# Fluctuation-Enhanced Sensing (FES): A Promising Sensing Technique
## 6 Reducing Errors and Enhancing the Sensory Information Channel Capacity
### Dedicated Instrumentation
The noise introduced by the feedback resistance (\(R_{R}\)) is due to its thermal noise and its power spectra density \(S_{V\!R}\) is given by:  
\[S_{V\!R}=4 kTR_{R} \tag{12}\]  
For resistances \(R_{R}\) in excess of 1 M\(\Omega\), the thermal noise of the resistance is much larger than \(1\times 10^{-14}\) V\({}^{2}\)/Hz (or 100 nV/\(\sqrt{\text{Hz}}\)) so that, unless we need to work at very low frequencies (\(f<<1\)Hz), the contribution of \(e_{Vn}\) to the equivalent input noise \(i_{n}\) can be neglected (the PSD of \(e_{Vn}\) for the TLC  
Figure 12: Ideal circuit for separating the current fluctuations entering the Low Noise Transimpedance Amplifier (LN-TIA) from the DC bias current (ideally flowing through the inductance LA). 2201 MOSFET input OA is in the order \(3.6\times 10^{-15}\) V\({}^{2}\)/Hz at 1 Hz and \(43.6\times 10^{-16}\) V\({}^{2}\)/Hz at 10 Hz). In this assumption we obtain, for the PSD \(S_{IN}\) of the equivalent input noise source \(i_{n}\):  
\[S_{IN}\approx\frac{4 kT}{R_{R}} \tag{13}\]  
Equation (13) is particularly interesting since it shows that in order to reduce the equivalent input noise of a LN-TIA realized as in Figure 13 the feedback resistance must be made as large as possible. Unfortunately, since also the bias current flows through the feedback resistance, there is a limit at the maximum value of \(R_{R}\) that can be used. If \(I_{B}\) is DC bias current sourced by \(V_{B}\), the DC output voltage \(V_{ODC1}\) at the output \(V_{O1}\) in Figure 13 is:  
\[V_{ODC1}=-R_{R}I_{B} \tag{14}\]  
For the OA in Figure 13 to remain in linearity, \(V_{ODC1}\) must remain within the power supply voltages, typically a few V. Therefore, in performing current noise measurements we are in a situation where we could, in principle, obtain an equivalent input current noise as low as desired, by increasing the feedback resistance. However, increasing \(R_{R}\) reduces proportionally the maximum current that can flow through the DUT. The fact that the DC output voltage is proportional to the bias current flowing through the DUT has however one useful outcome: by measuring the DC voltage at the output \(V_{O1}\), we obtain the value of the DC current through the DUT for any given bias voltage \(V_{B}\). The AC filter \(C_{B}R_{B}\) in Figure 9 is used to remove the (large) DC component and to allow the amplification of the noise only up to a level compatible with the dynamic of the signal acquisition and elaboration system used for spectral estimation. In order to evaluate the overall noise contribution to the output and to analyze to what extent the output noise in the circuit in Figure 13 can be regarded as essentially due to the DUT we must also take into account the effect of the noise introduced by the bias system (\(e_{Bn}\)) and the noise introduced by the equivalent input voltage noise \(e_{Vn}\) when regarded as responsible for the equivalent input voltage noise \(e_{n}\). The voltage noise \(V_{On}\) at the output \(V_{O1}\) can be obtained as:  
\[V_{On}=-e_{Bn}\frac{R_{R}}{Z_{DUT}}+e_{Vn}\bigg{(}1+\frac{R_{R}}{Z_{DUT}} \bigg{)}+e_{Rn}-i_{DN}R_{R} \tag{15}\]  
The PSD of \(V_{On}\) can be then obtained as (the noise sources in Equation (15) are uncorrelated):  
\[S_{VO}=S_{VB}\bigg{|}\frac{R_{R}}{Z_{DUT}}\bigg{|}^{2}+S_{VV}\bigg{|}1+\frac{ R_{R}}{Z_{DUT}}\bigg{|}^{2}+4 KTR_{R}+S_{ID}R_{R}^{2} \tag{16}\]  
Figure 13: Reference circuit for current noise measurements. By dividing Equation (16) by the transresistance gain squared (\(R_{R}{}^{2}\)) we can interpret the noise at the output as due to the PSD \(S_{IN}\) of a current source at the input of a noiseless TIA with the same gain:  
\[S_{IN}=\frac{S_{VB}}{\left|Z_{DUT}\right|^{2}}+S_{VV}\bigg{|}\frac{1}{R_{R}}+ \frac{1}{Z_{DUT}}\bigg{|}^{2}+\frac{4 kT}{R_{R}}+S_{ID} \tag{17}\]  
Clearly we can interpret \(S_{IN}\) as due to the DUT (\(S_{ID}\)) as long as all other contributions can be neglected.