## 2 Design of Low-Pass Filter
### Operational Amplifier  
Linear active stages are critical for any high linearity filter, since they are the most contributing sources of nonlinearity [21]. The structure of two-stage fully differential operation amplifier (FDOA), used for both LPF topologies, is shown in Figure 7. The FDOA is composed of two gain stages, an input differential pair stage (M\({}_{1}\)-M\({}_{5}\)) and an output stage (M\({}_{13}\)-M\({}_{22}\)), both of which are controlled by two common mode feedback (CMFB) loop amplifiers. The biasing of the input stage and the first CMFB loop amplifier (M\({}_{6}\)-M\({}_{10}\)) can be controlled by changing the value of current source I\({}_{\text{bias1}}\), which is mirrored and scaled by M\({}_{11}\). In a similar way, biasing of the output stage and second CMFB loop amplifier (M\({}_{23}\)-M\({}_{27}\)) can be controlled by I\({}_{\text{bias2}}\), which is mirrored and scaled by M\({}_{12}\). The first CMFB loop amplifier controls the bias point of the output stage via M\({}_{5}\).  
The output stage is designed to work in class-A configuration, where each single-ended output leg is driven by the differential output of the input stage. The voltage of the common mode output of the FDOA is set by V\({}_{\text{CMO}}\) and controlled by the second CMFB  
Figure 6: Structure of the resistor bank calibration.  
Figure 7: Schematic of the proposed fully differential operational amplifier for 2 nd and 4 th order low-pass filter topologies.  
loop amplifier via M\({}_{21}\)-M\({}_{22}\) in the output stage. In the designed filter, the V\({}_{\text{CMO}}\) voltage is fixed to half of the supply voltage.  
Passive RC common signal sense elements (R\({}_{\text{cm1}}\), R\({}_{\text{cm2}}\), C\({}_{\text{cm1}}\), C\({}_{\text{cm2}}\)) are used between the input and output stage sensing nodes and the corresponding CMFB amplifier input for improved linearity performance, as opposed to using active elements. Miller compensation elements R\({}_{\text{CR}}\) and C\({}_{\text{CR}}\) are used between the input and output stages. The values of the resistor and capacitor are controlled using resistor and capacitor tuning banks, which have same structures as shown in Figure 5a. The corresponding control word of the LPF capacitor tuning bank is used to automatically select the required compensation values of the R\({}_{\text{CR}}\) and C\({}_{\text{CR}}\).  
When FDOA is powered down (circuitry is not show in Figure 7), the output stage is in a high-impedance state, where the impedance is mainly defined by the CMFB common signal sense and Miller compensation elements, if terminating switches are not present.  
Control of I\({}_{\text{bias1}}\) and I\({}_{\text{bias2}}\) allows to change the GBWP of the FDOA at the expense of increased power consumption, without significantly affecting other FDOA parameters, such as gain and linearity. Thus, bias control enables power optimization of the FDOA at different filter bandwidth configurations, over a wide operation range.  
LP and HP stages use same structure of the FDOA, but transistor sizing and driving strength are different to meet the different requirements for them. The FDOA in the 2 nd and 4 th order stages on the same path (LP or HP) are identical, only having subtle variations in the values of the R\({}_{\text{CR}}\) and C\({}_{\text{CR}}\) compensation bank.