## 3 Circuit Design
### Analog Front-End
#### 3.1.3 High-Pass Filter Design and Optimization  
Let us now consider the noise due to the MOS-bipolar pseudo-resistors that was neglected in Equation (2). It was experimentally verified that the noise spectral density of the pseudo-resistors complies with the usual equation:  
\[\overline{S^{2}}_{R}=\frac{4 kT}{R} \tag{9}\]  
where \(R\) is the small signal incremental resistance. This noise contribution to the OTA input-referred noise power spectral density follows a \(1/f^{2}\) law and is given by:  
\[\overline{E^{2}}_{n_{R}}=2\frac{4 kT}{R}\frac{1}{(C_{1}2\pi f)^{2}}=\frac{8 kTf _{1}}{2\pi G_{1}^{2}C_{2}f^{2}} \tag{10}\]  
where \(f_{1}\) is the cut-off frequency of the high-pass filter set by the first amplifier stage. The factor of \(2\) in Equation (10) accounts for the two pseudo-resistors, one in the feedback path and one connected to the positive input terminal of the operational amplifier. For simplicity, let us now neglect the presence of the selective high-pass filter following the first stage. By integrating Equation (10) from \(f_{1}\) to the low-pass cut-off frequency of the overall amplifier, \(f_{LP}\), we get:  
\[\int_{f_{1}}^{f_{LP}}\overline{E^{2}}_{n_{R}}df\cong\frac{8 kT}{2\pi G_{1}^{2 }C_{2}} \tag{11}\]  
which is independent of the cut-off frequency \(f_{1}\). From Equation (11) the contribution of the pseudo-resistors to the OTA input-referred noise is about \(2.79\)\(\mu\)V\({}_{rms}\). This value is not negligible and explains why it is not convenient to implement the selective high-pass filter (_i.e._, the \(300\) Hz cut-off frequency high-pass filter) in the first stage, as this solution would degrade the noise performance. The \(G_{M}\)-\(C\) high-pass filter is instead able to cut off most of the noise from the pseudo-resistors of the first stage, but a careful choice of the filter capacitor is needed to reduce its noise contribution. We will discuss this point in detail. Let us now suppose that \(f_{1}<<f_{HP}\), where \(f_{HP}\) is the cut-off frequency of the \(G_{M}\)-\(C\) high-pass filter. For frequency above \(f_{1}\), the input-referred power spectral density due to both the pseudo-resistors of the first stage and to the \(G_{M}\)-\(C\) filter can be written as:  
\[\overline{E^{2}}_{eq(1/f^{2})}\cong\frac{8 kT}{R}\frac{1}{(C_{1}2\pi f)^{2}}+ \overline{S_{nI}^{2}}\ \frac{1}{(G_{1}C_{HP}\ 2\pi f)^{2}} \tag{12}\]  
where \(\overline{S_{nI}^{2}}\) is the output current noise of the high-pass filter. For simplicity, let us assume that the current noise generated by the \(G_{M}\) cell (\(G_{M}=1/R_{HP}\)) can be written as \(4 kT/R_{HP}\). Equation (12) becomes:  
\[\overline{E^{2}}_{eq(1/f^{2})}=\frac{8 kTf_{1}}{2\pi G_{1}^{2}C_{2}f^{2}}+ \frac{4 kTf_{HP}}{2\pi G_{1}^{2}C_{HP}f^{2}} \tag{13}\]  
By integrating Equation (13) over the amplifier band, _i.e._, from \(f_{HP}\) to \(f_{LP}\), we obtain:  
\[\int_{f_{HP}}^{f_{LP}}\overline{E^{2}}_{eq(1/f^{2})}\ df=\frac{8 kT}{2\pi G_{1 }^{2}C_{2}}\frac{f_{1}}{f_{HP}}+\frac{4 kT}{2\pi G_{1}^{2}C_{HP}} \tag{14}\]  
The result suggests to reduce the first term by minimizing the \(f_{1}/f_{HP}\) ratio while a large capacitor value \(C_{HP}\) is needed to reduce the second term. In practice, the \(G_{M}\) cell is a simple differential stage(see Figure 3) whose bias current can be externally tuned. Because a small bias current (\(I\cong 1\) nA) is needed to synthesize a high value resistance, all the transistors work in weak-inversion region and their transconductance is \(G_{M}=g_{m1}\cong\kappa I/U_{T}\). The current noise of this configuration is therefore:  
\[\overline{S_{nI}^{2}}=4\frac{2 kT}{\kappa}g_{m1}=\frac{8 kT}{\kappa R_{HP}} \tag{15}\]  
The current noise of the \(G_{M}\) cell is therefore a factor of \(2/\kappa\) larger than that of a resistor \(R_{HP}\) and this factor has to be added in the second term of Equation (14).