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Synchrosqueezing transforms: From low- to

high-frequency modulations and perspectives

Sylvain Meignen, Thomas Oberlin, Duong-Hung Pham

To cite this version:

Sylvain Meignen, Thomas Oberlin, Duong-Hung Pham. Synchrosqueezing transforms: From low- to

high-frequency modulations and perspectives. Comptes Rendus Physique, Centre Mersenne, 2019, 20

(5), pp.449-460. �10.1016/j.crhy.2019.07.001�. �hal-02573644�

a publisher's

https://oatao.univ-toulouse.fr/26560

https://doi.org/10.1016/j.crhy.2019.07.001

Meignen, Sylvain and Oberlin, Thomas and Pham, Duong-Hung Synchrosqueezing transforms: From low- to

high-frequency modulations and perspectives. (2019) Comptes Rendus Physique, 20 (5). 449-460. ISSN 1631-0705

C. R. Physique 20 (2019) 449–460

Contents lists available atScienceDirect

Comptes

Rendus

Physique

www.sciencedirect.com

Fourier and the science of today / Fourier et la science d’aujourd’hui

Synchrosqueezing

transforms:

From

low- to

high-frequency

modulations

and

perspectives

La

transformée

synchrosqueezée,

adaptation

aux

signaux

fortement

modulés

et

perspectives

Sylvain Meignen

a

,

∗

,

Thomas Oberlin

b

,

Duong-Hung Pham

c

a_{LaboratoireLJK,bâtimentIMAG,UniversitéGrenobleAlpes,700,avenueCentrale,CampusdeSaint-Martin-d’Hères,38401Domaine}

universitairedeSaint-Martin-d’Hères,France

b_{IRIT,ToulouseINP–ENSEEIHT,2,rueCharles-Camichel,BP7122,31071Toulousecedex7,France}

c_{ICube,UMR7357,Laboratoiredessciencesdel’ingènieur,del’informatiqueetdel’imagerie,300,bdSébastien-Brant,CS10413,67412Illkirch}

cedex,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline11July2019

Keywords: Time-frequencyanalysis Multicomponentsignals Reassignmenttechniques Mots-clés : Analysetemps-fréquence Signauxmulticomposantes Techniquesderéallocation

Thegeneralaimofthispaperistointroducetheconceptofsynchrosqueezingtransforms (SSTs) that was developed tosharpen lineartime–frequency representations (TFRs), like the short-time Fourier or the continuous wavelet transforms, in such a way that the sharpenedtransformsremaininvertible.Thispropertyisofparamountimportancewhen oneseekstorecoverthemodesofamulticomponentsignal(MCS),corresponding tothe superimpositionofAM/FMmodes,amodeloftenusedinmanypracticalsituations.After havingrecalledthebasicprinciplesofSSTandexplainedwhy,whenappliedtoanMCS,it workswellonlywhenthemodesmakingupthesignalareslightlymodulated,wefocus on howto circumvent thislimitation. We then give illustrations inpractical situations eitherassociatedwithgravitationalwavesignalsormodeswithfastoscillatingfrequencies anddiscusshowSSTcanbeusedinconjunctionwithademodulationoperator,extending existingresultsinthatmatter.Finally,welistaseriesofdifferentperspectivesshowingthe interestofSSTforthesignalprocessingcommunity.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r

é

s

u

m

é

Dans cet article, nous présentons le principe de la transformée synchrosqueezée (SST) développée pour améliorer, en utilisant des techniques de réallocation, la qualité des représentations linéaires temps–fréquence, comme les transformées de Fourier à court terme ou en ondelettes. Les transformées réallouées demeurent inversibles, ce qui est fondamental pour l’étude des signaux multicomposantes(MCS), i.e. lasuperposition de modes modulés à la fois en amplitude et en fréquence, utilisés comme modèle dans de nombreuses applications. Cependant, la SST, dans sa formulation initiale, n’est pas adaptée auxsignaux composés de modesfortement modulés,etdes améliorations,que nous présentonsici, ontrécemment étéproposées pour pallier cedéfaut. Pourillustrer

*

Correspondingauthor.

E-mailaddresses:sylvain.meignen@univ-grenoble-alpes.fr(S. Meignen),thomas.oberlin@enseeiht.fr(T. Oberlin),dhpham@unistra.fr(D.-H. Pham).

https://doi.org/10.1016/j.crhy.2019.07.001

1631-0705/©2019Académiedessciences.PublishedbyElsevierMassonSAS.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

cesnouveaux développements,nousétudieronslecasdesondesgravitationnellesetdes modes à phase oscillante. Dans un second temps, nous montrerons comment utiliser conjointementlaSSTetladémodulationpouraméliorerlareconstructiondesmodesd’un MCS,ettermineronsenévoquantdifférentesperspectivesactuellesautourdelaSST.

©2019Académiedessciences.PublishedbyElsevierMassonSAS.Thisisanopenaccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Sinceitsfoundation,thefieldofsignalandimageprocessinghasalwaysmadeintensiveuseoffrequencyrepresentations, and today,Fourieranalysis, ormoregenerally speakingharmonic analysis, isthe fundamental tool to analyzeorprocess signalsorimages[1].ManyreasonscanexplainwhytheuseofFouriertheoryissowidespreadinsidethesignalprocessing community. First, the notion of frequency is essential to describe oscillatory signals, as for instance sounds.Second, the Fourier transform(FT)isintrinsically relatedtolinear time-invariantfiltersoftenused tomodel theresponse ofphysical systems or sensors. Last, when considering random measurements, FT allows usto easily model or simulate stationary Gaussian random fields.Inthispaper, we willparticularlyfocus onthe firstpoint, thatis todesign afrequency analysis framework,butadaptedtonon-stationary signals.

Indeed,themainlimitationofFTisthatitdoesnotlocateintimethefrequencyinformationcomputedonagivensignal, while mostsignalsfromthephysicalworld areintrinsically non-stationary,inthesense that theirfrequencies varyalong time. Analyzingorprocessingthesesignalsefficientlythusrequirestheuseoflocal transformations,calledtime–frequency representations (TFRs), among whichthe mostpopular isprobablythe short-timeFouriertransform (STFT), whichbasically windows the signal around the time ofinterest before applying FT.Note that thewindow cannot be of arbitrarylength since,accordingtotheHeisenberg–Gaboruncertaintyprinciple[1],asmalltemporalwindow(associated withagoodtime localization) leads to a bad frequencyresolution, andvice-versa.Another popularTFR is thecontinuouswavelettransform

(CWT)[2],whichsharesmanypropertieswithSTFT,butisbasedonadifferentfrequencyresolution.Whilebothtransforms are invertible and allow one to process non-stationary signals, they suffer from a strong limitation, that is, the time– frequency(TF)resolutionisconstrainedbythechoice ofwindow orwavelet, limitingboththeadaptivityoftheproposed analysisandthereadabilityoftheTFR.

Many worksinthe pastdecades havetackledthislimitation, by using,forinstance,quadraticTFRs, e.g., Wigner–Ville distributions [3], which are not constrained by the uncertainty principle, but exhibit strong interference hampering the representation andare not invertible.Anotherattempt, called thereassignmentmethod (RM), datingback to the workby Kodera et al. [4] in the 1970s and then further developed in [5], essentially proposed a means of improving the TFR readability. Unfortunately,thereassignedrepresentationwasalsonolongerinvertible.Thisinparticularmeansthat,when applied totheTFR ofamulticomponent signal(MCS)madeofAM/FMmodes,RMdoesnotallowforan easyretrieval of thecomponentsoftheMCS.Otherworks,liketheEmpiricalModeDecomposition(EMD)[6],preciselyfocusedonthislatter aspectandconsistsofasimplealgorithmtoadaptivelydecomposean MCSintomodulatedAM/FMwaves.Whileitproved to beinterestinginmanypracticalapplications,itlacks mathematicalfoundationsandbehaves likeafilterbank resulting in mode mixing [7]. To partially overcome thesedrawbacks, recentworks tried to mimic EMD butusing a more stable framework,eitherbasedonwavelettransforms[8] orconvexoptimization[9,10].

Synchrosqueezingtransform(SST)isyetanotherapproachwhoseinitialgoalwastoimprovethereadabilityoftheTFR givenby CWT[11] using areassignmentprocedure, insucha waythat thereassignedtransformwasstill invertible.This has theniceconsequencethat theretrievalofthe modesofan MCSisthen possiblefromthereassignedtransform[12], whichisofgreatpracticalinterest.Indeed,real-lifesignalsareoftenmodeledby meansofMCS, foundinawiderangeof applicationssuchasaudiorecordings,structuralstability[13,14],orphysiologicalsignals[15,16].Because,intheseinstances, themodesareessentiallyAM/FM,TFRsarewelladaptedtorepresentsuchnon-stationarysignalsandSSTprovedtobean efficienttechniquetoobtainimprovedTFRswhileallowingformoderetrieval.

SST has recently been extended in many ways: it was adapted to STFT, known asSTFT-basedSST (FSST) in [17], for whichmathematicalanalysesareavailablein[18–20],abidimensionalextensionwasproposedbymeansofthemonogenic wavelettransform[21,22],anda multivariateextension adaptedto quasi-circularmodulatedsignalswas alsoproposed in the context ofbrainelectrical recordings[23]. Inanother direction,a multi-taperapproach was proposed toincrease the resolution andtherobustness ofSST,asintheConceFT method[24], andthecaseofnon-harmonicwavesinvestigatedin [25].Inspiteofalltheseadvances,onemajorproblemassociatedwithSSTinitsoriginalformulationisthatitcannotdeal withMCSscontainingmodeswithstrongfrequencymodulation,whichareverycommoninmanyfieldsofpracticalinterest, asfor instanceradar[26],speechprocessing [27],gravitationalwaves [28,29], orinthe analysisofotoacousticemissions [30].Inthisregard,an adaptionofFSSTtobetter handlethattype ofsignals, knownasthe second-ordersynchrosqueezing transform (FSST2), was introduced in[20], andits theoreticalfoundations settled in[31]. Such anewtransform was also usedinademodulationalgorithm[32] andextendedtobetterdealwithmodeswithfastoscillatingphase[33].

S. Meignen et al. / C. R. Physique 20 (2019) 449–460 451

Inthis paper,we willbriefly recallsome usefuldefinitions inSection 2, andthen thoseof SSTin theSTFTandCWT settingsinSection3,beforeputtingtheemphasisonthedifferentbehaviorsofthesetwotransforms.Then,wewillpresent anoverviewofrecentdevelopmentsonSST,includingthoseonhigher-orderSST[31,33] andonthedesignofthe demodu-lationalgorithmintheSSTcontext[32].Then, wewillgivedetails onpracticalimplementation,anaspectoftenleftapart intheliterature.

2. Background

Considerasignal f

∈

L1

_(R)

_{,itsFouriertransform correspondsto:}



f

(ξ )

=

F

{

f

}(ξ) =



R

f

(

t

)

e−2πiξtdt (1)

anditsshort-timeFouriertransform (STFT)isdefinedusinganyslidingwindow g

∈

L∞

(R)

by: Vg_f

(

t

, ξ )

=



R f

(

τ

)

g

(

τ

−

t

)

e−2πiξ(τ−t)d

τ

=



R f

(

t

+

τ

)

g

(

τ

)

e−2πiξτd

τ

(2)

If f ,

ˆ

f , g andg are

ˆ

allinL1

_(R)

_{, f canbereconstructedfromitsSTFTassoonasg isnon-zeroat0:}

f

(

t

)

=

1 g

(

0

)



R

Vg_f

(

t

, ξ )

d

ξ

(3)

where Z denotesthecomplexconjugateofZ .

Inthispaper,wewillintensivelystudymulticomponent signals(MCSs)definedasasuperimpositionofAM/FM compo-nentsormodes: f

(

t

)

=

K



k=1 fk

(

t

)

with fk

(

t

)

=

Ak

(

t

)

e2πiφk(t) (4)

forsomefiniteK

∈ N

, Ak

(

t

)

and

φ

k

(

t

)

beingrespectivelytheinstantaneousamplitude(IA)andfrequency(IF) fksatisfying: Ak

(

t

)

>

0,

φ

_k

(

t

)

>

0 and

φ

_k₊₁

(

t

)

> φ

_k

(

t

)

forallt.SuchasignaladmitsanidealTF(ITF)representationdefinedas:

TIf

(

t

,

ω

)

=

K



k=1 Ak

(

t

)δ



ω

− φ

k

(

t

)



(5)

where

δ

denotestheDiracdistribution.Inpractice,theIFofthemodescannotbe recoveredby estimatingthe IFof f as

isdoneinthetheoryofanalyticalsignals, andtolocatethemodesintheTFplaneisessentialbeforecomputingtheIFof eachmode:thisisoneofthegoalsofSST,whoseconstructionisrecalledhereafter.

3. Thesynchrosqueezingtransform

SSTpursuestwomainobjectives:firsttosharpenagivenlinearTFR,andthen,whenappliedtoanMCS,toseparateand retrievethemodes.The transformisbasedonan estimation,fromtheTFR,oftheIFofeachmode,then usedtosharpen theenergyoftherepresentationaroundso-calledridges,approximatingthecurves

(

t

,

φ

_k

(

t

))

.SSToperatingonalinearTFR, e.g.,STFTorCWT,alsoallowsformoderetrievalexploitingthesynthesisformula(3),aswillberecalledlater.SSTwasfirst introducedin[11,12] intheCWTsetting,andthenadaptedtoSTFTin[18–20].Inthispaper,wemainlyfocusonSTFT-based SST,andwewillthereforeonlymention,whennecessary,theonebasedonCWT.

3.1. STFT-basedsynchrosqueezingtransform

STFTbasedsynchrosqueezing (denoted byFSST inthe sequel)is basedonthe definitionofthe localinstantaneous fre-quencyestimate



ω

f [5]:



ω

f

(

t

, ξ )

=

∂

arg Vg_f

(

t

, ξ )

∂

t

= 



1 2

π

i

∂

tVg_f

(

t

, ξ )

Vg_f

(

t

, ξ )



,

wherever Vg_f

(

t

, ξ )

=

0 (6)

The FSST of f with threshold

γ

is then obtained by moving anycoefficient Vg_f

(

t

,

ξ )

with magnitude larger than

γ

to location

(

t

,



ω

f

(

t

,

ξ ))

.Thiscanbeformallywrittenas:

Tγ_f

(

t

,

ω

)

=



|Vg_f(t,ξ )|>γ

Vg_f

(

t

, ξ )δ(

ω

− 

ω

f

(

t

, ξ ))

d

ξ

(7)

Any mode fk canthenbe reconstructedbysummingFSST coefficientsaround thekth ridge, whichamountstomodifying

thesynthesisformula(3) toselectonlythecoefficientsrelatedtothekth mode,namely:

fk

(

t

)

≈

1 g

(

0

)



|ω−ϕk(t)|<d Tγ_f

(

t

,

ω

)

d

ω

(8)

where

ϕ

k isanestimateof

φ

_k,andparameterd isusedtocompensateforerrorsinIFestimation;itshouldbekeptsmall

enoughtoavoidmode-mixing,anditsinfluencewillbediscussedlater.

3.2. Wavelet-basedsynchrosqueezingtransform

Let

ψ

beawavelet;wedefine,attimet andscalea

>

0,thecontinuouswavelettransform (CWT)of f by:

Wψ_f

(

t

,

a

)

=

1 a



R f

(

τ

)ψ

τ

−

t a

d

τ

(9)

If f and

ˆ

f areinL1

(R)

and f or

ψ

isanalytic (whichmeans,notcontaining anynegative frequency),andassumingthat

C_ψ

=



₀+∞

ψ(ξ )



d_ξξ

<

∞

,thewavelet-basedSST(WSST)localinstantaneousfrequencyestimateisdefinedby:



ω

f

(

t

,

a

)

= 

⎧

⎨

⎩

1 2

π

i

∂

tWψ_f

(

t

,

a

)

Wψ_f

(

t

,

a

)

⎫

⎬

⎭

(10)

WSSTthenconsistsinverticallymovingthecoefficientsaccordingtothemap

(

t

,

a

)

→ (

t

,



ω

f

(

t

,

a

))

:

Sγ_f

(

t

,

ω

)

=



|Wψ_f(t,a)|>γ Wψ_f

(

t

,

a

)δ



ω

− 

ω

f

(

t

,

a

)



da a (11)

Thekth modecanthenbeapproximatelyreconstructedbyintegratingSγ_f

(

t

,

ω

)

alongthefrequencyaxisasfollows: fk

(

t

)

≈

1 C_ψ



|ω−ϕk(t)|<d Sγ_f

(

t

,

ω

)

d

ω

(12)

3.3. ApproximationresultsforFSSTandWSST

As pointedout intheintroduction,thesuccessofSSTispartlyduetothenicetheoreticalresultspioneeredin[12],of whichweheregiveaflavor(referringthereaderto[12] and[20] fordetailsonWSSTandFSST,respectively).Theseresults considerMCSsasinequation(4),assumingthat:

(i) Each fkisaperturbationofapurewave,i.e.itsIAAkandIF

φ

_k areslowvarying.

(ii) Thedifferentmodesareseparated intheTFplane,i.e.foreachtimet

φ

_k

(

t

)

fork

=

1

· · ·

K aresignificantlydifferent. EventhoughFSSTandWSSTareverysimilarinnature,theybehavewellondifferenttypesofMCSs.Indeed,whileboth techniquesenableaperfectrepresentationandreconstructionofpurelyharmonicmodes[12,20],thisisnolongerthecase whenthemodesaremodulated.

The main difference between the two transforms is related to the frequency resolution: STFT has a fixed frequency resolution,whereastheresolutionofCWTdependsonthescale(orfrequency).ThustheslowvariationforA_k

(

t

)

and

φ

_k

(

t

)

andseparationassumptionsfortheIFofthemodesintheCWTsettingarerelativetotheIF

φ

_k,i.e.

|

A_k

(

t

)

|

,

|φ

_k

(

t

)

|

 φ

_k

(

t

)

and φk+1(t)−φk(t)

φ_k₊₁(t)+φ

k(t)

≥ 

(



being thefrequency bandwidth ofthe wavelet), respectively, whereas inthe STFT settingthese are constantallover theTFplane,i.e.

|

A_k

(

t

)

|

,

|φ

_k

(

t

)

|



1 and

φ

_k₊₁

(

t

)

− φ

_k

(

t

)

≥

2



(



beingthefrequencybandwidthof the window g), respectively. Asa firstsimple exampleto illustratethesedifferences, one mayconsidera signal madeof two parallellinearchirps(i.e.withlinearIFs),forwhichtheseparationassumption ismorestringentathighthanatlow frequencieswhenusingWSSTbecauseofthedivisionby

φ

₂

(

t

)

+ φ

₁

(

t

)

,butsuchisnotthecasewithFSST.

Asecondexampleisthe3-modesignalshowninFig.1,madeofapureharmonicwave,alinearchirpandahyperbolic chirp(i.e.withIFsatisfying

φ



(

t

)

=

C st

φ



(

t

)

2_{).WefirstnotethattheFSSTandWSSTofthepurelyharmonicmodearevery}

S. Meignen et al. / C. R. Physique 20 (2019) 449–460 453

Fig. 1. Illustration of STFT, CWT, FSST and WSST on a synthetic signal.

much thesame,then regardingthe representationof thelinearchirp, we observethat since the resolutionwithCWTis muchbetter atlowfrequencies, therepresentationassociatedwithWSSTis lesssharpintheseinstances (closetot

=

0), andthereverseisalsotrue,i.e.therepresentationissharperwhenthefrequencyincreases.Onthecontrary,thesharpness oftherepresentationremainsthesamewhateverthefrequencywhenoneconsidersFSST.Regardingtheexponentialchirp, since itis locatedin thehigh-frequency partofthe TFplane, thefrequencyresolution withWSST ispoor, resultingina sharprepresentation,whileforFSSTtheresolutionbeingthesamewhateverthefrequency,therepresentationbecomesless andlesssharpwhenthemodulationincreases.AlastinterestingpointregardingtheanalysisofFig.1isthattheseparation ofthehyperbolicchirp fromthelinearoneseems muchsimplerwithFSSTthanwithWSST,asaconsequenceofthefact thatthesemodesathighfrequenciesarecloserinthetime-scale(TS)thanTFplanes.

4. Higher-orderSST

4.1. Secondordersynchrosqueezingtransforms

Asjustexplained,theapplicabilityofFSSTisrestrictedtoaclassofMCSscomposedofslightlyperturbedpurelyharmonic modes.Toovercomethislimitation,anextension ofFSSTwas introducedbasedona moreaccurateIFestimate,thenused todefineanimprovedsynchrosqueezingoperator,calledsecond-orderSTFT-basedsynchrosqueezingtransform (FSST2)[34,31]. Moreprecisely,onefirstdefinesasecond-orderlocalmodulationoperator,whichisthenusedtocomputethenewIFestimate. To doso, one introduces complex reassignment operators

ω

˜

f

(

t

,

ξ )

=

∂tVgf(t,ξ )

2πiVg_f(t,ξ ) andt

˜

f

(

t

,

ξ )

=

t

−

∂ξVgf(t,ξ )

2πiVg_f(t,ξ ), andthen one definesacomplexfrequencymodulationoperatoras[34]:

˜

qf

(

t

, ξ )

=

∂

t

ω

˜

f

(

t

, ξ )

∂

tt

˜

f

(

t

, ξ )

=

∂

t

∂tVgf(t,ξ ) Vg_f(t,ξ )

2

π

i

− ∂

t

∂ξVgf(t,ξ ) Vg_f(t,ξ )

(13)

Thesecond-orderlocalmodulationoperatorthencorrespondsto





q

˜

f

(

t

, ξ )



,andthesecondordercomplexIFestimateof

f isdefinedby:

˜

ω

[_f2]

(

t

, ξ )

=



˜

ω

f

(

t

, ξ )

+ ˜

qf

(

t

, ξ )(

t

− ˜

tf

(

t

, ξ ))

if

∂

t

˜

tf

(

t

, ξ )

=

0

˜

ω

f

(

t

, ξ )

otherwise

,

(14)

andwe thenput

ω

ˆ

[_f2]

(

t

,

ξ )

= 



˜

ω

[_f2]

(

t

, ξ )



.Itwas provenin[31] that

ω

ˆ

[_f2]

(

t

,

ξ )

= φ



(

t

)

,when f isa Gaussianmodulated linearchirp.Itisalsoworthmentioningherethatq

˜

f

(

t

,

ξ )

canbecomputedbymeansoffivedifferentSTFTs.Finally,FSST2

isdefinedbyreplacing

ω

ˆ

f

(

t

,

ξ )

by

ω

ˆ

[_f2]

(

t

,

ξ )

in(7),toobtaintheso-calledT₂γ_,f,andmodereconstructionisthenperformed

byreplacingTγ_f by T₂γ_,f in(8).

Similarly,complexreassignmentoperators

ω



f

(

t

,

a

)

and



τ

f

(

t

,

a

)

intheCWTcontextarerespectivelydefined,forany

(

t

,

a

)

s.t. Wψ_f

(

t

,

a

)

=

0,by:



ω

f

(

t

,

a

)

=

1 2

π

i

∂

tWψ_f

(

t

,

a

)

Wψ_f

(

t

,

a

)

and



τ

f

(

t

,

a

)

=



R

τ

f

(

τ

)

1 a

ψ (

τ− t a

)

d

τ

Wψ_f

(

t

,

a

)

=

t

+

a Wt_fψ

(

t

,

a

)

Wψ_f

(

t

,

a

)

(15)

The second-orderlocalcomplexmodulationoperatorcorrespondsto



qf

(

t

,

a

)

=

∂tωf( t,a)

∂tτf(t,a) andthedefinitionoftheimproved

complexIFestimateassociatedwithCWTisderivedas:



ω

[_f2]

(

t

,

a

)

=





ω

f

(

t

,

a

)

+

qf

(

t

,

a

)(

t

− 

τ

f

(

t

,

a

))

if

∂

t



τ

f

(

t

,

a

)

=

0



ω

f

(

t

,

a

)

otherwise (16)

whoserealpart



ω

[_f2]

(

t

,

a

)

= {

ω

[_f2]

(

t

,

a

)

}

isthedesiredIFestimate.Itwasshownin[35] that







qf

(

t

,

a

)



= φ



₍

_t

₎

_{when f is}

aGaussianmodulatedlinearchirp,i.e. f

(

t

)

=

A

(

t

)

e2πiφ (t)wherebothlog

(

A

(

t

))

and

φ (

t

)

arequadraticandthat

{

ω

[_f2]

(

t

,

a

)

}

isanexactestimateof

φ



(

t

)

forthatkindofsignals.ForamoregeneralmodewithGaussianamplitude,its IFcanbe esti-matedby

{

ω

[_f2]

(

t

,

a

)

}

.AsforFSST2,

ω



f

(

t

,

a

)

and



qf

(

t

,

a

)

canbecomputedbymeansofonlyfiveCWTs. Thesecond-order

WSST(WSST2)isthendefinedbysimplyreplacing



ω

f

(

t

,

a

)

by



ω

[_f2]

(

t

,

a

)

in(11):

S₂γ_,f

(

t

,

ω

)

:=



|Wψ_f(t,a)|>γ Wψ_f

(

t

,

a

)δ



ω

− 

ω

[_f2]

(

t

,

a

)



_da a (17)

and fk isfinallyretrievedbyreplacing Sγ_f

(

t

,

ω

)

withSγ_2,f

(

t

,

ω

)

in(12). 4.2. Higher-ordersynchrosqueezingtransforms

DespiteFSST2definitelysharpenstheTFRitisbasedon,itisprovedtoprovideatrulysharpTFRonlyforperturbations of linearchirps withGaussian modulatedamplitudes.To handlesignalscontaining more generaltypesof AM/FMmodes having non-negligible

φ

_k(n)

(

t

)

forn

≥

3, especiallythose withfastoscillating phase, one definesnewSST operators based on third- or higher-order approximationsof both the amplitude andphase [33]. Tointroduce the technique, we restrict ourselves to the STFT context, but the technique presented hereafter could easily be extended to the CWT setting. Let

f

(

τ

)

=

A

(

τ

)

e2πiφ (τ)withA

(

τ

)

(resp.

φ (

τ

)

)beingequaltoitsLth-order(resp.Nth-order) Taylorexpansionfor

τ

closetot,

namely: log

(

A

(

τ

))

=

L



k=0

[

log

(

A

)

]

(k)

(

t

)

k

!

(

τ

−

t

)

k _and

_{φ (}

_τ

₎

₌

N



k=0

φ

(k)

(

t

)

k

!

(

τ

−

t

)

k ₍₁₈₎

where Z(k)

₍

_t

₎

_{denotesthekth derivativeofZ evaluatedatt.Suchamode,withL}

_≤

_{N,canbewrittenas:} f

(

τ

)

=

exp



_N



k=0 1 k

!



[

log

(

A

)

]

(k)

(

t

)

+

2

π

i

φ

(k)

(

t

)



(

τ

−

t

)

k



(19)

since

[

log

(

A

)

]

(k)

(

t

)

=

0 if L

+

1

≤

k

≤

N.ItscorrespondingSTFTwrites: Vg_f

(

t

, ξ )

=



R f

(

τ

+

t

)

g

(

τ

)

e−2πiξτd

τ

=



R exp



_N



k=0 1 k

!



[

log

(

A

)

]

(k)

(

t

)

+

2

π

i

φ

(k)

(

t

)



τ

k



g

(

τ

)

e−2πiξτd

τ

Bytakingthepartialderivativeof V_fg

(

t

,

ξ )

withrespecttot andthendividingby2

π

iVg_f

(

t

,

ξ )

,thelocalcomplex reassign-mentoperator

ω

˜

f

(

t

,

ξ )

definedinSection4.1canbewritten,when Vg_f

(

t

,

ξ )

=

0,as:

˜

ω

f

(

t

, ξ )

=

N



k=1 rk

(

t

)

Vt_fk−1g

(

t

, ξ )

Vg_f

(

t

, ξ )

=

1 2

π

i

[

log

(

A

)

]



₍

_t

₎

_{+ φ}



₍

_t

₎

₊



N k=2 rk

(

t

)

Vt_fk−1g

(

t

, ξ )

Vg_f

(

t

, ξ )

(20)

S. Meignen et al. / C. R. Physique 20 (2019) 449–460 455 where rk

(

t

)

=

(k−11)!

1 2

π

i

[

log

(

A

)

]

(k)

₍

_t

₎

_{+ φ}

(k)

₍

_t

₎

. It is clearthat to get an exact IF estimate for the studied signal, one

needstosubtract



⎧

⎨

⎩

N



k=2 rk

(

t

)

Vt_fk−1g

(

t

, ξ )

Vg_f

(

t

, ξ )

⎫

⎬

⎭

to





˜

ω

f

(

t

, ξ )



,whichrequiresthecalculationofrk

(

t

)

fork

=

2

,

. . . ,

N.Forthat

purpose,onederivesafrequencymodulationoperatorq

˜

[_fk,N]

(

t

,

ξ )

,equaltork

(

t

)

forthetypeofmodesjustintroduced,and

obtainedbydifferentiatingdifferentSTFTswithrespectto

ξ

(todifferentiatewithrespectto

ξ

ratherthant leadstomuch simplerexpressions).q

˜

[_fk,N]

(

t

,

ξ )

for2

≤

k

≤

N canbederivedrecursively,following[33]:

˜

q[_fN,N]

(

t

, ξ )

=

yN

(

t

, ξ )

andq

˜

[_fj,N]

(

t

, ξ )

=

yj

(

t

, ξ )

−

N



k=j+1 xk,j

(

t

, ξ )

q

˜

[fk,N]

(

t

, ξ ),

N

−

1

≥

j

≥

2 (21)

where yj

(

t

,

ξ )

andxk,j

(

t

,

ξ )

aredefinedasfollows:

y1

(

t

, ξ )

= ˜

ω

f

(

t

, ξ )

and xk,1

(

t

, ξ )

=

Vt_fk−1g

(

t

, ξ )

Vg_f

(

t

, ξ )

,

1

≤

k

≤

N

,

yj

(

t

, ξ )

=

∂

ξyj−1

(

t

, ξ )

∂

ξxj,j−1

(

t

, ξ )

and xk,j

(

t

, ξ )

=

∂

ξxk,j−1

(

t

, ξ )

∂

ξxj,j−1

(

t

, ξ )

,

2

≤

j

≤

N and j

≤

k

≤

N

ThedefinitionoftheNth-orderIFestimatethenfollows[33]:

˜

ω

[_fN]

(

t

, ξ )

=

⎧

⎪

⎨

⎪

⎩

˜

ω

f

(

t

, ξ )

+

N



k=2

˜

q[_fk,N]

(ξ,

t

)



−

xk,1

(

t

, ξ )



,

if Vg_f

(

t

, ξ )

=

0

,

and

∂

ξxj,j−1

(

t

, ξ )

=

0

,

2

≤

j

≤

N

˜

ω

f

(

t

, ξ )

otherwise

Therealpart

ω

ˆ

[_fN]

(

t

,

ξ )

= { ˜

ω

[_fN]

(

t

,

ξ )

}

isthedesiredIFestimate,whichis,byconstruction,exactfor f satisfying(19).As for FSST2,the Nth-order FSST (FSSTN) isdefined by replacing

ω

ˆ

f

(

t

,

ξ )

by

ω

ˆ

[fN]

(

t

,

ξ )

in (7) to obtain T

γ

N,f

(

t

,

ω

)

andthe

modesoftheMCScanbereconstructedbyreplacingTγ_f

(

t

,

ω

)

by Tγ_N,f

(

t

,

ω

)

in(8).

4.3. Illustration

Toillustrate the benefits broughtby higherorder FSSTs,we consider a synthetic mono-component signal definedas:

f

(

t

)

=

A

(

t

)

e2πiφ (t)_withA

₍

_t

₎

₌

₁

₊

_3t2

₊

₄

₍

₁

₋

_t

₎

7_and

_{φ (}

_t

₎

₌

_240t

₋

_2exp

₍

₋

_2t

₎

_sin

₍

₁₄

_π

_t

₎

_fort

_{∈ [}

₀

_,

₁

_]

_{,whichcorrespondsto} adamped-sinefunctionsignalcontainingverystrongnonlinearsinusoidalfrequencymodulationsandhigh-orderpolynomial amplitudemodulations.Such afunctionissampledatarate N

=

1024 Hzon

[

0

,

1

]

.TheSTFTof f isthencomputedwith the L1-normalized Gaussian window g

(

t

)

=

σ

−1_e−π_σt22_,where

_σ

_{is the optimal value determined by the Rényi entropy} technique introduced in [36]. To use thisoptimal value for the window length parameter is crucial, since it allows for a good tradeoff betweentime and frequencyresolutions andalso a good mode retrieval performance [33]. In Fig. 2(a) and (b), therealpartofthesignal f superimposedwith A anditsSTFTmodulusare respectivelydisplayed. Then,inthe secondrowofFig.2,we showclose-upsofSTFTandofreassignedrepresentationsgivenby FSSTsofthefirstfourorders. ItisclearthatthehighertheorderofFSSTthesharper theTFR,especiallywhentheIFofthemodehasanon-negligible

curvature

φ



(

t

)

.

AsecondillustrationconsidersthegravitationalwavesignalGW151226[37],producedbythecollisionoftwoblackholes andrecordedby theLIGOandVirgoScientificCollaborations. Itmainlyconsistsofachirp whoseIFisincreasinguntilthe mergingphase.Thissignal hasbeenpre-processedasspecifiedintheGravitationalWave OpenScienceCenter.1 However, such pre-processing doesnot remove all the noise, which makes the chirp hardly distinguishable (Fig. 3 (b) (bottom)). We applied WSST2, which seems to effectivelysharpen the TFR given by CWT(Fig. 3 (a)). Thisenabled the retrieval of themain mode,displayed inFig.3(b)(top) alongwiththetheoretically predictedgravitationalsignal.Wecan observea goodcorrespondenceuntilt

=

0,afterwhichthesignalisnolongeran AM/FMmode.Itisworthremarkingherethatthe reassignment operationis directlyperformedon thenoisy CWTandthat no extradenoisingprocedure isneeded, which demonstratesthatSSTissomewhatrobust,eventhoughlittlehasbeendonesofartoquantifythisrobustness.

Fig. 2. Illustrationofthedifferencebetweensecond-,third-,andfourth-orderFSSTs:(a)realpartoff withA superimposed;(b)modulusoftheSTFToff ;

(c)STFTzoomofasmallTFpatch(delimitedbyaredrectangle)extractedfrom(b);(d)FSSTzoomcarriedoutontheSTFTshownin(c);from(d)to(g), sameas(d)butforFSST2,FSST3,andFSST4,respectively.

Fig. 3. Illustration of the second-order WSST2 on a gravitational wave signal.

5. Practicalimplementation 5.1. Settingsparameters

Mostofthe theoreticalworkson SSTstudiedthe continuoustime andfrequencysettings,while inpracticebothtime andfrequencyarediscretizedandthisrequiresspecialattention.InparticularandasalreadyshowninFig.1,thefrequency resolutiondirectlyimpactsthesharpnessofthereassignedtransform.Furthermore,SSTsdependontwomainparameters: thechoice ofwindoworwaveletdefiningtheTFR,andthethreshold

γ

usedinthedefinitionoftheSSTs.Althoughthere is noidealwayto determinethe windowlength, acommonpractice istoconsider thewindow (orwavelet) lengththat

S. Meignen et al. / C. R. Physique 20 (2019) 449–460 457

minimizestheRényientropyoftheconsideredTFR [38,32]. Regardingthedeterminationofthethreshold

γ

,asmall

γ

is clearlyfavoredtoavoidsomedataassociatedwiththesignaltoberemoved.

As far as mode retrieval is concerned, a critical issue is the computation of the estimation

ϕ

k of the IF of mode k,

usedin(8) andcarriedoutbyridgeextraction,performedbyfittingsmoothcurveswiththelocationsofthehighestenergy coefficientsoftheconsideredSTFT.Thisisusuallydonebyconsideringthenon-convexvariationalproblem,whoseoptimal solutioncanbeapproximatedusingstochasticsamplingorheuristictechniques[39,12,19].Thisdifficultproblemsomehow reducesthescopeofthetheoreticalresults,sincethelatterassumesthattheridgeinformationisknown.

5.2. Influenceoffrequencyresolution

We here discuss the influence of the frequencyresolution on FSST. Indeed, assume f is with finite length, typically definedontheinterval

[

0

,

T

]

,discretizedinto f

(

nT_N

)

n=0,··· ,N−1,andg supportedon

[−

LT_N

,

LT_N

]

,withL

<

N

/

2,theSTFTof f isthencomputedasfollows:

Vg_f

(

t

, ξ )

=

LT N



−LT N f

(

t

+

τ

)

g

(

τ

)

e−2πiτξd

τ

≈

T N L



n=−L f

(

t

+

nT N

)

g

(

nT N

)

e −2πinT_Nξ ₍₂₂₎

fromwhichweinferthat,for0

≤

p

≤

N

−

1: Vg_f

(

qT N

,

pN M T

)

≈

T N L



n=−L f

(

(

q

+

n

)

T N

)

g

(

nT N

)

e −2πinp_M

forsome M

≥

2L

+

1.Thelast sumiscomputedbymeansofadiscreteFT,andthefrequencyresolutionequals _{M T}N ,which hastheconsequencethattheTFRassociatedwithFSSTisallthemorecompactthatthefrequencyresolutionislow.

6. Modereconstructionforstrongmodulations

In thissection, weare interested in assessing thequality ofmode reconstruction givenby formula oftype (8), when

d

=

0 andfordifferentfrequencyresolutions,toclearly statetheimportanceofthelatterinmodereconstructionbasedon FSST2evaluatedontheridge.Furthermore,weexplainhowFSSTscanbeusedinademodulationalgorithmforthepurpose ofmodereconstruction,theprincipleofwhichisrecalledhereafterandwhoseperformancesare thencomparedtoFSST2.

6.1. DemodulationalgorithmbasedonFSSTformodereconstruction

We briefly recall how FSST (or FSST2) was combined in [32] with a demodulation algorithm for the purpose of mode reconstruction. Assume that for f

(

t

)

=

K



k=1

Ak

(

t

)

e2iπφk(t), one has computed, from FSST (or FSST2), K estimates

(

ϕ

k

(

t

))

k=1,··· ,K of the IFs ofthe modes. Then, associated with each of theseestimates,one defines K demodulation

op-eratorse−2πi(0tϕk(x)dx−ϕ0t)_{,in which}

_ϕ

0 issome positive constantfrequency.One then remarksthat, if

ϕ

k

(

t

)

is agood IF

estimateformodek, thesignal fD,k

(

t

)

=

f

(

t

)

e−2πi(

t

0ϕk(x)dx−ϕ0t) _{shouldcontainan almostpurelyharmoniccomponentat} frequency

ϕ

0,so thatcomputingFSST for fD,k andthenextractingtheinformationcontainedinthevicinity offrequency

ϕ

0,onecanrecover fk through

fk

(

t

)

≈

⎛

⎜

⎝



|ω−ϕ0(t)|<d Tγ_f D,k

(

t

,

ω

)

d

ω

⎞

⎟

⎠

e2πi(0tϕk(x)dx−ϕ0t) ₍₂₃₎

In[32],

(

ϕ

k

(

t

))

k=1,··· ,K wereestimatedasthepiecewiseconstantcurvesassociatedwiththeridgesextractedfromFSST

(orFSST2), andwere thereforegreatlydependentonthe frequencyresolution M. Weproposehereamuch morerelevant techniquetocompute IFsestimatesthantheonebasedoncruderidgeextractionandwhichwillprovetobe onlyslightly dependent on the frequencyresolution. Indeed,while performing FSST (or FSST2), one computes



ω

f

(

t

,

ξ )

(or



ω

[f2]

(

t

,

ξ )

),

which,evaluatedontheridgeassociatedwiththekth mode,i.e.



ω

f

(

t

,

ϕ

k

(

t

))

(or



ω

[_f2]

(

t

,

ϕ

k

(

t

))

),leadstoa muchsmoother

IF estimate than the one proposed in [32]. Thisis due to the fact that the former is not constrained by the frequency resolution.Inwhatfollows,wedenote,forthesakeofsimplicity,

ω



f

(

t

,

ϕ

k

(

t

))

(or

ω



[f2]

(

t

,

ϕ

k

(

t

))

)by

ϕ

k

(

t

)

,dependingonthe

Fig. 4. (a)FSST2ofthetwo-chirpssignalwhenM=N (N beingthesignallength);(b)sameas(a)whenM=N/8;(c)reconstructionresultsformodef1 usingFSST2orthedemodulationalgorithmbasedonFSST2orFSST(withd=0,fordifferentinputSNRs,andwhen f(t)isthetwochirpsignal);(d)same as(c)butformodef2;(e)FSST2ofthetwomodeswithcosinephasewhenM=N,(f)sameas(e)butwhenM=N/8,(g)reconstructionresultsformode

f1usingFSST2orthedemodulationalgorithmbasedonFSST2orFSST(withd=0,fordifferentinputSNR,and f(t)isthesignalmadewithtwomodes withcosinephase);(h)Sameas(g)butformodef2.

6.2. PerformanceofthedemodulationalgorithmbasedonFSST,comparisonwithFSST2

We here consider two types of signal to illustrate the benefits of the demodulation procedure based on FSST (or FSST2) introducedin theprevious section:the firsttested signal is f

(

t

)

=

f1

(

t

)

+

f2

(

t

)

madeof thelinearchirp f1

(

t

)

=

2e2πi(130t+100t2₎

andthequadraticchirp f2

(

t

)

=

2e2πi(250t+50t 3₎

,fort

∈ [

0

,

1

]

sampledatN

=

1024 Hz.TheSTFTiscomputed withthesameL1_{-normalizedGaussianwindowaspreviously,forwhichtheoptimal}

_σ

_{leadstoafilterlength2L}

₊

₁

₌

_123. The representations ofthe FSST2 ofthesesignals when M

=

N and M

=

N

/

8

=

128 aredisplayed inFigs. 4(a) and (b), whentheinputSNRequals10dB.TheninFigs.4(c)and(d),wedisplaythereconstructionresultsassociatedwithmodes

f1 and f2, respectively, when d

=

0 (meaning we only consider the information on the ridge). We either consider the direct reconstruction based on FSST2 displayed in (a) and (b) to show that a more concentrated representation results in a better reconstruction fromthe information on the ridge. Furthermore, we remark that using FSST or FSST2 in the demodulation process (with M

=

N

/

8 in the definition of STFT) doesnot make any difference for that type of modes (these casesare denoted by “FSST demod” and“FSST2 demod” in the considered figures) and that the results obtained with these latter techniques are significantly better than those obtained using FSST2 only. As a second illustration, we consider the signal f

(

t

)

=

f1

(

t

)

+

f2

(

t

)

, inwhich themodes havecosine phases, namely f1

(

t

)

=

2e2πi(190t+9 cos(3πt)) and

f2

(

t

)

=

2e2πi(330t+16 cos(3πt)).Aspreviously,t belongsto

[

0

,

1

]

,N

=

1024,andthewindow g remainsthesame(theoptimal

σ

givenbytheRényientropybeingverysimilarinbothcases).TheFSST2ofthissignalcomputedwithM

=

N orM

=

N

/

8 are displayed in Figs. 4(e) and(f),respectively. Then, we againinvestigatethe quality ofthe reconstruction process, as-sumingd

=

0 and wheneitherreconstruction fromFSST2isconsideredorthedemodulationprocessbasedoneitherFSST orFSST2.Asinthepreviouscase,thedemodulationalgorithmbasedonFSST2stillbehaves betterthanFSST2alone,as re-portedinFigs.4(g)and(h).Furthermore,weremarkthatwhileusingFSSTorFSST2tocomputethedemodulationoperator doesnotmakeanydifferenceformode f1(Fig.4(g)),forthemoremodulatedmode f2,takingintoaccountthemodulation inthedemodulationoperatorbymeansofFSST2leadstomuchbetterresults(Fig.4(h)).Toconclude,thispleadsinfavor ofusingdemodulationbasedonFSST2toreconstructthemodesratherthanFSST2only.

7. Conclusionandperspectives

The synchrosqueezingtransformentersnowits matureage,andhasproventobe usefulwhenanalyzingawide range ofreal-lifesignals.Manyvariantshavebeenproposedtoextenditsdomainofapplication,butopenquestionsstillremain.

(i) Itisstill notclearwhetherSSTis theperfecttoolformode decomposition.Themainreasonisthat thelatterhighly dependsonthe ridgeestimationstep,which isadifficulttask:extractingtheridgesassociatedwiththemodesfrom thereassignedtransformseemstobeabetteroptionthandoingthesamethingonSTFTs,butsometheoretical devel-opmentsstill needtobecarriedoutonthatmatter[40].Furthermore,SSTwhenusedformodereconstructioncannot operateonaSTFTdownsampledintime,whichisastrongconstraintwhenonehastodealwithlong,high-ratesignals.

S. Meignen et al. / C. R. Physique 20 (2019) 449–460 459

Tocircumventthislastlimitation,furtherworksarethereforeneeded,and,inthisregard,thecomparisonoftheresults obtainedwithSSTandalternativerepresentationsinthecontextofaudiosourceseparationisafirststep[41]. (ii) SSTsuffersfromtheintrinsiclimitationthatitoperatesonalinearTFR,associatedwithafixedTFresolutiongivenby

aglobalwindoworwavelets.Thiscanbe mitigatedbyusingmulti-tapermethods[24],thoughthere isstilla lackof flexibility.Morecritically,anytechnique basedonlinearTFRrequires TFseparationforthecomponents,whichisnot thecaseinmanyapplications.Some recentworksconsideredinterferingorevencrossing components[42], butwith limitedgenerality.

TheselimitationsofSSTmakesomeresearchersquestiontheusefulnessofSST[43],butinthat papertheanalysiswas restrictedto theoriginal SSTandthe numericalinvestigations werecarried outon simpleslightlymodulatedmodes. We definitelythinkthatrecentextensionsofSSTbroughtabouttwostrongandclearcontributions:

– thekindofresultsshownin[12,20],byusinglocalslow-variationandseparationassumptions,canbeeasilyextended tootheradaptivedecompositionsbasedonSTFTorwavelets;

– thelocalinstantaneousfrequencyanditsextensionsbasedonhigher-orderphasederivativesareperfectlyadaptedfor defining a meaningful instantaneous frequency for real-life signals. Furthermore,this quantity also enables fastand accuratepost-processingoperations,suchasridgedetection,reassignment,andmoderetrieval.

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