# Design of Shadow Filter Using Low-Voltage Multiple-Input Operational Transconductance Amplifiers
## 2 Proposed Circuit
### Proposed Second Shadow Filters  
The shadow filter depicted in Figure 4 employs an external amplifier solely to adjust the quality factor. As illustrated in the block diagram of the shadow filter in Figure 5a [7], the output signals BPF and LPF are amplified by amplifiers A\({}_{1}\) and A\({}_{2}\), respectively, and then combined with the input signal. This design enables independent control of the quality factor and natural frequency using A\({}_{1}\) and A\({}_{2}\). However, modifying the natural frequency can influence the passband gain of both the BPF and LPF, potentially causing it to either increase or decrease [8; 25; 26; 31; 32]. To address this issue, the updated block diagram of the shadow filter, shown in Figure 5b, introduces A\({}_{1}\) and A\({}_{2}\), and additional inputs \(V_{in2}\) and \(V_{in3}\) to stabilize the passband gain. For an HPF configuration, the input signal is applied to \(V_{in1}\), with \(V_{in2}\) and \(V_{in3}\) grounded. In contrast, for the BPF and LPF configurations, \(V_{in1}\), \(V_{in2}\), and \(V_{in3}\) can be driven by the same input signal under appropriate conditions. Figure 6 presents the proposed second shadow filter using MI-OTAs. The circuit comprises five MI-OTAs, two grounded capacitors, and one grounded resistor. Leveraging the multiple-input capability of the OTAs, this design facilitates t  
Figure 5: Block diagram of (**a**) second shadow filter in [2] and (**b**) shadow filter with improved passband gain. mentation of various filter functions. By grounding \(V_{in2}\) and \(V_{in3}\) and applying Equation (1) with nodal analysis, the output voltage of the HPF can be expressed as follows:  
\[V_{HPF}=\frac{s^{2}C_{1}C_{2}(V_{in1}-V_{in4})}{s^{2}C_{1}C_{2}+sC_{2}g_{m1}(1-g _{m3}R_{1})+g_{m1}g_{m2}(1-g_{m4}R_{1})} \tag{14}\]  
If \(V_{in1}\) is the input signal, a non-inverting HPF is obtained. Alternatively, applying the input signal to \(V_{in4}\) results in a non-inverting BPF. **Case 1. **: _When \(V_{in2}\) is grounded and \(V_{in3}\) is connected to either \(V_{in1}\) or \(V_{in4}\), nodal analysis gives the BPF output voltage as follows:_  
\[V_{BPF(1)}=\frac{sC_{2}g_{m1}(1-g_{m4}R_{1})(V_{in4}-V_{in1})}{s^{2}C_{1}C_{2 }+sC_{2}g_{m1}(1-g_{m3}R_{1})+g_{m1}g_{m2}(1-g_{m4}R_{1})} \tag{15}\]  
_If \(V_{in1}\) is the input signal (with \(V_{in1}\) also connected to \(V_{in2}\)), an inverting BPF is obtained with \(V_{in4}\) grounded. Conversely, if \(V_{in4}\) is the input signal (with \(V_{in4}\) also connected to \(V_{in2}\) and \(V_{in1}\) grounded), a non-inverting BPF is achieved. In this configuration, the BPF gain remains constant when the natural frequency is adjusted by amplifier \(A_{2}\)._  
**Case 2. **: _When \(V_{in3}\) is grounded and \(V_{in2}\) is connected to either \(V_{in1}\) or \(V_{in4}\), the BPF output voltage is as follows:_  
\[V_{BPF(1)}=\frac{sC_{2}g_{m1}(1-g_{m4}R_{1})(V_{in4}-V_{in1})}{s^{2}C_{1}C_{2 }+sC_{2}g_{m1}(1-g_{m3}R_{1})+g_{m1}g_{m2}(1-g_{m4}R_{1})} \tag{16}\]  
_The conditions for obtaining a BPF in this case are similar to those for Case 1, except that \(V_{in2}\) is connected to either \(V_{in1}\) or \(V_{in4}\). Here, the BPF gain remains constant when the quality factor is adjusted using amplifier \(A_{1}\), while \(A_{2}\) is fixed._  
Figure 6: Proposed second shadow filter using MI-OTAs. By grounding \(V_{in2}\) and connecting \(V_{in3}\) to either \(V_{in1}\) or \(V_{in4}\), the LPF output voltage is expressed as follows:  
\[V_{LPF}=\frac{g_{m1}g_{m2}(1-g_{m4}R_{1})(V_{in1}-V_{in4})}{s^{2}C_{1}C_{2}+sC_{2 }g_{m1}(1-g_{m3}R_{1})+g_{m1}g_{m2}(1-g_{m4}R_{1})} \tag{17}\]  
If \(V_{in1}\) is the input signal (with \(V_{in1}\) also connected to \(V_{in3}\)), a non-inverting LPF is obtained, with \(V_{in4}\) grounded. Conversely, if \(V_{in4}\) is the input signal (with \(V_{in4}\) also connected to \(V_{in3}\) and \(V_{in1}\) grounded), an inverting LPF is realized.