Research Article. Noise Separation from the Weak Signal Chaotic Detection System. Hanjie Gu * and Fan Wu

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1 Jesr Journal o Engineering Science and Technology Review 8 () (015) Special Issue on Synchronizaion and Conrol o Chaos: Theory, ehods and Applicaions Research Aricle JOURNAL OF Engineering Science and Technology Review Noise Separaion rom he Weak Signal Chaoic Deecion Sysem Hanjie Gu * and Fan Wu Insiue o Inormaion Technology, Zhejiang Shuren Universiy, Hangzhou, , Zhejiang, China. Received 7 Sepember 014; Revised 14 Ocober 014; Acceped 18 November 014 Absrac The radiional weak signal chaoic deecion sysem sill resrics some echnical issues in he siuaion o he signal wih noise, such as poor denoising abiliy and low deecion precision. In his paper, we propose a novel weak signal chaoic deecion sysem based on an improved wavele ransorm algorihm. Firs, he radiional wavele ransorm algorihm domain variables have been ransormed and discreized o eliminae he redundan ransorm. Then, based on he discree opimizaion, he wavele coeiciens have been opimized by hreshold compromise sraegy. The improved wavele ransorm algorihm is applied in he weak signal chaoic deecion sysem. The noise signal aer inie discree processing is reaed as a perurbaion o cycle power and pu ino a chaoic sysem or deecing weak signal under he noise condiions. The simulaion experimens show ha he proposed improved wavele ransorm algorihm has a beer denoising eec han he radiional wavele ransorm algorihm. oreover, he improved algorihm shows beer accuracy and higher robusness in he weak signal chaoic deecion sysem. Keywords: Weak Signal Deecion, Chaoic Sysem, Noisy Signals, Improved Wavele Transorm Algorihm, Discree Opimizaion, Threshold Compromise Sraegy. 1. Inroducion In he long-erm producion pracice and scieniic researches, people would like o know how o exrac ineresed signals rom noise background [1]. The problem is summed up as weak signal deecion and esimaion. The echnology and mehods o exrac useul signal rom noise is he main conen o he sudy o weak signal deecion []. I uses he inormaion heory, elecronics, physics and mahemaical saisics mehod, ec. o analyze he characerisics and regular paerns o noise and valuable inormaion. Then, i uses appropriae signal processing mehods o exrac he useul inormaion rom srong noise. I can also be used or deecing he he indicaors o signal and he noise [3]. Thereore, weak signal deecion echnology plays an imporan role in he developmen o high and new echnology, exploraion and discovery o new naural law, promoing he developmen o science and echnology and he producion [4]. The commonly used convenional signal deecion mehods include power specral analysis, wavele ransorm, adapive iler, he relevan es, synchronous accumulaive mehod and signal o noise space decomposiion. However, hese mehods are mainly based on linear sysem. The arge signal exracion is oen accomplished by ampliicaion o useul signal hrough suppressing noise condiions [5]. Klein elecronic echnology mehod accomplished digial * guhanjie8@yeah.ne ISSN: Kavala Insiue o Technology. All righs reserved. average and inegral sampling or he irs ime. However, he mehod has o know he signal requency beore he algorihm implemenaion. Thus, i will no be able o realize he signal cycle division i he signal requency is unknown. I also will no be able o ake average or digial sample o signals under he same emporal inerval wih dieren periods [6]. Weinred repored a deecion mehod which implemens he process o periodic signal exraced rom random noise background. This mehod only requires a small amoun o calculaion wih simple principles. I has been widely used in he engineering pracice. The drawback o his mehod is ha i loses signal phase inormaion aer signal processing [7]. Fourier proved ha any periodic signal is a developable orm ino a series o sine uncions, which makes Fourier ransorm mehod is widely used in he ield o weak signal deecion and esimaion. However, his mehod has poor perormance under he low signal noise raio, which prone o make requency specrum leakage and ence eec, leading o he non-accurae deecion [8]. Because chaoic sysem has a high sensiiviy o he ininiesimal changes in he iniial value and perurbaion, so i is widely used in he weak signal deecion [9]. Li [10] improved he Duing oscillaor and used he elnikov uncion o analyze he principle and mehod o he phase ransiion o he chaoic sysem, which reaches o he deecion o limi o dB under colored noise background. Lu e al. [11] repored a weak signal ampliude esimaion mehod regards o he sensiive dependence or parameers o chaoic sysem. This mehod uses he leas square mehod o ind he relaionship

2 beween he weak signal ampliude and Duing under he condiion o known signal requency, and gives he mahemaical expression o he ampliude calculaion. Chen e al. [1] combined he cross-correlaion algorihm wih chaoic deecion mehod, proposing a weak signal deecion mehod based on he cross-correlaion mehod and chaos heory. The experimenal resuls show ha wih he combinaion o cross-correlaion algorihm makes chaos deecion obain a beer perormance. Li e al. [13] proposed a weak signal deecion mehod based on he wavele ransormaion and chaos heory. This mehod pus he prereaed signals by wavele ransorm ino chaos deecor and esimaes he ampliude o he sine signal under he signal-o-noise raio o -40 db. This mehod has been used or collecing he elecric shock signal deecion. Wang e al. [14] analyzed he basic principle and easibiliy o chaoic oscillaor deecion o a weak signal under srong noise background and denally sudied he eaures o noise. The resuls indicae he advanages o using chaos algorihm o deermine he signal. Chance and Sco [15] used he perurbaion o chaoic rajecory conrol he weak signal deecion o he microwave sysem. Alhough, he above mehods can deec he weak signal, bu under he srong noise or he condiion o desroyed chaoic phase space srucure, hese mehods remain he deecs o accuracy. This aricle in view o he exising problems o weak signal chaoic deecion sysem aer adding noise, proposes a chaoic deecion sysem based on improved wavele ransorms. This mehod uses discree opimizaion and compromise sraegies or opimizing hreshold wavele ransorm algorihm and improves he denoising abiliy o weak signal deecion chaoic sysem.. Weak Signal Chaoic Deecion Sysem under Noise Condiion Proposed Circui The Duing oscillaor chaoic sysem has been widely used in weak signal deecion due o is simple equaion srucure and low order. Duing equaion can be shown as: x()+ kx() x()+ x 3 ()= acosω (1) where k is damping raio, α is cycle power ampliude, x + x 3 is nonlinear elasic orce, ω is cycle policy dynamics. The ormula (1) can be convered o he irsorder dierenial equaions. x()= dx() d = dx(ωλ) d(ωλ) or = 1 ω x(ωλ) x()= d 1 ω x(ωλ) /d(ωλ) (6) = 1 ω x(ωλ) Pu Εqs. (4) and (5) ino Duing equaion, hen: 1 ω x(ωλ)+ k ω x(ωλ) x(ωλ)+ x 3 (ωλ)= γ cos(ωλ) (7) Equaion (7) is a sysem equaion based on he independen variable λ. The equaion o sae can be changed as ollows: x = ωkx +ω (x x 3 +γ cosωλ) (8) Resuls derived rom he equaion show ha he hreshold o Duing chaoic sysem only relaes wih ω. When he signal requency changes, he only value should be changed is ω. The border cycle rack o weak signal chaos deecion sysem under noise condiion is rough. Use Δx() as he microvariaions o x(), hus i is concluded ha in he dierenial equaion o he sysem under he noise condiion. Eq. (3) can be expressed as: (x + Δx)+ k( x + Δx) (x + Δx) 3 +(x + Δx) 5 = = γ cos(ω)+ n() where n() is noise, E{n()} = 0. Equaion (9) minus Equaion (3), because he value o Δx is iny, i can omi is high-order: Δx + kδx +3x Δx 5x 4 Δx = n() (10) (9) (5) x = y y = x x 3 + acosω ky () Le c()= 5x 4 3x hen, Δx + kδx c()δx = n() (11) In order o measure any requency periodic signal, he Duing equaion has o be improved because he requency o he periodic signal under weak signal deecion is changing. For equaion o Duing chaoic sysems: x + kx x + x 3 = γ cosω (3) Le = ωλ, hen x()= x(ωλ). Then: x()= x(ωλ) (4) and Express Eq. (11) o vecor dierenial equaion: X()= A()X()+ N() (1) and X()= A()= N()= x 1 x = Δx() Δx() 0 1 c() k 0 n() (13) 38

3 Is soluion is X()= Φ(, 0 )X 0 + Φ(,u)N(u)du, where 0 Φ is he sysem sae ransiion marix. Due o he irs soluion is he ransien soluion, which will soon decay o zero, so only he second soluion needs o be considered. Thereore: X() = Φ(,u)N(u)du 0 E{X()} == Φ(,u)E{N(u)}du = 0 0 (14) In he sense o saisics, any zero-mean noise will no change he original rajecory o he sysem, hey will only make he sysem rajecory become rough and swing near he ideal rajecory. 3. Opimizaion o Weak Signal Chaoic Deecion Sysem 3.1. Discree Opimizaion o Wavele Transorm Algorihm Wavele analysis is he expansion o Fourier analysis. Le () L (R), he wavele ransorm o () can be deined as: WT (a,b)= a 1/ ()ψ b a d,a 0 (15) Or use he inner produc orm: WT (a,b)=,ψ a,b (16) In he equaion: ψ a,b ()= a 1/ ψ b a (17) To make he inverse ransormaion, saisied: ψ () has o be ψ ˆ (ω ) C ψ = dω < (18) ω In he equaion, ψ ˆ (ω ) is he Fourier ransorm o ψ(). The inverse ransormaion is: 1 ()= C ψ ψ a,b ()WT (a,b)db da (19) a Consan C ψ resrics he species o L (R) o ψ uncion (as moher wavele). I he ψ is a window uncion, hus he ψ should belong o L1 (R). So: ψ () d < (0) ψ ˆ (ω ) is a coninuous uncion o R. From equaion (18), he ψˆ value is 0 a he origin. Then: ψ ˆ (0)= ψ ()d = 0 (1) The equaion (0) indicaes he wavele uncion has ineviable oscillaion. For he coninuous wavele ransorm is redundan, ransorm domain variables α and b has o be discreized o remove he redundan in he ransormaion. In pracice, he value o variables are usually chosen as and j,k Z. Then: b = k, j a = 1 j ψ a,b ()=ψ 1 ()= j/ ψ ( j k) (), k j j Oen abbreviaed o: ψ (). The ransormaion can be expressed as: 1 WT, k j j =,ψ. In order o reconsruc he signal (), {ψ } has o z be a Riesz base o L (R). Funcion ψ L (R) is ar uncion, i {ψ } z is a Riesz base in he ollowing deiniion: i {ψ } z has a dense linear r. vishny in L (R) and here are normal numbers A and B (0 < A B < ). Then: A {c } l c ψ B {c } l j k (3) For all he double ininie square and sequence {c } can be esablished, hen i is o he {c } l = c <. j k Suppose ψ is a uncionr, hen here is only one Riesz base {ψ } z in L (R). ψ,ψ l,m = δ j,l δ k,m, j,k,l,m Z (4) I he equaion is he dual o {ψ }, hen every () L (R) has only one series according o he equaion (1): ()=,ψ ψ () (5) j= k= Especially, i {ψ } z is L (R) sandard orhogonal basis, hen ψ =ψ. The reconsrucion ormula can be expressed as: 39

4 ()=,ψ ψ () (6) j= k= ŵ sign(w )( w αλ), w λ 0, w < λ,(0 α 1) (31) 3. Threshold Opimizaion o Discree Wavele Transorm Algorihm Based on he discree wavele ransorm algorihm opimizaion, in his paper, we propose a mehod or urher opimizaion o is hreshold. For he one-dimensional signal (), we irs ake discree sample and collec discree signal (n),n = 0,1,...,N 1 rom N poins. Is wavele ransorm is: N 1 WT ( j,k)= j/ (n)ψ ( j n k) (7) n=0 where WT ( j,k) is wavele coeicien. I is complicaed i use Eq. (7) direcly or calculaion in he real siuaion. Also, ψ() usually has no expression. Thereore, based on he double dimension equaion, he wavele ransorm mehod o recursive implemenaion can be expressed as: SF( j +1,k)= SF( j,k)*h( j,k) (8) WT ( j +1,k)= SF( j,k)* g( j,k) (9) where h and g is low-pass and high-pass iler o scaling uncion φ(x) and wavele uncion ψ(), respecively. SF(0,k) is original signal o (k). SF( j,k) is scale acor. WT ( j,k) is wavele coeicien. The wavele ransorm reconsrucion ormula can be expressed as: SF( j 1,k)= SF( j,k)* h( j,k)wt ( j,k)* g( j,k) (30) For convenience, wavele coeicien WT ( j,k) has been denoed as w. From he linear properies o wavele ransorm, here are wo pars exisence a wavele coeicien w aer discree wavele ransorm o (k)= s(k)+ n(k). One par is he s(k) s wavele coeicien Ss( j,k) denoed as u. Anoher par is n(k) s wavele coeicien Sn( j,k) denoed as v. The wavele hreshold value can be divided ino so hreshold and hard hreshold. In hard hreshold, because he is disconinuous a λ, he reconsruced signal may ŵ produce some oscillaion. The induced rom ŵ so hreshold has overall good coninuiy, bu when w λ, here is a consan dierence beween ŵ and w, which could direcly aec he reconsruced signal and real signal approximaion degree. Thereore, his aricle proposes a sraegy or hreshold compromise opimizaion. Se he wavele coeicien as: The absolue value o induced rom so hreshold ŵ has less han w or λ, so reducion o he deviaion is necessary. However, i reduce his value o 0 (as hard hreshold) may also ace some problems. Because he w is consruced by u and v, w > u may happen by he inluence o v. However, our aim is o obain he minimum value o ŵ u. Thereore, pu ŵ beween he w λ and w may resul a wavele coeicien more close o he ŵ u. Based on his idea, we add α acor in hreshold esimaor. A beer perormance can be obained by appropriae adjusmen o α beween 0 and 1. In his case, we use α= Signal Denoising Based on Improved Wavele Transorm The denoising processing o weak signal chaoic deecion sysem is based on he improved wavele ransorm algorihm. Se he sampling signal as s(n)= acos(ωn/ +ϕ)+ z(n), where s is he sample s µ requency. The deail o denoising processing can be explained as ollows: Ge signal Ps(n) aer wavele prereamen o he sampled signal s(n). Then Ps(n) has been N inie discree decomposiion by improved wavele ransorm algorihm o obain he high requency coeicien marix: C N D N D N 1 D 1 = L N L N 1 LL 1 H N L N 1 LL 1 H N 1 L N LL 1 H 1 Ps(n) (3) where P is improved wavele prereamen marix. D j is j s high requency coeicien marix generaed by wavele ransorm algorihm. H j and L j is j s high requency ransormaion marix and low requency ransormaion marix. Se D j (k,i) as he i elemen in he row o k in marix D j, represening coeicien k in he high requency subband o i in he wavele ransorm j. Considering he keep o signal energy, denoising hreshold compromise sraegy has been applied or D j (i,k) o ge he reaed wavele ransorm coeicien ˆD j (i,k) : D j (i,k), D j j (i,k) λ ˆD j (i,k)= i 0, D j j (i,k) < λ i (33) 40 where he value o λ i j sraegy. is decided by hreshold compromise

5 Denoising signal ŝ(n) can be obained by wavele reconsrucion o hreshold processed wavele coeicien ˆD j wih N: ŝ(n)= Q L N L N 1 LL 1 H N L N 1 LL 1 H N 1 L N LL 1 H 1 C N ˆD N ˆD N 1 ˆD 1 (34) In he equaion, Q is pos-processed wavele ransorm marix. The denoising signal ŝ(n) aer auocorrelaion arihmeic: Rŝŝ (k)= ŝ(n)ŝ(n k) (35) n In order o aciliae subsequen ieraion process, denoe R (k)= R (n). ŝŝ ŝŝ Se Duing vibraor parameers. Adjus he requency o inernal signals as ω, arge deec requency as ω 0. Se he cycle o he oscillaor power ampliude hreshold value or he sysem as r, leing he maximum Lyapunov index λ 1 mees λ The ollowing Duing oscillaor is consruced: x = ω 0 y y = ω 0 (x 3 x 5 ky + r 1 cos(ω 0 )) (36) Pu he signal R (n) ino discree Duing oscillaor: ŝŝ x(n/ s ) = ω 0 y(n/ s ) y(n/ s ) = ω 0 (x 3 (n/ s ) x 5 (n/ s ) ky(n/ s ) + r 1 cos(ω 0 (n/ s ) + Rŝŝ (n))) (37) Fig. 1. Comparison o wo algorihms de-noising error. Frequency SNR/dB Relaive Time/s (rad/s) Wavele - Wavele error Wavele - Wavele % % % % % % % % % % Tab. 1. Wavele ransorm algorihm and improved conras wavele ransorm algorihm. Then he improved wavele ransorm algorihm is used in he weak signal deecion chaoic sysem. Fig. shows he weak signal wihou noise. Fig. 3 shows he weak signal wih noise. Fig. 4 shows he denoising resuls o weak signal deecion chaoic sysem based on he wavele ransorm. Fig. 5 shows he denoising resuls o weak signal deecion chaoic sysem based on he improved wavele ransorm. The saisical resuls are shown in Table. where s is sampling requency o s(n). 4. Algorihm Perormance Simulaion In order o veriy he eeciveness o he improved algorihm proposed in his paper, simulaion experimens are conduced in he PC o memory wih GB, and CPU as Penium (R) Dual - Core T GHZ. The resuls o denoising processing using radiional wavele ransorm algorihm and improved wavele ransorm algorihm are presened in Table 1 and Fig. 1. Fig.. The weak signal-o-noise-ree. 41

6 Fig. 3. Conaining a weak signal-o-noise. Fig. 5. Denoising resul o chaoic deecion sysem based on improved wavele ransorm. I can be seen rom he simulaion resuls ha he discree wavele ransorm algorihm o opimizaion and compromise hreshold sraegy presened in his paper show beer denoising eec han he original algorihm. I improves he denoising eec when applying o he weak signal chaoic deecion sysem. 5. Conclusion Fig. 4. Denoising resul o chaoic deecion sysem based on wavele ransorm. Frequency SNR/dB (rad/s) Wavele I- Wavele Relaive error % % % % % % % % % % Weak signal deecion (WSD) is a new echnique rising in recen years, which overcomes he problem o high inpu SNR hreshold in radiional linear signal deecion echnology. This paper proposes a weak signal chaoic deecion sysem based on wavele ransorm algorihm, opimizing he wavele ransorm algorihm wih discree opimizaion and hreshold compromise sraegy. I can be seen rom he simulaion resuls comparing wih radiional wavele ransorm ha he improved sraegy o his paper increases he deecion accuracy o weak signal wih noise and shows he beer denoising eec han he original algorihm. Tab.. Error analysis o weak signal denoising. Reerences 1. H. Xing, Dissipaion ype synchronous weak periodic signal deecion and noise impac analysis, Journal o Jilin Universiy, vol. 4, pp (014).. A. Arneodo, P. Coulle, and C. Tresser, Possible new srange aracors wih spiral srucure, Communicaions in ahemaical Physics, vol. 79, pp (1981). 3. L. Pang, Research and developmen o he daa collecor or ime domain aeronauical elecromagneic weak signal, Process Auomaion Insrumenaion, vol. 35, pp (014). 4. W. Liu, A general ampliier design or low requency weak signal acquisiion, Journal o UEST o China, vol., pp. - 5 (014). 5. C. Zhang, Analyses based on he echnology o high-speed weak phooelecric deecion relaed research, Science Technology and Engineering, vol. 8, pp (014). 6. Klein, Research on weak signal deecion in ime domain based on phase locked loop and Duing oscillaor, Science Technology and Engineering, vol. 6, pp (014). 4

7 7. Weinred, Research on chaoic characerisics o a deormable Rossler sysems and is usage or weak signal deecion, Compuer easuremen & Conrol, vol. (), pp (014). 8. Fourier, Weak signal deecion sysem or marine environmen based on LabVIEW, Insrumen Technique and Sensor, vol. 11, pp (013). 9. Wang, Weak sine signal deecion based on improved double dierenial oscillaors, Journal o Henan Normal Universiy (Naural Science), vol. 6, pp (013). 10. Y. Li, Chaos weak signal deecion based on GPU criical hreshold is deermined, Applicaion Research o Compuers, vol. 31(4), pp (014). 11. P. Lu, Chaos algorihm and subspace algorihm applied in weak signal deecion, Science Technology and Engineering, vol. 1, pp (014). 1. W. Chen, Research on weak signal deecing in disribued FBG sensing sysem, Journal o Xian Universiy o Technology, vol. 9(4), pp (014). 13. C. Li, Based on orle iler and he paricle swarm algorihm o weak signal deecion, Compuer easuremen & Conrol, vol. 1(10), pp (013). 14. G. Wang, Weak signal inelligen deecion sysem based on sochasic resonance and ariicial ish swarm algorihm, Chinese Journal o Scieniic Insrumen, vol. 34(11), pp (013) Chance Glenn and S Hayes, The design o weak signal deecion sysem in Srong magneic ield, Journal o Transducion Technology, vol. 6(6), pp (013). 43