## 2 Description of the Proposed Method  
The port relations of an ideal DDCC can be characterized by (1), where the plus and minus signs indicate whether the current conveyor is DDCC\({}^{+}\) or DDCC\({}^{-}\). Figure 1 shows the symbolic pathological representations of DDCCs. It is clear that each of terminal-Y\({}_{i}\) (\(i\) = 1-3) possesses high input impedance. In the NAM expansion process, the addition of row and column of zero terms and infinity-variable terms with a common node on the primary diagonal of the admittance matrix is needed to transform the admittance terms to their correct form. The correct form set of admittance terms includes a unique positive term on the primary diagonal realizing a grounded admittance and a group of four terms with two positive terms on the primary diagonal and two negative terms on the off-diagonal representing a floating admittance. Therefore, in the synthesized circuits, pairs of pathological elements with a common node can be realized by the proper type of CCIs [19]. The DDCC is similar to a CCII but with two more additional terminals, _i.e._, terminal-Y\({}_{2}\) and terminal-Y\({}_{3}\). Thus, the DDCC- and DDCC+ can be respectively represented by infinity-variables notation given by Equations (2) and (3) in NAM expansion with the common node on the primary diagonal assigned to terminal-X. The signs of infinity-variables of terminal-Y\({}_{1}\) and terminal-Y\({}_{3}\) are different to that of terminal-X, while the signs of infinity-variables of terminal-X and terminal-Y\({}_{2}\) are identical. Based on the infinity-variables notation in (2) and (3), it can be seen that the connection between terminal-X and terminal-Y\({}_{1}\) or terminal-Y\({}_{3}\) corresponds to a nullator, and the connection between terminal-X and terminal-Y\({}_{2}\) corresponds to a voltage-mirror (VM). These relationships are very important for deriving the numerator of the transfer function of the synthesized filters. \[ \begin{array}{cccc}\mathrm{Y}_{3}&\mathrm{Y}_{2}&\mathrm{Y}_{1}&\mathrm{X} \\ \left[\begin{array}{cccc}-\infty_{\mathrm{i}}&\infty_{\mathrm{i}}&-\infty_{ \mathrm{i}}&\infty_{\mathrm{i}}\\ \infty_{\mathrm{i}}&-\infty_{\mathrm{i}}&\infty_{\mathrm{i}}&-\infty_{ \mathrm{i}}\end{array}\right]&\mathrm{X}\\ \mathrm{Y}_{3}&\mathrm{Y}_{2}&\mathrm{Y}_{1}&\mathrm{X}\\ \left[\begin{array}{cccc}-\infty_{\mathrm{i}}&\infty_{\mathrm{i}}&-\infty_{ \mathrm{i}}&\infty_{\mathrm{i}}\\ -\infty_{\mathrm{i}}&\infty_{\mathrm{i}}&-\infty_{\mathrm{i}}&\infty_{ \mathrm{i}}\end{array}\right]&\mathrm{Z+}\\ \end{array}  \tag{2}\]  
Figure 1: The symbolic pathological representations of DDCCs. It is known that to synthesize filter circuits using NAM expansion, the denominator D(s) of a transfer function of a filter should be expressed as an admittance matrix in NAM equations shown in Equation (4). This matrix can be used as a starting matrix to find the circuit configuration with no input signals. Then, the equivalent circuit of an input voltage source as shown in Figure 2 can be injected to the obtained circuit represented by expanded NAM of (4) to obtain the voltage-mode filters [19]. \[\left[ \begin{array}{cccccc}\mathrm{y}_{1,1}&\mathrm{y}_{1,2}&\cdots&\mathrm{ y}_{1,\mathrm{j}}&\cdots&\mathrm{y}_{1,\mathrm{n}}\\ \mathrm{y}_{2,1}&\mathrm{y}_{2,2}&\cdots&\mathrm{y}_{2,\mathrm{j}}&\cdots& \mathrm{y}_{2,\mathrm{n}}\\ \vdots&\vdots&\cdots&\vdots&\cdots&\vdots\\ \mathrm{y}_{i,1}&\mathrm{y}_{i,2}&\cdots&\mathrm{y}_{i,\mathrm{j}}&\cdots& \mathrm{y}_{i,\mathrm{n}}\\ \vdots&\vdots&\cdots&\vdots&\cdots&\vdots\\ \mathrm{y}_{\mathrm{n},1}&\mathrm{y}_{\mathrm{n},2}&\cdots&\mathrm{y}_{ \mathrm{n},\mathrm{j}}&\cdots&\mathrm{y}_{\mathrm{n},\mathrm{n}}\\ \end{array} \right] \tag{4}\]  
To synthesize DDCC-grounded capacitor-based filters using NAM expansion, we firstly use the NAM stamps of CCIIs [15] to expand the starting matrix in (4). The expanded NAM of (4) includes the correct form of admittance elements and pairs of infinity-variables represented terminal-X, terminal-Z and one terminal-Y\({}_{i}\) of DDCCs.