# 0.5-V High-Order Universal Filter for Bio-Signal Processing Applications
## 2 Proposed Circuit
### Proposed High-Order Universal Filter  
The proposed \(n\)th-order universal filter using MI-OTA is shown in Figure 5. It uses n + 1 MI-OTAs and n grounded capacitors. Note that the input terminals of the universal filter are connected to the high-impedance terminals of the OTA; hence, the filter has a high input impedance. Signals can therefore be applied to the filter inputs without the requirement for buffer circuits. The universal filter using multiple-input OTAs (including plus/minus terminals) can easily provide the non-inverting and inverting transfer functions of a low-pass filter (LPF), high-pass filter (HPF), band-pass filter (BPF), band-stop filter (BSF), and all-pass filter (APF) without the additional requirements of an inverting input amplifier. The use of a grounded capacitor facilitates compensation for parasitic circuit capacitances. Using (1) and nodal analysis, the output voltage of the filter in Figure 5 can be expressed by  
\[V_{out(nth)}=\frac{N(s)}{D(s)} \tag{14}\]  
\[ \begin{split} N(s)=\left(V_{in(n+1)+}-V_{in(n+1)-}\right)\prod_{i= 1}^{n}\left(\frac{\delta_{mi}}{C_{i}}\right)\\ +\sum_{i=1}^{n}\Bigg{(}\left(V_{in(n+1-i)+}+V_{in(n+1-i)-}\right) s^{i}\prod_{k=j}^{n-1}\left(\frac{\delta_{mk}}{C_{k}}\right)\Bigg{)}\end{split}  \tag{15}\]  
\[D(s)=\prod_{i=1}^{n}\left(\frac{\delta_{mi}}{C_{i}}\right)\sum_{i=1}^{n}s^{i} \prod_{k=j}^{n-1}\left(\frac{\delta_{mk}}{C_{k}}\right) \tag{16}\]  
Figure 5: Proposed nth-order universal filter using MI-OTAs. To represent the operation of the system in Figure 5, the first-, second-, and third-order universal filters have been designed as an example. Figure 6 shows the first-order universal filter that employs two OTAs and one grounded capacitor. The output voltage can be expressed by  
\[V_{out(1 st)}=\frac{s(V_{in21}-V_{in22})+\frac{\delta\pi 1}{\mathbb{C}_{1}}(V_{in11}-V_{in12})}{s+ \frac{\delta\pi 1}{\mathbb{C}_{1}}} \tag{17}\]  
The pole frequency of the filter can be given by  
\[\omega_{o}=\frac{g_{m1}}{C_{1}} \tag{18}\]  
With the first-order universal filter using MI-OTA, variant filter functions LPF, HPF, and APF can be obtained under suitable conditions as shown in Table 1. Inputs to which no input signal is applied should be connected to ground to prevent noise injection. The second-order universal filter that employs three OTAs and two grounded capacitors is shown in Figure 7. The output voltage can be expressed by  
\[V_{out(2 nd)}=\frac{s^{2}(V_{in31}-V_{in32})+s\frac{g_{m1}}{\mathbb{C}_{1}}(V_{ in21}-V_{in22})+\frac{g_{m1}g_{m2}}{\mathbb{C}_{1}\mathbb{C}_{2}}(V_{in11}-V_{in12})}{s ^{2}+s\frac{g_{m1}}{\mathbb{C}_{1}}+\frac{\delta\pi 1}{\mathbb{C}_{1}}s^{2}+\frac{ \delta\pi 1}{\mathbb{C}_{1}}s^{2}} \tag{19}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Filtering Function**} & **Input** & **Transfer Function** \\ \hline \multirow{3}{*}{LPF} & Non-inverting & \(V_{in11}=V_{in}\) & \(H(s)=\frac{\frac{\xi_{\text{ml}}}{\mathbb{C}_{1}}}{D(s)}\) \\ \cline{2-4}  & Inverting & \(V_{in12}=V_{in}\) & \(H(s)=-\frac{\frac{\xi_{\text{ml}}}{\mathbb{C}_{1}}}{D(s)}\) \\ \hline \multirow{3}{*}{HPF} & Non-inverting & \(V_{in21}=V_{in}\) & \(H(s)=\frac{s}{D(s)}\) \\ \cline{2-4}  & Inverting & \(V_{in22}=V_{in}\) & \(H(s)=-\frac{s}{D(s)}\) \\ \hline \multirow{3}{*}{APF} & Non-inverting & \(V_{in12}=V_{in21}=V_{in}\) & \(H(s)=\frac{s-\frac{\delta\pi 1}{\mathbb{C}_{1}}}{D(s)}\) \\ \cline{2-4}  & Inverting & \(V_{in11}=V_{in22}=V_{in}\) & \(H(s)=-\frac{s-\frac{\delta\pi 1}{\mathbb{C}_{1}}}{D(s)}\) \\ \hline \hline \end{tabular} Note: \(D(s)=s+\frac{g_{m1}}{\mathbb{C}_{1}},H(s)=\frac{V_{out(in)}}{V_{in}}\).  
\end{table} 
Table 1: Obtaining variant filtering functions of the first-order universal filter. Figure 6: First-order universal filter.