Exploiting vibrational resonance in weak-signal detection

In this paper, we investigate the first exploitation of the vibrational resonance (VR) effect to detect weak signals in the presence of strong background noise. By injecting a series of sinusoidal interference signals of the same amplitude but with different frequencies into a generalized correlation detector, we show that the detection probability can be maximized at an appropriate interference amplitude. Based on a dual-Dirac probability density model, we compare the VR method with the stochastic resonance approach via adding dichotomous noise. The compared results indicate that the VR method can achieve a higher detection probability for a wider variety of noise distributions.

Figure 1

Generalized correlation detector with $M$ high-frequency sinusoidal interfering signals.

Figure 2

Theoretical detection probability $P_{\mathrm{D}}$ versus the interference amplitude $A_{\eta }$ for different numbers $M=1,2,5,10$, and $\infty$ of nonlinear elements. Here, the detection probability $P_{\mathrm{D}}$ in Eq. (11) can be theoretically computed by the output SNR $R_{\mathrm{out}}$ of Eq. (10) for finite numbers of elements $M=1,2,5$, and 10 (black lines from the bottom up). While, for the infinite number $M=\infty$ of elements, the output SNR $R_{\mathrm{out}}$ of Eq. (10) needs to be evaluated by Eqs. (13) and (14). Accordingly, the theoretical detection probabilities $P_{\mathrm{D}}$ are plotted versus the interference amplitude $A_{\eta }$ for $M=\infty$. The red solid line corresponds to $f_{\ell }=29f_s$ and $f_k=43f_s$, while the blue dashed line represents the theoretical detection probabilities $P_{\mathrm{D}}$ obtained as $f_{\ell }=21f_s$ and $f_k=77f_s$. These two lines almost coincide with each other. Here the decay exponent of noise $\alpha =8$, the threshold $\mathrm{\Theta }=1.6$, the observation length $N=1000$, the ratio $f_s/f_{sa}={10}^{-3}$, the false alarm probability $P_{\mathrm{FA}}=0.01$, and the input SNR $R_{\mathrm{in}}=-25$ dB.

Figure 3

Detection probability $P_{\mathrm{D}}$ versus interference amplitude $A_{\eta }$ and threshold $\mathrm{\Theta }$. Here the number $M=\infty$ of elements, and the interference frequencies $f_{\ell }=29f_s$ and $f_k=43f_s$. The other parameters are the same as in Fig. 2.

Figure 4

Maximum of detection probability $P_{\mathrm{D}}$ as a function of $R_{\mathrm{in}}$ for three threshold values of $\mathrm{\Theta }=1$ (black dotted line), 1.6 (red solid line), and 2.5 (blue dashed line) via optimizing the interference amplitude $A_{\eta }$. The other parameters are the same as in Fig. 3.

Figure 5

Theoretical maximum detection probability $P_{\mathrm{D}}$ versus the decay exponent $\alpha$ for a given input SNR $R_{\mathrm{in}}=-25$ dB and the threshold $\mathrm{\Theta }=1.6$. Here the blue asterisks represent the maximum $P_{\mathrm{D}}$ achieved by the VR method, while the black squares represent the maximum $P_{\mathrm{D}}$ via the stochastic resonance method. The detection probabilities of the LOD (red solid line) and the LCD (magenta dashed line) are also plotted as a function of the decay exponent $\alpha$, respectively. The other parameters are the same as in Fig. 3.

Figure 6

Detection probability $P_{\mathrm{D}}$ as a function of threshold $\mathrm{\Theta }$ and the dichotomous noise level ${\sigma }_{\eta }$. Here the dichotomous noise components, instead of high-frequency sinusoidal interfering signals, is added to the generalized correlation detector of Eq. (6). The other parameters are the same as in Fig. 3.

Figure 7

Numerical and theoretical detection probabilities $P_{\mathrm{D}}$ as a function of the number $M$ of nonlinear elements. For an infinite number $M=\infty$, the theoretical maximum detection probability $P_{\mathrm{D}}=0.65$ of Eq. (11) (magenta dashed line) is calculated by the output SNR $R_{\mathrm{out}}$ of Eq. (10) based on Eqs. (13) and (14). For a finite number $M\ge 2$ of elements, the theoretical detection probability $P_{\mathrm{D}}$ of Eq. (11) calculated by Eqs. (8) and (9) is plotted (blue solid line). In numerical experiments, the values of the detection probability (black upper triangles) are achieved by the Monte Carlo simulation method [27]. Based on Eq. (14) and Eq. (2.37) of Ref. [27], the decision threshold $\gamma =Q^{-1}\left(P_{\mathrm{FA}}\right)/\mathrm{var}\left({\mathrm{T}}_{\mathrm{HF}}\right|{\mathrm{H}}_0)$ can be obtained for a given false alarm probability $P_{\mathrm{FA}}=0.01$. Then, for the observation size $N=1000$ and generating $N$ independent random variables $Z_n$, we compare the statistic $T_{\mathrm{HF}}$ in Eq. (6) with the obtained threshold $\gamma$ for a number of realizations of ${10}^5$, and estimate the detection probability ${\widehat{P}}_{\mathrm{D}}$ as the ratio of the number of $T_{\mathrm{HF}}>\gamma$ over the ${10}^5$ realizations. From Eq. (2.38) of Ref. [27], the number of realizations needs to be larger than ${\left[Q^{-1}(\beta /2)\right]}^2(1-P_{\mathrm{D}})/{\epsilon }^2{P}_{\mathrm{D}}$. Here the number of realizations ${10}^5$ is large enough for evaluating the detection probability $P_{\mathrm{D}}$ with a relative absolute error $\epsilon =0.01$ and the confidence level $100(1-\beta )\%$ ($\beta =0.05$). Here, the threshold value of the nonlinearity is $\mathrm{\Theta }=1.6$, the input SNR is $R_{\mathrm{in}}=-25$ dB, the interference amplitude $A_{\eta }=0.2$, and the other parameters are the same as in Fig. 3.