## 3 Coupled Cavities Filter in Rectangular Waveguide  
The performance of the fast frequency technique proposed in this work was first tested with the analysis of an all-metallic coupled cavities filter in rectangular waveguide technology. The layout of this filter is shown in Figure 1. The filter was designed for a Chebyshev frequency response centered at 15 GHz, with a bandwidth of 450 MHz, and a ripple of 0.02 dB in the pass band. To test the accuracy and speed of the fast frequency technique for different sizes of the whole device, the filter was designed for different numbers of resonators \(R\), with \(R\) varying within 3-9. 1. **R = 3:**\(\mathrm{w}_{1}=\mathrm{w}_{4}=8.314\) mm, \(\mathrm{w}_{2}=\mathrm{w}_{3}=5.787\) mm, \(\mathrm{l}_{1}=\mathrm{l}_{3}=9.990\) mm, \(\mathrm{l}_{2}=11.201\) mm
2. **R = 4:**\(\mathrm{w}_{1}=\mathrm{w}_{5}=8.178\) mm, \(\mathrm{w}_{2}=\mathrm{w}_{4}=5.551\) mm, \(\mathrm{w}_{3}=5.124\) mm, \(\mathrm{l}_{1}=\mathrm{l}_{4}=10.152\) mm, \(\mathrm{l}_{2}=\mathrm{l}_{3}=11.503\) mm  
Figure 1: Layout of an all-metallic coupled cavities filter in rectangular waveguide. The filter has been design for different number of resonant cavities \(R\). In all cases, the width of the guide is \(a=15.7988\) mm, the height is \(b=7.8994\) mm, and the length of the inductive irises is \(t=2\) mm. Below, the remaining dimensions for different filter order \(R\) are listed. \(\mathbf{R=5}\): w\({}_{1}\) = w\({}_{6}\) = 8.114 mm, w\({}_{2}\) = w\({}_{5}\) = 5.457 mm, w\({}_{3}\) = w\({}_{4}\) = 4.959 mm, l\({}_{1}\) = l\({}_{5}\) = 10.217 mm, l\({}_{2}\) = l\({}_{4}\) = 11.581 mm, l\({}_{3}\) = 11.733 mm
4. \(\mathbf{R=6}\): w\({}_{1}\) = w\({}_{7}\) = 8.072 mm, w\({}_{2}\) = w\({}_{6}\) = 5.404 mm, w\({}_{3}\) = w\({}_{5}\) = 4.887 mm, w\({}_{4}\) = 4.813 mm, l\({}_{1}\) = l\({}_{6}\) = 10.258 mm, l\({}_{2}\) = l\({}_{5}\) = 11.618 mm, l\({}_{3}\) = l\({}_{4}\) = 11.793 mm
5. \(\mathbf{R=7}\): w\({}_{1}\) = w\({}_{8}\) = 8.043 mm, w\({}_{2}\) = w\({}_{7}\) = 5.370 mm, w\({}_{3}\) = w\({}_{6}\) = 4.850 mm, w\({}_{4}\) = w\({}_{5}\) = 4.753 mm, l\({}_{1}\) = l\({}_{7}\) = 10.285 mm, l\({}_{2}\) = l\({}_{6}\) = 11.639 mm, l\({}_{3}\) = 1\({}_{5}\) = 11.818 mm, l\({}_{4}\) = 11.845 mm
6. \(\mathbf{R=8}\): w\({}_{1}\) = w\({}_{9}\) = 8.030 mm, w\({}_{2}\) = w\({}_{8}\) = 5.349 mm, w\({}_{3}\) = w\({}_{7}\) = 4.829 mm, w\({}_{4}\) = w\({}_{6}\) = 4.725 mm, w\({}_{5}\) = 4.703 mm, l\({}_{1}\) = l\({}_{8}\) = 10.298 mm, l\({}_{2}\) = l\({}_{7}\) = 11.652 mm, l\({}_{3}\) = l\({}_{6}\) = 11.832 mm, l\({}_{4}\) = l\({}_{5}\) = 11.866 mm
7. \(\mathbf{R=9}\): w\({}_{1}\) = w\({}_{10}\) = 8.028 mm, w\({}_{2}\) = w\({}_{9}\) = 5.343 mm, w\({}_{3}\) = w\({}_{8}\) = 4.820 mm, w\({}_{4}\) = w\({}_{7}\) = 4.710 mm, w\({}_{5}\) = \({}_{6}\) = 4.678 mm, l\({}_{1}\) = l\({}_{9}\) = 10.300 mm, l\({}_{2}\) = l\({}_{8}\) = 11.655 mm, l\({}_{3}\) = l\({}_{7}\) = 11.837 mm, l\({}_{4}\) = l\({}_{6}\) = 11.875 mm, l\({}_{5}\) = 11.883 mm  
To implement the fast frequency technique, the filter had to be segmented into simple building blocks, as shown in Figure 2. In this case, these building blocks were either empty sections of rectangular waveguide (labeled as _Line i_ in Figure 2) or H-plane inductive irises (labeled as _Iris i_ in Figure 2). The generalized scattering matrix (GSM) of the empty line sections could be computed analytically [23], and the GSM of the irises had to be obtained using a numerical method. In this case, the mode matching method described in [24] was used, although any other method could also have been used. The fast frequency sweep procedure was not affected by the particular choice of analysis methods used for computing the scattering matrices of the building blocks, thus its performance would be the same if other simulators were used for obtaining these matrices.