## 1 Introduction  
Impedance transformers (ITs) have been widely used in microwave circuits and systems as a basic circuit component [1]. A diagram of the typical end-fed antenna system is shown in Figure 1, where the IT is used to match the antenna impedance to the coax line impedance (usually 50 \(\Omega\)). Many IT and filter structures have been developed in recent years [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], and multi-function IIs which are integrated by filters [14, 15, 16, 17, 18, 19, 20], phase shifters [21], power dividers [22], and antennas [23] have also been widely researched.  
In [2, 3], a quarter-wave IT is presented using transmission lines (TLs) for single-band applications with a limited operating range. Several IT structures have been designed to  
Figure 1: The diagram of typical end-fed antenna system.  
meet the requirements of multi-band communication systems [4; 5; 6; 7; 8; 9], in which multi-section structures are an effective way to design multi-band systems and match impedance [4; 5; 6; 7]. Inserting open/short stubs is also available to realize impedance matching [8; 9].  
In wideband IT applications, filter-integrated ITs have been proposed and developed in recent years to enhance the out-band suppression and reduce the circuit size [14; 16; 17; 18; 19; 20]. In [14], the lumped elements are used to realize wideband response. However, lumped elements are limited in high-frequency applications, and the soldering will also involve undesirable parasitic effects. Instead of using lumped components, open/short stubs can be used to provide better high-frequency performance for the filtering ITs [16; 17; 18; 19; 20]. The equal-ripple response can be realized by lumped elements and J-inverters [24; 25; 26; 27]. A Chebyshev polynomial is also used to constrain the equal-ripple response in filter and IT designs [28; 29; 30]. To realize DC block and reduce the circuit size, coupled line sections (CLSs) are widely used in wideband ITs and filters [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45]. For example, two CLSs are added to each terminal of the stepped-impedance TLs symmetrically to integrate the filtering function [30].  
To the best of the author's knowledge, there still are some relationships among these transformers and filters that are not mentioned. For example, in [26; 28; 29; 30; 33], filters and ITs with a Chebyshev response are constituted by cascaded CLSs and TLs. Although the similar design synthesis was reported, the numbers and positions of CLSs and TLs were not clear.  
In this paper, a novel synthetization approach is presented for filter-integrated multi-section wideband ITs. From the point of view of the number of variable parameters, the wideband IT, which consists of \(N\) cascaded CLSs, is provided with a Chebyshev performance, where \(N+2\) design conditions are consumed, and \(N-2\) variable parameters remain. To further optimization, CLSs are newly substituted by TLs to consume the remaining variable parameters and simplify the circuit topology. Therefore, there are \(2^{N}-N-2\) substituting possibilities in total for \(N\)-section ITs, and the design of the IT can be more flexible for both symmetric and asymmetric structures with the same wideband filtering performance.