## 2 Description of the Proposed Method
The remained Y\({}_{i}\)-terminals can be used to inject the input voltage source for deriving high input impedance filters. The procedure can be summarized as follows. 1. Of the desired transfer function of the synthesized filter in the form of the Matrix (4). It must be noted that each capacitor in the denominator must be arranged to have only a single position on the primary diagonal to obtain circuits with grounded capacitors. 2. Introduce a row and a column of zeros to row 1 and column 1 and place a unity resistor to position (1,1) of (4). The existing columns and rows are moved to the right and to the bottom, as given by (5). Then add infinity variables and zero terms to realize a nullator between column 1 and column 2 and a norator between row 2 and ground. The Step 2 corresponds to the adding of the equivalent circuit of voltage source in Figure 2[19]. So, Matrix (5) becomes (6). Figure 2: R-nullor equivalent circuit of a voltage source. \[ \begin{bmatrix}1&0&0\\ 0&\mathrm{G}_{1}+\mathrm{sC}_{1}&-\mathrm{G}_{2}\\ 0&\mathrm{G}_{3}&\mathrm{sC}_{2}\end{bmatrix}  \tag{12}\]  
\[ \begin{bmatrix}1&0&0&0\\ -\infty_{{}_{1}}&\infty_{{}_{1}}&0&0\\ 0&0&\mathrm{G}_{1}+\mathrm{sC}_{1}&-\mathrm{G}_{2}\\ 0&0&\mathrm{G}_{3}&\mathrm{sC}_{2}\end{bmatrix}  \tag{13}\]  
Using Step 3, three columns and rows of zero terms are added and pairs of nullor-mirror elements represented by \(\varpi_{2}\), \(\varpi_{3}\) and \(\varpi_{4}\) are introduced to the right and bottom of Matrix (13). So, Matrix (13) can be expanded as (14). There are eight alternative cases (cases 1-8) in expanding the Matrix (13), as shown in Table 1.  
\[\begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{{}_{1}}&\infty_{{}_{1}}&0&0&0&0&0\\ 0&0&\mathrm{G}_{1}+\infty_{{}_{2}}&-\infty_{{}_{3}}&-\infty_{{}_{2}}&\infty_ {{}_{3}}&0\\ 0&0&\infty_{{}_{4}}&\mathrm{sC}_{2}&0&0&-\infty_{{}_{4}}\\ 0&0&-\infty_  
\[ \begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{1}&\infty_{1}&0&0&0&0&0\\ 0&-\infty_{2}&\mathrm{G}_{1}+\infty_{2}&-\infty_{3}&-\infty_{2}&\infty_{3}&0\\ 0&0&\infty_{4}&\mathrm{SC}_{2}&0&0&-\infty_{4}\\ 0&\infty_{2}&-\infty_{2}&0&\mathrm{sC}_{1}+\infty_{2}&0&0\\ 0&0&0&-\infty_{3}&0&\mathrm{G}_{2}+\infty_{3}&0\\ 0&0&-\infty_{4}&0&0&0&\mathrm{G}_{3}+\infty_{4}\end{bmatrix}  \tag{15}\]  
\[ \begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{1}&\infty_{1}&0&0&0&0&0\\ 0&\infty_{2}&\mathrm{G}_{1}+\infty_{2}&-\infty_{3}&-\infty_{2}&\infty_{3}&0\\ 0&0&\infty_{4}&\mathrm{sC}_{2}&0&0&-\infty_{4}\\ 0&-\infty_{2}&-\infty_{2}&0&\mathrm{sC}_{1}+\infty_{2}&0&0\\ 0&0&0&-\infty_{3}&0&\mathrm{G}_{2}+\infty_{3}&0\\ 0&0&-\infty_{4}&0&0&0&\mathrm{G}_{3}+\infty_{4}\end{bmatrix}  \tag{16}\]  
Similarly, by applying the input voltage source to terminal-Y\({}_{2}\) of DDCC(2), term G\({}_{2}\) will appear in the position (2,1) of (12), then a bandpass function at V\({}_{\text{out1}}\) and a lowpass function at V\({}_{\text{out2}}\) can be obtained. The admittance matrix can be given in (17) and the derived circuit is shown in Figure 3a with moving the injected voltage source to node V\({}_{\text{in3}}\) and grounding nodes V\({}_{\text{in1}}\), V\({}_{\text{in2}}\), V\({}_{\text{in4}}\), V\({}_{\text{in5}}\) and V\({}_{\text{in6}}\). \[ \begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{1}&\infty_{1}&0&0&0&0&0\\ 0&\infty_{3}&\mathrm{G}_{1}+\infty_{2}&-\infty_{3}&-\infty_{2}&\infty_{3}&0\\ 0&0&\infty_{4}&\mathrm{sC}_{2}&0&0&-\infty_{4}\\ 0&0&-\infty_{2}&0&\mathrm{sC}_{1}+\infty_{2}&0&0\\ 0&\infty_{3}&0&-\infty_{3}&0&\mathrm{G}_{2}+\infty_{3}&0\\ 0&0&-\infty_{4}&0&0&0&\mathrm{G}_{3}+\infty_{4}\end{bmatrix}  \tag{17}\]  
By applying the input voltage source to terminal-Y\({}_{3}\) of DDCC(2), the term -G\({}_{2}\) will appear in position (2,1) of (12), then we can obtain the admittance matrix in (18) and the filter with identical  
Figure 3: The pathological representations of derived type-A filters (cases 1–4). transfer functions as (17) but with different signs at nodes V\({}_{\text{out1}}\) and V\({}_{\text{out2}}\).