## 5 Filter Design, Simulation, and Experimental Results
### Filter Design, 4-0-2 Topology  
The design of an equalization-zero filter implementation has been successfully developed and is presented in Figure 33a, with the corresponding graph diagram shown in Figure 33b.  
The electromagnetic simulation results from ANSYS HFSS(r) 2024 R1 appear in Figure 34a, with corresponding experimental measurements in Figure 34b. The frequency response demonstrates a reflection coefficient \(S_{11}<-20\) dB across 82% of the passband (minimum \(-16.5\) dB), featuring symmetric adjustable equalization zeros that achieve 0.28 dB peak-to-peak insertion loss variation across the operational bandwidth.  
Figure 31: Group delay response, 4-2-0 topology.  
Figure 32: Frequency response in specific regions, 4-2-0 topology.  
Measured insertion loss increases from 0.7 dB (simulated) to 1.9 dB while maintaining 1 GHz spurious-free operation from band edges. This configuration's quality factor (approximately 4000) exceeds that of the transmission-zero-based topology, underscoring the importance of characterizing dominant quality factor degradation mechanisms as previously analyzed.  
As shown in Figure 35, the equalization zeros behave as expected, maintaining a flatter in-band response, particularly at the band edges.  
The in-band and out-of-band performance are characterized in Figure 36a,b. Notably, the slope at the lower band edge (Figure 36a) results from the equalization zero position, which aligns with the group delay response shown in Figure 35b, at lower frequencies within the passband.  
Figure 34: Frequency response, 4-0-2 topology.  
Figure 35: Group delay response, 4-0-2 topology.  
Regarding far-band behavior, a resonant mode appears at 11 GHz (twelve times the filter bandwidth), sufficiently distant from the passband. Above the passband, another mode emerges at 12.40 GHz (ten times the bandwidth, centered at 11.8 GHz), maintaining \(-32\) dB suppression in transmission with low quality factor.  
The measured coupling matrix in (19) exhibits a positive M\({}_{41}\) coefficient, confirming successful implementation and control of equalization zero pairs.  
\[M_{\text{norm}}=\left[\begin{array}{cccccc}0&1.02&0&0&0&0\\ 1.02&0&0.86&0.03&0.08&0\\ 0&0.86&0&0.68&0&0\\ 0&0.03&0.68&0&0.90&0\\ 0&0.08&0&0.90&0&1.04\\ 0&0&0&0&1.04&0\end{array}\right] \tag{19}\]  
Finally, the summary of the experimental results obtained for both filter designs are shown in Tables 4 and 5.  
In addition, a comparative analysis of the three technologies is presented in Table 6. The results demonstrate successful positioning of DRMFs as an intermediate solution between conventional microstrip filters and DRCFs. The proposed DRMF technology achieves balanced electrical performance, making it a viable candidate for space system applications.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value/Type** \\ \hline Bandwidth (BW) & 45.0 MHz \\ Insertion Loss (IL) & 2.4 dB \\ Quality Factor (\(Q_{L}\)) & 3000 \\ Return Loss (RL) & 18.0 dB \\ Transmission zeros & \(-30\) dBc \\ Group Delay Variation & 16 ns \\ \hline
**Tuning Type** & **Implementation** \\ \hline I/O Coupling & C-TL feeder, screw over TL \\ Inter-resonator Coupling & Screw between resonators \\ Frequency Tuning & Screw over resonators \\ TL (length in multiples of N\(\lambda/4\), with \(N\in\{2,4,6,\dots\}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 4: 4-2-0 Topology with Transmission Zeros (TZs)  
Figure 36: Frequency response in specific regions—4-0-2 topology.