# Design of Shadow Filter Using Low-Voltage Multiple-Input Operational Transconductance Amplifiers
## 3 Simulation Results  
The proposed CMOS circuit and filter applications were designed and simulated using the Cadence Virtuoso Design Environment, utilizing TSMC's 0.18 \(\upmu\)m CMOS technology. The circuit is capable of operating with either a single supply voltage of 450 mV or dual supply voltages of \(\pm\)225 mV, as utilized in the presented simulations. The transistor aspect ratios are detailed in Table 1. Notably, relatively long channel lengths (L = 1 \(\upmu\)m) were selected for the devices to achieve both higher voltage gain and larger device areas, which help mitigate flicker noise and mismatch effects. The channel widths of M\({}_{1}\), M\({}_{2}\), and M\({}_{\rm ic}\) (i = 5-13) were determined during the simulation phase to provide quiescent V\({}_{\rm CS}\) voltages of the above transistors equal to around V\({}_{\rm DD}\)/3, which provides the maximum headroom for possible PVT variations and signal swing. The aspect ratios of M\({}_{3}\) and M\({}_{4}\) were half of the ones for M\({}_{1}\) and M\({}_{2}\), which provides m = 0.5, required for optimal linearity. The channel widths of M\({}_{\rm i}\) were twice as large as those for M\({}_{\rm ic}\), which provided V\({}_{\rm DS}\) of M\({}_{\rm ic}\) equal to around 25 mV--a good compromise between voltage gain and signal swing of the OTA. The input metal-insulator-metal (MIM) capacitor \(C_{\rm Bi}\) was set to 0.5 pF, and the bias voltage \(V_{\rm B}=-50\) mV. It is important to note that the simulation and experimental results for the MI-OTA with two inputs are provided in [33]. Consequently, this section focuses on the simulation results for filter applications. For the first shadow filter, with \(C_{1,2}\) = 18 pF, \(R_{1}\) = 5 M\(\Omega\), \(I_{\rm set3}\) = 12 nA, and \(I_{\rm set1,2,4}\) = 6 nA, the frequency responses of the LPF, BPF, and BPF are illustrated in Figure 7. The cutoff frequency was determined to be 106 Hz, with a total power consumption of 81 nW. Figure 8 depicts the frequency responses of the first shadow filter with a constant \(I_{\rm set3}\) = 12 nA and varying \(I_{\rm set}=I_{\rm set1,2,4}\) = [1.5, 3, 6, 12, 24, 48] nA. The corresponding cutoff frequencies were [26.6, 53, 106, 199.5, 398.1, 708] Hz, respectively. This figure validates the filter's tunable natural frequency, as described in Equation (9). With a frequency range spanning from subhertz to 10 kHz, the proposed filter is suitable for bio-signal processing applications. This work introduces a universal filter that integrates low-pass (LPF), high-pass (HPF), and band-pass (BPF) functionalities within a single topology. The specific application depends on the user's requirements. For instance, a low-pass filter can be utilized to remove interference noise, while a band-pass filter can be employed to isolate the desired frequency range. \begin{table}
\begin{tabular}{c c} \hline
**Transistor** & **W/L (\(\upmu\)m/\(\upmu\)m)** \\ \hline M\({}_{1}\), M\({}_{2}\) & \(2\times 15/1\) \\ \hline M\({}_{3}\), M\({}_{4}\) & 15/1 \\ \hline M\({}_{5}-\) M\({}_{8}\) & \(2\times 10/1\) \\ \hline M\({}_{5c}-\) M\({}_{8c}\) & 10/1 \\ \hline M\({}_{9}-\) M\({}_{13}\) & \(2\times 15/1\) \\ \hline M\({}_{9c}-\) M\({}_{13c}\) & 15/1 \\ \hline M\({}_{\rm L}\) & 5/4 \\ \hline \end{tabular}
\end{table} 
Table 1: Transistor aspect ratios of the proposed MI-OTA. Figure 9 presents the frequency responses of the first shadow filter with \(R_{1}=5\) M\(\Omega\), a constant \(I_{\text{set1,2,4}}=6\) nA, and varying \(I_{\text{set}}=I_{\text{set3}}=\) [1.5, 3, 6, 12, 24, 48] nA. This figure demonstrates the tunability of the filter's quality factor \(Q\), as described in Equation (10). The filter can achieve broader tunability of the quality factor \(Q\), if required, by adjusting \(R_{1}\), as demonstrated in Figure 10. For this analysis, \(I_{\text{set1,2,4}}=6\) nA, \(I_{\text{set3}}=48\) nA, and various \(R_{1}=\) [2.5, 5, 10, 20] M\(\Omega\) were used.