## 6 Discussion  
In this work, a new fast frequency sweep technique is presented. It consists in segmenting the device under analysis into simple building blocks, and characterizing each block with its generalized circuital matrix. Since each element of that matrix presents a simple and flat response over the frequency, it is approximated with very few points with natural cubic splines. The spline approximation provides accurate response of the block along the frequency band with as many points as desired in a very efficient way, thus saving computational time. The circuital matrices of all the blocks are then cascaded, and the scattering parameters of the whole device are finally obtained. To test the performance of the technique with devices that contain a wide variety of possible building blocks, several types of devices were analyzed. All metallic and dielectric loaded filters in rectangular waveguide, as well as a filter in substrate integrated waveguide were analyzed. Therefore, the analyzed building blocks were empty lines, irises and dielectric posts in rectangular waveguide, as well as metallized circular vias. Depending of the block, the complexity of the frequency response differs, and a different number of points was needed for the spline approximation, but, in the worst case, it was enough to use five points. As a result, the new fast frequency provided accurate results with a time reduction that ranged from 9% to 19%, depending on the device, when compared with the discrete frequency sweep with the same electromagnetic solver. Figure 16: Comparison of discrete sweep of this work with fast sweeps of AWE [2] and CFH [8]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Method** & **CPU Time** \\ \hline This work & 4.004 s \\ AWE & 4.8321 s \\ CFH & 36.8805 s \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of fast frequency methods. When compared to other fast frequency sweep techniques, such as AWE or CFH, the method of this work performed much better. It can be concluded that, if the structure is divided into the smallest possible building blocks (such as empty lines, irises, dielectric posts, or metallic vias), the method is accurate and fast. The smaller are the building blocks, the simpler is the frequency response of each individual block, and the greater is the final accuracy using this fast frequency method. At the same time, when the building blocks are very small, their scattering matrices can be reused for the same building block appearing in another part of the structure, and the total computation time is reduced. Although the fast frequency method was applied here to inline structures, it could also easily be applied to more complex structures such as cross-coupled filters or folded dual-mode filters. The difference is that, in those more complex structures, some individual blocks should be three- or four-port devices (such as a three port T-junction). However, even a T-junction has a simple frequency response that can be easily approximated with splines with a small number of frequency points. Conceptualization, A.B. ; Methodology, A.B., J.A.M. and H.E. ; Software, A.B., J.A.M. and H.E. ; Formal analysis, A.B., J.A.M. and H.E. ; Validation, A.B., J.A.M. and H.E. ; Investigation, A.B., J.A.M. and H.E. ; Resources, A.B. ; Writing--Original Draft Preparation, J.A.M. ; Writing--Review and Editing, H.E. ; Visualization, A.B., J.A.M. and H.E. ; Supervision, A.B. and H.E. ; Project Administration, A.B. ; and Funding Acquisition, A.B. and H.E. This research was funded by _Ministerio de Economia, Industria y Competitividad_, Spanish Government, under Research Projects TEC2016-75934-C4-3-R and TEC2016-75934-C4-1-R. The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results. The following abbreviations are used in this manuscript: