## 3 Circuit Design
### Analog Front-End
#### 3.1.1 Noise Analysis and First Stage Sizing
Its value can be estimated from the EKV model [13], valid in all regions of inversion:  
\[g_{m}\cong\frac{\kappa I_{D}}{U_{T}}\frac{2}{1+\sqrt{1+4\:IC}} \tag{4}\]  
\(U_{T}\) and \(IC\) being the thermal voltage and the inversion coefficient [12], respectively. The latter parameter is defined as:  
\[IC=\frac{\kappa I_{D}}{2\mu C_{ox}^{\prime}\left(\frac{W}{L}\right)U_{T}^{2}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Transistor** & \(\mathbf{V_{GS}-V_{T}}\) [mV] & **IC** & \(\mathbf{g_{m}}\) [\(\mathbf{\mu\)A/V]} & \(\mathbf{r_{o}}\) [M\(\Omega\)] \\ \hline \(M_{p}\) & 110 & 0.075 & 52.3 & 5.1 \\ \(M_{n}\) & 490 & 51 & 7.76 & 108 \\ \(M_{cas}\) & 24 & 0.36 & 43.7 & 8.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Transistor operating points. and is much less than \(1\) in weak-inversion or subthreshold region. ?From Equation (4), the overall-amplifier input-referred thermal noise power spectral density results:  
\[\overline{E^{2}}_{n_{th}}=\frac{8 kTU_{T}}{\kappa^{2}I_{bias}} \tag{6}\]  
with \(I_{bias}\) being the bias current of the telescopic cascade amplifier. Assuming a first-order roll-off of the frequency response, the noise bandwidth is \(\frac{\pi}{2}\cdot 10\) kHz \(\cong 15.7\) kHz. For an upper limit of \(3\)\(\mu\)V\({}_{rms}\) for the thermal noise contribution in this noise band, a minimal \(I_{bias}\) value of about \(3\)\(\mu\)A is required; to be conservative, we set \(I_{bias}=4\)\(\mu\)A. With regard to the flicker noise, its contribution can be reduced by increasing the PMOS transistor area (\(W_{p}\times L_{p}\)). Considering that for the adopted technology \(K_{p}^{(1/f)}\approx 2\times 10^{-26}\) V\({}^{2}\)F and \(C_{ox}^{{}^{\prime}}\approx 5\) fF/\(\mu\)m\({}^{2}\) and setting a noise corner frequency lower than \(100\) Hz, the input PMOS transistors were sized with \(W_{p}\times L_{p}=400\times 1\)\(\mu\)m\({}^{2}\). This sizing results into a stray OTA input capacitance \(C_{p}\) of approximately \(500\) fF, which does not excessively impair the equivalent input noise. Moreover, in order to fulfill the requirement \((W/L)_{p}\gg(W/L)_{n}\), the settings \(W_{n}=5\)\(\mu\)m and \(L_{n}=40\)\(\mu\)m were applied, thus forcing these devices to work in the strong inversion region (see Table 1). For this choice of the transistor lengths, cascading of the input differential pair is needed to not to degrade the output resistance and, thus, to not to lower the amplifier gain. The introduction of \(M_{cas}\) guarantees an output resistance \(R_{out}=r_{0n}||(g_{meas}r_{0 cas}\,r_{0p})\approx r_{0n}\) and an amplifier gain \(g_{mp}r_{0n}\approx 75\) dB. In summary, by careful transistor sizing, the noise within the amplifier band may be reduced to the sole thermal noise. Since in a generic amplifier a trade-off exists between noise and current consumption, its efficiency may be evaluated referring to the noise efficiency factor (\(NEF\)), which is a parameter commonly adopted in literature, proposed in [14] and defined as:  
\[NEF=V_{in,rms}\sqrt{\frac{2I_{tot}}{\pi U_{T}\,4 kT\,BW}} \tag{7}\]  
where \(V_{in,rms}\) is the input-referred rms noise, \(I_{tot}\) is the total supply current and \(BW\) is the bandwidth of the amplifier. In this limit, the minimum \(NEF\) achievable with a simple differential stage can be estimated from Equations (6) and (7) leading to a value of \(\sqrt{2}/\kappa=2.02\)[15]. However, in our design the contribution to the thermal noise of the current mirror transistors cannot be completely neglected. In fact, even if \(M_{p}\) transistors work in weak-inversion and \(M_{n}\) FETs are biased in strong inversion region, the transconductance ratio is only about \(7\) (see Table 1).