## 4 Results and Discussion  
A high selectivity with eight transmission zeros triband bandpass filters is designed in this study. The proposed prototype combines low impedance section \(Z_{1}\) with one end open circuited and high impedance section \(Z_{2}\), which is folded to introduce coupling phenomena and size reduction with metallic via at lower edges. Later, a half-wavelength L-shaped resonator is further utilized for multi-passbands. The first two passbands for the applications of GPS and WiMAX are obtained due to the quarter wavelength step impedance resonator with electrical parameters \(Z_{0}=50\)\(\Omega\), \(Z_{1}=71\)\(\Omega\), \(Z_{2}=50\)\(\Omega\), \(\theta_{1}=77\), \(\theta_{2}=78.94\), \(\theta_{3}=88.6\), and \(\theta_{4}=15\) and the resonance frequency ratio \(\mathrm{K_{x}}=\mathrm{f_{2}}/\mathrm{f_{1}}\) which is 2.22 in this case. The electrical lengths of the proposed SIR can be changed to observe the first and second passbands. Consider Figure 4, which shows the variation of the first and second passband for electrical stub lengths \(\theta_{1}\), where the upper cutoff frequency of the first band decreases with an increase of stub length \(\theta_{1}\) and the upper and lower cutoff frequencies of the second band varied with no effect on third passband. Figure 5 shows that with the increase of electrical stub length \(\theta_{4}\), only the lower cutoff frequency of the first passband and upper cutoff frequency of the second passband decreases with no effect on the third passband. This reveals that the first and second passbands are generated with the ratio of impedances and electrical stub lengths of the SIR. The third passband for satellite application is obtained due to the L-shaped structure embedded in the SIR. Thus, a triple band filter with multiple T.Z.s at different locations is obtained with the central frequencies 1.57 GHz, 3.57 GHz, and 4.2 GHz. The simulated results of the proposed prototype with and without the embedded L-shaped structure are illustrated in Figures 6 and 7, where it is observed that the third passband is obtained due to the half-wavelength L-shaped resonator without affecting the first two passbands. Moreover, the third passband is controllable, and by varying the resonator length L\({}_{4}\), the upper cut-off frequency is moved downward compared to the lower cut-off frequency without affecting the remaining passband cut-off frequencies, as shown in Figure 8, respectively. Figure 3: (a) Proposed triband geometry. (b) Equivalent circuit model. To improve the selectivity of the filter, the structure is folded in a way that it can produce a coupling phenomenon; thus, a total of eight T.Z.s, including 1.22 (TZ\({}_{1}\)), 1.42 (TZ\({}_{2}\)), 1.98 (TZ\({}_{3}\)), 3.18 (TZ\({}_{4}\)), 3.82 (TZ\({}_{5}\)), 3.98 (TZ\({}_{6}\)), 4.38 (TZ\({}_{7}\)), and 4.53 (TZ\({}_{8}\)) have been observed between the passbands as shown in Figure 9. Due to these T.Z.s, the roll-off rates (\(\xi\)) of the three passbands increased. A higher roll-off rate indicates better selectivity, and it can be defined by regarding \(-\)20 dB attenuation and \(-\)3 dB attenuation as two cut-off points for the passband. It is equal to the rate of 17 dB with the bandwidth of roll-off [45]. Thus, the roll-off rates of the three passbands reach up to 294 dB/GHz (\(\xi_{1}\)), 170 dB/GHz (\(\xi_{2}\)), 106 dB/GHz (\(\xi_{3}\)), 242 dB/GHz (\(\xi_{4}\)), 106 dB/GHz (\(\xi_{5}\)), and 121 dB/GHz (\(\xi_{6}\)), respectively. The TZ\({}_{1}\) and TZ\({}_{6}\) are generated due to the mixed-coupling of the SIR, TZ\({}_{2}\) and TZ\({}_{5}\) are generated by the SIR itself, TZ\({}_{3}\) and TZ\({}_{4}\) are produced due to the metallic hole etched to the lower end of the high-impedance section of the SIR, while TZ\({}_{7}\) and TZ\({}_{8}\) are generated by the embedded L-shaped structure coupled to the low impedance of the SIR, respectively. Moreover, the proposed tri-band BPF has a maximum roll-off rate of up to 294 dB/GHz, demonstrating its great selectivity and abrupt roll-off.