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In this paper, an approach is presented to detect faint signals with strong noises in sensors by stochastic resonance (SR). We adopt the power spectrum as the evaluation tool of SR, which can be obtained by the fast Fourier transform (FFT). Furthermore, we introduce the adaptive filtering scheme to realize signal processing automatically. The key of the scheme is how to adjust the barrier height to satisfy the optimal condition of SR in the presence of any input. For the given input signal, we present an operable procedure to execute the adjustment scheme. An example utilizing one audio sensor to detect the fault information from the power supply is given. Simulation results show that the modified stochastic resonance scheme can effectively detect fault signal with strong noise.

Noises usually play a negative role on the detection of useful signals, and faint signal detection is a challenge work, especially detecting faint signals submerged in strong noise. Traditional filter is used to separate the signals of interest and their noise by restraining noises rather than using of them. For example, Fourier transforms and Wavelet transforms are the widely used idea which performs the filtering operation by projecting the signal into another space, then transforms back into the original space. But the approaches from this idea are effective only when the signals and the noise are not strongly mixed with each other in frequency spectrum. The phase-locked loop (PLL) is a typical approach to detect the faint signals by coherency modulating and demodulating. However, there is the limitation to this approach that is required to provide the reference signal with the same frequency as the detected signal. Other approaches are the application of statistical ideas to filtering problems, such as Kalman filter can make use of imprecise data on a linear (or nearly linear) system with Gaussian errors to continuously update the best estimate of the system's current state[

However, noises are not always suppressed passively. They can be utilized to suppress themselves. In some systems increasing the amount of ambient noise actually enhances (up to a certain point) the SNR through a complicated nonlinear cooperation between the system and detector. This effect, known to operate in neurons [

For SR, most of the studies were carried out using a dynamical system with bistability, modelled by a double well potential. Here SR is realised due to the shuttling between the two stable states at the frequency of the subthreshold signal (i.e. faint signal) with the help of noise. When the signal (faint signal with noises) is applied into a SR system, the potential barrier height of bistable system is adjusted to meet the condition of SR phenomenon. If SR occurs, the output spectrum of SR system would produce a sharp peak at the faint signal frequency superimposed on a smooth background spectrum [

This paper discusses how to use SR to detect faint signals with strong noises. Based on the bistable system, we combine SR phenomenon and Fast Fourier Transform (FFT) spectrum analysis together to detect faint signal mixed with noises. Note that the approach presented here is different from the aforesaid adaptive filter in design idea. Though those adaptive filters are in reality a non-linear system and can solve the nonlinear filtering problem, the nonlinear system need be translated into the linear system by various skills in the actual application process. The approach in this paper uses directly the nonlinear mechanism to construct filter and avoids the linearization process. Finally, an adaptive filtering procedure is added to the detection scheme to provide the possibility of application. The procedure resembles an executable program and its performance only depends on the adjustment strategy and evaluation index for SR from the angle of real application. In this sense, our approach is not a well-defined numerical algorithm indeed and is only an application scheme.

Occurrence of SR phenomenon need three indispensable conditions: (1) nonlinear system, e.g., bistable (or multistable) nonlinear system; (2) signals to be detected; (3) noises. The simplest SR system is the bistable system which is defined by nonlinear Langenvin equation:
_{1}_{2}_{1}-_{2}_{1}_{2}

Suppose

^{2}/4_{c}_{c}

According to the above discussion, the system described by dynamics equation

The corresponding simulation example as follows is given to validate the effect of SR on detecting the electrical signal with the strong noise.

^{2}/4

As shown in

For measurement of real signals, the pre-knowledge about signals is often lack; hence the above-mentioned SR system can not be directly used. The practicable approach requires that the SR can adapt different input signals. Therefore, an adaptive stochastic resonance scheme based on bistable system is presented as follows.

We adopt the power spectrum as the evaluation tool of SR. Here the power spectrum can be obtained by the fast Fourier transform (FFT). Furthermore, we introduce the adaptive filtering scheme to realize signal processing automatically. The key of the scheme is how to adjust the barrier height to satisfy the optimal condition of SR in the presence of any input. A simple idea is to estimate the shift and rise of the power spectrum peak when the barrier height is modulated. If the spectrum peak at a certain frequency is significant relative to the rest frequency components, then we can conclude that SR occurs and the detected signal takes on this frequency feature.

The signal noise ratio (SNR) is defined as the ratio of the spectrum value at a frequency _{i}_{i}_{1}), …, _{i}

Our approach is not a well-defined numerical algorithm indeed and does not exist cost function or transfer function, although it resembles the control algorithm in the form of block diagram. Optimal condition is set up not according to mathematical deduction but according to phyical property of SR systmen. Therefore, we present only an application scheme as follow.

For the given input signal, we will present an operable procedure to execute the adjustment scheme. During this procedure, the optimal condition will be transformed into two in

The estimation value for a certain frequency is expressed by the parameters _{fi}_{mean}_{i}_{i}_{fi}_{fi}_{i}_{i}

The detail of the procedure is described as follows:

For conveniently regulating the SR system, we change one of parameters in the system and fix the rest parameters. It is assumed that the parameter

We take parameter _{1…}_{m}_{0} , _{1} ,…, _{n}

Choose the appropriate threshold

For any frequency _{i}_{1}_{2}_{m}_{fi}

Find out the parameter _{i}_{i}

If the threshold

For the parameter

In normal work status, a power supply emits the audio signal with range of frequency from 0 to 4000Hz. But the frequencies of fault signals, which directly relates with the degree of fault, are about tens of hertz. Obviously, “noise” is mixed with the fault signal to be detected.

According to the abovementioned procedure, the fault signal with noise is considered as the input of the system in

An application method of SR was presented by theoretical analysis and example simulation in this paper. We demonstrated the advantage of SR by comparing the signal processing results with and without SR. According to this idea, the adaptive filtering approach based on SR was given to improve signal processing conveniently. Correspondingly, the bistable equation was used as SR model, and power spectrum information was introduced as the evaluation index of the filter. An example utilizing audio sensor was used to detect the fault information from power supply. From frequency spectrum of the output, we can find the amplified spectrum feature of fault signal relative to the noise. This result indicates feasibility of the method. It is worth noting that if one SR system cannot realize the complex signal detection, i.e., optimal SR spectrum does not occur, second or more SR system may be used to obtain the optimal effect.

This work was partially supported by the National Natural Science Foundation of China (projects No. 30470470) and the Project of Zhejiang Province Education Department of China (20050907).

Input signal spectrum (upper figure) and output response spectrum (lower figure) of SR system when signal amplitude

Input signal spectrum (upper figure) and output response spectrum (lower figure) of SR system when signal amplitude A=0.1V, frequency

Scheme of signal detecting with SR system.

Audio signal of intermediate frequency power supply.

spectrum of input signal.

Frequency spectrum of output. The parameter is: