Stochastic Exceptional Points for Noise-Assisted Sensing

Noise is a fundamental challenge for sensors deployed in daily environments for ambient sensing, health monitoring, and wireless networking. Current strategies for noise mitigation rely primarily on reducing or removing noise. Here, we introduce stochastic exceptional points and show the utility to reverse the detrimental effect of noise. The stochastic process theory illustrates that the stochastic exceptional points manifest as fluctuating sensory thresholds that give rise to stochastic resonance, a counterintuitive phenomenon in which the added noise increases the system’s ability to detect weak signals. Demonstrations using a wearable wireless sensor show that the stochastic exceptional points lead to more accurate tracking of a person’s vital signs during exercise. Our results may lead to a distinct class of sensors that overcome and are enhanced by ambient noise for applications ranging from healthcare to the internet of things.

Figure 1

Noise-assisted sensing at stochastic exceptional points (EPs). (a) Illustration of the system comprising of a pair of coupled resonators, one with gain and the other with loss. The EP separates two phases of the system, one which is insensitive to the input and the other which is sensitive. $\kappa$, normalized coupling rate of two resonances; $\gamma$, loss rate; ${\omega }_{\pm }$, eigenfrequencies. (b) Illustration of noise-assisted sensing at the stochastic EPs. Considering a periodic input in the weak coupling (i.e., broken PT phase), stochastic EPs assist the input to randomly cross the threshold, resulting in a series of pulses of random PT-phase transitions. (c) Signature of stochastic resonance. The addition of a certain level of noise optimizes the signal-to-noise ratio.

Figure 2

Theory of stochastic EPs. (a) Surface plot of the eigenfrequencies ${\omega }_{\pm }$ as a function of $\kappa /\gamma$ and detuning ${\omega }_1-{\omega }_2$. The EP is located at $\kappa =\gamma$ and ${\omega }_1={\omega }_2$. $\kappa$, coupling rate; $\gamma$, loss rate; ${\omega }_n$, resonant frequencies. (b) Example of the system response. The input ${\kappa }_{\mathrm{s}}(t)$ is a sinusoidal signal. The stochastic EPs have white noise $\xi (t)$ with standard deviation $\sigma$. The EP is initially set to $\kappa =\gamma =0.2$ and the system is biased at ${\kappa }_0=0.175$. The response displays a series of pulses of noise-assisted PT-phase transitions, i.e., SR at stochastic EPs. (c) SNR as a function of $\sigma$. The system reaches a maximal SNR as indicated by the star. The dashed curve shows the theory fit by Eq. (3). (d),(e) SNR as a function of $\sigma$ and $\gamma$ when the resonant frequencies are matched ${\omega }_1={\omega }_2$ (d) and when they are slightly detuned ${\omega }_1-{\omega }_2=5\times {10}^{-4}$ (e). The dashed black line shows the evolution of SR.

Figure 3

Sensor characterization. (a) Sensor controlled by a linear stage. The system response has a sensory threshold at EP located at $d=13.74\mathrm{mm}$. (b) Response of a sinusoidal input under the stochastic EPs. The input signal has a sub-threshold amplitude of 1 mm while the noise has a standard deviation of $\sigma =0.5\mathrm{mm}$. (c) SNR as a function of $R_2$ and $\sigma$. (d) SNR as a function of $\sigma$ for the values of $R_2$ indicated by the dashed lines in (c) and the comparison of sensors with and without stochastic EPs. The reference sensor consists of directly measuring the resonance with a network analyzer. Solid lines show theory fit by Eq. (3). Error bars show mean $\pm \mathrm{s}.\mathrm{d}.$ ($n=3$ technical trials).

Figure 4

Ambient-noise-enhanced wireless sensing at stochastic EPs. (a) Experimental results for an 18-min exercise protocol with treadmill speeds of 0, 1.5, 2.5, 3.5, 4.5, $5.5\mathrm{km}/\mathrm{h}$ for each 3 min, respectively. $f$, sensor resonant frequency; RR, respiratory rate. (b) SNR optimized by additional noise during physiological exercise. Data points correspond to 30 sec at different exercise speeds. (c) Representative sensor output waveforms for motion speeds of 1.5, 4.5, and $5.5\mathrm{km}/\mathrm{h}$ (upper row), and deviation of the RR measured by the sensor compared to the reference (Bland-Altman plot, lower row). Solid lines show mean and dashed lines show $\pm 2$ standard deviation (SD).