Nonlinear sensors: An approach to the residence time detection strategy

The monitoring of the residence time difference in bistable sensors has been recently proposed as a valid scheme for improving the detection capabilities of sensors as diverse as fluxgate magnetometers, ferroelectric sensors and mechanical sensors. In this paper we propose an approach to the residence time based detection strategy based on the measurement of the slope $m$ of the sensor output integral. We demonstrate that such a method, far from degrading the detection performances can provide an easier way to realize fast and reliable sensors without the computationally demanding task related with the computation of the residence time difference. We introduce the receiver operating characteristic curve as a quantitative estimator for the comparison of the two methods and show that the detector performances increase with increasing the periodic bias amplitude $A$ up to a maximum value. This condition has potentially relevant consequences in the future detectors design.

Figure 1

Schmitt trigger model in the deterministic condition [$A\left(t\right)$ is a triangular wave function with the bias amplitude $A$ larger than the barrier] for the case where no dc signal is present (upper) and for the case where there is a signal with constant amplitude $\epsilon$ (lower). In the two figures the input and the output time series are represented. Apparently, in the upper panel the residence times are equal, while in the lower panel the two residence times are different.

Figure 2

$I\left(\tau \right)$ versus $\tau$, with $\tau =t/T_0$, for four different values of the target signal $ϵ$. With dots we have plotted the average value $⟨I\left(\tau \right)⟩$. These values are used to compute the linear regression and to extract $⟨m⟩$.

Figure 3

Slope $⟨m⟩$ versus the noise standard deviation ${\sigma }_{\zeta }$, for four different values of the bias signal amplitude $A$ with the same frequency ${\nu }_0=1/T_0$. $b=70$, $\epsilon =1$, $T_0=5$, and ${\tau }_0=0.5$. The horizontal lines represent the theoretical prediction for the zero noise limit in the suprathreshold case.

Figure 4

Slope $⟨m⟩$ versus the amplitude $A$ of the triangular forcing bias for three fixed values of the noise standard deviation ${\sigma }_{\zeta }$. The horizontal lines represent the theoretical prediction (mean first passage time) for the case where only noise is present ($A=0$, i.e., zero bias signal amplitude limit). The dashed line is the theoretical prediction for the deterministic limit. The relevant parameters value $b$, $\epsilon$, $T_0$, and ${\tau }_0$ are as in Fig. 3.

Figure 5

Probability density for $x\left[n\right]$ and $\epsilon =0.5$ under $H_0$ and $H_1$.

Figure 6

ROC curves for the two cases: $\delta$-correlated noise (continuous line) and exponentially correlated noise (dashed line) for three values of the SNR: 6, 10, and 20. Here $\lambda =1$ for all the curves.

Figure 7

ROC area versus SNR for the $\delta$-correlated case.

Figure 8

ROC of the optimal detector compared with the ROC for the Integral Detector (dotted lines). Here $\epsilon =1$, $\sigma =10$, $b=70$, $A=100$, and ${\omega }_0=0.2$. $⟨m⟩$ is computed from a time series of $T=60$ averaged ${10}^4$ times.

Figure 9

(Color online) ROC area percentage difference versus the amplitude $A$ of the periodic force for different values of the noise intensity $\sigma$. The other parameter values are $\epsilon =1$ and $b=70$.

Figure 10

ROC area for $⟨m⟩$ versus the amplitude $A$ of the periodic force for different values of the noise intensity $\sigma$. The other parameter values are $\epsilon =1$, $b=70$, and ${\omega }_0=0.2$. $⟨m⟩$ is computed from a time series of $T=60$ averaged ${10}^4$ times.

Figure 11

(Color online) ROC area for $⟨\Delta T⟩$ versus the amplitude $A$ and $\sigma$. The other parameter values are $\epsilon =1$, $b=70$, and ${\omega }_0=2.0$. $⟨\Delta T⟩$ is computed from a time series of $T=120$ averaged 5000 times.

Figure 12

(Color online) ROC area versus the amplitude $A$ and $\sigma$. The same parameter values as in Fig. 11.