# Fluctuation-Enhanced Sensing (FES): A Promising Sensing Technique
## Fundamental FES Resolution Limits
### Information Channel Capacity in Fluctuation-Enhanced (Stochastic) Sensing  
Current understanding of related fluctuations implies that the spectrum \(S\)(\(f\)) is the superposition of elementary Lorentzian spectra \(s_{j}\)(\(f\)). Physically, each of such a Lorentzian fluctuators has an underlying exponential relaxation process with a single time constant \(\tau_{j}\) determined by microphysical parameters:  
\[s_{j}(f)=\frac{c_{j}}{1+2\pi f\tau_{j}}\text{, }S(f)=\sum_{j}s_{j}(f) \tag{5}\]  
where \(s_{j}\)(\(f\)) is the spectrum originating from the \(j\)-th fluctuator, \(C_{j}\) is its strength, and \(\tau_{j}\) is its time constant.  
Suppose the power density spectrum of the resistance fluctuations in a FES sensor has \(K\) different frequency ranges, in which the dependence of the response on the concentration of the chemical species is different from the response in the other ranges, one can write [43; 44; 45]:  
\[\begin{split}\Delta S(f_{1})&=A_{1,1}C_{1}+A_{1,2}C_{ 2}+\cdots+A_{1,N}C_{N}\\ &\vdots\\ \Delta S(f_{K})&=A_{K,1}C_{1}+A_{K,2}C_{2}+\cdots+A_{K,N}C_ {N}\end{split} \tag{6}\]  
where \(\Delta S(f_{i})\) is the change of the power density spectrum of resistance fluctuations at the \(i\)-th characteristic frequency band, and the \(A_{ij}\) quantities are calibration constants in the linear response limit. Thus, a single sensor is able to provide a set of independent equations to determine the gas composition around the sensor. The number \(K\) of different applicable frequency ranges has to be greater than or equal to the number \(N\) of chemical species, i.e., \(K\geq N\). To meet the ideally required requirements, the sensors should exhibit linear responses. Unfortunately, in [55] it has been shown that commercial Taguchi sensors do not satisfy such a condition (see Figure 4, which has been reported here for the sake of clarity). It is however reasonable to speculate that nanoscale devices may meet this requirement because of the linear superposition of elementary fluctuations.  
In [73], the information channel capacity of power density spectrum based FES was estimated by assuming equal frequency bands and supposing that the relative error of the spectrum is much less than unity, then the information channel capacity (roughly) scales as:  
\[C\propto\frac{t_{m}t_{w}f_{s}^{2}}{\Delta f} \tag{7}\]  
where \(t_{w}\) is the duration (time window) of a single data sequence (for a single Fourier transformation), \(t_{m}\) is the total measurement time (the elementary power spectra are averaged over that), and \(f_{s}\) is the sampling frequency. It is supposed that the FES measurement starts after the sensor reached a stationary state in the test gas and that \(t_{m}\) is much longer than the time needed to reach the stationary state. Moreover, the condition of thin sensor film must be verified, just like in the case of the considerations for a classical sensor above.  
In such a case, the most important conclusion of Equation (6) is that, in resistive FES applications, the sensor surface can be very small without limiting the performance (as large as enough Lorentzian  
Figure 4: Experimental proof that the response in Taguchi sensors is not additive (nonlinear) [55]. Both in the case of the sensor NAP-11 AS (**a**) and of the sensor TGS 2610 (**b**), the response to a mixture of two chemicals (ethanol and S\({}_{2}\)O) is not the sum of the responses obtained when each chemical is alone in the test chamber.  
fluctuators are present for each frequency band) because measurement time related statistical inaccuracies limit the information channel capacity, not the background noise.  
In [73], similar conclusions are obtained for bispectrum based FES sensor systems.