Synchrosqueezing transforms: From low- to high-frequency modulations and perspectives

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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

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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_Laboratoire_LJK,_bâtiment_IMAG,_Université_Grenoble_Alpes,_700,_avenue_Centrale,_Campus_de_{Saint-Martin-d’Hères,}₃₈₄₀₁_Domaine

universitairedeSaint-Martin-d’Hères,France

b_IRIT,_Toulouse_INP_–_ENSEEIHT,_2,_rue_{Charles-Camichel,}_BP_7122,₃₁₀₇₁_Toulouse_cedex_7,_France

c_ICube,_UMR_7357,_Laboratoire_des_sciences_de_{l’ingènieur,}_de_{l’informatique}_et_de_{l’imagerie,}_300,_bd_{Sébastien-Brant,}_CS_10413,₆₇₄₁₂_Illkirch

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.

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

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

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,butadaptedto

non-stationary signals.

Indeed,themainlimitationofFTisthatitdoesnotlocateintimethefrequencyinformationcomputedonagivensignal, while mostsignalsfromthephysicalworld areintrinsically non-stationary,inthesense that theirfrequencies varyalong time. Analyzingorprocessingthesesignalseﬃcientlythusrequirestheuseof

local transformations,

calledtime–frequency representations (TFRs), among whichthe mostpopular isprobablythe short-time Fourier transform (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 the

continuous wavelet transform

(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 the

reassignment method (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-based SST (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,asinthe

ConceFT 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-order synchrosqueezing transform (FSST2), was introduced in[20], andits theoreticalfoundations settled in[31]. Such anewtransform was also usedinademodulationalgorithm[32] andextendedtobetterdealwithmodeswithfastoscillatingphase[33].

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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)

_,_its

_{Fourier transform corresponds}

_to:

f

(ξ )

=

F

{

f

}(ξ) =

R

f

(

t

)

e−2πiξtdt (1)

andits

short-time Fourier transform (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 and g are

ˆ

allin

L

1

_(R)

_, _{f can}_be_{reconstructed}_from_its_STFT_as_soon_as

_{g is}

_non-zero_at_0: f

(

t

)

=

1

g

(

0

)

R

Vg_f

(

t

, ξ )

d

ξ

(3)

where Z denotes thecomplexconjugateofZ .

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

(

t

)

=

K

k=1 fk

(

t

)

with fk

(

t

)

=

Ak

(

t

)

e2πiφk(t) (4)

forsomeﬁnite

K

∈ 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

)

forall

t.

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-called

ridges,

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-based synchrosqueezing transform

STFTbasedsynchrosqueezing (denoted byFSST inthe sequel)is basedonthe deﬁnitionofthe local instantaneous fre-quency estimate

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:

(5)

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 the

kth ridge,

whichamountstomodifying

thesynthesisformula(3) toselectonlythecoeﬃcientsrelatedtothe

kth mode,

namely:

fk

(

t

)

≈

1 g

(

0

)

|ω−ϕk(t)|<d Tγ_f

(

t

,

ω

)

d

ω

(8)

where

ϕ

k isanestimateof

φ

_k,andparameter

d is

usedtocompensateforerrorsinIFestimation;itshouldbekeptsmall

enoughtoavoidmode-mixing,anditsinﬂuencewillbediscussedlater. 3.2. Wavelet-based synchrosqueezing transform

Let

ψ

beawavelet;wedeﬁne,attime

t and

scale

a

>

0,the

continuous wavelet transform (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 are inL1

(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)

The

kth mode

canthenbeapproximatelyreconstructedbyintegrating

S

γ_f

(

t

,

ω

)

alongthefrequencyaxisasfollows:

fk

(

t

)

≈

1 C_ψ

|ω−ϕk(t)|<d Sγ_f

(

t

,

ω

)

d

ω

(12)

3.3. Approximation results for FSST and WSST

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 are

slow varying.

(ii) Thedifferentmodesare

separated in

theTFplane,i.e.foreachtime

t

φ

_k

(

t

)

for

k

=

1

· · ·

K are signiﬁcantlydifferent. 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_)._We_ﬁrst_note_that_the_FSST_and_WSST_of_the_purely_harmonic_mode_are_very

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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 (closeto

t

=

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. Second order synchrosqueezing transforms

Asjustexplained,theapplicabilityofFSSTisrestrictedtoaclassofMCSscomposedofslightlyperturbedpurelyharmonic modes.Toovercomethislimitation,anextension ofFSSTwas introducedbasedona moreaccurateIFestimate,thenused todeﬁneanimprovedsynchrosqueezingoperator,called

second-order STFT-based synchrosqueezing transform (FSST2)

[34,31]. Moreprecisely,oneﬁrstdeﬁnesa

second-order local modulation

operator, whichisthenusedtocomputethenewIFestimate. To doso, one introduces complex reassignment operators

ω

˜

f

(

t

,

ξ )

=

∂tVg_f(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

∂tVg_f(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 is deﬁnedby:

˜

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)

(7)

andwe thenput

ω

ˆ

[_f2]

(

t

,

ξ )

=

˜

ω

[_f2]

(

t

, ξ )

.Itwas provenin[31] that

ω

ˆ

[_f2]

(

t

,

ξ )

= φ

(

t

)

,when f is a Gaussianmodulated linearchirp.Itisalsoworthmentioningherethatq

˜

f

(

t

,

ξ )

canbecomputedbymeansofﬁvedifferentSTFTs.Finally,FSST2

isdeﬁnedbyreplacing

ω

ˆ

f

(

t

,

ξ )

by

ω

ˆ

[_f2]

(

t

,

ξ )

in(7),toobtaintheso-called

T

₂γ_,_f,andmodereconstructionisthenperformed

byreplacing

T

γ_f by

T

₂γ_,_f in(8).

Similarly,complexreassignmentoperators

ω

f

(

t

,

a

)

and

τ

f

(

t

,

a

)

intheCWTcontextarerespectivelydeﬁned,forany

(

t

,

a

)

s.t.

W

ψ_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

)

withSγ₂_,_f

(

t

,

ω

)

in(12).

4.2. Higher-order synchrosqueezing transforms

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

)

)beingequaltoits

Lth-order

(resp.

Nth-order)

Taylorexpansionfor

τ

closeto

t,

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

₎

_denotes_the

_{kth derivative}

_of_{Z evaluated}_at

_t.

_Such_a_mode,_with_L

_≤

_N,_can_be_written_as:

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 )

withrespectto

t and

thendividingby2

π

iVg_f

(

t

,

ξ )

,thelocalcomplex reassign-mentoperator

ω

˜

f

(

t

,

ξ )

deﬁnedinSection4.1canbewritten,when

V

g_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)

(8)

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 )

,whichrequiresthecalculationof

r

k

(

t

)

for

k

=

2

,

. . . ,

N.

Forthat

purpose,onederivesafrequencymodulationoperatorq

˜

[_fk,N]

(

t

,

ξ )

,equalto

r

k

(

t

)

forthetypeofmodesjustintroduced,and

obtainedbydifferentiatingdifferentSTFTswithrespectto

ξ

(todifferentiatewithrespectto

ξ

ratherthan

t leads

tomuch simplerexpressions).q

˜

[_fk,N]

(

t

,

ξ )

for2

≤

k

≤

N can bederivedrecursively,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 )

and

x

k,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

modesoftheMCScanbereconstructedbyreplacing

T

γ_f

(

t

,

ω

)

by

T

γ_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)_with

_A

₍

_t

₎

₌

₁

₊

_3t2

₊

₄

₍

₁

₋

_t

₎

7_and

_{φ (}

_t

₎

₌

_240t

₋

₂_exp

₍

₋

_2t

₎

_sin

₍

₁₄

_π

_t

₎

_for

_t

_{∈ [}

₀

_,

₁

_]

_,_which_corresponds_to

adamped-sinefunctionsignalcontainingverystrongnonlinearsinusoidalfrequencymodulationsandhigh-orderpolynomial amplitudemodulations.Such afunctionissampledatarate N

=

1024 Hzon

[

0

,

1

]

.TheSTFTof f is thencomputedwith 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 superimposed with A and itsSTFTmodulusare 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 goodcorrespondenceuntil

t

=

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

(9)

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. Settings parameters

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

(10)

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) andcarriedoutby

ridge extraction,

performedbyfittingsmoothcurveswiththelocationsofthehighestenergy coefficientsoftheconsideredSTFT.Thisisusuallydonebyconsideringthenon-convexvariationalproblem,whoseoptimal solutioncanbeapproximatedusingstochasticsamplingorheuristictechniques[39,12,19].Thisdifficultproblemsomehow reducesthescopeofthetheoreticalresults,sincethelatterassumesthattheridgeinformationisknown.

5.2. Inﬂuence of frequency resolution

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,and

g supported

on

[−

LT_N

,

LT_N

]

,with

L

<

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. Demodulation algorithm based on FSST for mode reconstruction

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

estimateformode

k,

thesignal fD,k

(

t

)

=

f

(

t

)

e−2πi( t

0ϕk(x)dx−ϕ0t) _should_contain_an _almost_purely_harmonic_component_at

frequency

ϕ

0,so thatcomputingFSST for fD,k andthenextractingtheinformationcontainedinthevicinity offrequency

ϕ

0,onecanrecover fk through

fk

(

t

)

≈

⎛

⎜

⎝

|ω−ϕ0(t)|<d Tγ_fD ,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,evaluatedontheridgeassociatedwiththe

kth 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

(11)

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. Performance of the demodulation algorithm based on FSST, comparison with FSST2

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₎

,for

t

∈ [

0

,

1

]

sampledat

N

=

1024 Hz.TheSTFTiscomputed withthesame

L

1_-normalized_Gaussian_window_as_previously,_for_which_the_optimal

_σ

_leads_to_a_ﬁlter_length_2L

₊

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 belongs

to

[

0

,

1

]

,

N

=

1024,andthewindow g remains thesame(theoptimal

σ

givenbytheRényientropybeingverysimilarinbothcases).TheFSST2ofthissignalcomputedwith

M

=

N or M

=

N

/

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

d

=

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.

(12)

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.

References

[1]S.Mallat,AWaveletTourofSignalProcessing,ThirdEdition:TheSparseWay,3rdedition,AcademicPress,2008.

[2]A.Grossmann,J.Morlet,Decomposition ofhardy functionsintosquare integrablewaveletsofconstantshape,SIAMJ.Math. Anal.15 (4)(1984) 723–736.

[3]P.Flandrin,Time-Frequency/Time-ScaleAnalysis,vol.10,AcademicPress,1998.

[4]K.Kodera,R.Gendrin,C.Villedary,Analysisoftime-varyingsignalswithsmallbtvalues,IEEETrans.Acoust.SpeechSignalProcess.26(1978)64–76.

[5]F.Auger,P.Flandrin,Improvingthe readabilityoftime–frequency andtime-scalerepresentationsbythereassignment method,IEEETrans.Signal Process.43 (5)(1995)1068–1089.

[6]N.E.Huang,Z.Shen,S.R.Long,M.C.Wu,H.H.Shih,Q.Zheng,N.-C.Yen,C.C.Tung,H.H.Liu,TheempiricalmodedecompositionandtheHilbertspectrum fornonlinearandnon-stationarytimeseriesanalysis,Proc.R.Soc.Lond.,Ser.A,Math.Phys.Eng.Sci.454 (1971)(1998)903–995.

[7]P.Flandrin,G.Rilling,P.Goncalves,Empiricalmodedecompositionasaﬁlterbank,IEEESignalProcess.Lett.11 (2)(2004)112–114.

[8]J.Gilles,Empiricalwavelettransform,IEEETrans.SignalProcess.61 (16)(2013)3999–4010.

[9]M.Kowalski,A.Meynard,H.tiengWu,Convexoptimizationapproachtosignalswithfastvaryinginstantaneousfrequency,Appl.Comput.Harmon. Anal.44(2018)89–122.

[10]N.Pustelnik,P.Borgnat,P.Flandrin,Empiricalmodedecompositionrevisitedbymulticomponentnon-smoothconvexoptimization,SignalProcess.102 (2014)313–331.

[11]I.Daubechies,S.Maes,Anonlinearsqueezingofthecontinuouswavelettransformbasedonauditorynervemodels,in:WaveletsinMedicineand Biology,1996,pp. 527–546.

[12]I.Daubechies,J.Lu,H.-T.Wu,Synchrosqueezedwavelettransforms:an empiricalmodedecomposition-liketool,Appl.Comput.Harmon.Anal.30 (2) (2011)243–261.

[13]M.Costa,A.A.Priplata,L.A.Lipsitz,Z.Wu,N.E.Huang,A.L.Goldberger,C.-K.Peng,Noiseandpoise:enhancementofposturalcomplexityintheelderly withastochastic-resonance-basedtherapy,Europhys.Lett.77(2007)68008.

[14]D.A.Cummings,R.A.Irizarry,N.E.Huang,T.P.Endy,A.Nisalak,K.Ungchusak,D.S.Burke,Travellingwavesintheoccurrenceofdenguehaemorrhagic feverinThailand,Nature427(2004)344–347.

[15]Y.-Y.Lin,H-t.Wu,C.-A.Hsu,P.-C.Huang,Y.-H.Huang,Y.-L.Lo,Sleepapneadetectionbasedonthoracicandabdominalmovementsignalsofwearable piezoelectricbands,IEEEJ.Biomed.HealthInform.21 (6)(2017)1533–1545.

[16]C.L.Herry,M.Frasch,A.J.Seely,H.-T.Wu,Heartbeatclassiﬁcationfromsingle-leadecgusingthesynchrosqueezingtransform,Physiol.Meas.38 (2) (2017)171–187.

[17]G. Thakur,H.-T.Wu,Synchrosqueezing-basedrecovery ofinstantaneous frequencyfromnonuniformsamples, SIAMJ. Math. Anal.43 (5)(2011) 2078–2095.

[18]H.-T.Wu,AdaptiveAnalysisofComplexDataSets,PhDthesis,Princeton,NJ,USA,2011.

[19]F.Auger,P.Flandrin,Y.-T.Lin,S.McLaughlin,S.Meignen,T.Oberlin,H.-T.Wu,Time-frequencyreassignmentandsynchrosqueezing:anoverview,IEEE SignalProcess.Mag.30 (6)(2013)32–41.

[20]T.Oberlin,S.Meignen,V.Perrier,TheFourier-basedsynchrosqueezingtransform,in:2014IEEEInternationalConferenceonAcoustics,Speechand SignalProcessing(ICASSP),2014,pp. 315–319.

[21]M.Unser,D.Sage,D.VanDeVille,MultiresolutionmonogenicsignalanalysisusingtheRiesz–Laplacewavelettransform,IEEETrans.ImageProcess. 18 (11)(2009)2402–2418.

[22]M.Clausel,T.Oberlin,V.Perrier,Themonogenicsynchrosqueezedwavelettransform:atoolforthedecomposition/demodulationofam-fmimages, Appl.Comput.Harmon.Anal.39(2015)450–486.

[23]A.Ahrabian,D.Looney,L.Stankovi ´c,D.P.Mandic,Synchrosqueezing-basedtime–frequencyanalysisofmultivariatedata,SignalProcess.106(2015) 331–341.

[24]I.Daubechies,Y.G.Wang,H.-T.Wu,Conceft:concentrationoffrequencyandtimeviaamultitaperedsynchrosqueezedtransform,Philos.Trans.R.Soc. A,Math.Phys.Eng.Sci.374(2016).

[25]H.-T.Wu,Instantaneousfrequencyandwaveshapefunctions(i),Appl.Comput.Harmon.Anal.35(2013)181–199.

[26]M.Skolnik,RadarHandbook,McGraw-HillEducation,2008.

[27]J.W.Pitton,L.E.Atlas,P.J.Loughlin,Applicationsofpositivetime–frequencydistributionstospeechprocessing,IEEETrans.SpeechAudioProcess.2 (4) (1994)554–566.

[28]E.J.Candes,P.R.Charlton,H.Helgason,Detectinghighlyoscillatorysignalsbychirpletpathpursuit,Appl.Comput.Harmon.Anal.24 (1)(2008)14–40.

[29]B.P.Abbott,R.Abbott,T.Abbott,M.Abernathy,F.Acernese,K.Ackley,C.Adams,T.Adams,P.Addesso,R.Adhikari,etal.,Observationofgravitational wavesfromabinaryblackholemerger,Phys.Rev.Lett.116 (6)(2016)061102.

(13)

[31]R.Behera,S. Meignen,T.Oberlin, Theoreticalanalysisofthe second-order synchrosqueezingtransform,Appl.Comput. Harmon.Anal. 45(2018) 379–404.

[32]S.Meignen,D.-H.Pham,S.McLaughlin,Ondemodulation,ridgedetectionandsynchrosqueezingformulticomponentsignals,IEEETrans.SignalProcess. 65 (8)(2017)2093–2103.

[33]D.-H.Pham,S.Meignen,High-ordersynchrosqueezingtransformformulticomponentsignalsanalysis–withanapplicationtogravitational-wavesignal, IEEETrans.SignalProcess.65(2017)3168–3178.

[34]T.Oberlin,S.Meignen,V.Perrier,Second-ordersynchrosqueezingtransformorinvertiblereassignment?Towardsidealtime–frequencyrepresentations, IEEETrans.SignalProcess.63(2015)1335–1344.

[35] T. Oberlin,S. Meignen,Thesecond-orderwaveletsynchrosqueezingtransform,in:Proc.42thInternationalConferenceonAcoustics,Speech,andSignal Processing(ICASSP),NewOrleans,LA,USA,5–9March2017.

[36]L.Stankovi ´c,Ameasureofsometime–frequencydistributionsconcentration,SignalProcess.81 (3)(2001)621–631.

[37]B.P.Abbott,R.Abbott,etal.,GW151226:observationofgravitationalwavesfroma22-solar-massbinaryblackholecoalescence,Phys.Rev.Lett.116 (24) (2016)241103.

[38]R.G.Baraniuk,P.Flandrin,A.J.Janssen,O.J.Michel,Measuringtime–frequencyinformationcontentusingtherényientropies,IEEETrans.Inf.Theory 47 (4)(2001)1391–1409.

[39]R.Carmona,W.Hwang,B.Torresani,Characterization ofsignalsbytheridgesoftheirwavelettransforms, IEEETrans.SignalProcess.45(1997) 2586–2590.

[40]S.Meignen,T.Oberlin,P.Depalle,P.Flandrin,S.McLaughlin,Adaptivemultimodesignalreconstructionfromtime–frequencyrepresentations,Philos. Trans.R.Soc.A374 (2065)(2016)20150205.

[41] D. Fourer,G. Peeters,FastandadaptiveblindaudiosourceseparationusingrecursiveLevenberg-Marquardtsynchrosqueezing,in:Proc.IEEE Interna-tionalConferenceonAcoustics,SpeechandSignalProcessing(ICASSP),Calgary,Canada,15–20April2018,pp. 766–770.

[42] V. Bruni,M. Tartaglione,D. Vitulano,Onthetime–frequencyreassignmentofinterferingmodesinmulticomponentfmsignals,in:Proc.26thEuropean SignalProcessingConference(EUSIPCO),Rome,3–7September2018,pp. 722–726.

[43]D.Iatsenko,P.V.McClintock,A.Stefanovska,Linearandsynchrosqueezedtime–frequencyrepresentationsrevisited:overview,standardsofuse, resolu-tion,reconstruction,concentration,andalgorithms,DigitalSignalProcessing42(2015)1–26.