## 2 Design of the Wideband GGW Bandpass Filter
### Filter Design  
The design of the proposed fifth-order GGW bandpass filter (Figure 1) is performed by combining the high-pass behavior of the GGW with a low-pass Chebyshev filter synthesis method based on stepped impedances [46; 47], thus allowing us to control its bandwidth. The lower cut-off frequency (\(f_{c,L}\)) of this filter was previously adjusted to the desired value (Table 1), optimizing the GGW dimensions as in Section 2.1. Following this, the synthesis method based on stepped impedances was applied to obtain the transfer function of the fifth-order Chebyshev filter according to the design specifications (\(f_{c,H}\), \(RL\), \(\theta_{e}\)) required by Table 1. The five characteristic impedances (Figure 5a) of the filter are extracted by applying an iterative algorithm to the [ABCD] matrix and the previously determined transfer function [47]. To achieve a practical implementation of the filter as in Figure 1, the five stepped impedance transmission lines (\(Z_{i},i=1,\ldots,5\)) with electrical length (\(\theta_{e}\)) in Figure 5a are transformed into six sections (Figure 5b) of same impedance (\(Z_{0}\)) and electrical length (\(\theta_{e}\)) by means of impedance inverters (\(K_{i},i=1,\ldots,6\)). In Figure 5b, the impedance inverter \(K_{i}\) corresponds to the metallic pin \(t_{xi}\times t_{yi}\ \times\ t_{zi}\) of the \(i\)th-section of the filter (Figure 3), while the electrical length \(\theta_{e}\) is related with the physical section length \(L_{i}\). The normalized impedance inverter \(K_{i}\) and \(S_{i}\)-parameter are given by:  
\[K_{i}=\frac{1}{\sqrt{Z_{i-1}^{\prime}Z_{i}^{\prime}}}\ \text{and}\ i=1,\ldots,n+1, \tag{1}\]  
\[S_{11,i}=S_{22,i}=\frac{K_{i}/Z_{0}-Z_{0}/K_{i}}{K_{i}/Z_{0}+Z_{0}/K_{i}}e^{- i\theta_{e}}, \tag{2}\]  
\[S_{12,i}=S_{21,i}=\frac{2}{j(K_{i}/Z_{0}+Z_{0}/K_{i})}e^{-j\theta_{e}}, \tag{3}\]  
where \(n\) corresponds to the order of the filter, \(Z_{i}^{{}^{\prime}}=Z_{i}\) if \(i\) is odd or \(Z_{i}^{{}^{\prime}}=1/Z_{i}\) if \(i\) is even, and \(Z_{0}=Z_{S}=Z_{L}=1\). Figure 4: \(|S_{21}|\) of the simulated \(i\)th-section (Figure 3) as a function of the dimensions of the metallic pins: \(t_{xi}\) (\(t_{zi}\) = 2 mm and \(t_{yi}\) = 4 mm fixed), \(t_{yi}\) (\(t_{xi}\) = \(t_{zi}\) = 2 mm fixed) and \(t_{zi}\) (\(t_{xi}\) = 2 mm and \(t_{yi}\) = 4 mm fixed). Table 3 gives the element values (\(K_{i}\), \(|S_{21,i}|\)) of the six sections of the equivalent circuit (Figure 5b), which were obtained from the stepped impedance Chebyshev synthesis method and for the design specifications given in Table 1. All sections are selected to have the same phase shift value: \(\angle S_{21,i}=-\theta_{e}-90^{\circ}=-120^{\circ}\). The physical \(t_{yi}\) and \(L_{i}\) dimensions of each section are optimized by means of an EM simulator (HFSS) to match the \(|S_{21,i}|\)-magnitude and \(\angle S_{21,i}\)-phase values included in Table 3. For this purpose, the same GGW dimensions included in Table 2 are used, and the widths \(t_{xi}\) and \(t_{zi}\) of the metallic pins are set at 2 mm. The GGW filter is built with the different previously designed sections and a final optimization is performed to compensate for the couplings. Table 4 includes the final dimensions of the fifth-order GGW Chebyshev filter shown in Figure 1 obtained from this design approach to achieve the design specifications given in Table 1. The size of the filter is: 47.48 mm (length) \(\times\) 47.05 mm (width) \(\times\) 9.525 mm (height). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline \(t_{y1}=t_{y6}\) & 3.09 & \(L_{1}=L_{6}\) & 8.985 \\ \(t_{y2}=t_{y5}\) & 4.19 & \(L_{2}=L_{5}\) & 7.835 \\ \(t_{y3}=t_{y4}\) & 4.56 & \(L_{3}=L_{4}\) & 6.96 \\ \(t_{xi}=t_{zi}\) & \(i=1,\ldots,6\) & 2 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Dimensions of the fifth-order wideband GGW Chebyshev filter (Figure 1) for the design specifications included in Table 1.  
Figure 5: Equivalent circuits of a fifth-order stepped impedance filter.