## 2 Description of the Proposed Method
### Synthesis of Type-B Universal Filters
 \tag{32}\]  
Injecting the input voltage source to terminal-Y\({}_{3}\) of DDCC(2), the obtained transfer functions at nodes V\({}_{\mathrm{out1}}\) and V\({}_{\mathrm{out2}}\) are identical to that obtained in (32) but with different signs, as shown in (33). The derived filter is shown in Figure 6a by moving the injected voltage source equivalent circuit to node V\({}_{\mathrm{in4}}\) with grounding nodes V\({}_{\mathrm{in1}}\), V\({}_{\mathrm{in2}}\), V\({}_{\mathrm{in3}}\), V\({}_{\mathrm{in5}}\) and V\({}_{\mathrm{in6}}\). Besides, a lowpass function at V\({}_{\mathrm{out1}}\) can be achieved by applying the input voltage source to terminal-Y\({}_{2}\) of DDCC(3). This operation corresponds to the inserting of terms \(\infty_{4}\) to the second column of (29), as given in (34). The obtained circuit is shown in Figure 6a by moving the injected voltage source equivalent circuit to node V\({}_{\mathrm{in5}}\) and grounding nodes V\({}_{\mathrm{in1}}\), V\({}_{\mathrm{in2}}\), V\({}_{\mathrm{in3}}\), V\({}_{\mathrm{in4}}\) and V\({}_{\mathrm{in6}}\). \[\begin{bmatrix}1&0&0&0&0&0&0\\ \begin{bmatrix}-\infty_{{}_{1}}&\infty_{{}_{1}}&0&0&0&0&0\\ 0&\infty_{{}_{3}}&\mathrm{G}_{{}_{1}}+\infty_{{}_{2}}&\infty_{{}_{3}}&-\infty\[ \begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{{}_{1}}&\infty_{{}_{1}}&0&0&0&0&0\\ 0&0&\mathrm{G}_{1}+\infty_{{}_{2}}&\infty_{{}_{3}}&-\infty_{{}_{2}}&-\infty_{{} _{3}}&0\\ 0&\infty_{{}_{4}}&-\infty_{{}_{4}}&\mathrm{SC}_{{}_{2}}&0&0&\infty_{{}_{4}}\\ 0&0&-\infty_{{}_{2}}&0&\mathrm{s}\mathrm{C}_{{}_{1}}+\infty_{{}_{2}}&0&0\\ 0&0&0&-\infty_{{}_{3}}&0&\mathrm{G}_{{}_{2}}+\infty_{{}_{3}}&0\\ 0&\infty_{{}_{4}}&-\infty_{{}_{4}}&0&0&0&\mathrm{G}_{{}_{3}}+\infty_{{}_{4}} \end{bmatrix}  \tag{34}\]  
Injecting the input voltage source to terminal-Y\({}_{3}\) of DDCC(3), the obtained transfer function at node V\({}_{\mathrm{out1}}\) is identical to the circuit represented by (34) with reverse signs. The obtained matrix is shown in (35) and the circuit is shown in Figure 6a by moving the injected voltage source equivalent circuit to node V\({}_{\mathrm{in6}}\) with grounding nodes V\({}_{\mathrm{in1}}\), V\({}_{\mathrm{in2}}\), V\({}_{\mathrm{in3}}\), V\({}_{\mathrm{in4}}\) and V\({}_{\mathrm{in5}}\) grounded. \[ \begin{bmatrix}1&0&0&0&0&0&0\\ -\infty_{{}_{1}}&\infty_{{}_{1}}&0&0&0&0&0\\ 0&0&\mathrm{G}_{{}_{1}}+\infty_{{}_{2}}&\infty_{{}_{3}}&-\infty_{{}_{2}}&- \infty_{{}_{3}}&0\\ 0&-\infty_{{}_{4}}&-\infty_{{}_{4}}&\mathrm{s}\mathrm{C}_{{}_{2}}&0&0&\infty_{ {}_{4}}\\ 0&0&-\infty_{{}_{2}}&0&\mathrm{s}\mathrm{C}_{{}_{1}}+\infty_{{}_{2}}&0&0\\ 0&0&0&-\infty_{{}_{3}}&0&\mathrm{G}_{{}_{2}}+\infty_{{}_{3}}&0\\ 0&-\infty_{{}_{4}}&-\infty_{{}_{4}}&0&0&0&\mathrm{G}_{{}_{3}}+\infty_{{}_{4}} \end{bmatrix}  \tag{35}\]  
Different type-B filter functions at V\({}_{\mathrm{out1}}\) and V\({}_{\mathrm{out2}}\) can be obtained by using similar method as mentioned in Section 3.1. The notch function at V\({}_{\mathrm{out1}}\) can be achieved by injecting the input voltage  
Figure 6: The pathological representations of type-B filters. source to terminal-Y\({}_{2}\) of DDCC(1) and terminal-Y\({}_{2}\) of DDCC(3). The obtained filter is shown in Figure 6a by moving the injected voltage source equivalent circuit to the merged node of V\({}_{\text{in1}}\) and V\({}_{\text{in5}}\) and grounding nodes V\({}_{\text{in2}}\), V\({}_{\text{in3}}\), V\({}_{\text{in4}}\) and V\({}_{\text{in6}}\). Another notch function at V\({}_{\text{out1}}\) in Figure 6a can be also obtained by injecting the input voltage source to the merged node of V\({}_{\text{in2}}\) and V\({}_{\text{in6}}\) with nodes V\({}_{\text{in1}}\), V\({}_{\text{in3}}\), V\({}_{\text{in4}}\) and V\({}_{\text{in5}}\) being grounded. Similarly, an allpass function at V\({}_{\text{out1}}\) (with G\({}_{2}\) = G\({}_{1}\)) can be achieved by injecting the input voltage source to terminal-Y\({}_{2}\) of DDCC(1), terminal-Y\({}_{3}\) of DDCC(2) and terminal-Y\({}_{2}\) of DDCC(3).