## 4 Design and Electrical Performance Characterization
### Input, Inter-Resonator, and Output Coupling Characterization
#### 4.4.3 Transmission and Equalization Zeros' Characterization
Notably, the model exhibits symmetry, making the behavior equivalent, regardless of rotation direction, as long as the relative rotation or phase relationship between resonators is maintained. To cross-validate this behavior, a cross-coupling between DR1 and DR3 has been forced, as this configuration introduces circuit asymmetry, generating an upper-band transmission zero when the phase relationship is positive, by convention, or a lower-band transmission zero when negative. Figure 24 presents the validation filters demonstrating coupling matrix sign change through asymmetric coupling generation in Figure 25. The directional reversal of arc \(\theta\) successfully converts the response between transmission-zero and equalization-zero behavior. Figure 25 presents the frequency responses of both electromagnetically simulated filters and their comparison with the ideal coupling matrix obtained from generalized coupling matrix synthesis. The results demonstrate the sign inversion behavior in the coupling matrix achieved through a simple rotation that reverses the TE\({}_{01,6}\) mode's electric field distribution, as described by the CM in (17). The key difference between both  
Figure 23: Generalized Chebychev matrix, cross-coupling model. (**a**) Negative sign in M\({}_{41}\)—cross-coupling w/o offset. (**b**) Positive sign in M\({}_{41}\)—cross-coupling with offset \(L_{\text{offset,1}}=\lambda/4\). (**c**) Negative sign in M\({}_{41}\)—cross-coupling with offset \(L_{\text{offset,2}}=\lambda/2\). Figure 24: Cross-coupling \(M_{31}\) characterization design. (**a**) Negative sign in the coupling matrix in coefficient \(M_{31}\). (**b**) Negative sign in the coupling matrix in coefficient \(M_{31}\). (**c**) Positive sign in the coupling matrix in coefficient \(M_{31}\). (**d**) Positive sign in the coupling matrix in coefficient \(M_{31}\). Figure 23: Generalized Chebychev matrix, cross-coupling model. (**a**) Negative sign in M\({}_{41}\)—cross-coupling w/o offset. (**b**) Positive sign in M\({}_{41}\)—cross-coupling with offset \(L_{\text{offset,1}}=\lambda/4\). (**c**) Negative sign in M\({}_{41}\)—cross-coupling with offset \(L_{\text{offset,2}}=\lambda/2\). coupling matrices lies in the cross-coupling factor M\({}_{13}\) (sign reversal), which verifies the effectiveness of the proposed coupling method. Furthermore, the results highlight common parameter values between both matrices, where self-resonance terms M\({}_{ii}\) are also adjusted to compensate for matrix asymmetry, and consequently, the frequency response asymmetry. The coupling matrix elements that vary between the two frequency responses are as follows: M\({}_{31}\) = M\({}_{13}\) = 0.32, M\({}_{11}\) = \(-\)0.02, M\({}_{22}\) = 0.21, and M\({}_{33}\) = \(-\)0.04. \[M= \begin{bmatrix}0&1.00&0&0&0&0\\ 1.00&M_{11}&0.79&\pm M_{13}&0&0\\ 0&0.79&M_{22}&0.60&0&0\\ 0&\pm M_{31}&0.60&M_{33}&0.85&0\\ 0&0&0&0.85&M_{44}&1.00\\ 0&0&0&0&1.00&0\end{bmatrix}  \tag{17}\]  
The results demonstrate that the principal design limitation for filters combining both transmission zeros and equalization zeros--namely the increased physical length requirement--has been successfully addressed through a simpler, more easily implemented solution. Leveraging the established sign-dependent behavior of the coupling matrix, this work proposes a novel cross-coupling evolution technique as follows: maintaining identical transmission-line lengths while physically reversing the arc orientation (clockwise to counter-clockwise or vice versa). This straightforward PCB-level modification effectively inverts the matrix sign while preserving the filter response, offering significant advantages from a manufacturability perspective.