## 1 Introduction  
Multiband bandpass filters are crucial in modern and emerging multi-mode wireless communication systems. These filters enable the simultaneous operation of wireless devices in multiple frequency bands, allowing for integrating previous wireless standards with new ones and supporting the increasing demand for wireless applications, including 5G and IoT [1]. The characteristics of low loss, sharp selectivity, compact size, and high inter-band isolation make multiband bandpass filters highly desirable for modern wireless communication systems. Low-loss filters help minimize signal loss during transmission, while compact size is essential for space-constrained devices [2, 3]. Overall, the demand for multiband bandpass filters will continue to grow in the coming years as the world becomes more wireless and the need for multi-mode wireless communication systems increases [4, 5]. The role of a bandpass filter in a communication system, as given below in Figure 1, is essential as it helps select the wanted signal while stopping unwanted signals. In portable terminals, miniaturization of the BPF is critical because BPFs typically occupy a space of 40% of the R.F. circuits. Reducing their size can help improve the device's overall form factor and usability [6; 7; 8]. In addition to miniaturization and low insertion loss, a good passband with sharpened rejection skirts is also necessary for effective filter design. This is because, in multiband operation, there are often several wireless systems operating in proximity, such as GSM, Wi-Fi, and WiMAX systems, which can cause interference. The filter design must have sharp rejection skirts to reject these interferences while allowing the desired signal to pass through. One way to achieve the necessary sharpness in filter design is to introduce transmission zeros (TZ) between the passbands. These TZs create deep notches in the filter response, which can help improve the rejection of unwanted signals. Therefore, designing a high-quality multiband bandpass filter involves a careful balance of miniaturization, passband performance, and rejection of unwanted signals with sharp attenuation levels [9; 10]. An essential consideration in the design of multiband bandpass filters is the creation of TZs. In [11], [PERSON] introduced the concept of TZ using the cross-coupling method, one of the first methods developed in 1954, and it was widely employed in three cavity waveguide filters until 1965 [12]. TZs are also produced by coupling resonators that are non-adjacent to one another. Popular cross-coupling structures that may independently generate one or two TZs are the cascaded triplet (C.T.) and cascaded quadruplet (C.Q.) topologies, which are appropriate for large-scale manufacturing and are simple to troubleshoed [13; 14; 15; 16]. The source/load (S/L) coupling, which [PERSON] initially suggested in 1974, is another crucial TZ-generating mechanism. S/L coupling has been employed extensively in filter design and can produce up to N TZs [17; 18]. As an example, a BPF was developed in [19] using splitting DGS and stepped impedance hairpin resonators to add two TZs on the lower stopband. Using the same source-load coupling method that [PERSON] discussed in [7], the authors of [20] inserted TZs on the right side of the passband. Another technique to create TZs is to let the signal go through two different pathways with almost equal magnitudes but phase differences of (2n + 1), which will cause them to cancel out the effect and create T.Zs. To create several T.Z.s, [21] employed this idea. The authors of [22] designed another BPF by integrating a shunt inductor and a gap-coupled capacitor to produce two TZs between the passbands. In the construction of filters, microstrip transmission lines with loaded stubs also produce TZs at certain frequencies. For instance, a multi-mode stub-loaded resonator was proposed in [23], and a dual-band BPF with the dual-mode operation was built in [24] employing stubs-loaded UIRs.