## 5 Filter Design, Simulation, and Experimental Results
### Filter Design, 4-2-0 Topology  
The design of a transmission-zero filter implementation has been successfully developed and is presented in Figure 29a, with the corresponding graph diagram shown in Figure 29b.  
The electromagnetic simulation results are presented in Figure 30a, with the corresponding experimental measurements shown in Figure 30b. The near-band frequency response demonstrates strong correlation between simulation and measurement.  
The reflection coefficient \(S_{11}\) performs better than \(-20\) dB across 90% of the passband (worst-case \(-18\) dB), with symmetrically adjustable transmission zeros achieving over 35 dB rejection in simulation and 30 dB in measurement. The transmission response demonstrates successful implementation and precise control of transmission zeros, marking a significant advancement in dielectric resonator microstrip filter (DRMF) technology for demanding applications.  
Figure 28: Filter design and filter prototype.  
A measurable degradation occurs in both insertion loss and loaded quality factor \(Q_{L}\). The insertion loss increases from 0.7 dB (simulated) to 2.4 dB (measured), corresponding to a \(Q_{L}\) reduction to approximately 3000. Despite this, the measured S-parameters validate the simulated behavior, showing excellent in-band symmetry and flatness--particularly when accounting for bandwidth and insertion loss limitations imposed by input-output coupling constraints.  
This performance degradation stems from multiple factors: the finite conductivity \(\sigma\) of uncoated aluminum cavity surfaces, dielectric losses, the reduced resonator aspect ratio (including \(\pm\)0.1 mm hand-filing tolerances), adhesive uncertainties at operational frequencies (including parasitic effects from adhesive overflow and ground plane voids [29]), deep penetration of tuning screws near resonators and transmission lines, and PCB bonding losses to the carrier. Additional considerations include the test setup measurement error (\(\pm\)0.15 dB) and phase deviations between the C-TL and stub relative to dielectric resonator energy coupling. These factors collectively degrade the overall quality factor.  
The group delay results (Figure 31a for simulation and Figure 31b for measurement) show both total variation and absolute value. The measured filter, affected by the aforementioned factors, exhibits a degraded ground plane (reduced \(Q_{L}\)) and limited input-output coupling, resulting in slightly narrower bandwidth than simulated. Crucially, it confirms proper and controlled transmission-zero operation.  
Finally, both in-band (Figure 32a) and out-of-band behaviors are examined in Figure 32b. For frequencies above the passband, the response remains spurious-free over a 1 GHz span. Below the passband, a controlled peak below \(-\)40 dB appears at 11.15 GHz--even times the filter bandwidth. This frequency, which corresponds to the response  
Figure 30: Frequency response, 4-2-0 topology.  
Figure 29: Filter design, 4-2-0 topology.  
shown in Figure 27a, verifies the quality factor of the weakly excited resonant mode--specifically, the over-penetration mode of a tuning screw within the housing structure for mechanical adjustment.  
The measured coupling matrix in (18) demonstrates a negative \(M_{41}\) coefficient, verifying the standard-compliant cross-coupling behavior for the transmission-zero implementation.  
\[M_{\text{norm}}=\left[\begin{array}{cccccc}0&1.06&0&0&0&0\\ 1.06&0&0.77&-0.07&-0.12&0\\ 0&0.77&0&0.68&0&0\\ 0&-0.07&0.68&0&0.77&0\\ 0&-0.12&0&0.77&0&1.04\\ 0&0&0&0&1.04&0\\ \end{array}\right] \tag{18}\]