# Design of Shadow Filter Using Low-Voltage Multiple-Input Operational Transconductance Amplifiers
## 3 Simulation Results
Figure 11, showing the frequency responses of the first shadow filter with \(C_{1,2}=18\) pF, \(R_{1}=5\) M\(\Omega\), \(I_{\text{set3}}=12\) nA, and \(I_{\text{set1,2,4}}=6\) nA, was repeated using Monte Carlo (MC) analysis, which included mismatch and process variations over 200 runs. As shown in Figure 11, the filter circuit exhibited an acceptable variation across these runs. Figure 8: Frequency responses of the first shadow filter with varying \(I_{\text{set}}=I_{\text{set1,2,4}}\) for (a) LPF, (b) HPF, and (c) BPF. Figure 7: Frequency responses of the first shadow filter with \(I_{\text{set3}}=12\) nA and \(I_{\text{set1,2,4}}=6\) nA. Figure 10: Frequency responses of the first shadow filter with various \(R_{1}\) for (**a**) LPF, (**b**) HPF, and (**c**) BPF. Figure 9: Frequency responses of the first shadow filter with varying \(I_{\text{set}}=I_{\text{set5}}\) for (**a**) LPF, (**b**) HPF, and (**c**) BPF. Figure 12 was also repeated using process, voltage, and temperature (PVT) analysis, which included the following conditions: fast-fast, slow-fast, fast-slow, and slow-slow transistor process corners, \(\pm\)10% variations in supply voltage, and temperature corners of -10 \({}^{\circ}\)C and 60 \({}^{\circ}\)C. As shown in Figure 12, the filter circuit again demonstrated an acceptable variation across these PVT corners. The transient performance of the LPF in the first shadow filter, configured as depicted in Figure 13, was evaluated. As shown in Figure 13a, the input signal featured a 300 mV peak-to-peak amplitude at a frequency of 10 Hz. The results show that the input and output waveforms closely overlapped, with a total harmonic distortion (THD) of 1.8%. Figure 12b illustrates the relationship between the THD and the peak-to-peak amplitude of the input signal at 10 Hz. Notably, the circuit maintained a THD of 1% for a 200 mV peak-to-peak input, despite operating at a low supply voltage of 450 mV. Figure 14 presents the frequency response of the output noise for the LPF in the first shadow filter. The integrated noise within the frequency range of 1 Hz to 106 Hz was determined to be 264 \(\upmu\)V, yielding a dynamic range (DR) of 54.5 dB for 1% THD. Figure 15 presents the frequency responses of the second first shadow filter with \(C_{1,2}\) = 18 pF, \(R_{1}\) = 20 M\(\Omega\), a constant \(I_{\text{set1},2,5}\) = 6 nA, \(I_{\text{set4}}\) = 12 nA, and varying \(I_{\text{set}}\) = \(I_{\text{set5}}\) = [1.5, 3, 6, 12, 24, 48] nA. This figure demonstrates the tunability of the filter's quality factor \(Q\), as described in Equation (16). Figure 11: Frequency responses of the first shadow filter with MC for (**a**) LPF, (**b**) HPF, and (**c**) BPF. Figure 14: Frequency response of the output noise for the LPF in the first shadow filter. Figure 12: Frequency responses of the first shadow filter with PVT for (**a**) LPF, (**b**) HPF, and (**c**) BPF. Figure 13: (**a**) Transient response of the LPF in the first shadow filter and (**b**) the THD. The proposed shadow filter was compared with previously reported shadow filters, as summarized in Figure 10.0 Recent shadow filters from [8; 18; 25; 27; 29], which provide voltage-mode transfer functions, were selected for the comparison. It is evident that the proposed shadow filter allows the parameters \(\omega_{o}\) or \(Q\) to be adjusted without affecting the passband magnitude of the transfer functions. Although the circuit in [27] can compensate for the passband magnitude using amplifiers, the adjustment must be performed manually. This underscores the novel solution provided by the proposed circuit. The shadow filter demonstrated the lowest supply voltage and power consumption among its counterparts, as shown in comparison with [8; 18; 25; 27; 29]. Additionally, compared to [8], the proposed circuit offers the added benefit of electronic control.