## 2 Proposed Circuit
### Proposed Mixed-Mode Universal Filter  
Figure 3 shows the proposed second-order mixed-mode universal filter employing three DDCCTAs, two grounded capacitors, and three grounded resistors. The \(V_{1}\) to \(V_{7}\) are the input voltages, \(I_{1}\) to \(I_{5}\) are the input currents, \(V_{01}\) to \(V_{07}\) are the output voltages, and \(I_{01}\) to \(I_{06}\) are the output currents. Thanks to the multiple-input and the multiple-output of the DDCCTA, a multiple-input and a multiple-output of the mixed-mode universal filter can be achieved. It should be noted that the inputs \(V_{1}\) to \(V_{7}\) are applied to the high impedance level of the DDCCTAs while the outputs \(I_{01}\) to \(I_{06}\) are supplied from the high impedance level of the DDCCTAs. Thus, the proposed mixed-mode universal filter offers high-input impedance of voltage input terminals, and high-output impedance of current  
output terminals, which is required for VM and CM circuits. Under appropriate conditions, many filtering functions can be obtained. If \(V_{1}\) to \(V_{7}\) are the input voltages and \(V_{o1}\) to \(V_{o7}\) are the output voltages, the circuit can work as a VM filter. Using nodal analysis and (1), the outputs \(V_{o1}\) to \(V_{o7}\) can be expressed, respectively, by  
\[V_{o1}=\frac{\left\{ \begin{matrix}(sC_{1}g_{m3}+g_{m1}g_{m2})(V_{1}-V_{2})+sC_{ 1}g_{m3}(V_{4}-V_{3})\\ +sC_{1}g_{m2}(V_{6}-V_{5})+g_{m1}g_{m2}V_{7}\\ \end{matrix} \right\}}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{9}\]  
\[V_{o2}=\frac{\left\{ \begin{matrix}s^{2}C_{1}C_{2}(V_{2}-V_{1})+\left(s^{2}C_{1 }C_{2}+g_{m1}g_{m2}\right)(V_{3}-V_{4})\\ +sC_{1}g_{m2}(V_{6}-V_{5})+g_{m1}g_{m2}V_{7}\\ \end{matrix} \right\}}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{10}\]  
\[V_{o3}=\frac{\left\{ \begin{matrix}sC_{2}g_{m1}(V_{1}-V_{2})+g_{m1}g_{m2}(V_{3} -V_{4})\\ +\left(s^{2}C_{1}C_{2}+sC_{1}g_{m3}\right)(V_{6}-V_{5})+(sC_{2}g_{m1}+g_{m1}g_ {m3})V_{7}\\ \end{matrix} \right\}}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{11}\]  
\[V_{o4}=\frac{\left\{ \begin{matrix}sC_{2}g_{m1}(V_{2}-V_{1})+g_{m1}g_{m3}(V_{4 }-V_{3})+g_{m1}g_{m2}(V_{6}-V_{5})\\ -\left(sC_{2}g_{m1}+g_{m1}g_{m3}\right)V_{7}\\ \end{matrix} \right\}}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{12}\]  
\[V_{o5}=\frac{\left\{ \begin{matrix}s^{2}C_{1}C_{2}(V_{2}-V_{1})+sC_{1}g_{m3}(V_ {4}-V_{3})+sC_{1}g_{m2}(V_{6}-V_{5})\\ +g_{m1}g_{m2}V_{7}\\ \end{matrix} \right\}}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{13}\]  
Figure 3: Proposed mixed-mode universal filter using DDCCTAs. \[I_{06}=\frac{\left\{-s^{2}C_{1}C_{2}g_{m3}R_{1}I_{1}+sC_{1}g_{m3}(I_{2}-I_{3})-g_{ m1}g_{m2}I_{4}\right\}}{-\left(s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}\right.} \tag{27}\]  
If \(I_{1}\) to \(I_{5}\) are the input currents, and \(V_{o1}\) to \(V_{o7}\) are the output voltages, the circuit can work as a TIM filter. Using nodal analysis and (1), the outputs \(V_{o1}\) to \(V_{o7}\) can be expressed, respectively, by  
\[V_{o1}=\frac{-(sC_{1}g_{m3}R_{1}+g_{m1}g_{m2}R_{1})I_{1}+sC_{1}(I_{3}-I_{2})+g_ {m2}(I_{4}-I_{5})}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{28}\]  
\[V_{o2}=V_{o5}=\frac{s^{2}C_{1}g_{m3}R_{1}I_{1}+sC_{1}(I_{3}-I_{2})+g_{m3}(I_{4 }-I_{5})}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{29}\]  
\[V_{o3}=\frac{-sC_{1}g_{m3}R_{1}I_{1}+g_{m1}(I_{2}-I_{3})+(sC_{2}+g_{m3})(I_{4 }-I_{5})}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{30}\]  
\[V_{o4}=\frac{sC_{1}g_{m3}R_{1}I_{1}+g_{m2}(I_{3}-I_{2})+(sC_{2}+g_{m3})(I_{5}- I_{4})}{s^{2}C_{1}C_{2}+sC_{1}g_{m3}+g_{m1}g_{m2}} \tag{31}\]  
\[V_{o6}=\left\{ \begin{array}{c}\left(s^{2}C_{1}C_{2}g_{m3}R_{1}R_{2}+sC_{2}g _{m1}g_{m2}R_{1}R_{2}\right)I_{1}\\ -(sC_{1}g_{m3}R_{2}+g_{m1}g_{m2}R_{2})I_{2}\\ -s^{2}C_{1}C_{2}R_{2}I_{3}+sC_{2}g_{m2}R_{2}(I_{5}-I_{4})\end{array} \right\} \tag{32}\]  
\[V_{o7}=\frac{\left\{ \begin{array}{c}-s^{2}C_{1}C_{2}g_{m3}R_{1}R_{3}I_{1}+sC _{1}g_{m3}R_{3}(I_{2}-I_{3})\\ +g_{m2}g_{m3}R_{3}(I_{5}-I_{4})\end{array}