Research Article Research on Stochastic Resonance Signal s Recovery

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1 Mathematical Problem in Engineering Volume 23, Article ID 37269, 7 page Reearch Article Reearch on Stochatic Reonance Signal Recovery Hao Wang, Jie Cao, and Xiang Wei 2 School of Economic and Management, Nanjing Univerity of Information Science & Technology, Nanjing 244, China 2 College of Electrical and Information Engineering, Nanjing Univerity of Information Science & Technology, Nanjing 244, China Correpondence hould be addreed to Jie Cao; cj@nuit.edu.cn Received 2 January 23; Accepted 2 March 23 Academic Editor: Critian Toma Copyright 23 Hao Wang et al. Thi i an open acce article ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited. Uing tochatic reonance to detect weak periodic ignal ha been widely ued in variou field of cience, which attract much attention of reearcher due to it advantage of revealing receive periodic law. Thi paper utilized thi method to eek the underlying rule of etting weather index, o we can find that how to obtain the accurate expreion of original periodic law by further invetigation. Thi paper deal with the noie-contained ignal retoring on the bai of the etablihed ytem coupling the inverion ytem and bitable ytem. The imulation how that thi ignal recovery method inverion effect i better and the application range i wider.. Introduction A a new type of inurance project, weather index inurance i a hot iue in the current weather economy reearch. However, there till exit much difficulty in the weather index inurance reearch; particularly, the etting of weather index i cloely related to the hidden rule obtained by uing advanced cientific technology and method. The etting of weather index cannot do without plenty of meteorological data. Epecially mining the potential rule of meteorological data help to define the critical point of meteorological index, which contribute to find reaonable effective method for etting the meteorological index. The application of tochatic reonance method [] provide a new idea for exploring the underlying rule of data. Since the concept of tochatic reonance (SR) ha been propoed, it ha attracted reearcher interet, and it ha been tudied in phyic, chemical, biological, economic, and other field [2 7]. The idea and method of SR have broken the old mode of thinking that noie wa harmful and profitle in the proceing of ignal. It identified the unknown weak periodic ignal by uing the ynergie between the noie, ignal, and nonlinear ytem [8]. Some periodic law are hidden from meteorological data, which are hard to be found due to variou complicated natural factor. Even if the acquiition equipment preciion i improved, only the diturbance of noie i reduced, and the reliability of data i enhanced. Via tochatic reonance method, it i poible to mine the potential rule of meteorological data. In the proce of meteorological data ignal acquiition, due to the complex and volatile natural factor and the inevitable noie diturbance, it i ure that there will appear ignal lo or ditortion. How to identify and recover the meteorological data ignal? On one hand, we can improve the accuracy of the collection equipment, reduce noie, and amplify data ignal. On the other hand, we can improve the ignal proceing method; for example, according to tochatic reonance, the diturbed data ignal recovery can be enhanced by multifactor ynergim. Thi contribute to the identification of the unknown weak periodic weather ignal. Initially,wearecommittedtoenhanceunknownweak periodic ignal by adding noie in the tudy of SR. Obviouly, thi method ha trong contingency. In 992, Anihchenko et al. [9]foundthatSRcanalobeachievedbychangingthe ytem parameter, and the range of ignal recognition wa expanded. Then, Bulara and Gammaitoni [] alo proved the importance of adjut ytem parameter on the application of SR. Profeor Leng et al. [] propoedatheoryof SR that baed on parameter adjutment, analyzed the law of adjuting the parameter of SR, and believed that the two method of SR are the ame eentially. In the proce of the realization of the SR, affected by the nonlinearity of the ytem, ignal waveform i ditorted at the inflection point in the ytem potential function, and becaue

2 2 Mathematical Problem in Engineering SR will amplify the ignal, we cannot get the accurate amplitude of the ignal, and the reearch of output waveform ha been limited. Therefore, it uually identified the frequency characteritic of periodic input ignal in the frequency domain by uing the tochatic reonance technology. However, uch identification method cannot meet the requirement in dealing with nonperiodic ignal or when you need to obtain more valuable information, a recovery proceing of the ignal in the time domain i needed. In [2], an inverion formula which i baed on the bitable nonlinear ytem wa propoed to recover output waveform in the time domain. In [3], a detailed and reaonable explanation of the periodic ignal inverion principle wa given by the particle Kinematic. In [4 6], Profeor Leng et al. tudied the reaon of the ignal ditortion in inverion proce, repectively, the law of reonance and inverion of ignal with different ignal amplitude and ytem parameter, and then propoed two inverion method for different ignal. With the deepening of the tudy, the combination of the inverion ytem and bitable ytem will become a new ignal detection method. In [7 9], it analyzed the influence of ytem parameter, the intenity of noie, the ignal amplitude, and frequency to the SR phenomenon in the nonlinear bitable ytem SR model. Wu et al. propoed an adaptive trategy to adjut the parameter of the ytem [2]. Whether the elected ytem parameter can reach the optimal tate of SR, which will affect the effect of ignal recovery. Therefore, how to chooe the ytem parameter become one of the important topic of tudy SR. Signal waveform will generate pule ditortion at the inflection point of the ytem in the inverion ytem [2 6].Toretoretheignal,firtlyweneedtofindtheinflection point, and to proce the waveform of the inflection point, then to take the nonlinear interpolated in eriou ditortion part; finally, polynomial fitting hould be done for the ignal. Thi method i very complex, and too many parameter need to be adjuted. Therefore, it i difficult to carry out, and the etimate of the ignal amplitude i not accurate. In thi paper, according to the reaon of the waveform ditortion in the proceing of bitable SR and ignal inverion, a bitable recovery ytem combine the inverion ytem, and bitable ytem i etablihed directly to recovery denoie ignal. At the ame time, it i utilized the relationhip between the bitable ytem parameter b and noie variance, and ued an adaptive optimization method. Firtly, to ample noiy periodic ignal and etimate the noie variance and the ignal frequency and then regard maximum SNR a the optimization target to determine the optimal ytem parameter, at lat, a cacade recovery ytem wa ued to further filtering ignal; the imulation reult how that the effect i good. 2. The Bitable Stochatic Reonance Sytem and the Inverion Sytem SR can be conidered a a proce that amplified excitation ignal containing noie H(t) by the bitable ytem with potential function U(x), the ytem equation i a follow: du (x) x+δ x= +H(t), () dx where x i an infiniteimal and wa ignored, and et δ=,o thelangevinequationiafollow: du (x) x= +H(t). (2) dx In the claical model of the nonlinear bitable tochatic reonance, an excitation ignal containing noie H(t) conit of a random noie n(t) and external periodic ignal (t); n(t) i a Gauian white noie with zero mean and autocorrelation E n(t) =, n(t) = σε(t); ε(t) i a white noie with zero mean and it variance equal to. The external periodic ignal i a ine ignal that (t) = A in(2πft). Auming that the nonlinear bitable ytem U(x) would have the following bitable potential function: U (x) = 2 ax2 + 4 bx4, (3) where a, b are conidered a the ytem parameter, and the Langevin equation can be implified a follow: x=ax bx 3 +(t) +n(t). (4) SR i mainly ued for the mall parameter ignal, o that the amplitude of the periodic input ignal and noie intenity arerequiredtobemuchlethan.whentheignaldoenot meet the parameter required, we can proce after that the ignal i converted to a mall parameter by the ubampling method [2]. So it i aumed that the noie i far le than. Becaue the frequency of the ignal i very mall, x(t) can be approximated to zero, at thi time, (4) i atified: ax (t) bx 3 (t) +(t) +n(t). (5) The above equation can be implified to (t) +n(t) ax(t) +bx 3 (t). (6) It i averaged on both ide of (6), becaue the mean of the noie E[n(t)] =, owecanobtainthefollowingequation: E [ (t)] a(e [x (t)]) +b(e [x (t)]) 3. (7) Signal inverion equation obtained from the tatitical average ene [2]i (t) = ax(t) +bx 3 (t). (8) In [2], it propoed an inverion ytem in accordance with the above inverion equation: h (x) = ax+bx 3. (9) Figure (a) i a weak ine ignal (t) =.3 in(2π.t) containing a white noie with zero mean, and it variance i equal to.6, which i unable to ditinguih a ine ignal by thenoie.srofthemeaurednoiyignaligeneratedinthe bitable nonlinear ytem, which ytem parameter a=, b=, and the output ignal i hown in Figure (b) and (c);

3 Mathematical Problem in Engineering 3 (db) (a) t () t () 5 5 (b) (c) f (Hz) Figure : SR ignal. it i hown that SR enhance the ignal amplitude, o that the weak ignal i prominent, and the frequency of the ine ignal ha been detected, but the time domain waveform of the ina ignal ha become a trapezoid wave, and the waveform i ditorted the waveform cannot accurately how the input ignal. In order to obtain a more realitic ignal, Figure 2 how that an inverion ignal obtained the output ignal from the bitable tochatic reonance by the inverion ytem (9); the parameter of inverion ytem are a=, b=. Figure 2 howthattheamplitudeofinverionignali reduced to ome extent, and the waveform i much cloer to the input ignal, but the ditortion i large, o that the inverion i ineffective. Auming that the noie intenity i zero, (6) canbe approximated to (t) = ax+bx 3. () t () Figure 2: The inverion ignal in the time domain. Derivative with repect to t on both ide of the equation: (t) = a x (t) +3bx 2 x (t). () The above equation can be implified to (t) x (t) = a + 3bx 2. (2) Obviouly, the ytem outputignalx(t) no longer trictly follow input ignal (t), and when the ignal i in the ytem inflection x=± (a/3b), x, the output ignal waveform ha a pule ditortion. To make the output ignal approximated to the periodic input ignal (t), both ide of the formula (2) multiply a + 3bx 2 : x ( a + 3bx 2 )= (t). (3) Therefore, we find a recovery ytem: h (x) = a +3b x 2. (4) Then, the bitable recovery ytem combine the inverion ytem, and bitable ytem i etablihed; the ytem equation i a follow: x=(ax bx 3 +(t) +n(t)) ( a +3b x 2 ). (5) Baedontheforegoingaumptionthatnoieifarle than, o we adjut the parameter of the inverion ytem being the ame a the parameter of bitable ytem, that i, a =a, b =b. If the noie i too large, we hould reduce the noie before recovering the ignal. Figure 3 i the block diagram of bitable recovery ytem. Set the parameter of the bitable recovery ytem a=, b=, the inverion ignal i obtained from noiy periodic ignal in Figure by bitable recovery ytem, which i hown in Figure 4. The graph of comparing the recovery ignal with a inuoidal ignal to be meaured i hown in Figure 4(a); wecan eethattheretoredignalhadaphaelagabout8. In order to oberve whether the recovery waveform of the ignal i cloer to a inuoidal ignal to be meaured, we delay the phae of the ine ignal 8 before comparing with recovery ignal in Figure 4(b).

4 4 Mathematical Problem in Engineering (t) + n(t) Sum Multiply / x.4.2 a + 3bx 2 ax + bx 3 Figure 3: The bitable ytem recovery ytem. SNR t () t ().5.5 (a) (b) t () Figure 4: The recovery ignal in the time domain. Comparing Figure 4(b) with Figure 2, itiobviouthat the reult of bitable recovery ytem i much better; the amplitude and waveform of the recovery ignal are more cloed to the input ignal. 3. Adaptive Parameter Adjutment 3.. The Parameter of SR Sytem. The amplitude and frequency of the ignal, noie intenity, and the ytem parameter may contrain and influence SR phenomenon in Bitable tochatic reonance ytem. In order to reach the optimum a =.2, b =.2 a =.5, b =.5 a =.5, b =.5 σ 2 a =, b = a = 2, b = 2 Figure 5: The output curve of the SNR with the noie variance with different a, b. tate and take the maximum value of the output SNR, how to adjut the variou parameter are eential. OnaccountofthecharacteriticofGauianwhitenoie E n(t) =, n(t) = σε(t),thelangevinequation(4)iatranformed ubtitution, that y=x/σ, b =bσ 2, = /σ, and i related with SNR. In thi cae, the Langevin equation become y=ay b y 3 + (t) +ε(t). (6) ε(t) i a white noie, it mean i, variance i, and intenity i invariant, o the main factor i the ytem parameter a and b. When b =bσ 2, the bitable ytem parameter b and the noie variance σ 2 playtheameroletob,whichmeanthat adding noie or change the ytem parameter b caue the ame effect The Method of Adaptive Adjuting the Sytem Parameter. According to the adiabatic approximation theory [7], we can get the output SNR of formula (4) a follow: SNR = 2a 2 A 2 bσ 4 e a 2 /2bσ 2. (7) Figure 5 howthecurvegraphoftheoutputsnrchanging with the noie variance σ 2,whentakinga and b with different value. Obviouly, increaing noie variance, SNR firtly increae and then decreae, which generate peak, and the poition of peak (ie, the optimal noie variance) i different when the SR ytem parameter i changed. From the energy converion, the way to increae SNR by adding noie or input periodic ignal will increae the energy

5 Mathematical Problem in Engineering 5 (t) + n(t) (t) + n(t) Sampling f =5 2 Sampling f = 5f Add σ 2 SR (a =, b = ) Etimate noie variance σ 2 N Maximum SNR? Y Detect the frequency of the ignal f Etimate noie variance σ 2 a=; b=σ 2 /σ2 Bitable recovery ytem Output ignal 2 End Figure 6: The adaptive parameter adjutment recovery ytem. of the output ignal, which mean that the amplitude of the output ignal will be greatly increaed, o it i unfavourable to retored ignal. So that we utilize the adaptive optimization method to change the parameter by the relationhip between the bitable ytem parameter b and noie variance. The flow diagram i hown in Figure 6, it i divided into the following three tep: () Set the bitable ytem parameter a=and b=, by adding noie to the ytem to achieve the bet tate of tochatic reonance (SNR i maximum), then to etimate the total noie variance σ 2. (2) Detect the frequency of the ignal f at the tate of the optimum SR, and ampling the received ignal with the ampling frequency f = Mf (generally, M = 5). (3) Etimated the noie variance σ 2 in the input ignal, calculate optimal ytem parameter b=σ 2 /σ2,and configured the bitable recovery ytem parameter a=and b=, then recover to input ignal The Simulation of Adaptive Regulation Sytem. Auming noiy periodic input ignal H(t) = A in(2πft) + n(t),where the amplitude of the ine ignal A =.3, thefrequencyf=. Hz, noie n(t) i a Gauian white noie with the mean of andvarianceof.6. The firt tep i etting the bitable ytem parameter a=, b=,whensnrithelarget,andtoetimatenoie variance σ 2, the econd tep i detecting the frequency of the ignal f at the tate of the optimum SR, and determining the ampling frequency f = 5f. The imulation reult are hown in Figure 7, the bet noie variance σ 2 =.9, SNR max = , the frequency of the ignal f =.,and the ampling frequency f =5. The third tep i calculating optimal ytem parameter b = in accordance with b=σ 2 /σ2,configuringthe bitable recovery ytem parameter a =, b = Figure 8 how the recovery ignal by the bitable recovery ytem. A hown in Figure 8, although the recovery waveform of ignal had a certain degree of ditortion, it i not trictly agreed to the variation law of the ine ignal. But the amplitude of the recovery ignal i very cloe to the amplitude of the input ignal, o to ignore thi light error, we can extract the amplitude of the ignal. Meanwhile, the frequency

6 6 Mathematical Problem in Engineering SNR SNR σ σ-snr f (Hz) Figure 7: The curve of the SNR with the noie variance and the output ignal in the frequency domain Figure 8: The comparion of adaptive recovery ignal with input ignal. of the ignal had been meaured on the firt tep, we find the frequency f and amplitude A to the input ignal A in(2πft), then we can determine thi ine ignal. 4. Cacade Recovery It known that cacade SR ha a good denoie filtering effect [22]. Simultaneouly, cacade recovery ytem can alo achieve the effect of denoie filtering. Figure 9 hown the output waveform of three cacaded recovery ytem. We can findthatthepreviououtputihapedbytherecoveryytem, o that the final output waveform contour of the ytem become moother. 5. Concluion By etablihing the coupling relationhip of the inverion ytem and the bitable ytem, we can proce noiy periodic t Figure 9: Time-domain waveform diagram of three cacaded recovery ytem. input ignal directly and ue the auxiliary adaptive optimization method and the cacade ignal recovery method, the recovery ignal i much cloer to the actual weak periodic ignal;theimulationreulthowthatthimethodhabetter peed and accuracy. In order to improve the preciion of etting weather index, we will convert the meteorological data to the information flow and thereby ue thi technique to identify and recover the periodic ignal hidden from the meteorological data. Acknowledgment ThiworkipartiallyupportedbyNationalNaturalScience Foundation of China (no , 6462, and ), Jiangu Qing Lan Project and PAPD, Chinee Soft Science (2GXS5B47), and the National Public Sector (weather) Special Fund (GYHY269). Reference [] R.Beniz,A.Sutera,andA.Vulplani, Themechanimoftochatic reonance, Phyic A,vol.4,no.,pp , 98. [2] H. Deng and L. Zhang, Study on the application of weak ignal recovery ytem baed on the tochatic reonance, Key Engineering Material, vol , pp , 2. [3] X. Godivier and F. Chapeau-Blondeau, Stochatic reonance in the information capacity of a nonlinear dynamic ytem, International Bifurcation and Chao in Applied Science and Engineering,vol.8,no.3,pp ,998. [4] A. Krawiecki, Stochatic reonance in a ytem of coupled chaotic ocillator., Acta Phyica Polonica B, vol.3,no.8,pp , 999. [5] P. S. Burada, G. Schmid, D. Reguera, M. H. Vaintein, J. M. Rubi, and P. Hänggi, Entropic tochatic reonance, Phyical Review Letter, vol., no. 3, Article ID 362, 4 page, 28. [6] H. Deng, E. Shang, B. Xiang et al., Application of the tochatic reonance algorithm to the imultaneou quantitative determination of multiple weak peak of ultra-performance liquid chromatography coupled to time-of-flight ma pectrometry, Rapid Communication in Ma Spectrometry,vol.25,no.5,pp , 2. t

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