## 4 Design and Electrical Performance Characterization
### Input, Inter-Resonator, and Output Coupling Characterization
#### 4.4.3 Transmission and Equalization Zeros' Characterization  
This section presents a comprehensive analysis of a novel cross-coupling implementation, beginning with a baseline fourth-order filter employing only direct couplings, designed using the characterization charts previously discussed in Section 4.4.2. The study includes complete coupling matrix analysis and its progressive refinement to achieve enhanced transmission zero control and response equalization. Two distinct approaches were investigated. The first method, well-established in dielectric resonator filter design (particularly in DRCFs), utilizes an iris configuration [16] (Figure 6a), where the aperture functions as a controlled energy window to generate the required cross-coupling. The second approach employs C-TL coupling, which--while limited for direct path coupling as shown in Figure 18b--proves entirely suitable for cross-coupling applications. Regarding potential Q-TEM mode excitation at specific transmission line lengths, the substantially lower coupling factor compared to cross-coupling paths inherently reduces resonant mode impact. For robust operation, the C-TL electrical length is carefully optimized to prevent interference between TE\({}_{01\delta}\) and Q-TEM modes in both near-band and out-of-band regions. As established earlier, the coupling factor of C-TL along dielectric resonators exhibits phase dependence. To investigate cross-coupling behavior, we systematically varied transmission line lengths while maintaining a constant curvature angle \(\theta\), thereby creating different phase relationships between resonator couplings. Figure 19 illustrates the progressive extension of transmission line length for fixed \(\theta\). The characterization results (Figure 20) validate the predicted behavior as follows: the coupling factor varies with phase relationship and, due to the unique properties of this coupling method, undergoes sign reversal (transitioning between inductive and capacitive effects) at \(\lambda_{\text{eff}}/4\) intervals. This characteristic enables the selective implementation of either transmission zeros or equalization zeros. To complete the characterization of transmission and equalization zero behavior, Figure 21 shows their variation as the arc length \(\theta\) in C-TL increases, confirming control of both TZs and EZs through this novel coupling method. The simulation results are presented in Figure 22 for the case without additional transmission line length (Figure 22a) and with extra line length \(\ell_{\text{offset,1}}=\lambda_{\text{eff}}/4\) in Figure 22b. Figure 19: Cross-coupling transmission-line length sweep. This behavior is summarized for relevant cases in Figure 23 using the generalized Chebychev matrix approach, where each \(\lambda_{\text{eff}}/4\) multiple forces a sign reversal, enabling the implementation of both equalization and transmission zeros as required by the design specifications. While this effect is undoubtedly valuable for enabling the straightforward introduction of transmission and reflection zeros, it remains susceptible to near-band Q-TEM mode excitation. Consequently, despite its conceptual interest, each design requires ad hoc optimization to suppress Q-TEM resonant behavior. Given the established understanding  
Figure 21: Cross-coupling variation versus C-TL rotation angle \(\theta\)—simulation results. Figure 22: Electromagnetic cross-coupling characterization. (**a**) Response of the negative sign in the coupling matrix in coefficient M\({}_{41}\). (**b**) Response of the positive sign in the coupling matrix in coefficient M\({}_{41}\). Figure 20: Frequency response variation versus stub length \(\ell_{\text{offset}}\)—simulation results. of C-TL characteristics, an evolution method has been implemented to achieve sign reversal without physically extending the transmission line length--by reversing the arc direction, clockwise to counter-clockwise and vice versa--which directly introduces a sign change in the coupling matrix.