## 2 Proposed Structure: Analysis and Design Methodology
### Design Methodology  
When designing a dual-band filter, it is very important that the two passbands can be independently tuned in terms of both center frequency and fractional bandwidth (FBW). In the case at hand, the first point that should be noted is that the middle layer of the structure introduces the lower passband, centered at \(f_{01}^{d}=1.575\) GHz, the L1 GPS signal frequency, while the top layer resonates at the first Wi-Fi band frequency, \(f_{02}^{d}=2.45\) GHz. This means that the upper frequency band should be allocated by setting the top layer parameters while lower frequency band must depend, exclusively, on the middle layer geometry. Let's start by studying the resonance frequency of resonators in both layers. In Figure 5, the differential return loss, \(|S_{11}^{dd}|\), have been represented when a single couple of top and middle resonators are weakly excited. The dips in the return loss correspond to the resonance frequencies of both resonators. Several parameters from each resonator were varied to show the dependence of those resonance frequencies with the resonator's length. From the simulated results (Figure 5) the following conclusions can be inferred:  
1. If middle layer resonators length is kept constant, changing top layer resonators length only affects the upper band center frequency, with very slight effect on the lower band center frequency. 2. If top layer resonators length is kept constant, changing middle layer resonators length only affects the lower band center frequency with negligible effect on the upper band center frequency. Figure 3: Detailed view of the broadside capacitance involved in the feeding of middle layer resonators. From these results, it is clear that there is flexibility in locating the center frequencies of each passband as long as both frequencies are not very close to each other. In this latter case, resonators on both layers would interact significantly, thus leading to band-to-band dependence breaking the asynchronous behavior between the pair of resonators. This situation would not be desirable for designing purposes. Furthermore, the independence of the passbands is also related to the excitation mechanism of each resonator pair. As mentioned before, top layer resonators are mainly excited due to the electric field confined within the gap of width \(s_{2}^{t}\) (gap capacitance) while middle layer resonators are excited by means of the electric field concentrated between the feeding lines and the resonator conductor strips (broadside capacitance). This fact, together with the sufficient separation of the two center frequencies, leads to the desired independence exhibited when locating the center frequencies of the differential passbands. Once the independent control of the resonance frequencies has been demonstrated, the next electrical parameter to be considered in the design process is the correct excitation of the resonators to generate two properly matched differential passbands. In this regard, as mentioned above, the parameters that control top and middle layer resonators excitation are mainly \(s_{2}^{t}\) and \(l_{\text{over}}\), respectively. In order to investigate on the influence of such parameters on the matching level of both differential passbands, i.e., \(|S_{11}^{dd}|\), Figure 6 shows the differential return loss of the two differential passbands as a function of \(s_{2}^{t}\) and \(l_{\text{over}}\). From this figure it is clear that \(l_{\text{over}}\) mainly influences the reflection coefficient of the lower differential passband while \(s_{2}^{t}\) controls the reflection coefficient of the upper differential passband. Note that \(s_{2}^{t}\) slightly modifies the value of the upper center frequency. However, this problem can be solved by barely modifying some of the other dimensions of the top layers resonators in a very simple manner, as explained in [26]. Finally, the last step in the design process is to verify that the fractional bandwidth of the two differential passbands can be set independently.