UNIVERSITY OF SOUTHERN CALIFORNIA Department of Civil Engineering ESTIMATION OF INSTANTANEOUS FREQUENCY OF SIGNALS

Transcription

1 UNIVERSITY OF SOUTHERN CALIFORNIA Deprmen o Civil Engineering ESTIMATION OF INSTANTANEOUS FREQUENCY OF SIGNALS USING THE CONTINUOUS WAVELET TRANSFORM by M.I. Todorovsk Repor CE -7 December, (Revised Jnury, 4) Los Angeles, Cliorni

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3 ABSTRACT This repor reviews mehods nd lgorihms or esimion o insnneous requency o signls bsed on he Coninuous Wvele Trnsorm (CWT), wih emphsis on heir ccurcy, nd or he purpose o use in nlysis o nonliner response o soil-srucure sysems rom recorded erhquke response. The mehods reviewed re he Hilber Trnsorm mehod, he Mrseille mehod (using he phse o he CWT), nd he Crmon mehod (using he mpliude o he CWT), which works lso or signls wih considerble moun o ddiive noise, nd he simple mehod (using he mpliude o he CWT). The heory behind hese mehods nd lgorihms or implemenion is presened in deph. These mehods re pplied o signls wih well-deined requency chnges, mpliude nd requency moduled so h hey resemble pulses in he response o srucures o srong erhquke ground moion. Signls wih ddiive noise re lso considered. The revision ded Jnury 4 hs correced ypogrphicl miskes nd error in σ, (ws.63σ nd now is.7σ ) nd σ ω, (ws.63/σ nd now is.7/σ ) or he Morle wvele, which implies miniml chnges in Chper (eqns. (.7) (.3), (.4) (.4), nd (.44) (.46)). i

4 ACKNOWLEDGEMENTS Finncil suppor or his work ws provided by he Nionl Science Foundion (grn CMS-757 rom he POWRE progrm). ii

5 TABLE OF CONTENTS ABSTRACT.i ACKNOWLEDGEMENTS.ii. INTRODUCTION.... WAVELETS - THEORETICAL BACKGROUND Wveles nd Wvele Fmilies Loclizion in Time nd Frequency The Coninuous Wvele Trnsorm The Wvele Trnsorm s Time-Frequency Energy Disribuion The Morle Wvele The Prsevl Equliy nd Compuion o Wvele Trnsorm vi FFT....7 The Windowed Fourier Trnsorm nd he Gbor Trnsorm....8 Meningul Scles nd Frequencies or Compuing he Wvele Trnsorm nd he Gbor Trnsorm INSTANTENEOUS FREQUENCY OF SIGNALS Deiniion o Insnneous Frequency Hilber Trnsorm Anlyic Signl Associed wih Rel Signl nd Hilber Trnsorm Mehod or Deerminion o Insnneous Frequency Asympoic Signls Mehod o Sionry Phse or Approximion o Inegrls Esimion o Insnneous Frequency From he Phse o he Coninuous Wvele Trnsorm (Mrseille Mehod) Coninuous Wvele Trnsorm o Asympoic Signls The Ridge o he Coninuous Wvele Trnsorm nd Is Relion o Insnneous Frequency The Skeleon o he Coninuous Wvele Trnsorm Ridge Exrcion rom he Phse o he Coninuous Wvele Trnsorm Algorihm or he Mrseille Mehod or Insnneous Frequency Exrcion nd Signl Reconsrucion Ridge Exrcion rom he Modulus o he Coninuous Wvele Trnsorm- Crmon Mehod Ridge Esimion in Presence o Noise Simuled Anneling Algorihm Signl Reconsrucion The Simple Mehod Generl Remrks NUMERICAL EXAMPLES The Signls Insnneous Frequency nd Ridge nd Skeleon Esimion or Pulse-Like Signls wihou Noise Insnneous Frequency or Noisy Signls DISCUSSION AND CONCLUSIONS REFERENCES iii

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7 . INTRODUCTION Wvele nlysis hs is roos in isoled work o mhemicins crried ou mosly round 93s. The irs wvele bsis, he Hr s bsis, ws inroduced even erlier (in 9), bu he modern wvele nlysis, he wy we reer o i now, ws inroduced no so long go, in he 98s, by French geophysiciss (Morle e l., 98,b; Goupillud e l., 984/85; Grossmnn nd Morle, 984). For hisoricl review o is roos, rediscoveries nd more recen developmens, he reder is reerred o Meyer (993). Since hen, wvele nlysis is being used in mny ields o science nd engineering, probbly mos exensively in digil signl nd imge processing nd communicion, due o heir eiciency in d compression (Veerli nd KovceYLü,WLVDOVRXVHGLQDUWLILFLDO inelligence or deining conours o objecs in bimp imges nd or pern recogniion (Tng e l., 999), in sisics, or nonprmeric uncion esimion (Anonides nd Oppenheim, 995), nd in mny oher ields. The wvele rnsorm is priculrly suible or nlysis o rnsien signls nd ime vrying sysems, becuse i is loclized boh in ime nd requency. Is widespred use is lso due o he exisence o orhogonl nd bi-orhogonl bses, nd he vilbiliy o s nd ccure compuionl lgorihms or signl/imge rnsormion nd reconsrucion. Wveles were irs inroduced o mechnicl vibrion problems by Newlnd (993; 994,b). Ler, hey were used by Bsu nd Gup (997,b; 999) in sochsic nlysis o response o liner sysems subjeced o seismic exciion, by Ghnem nd Romeo () nd Pei e l. () in prmeric ideniicion o nonliner nd liner ime vrying dynmicl sysems, by Amrung nd Willims (993) nd Willims nd Amrung (995) in solving generl pril dierenil equions, s well s by Spnos nd Ro () in represenion o rndom ields. This repor reviews mehods nd lgorihms or esimion o insnneous requency o signls bsed on he Coninuous Wvele Trnsorm (CWT), wih emphsis on heir ccurcy, nd or he purpose o ler use in nlysis o nonliner response o soilsrucure sysems rom recorded erhquke response (Triunc e l.,,b). The mehods reviewed re he Hilber Trnsorm mehod, he Mrseille mehod (using he phse o he CWT; Deplr e l., 99), nd he Crmon mehod (using he mpliude o he CWT; Crmon e l., 997), which works lso or signls wih considerble moun o ddiive noise, nd he simple mehod (using he mpliude o he CWT). Chper presens heoreicl bckground on wveles s reled o nd consisen wih he convenions used in published work on hese mehods. For generl bckground on wveles, he reder is reerred o sndrd books on wveles, e.g. by Dubechies (99) nd by Veerli nd.rydfhylü &KDSWHU SUHVHQWV LQ GHWDLO WKH WKHRU\ WKHVH mehods re bsed on, nd lgorihms or heir implemenion. Chper 4 presens

8 numericl resuls or well-deined signls, mpliude nd requency moduled so h hey resemble pulses in he response o srucures o srong erhquke ground moion. Signls wih ddiive noise re lso considered. All in Forrn by he uhor (excep or he Fs Fourier Trnsorm, rndom number generor, nd Gussin noise generor subrouines which were ken rom Numericl Recipes; Press e l., 99), used o produce he resuls in Chper 3. Elecronic iles o he Forrn progrms cn be downloded rom he USC Srong Moion Reserch Group web sie This repor is lso inended o serve s educionl meril or civil engineering sudens on he subjec o esimion o insnneous requency o signls nd sysems.

9 . WAVELETS - THEORETICAL BACKGROUND This chper presens heoreicl bckground on wveles nd on he coninuous wvele rnsorm, s reled o nd ollowing he convenion o he mehods or deerminion o insnneous requency reviewed in his repor.. Wveles nd Wvele Fmilies A wvele is zero men wiggle (rel or complex), loclized boh in ime nd in requency (in oher words, he wvele is nonzero only ner ime = nd is Fourier rnsorm is lso nonzero ner some requency ω=ω ). For he zero men condiion (lso clled dmissibiliy condiion) o be sisied, i mus be oscillory - hence he nme wvele (Veerli nd Kovcevic, 995). Given prooype wvele ψ () L ( R), mily ψ b, () cn be consruced by elemenry operions consising o ime shis nd scling (i.e. dilion or conrcion). The mily o wveles is deined by ψ b ( b, )() = ψ, b R, > (.) where is he scling cor nd b is he ime shi. The prooype wvele is clled he moher wvele, nd i is he member o he mily corresponding o b= nd =. Scle cor > corresponds o dilion nd < o conrcion o he moher wvele. In eqn (.), / in ron o ψ is normlizing cor or he mpliude o he wvele, chosen so h ll wveles in he mily hve sme L norm ψb, () d = ψ() d (.) This convenion is sme s he one used by Crmon e l. (995; 997; 998) nd Deplr e l. (99), nd diers rom he convenion in mos books nd ppers on wveles where ll wveles in mily hve sme L norm.. Loclizion in Time nd Frequency The ime loclizion o wvele in mily o wveles, i.e. is cenrl ime nd spred round he cenrl ime, re deined vi he verge nd vrince o he ime in he expression or ψ, (), s ollows (Crmon e l., 998; Flndrin, 999) b 3

10 ( b, ) = ( b, ) ( b, ) ψ () d ψ () d (.3) σ, ( b, ) ( ) ( b, ) ψ( b, )() = d ψ ( b, ) () d (.4) Similrly, he cenrl requency nd he spred round he cenrl requency re deined s he verge nd sndrd deviion o he circulr requency, ω, in he Fourier Trnsorm o he wvele, ψˆ (, )( ω ), s ollows b ω ( b, ) = ωψˆ ( ω) dω ( b, ) ψˆ ( ω) dω ( b, ) (.5) σ ω, ( b, ) ( ) ˆ ( b, ) ( b, )( ) = ω ω ψ ω dω ψˆ ( ω) dω ( b, ) (.6) Le nd ω be he cenrl ime nd requency o he moher wvele, nd σ, nd σ ω, be he corresponding vrinces. Then eqns (.3) hrough (.6) imply he ollowing relionships = + b (.7) ( b, ) ω ω ( b, ) = (.8) σ, ( b, ) σ, = (.9) nd 4

11 σ ω, σ ω, ( b, ) = (.) Equions (7) hrough () imply h diled wvele ( > ) hs imes lower requency hn he moher wvele, nd is beer loclized in requency bu more poorly in ime. The bove semen is rue or ny uncion, nd is ormlly expressed by he Heisenberg s Unceriny Principle, which ses h uncion cnno be rbirrily well loclized boh in ime nd in requency, nd h he produc o he uncerinies in hose loclizions is consn. I he uncerinies re expressed vi he vrinces in eqns (.4) nd (.6), hen he Heisenberg s Unceriny Principle cn be expressed by σ σ ω = C (.) where he consn C is he smlles or Gussin uncions. Obviously, incresed loclizion in ime cn be chieved only he cos o decresed loclizion in requency. Two exreme uncions re he complex exponenil nd he Dirc Del-uncion. The complex exponenil is perecly loclized in requency (i.e. poin) bu no loclized ll in ime, while he Dirc Del-uncion is perecly loclized in ime bu is no loclized ll in requency (is Fourier Trnsorm is h o whie noise). A windowed complex exponenil is loclized in ime bu no perecly, nd he loclizion depends on he lengh o he window. For lrger window, he spred in ime increses bu he spred in requency decreses..3 The Coninuous Wvele Trnsorm The coninuous wvele rnsorm o uncion () L ( R) is deined s is inner produc wih mily o dmissible wveles ψ, (), i.e. b T ( b, ) =<, ψ > = () ψ () d (.) ( b, ) L ( b, ) where b nd re he ime nd scle vribles, nd he br over ψ indices complex conjuge. The inverse rnsorm is (Crmon e l., 998) ddb () T ( b, ) ψ ( b, )() C ψ = (.3) 5

12 where C ψω ˆ ( ) ψ = dω (.4) ω We noe h he reconsrucion ormul, eqs (.3), nd he expression or C ψ in eqn (.3) re such h hey re consisen wih he deiniion o he wvele mily in eqn (.). In he Wvele Trnsorm, he scling cor or he wvele mily is used s vrible clled scle. The scle is reled o requency. The requency corresponding o scle is he cenrl requency o he wvele divided by. i.e. ω = ω / (.5) Hence, scle is inversely proporionl o requency..4 The Wvele Trnsorm s Time-Frequency Energy Disribuion Becuse he wvele rnsorm T ( b, ) is essenilly he projecion o () ono wvele ψ ( b, ), which is nonzero only ner he cenrl ime o ψ ( b, ) nd he cenrl requency o ψ ( b, ), i describes he properies o he uncion ner hese locliies, nd is orm o ime-requency energy disribuion o (). The loclizion properies o he rnsorm re hose o he wvele mily, nd T ( b, ) represens he properies o () in he cell (, ( b, ) ) ( ω, ( b, ) ) ( ) ω σ ω, ( ) ( b, ) ± σ ω( b, ) ± σ = + b ± σ, ± (.6) o he ime-phse plne. Figure. shows hree such bens, or =, < nd >. I cn be seen h or requencies ω > ω ( < ) he ben is sreched long he ω-xis, bu compressed long he -xis, nd he opposie is rue or ω < ω ( > ). This mens h he lower requency componens re beer loclized in requency hn higher requency componens. An imporn propery o he Wvele Trnsorm is h while he spred in requency ω is vrible, he relive spred ω / ω is consn 6

13 > < Wvele Trnsorm ω (=ω /) ω ω = σ ω, / ω = / σ ω, ω =cons. σ, ω>ω σ ω, / σ, = ω σ ω, σ, ω<ω σ ω, / +b Fig.. Time-requency resoluion or he Wvele Trnsorm. 7

14 σ ω, ω σ = = ω, (.7) ω ω ω This propery implies h he Wvele Trnsorm is beer suied or ime requency nlysis o signls wih lrge vriions in requency (represened, e.g. on logrimic scle) hn ime-requency disribuions wih consn spred..5 The Morle Wvele For nlyses o phse reled properies o rel uncions (e.g. deerminion o insnneous requency), he complex Coninuous Wvele Trnsorm is more suible hn he rel Wvele Trnsorm or he discree Wvele Trnsorm. The mos commonly used moher wvele or such pplicions is he complex Morle wvele ψ = ω σ (.8) ( ) exp( i ) exp( / ) which is essenilly complex exponenil moduled by Gussin envelope. The mpliude is normlized so h ψ () =, ollowing he convenion o Crmon e l. (995; 997), nd σ is mesure o he spred in ime. The Morle wvele hs Fourier rnsorm ψ( ω) = π σ exp ( ω ω) σ (.9) where he Fourier Trnsorm o uncion () nd he inverse rnsorm re deined by ( ω) ( ) ˆ iω = e d (.) () ˆ iω = ( ω) e dω π (.) The Fourier Trnsorm o he Morle wvele is nonzero only or ω >. Such wveles re clled progressive, nd re priculrly convenien or nlyses o phse o rel vlued uncions, he Fourier Trnsorm o which or negive requencies is he complex conjuge o hose or posiive requencies (his simpliies he nlysis wihou loss o inormion). The subspce o L conining uncions wih such propery re clled Hrdy spces 8

15 { ˆ } H( R) = L ( R): =, ω < (.) The populriy o he Morle wvele s n nlysis ool is due o he c h i is described by n nlyic uncion, nd so is is Fourier Trnsorm. A no so good quliy is h i hs ininie suppor, bu his is no prcicl problem s is envelope rpidly decreses wy rom =. Also, he Morle wvele is no sricly speking n dmissible wvele, bu or ω >5 (i σ=), he dmissibiliy condiion is prciclly sisied ( ψ ( d ) ). The rigorous deiniion o he Morle wvele h sisies lierlly he dmissibiliy condiion is (Fououl-Georgiou nd Kumr, 994) ( ) ψ ( ) = exp iω exp ω / exp( / σ ) (.3) bu his orm is lmos never used. Figure. shows he Morle wvele nd is Fourier Trnsorm mpliude scles = nd, compued or σ= nd cenrl requency ω =π. Commonly used vlue in lierure re lso σ= ndω =5.6, which is requency or which he mpliude o he second lrges lobe o he wvele is hl o he mpliude o he lrges lobe. In his repor, ω =π will be used. The L norm o he moher wvele (nd o he enire mily, ccording o he convenion doped in his repor) is ψ () = ψ() d = πσ L (.4) The scled wvele nd is Fourier Trnsorm re b b ψ ( b, )() = exp iω exp /σ ( ) exp ψ (, ) ( ) ( ) b ω = iωb ψ ω (.5) (.6) The loclizion properies o he Morle wvele re s ollows. The cenrl ime o ψ () is =, he cenrl requency is ω, he vrinces in ime nd requency re 9

16 ^ ψ ( 6,.5) Rel pr Imginry pr ψ (,) ψ (6,) ψ (b,) ψ (b,) ψ (6,) ψ (,) - s ψ ( 6,.5) ω = π σ = =ω/π - Hz Fig.. Three wveles o he Morle mily wih ω = π nd he requency domin (boom). σ =, in he ime domin (op) nd in

17 σ, = σ σ, =.7σ (.7) σ ω, σ ω,.7 σ σ (.8) he Heisenberg inequliy becomes σ σ = (.9), ω, nd he relive loclizion in requency is σ ω, ω σ ω, = = = (.3) ω ω ω σω.6 The Prsevl Equliy nd Compuion o Wvele Trnsorm vi FFT The Prsevl equliy or he inner produc o wo uncions nd g is () gd () = ˆ( ) ˆ ( ) ω g ω d ω π (.3) nd i implies T (, b ) = () ψ () d ( b, ) = ˆ ( ω) ψˆ ( b, )( ω) dω π ˆ( ) ( ) iωb = ω ψ ω e dω π = { ˆ( ωψ ) ( ω ) } FT (.3) Then T (, b ) cn be compued eicienly using Fs Fourier Trnsorm, by irs compuing ˆ( ω ), nd hen compuing he inverse rnsorm o he produc ˆ ( ω) ψˆ ( ω ), where ψˆ ( ω ) is compued vi eqn (.9).

18 .7 The Windowed Fourier Trnsorm nd he Gbor Trnsorm A usul ool or ime requency nlysis hs been he Windowed Fourier Trnsorm. The Windowed Fourier Trnsorm o uncion, WFT, consiss o muliplying () by (usully rel) window uncion w shied in ime. I w() is prooype window, symmeric bou =, hen iω WFT (, b ω) = () w( b) e d (.33) The window is some pering uncion, e.g. recngulr box, box wih liner rmps he ends, ringulr (Brle), Hnning, Hmming, Blckmn nd oher windows cenered =. The Gbor Trnsorm is essenilly Windowed Fourier Trnsorm wih Gussin window. I is deined s he inner produc beween () nd mily o uncions g (b,ω) () clled Gbor uncions G (, b ω) = () g () d (.34) ( b, ω ) where g () ( ) i b ( b, ) g b e ω ω ( ) = (.35) nd g ( b) is shied Gussin window. A comprison o eqns (.33), (.34) nd (.35) shows h WFT (, b ω ) is essenilly equl o G ( b, ω ) excep or phse shi in w( b). Someimes, he Gbor Trnsorm is deined wihou his phse shi. In his repor, he deiniion in eqns (.34) nd (.35) is doped (Crmon e l., 998). Subsiuing in g (b,ω) () or Gussin window implies ( b) g( b, ω )() = exp exp iω ( b) σ (.36) which hs Fourier Trnsorm gˆ [ ] ( ξ) = exp iξb gˆ ( ξ) ( b, ω) (, ω) = exp[ iξ b] πσ exp ( ξ ω) σ (.37)

19 Numericlly, he Gbor Trnsorm cn lso be compued using he Prsevl equliy nd Fs Fourier Trnsorm, s ollows G (, b ω) = () g () d ( b, ω ) = ˆ ( ξ) gˆ ( b, ω) ( ξ) dξ π ˆ ibξ = ( ξ ) gˆ (, ω) ( ξ) e dξ π = { ˆ( ξ) ˆ(, ω) ( ξ) } FT g (.38) The similriy beween he Coninuous Wvele Trnsorm wih he Morle wvele nd he Gbor Trnsorm is sriking. To compre hese wo rnsorms mos direcly, we express scled wvele in erms o ω insed o scle, reclling h ω / = ω ψ ( b) ( σ ) () = exp exp iω( b) ( b, ) (.39) A comprison o eqns (.36) nd (.39) shows h ψ ( b, )() is essenilly g( b, ω )() bu wih vrible window widh, depending on he vlue o he scle. (There is lso normlizon cor or he mpliude bu his is no signiicn.) A scle =, hey re idenicl. A scles > (i.e. or smller requencies), he wvele rnsorm hs wider window, while he window widh or he Gbor Trnsorm is ixed or ll requencies. An implicion o his is h he wvele lwys conins sme number o wvelenghs, while he Gbor uncion conins smller number o wvelenghs or longer periods. As boh rnsorms re meningul priculr period only i here is les one wvelengh conined wihin he widh o he window (i.e. wihin pr o he window wih signiicn mpliudes), his implies grer lexibiliy or he Wvele Trnsorm, which cn be used or lrger spn o requencies or chosen moher wvele nd Gbor uncion. The loclizion properies o he Gbor Trnsorm re deermined by hose o he Gussin window. As he widh o he window is consn or ll requencies, he vrince o he ime loclizion will be he sme or ll requencies, nd, by he Heisenberg unceriny principle, he vrince o he requency loclizion will lso be he sme or ll requencies. These vrinces re sme s hose or he Morle wvele in eqns (.7) nd (.8) nd re hose or he enire mily o he Gbor uncions 3

20 Gbor Trnsorm ω (=ω /) ω = σ ω = cons. σ ω>ω σ ω σ ω σ ω σ ω<ω σ ω +b Fig..3 Time-requency resoluion or he Gbor Trnsorm. 4

21 σ = σ σ =.7σ (.4) σ = ω σ.7 ω σ = σ (.4) nd σ σω = (.4) Figure.3 shows he consn size loclizion cells in he ime-requency plne or he Gbor Trnsorm. Agin, he min dierence beween he Wvele nd Gbor Trnsorms is h he ormer oers sme relive ccurcy o loclizion in requency (sme ω/ω), while he ler oers sme bsolue ccurcy o loclizion in requency (sme ω). A comprison o lines 4 in eqns (.3) nd (.38) shows h he compuionl eors o compue boh, ollowing he Prsevl equliy nd using FFT, re he sme. We lso noe h or boh rnsorms, inverse FFT needs o be evlued or chosen grid o scles (requencies) wihin he rnge o ineres, nd ech run o inverse FFT gives he rnsorms or ll imes b. Alhough n rbirrily ine grid cn be preseleced, he ccurcy o loclizion in requency cnno be rbirrily smll. For he Gbor Trnsorm, he bsolue ccurcy o requency loclizion ( ω) is limied by σ o he Gussin window, nd i cn be incresed only by incresing σ. For he Wvele Trnsorm, he relive ccurcy o requency loclizion ( ω /ω) cn be incresed by decresing σ, by decresing ω, or by decresing boh (see eqn (.3)). I boh σ nd ω chnge bu heir produc remins he sme, ω/ω will no chnge..8 Meningul Scles nd Frequencies or Compuing he Wvele Trnsorm nd he Gbor Trnsorm The mximum requency, ω mx, or which boh he Coninuous Wvele Trnsorm nd he Gbor Trnsorm re meningul is deermined by he smpling re o he signl, nd is is he Nyquis requency, ω Nyqis, (Oppenheim nd Scher, 999) ω mx π < ωnyqis = (.43) 5

22 where is he smpling ime inervl, which implies minimum meningul scle or he Wvele Trnsorm min ω π > (.43) The minimum meningul requency (mximum scle) is deermined by he lengh o he window in wy h here should be les one period T = π / ω conined wihin he window o he wvele or o he Gbor uncion, nd ulimely by he lengh o he signl in he sense h he window lengh should lwys be shorer hn he lengh o he signl. For Gussin window (wih ininie suppor nd nonuniorm mpliude), his condiion is expressed in erms o σ, e.g. h here should be les one period conined in 6σ (Fouul-Georgiou nd Kumr, 994). This implies or he Gbor Trnsorm, s deined in his repor, ω min = π T > π π π 6σ = 6.7 > N mx ( σ) (.44) For he Wvele Trnsorm, he window lengh is lexible nd increses proporionlly wih he scle, nd once ω nd σ hve been se or he moher wvele so h he wvele does no viole much he dmissibiliy condiion, he upper bound or mximum scle is deermined by he lengh o he signl. For he Wvele Trnsorm wih he Morle wvele s deined in his repor,( b, ) mx, mx ( ) 6σ = 6 σ = 6.7σ < N (.45) which implies mx N N < = (.46) 6σ 6.7 ( σ ) 6

23 3. INSTANTENEOUS FREQUENCY OF SIGNALS This chper presens he heoreicl bckground reled o he deiniion o insnneous requency o signls (Hilber Trnsorm, nlyic signls ssocied wih rel signls, sympoic signls, nd principle o sionry phse), nd wo mehods or deerminion o insnneous requency o sympoic signls, one rom he phse (clled Mrseille mehod by Crmon e l., 997) nd he oher one rom he mpliude (reerred o s Crmon mehod in his repor) o he Coninuous Wvele Trnsorm 3. Deiniion o Insnneous Frequency Frequency is he derivive o phse wih respec o ime, nd physicl signls re rel vlued. Then, n inuiively resoning would sugges h insnneous requency ω () o signl () could be deined by irs wriing i s () = A()cos Φ () (3.) where boh A() nd Φ () re rel, nd hen diereniing Φ () dφ ω ( ) = (3.) d One problem wih his pproch is h here re ininiely mny wys o wriing () s in eqn (3.), nd ddiionl consrins re needed or such represenion o be unique. This cn be done wih he help o he Hilber Trnsorm, s ollows. 3.. Hilber Trnsorm The Hilber Trnsorm o rel uncion (x) is by deiniion H ( x) = P. ( x + y) π dy y, x R (3.3) where P. indices principl vlue o he inegrl (i.e. up o scling cor, he Hilber Trnsorm is he principl vlue o he convoluion o (x) wih / x ). Is Fourier Trnsorm hs very simple nd convenien orm, which describes is useulness Hˆ ( ω) = i sgn( ω) ˆ( ω), ω R (3.4) 7

24 Euion (3.4) implies h ˆ( ω) + ihˆ ( ω) is nonzero only or nonnegive requencies. This moives he deiniion o n nlyic signl ssocied wih rel signl. 3.. Anlyic Signl Associed wih Rel Signl nd Hilber Trnsorm Mehod or Deerminion o Insnneous Frequency The nlyic signl Z (x) ssocied wih rel signl (x) is by deiniion he complex signl he rel pr o which is (x) nd he imginry pr o which is he Hilber Trnsorm o (x) Z ( x) = ( x) ih ( x) (3.5) + Is Fourier Trnsorm is Zˆ ( ω) = ˆ( ω) + ihˆ ( ω) = ˆ( ω) + sgn( ω) ˆ( ω) ˆ( ω), ω =, ω < = ˆ( ω) Η( ω) (3.6) where Η (ω ) is he Heviside sep uncion. Equion (3.6) shows h Z (x) hs energy only in he nonnegive pr o he specrum (i.e. i is member o Hrdy spce) { ˆ } Z H( R) = L ( R): =, ω < (3.7) in which domin is Fourier Trnsorm is wice he Fourier Trnsorm o (x). This resul is very convenien or nlysis o rel nd cusl processes, s i oers he dvnges o he complex uncion spces wihou inroducing ddiionl inormion (or rel nd cusl processes, ll he inormion is conined in he domin ω >, nd negive requencies hve no physicl mening). Complex uncion Z (x) is n nlyic uncion in he upper complex plne (hence he ribue nlyic), nd hs unique polr coordine represenion Z () Z () where irg Z () = e (3.8) 8

25 Z [ ] + [ IZ () ] () RZ () = (3.9) () () IZ rg Z () = rcn (3.) RZ nd where R nd I indice rel nd imginry prs o complex number. Then unique represenion o he orm in eqn (3.) cn be deined, reerred o s he cnonicl represenion o rel signl ( ) = A( )cosφ( ) (3.) where A () RZ () = (3.) () Z () = (3.3) () = rg Z () Φ (3.4) The insnneous requency hen cn be deined s in eqn (3.), i.e. d ω ( ) = rg Z () (3.5) d This mehod or deerminion o insnneous requency is reerred o s he Hilber Trnsorm Mehod. I is he mos direc mehod or deerminion o insnneous requency, nd is esy o implemen. Also, besides he requency modulion, ω (), i lso gives he mpliude modulion o he signl, A (). However, his mehod hs he ollowing disdvnges: () ω () deined by his mehod does no lwys hve he physicl mening o requency, nd () numericl evluion o derivives is oen unsble, especilly or rel lie signls. 3. Asympoic Signls For some rel signls ( ) = A( )cosφ( ), he ssocied nlyic signl is Z ( ) iφ( ) = A( ) e, bu his is no rue in generl. Signls or which his is rue re clled sympoic signls, ormlly deined s ollows. 9

26 Deiniion: Given rel squre inegrble signl ( ) = A( )cosφ( ) where A ( ) nd [,π ] Φ( ) or ll R, () is sid o be sympoic i i is oscillory enough so h is ssocied nlyic signl is Z ( ) iφ( ) = A( ) e. Here oscillory enough mens h he vriions o () due o cosφ ( ) (i.e. chnge o phse) re much ser hn he vriions due o A () (i.e. mpliude modulion). An exmple o signl h is no sympoic is nlyic signl is Z ( ) = cos, or which he ssocied ( ) = ( ) + ih ( ) = sin + i cos = e i e. The bove propery o sympoic signls ollows rom he ollowing lemm. Lemm: Le ( ) = A( )cosφ( ), where λ >> (i.e. lrge posiive number), A( ) C ( R) (spce o wice coninuously dierenible uncions) nd Φ ( ) C4 ( R) (spce o our imes coninuously dierenible uncions). Then, s λ, he nlyic iλφ( ) signl ssocied wih () Z ( ) = A( ) e + O λ 3 / is ( ). i x i x 3.3 Mehod o Sionry Phse or Approximion o Inegrls The Mehod o Sionry Phse is bsed on he ollowing principle. The Principl o Sionry Phse (proposed by Lord Kelvin): Le b ( ) ( ) iλφ ( x) I x A x e dx = (3.6) where Az ( ) nd φ (z) re nlyic on he complex plne nd rel vlued on he rel line. Then, s λ (i.e. he inegrnd is very oscillory) he dominn erms in he sympoic expnsion o I(x) rise rom he immedie neighborhood o he end poins, nd inermedie poins which λφ ( x) is sionry, i.e. λφ ( x) =. The physicl inerpreion o his principle is s ollows. When λ nd λφ ( x), ω = λφ ( x) is very lrge, which mens h he signl is very oscillory, so oscillory h he mpliude A (x) chnges very lile during one cycle nd he inegrl in eqn (3.6) evlued over one cycle is zero. A poins x = x where φ ( x) = (i.e. φ (x) is sionry), his cncellion does no occur. So only conribuions rom he sionry poins nd he boundry poins dd o he vlue o he inegrl.

27 Mehod o Sionry Phse: Le A (z) nd φ (z) be nlyic complex uncions h re rel on he rel line. I hen ( ) Axe ( ) iλφ x hs only one sionry poin, x = x (o irs order), b iλφ ( x) limi( x) = lim A( x) e dx λ λ ( π /4) sgn φ ( x ) i π e iλφ ( x ) Ax ( ) e O = + λφ λ ( x ) (3.7) In cse o more sionry poins, he limi o he inegrl is he sum o he conribuions rom ll he sionry poins. Equion (3.7) shows h or lrge λ I (x) = (vlue o he inegrnd x = x ) (correcion erm) (3.8) The correcion erm depends only on φ ( x). Equion (3.7) holds or φ ( x) = bu φ ( x ), i.e. or sionry poins o irs order. For he generl cse o sionry ( poins o order n, i.e. when n) ( n ) φ ( x) = bu φ + ( x ), he correcion erm depends on ( n ) φ + ( x ) (see eqns (4.) nd (4.) in Depl e l., 99). 3.4 Esimion o Insnneous Frequency From he Phse o he Coninuous Wvele Trnsorm (Mrseille Mehod) This secion describes he mehod or esimion o insnneous requency o signls rom he phse o he coninuous wvele rnsorm proposed by Deplr e. l. (99). We will reer o i shorly s he Mrseille Mehod (Crmon l., 998). The heoreicl bckground or his mehod is irs described nd hen he lgorihm is presened Coninuous Wvele Trnsorm o Asympoic Signls The objecive is o deermine he insnneous requency o he sympoic signl s( ) = A ( ) cosφ ( ) (3.9) s s rom is Coninuous Wvele Trnsorm Le Z( ) iφs ( ) = As ( ) e (3.)

28 be is ssocied nlyic signl, nd le ) ( ), ( b ψ be mily o progressive wveles (i.e. wvele whose Fourier Trnsorm is nonnegive). The wvele rnsorm o () s is hen Hrdy uncion. We will use he Morle wvele bu he derivion o he mehod will be presened or generl progressive nd sympoic wvele ) ( ) ( ) ( i e A φ ψ ψ ψ = (3.) The coninuous wvele rnsorm o ) ( s is d b Z d b s d s s b T b b L s = = = > =< ψ ψ ψ ψ ) ( ) ( ) ( ) (, ), ( ), ( ), ( (3.) Subsiuion o eqns (3.) nd (3.) ino he ls line o (3.) gives d e b A A b T i s s b ) ( ), ( ) ( ), ( Φ = ψ (3.3) where = Φ b s b ψ φ φ ) ( ) ( ), ( (3.4) is he phse o ), ( b T s. I boh he signl nd wvele re sympoic, hen he inegrnd in eqn (3.3) is lso sympoic. Le = be sionry poin o irs order o his inegrnd, i.e. such h ) ( ) ( ), ( = = Φ b s b ψ φ φ (3.5) nd ) ( ) ( ), ( = Φ b s b ψ φ φ (3.6)

29 Then by he Mehod o Sionry Poins, T s ( b, ) cn be pproximed by ( π /4) sgn Φ ( ) i ( b, ) e b s ψ π Ts(, b ) A () A e Φ ( ) ( b, ) iφ( b, )() (3.7) which cn be wrien s he vlue o he inegrnd = correced by some cor, i.e. T s ( b, ) where b Z( ) ψ π corr( b, ) (3.8) corr / { [ ]} = b π Φ (3.8b) ( ) exp i( / 4) sgn ( ) ( b, ) Φ (, ) ( b, ) ( k) ( k+ ) I = is sionry poin o order k (i.e. Φ ( b, ) ( ) = nd Φ ( b, ) ( ) ), hen T s 4 ( b, ) Γ 6 3 / 3 b Z( ) ψ corr( b, ) (3.9) where [ ] ( k+ ) π corr( b, ) = Φ ( b, ) sgn ( b, ) ( k + ) /( k+ ) ( k+ ) ( ) exp i Φ ( ) (3.9b) Wihou loss o generliy, rom now on we ssume h = is unique sionry poin nd h i is o irs order. For composie signls, his mens h hey hve componens which do no inerc so he nlysis cn be resriced o domin where he wvele coeiciens o ll bu one componen re negligible The Ridge o he Coninuous Wvele Trnsorm nd Is Relion o Insnneous Frequency Deiniion: The ridge o he wvele rnsorm o s (), T s ( b, ), is he se o poins ( b, ) in he domin o he rnsorm where he phse o s ) ψ ( ) is sionry, i.e. he poins h sisy ( ( b, ) 3

30 ( b, ) = b (3.3) Le r (b) be he ridge. Subsiuing = b, = r (b) nd φ s ( ) = ω s ( ) = ω s ( b) in he sionry phse condiion (3.5) gives φ ( ) ω s ( b) = ψ (3.3) ( b) r Furher by subsiuion or φ ψ ( ) = ω or he Morle wvele gives ω ω s ( b) = (3.3) ( b) r Equion (3.3) shows h he insnneous requency o he signl ime b is he requency o he wvele scle r (b). So, once he ridge o he wvele rnsorm is deermined, he insnneous requency cn be deermined esily rom eqn (3.3). The ridge cn be deermined rom he mpliude o he wvele rnsorm or rom he phse, s i will be seen ler The Skeleon o he Coninuous Wvele Trnsorm Deiniion: The skeleon o he wvele rnsorm o s (), ( b, ), is he wvele rnsorm evlued on he ridge, i.e. T ( b, ( b)) Equion (3.8) implies h on he ridge T ( b, s r ( ) s r. π Z ( b) ψ ( b)) (3.33) corr( b, r ( b)) The correcion erm requires Φ ( b, ) evlued on he ridge. Equion (3.6) implies Φ ( b, r ( b)) b) = φ ( b) φ () (3.34) ( s [ r ( b) ] ψ For he Morle wvele, he second erm on he righ hnd side o eqn (3.34) is zero, nd φ (b) evlued using eqn (3.3) is s T s r ( b) ω φ ( b) = ω ( b) = s s [ ( b) ] r (3.35) 4

31 Then r ( b) ω Φ ( b, r ( b)) ( b) = (3.36) [ ( b) ] r which implies h he correcion erm in eqn (3.33) depends only on he ridge nd on he nlyzing wvele. Equion (3.33) cn be used o reconsruc Z (b) rom he skeleon o he wvele rnsorm s corr( b, r ( b)) Z ( b) Ts ( b, r ( b)) (3.37) π ψ () The insnneous mpliude o he signl is hen corr( b, r ( b)) A( b) = Z( b) Ts ( b, r ( b)) (3.38) π ψ () i.e. he mpliude o he skeleon imes correcion Ridge Exrcion rom he Phse o he Coninuous Wvele Trnsorm The ridge cn be exrced rom he mpliude or rom he phse o T s ( b, ). Theoreiclly les, exrcion rom he phse is more ccure; in prcice i my no be so becuse i involves diereniion o phse. Le Ψ ( b, ) be he phse o ( b, ). Then eqn. (3.7) implies h T s π Ψ( b, ) = rgts ( b, ) = i sgn Φ ( b, ) ( ) + Φ( b, ) ( ) 4 (3.39) nd he derivives o Ψ ( b, ) re essenilly hose o Φ ), deined by eqn (3.4). Diereniion o Ψ( b, ) wih respec o scle gives ( b, ) ( Ψ( b, ) b = Φ ( b, ) b (3.4) On he ridge (where = b ) 5

32 Ψ( b, ) = r ( b) = (3.4) Then, or given ime b, he ridge r (b) cn be ound by ierion, s he ixed poin o eqn (3.4). Similrly, he derivive o Ψ( b, ) wih respec o b evlued on he ridge is Ψ( b, ) b = ( b) r = ψ () (3.4) nd or he Morle wvele is Ψ( b, ) ω = b ( ) r ( b) = r b (3.43) Agin, he ridge r (b) cn be ound by ierion, s he ixed poin o eqn (3.43) Algorihm or he Mrseille Mehod or Insnneous Frequency Exrcion nd Signl Reconsrucion The lgorihm consiss o he ollowing seps. Se iniil vlue o (n rbirry vlue wihin he serch domin).. Do or ime seps ib=, nb. Find he ridge r (b) s he ixed poin o eqn (3.43) Ψ( b, ) ω = b ( ) r ( b) where = r b Ψ ( b, ) = rgt ( b, ) s ollows: s Ψ( b, ) Compue b vlues o nd b; using inie dierence scheme nd he curren 6

33 Compue new s (, b) = ω / Ψ b new Repe he ierion unil nd new re suicienly close. Se = new b. Compue he insnneous requency ω (b) rom eqn (3.3) ω ω s ( b) = ( b) r c. Compue he insnneous mpliude A (b) rom eqn (3.38) s A( b) = Z( b) T ( b, s r ( b)) corr( b, π ψ r () ( b)) where rom eqns (3.8b) nd (3.36) / { [ ]} corr( b, ) = Φ ( b, ) π Φ Φ r ( b) ω ( b, r ( b)) ( b) = () b exp i( / 4) sgn ( b) [ ( b) ] r ( b, ) d. Compue he reconsruced signl ime b [ ω ] srec( b) = A( b)cos ( b) b s End o do loop over ime seps ib. This lgorihm ws implemened in he FORTRAN progrm mrs_rdg.or. 7

34 3.5 Ridge Exrcion rom he Modulus o he Coninuous Wvele Trnsorm- Crmon Mehod This mehod ws proposed by Crmon e l. (997) nd is lso described in Crmon e l. (998). I ws used by Sszewski (998) in prmeric ideniicion o nonliner sysems. This mehod is less ccure or exrcion o he insnneous requency bu is more robus or signls h hve noise (hence or rel lie signls). A nice propery o he mehod is h i does no require knowledge o he noise (e.g. like he Klmn Filer), which is he cse in mos rel lie problems. I does however llow incorporion o prior inormion bou he ridge, e.g. is smoohness. Thereore, i cn be inerpreed s Byesin mehod. This mehod is bsed on he wvele Plncherel ormul T = c, L ( R) (3.44) ψ ccording o which he modulus squred o he wvele rnsorm cn be inerpreed s ime-scle energy densiy, nd on he ollowing lemm (Crmon e l., 998). Lemm: Le s( ) = A( )cosφ ( ) L ( R) be n sympoic signl wih wvele rnsorm ( b, ). Then T s iφ ( b) A φφ T = ( ) ˆ ( ( )) +, s ( b, ) A b e ψ φ b O (3.45) A φ Proo: A rigorous proo explining he reminder cn be obined by Tylor expnsion. The principl pr cn be explined s ollows. Le signl ssocied wih s (). Then Z ( ) s iφ ( ) = A( ) e be he nlyic 8

35 T (, b ) = s() ψ () d s = π ( b, ) sˆ( ωψ ) ˆ ( ω) dω ( b, ) ˆ iωb = Z ( ) ˆ sωψ( ω) e dω π ˆ( ( )) ˆ iωb ψ φ b Zs( ω) e dω π (3.46) = ψ ˆ( φ ( b )) Zs ( ) = Abe ψˆ φ b iφ( b) ( ) ( ( )) The pproximion leding rom line 3 o line 4 is bsed on he c h, becuse he signl is nrrow-bnd, wih energy concenred ner ω = ω s ( b) = φ ( b), he produc Zˆ s ( ωψ ) ˆ ( ω ) will be signiicn only ner his requency. Becuse ψˆ ( ω ) is concenred i ( ) ner ω = ω = he cenrl requency o he wvele, he energy densiy A( b) e φ b ψˆ ( φ ( b)) will be concenred long he -xis ner he curve φ ( b) = ω, which is excly he ridge o he wvele rnsorm ω ω r ( b) = = (3.47) φ ( b) ω ( b) s deined erlier in his chper. Using his ls rgumen, hen, he ridge cn be deermined, or ech ime b, by inding he scle where he modulus o he wvele rnsorm is he mximum. This is cerinly he simples nd mos direc pproch. For noisy signls, however, here is beer pproch described in wh ollows Ridge Esimion in Presence o Noise I he signl is conmined wih noise, is wvele rnsorm (modulus nd phse) will lso be conmined wih noise. The signl-o-noise rio however will be he lrges ner he ridge o he rnsorm, becuse mos o he energy o he signl is concenred ner he ridge (unless he noise is lso nrrow-bnd nd wih requency ner h o he signl). This is why using he rnsorm o esime he insnneous requency is beer hn using he signl direcly. Furher, exrcing he ridge rom he modulus o he rnsorm is more robus hn h rom he phse, becuse he exrcion rom he modulus does no involve diereniion o phse. Anoher dvnge o using he 9

36 modulus is h his procedure cn esily incorpore -priori inormion on he ridge s consrin, e.g. h he ridge is smooh. The problem hen reduces o solving consrined opimizion problem, o inding mong ll cndide curves r (b) he one h minimizes he ollowing penly uncion F s [ ( b) ] = T ( b, ( b) ) db + ( b) + µ ( b) db r s r r r λ (3.48) Here minimizion o he penly uncion is equivlen o mximizion o he modulus o he rnsorm long he ridge. Consns λ nd µ cn be chosen by he nlys depending on he problem nd on how much weigh is o be plced on he smoohness consrin (e.g. he lrger he vlues o hese consns, he lrger he weigh o he consrin). This opimizion problem cn be solved by ormuling nd solving he Euler equions o ind he locl exrem, bu more robus mehod or noisy signls is one bsed on simuled nneling. I he record is noisy, hen he modulus o he rnsorm will hve mny spurious locl exrem. The simuled nneling lgorihm llows jumping over hese spurious exrem nd serching urher or he globl exremum. The ollowing secion describes he principles on which he simuled nneling mehod is bsed nd n lgorihm or is implemenion Simuled Anneling Algorihm This mehod is combinoril opimizion mehod nd is pproprie o use when he spce o possible uncions h minimize he objecive uncion is discree nd very lrge so h exhusive serch is prohibiively expensive (Press e l., 994). In our problem, he region ( min, mx ) ( bmin, bmx ) is discreized nd he spce o cndide ridge uncions consiss o ll possible chins o elemens (,b) such h b increses monooniclly bu cn be ny one o he prescribed discree vlues in he domin. The simuled nneling mehod consiss o rndom seps bu in cooling environmen, nd is bsed on nlogy wih he wy hermodynmic sysems cool, i.e. liquids crysllize nd mels nnel. I he cooling is slow, he crysls/oms rech nurlly minimum energy se, in which billions o molecules/oms re perecly lined up. I he re o cooling is s, he sysem reches higher energy, polycryslline or morphous se. A high emperures, he molecules o liquid move reely wih respec o ech oher, nd heir mobiliy decreses wih cooling. I he cooling is slow, he molecules/oms hve ime o redisribue hemselves o minimum energy se. The mobiliy o hese nurl sysems is described by he Bolzmnn probbiliy disribuion o energy s uncion o he 3

37 bsolue emperure o he sysem, P(E ~ exp[ -E /( kt )] ) (higher energy ses re less likely, more so low emperures, bu re sill possible even low emperure). The principle o slow cooling nd nneling ws irs implemened in numericl lgorihms by Meropolis (953). According o his lgorihm, he probbiliy o hermodynmic sysem o chnge energy rom se E o se E is described by P(E, E ) ~ mx{,exp[ -( E E )/( kt )]}. This disribuion is such h whenever E > E, P(E, E ) = while when E < E, P(E, E ) <, i.e. he sysem will lwys go rom higher energy o lower energy se, nd someimes go rom lower energy se o higher energy se. For generl sysem, he lgorihm mus hve he ollowing elemens: () descripion o possible sysem conigurions, () generor o rndom chnges in he sysem, (3) n objecive uncion, nd (4) conrol prmeer like emperure nd n nneling schedule. The seps o his lgorihm re s ollows. Divide he domin (, ) ( b b ) where cndide ridges re deined in min mx min, mx discree grid o eqully spced poins, indexed (i,ib), i=,, n nd ib=,, nb nd compue he wvele rnsorm on his grid.. Deine vlues or prmeers λ nd µ, isme_mx or he sopping crierion (sep e), Cemp nd seeds or he rndom number generors (explined below). 3. Deine n iniil ril ridge. The ridge is deined by he rry i_r(ib), ib=,, nb where i_r(ib) is n index o he scle grid corresponding o index ib o he ime grid. Compue iniil vlue o he emperure s T=Cemp/ln where Cemp is consn, nd compue he iniil vlue o he objecive uncion F_obj. 4. Sr do loop over rndom seps or k= o nier. Compue n upded vlue o he emperure s per he ollowing schedule: T=Cemp/ln (+k) b. Choose neres neighbor ridge, i.e. new ridge h diers rom he curren ridge only ime poin ib=l. Firs selec l s rndom number beween nd nb-. Then selec rndom shi i ε = ±. Deine he new ridge nd check 3

38 wheher i is on he grid; i i is no, coninue choosing new neres neighbor ridge unil i is in he serch domin. This is he cndide ridge or sep k. c. Clcule he objecive uncion or he cndide ridge, F _ obj _ cnd. d. Decide wheher o upde he ridge s ollows. I he objecive uncion or he cndide ridge is less hn or equl o he one or he curren ridge, upde he ridge. I i is no, hen selec rndom number sigm uniormly disribued beween nd. Upde he ridge only i sigm exp F _ obj _ cnd F _ obj / T. Oherwise coninue. [ ( ) ] e. Check wheher he sopping crierion is sisied. Deine -priori number o seps isme_mx. I he ridge hs no chnged or he ls isme_mx seps, hen exi he do-loop. I no coninue serching (i.e. go o sep ).. End do-loop over rndom seps k. 5. Once he ridge hs been deermined, ind he insnneous requency using ω ω s ( b) = ( b) Smooh ω (b) i necessry. s r The bove lgorihm ws implemened in he Forrn progrm crm_rdg.or or ridge deecion nd insnneous requency esimion by Crmon s mehod Signl Reconsrucion Alhough he signl cn be reconsruced rom he skeleon o he wvele rnsorm, e.g. using eqn (3.45), or noisy signls he skeleon is no smooh. Crmon e l. (998) proposed o solve n opimizion problem o inding signl s r () such h is wvele rnsorm long he ridge is close enough o he one or he originl signl, nd i is lso smooh uncion. This mehod is no reviewed in his repor. 3.6 The Simple Mehod The mos srigh orwrd mehod or deermining he ridge o he Coninuous Wvele Trnsorm (nd hence he insnneous requency) o signl rom is mpliude is by inding or ech ime b he scle or which he mpliude o he rnsorm is mximum. In his repor, his mehod is reerred o s he simple mehod. This mehod does no llow or using ny priori inormion bou he ridge such s is smoohness in he 3

39 Crmon mehod, bu is much ser hn he Crmon mehod, which my require housnds (or hundreds housnds) o ierions, nd is more sble hn he Mrseille mehod becuse i dels wih he mpliude o he rnsorm insed o he phse. This mehod is bsed on he heurisic h in nrrow ime inervl he energy o he signl is concenred ner is insnneous requency or h ime inervl. The seps o his lgorihm re s ollows. Divide he domin (, ) ( b b ) where cndide ridges re deined in min mx min, mx discree grid o eqully spced poins, indexed (i,ib), i=,, n nd ib=,, nb nd compue he wvele rnsorm on his grid.. For ech ib=,, nb Find he index i_r(ib) such h ( ( ), ( )) ridge. 3. End o do-loop over rndom seps k. T b ib i is mximum. This deines he 5. Once he ridge hs been deermined, ind he insnneous requency using ω ω s ( b) = ( b) r This lgorihm ws implemened in Forrn progrm dir_rdg.or. s 3.7 Generl Remrks The Mrseille, Crmon nd Simple mehods cn lso be generlized o use he Gbor Trnsorm in he plce o he Wvele Trnsorm. The heory or he Mrseille mehod cn be ound in Derr e l. (99). For he Crmon nd Simple mehods, he procedure remins virully unchnged excep h he Gbor Trnsorm is precompued on grid insed o he Wvele Trnsorm. 33

40 4. NUMERICAL EXAMPLES This chper presens resuls or he mehods described in Chper 3 pplied o hree welldeined chirp signls, one wih consn requency, noher one wih piecewise consn requency nd wih shrp jump in he requency, nd hird one wih linerly incresing requency. The mpliude o he signls is moduled so h hey resemble 6 s pulses in srucurl response. The purpose o hese exmples is o es he codes, nd o ind ou how close he esimes o insnneous requency re o he exc vlues, nd how well hese mehods cn loclize jump in requency. 4. The Signls All he hree es signl hve N=6 poins, eqully smpled =. s, mpliude pered by Hnning window ( π ) ( ) w () =.5 cos / N (4.) nd uni mximum mpliude. Their nlyicl expressions re s ollows. Tes signl : s () = w ()cosπ, = Hz (4.) Tes signl : [ ] ( ] ( ) ( ] w ()cos π,, 4,6 s () = w ()cosπ,,4, = Hz (4.3) Tes signl 3: s () w ()cos.5 = π + N, = Hz, = 3 Hz/s (4.4) Figure 4. shows he signls (op) nd he mpliude o heir Coninuous Wvele Trnsorm s n imge (cener) nd s surce (boom). The signls were genered by progrm chirp.or which cn lso dd Gussin noise o he signls. 34

41 . Signl s() s 6 T s (b,) b - s 6 T s (b,) ω = π σ = b - s Fig. 4. Tes signl, wih uniorm requency nd mpliude pered by Hnning window. I is deined by s () = w ()cosπ wih = Hz nd w ( ) =.5( cos π/ N ). The signl hs N=6 poins, smpled =. s. Top: he signl s uncion o ime. Cener: mpliude o is wvele rnsorm s n imge wih ligher color represening lrger mpliude. Boom: mpliude o is wvele rnsorm s surce. 35

42 s(). Signl s 6 T s (b,) b - s T s (b,) ω = π σ = b - s Fig. 4.b Sme s Fig. 4. bu or es signl. I hs piecewise uniorm requency nd s () = w ()cos π,, 4,6 mpliude pered by Hnning window. I is deined by [ ] ( ] nd s () = w ()cosπ( ), (,4], wih = Hz. 36

43 s(). Signl s 6 T s (b,) b - s T s (b,) ω = π σ = b - s Fig. 4.c Sme s Fig. 4. bu or es signl 3. I hs uniormly incresing requency nd s ( ) = w ( ) cos π +.5 wih mpliude pered by Hnning window. I is deined by ( ) = Hz, =.5 Hz/s. 37

44 nd 3. The op prs o ll hese igures excep Fig. 4. show he signls (solid line) nd he skeleons o he Coninuous Wvele Trnsorm (dshed line) s n pproximion o he signl. The cenrl prs show he phses o he signls φ s ( ) (Fig. 4.) or he ridges r() (Figs 4.3 hrough 4.5), nd he boom prs show he insnneous requency s() esimed by he mehod (solid line or symbols) nd he exc vlue (dshed lines). Wherever he wvele rnsorm ws used, he moher wvele ws such h ω = π nd σ =, nd he shded boxes in he plos o ( ) show σ σ cells in he ime requency plne equivlen o hose shown in Fig.. (or hese exmples, σ =.63/ ) nd σ =.63 /( π). The resuls in Figs 4. hrough 4.5 were produced respecively by progrms hilb_rdg.or, mrs_rdg.or, crm_rdg.or nd dir_rdg.or. Figure 4. shows h he Hilber Trnsorm mehod is very ccure excep ner he beginning nd he end o he signl nd ner he jumps in requency or Signl (he rnsiion is grdul rher hn jump nd ( ) exhibis wek oscillions). Figure 4.3 shows h he Mrseille mehod is very ccure or he signl wih consn requency (signl ), less ccure or he signl wih grdue chnge in requency (Signl 3) nd he les ccure or he signl wih jumps in requency (Signl ). In c, or Signl, he lgorihm does no converge o he rue ridge or ll sring vlues o scle, bu converges o lrger scles (smller requencies), corresponding o he smller locl mximum in he mpliude o he wvele rnsorm (see Fig. 4.b, boom). Figure 4.4 shows h he Crmon mehod is lso very ccure or he consn requency signl (Signl ). For he signl wih jumps (Signl ), his mehod is more sble hn he Mrseille mehod. However, he depiced chnge in requency is grdul (wih spn o les one second) rher hn brup. Figure 4.5 shows h he resuls by he Simple mehod re prciclly he sme s hose by he Mrseille mehod. For such ypes o signls, he Mrseille mehod does no oer ny dvnges. Common observion in ll our igures is h, or Signl 3, he ccurcy is he poores ner he beginnings nd he ends o he signls, when he mpliudes re smll. Ner =, when = Hz, he error is lrger hn wihin σ. s s σ, while ner = 6 s, when = 3 i is 38

45 s () - Hz s() Hilber Trnsorm Mehod Signl Φ s () - rd exc s Fig. 4. Tes signl (op), nd is phse (cener) nd insnneous requency (boom) deermined by he Hilber Trnsorm mehod. 39

46 s () - Hz s() Hilber Trnsorm Mehod Signl Φ s () - rd exc s Fig. 4.b Tes signl (op), nd is phse (cener) nd insnneous requency (boom) deermined by he Hilber Trnsorm mehod. 4

47 s () - Hz s() Hilber Trnsorm Mehod Signl Φ s () - rd exc s Fig. 4.c Tes signl 3 (op), nd is phse (cener) nd insnneous requency (boom) deermined by he Hilber Trnsorm mehod. 4

48 s () - Hz r () s() "Mrseille" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.3 Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Mrseille mehod. 4

49 s () - Hz r () s() "Mrseille" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.3b Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Mrseille mehod. 43

50 s () - Hz r () s() "Mrseille" Mehod Signl 3. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.3c Tes signl 3 (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Mrseille mehod. 44

51 s () - Hz r () s() "Crmon" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.4 Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Crmon mehod. 45

52 s () - Hz r () s() "Crmon" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.4b Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Crmon mehod. 46

53 s () - Hz r () s() "Crmon" Mehod Signl 3. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.4c Tes signl 3 (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Crmon mehod. 47

54 s () - Hz r () s() "Simple" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.5 Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Simple mehod. 48

55 s () - Hz r () s() "Simple" Mehod Signl. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.5b Tes signl (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Simple mehod. 49

56 s () - Hz r () s() "Simple" Mehod Signl 3. 6NHOHWRQ ω = π σ = σw [σi exc s Fig. 4.5c Tes signl 3 (op), nd is ridge (cener) nd insnneous requency (boom) deermined by he Simple mehod. 5

57 s () - Hz s() "Crmon" mehod Signl 3 s()+n() s() T s (b,) b -s 6 "Simple" mehod "Crmon" mehod exc σw [σi ω = π n(): µ n =, σ n =. σ = s Fig. 4.6 Tes signl 3 wih uniormly incresing requency nd uni mpliude pered by Hnning window (see Fig.; 4.c), nd wih ddiive Gussin noise (wih zero men nd sndrd deviion.). Top: he signl s uncion o ime. Cener: mpliude o is wvele rnsorm s n imge wih ligher color represening lrger mpliude. Boom: is insnneous requency deermined by he Crmon mehod (symbols) nd by he simple mehod (solid line). 5

58 4.3 Insnneous Frequency or Noisy Signls Figure 4.6 shows resuls or Signl 3 wih ddiive Gussin noise, wih men zero nd sndrd deviion. (his implies rio beween he mpliude o he signl nd sigm o he noise equl o 5). The op pr o he igure shows he noisy signl, s well s he signl wihou noise, or reerence. The plo in he cener shows he modulus o he wvele rnsorm o he noisy signl, nd he plo in he boom shows he insnneous requency esimed by he Crmon mehod (symbols) nd by he Simple mehod (solid line). The prmeers λ nd µ or he simuled nneling scheme were se o λ =. nd µ =., he mximum number o ierions ws se o million, he sopping crierion ws se so h he objecive uncion should no chnge in 5 consecuive seps beore sopping. Also, he wvele rnsorm ws subsmpled so h every h poin in ime ws considered in he serch, nd i ws evlued or vlues o scle beween. nd.. Finer ime-scle grid requires much more ierion seps. The resul or his noisy signl shows h he insnneous requency is esimed wih similr ccurcy s or he signl wihou noise. Similr conclusion cn be drwn even or signl o noise rio much smller hn, e.g. /3 (no included in his repor). 5