\frac{c_{n}}{s-\overline{a_{n}}}+d+s\cdot h\ =\left(\sum_{j=1}^{N} \frac{\widetilde{c_{n}}}{s-\overline{a_{n}}}+1\right)f(s). \tag{20}\]  
A set of overdetermined equations contain the coefficients \(c_{n}\), \(d\), \(h\), and \(\widetilde{c_{n}}\) of different frequencies can be acquired, and the unknown coefficients can be obtained by the least squares method [18].  
In order to study the influence of the different incident angles on the transmission coefficients, transmission coefficients functions at different incident angles are obtained by using the B spline interpolation method. In the vertical incident TE wave situation, six complex poles are used for vector fitting the transmission coefficients of PM FSS. The results are compared with the FEM simulations. Figure 4 shows that the vector fitting function and the numerical calculation results are consistent within 2 to 25 GHz in both vertical and 50\({}^{\circ}\) incident angle, which means that our method satisfies the requirements of high frequency analysis.