## 4 Coupled Cavities Filter in Rectangular Waveguide Loaded with Circular Dielectric Posts  
To test the fast frequency technique with a device with more complex building blocks than in the all-metallic filter, a coupled cavities filter with dielectric resonators in rectangular waveguide was analyzed. The geometry of the filter is shown in Figure 7. Each resonant cavity was loaded with a cylindrical dielectric post. The dielectric posts reduced the size of the filter, at the expense of increasing the insertion losses. The filter was designed for a four-pole Chebyshev frequency response centered at 11 GHz and with a bandwidth of 300 MHz, thus the filter had four resonant cavities, and four dielectric posts. Figure 5: Computation time of discrete and fast frequency sweeps for the all-metallic coupled cavities filter for different filter orders \(R\). Figure 6: Percentage of computation time of the fast frequency sweep when compared with the discrete sweep for different filter orders \(R\). The segmentation of the filter into simple building blocks is illustrated in Figure 8. In this case, there were three types of building blocks. Besides sections of empty lines and irises, in this filter, there were also dielectric posts. As in the previous filter, the GSM of the empty lines were obtained analytically [23], and the GSM of the irises were obtained with mode matching [24]. The GSM of the dielectric posts could be computed using, for example, the hybrid mode matching method in [25]. Here, \(N=M=11\) modes were used to compute the GSM of all building blocks. The frequency response of the dielectric posts was not as flat as the response of the irises. For that reason, the adequate value of \(P\) might be different from in the previous filter. Table 3 shows the performance of the fast frequency sweep for different values of \(P\) for the dielectric loaded filter. As can be observed, the error was bigger than in the case of the all-metallic filter. It can also be observed that the percentage of computational time of the fast sweep when compared with the discrete sweep was around 9%, which is smaller than in the all-metallic filter. This was due to the higher computational cost of the dielectric posts, which means there was a greater gain in computational cost with the fast sweep. The error was reduced as \(P\) increased (and the computational time was slightly increased). The exact value of \(P\) from which the results were accurate can be better derived from the results in Figure 9. In that figure, the approximate frequency response of the element \(S_{1,1}^{(1,1)}\) of the GSM for the first dielectric post of the filter is depicted for different values of \(P\). That approximate frequency response was compared with the accurate result. It can be appreciated that the frequency response was more complex than that of the iris, and therefore in this case \(P=5\) points had to be used to get an accurate approximation with splines. Besides, as shown in Table 3, the error decreased very quickly for \(P=3\) and \(P=4\), but was stabilized for \(P\geq 5\), thus \(P=5\) was a good compromise between accuracy and efficiency. Figure 10 shows the frequency response of other elements of the GSM with \(P=5\), proving that with \(P=5\) all responses were well approximated. Figure 11a,b shows the frequency response of the whole filter with, respectively, \(P=4\) and \(P=5\). With four points for the spline approximation, the frequency response of the posts was not  
Figure 8: Segmentation of the dielectric loaded filter into simple blocks (empty lines, irises, and dielectric posts). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(P\) & **Fast** & **CPU Time Fast/Discrete** & **Error** \\ \hline
3 & 0.5527 & 8.2142\% & \(253.1607\times 10^{-3}\) \\
4 & 0.5882 & 8.7405\% & \(36.3357\times 10^{-3}\) \\
5 & 0.5985 & 8.8942\% & \(8.0251\times 10^{-3}\) \\
6 & 0.6082 & 9.0  
rectangular waveguide synthesized in a dielectric substrate with circular metallic vias that acted as the lateral walls of the waveguide.