## 2 Theoretical Analysis of the Filter  
The proposed filter comprises two symmetrical step impedance resonator configurations with one end open-circuited and the other short-circuited, as depicted in Figure 2b. It consists of high/low impedances with characteristic impedances \(Z_{2}/Z_{1}\) and electrical lengths \(\theta_{2}/\theta_{1}\), respectively, as shown in Figure 2a. W\({}_{1}\), L\({}_{1}\) and W\({}_{2}\), L\({}_{2}\) are the physical widths and lengths of the low and high impedances. The low impedance sections are attached with 50-ohm input/output ports of the filter, and the high impedance sections are parallel coupled with metallic via at the lower corner having a radius of 0.25 mm. The first two passbands are obtained due to the ratio of the two impedances (R = \(Z_{2}/Z_{1}\)) of the SIR, and the third passband is obtained due to the L-shaped structure coupled to the high impedance section. The resonators are folded in a compact way to introduce magnetic and electric coupling for the generation of multiple TZs in the passbands. The electric coupling, which is a weak coupling in this case, is observed due to space S\({}_{1}\) of the open-circuited stubsof the low-impedance section of the SIR. The magnetic coupling is observed due to gap G\({}_{1}\) of the parallel short-circuited high-impedance section of the SIR. The input admittance Y\({}_{\rm in1}\) (Y\({}_{\rm in}=1/Z_{\rm in}\)) of Figure 2a is calculated as [42; 43];  
\[\rm Y_{in1}=\frac{j\,R(\cot\theta_{1}-\tan\theta_{1}\ )+(\cot\theta_{2}-\tan \theta_{2}\ )}{Z_{1}\ [0.5(\cot\theta_{2}-\tan\theta_{2}\ )(\cot\theta_{1}-\tan \theta_{1})-2R]} \tag{1}\]  
where \(\rm R\ =\frac{Z_{2}}{Z_{1}}\) and \(\rm\alpha=\left(\frac{\theta_{2}}{\theta_{1}+\theta_{2}}\right)\)  
By tuning the impedance ratio R and electrical length ratio \(\alpha\), the first two passbands are obtained. Equation (1) can also be written as;  
\[\rm Y_{in1}=-\frac{jY_{1}R\ -\ tan\theta_{1}\ tan\theta_{2}}{\tan\theta_{1}+R \ tan\theta_{2}} \tag{2}\]  
where  
\[\theta_{1}=\beta_{1}l_{1}=\left(\frac{2\pi}{\lambda}\right)=\left(\frac{2\pi }{\rm v_{p}}\right)\rm fl_{1}\]  
and  
\[\theta_{2}=\beta_{2}l_{2}=\left(\frac{2\pi}{\lambda}\right)=\left(\frac{2\pi }{\rm v_{p}}\right)\rm fl_{2}\]  
By applying the resonance condition, Equation (2) becomes.  
\[\rm R\ -\ \tan\theta_{1}\ tan\theta_{2}=0 \tag{3}\]  
Figure 2: Configuration of resonators (**a**) Stepped impedance resonator (**b**) dual band filter topology (**c**) Equivalent structure of (**b**).  
In terms of resonance frequency ratio K, the unified resonant condition as derived in [44];  
\[\mathrm{R-\ tan\,k_{x}\theta_{1}\ tan\,k_{x}\theta_{2}=0} \tag{4}\]  
where  
\[\mathrm{K_{x}=\frac{f_{2}}{f_{1}}}\]  
The above analysis can be applied to Figure 2a. The same concept can also be applied to Figure 2b whose equivalent circuit is depicted in Figure 2c. So, the corresponding input admittance \(\mathrm{Y_{in2}}\) for Figure 2b is  
\[\mathrm{Yin_{2}=jY_{2}\bigg{[}\ \frac{tan\ (kx\theta_{3}+k\theta_{4})-R-tan\,k_{x }\theta_{1}\ tan\,k_{x}\theta_{2}}{tan\,k_{x}\theta_{1}+R\ \tan\,k_{x}\theta_{2}}\bigg{]}} \tag{5}\]  
Now apply the resonant condition  
\[R-\tan\,k_{x}\ \theta_{1}\ tan\,k_{x}\ (\theta_{2}+\theta_{3}\ +\theta_{4})=0 \tag{6}\]  
The electrical lengths \(\theta_{2}\) +\(\theta_{3}\) +\(\theta_{4}\) can be calculated easily by the required values of R and \(\mathrm{K_{x}}\). So, by adjusting the electrical lengths and impedance ratios, the first two passbands are obtained for GPS and WiMAX applications. Moreover, the final electrical lengths and impedances of Figure 2c on which the resultant SIR filter optimized are \(Z_{0}\) = 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.