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Sep 27, 2011 - Medium-Low-Frequency Signal Detection and Simulation Based on the. Principle of Stochastic Resonance. Wen

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Applied Mechanics and Materials ISSN: 1662-7482, Vols. 105-107, pp 1991-1994 doi:10.4028/www.scientific.net/AMM.105-107.1991 © 2012 Trans Tech Publications, Switzerland

Online: 2011-09-27

Medium-Low-Frequency Signal Detection and Simulation Based on the Principle of Stochastic Resonance Wenli Zhao1, a, Yuanping Yin1, b, Jin Liu1, c 1

College of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou 310018 a

[email protected], [email protected], [email protected]

Keywords: Stochastic resonance, Medium-low-frequency weak signal, MATLAB simulation;

Abstract. The principle of stochastic resonance in bistable system is introduced firstly. The medium-low-frequency periodic signal and multi-frequency harmonic signal (the large parameter signal) are common in mechanical failure, but it is difficult to achieve stochastic resonance in these signals detection. The signal modulation characteristic is used in this paper to transform the various frequency components into small parameter signals which satisfy the adiabatic approximation theory. On the basis of that, weak signal detection based on stochastic resonance theory is realized. Then a mixing circuit system based on stochastic resonance is designed, the circuit first makes a frequency selection processing with a mixer on the mixed signal between the measurable signal and a scanning signal, and then it is input to the nonlinear bistable system to realize signal detection based on stochastic resonance. At last, the MATLAB simulation result shows that the circuit can realize the stochastic resonance and detection of weak periodic signal in medium-low frequency from noise background. Introduction Stochastic resonance is a cooperative phenomenon. When the nonlinear system, random input and signal reach cooperative condition, the signal-to-noise ratio (SNR) of the output will increase remarkably by increasing input noise, so transforming the noise energy to the useful signal energy. The phenomenon provides a useful method to use stochastic resonance theory to detect the weak signal in the background of the strong noise. But a lot of research represented by the adiabatic approximation theory has shown that the advantage of the weak signal detected is distinct in small parameters conditions only (the signal amplitude, frequency and noise intensity are much less than 1) [1-3]. However, in engineering practice such as mechanical failure, the medium-low-frequency periodic signals, even periodic impulse signal, high harmonics signal, etc., are much more common and make detection more difficult. So how to apply stochastic resonance theory to detect the medium-low-frequency signal becomes one of the hot topics [4-8]. With the principles of signal modulation, in this paper, random input signal and scanning signal are mixed firstly. Then, it is input to the nonlinear bistable system to achieve the periodic signal detection theoretically based on stochastic resonance theory. Last, the simulation of MATLAB verifies that the method for the detection of low-frequency weak signal is feasible. The theory of stochastic resonance in bistable system The bistable system model. The simplest bistable system can be modeled with the Langevin equation of motion in the form (1) x (t ) = µx(t ) − x3 (t ) + A sin(Ωt + φ) + n(t ) In Eq. 1, µ is parameter of the system, µ>0. A is signal amplitude, and Ω is signal frequency respectively. In addition, there is an additive noise n(t) with intensity D. Typically, this is white Gaussian noise with mean and autocorrelation given respectively by

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 130.203.136.75, Pennsylvania State University, University Park, USA-10/05/16,18:46:18)

1992

Vibration, Structural Engineering and Measurement I

< n(t ) >= 0  < n(t )n(t + τ ) >= 2 Dδ (τ ) In Eq. 2, τ is the parameter of time delay. The nonlinear bistable system of Eq. 1 is x (t ) = µ x(t ) − x3 (t ) A simple symmetric bistable potential has the form of a standard quartic µ 1 U ( x) = − x 2 + x 4 2 4

(2)

(3) (4)

Fig. 1 The quartic bistable potential From the Eq. 3 and Eq.4, we know that there is a unstable state at x=0 and two stable states at xs = ± µ , separated by a barrier of height ∆U = µ 2 / 4 under static conditions, shown in Fig. 1. But the medium-low-frequency periodic signal and multi-frequency harmonic signal are common in mechanical failure, it doesn’t satisfy the condition of small parameters to detect difficultly. So, the signal modulation characteristic is used to transform the various frequency components into small parameter signals which satisfy the adiabatic approximation theory and can transfer the noise energy to the useful signal energy successfully.

Modulated stochastic resonance and its implementation The theory of modulated stochastic resonance. The modulation principle of periodic signal is shown in Fig. 2.The signal and noise are input to one end of the mixer. The scanning signal is input to the other end of the mixer. Then the mixer’s output is input to the nonlinear bistable system. So the system can realize the signal detection under the condition of stochastic resonance.

Fig. 2 The mixer and stochastic resonance principle Here the measurable signal containing noise can be described as ∞

xn (t ) = a0 + ∑ ( an cos nω0t + bn sinnω0t ) + n(t ) n =1

(5)



= a0 + ∑ An cos( nω0t + ϕ n ) + n(t ) n =1

In Eq. 5, An = an2 + bn2 , ϕn = −arctg

bn , n(t ) is white Gaussian noise. The scanning signal an

is y(t ) = cos ωct . The output signal of the mixer has been described as ∞

xm (t ) = xn (t ) y (t ) = [a0 + ∑ An cos(nω0t + ϕn ) + n(t )]cos ωc t n =1



1 1 ∞ = a0 cos ωc t + ∑ An cos[(nω0 − ωc )t + ϕ n ] + ∑ An cos[(nω0 + ωc )t + ϕn ] + n(t ) cos ωc t 2 n =1 2 n =1

(6)

Applied Mechanics and Materials Vols. 105-107

1993

In Eq. 6, xm1 (t ) = a0 cos ωct +

1 ∞ 1 ∞ An cos[(nω0 − ωc )t + ϕn ] + ∑ An cos[(nω0 + ωc )t + ϕ n ] ∑ 2 n =1 2 n =1

(7)

xm 2 (t ) = n(t ) cos ωc t So the autocorrelation of the xm2(t) is 1 (8) < xm 2 (t ) xm 2 (t − τ ) >= < n(t )n(t − τ ) > cos ωcτ = Dδ (τ ) cos ωcτ 2 Then the power spectrum of xm 2 (t ) is Fourier transform of its autocorrelation. It can be described as

Sm 2 (ω ) = ∫

+∞

−∞

D cos ωcτδ (τ ) e − jωτ d τ = D

(9)

So, it is easy to know that the noise via modulating is still white Gaussian noise. In the Eq. 7, ∆ω = ωc − nω0 . When we adjust the scanning frequency ωc close to nω0 gradually, ∆ω = ωc − nω0  1 , the system will satisfy the condition of the adiabatic approximation theory. Then the system can realize stochastic resonance. The circuit design of modulated stochastic resonance. Based on the theory, we has designed the circuit of the bistable system and the parameter ‘µ’ can be adjusted. Shown in Fig. 3, the mixer can achieve multiplication of the measurable signal and the scanning signal. Then the output signal contains the sum of frequency and the difference of frequency between the measurable signal and the scanning signal.

Fig. 3 Modulating circuit of stochastic resonance The integrator, the adder, the inverter, the multiplier and the amplifier are composed of the nonlinear bistable system. In the bistable system, the parameter ‘µ’ can be adjusted. This system can be expressed by the mathematical model as (10) x = ∫ [ µ x − k1k 2 x 3 + xn (t)] ⋅ y (t )]dt By the Eq. 10, the system can realize the adjustment of the parameter by changing the value of ‘µ’. In the practical application, firstly we should adjust the frequency of the scanning signal. When the frequency is close to that of the measurable signal, the output signal frequency characteristics can be found by observing the output spectrum of the system. Then, we can get the power spectrum peak of the output signal by combining fine-tune operation. From the peak, we can judge that there is stochastic resonance in the system and discover the frequency of the measurable signal. The simulation of the circuit system. When input the multi-frequency signal ∞ A A is the measurable signal amplitude and s (t ) = ∑ sin 2πf nt , n = 1, 2, 3... , where 2n − 1 n =1 2 n − 1 fn=(2n-1)f is the measurable signal frequency. For the multi-frequency signal, we can’t achieve the detection at one time. We have to detect gradually from lower frequency signal to higher. With the method, the first three frequency component f1, f2 and f3 of the measurable signal has been detected as shown in Fig. 4- Fig. 7.

1994

Vibration, Structural Engineering and Measurement I

As shown in picture, when the scanning frequency closed to the measurable signal frequency, it produces the spectrum peak. This has shown that when the difference value between the canning frequency fc and the measurable signal frequency fn is large, the system did not produce stochastic resonance. While the difference is smaller, the system can produce the phenomenon of stochastic resonance.

Fig. 4 f=10Hz, D=2 spectrum of stochastic resonance input waveform

Fig. 6 µ=0.8, f2=30Hz, D=2, fc=29.99Hz spectrum of stochastic resonance output waveform

Fig. 5 µ=0.8, f1=10Hz, D=2, fc=9.99Hz spectrum of stochastic resonance output waveform

Fig. 7 µ=0.8, f3=50Hz, D=2, fc=49.99Hz spectrum of stochastic resonance output waveform

Summary The medium-low-frequency periodic signal and multi-frequency harmonic signal are common in mechanical failure, but it is difficult to achieve stochastic resonance. The modulation characteristic can be used to transform the common measurable signal into small parameter signal thereby the weak signal detection based on stochastic resonance theory is realized. From simulation result of the MATLAB, we have verified that the method was feasible. But using the hardware system to implement the adjustment of the adaptive parameter remains to be further studied. Acknowledgements This paper is supported by the national natural science fund project (50875070). References [1] G. Hu: Stochastic forces and nonlinear systems (ShangHai Scientific & Technological education Publishing House, China 1994), p.220. [2] Fauve S and Heslot F. Stochastic resonance in a bistable system (Phys Lett, 1983), Vol. 97(1983), No.1 and No.2, p. 5 [3] F.R. Palomo,J.M. Quero and L.G. Franquelo: IEEE International Symposium(May 26-29,2002), Vol. 4 (2002), No.1, p. 505. [4] L. Gammaitoni, P. Hanggi and P. Jung. Stochastic resonance (Rev Modern Phys. 1998), p.70(1):223. [5] W.L. Zhao, L.H. Guo and F. Tian: Proceeding of the 2nd IEEE/ASME international conference on Mechatronic and Embedded Systems and Applications (Beijing, China, September 13-16, 2006). Vol. 10 (2006) No.5, 2006, p.1. [6] W.L. Zhao,L.Z. Wang and J.E. Cai: Conference Proceedings of ICEMI (Beijing, China August 6, 2005) Vol. (2005). [7] W.L. Zhao,F. Tian and L.D. Shao: Chinese Journal of Scientific Instrument, Vol. 28 (2007) No.10, p. 1787. [8] M. Lin and Y.M. Huang: ACTA PHYSICA SINICA, Vol. 55 (2006) No.07, p. 3277.

Vibration, Structural Engineering and Measurement I 10.4028/www.scientific.net/AMM.105-107

Medium-Low-Frequency Signal Detection and Simulation Based on the Principle of Stochastic Resonance 10.4028/www.scientific.net/AMM.105-107.1991

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