## 2 The Principle of the Synthetization Approach  
### The Number of Variable Parameters of Conventional N-Section IT  
Figure 2 shows the topology of \(N\)-section IT. It consists of \(N\) cascaded CLSs with the electrical length \(\theta\), where \(Z_{eqi}\) and \(Z_{adi}\) are the even-and odd-mode characteristic impedances of the \(i\)th CLS, and \(i\) = 1,2, \(\ldots\), \(N\). \(Z_{S}\) and \(Z_{L}\) are two different terminal loads, and its impedance ratio \(k\) is defined as \(k=Z_{S}/Z_{L}\), \(Z_{L}=1\) is defined in this paper; thus, \(k>1\). Based on [28], the \(ABCD\) matrix of the \(i\)th CLS can be summarized as  
\[\left[ \begin{array}{cc}A_{i}&B_{i}\\ C_{i}&D_{i}\end{array} \right]=\frac{\sin\theta}{T_{i}}\left[ \begin{array}{cc} qS_{i}&\frac{i}{2}&[T_{i}^{2}+q^{2}(T_{i}^{2}-S_{i}^{2})]\\ 2j&qS_{i}\end{array} \right] \tag{1}\]  
where \(q=\cot\theta\), and  
\[S_{i}=Z_{eqi}+Z_{adi} \tag{2}\]  
\[T_{i}=Z_{eqi}-Z_{adi} \tag{3}\]  
Figure 2: Topology of conventional \(N-\)section IT [28]. Then, the \(ABCD\) matrix of \(N\)-section filter-integrated \(\Pi\Gamma\) can be written as  
\[\left[ \begin{array}{cc}A_{T}&B_{T}\\ C_{T}&D_{T}\end{array} \right]=\left[ \begin{array}{cc}A_{1}&B_{1}\\ C_{1}&D_{1}\end{array} \right]\left[ \begin{array}{cc}A_{2}&B_{2}\\ C_{2}&D_{2}\end{array} \right]\cdots\left[ \begin{array}{cc}A_{i}&B_{i}\\ C_{i}&D_{i}\end{array} \right]\cdots\left[ \begin{array}{cc}A_{N-1}&B_{N-1}\\ C_{N-1}&D_{N-1}\end{array} \right]\left[ \begin{array}{cc}A_{N}&B_{N}\\ C_{N}&D_{N}\end{array} \right] \tag{4}\]  
where  
\[A_{T}=\sum a_{p}\cos^{p}\theta \tag{5}\]  
\[B_{T}=\frac{j}{\sin\theta}{\sum b_{q}\cos^{q}\theta} \tag{6}\]  
\[C_{T}=\frac{j}{\sin\theta}{\sum c_{q}\cos^{q}\theta} \tag{7}\]  
\[D_{T}=\sum d_{p}\cos^{p}\theta \tag{8}\]  
\(a_{p}\) and \(d_{p}\) are polynomial functions with degrees \(p\). \(b_{q}\) and \(c_{q}\) are polynomial functions with degrees \(q\). \(p\) and \(q\) defined in Equation (9). All polynomial functions can be determined by the even-and odd-mode characteristic impedances, respectively. \[\left\{ \begin{array}{c}p=1,3,5,\cdots,Nq=0,2,4,\cdots,N+1\text{ when $N$ is odd number}\\ p=0,2,4,\cdots,Nq=1,3,5,\cdots,N+1\text{ when $N$ is even number}\end{array} \right. \tag{9}\]  
\(S\)-parameters of the \(N\)-section IT are  
\[S_{11}=\frac{A_{T}Z_{L}+B-CZ_{S}Z_{L}-DZ_{S}}{A_{T}Z_{L}+B+CZ_{S}Z_{L}+DZ_{S}} \tag{10}\]  
\[S_{21}=\frac{2\sqrt{Z_{S}Z_{L}}}{A_{T}Z_{L}+B+CZ_{S}Z_{L}+DZ_{S}} \tag{11}\]  
The amplitude-squared transfer function can be expressed as  
\[|S_{21}^{circuit}(\theta)|^{2}=\frac{1}{1{+}|F_{circuit}|^{2}}=\frac{1}{1+ \left|(A_{T}+B_{T}-kC_{T}-kD_{T})/2\sqrt{k}\right|^{2}} \tag{12}\]  
where \(F_{circuit}\) is the characteristic function of the circuit:  
\[F_{circuit}=\sum(a_{p}-kd_{p})\cos^{p}\theta/2\sqrt{k}+\frac{j}{\sin\theta}{ \sum(b_{q}-kc_{q})\cos^{q}\theta}/2\sqrt{k} \tag{13}\]  
Conversely, based on the synthesis approach in [46], firstly, we define two variables: \(x=\cos\phi=\alpha\) cos\(\theta=\cos\theta/\cos\theta_{c}\), \(y=\cos\xi=x\sin\theta_{c}/\sin\theta\), where \(\theta_{c}\) is the electrical length at the lower cutoff frequency fc of the passband, and \(\alpha\) is a quantity used to determine the bandwidth of a bandpass filter. The Chebyshev transfer function can be written as  
\[|S_{21}^{formula}(\theta)|^{2}=\frac{1}{1+\left|F_{formula}\right|^{2}}= \frac{1}{1+\epsilon^{2}\cos^{2}(N\phi+\xi)} \tag{14}\]  
where \(\epsilon=\sqrt{10^{0.1L_{A}}-1}\), and \(L_{A}\) is the in-band ripple level factor. The characteristic function of Chebyshev formula \(F_{formula}\) can be rewritten as:  
\[F_{formula}=\epsilon\frac{\Sigma u_{q}\cos^{q}\theta}{\sin\theta} \tag{15}\]  
From \(F_{circuit}=F_{formula}\), the general design conditions of \(N\)-section IT are summarized as  
\[\left\{ \begin{array}{c}a_{p}-kd_{p}=0\\ (b_{q}-kc_{q})/2\sqrt{k}=\epsilon u_{q}\end{array} \right.