## 3 Proposed Mixed-Mode Universal Biquad Filter  
The proposed configuration, which is realized by two VDBAs, two resistors, and two capacitors, is shown in Figure 3. It is important to note that, in this realization, the resistors \(R_{1}\) and \(R_{2}\) are permanently grounded. From the proposed circuit in Figure 3, the universal biquadratic filter operated in all four possible modes is available as follows. For VM operation: Assuming ideal VDBA (i.e., \(\alpha\) =\(\beta\) = 1) and setting \(i_{in}\) = 0, the general voltage biquadratic transfer functions of this MISO filter can be obtained as follows. With \(v_{in}\) = \(v_{1}\) (input voltage) and \(v_{2}\) = \(v_{3}\) = \(v_{4}\) = 0 (grounded), then the LP response is realized as:  
\[T_{LP(VM)}\ =\ \frac{v_{out}}{v_{in}}\ =\ \frac{\left(\frac{g_{out}S_{in}2}{C_{1}C_{2 }}\right)}{D(s)}. \tag{3}\]  
Figure 3: Proposed mixed-mode biquadratic filter using VDBAs. Figure 2: Possible CMOS realization of the VDBA. \begin{table}
\begin{tabular}{c c} \hline \hline
**Transistors** & **W(\(\mu\)m)/L(\(\mu\)m)** \\ \hline M\({}_{1}\)-M\({}_{2}\), M\({}_{5}\), M\({}_{7}\)-M\({}_{8}\), M\({}_{12}\)-M\({}_{13}\) & 2.4/0.18 \\ M\({}_{3}\), M\({}_{9}\), M\({}_{14}\) & 5/0.18 \\ M\({}_{4}\), M\({}_{10}\) & 5.2/0.18 \\ M\({}_{6}\) & 3.25/0.18 \\ M\({}_{11}\) & 10/0.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Transistors’ aspect ratios in Figure 2.  
With \(v_{in}=v_{2}\), \(v_{1}=v_{3}=v_{4}=0\), and \(g_{m2}=1/R_{2}\), then the BP response is realized as:  
\[T_{BP(VM)}\ =\ \frac{v_{out}}{v_{in}}\ =\ \frac{\Big{(}\frac{g_{m2}}{C_{2}} \Big{)}s}{D(s)}. \tag{4}\]  
With \(v_{in}=v_{4}\), and \(v_{1}=v_{2}=v_{3}=0\), then the HP response is realized as:  
\[T_{HP(VM)}\ =\ \frac{v_{out}}{v_{in}}\ =\ \frac{s^{2}}{D(s)}. \tag{5}\]  
With \(v_{in}=v_{1}=v_{4}\), and \(v_{2}=v_{3}=0\), then the BS response is realized as:  
\[T_{BS(VM)}\ =\ \frac{v_{out}}{v_{in}}\ =\ \frac{s^{2}+\Big{(}\frac{g_{m1}S_{ m2}}{C_{1}C_{2}}\Big{)}}{D(s)}. \tag{6}\]  
With \(v_{in}=v_{1}=v_{3}=v_{4}\), \(v_{2}=0\), and \(g_{m2}=1/R_{2}\), then the AP response is realized as:  
\[T_{AP(VM)}\ =\ \frac{v_{out}}{v_{in}}\ =\ \frac{s^{2}-\Big{(}\frac{g_{m2}}{C_{2}} \Big{)}s+\Big{(}\frac{g_{m1}S_{m2}}{C_{1}C_{2}}\Big{)}}{D(s)}. \tag{7}\]  
where  
\[D(s)=s^{2}+\bigg{(}\frac{s}{R_{2}C_{2}}\bigg{)}+\bigg{(}\frac{g_{m1}S_{m2}}{C_ {1}C_{2}}\bigg{)}. \tag{8}\]  
Under appropriate conditions, the proposed circuit realizes all five generic biquadratic filter responses at \(v_{out}\), which are taken from the w-terminal of VDBA2. Thus, the voltage output of the circuit has a very low output impedance, which is suitable for VM cascadability. Moreover, in this MISO configuration, there is no requirement for negative and double input voltage signals to realize the desired filter responses. For CM operation: If \(v_{1}=v_{2}=v_{3}=v_{4}=0\) (grounded), the CM biquad transfer functions for this SIMO filter, attained from the circuit analysis of Figure 3, are given by  
\[T_{LP(CM)}\ =\ \frac{i_{LP}}{i_{in}}\ =\ \frac{-H_{0}\big{(}\frac{g_{m1}S_{ m2}}{C_{1}C_{2}}\big{)}}{D(s)}, \tag{9}\]  
\[T_{BP(CM)}\ =\ \frac{i_{BP}}{i_{in}}\ =\ \frac{\Big{(}\frac{g}{R_{2}C_{2}} \Big{)}}{D(s)}, \tag{10}\]  
and  
\[T_{HP(CM)}\ =\ \frac{i_{HP}}{i_{in}}\ =\ \frac{s^{2}}{D(s)}, \tag{11}\]  
where \(H_{0}\) is the passband gain equal to \(1/g_{m2}R_{1}\). Additionally, for \(R_{1}=1/g_{m2}\), the BS current response can be realized by connecting the appropriate output currents as \(i_{BS}=i_{HP}-i_{LP}\). In the same way, the AP response can also be obtained by the interconnection of LP, BP, and HP responses as \(i_{AP}=i_{HP}-i_{BP}-i_{LP}\). For TAM operation: With \(v_{in}=v_{3}\) and \(v_{1}=v_{2}=v_{4}=0\), then we obtain the TAM filter functions as follows:  
Figure 2.0  
Figure 2.1Figure 2.2  
Figure 2.3  
and  
Figure 2.4  
Equations (12)-(16) represent the TAM filter functions of the LP, BP, HP, BS, and AP responses with controllable passband gains, respectively.