## 3 Design  
To keep up with the ever-increasing complexity of these systems, engineers have been coming up with more creative and intricate designs. This is because diplexers are so integral to contemporary communication systems. There are specific requirements that diplexers must meet to be employed in multi-service and multi-band communications. Conversely, larger and more complex diplexers were produced by the conventional way of fabrication, which required a great number of external connectors [26; 27; 28; 29; 30]. This has led to the development of numerous different designs that can guarantee compliance with these regulations. Diplexers have an assortment of resonators, including meander lines, open-loop resonators, stepped impedance resonators (SIRs), spiral inductor resonators, and many more. This has been done to achieve both good performance and a reduction in the size of the circuit. Basically, the resonator is capable of storing electromagnetic energies under particular frequencies in L and C elements. Magnetic energy is stored in the inductor L, whereas electric energy is stored in the capacitor C.  
The construction of SIRs involves the utilization of a network of transmission lines that are interconnected with one another. Each transmission line is created with its specific impedance requirement. SIRs are frequently utilized in a wide variety of applications within the realm of radio frequency (RF) and microwave engineering. As a result of their diminutive size and remarkable Q-factor, they are very well-suited for use in oscillators and filters. They are incredibly beneficial for a wide variety of applications due to the extraordinary properties that they possess, which is the reason why this is the case. During the process of developing SIRs, design issues and the undesirable reactions of particular circuits were also taken into account by RF and microwave engineers. The development of SIRs was motivated by these limits. While lower SIRs can limit frequencies up to a certain point, larger UIRs are more commonly seen in RF and microwave devices. A wide range of additional uses also make use of UIRs. Figures 2 and 3 display multiple examples of feasible SIR designs. There is a physical difference between the SIR and UIR lines; the former is shorter [30]. This holds even though UIR and SIR are based on ABCD matrices. One possible approach to express the matrix is as follows:  
\[ \begin{bmatrix}A&B\\ C&D\end{bmatrix} = \begin{bmatrix}Cos\ \theta&j\ Z_{0}\ Sin\ \theta\\ j\ Z_{0}^{-1}Sin\ \theta&Cos\ \theta\end{bmatrix}  \tag{1}\]  
Figure 3: SIR varieties: (**a**) SSIR type; (**b**) ASIR type. Figure 2: (**a**) The quarter-wavelength type, and (**b**) the half-wavelength type for the SIR variants. Alternatively, \(Z_{0}\) is used to represent characteristic impedance and \(\theta\) is used to represent the length of the UIR line. There are more design options available in the SIR topology than in topologies that use uniform impedance resonators because it permits the insertion of three matrices at each node instead of the matrix. The reason behind this is that three matrices can be introduced into the system using the SIR topology. This matrix is an effective tool for SIR topologies to improve their overall performance. On the other hand, UIR topologies do not depend on anything other than line length to produce resonance. In SIRs, the resonance state is strongly affected by the length and impedance ratios that are considered. To fully comprehend the operation of SIRs in UIR structures, which are commonly used to build microwave filters, these characteristics are crucial. Regarding this specific case, the impedance ratio coefficient, denoted by \(K\), is equal to \(Z_{2}/Z_{1}\). In contrast, the electrical lengths in the SIR are denoted by the symbols \(\theta 1\) and \(\theta 2\). When the value of \(K\) is midway between 0 and 1, the electrical length of a resonator is known to be the shortest, and when the value of \(K\) is greater than 1, the electrical length is known to be the longest.