# Design of Shadow Filter Using Low-Voltage Multiple-Input Operational Transconductance Amplifiers
## 2 Proposed Circuit
### Proposed First Shadow Filters  
Referring to the block diagram of the shadow filter in Figure 3a [7], the outputs of the HPF and LPF are summed and amplified by amplifier A. The resulting amplified output is then summed with the input \(V_{in}\). In this shadow filter, the gain of amplifier A controls the quality factor (\(Q\)) of the filter. However, adjusting the quality factor can influence the passband gain of both the HPF and LPF, either increasing or decreasing it [8; 25; 31]. To address this, the shadow filter in Figure 3a [7] has been modified to maintain a constant passband gain for the HPF and LPF even as the quality factor is varied through the amplifier gain. The updated block diagram of the shadow filter is shown in Figure 3b. To keep the passband gain constant, a negative amplifier (\(-A\)) and an additional input, \(V_{in2}\), are introduced. If a band-pass filter (BPF) is needed, \(V_{in2}\) can be connected to ground. For the HPF and LPF configurations, both \(V_{in1}\) and \(V_{in2}\) can be driven by the same input signal (\(V_{in1}=V_{in2}=V_{in}\)). Figure 4 illustrates the proposed first shadow filter using MI-OTAs. The circuit consists of four MI-OTAs, two grounded capacitors, and one grounded resistor. Compared to Figure 3b, the components \(g_{m1}\) and \(R_{1}\) function as amplifiers \(A\) and \(-A\), with their roles slightly interchanged to facilitate easier control of the quality factor, i.e., \(1+A\). Thanks to the multiple inputs of the OTA, various filter functions can be easily realized. By connecting \(V_{in2}\) and \(V_{in4}\) to ground and using Equation (1) along with nodal analysis, the output voltage of the band-pass filter (BPF) can be expressed as follows:  
\[V_{BPF}=\frac{sC_{2}g_{m1}\{V_{in3}-V_{in1}\}}{s^{2}C_{1}C_{2}(1+g_{m3}R_{1})+ sC_{2}g_{m1}+g_{m1}g_{m2}(1+g_{m3}R_{1})} \tag{9}\]  
If \(V_{in1}\) is driven by the input signal, an inverting BPF is obtained, whereas applying the input signal to \(V_{in3}\) produces a non-inverting BPF. Using Equation (1) and applying nodal analysis, the output voltage of the LPF can be expressed as follows:  
\[V_{LPF}=\frac{g_{m1}g_{m2}\{(V_{in1}-V_{in3})+g_{m3}R_{1}(V_{in2}-V_{in4})\}} {s^{2}C_{1}C_{2}(1+g_{m3}R_{1})+sC_{2}g_{m1}+g_{m1}g_{m2}(1+g_{m3}R_{1})} \tag{10}\]  
Figure 3: Block diagram of (**a**) first shadow filter and (**b**) shadow filter with improved passband gain. If \(V_{in1}\) and \(V_{in2}\) are simultaneously driven by input signals while \(V_{in3}\) and \(V_{in4}\) are grounded, a non-inverting LPF is obtained. Conversely, if \(V_{in3}\) and \(V_{in4}\) are applied with input signals while \(V_{in1}\) and \(V_{in2}\) are grounded, an inverting LPF is achieved. Using Equation (1) and nodal analysis, the output voltage of the HPF can be expressed as follows:  
\[V_{HPF}=\frac{s^{2}C_{1}C_{2}\{(V_{in1}-V_{in3})+g_{m3}R_{1}(V_{in2}-V_{in4}) \}}{s^{2}C_{1}C_{2}(1+g_{m3}R_{1})+sC_{2}g_{m1}+g_{m1}g_{m2}(1+g_{m3}R_{1})} \tag{11}\]  
For a non-inverting HPF, \(V_{in1}\) and \(V_{in2}\) are applied with input signals together, while \(V_{in3}\) and \(V_{in4}\) are grounded. For an inverting HPF, \(V_{in3}\) and \(V_{in4}\) are applied with input signals while \(V_{in1}\) and \(V_{in2}\) are grounded. It should be noted that the shadow filter in Figure 4 allows the quality factors of the BPF, LPF, and HPF to be adjusted without affecting the passband gain of their transfer functions. Additionally, both non-inverting and inverting configurations of the BPF, LPF, and HPF can be achieved under appropriate conditions, facilitated by the MI-OTA-based circuit. Any unused input terminals can be connected to ground. If a band-stop filter (BSF) is required, it can be realized by summing \(V_{HPF}\) and \(V_{LPF}\). Similarly, an all-pass filter (APF) can be achieved by summing \(V_{HPF}\), \(V_{BPF}\), and \(V_{LPF}\). In both cases, an additional summing amplifier is necessary, which can be implemented using an MI-OTA in a negative feedback configuration similar to \(g_{m4}\).