THE SECOND-ORDER WAVELET SYNCHROSQUEEZING TRANSFORM. T. Oberlin, S. Meignen,

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1 THE SECOND-ODE WAVELET SYNCHOSQUEEZING TANSFOM T. Oberlin, S. Meignen, INP ENSEEIHT nd IIT, Universiy o Toulouse, Frnce Lboroire Jen Kunzmnn, Universiy o Grenoble, Frnce ABSTACT The pper dels wih he problem o represening nonsionry signls joinly in ime nd requency. We use he rmework o ressignmen mehods, h chieve shrp nd compc represenions. More precisely, we inroduce n enhnced version o he synchrosqueezed wvele rnsorm, which is shown o be more generl hn he sndrd synchrosqueezing, while remining inverible. Numericl experimens mesure he improvemen brough bou by using our new echnique on synheic d, while n nlysis o he grviionl wve signl recenly observed hrough he LIGO inereromeer pplies he mehod on rel dse. Index Terms synchrosqueezing; coninuous wvele rnsorm; mulicomponen signls; ime-requency; AM/FM; chirp deecion; grviionl wves. INTODUCTION For decdes, ime-requency (TF) nlysis hs imed designing new echniques or nlyzing non-sionry signls, i.e. whose requency vries cross ime. For insnce, he Shor-Time Fourier Trnsorm (STFT), he Coninuous Wvele Trnsorm (CWT) or he Wigner-Ville disribuion [, 2, 3] hve been inensively used, or insnce, in music or speech nlysis, geophysics, or elecricl engineering. STFT nd CWT, being he simples o liner TF represenions, re lso he mos populr. However, hey suer rom undmenl limiion: he Heisenberg-Gbor unceriny principle, which limis heir TF resoluion. To overcome his nd improve heir biliy o represen non-sionry signls, pioneer work proposed in he sevenies pos-processing echnique o shrpen TF represenions [4], which ws hen exended in [5] nd here nmed ressignmen (M). A very similr pproch ollowed in [6, 7] nd is known s synchrosqueezing (SST). The nice heoreicl resuls sed in [7] gve new impulse o he ield, nd mny new developmens round SST or M hve been crried ou, over hese ls yers; we cn menion he use o redundncy vi muliper pproches [8, 9] or wvele pckes [], he generlizion o imges [, 2], The uhors cknowledge he suppor o he French Agence Nionle de l echerche (AN) under reerence AN-3-BS3-2- (ASTES). nd he pplicion o non-hrmonic wves []. This led o new ineresing pplicions s, or insnce, in bio-medicl engineering [3] or r invesigion [4]. Ye, while boh SST nd M re similr echniques, heir im is quie dieren. The SST provides n inverible represenion similr o STFT or CWT, shrp or ny superposiion o moduled wves, s soon s he modulion remins negligible [7]. Insed, M provides only shrpened TF represenion, even or lrge requency modulion [5], bu no reconsrucion procedure. To generlize SST o non-negligible requency modulions, he uhors o his pper recenly proposed o improve he deiniion o he insnneous requency (IF) esime in he STFT seing [6]. The resuling rnsormion, clled second-order SST (SST2), ws deeply nlyzed in [7]. The im o his presen pper is o exend SST2 o he CWT seing, nd show, empiriclly, he ineres o his new rnsormion. To his end, we irs se some noion nd deiniions in Secion 2, beore inroducing our new SST in Secion 3. The numericl experimens presened in Secion 4 show he ineres o he new rnsormion on synheic nd rel d. 2. DEFINITIONS 2.. Coninuous wvele rnsorm We denoe by L () nd L 2 () he spce o inegrble, nd squre inegrble uncions. Consider signl L (), nd window g in he Schwrz clss, S(), he spce o smooh uncions wih s decying derivives o ny order; is Fourier rnsorm is deined by: (ξ) = F{}(ξ) = (τ)e ξτ dτ. () Le us consider n dmissible wvele ψ L 2 (), sisying < C ψ = 2 dξ ψ(ξ) ξ <. For ny ime nd scle >, he coninuous wvele rnsorm (CWT) o is deined by: W ψ (, ) = ( ) τ (τ)ψ dτ, (2) where z denoes he complex conjuge o z. We urher ssume h ψ is nlyic, i.e. Supp( ψ) [, [, nd h

2 sup ξ ˆψ(ξ) =. I rel-vlued, CWT dmis he ollowing synhesis ormul (Morle ormul): { } () = 2 W ψ (, )d, (3) C ψ where denoes he rel pr o complex number nd C ψ = ψ (ξ) dξ ξ Mulicomponen signl In his presen pper, we nlyze so-clled mulicomponen signls o he orm, () = K k (), wih k () = A k ()e φk(), (4) k= or some K, where A k () nd φ k () re uncions sisying A k () >, φ k () > nd φ k+ () > φ k () or ny nd k. In he ollowing, φ k () is oen clled insnneous requency (IF) o mode k nd A k () is insnneous mpliude (IA). One o he gol o TF nlysis is o recover he insnneous requencies {φ k ()} k K nd mpliudes {A k ()} k K, rom given signl. Noe h wih he nlyic wveles, considering complex modes or heir rel pr k () = A k () cos(2πφ k ()) is equivlen, s soon s φ () is lrge enough. The CWT o mulicomponen signl is known o exhibi ridge srucure: he inormion is concenred round ridges deined by φ k () = essignmen nd Synchrosqueezing A powerul pos-processing echnique ws inroduced in [5] o shrpen he sclogrm, ermed ressignmen mehod (M). I needs he so-clled ressignmen operors, deined wherever W (, ) by (ˆω, ˆτ ) = (( ω ), ( τ )) wih ω (, ) = τ (, ) = W (, ) W (, ) τ(τ) ψ ( τ W (, ) ) dτ. These operors loclly deine n IF nd group dely (GD), nd hey esime he posiion o ridge i i is close enough. Then, M consiss in moving he coeiciens o he sclogrm ccording o he mp (, ) (ˆω (, ), ˆτ (, )). We know, rom [5], h M perecly loclizes liner chirps wih consn mpliude. Alernively, SST ressigns he coeiciens o he CWT in he ime-scle plne ccording o he mp (, ) (ˆω (, ), ) [7]: T (, ) = 2 { C ψ W ψ {b, W ψ (b,) γ} (b, )δ(ˆω (, ) b) db b (5) }, (6) nd hen reconsrucion o k is perormed hrough: k () = T (, )d. φ k () <d (7) Since he (complex) coeiciens re moved only long he scle xis, he rnsorm remins inverible using ormul (3). Bu, since GD is ignored, he mehod cnno hndle lrge requency modulions: he perec loclizion propery is only ensured or purely hrmonic wves [6, 7]. 3. SECOND-ODE SYNCHOSQUEEZING On one hnd, M provides nice represenion or wide rnge o AM/FM or mulicomponen wves. On he oher hnd, SST is inverible, bu only suible or low-moduled wves ( φ () φ ()). SST2, inroduced in [6] in he STFT conex, ims o combine boh properies, by improving IF esime ˆω. 3.. An improved insnneous requency To sr wih, we loclly esime he requency modulion φ (), by mens o he ollowing operor: q (, ) = ω (, ) τ (, ), (8) which enbles us o deine new complex esime: { ω (2) (, ) = ω (, ) + q (, )( τ (, )) i (, ) ω (, ) oherwise, (9) nd hen ˆω (2) (, ) = ( ω(2) (, )), s new IF esime. The new synchrosqueezing operor is hen deined by replcing ˆω by ˆω (2) in (6). Noe h he deiniion is very similr in he STFT conex [6] Compuion CWT is compued scle by scle in he Fourier domin, hnks o he Plncherel heorem: (, ) := W ψ (, ) = ˆ(ξ) ˆψ(ξ) e ξ dξ. () We denoe by, W ξ ˆψ nd he CWTs corresponding, in he Fourier domin, o wveles ˆψ, ξ ˆψ nd ( ˆψ). We esily ge: (, ) = W ξ ˆψ (, ) () τ (, ) ˆψ (, ) = W (, ) + (, ).

3 Then, he ressignmen operors (5) cn be wrien: ω (, ) = W ξ ˆψ (, ) (, ) τ (, ) = + (, ) (, ). (2) By diereniing nd using equion () gin, we inlly obined where W ξ ˆψ q = 2 nd W ξ2 ˆψ W ξ 2 ˆψ + W ξ ˆψ ξ ˆψ (W )2 W ξ ˆψ, (3) denoe he CWTs compued using he wveles ξ ξ( ˆψ) nd ξ ξ 2 ˆψ, he vribles (, ) being omied o lighen he noion. 4. NUMEICAL EXPEIMENTS 4.. epresenion nd decomposiion o synheic signl We consider synheic 3-modes mulicomponen signl, mde o pure wve, n exponenil chirp nd n hyperbolic chirp, whose insnneous requencies re respecively consn, exponenil (φ () φ ()) nd hyperbolic (φ () φ () 2 ). CWT nd SST o our es-signl re displyed in Figure ; we cn clerly see on he represenion o SST he 3 ridges corresponding o he modes, bu he corresponding coeiciens re spred ou round hese ridges. This is becuse, excep or pure hrmonic wves, he requency modulion is no negligible nd should be ken ino ccoun in IF esimion A second-order SST We erm our new SST second-order, becuse i mnges o perecly loclize liner chirps: Theorem 3.. Le h() = A()e φ(), wih A() = Ae P (), where A >, P () > nd P nd φ re secondorder polynomils. Then we hve, wherever W h (, ), ˆω (2) (, ) = φ (). (4) Proo. We hve he ollowing expnsion or ny, τ : h(τ) = h()e (φ () P ())(τ )+(iπφ () P () 2 )(τ ) 2. (5) Then, we cn wrie W h (, ) = h() e (iπφ () P () ( )τ 2 τ ) 2 ψ e (φ () P ())τ dτ W h (, ) = [ φ () P () ] W h (, ) + [ φ () P () ] W h (, )( τ (, ) ). We inlly ge [ ω (, ) = φ () P ] [ () + φ () P ] () ( τ (, ) ). Dierening ech side wih respec o, we immediely ge: q (, ) = φ () P (), rom which we deduce, φ () = ( ω (, ) + q (, )( τ (, )) = ˆω (2) (, ) emrk: In [7], we proved sronger resul or he STFT, showing h he good loclizion propery sill holds or qusi-liner chirps, i.e. when we hve φ (). A similr resul should be vilble in he CWT conex or modes sisying φ () φ (), bu his is le or uure invesigions Fig.. CWT nd SST o he synheic signl. Now we consider noisy version o his es-signl (wih inpu SN = db), nd compre, in he ollowing Figure 2, he represenions given by SST, SST2 nd M. I is cler h boh SST2 nd M chieve compc represenion, lhough i is corruped by non-negligible noise. Now, i we reconsruc he dieren modes rom he synchrosqueezed represenions, we jus need o sum he vericl coeiciens round ridge, s expressed in (7). I we ke 3 coeiciens per ridge (prmeer d in (7)) nd per ime smple, we end up wih n ccurcy o 4 db or SST, nd 2 db or our SST2. To beer ssess he superioriy o SST2, we now compre he obined resuls wih he idel ime-requency represenion, by mens o he Erh Mover Disnce (EMD), s done in [8]. The EMD is sliced Wssersein disnce, commonly used in opiml rnspor, which llows or he comprison o wo disribuions. In Figure 3, we show EMD compued or he exponenil chirp o Figure, dieren inpu SNs nd dieren represenions (he lower he EMD, he beer). We see h he ressigned represenion (M) chieves he bes perormnce whever he inpu SN, nd lso h SST2 is relively close. In conrs, he represenion given by SST is quie poor. The improvemen brough bou by SST2 or M is priculrly signiicn low noise levels, bu remins whever he inpu SN.

4 I we reconsruc he chirp using boh represenions, we observe h SST2 gives resul closer o he expeced one h SST does: he SN is 6.45 db or SST nd 7.8 db or SST2. Time curves displyed in Figure 4 lso show h he ringdown pr o he chirp is beer recovered using SST2. CWT SST SST2 M Fig. 2. Time-requency represenions o he noisy synheic signl, wih inpu SN = db SST SST2 EMD CWT WSST WSST2 WM inpu SN WSST WSST2 numericl reliviy Fig. 3. EMD s uncion o Inpu SN, or dieren represenions Applicion on rel d We now illusre n pplicion o our echnique on he grviionl wve signl recorded ls yer [9], which hd considerble impc in he scieniic communiy. The signl recorded by he LIGO inereromeer is ypiclly chirp wih lrge requency modulion, or which he SST2 clerly improves he resul. We consider he pre-processed signl rom Hnord inereromeer, vilble online hps://losc.ligo.org/s/evens/ GW594/P594/ig-wveorm-H.x. We disply, in Figure 4, is TF represenions given by SST nd SST2, obined wih he complex Morle wvele wih σ =.8. Boh represenions show nice ridge wih incresing IF, bu SST2 is more concenred. We lso show he sme represenions, wih he esimed ridge superimposed. The resuls dier signiicnly, since only SST2 llows us o recover he complee ringdown, which is he pr o he signl emied er he usion o he wo blckholes ime (s) Fig. 4. From op o boom : TF represenion o he Hnord signl, he esimed ridge, nd he reconsrucion, or SST (le) nd SST2 (righ). 5. CONCLUSION We inroduced in his pper he second order SST, which is he counerpr or wveles o he STFT-bsed SST2 deined in [6]. Similr o he STFT conex, he new rnsormion ws shown o perecly loclize liner chirps, nd o remin eicien when he requency modulion is qusi-liner. Numericl experimens illusred he benei o SST2 over sndrd synchrosqueezing, nd reclled he min dierences beween STFT nd CWT or nlyzing mulicomponen signls. Fuure works should build heoreicl nlysis o SST2, o exend he resuls o [7]. On n pplicive poin o view, he nex sep would be o combine he mehod inroduced here wih oher recen improvemens o SST [9, ], which should led o signiicnly beer perormnce.

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