## 4 Band Pass Filter Based on the Second Iteration of the Minkowski Fractal Curve as DGS  
Figure 10 demonstrates the second iteration of the Minkowski curve with dimensions Li2 = Wi2 = 0.45 mm and Di2 = 0.37 mm. Figure 11 shows the structure of the proposed BPF with comb slots on the upper metallic layer and the second iteration fractal DGS on the ground plane.  
The equivalent circuit of the proposed design with parameter values is shown in Figure 12. The SIW structure acts as a two-wire transmission line; therefore, inductor L\({}_{\rm v}\) symbolizes metalized vias. A parallel resonator tank circuit is analogous to defective ground structures (DGSs) etched on the ground plane, which acts as a band stop filter [25]. Hence, the fractal DGS resonator is represented by Lr and C\({}_{\rm r}\), where the values of parameters Lr and Cr can be found using [26]. The electric and magnetic coupling due to comb-shaped slots at the input and output are represented by L\({}_{\rm c}\) and C\({}_{\rm c}\).  
The simulated scattering response of [PERSON]'s fractal-based filter is shown in Figure 13. It is clear from the response that the extended electrical length has caused the lower frequency to shift to 4.1 GHz. Furthermore, it can be shown that the proposed filter's 3 dB bandwidth of 1.05 GHz (4.1 GHz-5.15 GHz) has been increased by the inclusion of the DGS. Moreover, it attains a deep and wide-ranging stopband (\(>\)\(-\)28 dB) until approximately 2\(f_{0}\). The insertion loss is found to be 0.72 dB. The roll-off rate at the higher and lower edges of the passband is 68.7 dB/GHz and 73.4 dB/GHz, respectively. Also, it is evident that it produces three transmission zeros at 3.17 GHz, 6.2 GHz, and 6.9  
Figure 11: BPF based on the second iteration level of the Minkowski curve.  
Figure 12: Equivalent circuit of the proposed filter. The element values are Lc = 0.12 nH, Cc = 4.5 pF, Lv = 1.88 nH, Lr = 0.18 nH, and Cr = 7.22 pF.  
Figure 10: Second iteration of the Minkowski curve (Li2 = Wi2 = 0.45 mm, Di2 = 0.37 mm).  
augments selectivity and ameliorates the upper stopband performance. Sub-structures of the resonators come nearer, and the included length increases as the iteration degree rises while maintaining a constant side length. It produces capacitive coupling, which enhances the skirt properties of the filter at both edges [27]. This also results in an improvement in the filter's stopband performance [28].  
The equivalent circuit was validated using the Advanced Design System (ADS ver. 2019) software tool, which was used to design an equivalent circuit with RLC (as shown in Figure 12) and optimized it to obtain a nearly similar S parameter response, also shown in Figure 14, which compares the response of EM HFSS simulation and circuit simulation.  
Figure 15 depicts the S21 response of the comb-slotted MMR filter with a DGS under weak coupling. Figure 16 illustrates the mode chart of the projected MMR filter with a DGS. It is evident from the mode chart that the desired passband is formed by the resonant modes (at f = 4.1, 4.25, 4.8, and 5.2 GHz) that shift down to lower frequencies after DGS is incorporated.  
Figure 14: S parameters of EM simulation and circuit simulation.  
Figure 13: BPF frequency response based on the Minkowski curve’s second iteration level as the DGS.  
Figure 15: Simulated S21 response of a comb-slotted MMR filter with a DGS under weak coupling.  
Figure 17 portrays the current distribution in (a) the passband and (b) the stopband of the proposed BPF.