**1. Introduction**

Filters play a vital role in numerous microwave applications. A microwave bandpass filter (BPF), in general, is a class of filter that is utilized to operate on the frequency response within the range of frequencies lying between 300 MHz and 300 GHz and allowing the best signal transmission at desired frequencies (passband), while eliminating signals at redundant frequencies (stopband) [1]. Among various techniques to design a bandpass filter, substrate integrated waveguides (SIWs) [2] are becoming more popular recently. SIW is a planar structure that is fabricated by using two periodic rows of conducting cylindrical vias implanted in a dielectric substrate, as shown in **Figure 1**. Hence, it acts as a bridge between planar and nonplanar technology.

**Figure 1.**

*Conventional substrate integrated waveguide.*

To design efficient and well-performing wireless systems, there is a great need to design compact, lightweight microwave components. Over the past few years, various SIW miniaturization techniques have been proposed by researchers. Recently, [3] has reviewed the recent trends and various miniaturization techniques of SIW. Recently, folded SIW (FSIW) technique (C & T type FSIW) has been proposed by [4, 5]. Miniaturization was achieved using half mode SIW and Hilbert fractal for 5G applications [6]. Further, [7] proposed a ridge SIW to achieve miniaturization and suppress the harmonics.

From the design Equations [8] for a substrate integrated waveguide (SIW), d as the diameter of the vias and p as distance between the vias known as pitch, the equivalent width of dielectric-filled rectangular waveguide,

$$\mathcal{W}\_{EQ} = \frac{c}{2f\_c\sqrt{\varepsilon\_r}}\tag{1}$$

Width of SIW,

$$\mathcal{W}\_{SIW} = \mathcal{W}\_{EQ} + \frac{d^2}{0.95p} \tag{2}$$

Also, for choosing the value of *p* and d, the following inequalities should be satisfied.

$$p < 4d \quad \text{and} \ p < \frac{\lambda\_0}{2}\sqrt{\varepsilon\_r} \tag{3}$$
