Double-maximum enhancement of signal-to-noise ratio gain via stochastic resonance and vibrational resonance

This paper studies the signal-to-noise ratio (SNR) gain of a parallel array of nonlinear elements that transmits a common input composed of a periodic signal and external noise. Aiming to further enhance the SNR gain, each element is injected with internal noise components or high-frequency sinusoidal vibrations. We report that the SNR gain exhibits two maxima at different values of the internal noise level or of the sinusoidal vibration amplitude. For the addition of internal noise to an array of threshold-based elements, the condition for occurrence of stochastic resonance is analytically investigated in the limit of weak signals. Interestingly, when the internal noise components are replaced by high-frequency sinusoidal vibrations, the SNR gain displays the vibrational multiresonance phenomenon. In both considered cases, there are certain regions of the internal noise intensity or the sinusoidal vibration amplitude wherein the achieved maximal SNR gain can be considerably beyond unity for a weak signal buried in non-Gaussian external noise. Due to the easy implementation of sinusoidal vibration modulation, this approach is potentially useful for improving the output SNR in an array of nonlinear devices.

Figure 1

(a) Output-input SNR gain $G_N$ of Eq. (20) versus the internal uniform noise level ${\sigma }_{\eta }$ for different array sizes $N=1,2,5,10,20,100$, and $\infty$ (from the bottom up); (b) Roots of uniform noise RMS amplitudes ${\sigma }_{\eta }^{\left(k\right)}$ in Eq. (21) versus the array size $N$. The optimal noise levels ${\sigma }_{\eta }^{\left(1\right)}$ and ${\sigma }_{\eta }^{\left(3\right)}$ that locally maximize $G_N$ are marked by green asterisks ($*$) and red downward triangles ($▿$), respectively. While, the magenta squares ($\square$) represent ${\sigma }_{\eta }^{\left(2\right)}$ that correspond to the locally minimum of $G_N$. Here, the threshold $\lambda =1$ in Eq. (17), the external generalized Gaussian noise $\xi \left(t\right)$ is with RMS amplitude ${\sigma }_{\xi }=1/\sqrt{3}$ and exponent $\alpha =8$.

Figure 2

Variance terms ${\mathrm{E}}_z\left[g^2\left(z\right)\right]$ (solid line) and ${\mathrm{E}}_{\xi }\left\{{\text{E}}_{\eta }^2\left[g(\xi +\eta )\right]\right\}$ (dashed line) in the denominator of Eq. (14) versus the internal uniform noise RMS amplitude ${\sigma }_{\eta }$. Other parameters are the same as in Fig. 1.

Figure 3

Roots of uniform noise RMS amplitudes ${\sigma }_{\eta }^{\left(k\right)}$ of the denominator in Eq. (21) versus the decay exponent $\alpha$ of generalized Gaussian noise $\xi \left(t\right)$. The array size $N=100$ and the noise RMS amplitude ${\sigma }_{\xi }=1/\sqrt{3}$. Solid lines represent the optimal noise levels ${\sigma }_{\eta }^{(1,3)}$ that locally maximize the SNR gain $G_{100}$, while the dashed line indicates the corresponding noise level ${\sigma }_{\eta }^{\left(2\right)}$ that locally minimizes $G_{100}$.

Figure 4

The maximal SNR gains $G_{\infty }$ of Eq. (20) versus the generalized Gaussian noise decay exponent $\alpha$ at the corresponding optimal noise levels ${\sigma }_{\eta }^{\left(1\right)}$. The other parameters are the same as in Fig. 1. The solid line represents $G_{\infty }$, the black dashed line indicates the unity line and the magenta dashed line is the Fisher information of $I\left(f_{{\xi }_0}\right)$. At the exponent $\alpha =8$, the the corresponding Fisher information $I\left(f_{{\xi }_0}\right)=2.55$ marked by a square, and the asterisk mark indicates the maximum of SNR gain $G_{\infty }=2.36$ that obtained by the optimal noise level and the infinity of array size $N$ by the array of nonlinearities of Eq. (17).

Figure 5

Output-input SNR gain $G_N$ of Eq. (10) versus the sinusoidal vibration amplitude $A_{\eta }$ for different array sizes $N=1,2,5,10$, and $\infty$ (from the bottom up). For $N=\infty$, the stationary variance $\langle \mathrm{var}\left[y\right(t\left)\right]\rangle$ is calculated by Eq. (31). Here, the external generalized Gaussian noise $\xi \left(t\right)$ is with RMS amplitude ${\sigma }_{\xi }=1/\sqrt{3}$ and exponent $\alpha =8$. The sinusoidal vibration frequencies $f_n=(20+n)/T$ for $n=1,2,...,N$. The input signal $s\left(t\right)=0.01sin(2\pi t/T)$.