We study the design enhancement of the bistable stochastic resonance (SR) performance on sinusoidal signal and Gaussian white noise. The bistable system is known to show an SR property; however the performance improvement is limited. Our work presents two main contributions: first, we proposed a parallel array bistable system with independent components and averaged output; second, we give a deduction of the output signal-to-noise ratio (SNR) for this system to show the performance. Our examples show the enhancement of the system and how different parameters influence the performance of the proposed parallel array.

Stochastic resonance has attracted considerable attention over the past decades. SR is defined as a phenomenon that is manifest in nonlinear systems whereby generally feeble input information (such as a weak signal) can be amplified and optimized by the assistance of noise.

The physical mechanism of SR has been known since the initial work by Benzi et al. at the beginning of the 1980s [

SR can be envisioned as a particular problem of signal extraction from background noise. It is quite natural that a number of authors tried to characterize SR within the formalism of data analysis, most notably by introducing the notion of SNR [

The early study of SR system focused on nature nonlinear system to analyse its properties [

This paper is in fact inspired by traditional parallel system, proposes a new parallel array with independent sensors, and focuses on the output SNR performance. This is different from traditional parallel SR system since traditional system uses one receiving sensor and parallel array processing components so that input for each component is not independent in statistics. And it is also different from traditional array signal processing [

This paper is organized as follows. The framework of two-state model of bistable system is described in Section

We consider the overdamped motion of a Brownian particle in a bistable potential in the presence of noise and periodic forcing [

To simplify the problem in this paper we discuss two-state model [

Since the noise in the output of the system is no longer Gaussian, the definition

For the weak signal

In this section, we discuss the parallel array bistable system and its SR performance.

We consider the parallel array with

The parallel bistable array with

In the following of this section, we present the main results with respect to the parallel array bistable system. Two theorems form the SR performance analysis on output SNR. We utilize four lemmas for proving the theorems. The proofs of all the theorems and lemmas of this section are relegated to appendices.

For the parallel bistable system with

The theory is based on the following lemmas.

The pdf of

If

If

The power spectral density of the output of the parallel array bistable system with

For the weak signal (

We conclude this section with three remarks. Our first remark is about the simplified noise. The noise

Our second remark is to point out that the proposed array is different from the traditional SR array [

Our third remark is that Theorem

Output SNR as a function of noise variance with

We now provide examples to illustrate the properties of our proposed bistable parallel array system.

For illustration of the possibility of an SR in the output SNR, we consider two different systems based on the theory of (

SNR curve changes as noise power. (a)

At

The results of Figure

We also offered a validation by a Monte Carlo simulation of the proposed system in Figure

Output SNR with

We consider, in Figure

Output SNR as a function of noise variance with

In this example, we consider

Output SNR as a function of noise variance with

Figure

Output SNR as a function of noise variance. (a)

In Figure

We still adopt the definition for input and output SNR in (

Output SNR gain as a function of noise variance, with

In this work, we study the design of structure of bistable system aimed at enhancing the SR effect to improve the performance, driven by sinusoidal signal and Gaussian white noise. We first proposed a parallel array bistable system with

From the structure of the parallel array bistable system, the output of the system is

Based on Lemma

Since

It greatly simplifies in the stationary limit

If

It is obvious that the autocorrelation function depends on both times

Using Fourier transform of (

If

The Fourier transform of power spectrum density is

In conclusion, the power spectrum density of the output of the system is

In (

Then for the parallel bistable system with

The proof of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China, Grant no. 61102157, and the National Basic Research Program of China (973 Program), Grant no. 2013CB329003.