The Application of Duffing Oscillator in Weak Signal Detection

Transcription

1 The Appliction of Duffing Oscilltor in Wek Signl Detection 1 The Appliction of Duffing Oscilltor in Wek Signl Detection Abolfzl Jlilvnd 1 nd Hdi Fotoohbdi 2, Non-members ABSTRACT In this pper, method for identifying the chotic stte of duffing oscilltor is proposed where the duffing oscilltor is used for wek signl detection. This method is bsed on frequency spectrum nlysis nd filtering. Some reltive spects of this method for prcticl using re studied in detils too. The proposed method hs three properties; resonble clcultion complexity, robustness to moderte noise mount, nd cpbility of detection with short signl sequence. The proposed pproch hs good robustness, which is successfully shown in this pper. Keywords: Duffing Oscilltor, Wek Signl, Chos, Noise 1. INTRODUCTION Wek signl detection is chllenging tsk in signl detection nd lso in fult detection cses. In the fult detection cses, there will be some bnorml signls before fult breks out but the signl to noise (SNR) is very low, due to reltively wek chrcteristic signls s well s effects of trnsmission pth, trnsmission medi, reflection, refrction, etc. It is of gret vlue to study how to detect hrmonic signl of given frequency in noise. Chos hs potentil ppliction outlook in wek signl detection, becuse of the properties which re sensitive to certin signl nd immune to noise t sme time. Becuse of tiny perturbtion of prmeter might cuse n essentil chnge of the stte in non-liner chotic system, lrge number of reserchers used chotic oscilltions to detect wek signl in noisy environment. The duffing oscilltor is frequently used to detect wek signls [1-3]. In this method the key is to identifying the stte of the oscilltor. There re severl methods for identifying the chotic chrcter. Common theoreticl indictions of the duffing stte re Lypunov coefficients. However, Lypunov coefficients re not prcticlly pplicble since their evlution requires very lrge signl sequence [4]. In ddition Lypunov coefficients re very Mnuscript received on August 1, 2009 ; revised on Jnury 31, The uthor is with The Deprtment of Electricl Engineering, Znjn University, Znjn, Irn, E-mil: jlilvnd@znu.c.ir 2 The uthor is with The Islmic Azd University, Driun Brnch, Shirz, Irn, E-mil: fotohbdy hdi@yhoo.com sensitive to noise influence [5-6]. These drwbcks force the recent dvnce in development of relible chos detection mesures some of them re observtion of time- history, phse plne, Fourier spectrum nd utocorreltion, Poincre mps, frctl dimensions nd so on [5]. However, with these methods, either the complex clcultion is necessry, or it is not convenient for utomtic identifiction by computer, or it is cn not be used when the system involves noise. In this pper, we use method bsed on the nlysis of the frequency spectrum of the duffing oscilltor. The min dvntges of this pproch re resonble clcultion complexity nd robustness to moderte noise mount. Also its ppliction for evlution of the mesure requires short signl sequence. 2. FUNDAMENTAL PRINCIPLES FOR US- ING DUFFING OSCILLATOR IN SIG- NAL DETECTION Generlly, nonliner dynmic system hs four sttes: The fixed point, the smll periodic motion, the chotic motion nd the qusi-periodic motion (lrge periodic motion). When the system is in the criticl stte, smll perturbtion of the system prmeters my led to the qulittive chnge of the system stte [1]. The bsic ide of the signl detection scheme bsed on chotic oscilltor is tht smll periodic signl in noise cn be detected by duffing oscilltor vi trnsition from chotic motion to periodic motion s clssic nonliner system [2]. Generlly, the chotic system is constructed by the duffing oscilltor. The norml form of the duffing eqution is shown s: d 2 x dt 2 + δ dx dt x + x3 = γ cos(t) (1) Where δ is the rtio of dmping, γ cos(t) is the periodic driving force nd x + x 3 is the nonliner restoring force. Assuming y = dx dt = ẋ, then we hve: ẋ = y ẏ = δy + x x 3 + γ cos(t) (2) If we keep δ fixed then s γ vries from smll to big, the system stte vries from smll periodic motion (Fig. 1), to chotic motion (Fig. 2), nd, t lst, to gret periodic motion (Fig. 3).

2 2 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.1 Februry 2011 Fig.1: () phse plne digrm of smll periodic motion (b) time series digrm of smll periodic motion Fig.3: () phse plne digrm of gret periodic motion (b) time series digrm of gret periodic motion If we fix γ = γ c (γ c refers to the criticl vlue), so the system is put into the criticl stte (chos, but on the verge of chnging to the periodic motion). The to-be-detected signl cn be viewed s perturbtion of the min sinusoidl deriving force γ cos(t) (the reference signl). Although noise my be intensive, it cn only ffect the locl trjectory on phse plne digrm, without cusing ny phse trnsition. To detect wek signls with different frequencies by pplying (2), we must do some frequency trnsformtion. Defining t = ωτ, we obtin: x(t) = x(ωt) = x (t) dx(t) = 1 dx(ωτ) = 1 dx (τ) dt ω dτ ω dτ d 2 x(t) dt 2 = 1 d 2 x(ωτ) ω 2 dτ 2 = 1 d 2 x (τ) ω 2 dτ 2 (3) Substituting (3) into (2), omitting the superscript of x, nd dding the input signl we obtin: ẋ = ωy ẏ = ω(δy + x x 3 + γ cos(ωt) + Input) (4) Fig.2: () phse plne digrm of chotic motion (b) time series digrm of chotic motion where Input = s(τ) + σ(τ) + cos((ω + ω)τ + ϕ) + σ(τ),σ(τ)is the Gussin noise, ω is the frequency difference nd ϕ is the primry phse difference. By chnging the vlue of ω in (4) the wek

3 The Appliction of Duffing Oscilltor in Wek Signl Detection 3 signl with different frequencies cn be detected using bove mentioned principles. In this pper, we used the fourth-order Runge- Kutt lgorithm to solve the duffing eqution. Therefore, the system is discrete dynmic system by nture, slightly different from the originl continues system bsed on the chosen step size. As we know, there is trunction error (lso known s discrimintion error) involved in Runge-Kutt lgorithm [1]. Trunction error depends on the step size used, nd the dependence is especilly distinct when the system is strongly non-liner. As fr s our system is concerned, if the step size used is different, the trunction error will bring bout distinct discrepncy of the criticl vlue γ c. Whtever the vlue of step size is, the phse trnsition itself is cler nd distinct; wht mkes the different is just the vlue of γ c. The trunction error dose not mens tht the step size is required to be very smll to detect chos onset ccurtely. In this pper, we choose δ = 0.5 nd h = (step size) fixed. The vlue of γ c is different, depends on the system conditions nd the reference signl frequency. criterion for identifying the stte of the duffing oscilltor. 3. QUANTITATIVE DESCRIPTION OF DUFF- ING OSCILLATOR STATE The duffing oscilltor stte hs been described by observing the trjectory (phse plne digrm) which is not suitble for utomtic recognition. Hence, we need to quntittively judge the duffing oscilltor stte using the method bsed on frequency spectrum nd filtering. Anlyzing the frequency spectrum of the output of the duffing system, we found tht in periodic motion, it only includes the fundmentl wve nd its odd hrmonics, while in chotic motion; vrious components re in the spectr of the output of the duffing system, especilly the components with the frequency lower thn fundmentl component. The frequency spectrum of the output of the duffing system cos(100πt) is shown in Fig.4. As Fig. 4 indictes, informtion on the system stte cn be extrcted from the frequency spectrum of the output of the duffing system. Therefore, we designed low pss filter (its cut-off frequency is lower thn the reference frequency) to filter the output of the duffing oscilltor, nd then with the use of the Root Men Squre (RMS) vlue of the remining components; we estimte the stte of the duffing oscilltor. The specifictions of this filter for bove oscilltor re F pss = 35Hz nd F stop = 45Hz. After using the desired filter nd clculting the RMS vlue of the output of the duffing oscilltor in specific rnge (RMS f<50 ), we cn see, in chotic motion, RMS f<50 = nd in periodic motion RMS f<50 = Therefore RMS f<50 in the output of the oscilltor cn be considered s distinct Fig.4: () frequency spectrum of chotic motion (b) frequency spectrum of periodic motion 4. FURTHER DISCUSSION In this section we discuss problems relted to solutions of the duffing eqution nd propose method for detecting the wek signl bsed on the chrcteristics of the solutions. Assume tht our purpose is the detection of wek signl with frequency by duffing oscilltor. Then we obtin: ẍ + 0.5ẋ x + x 3 = cos(ωτ) (5) At this moment the system is in criticl stte. Now if n externl wek signl (for exmple s(τ) = 0.001cos(ωt)), with the sme frequency s the reference signl, is merged into duffing oscilltor, we obtin: ẍ+0.5ẋ x+x 3 = cos(ωτ) cos(ωτ) (6) The ultimte stte of the oscilltor is the periodic (Fig. 5). Although the wek signl (such s cos(100πt)) cn be detected but in prctice we must first solve series of problems.

4 4 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.1 Februry The Influence of the Noise If the externl exciting signl is Gussin noise S(t) = ε(t), then the solutions of Eq. (6) will be s shown in Fig. 6. The orbits re chotic, mens the orbits cn keep the stte of the motion stedily under the influence of the noise. Fig.6: () phse plne digrm σ = 1 (b) phse plne digrm σ = 4 Fig.5: () phse plne digrm before merging the input signl (b) time series digrm fter merging the input signl 4.2 Influence of to-be-detected Signl Pdded Noise It is ssumed tht the exciting signl consists of the noise nd wek signl with the sme frequency s the reference signl (S(t) = 0.01cos(100πτ) + σε(t)), where σ is 0.03, 0.05, 0.08, 0.1nd 0.2 respectively. The results show tht only when σ 0.08 the wek signl is identified relibly (Fig.7). Therefore, we cn conclude tht the threshold of the signl to noise rtio of the wek signl should be greter thn: SNR = ) ) 10 log (0.5 2 σ 2 = 10 log ( (7) = db Otherwise, it is not detectble by the mentioned method. Fig.7: () phse plne digrm σ = 0.08 (b) phse plne digrm σ = 0.02

5 The Appliction of Duffing Oscilltor in Wek Signl Detection Influence of the Different Frequency Signl When the frequency of the externl signl is different from tht of the reference signl, for exmple S(t) = 0.8 cos((ω ± ω)τ) nd ω = [ ] we gin solve Eq (6). The result shows tht the orbits keep chotic stte nd the stte of motion isn t shifted (Fig.8). γ(τ) = γ 2 c cos(ϕ) (10) ( ) sin(ϕ) ϕ(τ) = rctn cos(ϕ) + γ c (11) It cn be seen tht the phse shift is relted to the difference of phse between the externl signl nd the reference signl. ( ) When π rccos 4 π + ( rccos ),γ c γ(τ) the orbit trnsition will not occur. (2): if then: A(t) = [ γ c + cos(ϕ)] cos(ωτ) sin(ϕ) sin(ωτ) = γ(τ) cos(ωτ + ϕ(τ)) (12) γ(τ) = γ 2 c + 2 cos(ϕ) (13) ( ) sin(ϕ) ϕ(τ) = rctn (14) cos(ϕ) γ c ( ) When π rccos 4 π + ( rccos ),γ c γ(τ) the orbit trnsition will occur. The results of theoreticl computtion show tht if S(τ) = 0.01 cos(ωτ + ϕ) nd the referenced signl is cos(ωτ), ω = 100π[rd/sec], then the phse trnsition from chos to orbit will occur when ϕɛ[1.5769, In prctice, becuse of the discretiztion error, rnge of the ϕ, which in it the phse trnsition will occur, is nrrowed (ϕɛ[1.67, 4.61] ). Fig.8: () phse plne digrm ω = (b) phse plne digrm ω = Influence of the Initil Phse To consider the difference of initil phse between the externl signl nd the reference signl, ssuming the reference signl be γ c cos(ωτ +θ)) nd the wek signl be cos(ωτ + ϕ)), then we cn write the totl periodic exciting force s: A(t) = γ c cos(ωτ + θ) + cos(ωτ + ϕ) (1): if θ = 0 then: = γ c [cos(ωτ) cos(θ) sin(ωτ) sin(θ)] (8) + [cos(ωτ) cos(ϕ) sin(ωτ) sin(ϕ)] A(t) = [γ c + cos(ϕ)] cos(ωτ) sin(ϕ) sin(ωτ) = γ(τ) cos(ωτ + ϕ(τ)) (9) 5. CONCLUSION This pper presented simple method for signl detection, bsed on identifying chos in duffing oscilltor. The dvntges of this method re: evlution of the mesure requires short intervl (window width); resonble clcultion complexity; robustness to moderte noise mount. Furthermore the relted problems tht my occur when we use the duffing oscilltor for wek signl detection hve been discussed. References [1] W.Gunyu, G.Djun, nd X.Chen, The ppliction of chotic oscilltor to wek signl detection, IEEE Trnsction on Industril Electronics, vo1. 46 No. 2, 1999, pp [2] W.Gunyu nd S.He, A Quntittive study on detection nd estimtion of wek signls by using chotic duffing oscilltor, IEEE Trns. Circuits nd systems-i: Fundmentl theory nd pplictions, vo1. 50, No. 7, 1999, pp [3] L.Yue, Y. Bojun nd S.Y.Wu, Chos- bsed wek signl sinusoidl signl detection pproch

6 6 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.1 Februry 2011 under colored noise bckground, ct physic sinic 52 (3), 2003, pp [4] V. Rubezic, I. Djurovic nd M. Dkovic, Time-frequency representtions-bsed detector of chos in oscilltory circuits, Elsevier Signl Processing, 86(9), 2006, pp [5] C. S. Poon, M. Brhon, Titrtion of chos with dded noise, Proc. Ntl. Act. Sci, Vol 98, No. 13, 2001, pp [6] M. I. Rosenstein, J. J. Collins nd C.J. Deluc, A prcticl method for clculting lrgest Lypunov exponents for smll dt sets, Physic D, 65, 1993, PP Abolfzl Jlilvnd received B. Sc. in Electricl nd Electronic Engineering from Electricl nd Computer Engineering Fculty of Shhid Beheshti University, Irn, in He received M. Sc. nd PhD degrees from Electricl nd Computer Engineering fculty of Tbriz University, Irn, in Power Engineering nd Control Engineering in 1998 nd 2005, respectively. In 2006 he joined the Electricl Engineering Deprtment of Znjn University, Irn, s n ssistnt professor where he is hed of deprtment now. His min reserch interests include the Hybrid Control Systems, Petri Nets, Intelligent Control, Modeling nd Control of Power Electronic Converters, Control nd Stbiliztion of Power Systems, Appliction of Intelligent methods in Power Systems nd so on. He hs over 50 ppers in journls nd conferences. Hdi Fotoohbdi ws born in 1982 in Shirz, Irn. He ws received B.Sc. nd M.Sc. Degree In 2006 And 2009 from Islmic Azd University (Kzeroun Brnch) nd Znjn University, Irn, respectively. In 2009 he hs joined to Islmic Azd University, Driun Brnch, Shirz, Irn. His reserch interests include Power System Protection, Appliction of Artificil Intelligent in Power Systems nd Power Qulity. He hs severl ppers in interntionl conferences.