## 3 Band Pass Filter Based on the First Iteration of the Minkowski Fractal Curve  
Fractal geometries were first proposed by [PERSON] in 1953 [PERSON] in 1953 [PERSON], and since then, they have been broadly used in diverse engineering domains. A fractal is a geometric structure that fills up space and has the same statistical properties throughout all of its parts. Fractal shapes can deliver big surface areas in a restricted amount of space as they are created through a recursive process. A generator curve can be designed using fractal geometry using the Iteration Function System (IFS) [PERSON] (1953). It is an elementary unit for creating the fractal structure's subsequent iterations. The IFS can be written using matrices given in (1).  
\[\mathrm{H}\begin{bmatrix}\mathrm{X}\\ \mathrm{Y}\end{bmatrix}=\begin{bmatrix}\mathrm{a}&\mathrm{b}\\ \mathrm{c}&\mathrm{d}\end{bmatrix}\begin{bmatrix}\mathrm{X}\\ \mathrm{Y}\end{bmatrix}+\begin{bmatrix}\mathrm{e}\\ \mathrm{f}\end{bmatrix} \tag{1}\]  
Two elements, \(\mathrm{e}\) and \(\mathrm{f}\), control the linear translation, while four elements, \(\mathrm{a}\), \(\mathrm{b}\), \(\mathrm{c}\), and \(\mathrm{d}\), control the scaling and direction. Presume that the sequences \(\mathrm{H}1\), \(\mathrm{H}2\), \(\ldots\), \(\mathrm{H}\mathrm{n}\) are linear affine transformations. Through a series of transformations on \(\mathrm{H}\) (A), it can be represented by means of Equation (2). \(\mathrm{H}\) is called the Hutchinson operator and is used to obtain the fractal geometry.  
\[\mathrm{H}\;(\mathrm{A})=\bigcup_{\mathrm{n=1}}^{\mathrm{N}}\;\mathrm{H}_{ \mathrm{n}}\;(\mathrm{A}) \tag{2}\]  
Figure 6 illustrates the first three iterations of the Minkowski fractal curve, and Figure 7 shows the first iteration of the Minkowski curve with dimensions as \(\mathrm{Li}=\mathrm{Wi}=1.45\;\mathrm{mm}\) and \(\mathrm{Di}1=1.25\;\mathrm{mm}\). Figure 8 depicts a comb-slotted HMSIW BPF with the first iteration  
Figure 4: Mode chart of a comb-slotted MMR filter for different values of \(\mathrm{L}\).  
Figure 5: BPF frequency response of the comb-slotted HMSIW.  
of the Minkowski curve on the ground plane, which acts as the defected ground structure (DGS). A DGS [21] is a defect or slot in the ground plane of a microwave planar circuit that can be used to enhance its performance.  
Figure 9 elucidates the frequency response of the BPF with the first iteration level of the Minkowski curve. With a maximum return loss of 20.4 dB, the BPF resonates at 5.85 GHz. The insertion loss is 0.58 dB with a bandwidth of 1.08 GHz (5.32 GHz-6.40 GHz). It is evident from the S parameter response that the addition of a DGS caused the lower frequency to shift to 5.3 GHz. The slow wave effect [23] is the reason behind the increase in the resonator's effective electrical length that lowers the cutoff frequency causing size diminishment by 21.4%. In other words, the defect etched into the ground plane of a microwave circuit disturbs its current distribution, affecting the transmission line's capacitance and inductance. The augmented capacitance and inductance cause a slow-wave effect, enhancing the effective electrical length of the patch. This results in a decrease in the resonant frequency causing miniaturization [23]. Also, it is evident that there is a slight enhancement in the bandwidth as parasitic capacitance is produced by the existence of the ground plane defect as it upsurges the fringing field. This parasitic capacitance is the reason for the increased coupling amid the metallic top plane and the bottom ground plane, which produces a wider bandwidth [24].  
Figure 8: BPF using the Minkowski curve’s first iteration level as DGS.  
Figure 6: First three iterations of the Minkowski fractal curve [22].  
Figure 7: First iteration of the Minkowski curve (Li1 = Wi1 = 1.45 mm, Di1 = 1.25 mm).  
Figure 9: BPF frequency response based on the Minkowski curve’s first iteration level as the DGS.