## 2 Filter Design  
The proposed dual notched ultra-wideband bandpass filter using a T-shaped resonator and L-shaped defect microstrip structure is designed on a bandpass filter in the literature [5]. The structure of a ultra-wideband bandpass filter without a T-shaped resonator and L-shaped defect microstrip structure is shown in Figure 1a. Figure 1b lays out the simulation results of a basic ultra-wideband bandpass filter by using high frequency structure simulator (HFSS) software (ANSYS, Inc., Canonsburg, PA, USA). To satisfy the requirements for ultra-wideband performance and improve edge steepness of the passband, four short-circuited stubs (length of \(\lambda/4\)) are used. Figure 1b indicates that the bandwidth of the bandpass filter is from 2.632 to 10.623 GHz and the fractional bandwidths (FBW) is 123.9%. The insertion loss of the ultra-wideband bandpass filter is less than 0.8 dB, while the simulated return loss is greater than 16 dB within the passband. Figure 2 shows the design process of the dual notched bands filter. The basic filter can form the first notched band by using the T-shaped resonator and the second notched band by using the L-shaped defect microstrip structure. Figure 3a,b shows the circuit configuration of the proposed filter and its S parameters, which are calculated by HFSS. The proposed filter achieves dual notched bands, whose center frequencies are at the 3.5 and 7.5 GHz, respectively. On notched frequencies, the corresponding reflection coefficient is \(-\)28.6 dB and \(-\)25.1 dB with a the fractional bandwidths (FBW) of 15.76% and 4.96%, respectively. Figure 4 shows the structure of the T-shaped resonator and further analyzes its even-mode and odd-mode equivalent circuit. By using odd-mode and even-mode analysis method [6], the input impedance of even-mode and odd-mode can be expressed as the Equations (1) and (2), where \(\theta_{1}\) and \(\theta_{2}\) are the electrical length of the open-circuited stub and the short-circuited stub, respectively. \[Z_{\text{ine}}=jZ_{6}\frac{Z_{6}\tan\theta_{6}+2Z_{5}\tan\theta_{5}}{Z_{6}-2Z_ {5}\tan\theta_{6}\tan\theta_{5}} \tag{1}\]  
\[Z_{\text{ino}}=jZ_{6}\tan\theta_{6} \tag{2}\]  
Figure 1: (**a**) Structure of ultra-wideband bandpass filter without T-shaped resonator and L-shaped defect microstrip structure; (**b**) simulation results of the ultra-wideband bandpass filter without T-shaped resonator and L-shaped defect microstrip structure. Figure 3: (**a**) The circuit of the proposed filter; (**b**) simulation results of the proposed filter. Figure 2: Design process of the dual notched bands filter. \(\theta_{6}\) is defined as the electrical length of the open-circuited stub, \(\theta_{5}\) is defined as the electrical length of the short-circuited stub, which can be expressed as:  
\[\theta_{5}=\beta L_{6},\ \ \ \theta_{6}=\beta L_{6}/2 \tag{3}\]  
The even-mode resonant frequency \(f_{\text{even}}\) and odd-mode resonant frequency \(f_{\text{odd}}\) can be expressed as shown in Equations (4) and (5). If the two signals are in the same phase, they are stacked together to form a pass band. However, in this frequency range, the two signals' phase is opposite, and the stop band is formed. In order to further reduce the difficulty of design, the parameter is made \(L_{6}=2L_{5}\). In this way, the resonant frequency of odd mode and even mode are only related to \(L_{6}\), and the first resonance frequency is adjusted by adjusting \(L_{6}\). \[f_{\text{even}}=\frac{c}{4(L_{6}/2+L_{5})\sqrt{\epsilon_{\text{eff}}}}=\frac{ c}{4(L_{6}/2+L_{6}/2)\sqrt{\epsilon_{\text{eff}}}}=\frac{c}{4L_{6}\sqrt{ \epsilon_{\text{eff}}}} \tag{4}\]  
\[f_{\text{odd}}=\frac{c}{2L_{6}\sqrt{\epsilon_{\text{eff}}}} \tag{5}\]  
Figure 5a illustrates the structure of the L-shaped defect microstrip structure, whose width is W\({}_{7}\) and length is the sum of \(L_{7}\) and \(L_{8}\).