## 2 Design and Analysis of the SUWB-BPF  
The architecture of the basic SUWB-BPF is shown in Figure 2. It is constructed using a UTL loaded with three folded \(\lambda/4\) short-circuited shunt stubs with pads. Out of this, one stub is placed in the middle section of one side of the UTL, and the other two are placed at the symmetrical position on the opposite side of the UTL. The analysis of the uniform impedance resonator is carried out by considering a half-wavelength resonator connected with open- or short-circuited stubs in the middle, as shown in Figure 3a,b. Assuming no losses, the network resonance condition can be determined with the transfer matrix [ABCD]. The expression of the input admittance is [16; 17]:  
\[Y_{in}=\frac{1}{Z_{11}}=\frac{C}{A}=0 \tag{1}\]  
Figure 1: Design flow chart. where \(Z_{11}\) is the input impedance of the resonator. The resonance Equation (1) is identical to the requirement for resonators when the zeros of elements \(C\) and \(A\) in the transfer matrix do not coincide and \(A\) does not contain poles that are different from those in \(C\). \[C=0 \tag{2}\]  
The output admittance (\(Y_{out}\)) also becomes zero when the condition in Equation (2) is satisfied. Therefore,  
\[Y_{out}=\frac{1}{Z_{22}}=\frac{C}{D}=0 \tag{3}\]  
The generalized transfer matrix [\(ABCD\)] in the left portion of Figure 3a,b in terms of transfer matrices of the elements \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), \(D^{\prime}\) is [18]:  
\[ \begin{pmatrix}A&B\\ C&D\end{pmatrix} = \begin{pmatrix}1+2B^{\prime}C^{\prime}&2A^{\prime}B^{\prime} \\ 2C^{\prime}D^{\prime}&1+2B^{\prime}C^{\prime}\end{pmatrix}  \tag{4}\]  
Equation (4) represents the resonance condition of Equation (2) that satisfies the following two conditions:  
\[C^{\prime}=0 \tag{5}\]  
\[D^{\prime}=0 \tag{6}\]  
The above two conditions are simpler to perform in analysis than the traditionally used \(Y_{in}=0\). Using the resonators depicted in Figure 3a,b, let us construct the resonance equations. At the point where the stub connects to the TL (transmission line) segment, we  
Figure 3: Stubs configurations: (**a**) open stub connected to UTL, (**b**) short-circuited stub connected to UTL, (**c**) conventional stub, and (**d**) the proposed shunt stub. Figure 2: Proposed SUWB-BPF architecture. utilize the input admittance \(Y\)s of the stub and the transfer matrix [_abcd_] of the left half of the TL segment. The matrix [_A_'_'_B_'C_'_D_'] is clearly seen as the result of two matrices [19; 20]:  
\[ \begin{pmatrix}A^{\prime}&B^{\prime}\\ C^{\prime}&D^{\prime}\end{pmatrix} = \begin{pmatrix}a&b\\ c&d\end{pmatrix}\begin{pmatrix}1&0\\ \frac{Y}{2}&1\end{pmatrix} = \begin{pmatrix}a+\frac{bY_{s}}{2}&b\\ c+d\frac{Y_{s}}{2}&d\end{pmatrix}  \tag{7}\]  
The above chain equation follows the expression (3), i.e.,:  
\[d=0Y_{s}=\frac{-2c}{d}\]  
The matrix [_abcd_] may generally represent a portion of a non-uniform TL. When a segment with a uniform TL is used, its electrical length and characteristic impedance \(Z_{0}\) are:  
\[ \begin{pmatrix}a&b\\ c&d\end{pmatrix} = \begin{pmatrix}\cos\frac{\theta}{2}&jZ_{o}\sin\frac{\theta}{2} \\ Z_{o}{}^{-1}j\sin\frac{\theta}{2}&\cos\frac{\theta}{2}\end{pmatrix}  \tag{8}\]  
Applying resonance conditions on the above equation:  
\[\cos\frac{\theta}{2} \tag{9}\]  
\[Y_{s}=-jZ_{o}{}^{-1}\tan\frac{\theta}{2} \tag{10}\]  
The resonant electrical length of Equation (9) is:  
\[\theta_{n}=\pi,3\pi,5\pi\]  
The transcendental Equation (10) determines the other component of the resonant electrical lengths, which is dependent on the stub parameters. The same analysis can be used for the short-circuited stub of length \(\theta\)s \(<<\theta/2\) in Figure 2(b), but only Equation (10) takes the below form, while Equation (9) is still applicable:  
\[-Z_{s}{}^{-1}\cot\theta_{s}=-2Z_{o}{}^{-1}\tan\frac{\theta}{2}\]  
where \(\theta\) = \(\beta\)L shows the electrical length of the stubs with physical length L and propagation constant \(\beta\).