## 4 Coupled Cavities Filter in Rectangular Waveguide Loaded with Circular Dielectric Posts
The layout of the SIW filter is shown in Figure 12. In this case, the building blocks were made of metallic vias, as depicted in Figure 13. Some blocks were constituted by vias that form an empty SIW line, and others by vias that form a coupling iris. The filter was designed to have a Chebyshev response centered at 11 GHz and a bandwidth of 1 GHz. The filter hat eight resonant cavities. In this case, the GSM of the building blocks was obtained with the numerical method described in [1], which uses a hybrid method of moments and mode matching method. Here, \(N=M=31\) modes were used. To find the value of \(P\) that provides with accurate results, a study was performed, as in the previous examples, of the variation of the computational time and the error between fast sweep and discrete sweep responses as a function of \(P\). The result is shown in Table 4. It can be observed in this case that the computational times were significantly greater than in the previous examples, as the structure under analysis was much more complex (many metallic vias). The time reduction of the fast sweep when compared to the discrete sweep was smaller than in the first example, and similar to the second one, probably due to the greater complexity of the frequency responses of the building blocks, similar to the second example. The error was stabilized for \(P\geq 5\). Thus, this could be a good choice for \(P\). However, after studying the frequency response of the building blocks, it was observed that with \(P=4\) the accuracy was good enough to properly approximate those responses with splines, although \(P=5\) could also be a good choice. Figure 14 shows the frequency response of various elements of the GSM of various blocks. In all cases, the response was well approximated with four points. The response of the whole SIW filter for fast and discrete sweep is compared in Figure 15. A can be observed, both responses were in good agreement, thus using four points was enough for this filter. In this case, the discrete sweep took 88 s, and the fast sweep 6.8 s. Thus, the use of the fast sweep reduced the computational time to the 7.7% of the time required by the discrete sweep. Figure 5:0 Eight coupled-cavity SIW filter. Segmentation into constitutive blocks (empty lines, and irises). Figure 5:1 Comparison between accurate and approximate scattering parameters of different blocks of the SIW coupled cavities filter, and different combinations of input and output ports and modes. Big dots show the points used for spline approximation. In this case, using four points was enough for high accuracy results. Figure 5:2 Comparison of fast and discrete sweep for the eight-pole SIW filter \begin{table}
\begin{tabular}{c c c c} \hline \hline
**P** & **Fast** & **CPU Time Fast/Discrete** & **Error** \\ \hline
3 & 6.8111 & 7.6923\% & 206.5028 \(\times\)\(10^{-3}\) \\
4 & 7.4745 & 8.4415\% & 48.5913 \(\times\)\(10^{-3}\) \\
5 & 7.7130 & 8.7109\% & 3.2926 \(\times\)\(10^{-3}\) \\
6 & 7.2574 & 8.1963\% & 0.3573 \(\times\)\(10^{-3}\) \\
7 & 7.9938 & 9.0280\% & 0.1536 \(\times\)\(10^{-3}\) \\
8 & 8.9452 & 10.1025\% & 0.0720 \(\times\)\(10^{-3}\) \\
9 & 9.3161 & 10.5214\% & 0.0326 \(\times\)\(10^{-3}\) \\
10 & 9.5422 & 10.7767\% & 0.0134 \(\times\)\(10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Error and CPU time evolution for the SIW filter as the number of frequency points for the fast sweep (_P_) increases. Discrete sweep used 500 points and took 88.54 s.  
The performance of the fast sweep technique presented in this work was compared to the performance of the previous fast sweep techniques implemented in our research group (AWE [2] and CFH [8]). Figure 16 shows the reflection coefficient of the SIW filter of Figure 12. This reflection coefficient was calculated using the three fast sweeps. It can be observed that the AWE technique approximated the response correctly near the central frequency, but it failed to do so in the extremes of the frequency band.