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To cite this version:
Behera, Ratikanta and Meignen, Sylvain and Oberlin, Thomas Theoretical analysis of the second-order synchrosqueezing transform. (2018) Applied and Computational Harmonic Analysis, 45 (2). 379-404. ISSN 1063-5203.
Official URL:
https://doi.org/10.1016/j.acha.2016.11.001
Theoretical
analysis
of
the
second-order
synchrosqueezing
transform
Ratikanta Beheraa, Sylvain Meignena,∗, Thomas Oberlinb
a
JeanKuntzmannLaboratory,UniversitéGrenobleAlpes,700AvenueCentrale,38401 Domaine UniversitairedeSaint-Martin-d’Hères,France
bINP–ENSEEIHTToulouse,2 rueCharlesCamichel,B.P.7122,31071 ToulouseCedex 7,France
a b s t r a c t
Keywords:
Time-frequencyanalysis Synchrosqueezingtransform Multicomponentsignals
We consider in this article the analysis of multicomponent signals, defined as superpositionsofmodulatedwavesalsocalledmodes.Moreprecisely,wefocuson theanalysisofavariantofthesecond-ordersynchrosqueezingtransform,whichwas introduced recently,to dealwithmodes containingstrongfrequency modulation. Beforegoingintothisanalysis,werevisit thecasewherethemodes areassumed tobe withweakfrequency modulation as intheseminalpaper of Daubechieset al.[8],to showthattheconstraintonthecompactnessof theanalysiswindowin theFourierdomaincan bealleviated.Wealso explainwhythehypothesesmade on themodesmakingup themulticomponent signalmust be differentwhen one considers either waveletor short-time Fouriertransform-based synchrosqueezing. Therestof thepaper isdevotedto thetheoreticalanalysisof thevariant ofthe secondordersynchrosqueezingtransform[16]andnumericalsimulationsillustrate theperformanceofthelatter.
1. Introduction
Multicomponent signals (MCs) are encounteredin anumber of fields of practicalinterest such as me-teorology, structural stability analysis, and medical studies – see, e.g., [6,7,11,12]. Linear time–frequency (TF) analysistechniques havebeenextensivelyused toanalyzeandprocess thesesignals [14], themethod tobe useddepending onthenatureofthemodes makingupthesignal.Standard linearTFmethodssuch as theshort-time Fouriertransform (STFT) andthe continuouswavelet transform (CWT)are commonly usedtoanalyzesuchsignals.Inthatcontext,thereassignmentmethod(RM)wasdeveloped[1]toimprove thereadabilityoftheselinearTFrepresentations.Unfortunately,sinceRMappliestothemagnitudeofthe
* Correspondingauthor.
E-mailaddresses:ratikanta.maths@gmail.com(R. Behera),sylvain.meignen@imag.fr(S. Meignen),
studiedTFtransform,themethodresultsinalossof informationandthereassignedrepresentationis not sufficient torecovertheoriginalsignal.
Recently, Daubechies et al. [8] showed interesting results on the so-called synchrosqueezing transform (SST)to representMCs,amethodintroducedinthemid-1990sforaudiosignalanalysis[9].SSTcombines the localization and sparsity properties of RM with the invertibility of a traditional TF representation. Originally proposed as apost-processing method appliedto the CWT, SST can alternatively be applied to STFT with minor changes [15], to obtain the so-called FSST. Since the seminal paper of Daubechies et al. [8], manynew developments have been carried out invarious directions.First, an extension to the bidimensionalcasewasproposedin[5],whileageneralizationofthewaveletapproachbymeansofwavelet packets decompositionfor both onedimensionaland bidimensionalcases isavailable in[26,25]. Finally,it is also worth noting thatastudy of synchrosqueezing applied to a more generalclass of multicomponent signalswasdonein[22].
In the present paper, we focus on what essentially limits the applicability of SST which are, on the one hand, the hypotheses of weak frequency modulation for the modes making up the signal and, on the other hand, the compactness of the frequency support of the analysis window. To better take into account the frequency modulation, in the FSST context, a first approach based on the definition of a demodulationoperatorwasproposedin[13],while anewsynchrosqueezing operatorbasedonanewchirp rateestimatewasdefinedin[16,2].However,intheselasttwopapers,notheoreticalanalysisoftheproposed newsynchrosqueezing transformwasprovided.Todealwiththis issueisthemain aimofthispaper.
While pursuing this goal, we will pay particular attentionto use non-compactly supported windowin the frequency domain.Indeed, the main problem induced by using windows compactly supported inthe frequencydomainisthattheyarenotadaptedtodealwithreal-timecomputations.Todealwiththisissue, anewwavelet-basedSSTdefinedusing waveletwithsufficientlymanyvanishing momentsand aminimum supportinthetimedomainwasproposedin[4],butthemathematicalstudyofthecorresponding instanta-neousfrequency(IF)estimatesisstillunderdevelopment.Inthemeantime,anextensionofsynchrosqueezing based on wavelet packets not compactly supportedinthe frequency domain along with its mathematical analysiswasdevelopedin[24].Inthecontextofthispaper,thefocusisputontheFourier-basedSST,when theanalysiswindowisnotcompactlysupported.Wefirstrevisitthecaseofweakmodulationforthemodes using such atype ofwindow,and then proveanewapproximation theoremon aslightlydifferentversion of thesynchrosqueezing transformproposedin[16].
Theoutlineofthepaperisas follows:inSection2,werecallsomenotationanddefinitions.InSection3, we introduceFSST and wavelet-based SST (WSST) along with the corresponding approximation results and explain whythehypotheses onthephaseof themodes havetobe different. Then,after having intro-duced somenecessary ingredients,wederive,inSection4,theapproximationtheorem forthesecond order synchrosqueezing transform, showing that this new transform is fully adaptedfor analyzing modes with strongfrequencymodulations.Finally,numericalillustrationsconcludethepaper.
2. Definitions
2.1. Short-timeFouriertransform
We denote by L1(R) and L2(R) the space of integrable, and square integrable functions. Consider a signalf ∈ L1(R),andtakeawindowg intheSchwartzclass,S(R),thespaceofsmoothfunctionswithfast
decayingderivativesofanyorder; itsFouriertransformisdefinedby:
f (η) =F{f}(η) =
R
The need for time-localized frequency descriptors leads to short-timeFourier transform (STFT) which is obtainedthroughtheuseofaslidingwindowg∈ L2(R) definedby:
Vfg(η, t) = R f (τ )g(τ− t)e−2iπη(τ−t)dτ (2) = R
f (t + τ )g(τ )e−2iπητdτ =A(η, t)e2iπΦ(η,t). (3)
The representation of |Vfg(η,t)|2 in the TF plane is called the spectrogram of f . The STFT admits the
followingsynthesisformula(providedthewindowg doesnotvanishand iscontinuousat0): f (t) = 1
g(0)∗
R
Vfg(η, t)dη. (4)
Further,ifthesignalisanalytic(i.e.η≤ 0⇒ ˆf (η)= 0)thentheintegraldomain forη isrestrictedtoR+.
2.2. Continuouswavelet transform
Letusconsideranadmissiblewaveletψ∈ L2(R),satisfying0< Cψ=0∞| ψ(ξ)|2 dξξ <∞ andthendefine foranytime t andscalea> 0, thecontinuous wavelet transform (CWT)off by:
Wfψ(a, t) =1 a R f (τ )ψ τ− t a ∗ dτ, (5)
where z∗ denotes the complex conjugateof z. Assuming that ψ is analytic, i.e. Supp( ψ) ⊂ [0,∞[, and f real-valued, theCWTadmits thefollowing synthesisformula (Morletformula):
f (t) = 2R ⎧ ⎨ ⎩ 1 Cψ ∞ 0 Wfψ(a, t)da a ⎫ ⎬ ⎭, (6)
whereR denotestherealpartofacomplexnumberandCψ = ∞ 0 ψ∗(ξ)dξ ξ . 2.3. Multicomponent signal
Inthepresentpaper,weanalyzeso-calledmulticomponentsignals of theform,
f (t) = K k=1
fk(t), with fk(t) = Ak(t)e2iπφk(t), (7)
forsomeK,where Ak(t) and φk(t) aretime-varyingamplitudeandphasefunctionsrespectivelysuchthat Ak(t) > 0, φk(t) > 0 and φk+1(t) > φk(t) for all t. In the following, φk(t) is often called instantaneous frequency (IF) of mode k andAk(t) its instantaneous amplitude (IA).One of the goal of TF analysis is to recover the instantaneous frequencies {φk(t)}1≤k≤K and amplitudes {Ak(t)}1≤k≤K, from a given TF
representationoff .Notethat,ifwearegivenreal-valued functions,i.e.fk(t)= Ak(t)cos(2πφk(t)),wecan stillderivethesameanalysisbyconsideringanalyticwindowsor wavelets,assoonas φ1(t)> Δ∀t.
3. Synchrosqueezing transform
Thesynchrosqueezingtransform(SST)isanapproachoriginallyintroducedinthecontextofaudiosignals
[9], whose theoreticalanalysis wasrecently carriedoutin[8].It belongsto thefamily ofTFreassignment methods and corresponds to a nonlinear operator that sharpens the TF representation of a signal while enabling mode reconstruction.Moreover, SSTcombines thelocalizationand sparsity propertiesofTF re-assignment withtheinvertibilityproperty oftraditionalTFtransforms,andisrobusttoavarietyofsignal perturbations [18]. The key ingredient to SST is an IF estimate computed from the TF representation, whichweintroducenow.Theresultsenouncedinthenexttwosubsectionsarenotnew,butwerecallthem forthesakeofconsistencyandalsobecausetheyeasetheunderstandingofthefollowingpartsofthepaper. Theywereoriginallystatedin[8]inthewaveletcase,andin[15,21]forthatofSTFT.
3.1. ComputationofIFestimate in theTFplane
In the STFT framework, the IF of signal f at time t and frequency η can be estimated, wherever Vfg(η,t)= 0,by[1]: ωf(η, t) := 1 2π∂targV g f(η, t), = ∂tΦ(η, t) defined in(2), = R 1 2iπ ∂tVfg(η, t) Vfg(η, t) . (8)
Note thatthequantityofwhichwetaketherealpart:
˜
ωf(η, t) = ∂tV g f(η, t)
2iπVfg(η, t), (9)
is useful to moretheoretical investigations butshould notbe used to estimate real quantities suchas IF. Similar quantities can be derived in the CWT case as follows: ωf(a,t) = R(ωf(a, t)), with ω˜f(a,t) =
1 2iπ
∂tWfψ(a,t)
Wfψ(a,t) .
Remark 1.In[8], theestimate ωf(a,t) isused instead of ωf(a,t). However,most theorems arevalid only forarealestimateas willbeshownlater.
3.2. STFT-basedsynchrosqueezingtransform
Definition1.Letε> 0 andΔ> 0.ThesetBΔ,εofmulticomponentsignalswithmodulationε andseparation
Δ isthesetofallmulticomponent signalsdefinedin(7)satisfying: Ak ∈ C1(R) ∩ L∞(R), φk ∈ C2(R), sup t∈Rφ k(t) <∞, φk(t) > 0, Ak(t) > 0, ∀t |A k(t)| ≤ ε, |φk(t)| ≤ ε ∀t ∈ R. (10)
Further,thefksareseparatedwithresolution Δ,i.e., forallk∈ {1,...,K− 1} andallt,
Definition2.Leth beaL1-normalizedpositive functionbelongingtoC∞
c (R),thespace ofcompactly sup-portedsmoothfunction,andpickγ,λ> 0.TheSTFT-basedsynchrosqueezing (FSST)off withthreshold γ andaccuracyλ isdefinedby:
Tfλ,γ(ω, t) = 1 g(0) |Vg f(η,t)|>γ Vfg(η, t)1 λh ω− ωf(η, t) λ dη. (12)
Ifwemakeλ andγ tendtozero,thenTfλ,γ(ω,t) tendsto somevaluewhichweformallywrite as:
Tf(ω, t) = 1 g(0) R Vfg(η, t)δ(ω− ωf(η, t))dη, (13)
called FSSTinthesequel,andwhere δ istheDiracdistribution.
Theorem1.Consider f ∈ BΔ,ε andputε = ε˜ 1/3.Letg∈ S(R),theSchwartzclass, be suchthat supp(g)⊂
[−Δ,Δ]. Then,if ε issmall enough,thefollowingholds:
(a) |Vfg(η,t)|> ˜ε onlywhenthereexistsk∈ {1,...,K} suchthat(η,t)∈ Zk :={(η,t), s.t. |η −φk(t)|< Δ}. (b) For allk∈ {1,...,K} andall(η,t)∈ Zk suchthat |Vfg(η,t)|> ˜ε,wehave
|ωf(η, t)− φk(t)| ≤ ˜ε. (14)
(c) Forallk∈ {1,...,K}, thereexistsaconstant C suchthat forallt∈ R, lim λ→0 ⎛ ⎜ ⎝ |ω−φ k(t)|<˜ε Tfλ,˜ε(ω, t)dw ⎞ ⎟ ⎠ − fk(t) ≤ C ˜ε. (15)
Proof. ThistheoremgivesastrongapproximationresultfortheclassBΔ,ε,sinceitensuresthatthenon-zero
coefficientsoftheFSSTarelocalizedaroundthecurves(t,φk(t)),andthatthereconstructionofthemodes iseasilyobtainedfromtheconcentratedrepresentation.Themainstepsoftheproofaredetailedhereafter. (a) Forany (η,t)∈ R2, azeroth orderTaylor expansionoftheamplitudeand firstorder expansionofthe
phaseofthemodes leadto:
|Vg f(η, t)− K k=1 fk(t)ˆg(η− φk(t))| ≤ εΓ1(t), (16) where Γ1(t)= KI1+ πI2 K k=1 Ak(t), andIn = R|x|n|g(x)|dx.If ˜ ε≤ Γ1− 1 2 ∞ , (17)
and sinceg isˆ compactly supportedin[−Δ,Δ],and theIFs ofthe modes areseparatedby morethan 2Δ, wehave,forany (η,t)∈/
K k=1 Zk: |Vg f(η, t)| ≤ εΓ1(t)≤ ˜ε. (18)
(b) Now,wederivethesamekindofestimatefor∂tVfg.SinceV g
f(η,t) isdifferentiablewithrespecttoboth variables,andthatforany(η,t)∈ R2:
∂tVfg(η, t) = 2iπηVfg(η, t)− Vfg(η, t). (19) Thecounterpartofequation (16)isthen:
|∂tVg f(η, t)− 2iπ K k=1 fk(t)φk(t)ˆg(η− φk(t))| ≤ ε (Γ2(t) + 2π|η|Γ1(t)) , (20) whereΓ2(t)= KI1 + πI2 K k=1 Ak(t),withIn = R|t|n|g(t)|dt.
Now,wecanremarkthat,sincethesetsZk aredisjoint,thereisatmostonenon-zeroterminthesums involvedinequations(16)and(20).Wecanthuswrite forany(η,t)∈ Zk:
|ˆωf(η, f )− φk(t)| ≤∂tV g f(η, t)− 2iπφk(t)fk(t)ˆg(η− φk(t)) 2iπVfg(η, t) + φk(t)(fk(t)ˆg(η− φk(t))− V g f(η, t)) Vfg(η, t) ≤ ε (Γ2(t) + 2π|η|Γ1(t)) 2π ˜ε + φ k(t) εΓ1(t) ˜ ε ≤ ˜ε2 Γ2(t) 2π + (|η| + φ k(t))Γ1(t) .
Hence,ifε issufficiently small,i.e.ifitsatisfiesforallt
˜ ε≤ Γ2(t) 2π + (2φ k(t) + Δ)Γ1(t) −1 , (21) oneobtains |ˆωf(η, t)− φk(t)| ≤ ˜ε. (22)
Remark2.Notethattheproofofpoints(a)and(b)arealsoavailablein[19],butwithdifferenthypotheses. Indeed, in thatpaper, the constraints on the modes are non-local, i.e. Ak∞ ≤ εφk∞ and φk∞ ≤ εφk∞(Definition 3.1in[19]).Asweshallseelateron,whenanalyzingthebehaviorofωˆf(η,t) onalinear chirp,thequalityoftheestimateisonlyrelatedtotheamplitudeofφ,whichwilljustifythehypothesiswe putonφ.A moreimportantdifferencewith[19]involvestheseparationconditionsatisfiedbythedifferent modes makingupthesignal(Definition 3.2in[19]):
inf t φ k(t)− sup t φk−1(t) > 2Δ, (23)
which is muchstrongerthan condition(11). Forinstance,the simplesignals whose STFTare depicted in
Fig. 1 (firstcolumn) donotsatisfyhypothesis(23).
(c) Lett∈ R,usingthesametypeoftechniqueasin[8](Estimate 3.9),onecanwrite: lim λ→0 |ω−φ k(t)|<˜ε Tfλ,˜ε(ω, t) dω = 1 g(0)∗ {|Vg f(η,t)|>˜ε {|ˆω(η,t)−φ k(t)|<˜ε} Vfg(η, t) dη.
STFT FSST newVSST 500 500 500 400 400 400 // 300 300 300
--
� "' .,. "' ---200 200 200 ---100 100 100 0 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 1 STFT FSST newVSST 500 500 500 400 400 400 300 300 300 "' "' "'r,
200 200 200 \_ '-.._/ ,___.,. 100 100 100 0 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8Fig. 1. First row: magnitude of STFT, FSST and new VSST for test-signal 1; second row: same computation but for test-signal 2. \Ve now prove that the integration interval of the last equation, denoted by X = {IV/ ( 'I}, t) 1
>
i'} nn�,(11,t) -4>k(t)I < ê}, equals the following set:Y= {IVJ(ry,t)I
> ê} n{
111-4>k(t)I < �}.If 77 EX, it is such that IV/(11,t)I
> g
and lw,(77,t)- <Pk(t)I<
Ë. Also, from (a) there exists a unique lsuch that (77,t) E Z1. If l
=f
k, we would have:lw
1(11,t) -4>k(t)I � l4>i(t)-4>k(t)I-lw1(11,t)-<t>i(t)Ibecause of (b). Taking Ë � �, we get lw 1(11, t) -<Pk(t)I � f, which contradicts 77 EX. Hence, l = k and X C Y. Conversely, if 77 E Y, equation (26) immediately shows 77 E Y, hence X = Y.
Finally, g(�)*
J
Vf(11,t)d17-fk(t) {IV/<11,t)l>t} n{l11-,p1c(t)l<.ô.} Vf(11,t)d17-fk(t)- g(�)*J
{111-,p� (t)l<.ô.} n{IV/ (11,t)I �t} Vf (11,t) d17≤ 1 g(0)∗ {|η−φ k(t)|<Δ} Vfg(η, t) dη− fk(t) + 2Δ |g(0)|ε˜ ≤ 1 g(0) {|η−φ k(t)|<Δ} fk(t)ˆg(η− φk(t)) dη− fk(t) + 1 |g(0)| {|η−φ k(t)|<Δ} Vg f(η, t)− fk(t)ˆg(η− φk(t)) dη + 2Δ |g(0)|ε˜ ≤ 0 + |g(0)|2Δ εΓ1(t) + 2Δ |g(0)|ε˜ ≤ |g(0)|4Δ ε.˜ 2
3.3. Generalizationtonon-compactlysupportedˆg
In the abovesection, we assumed thatˆg was compactly supported.We hereexplain thatwe canstate thesameapproximationresultwhilerelaxingthisstrongcondition.
Theorem2.Assumethat thereexistconstantsM1,M2> 0 such thatg∈ S(R) satisfies:
For all|η| > Δ, |ˆg(η)| ≤ M1ε,
and |η|>Δ
|ˆg(η)| dη ≤ M2ε.˜
Then allconclusions ofTheorem 1 remain valid. Proof.
(a) We firstremarkthatequation(16)remainsvalidforany(η,t):
|Vg f(η, t)− K k=1 fk(t)g(η − φk(t))| ≤ εΓ1(t).
Ifweassumethatforany t,
˜ ε≤ min ⎡ ⎣1 2 M1 K k=1 Ak(t) −1 ,1 2Γ1(t) −1 2 ⎤ ⎦ , (24)
thenweget,forany (η,t)∈/ K k=1 Zk: |Vg f(η, t)| ≤ ε M1 K k=1 Ak(t) + Γ1(t) ≤ ˜ε. If(η,t)∈ Zk,|Vfg(η,t)− fk(t)g(η − φk(t))|≤ ε M1 l=kAl(t) + Γ1(t) ≤ ˜ε2.
(b) Similarly, we still have estimate (20) forany (η,t). If we assumethat for all k ∈ {1,· · · ,K} andfor all t, ˜ ε≤ min ⎡ ⎢ ⎣12 ⎛ ⎝M1 l=k φl(t)Al(t) ⎞ ⎠ −1 ,1 2 Γ2(t) 2π + (2φ k(t) + Δ)Γ1(t) −1 ⎤ ⎥ ⎦ , (25)
then wegetforany(η,t)∈ Zk suchthat|Vfg(η,t)|≥ ˜ε:
|ˆωf(η, f )− φk(t)| ≤ ˜ε. (26)
(c) Regardingpoint (c) ofTheorem 1, theonlydifference when oneconsidersanon-compactly supported window appearsat thevery end of theproof. Moreprecisely, the following termis notzero anymore, butcanbeboundedinthefollowingway:
1 g(0) |η−φ k(t)|<Δ fk(t)ˆg(η− φk(t)) dη− fk(t) ≤ Ak(t) |g(0)| |η|>Δ |ˆg(η)| dη ≤ Ak(t)M2 |g(0)| ε.˜ Thus, point(c) ofthetheoremstill holds,butwith constant
C = 4Δ + maxkAk∞M2
|g(0)| . 2
Remark3.Letusremarkthatthisextendedresultrequiresstrongerassumptionsonε:˜ conditions(17)and
(21)arereplacedby(24)and (25),respectively.
Remark 4. It is interesting to remark also that fk(t)ˆg(η − φk(t)) = V g f1
k(η,t) with f
1
k(τ ) = Ak(t)× e2iπ(φk(t)+(τ−t)φk(t)). Then, the right hand side of equations (16) and (20) can be rewritten respectively as|Vfg(η,t)− Vfg1 k(η,t)| and|∂tV g f(η,t)− ∂tV g f1
k(η,t)| sothatTheorem 1isbasedontheSTFTofaconstant amplitudeandfirstorderphaseapproximationofthekth mode.
3.4. Wavelet basedsynchrosqueezingtransform[8]
WenowrecallthedefinitionofSSTinthewaveletframework.
Definition 3. Let ε > 0 andΔ ∈ (0,1). The set AΔ,ε of multicomponent signals with modulation ε and
separationΔ correspondstosignalsdefinedin(7)withfk satisfying: Ak∈ C1(R) ∩ L∞(R), φk ∈ C2(R), inf t∈Rφ k(t) > 0, sup t∈Rφ k(t) <∞, Ak(t) > 0, |A k(t)| ≤ εφk(t),|φk(t)| ≤ εφk(t) ∀t ∈ R. (27)
Further,thefksareseparatedwithresolutionΔ,i.e.,forallk∈ {1,· · · ,K− 1} andallt |φ
Definition 4.Let h be a positiveL1-normalized window belonging to C∞
c (R), and consider γ,λ > 0, the wavelet-based synchrosqueezing(WSST)off withthresholdγ andaccuracy λ isdefinedby:
Sfλ,γ(ω, t) := Cψ −1 |Wψ f(a,t)|>γ Wfψ(a, t)1 λh ω− ˆωf(a, t) λ da a. (29)
Theorem3.Letf ∈ AΔ,ε,andsetε = ε˜
1
3.Letψ be awavelet belongingtoS(R) such thatψ is supportedin
[1− ,1+]. Then, providedε issufficiently small,thefollowinghold:
(a) |Wfψ(a,t)| ≥ ˜ε only when, there exists k ∈ {1,....,K}, such that for each pair (a,t) ∈ Zk := {(a,t), s.t. |aφk(t)− 1|< Δ}.
(b) For each k∈ {1,...,K} andthen forall(a,t)∈ Zk forwhichholds |Wfψ(a,t)|> ˜ε,wehave
|ωf(a, t)− φk(t)| ≤ ˜ε. (30)
(c) Moreover, foreach k∈ {1,...,K},there existsaconstant C suchthat foranyt∈ R lim λ→0 |ω−φ k(t)|<˜ε Sfλ,˜ε(ω, t)dω− fk(t) ≤ C ˜ε. (31)
Proof. This theorem is proved in[8] and gives a strong approximation result for the class AΔ,ε, since it
ensuresthatthenon-zerocoefficientsoftheWSSTarelocalizedaroundthecurves(t,φ1
k(t)) inthetime-scale space,andthatthereconstructionofthemodesiseasilyobtainedfromtheconcentratedrepresentation. 2 Remark 5. It is worth mentioning here that to consider compactly supported WSSTs in the frequency domain is crucial to ensure accurate estimations. However, these TF representations are not compatible with real-time applications,therefore new approacheshave been proposedinthat direction.Forinstance, in[4],waveletsaredesignedwithsufficientlymanyvanishingmomentsandaminimumsupportinthetime domain,themathematicalanalysisof suchanewmodelbeing stillunderdevelopment.
Remark 6. Pure waves obviously satisfy the assumptions of Theorems 1 and 3 for any ε. However, for modulated waves the assumptions for FSST and WSST are different because, in the FSST context, the assumption onthe frequencymodulationφ(t) does notdepend ontheIF. This isdue tothe factthatin the STFTframework, thefrequency resolution doesnot depend onthefrequency, whereasin thewavelet caseitdependsonthescale.
To makethisclearerintheSTFTcontext,weconsider theIFestimatecomputedonalinearchirp, i.e., a wavewith linear instantaneous frequency.To carry out this study, we compute STFTwith a Gaussian windowg(t)= e−πt2.Tochoosesuchawindowisinterestingforourpurpose,becauseonehasaclosedform expression fortheSTFTof alinearchirp,asrecalledhereafter.
Thestudy oftheIFestimateωfˆ (η,t) for alinearchirprequiresthefollowinglemma:
Lemma 1.Consider u(t)= e−πzt2, wherez = reiθ with cos θ > 0,so that u isintegrable. ThenitsFourier transformreads
ˆ
u(ξ) = r−12 e−iθ2e−reiθ−πξ
2
Proof. Thisresultisstraightforwardas onecanproceedas inthecasez real. 2
Then,consider thelinearchirphc(τ )= Ae2iπφ(τ ), whereφ is aquadratic polynomial.To startwith,we
remarkthathc(τ ) canbewritteninthefollowing form:
hc(τ ) = hc(t)e2iπ[φ
(t)(τ−t)+1
2φ(t)(τ−t) 2
]. (33)
Then,wehavethefollowingresult.
Proposition 1. The STFT of hc, computed using the Gaussian window g(t) = e−πt2 admits the following closed-formexpression: Vhcg(η, t) = hc(t)r− 1 2e−i θ 2e− π(1+iφ (t))(η−φ(t))2 1+φ (t)2 . (34)
Proof. Weremark that
Vhcg(η, t) = hc(t)F{e−π(1−iφ (t))τ2
}(η − φ(t)) = h
c(t) e−πzτ2(η− φ(t)),
wherez = 1− iφ(t)= reiθ.Lemma 1thengives
Vhcg(η, t) = hc(t)z− 1 2e−π (η−φ(t))2 z = h c(t)z− 1 2e−π (η−φ(t))2 r e−iθ. 2 (35)
Fromthis, weimmediatelygetthat:
ˆ
ωhc(η, t)− φ(t) = φ (t)2
1 + φ(t)2(η− φ
(t)). (36)
ThisshowsthattheIFestimateisnotexactforlinearchirp.Sonow,if|η − φ(t)|< ε,weobtain:
|ˆωhc(η, t)− φ(t)| < ε|1 − 1 1 + φ(t)2|.
Here we see that the quality of the estimate ωhc(η,t) close to the curve (t,φ(t)) only depends on the magnitudeofφ(t):ifthelattergoestozero,theestimatetendsto φ(t).Thisjustifieswhy,inTheorem 1, weassumethemodessatisfyφ(t)≤ ε:whatmattersishowsmallthemodulationis,φ(t) playingnorole. 4. NewFSSTbasedonsecondorderapproximation ofthephase
TheoriginalFSSTassumesφ(t) isnegligiblebut,inmanysituations,thesignalexhibitshighfrequency modulation,whichreducestheapplicabilityoftheresultsofTheorems 1 and3.Manydifferentapproaches havebeencarriedoutto takeintoaccount themodulationinthesynchrosqueezingcontext,as forinstance byintroducingademodulationoperatorbeforeapplyingthesynchrosqueezingtransform[13] [20].Inanother direction,anextensionofFSST,basedonasecondorderapproximationofthephase,wasrecentlyproposed in[16],but,to ourknowledge,notheoretical analysisoftheproposedestimate isavailable.Weproposeto bridgethatgap inthis section.
4.1. Computationofthenew IFestimate Remember ω˜f(η,t)=
∂tVfg(η,t)
2iπVfg(η,t) andthenintroduce:
˜
tf(η, t) = t−
∂ηVfg(η, t)
2iπVfg(η, t). (37)
Similarly to ωf(η,t) = R(˜ωf(η,t)), we define tf(η,t) = R(˜tf(η,t)). First, we recall an estimate of the frequencymodulationintroducedin[16].
Definition 5.Letf ∈ L2(R) andconsider whenVg
f(η,t)= 0 and∂t "∂ηVg f(η,t) Vg f(η,t) #
= 2iπ thequantity
˜ qf(η, t) = ∂tω˜f(η, t) ∂ttf˜(η, t) = ∂t "∂tVg f(η,t) Vfg(η,t) # 2iπ− ∂t"∂ηV g f(η,t) Vfg(η,t) #. (38)
Anestimateofthefrequencymodulationisthendefinedby ˆ
qf(η, t) =R (˜qf(η, t)) . (39)
HereweproposetofocusonthestudyofIFestimateassociatedwithTFrepresentationgivenbySTFT. In this regard, westudy aslightlydifferent IF estimate from the oneintroduced in[16], which allowsfor mathematical study.
Definition 6.Letf ∈ L2(R),wedefine thesecond orderIFcomplexestimateoff as:
˜ ω(2)f (η, t) = $ ˜ ωf(η, t) + ˜qf(η, t)(t− ˜tf(η, t)) if ∂t˜tf(η, t)= 0 ˜ ωf(η, t) otherwise, (40)
and thenitsrealpartωˆf(2)(η,t)=R(˜ωf(2)(η,t)).
Remark7.Theestimate(39)wasusedin[16]whereitwasprovedthatqfˆ(η,t)= φ(t),whenf isaGaussian modulatedlinearchirp,i.e.achirpwherebothφ andlog(A) arequadratic.Further,anewestimateofφ(t) wasalsoderivedthere,namely:
φ(t)≈ ωf(η, t) + ˆqf(η, t)(t− ˆtf(η, t)), (41) whichisexactforconstantamplitudelinearchirp.
Proposition 2. Let f ∈ L2(R), then the IFestimate ω(2)
f (η,t) can be expressed by means of five different STFTs, since wehave: ˜ ωf(η, t) = η− 1 2iπ Vfg(η, t) Vfg(η, t), ˜ qf(η, t) = 1 2iπ Vfg(η, t)Vfg(η, t)− (Vfg(η, t))2 Vftg(η, t)Vfg(η, t)− Vftg(η, t)Vfg(η, t), t− ˜tf(η, t) =− Vftg(η, t) Vfg(η, t).
Proof. The expressions forω˜f(η,t) andt− ˜tf(η,t) arestraightforward. Indeed,since g isinthe Schwartz class,theSTFToff belongstoC∞(R),and wehave:
∂ηVfg(η, t) =−2iπVftg(η, t) ∂tVfg(η, t) = 2iπηV g f(η, t)− V g f (η, t). (42)
Basedontheseequalities, theexpressionforqf˜(η,t) iseasyto obtain. 2
In the following section, we use IF estimate ωˆf(2)(η,t) to define a new synchrosqueezing transform for whichweproveapproximationresults.
Remark8.ItisworthnoticingherethatbyexploitingthepropertiesofSTFTregardingderivation, Propo-sition 2tellsusthatonedoesnotneedtousefinite differencestocomputeqf˜(η,t) andωf˜ (η,t).
4.2. Definition of thenewFSST andapproximation results
Inthissection,wedefineanotherclassofchirp-likefunctionslargerthanBΔ,εandshowthattheycanbe
successfullydealtwithbymeansof secondorder FSST,whichisdefinedinthissection. Sofirst,wedefine thenewsetofmulticomponent signalswearestudying:
Definition 7.Let ε> 0 and Δ> 0.Theset B(2)Δ,ε ofmulticomponent signalswith secondorder modulation ε andseparationΔ correspondstothesignalsdefinedin(7)satisfying:
(a) functionfk issuchthatAk andφk satisfythefollowing conditions: Ak(t)∈ L∞(R) ∩ C2(R), φk(t)∈ C3(R), φk(t), φk(t), φk(t)∈ L∞(R), Ak(t) > 0, inf t∈Rφ k(t) > 0, sup t∈Rφ k(t) <∞, |A k(t)| ≤ ε, |Ak(t)| ≤ ε, and |φk (t)| ≤ ε, (b) functionsfkssatisfythefollowing separationcondition
φk+1(t)− φk(t) > 2Δ, ∀t ∈ R, ∀k ∈ {1, · · · , K − 1}. (43) Now,letusdefine thesecond orderFSSTas follows
Definition 8. Let h be apositive L1-normalized windowbelonging to C∞
c (R), and consider γ,λ> 0, the secondorder FSSToff withthresholdγ andaccuracyλ isdefinedby:
Tfλ,γ(ω, t) = 1 g(0) |Vg f(η,t)|>γ Vfg(η, t)1 λh ω− ˆωf(2)(η, t) λ dη. (44)
InSection3, weshowedthat,for functionsf ∈ BΔ,ε,agoodIF estimatewasgiven byωˆf(η,t) and the approximationtheorem followed.Here,toassesstheapproximationpropertyof thesecondorderFSST we have just introduced, we consider a function f ∈ B(2)Δ,ε for which we are going to prove that a good IF estimateisprovidedbyωˆf(2)(η,t). Theapproximation theoremisasfollows:
Theorem 4.Consider f ∈ B(2)Δ,ε, setε = ε˜ 1/6. Letg be awindow satisfying, forallt, allk = 1,· · · ,K and
r ∈ {0,1,2} that, if |η| ≥ Δ, |F{τrg(τ )eiπφk(t)τ2}(η)| ≤ Krε, where K
r is some constant. Furthermore if, for all k = 1,· · · ,K,
|η|>Δ
|F{g(τ)eiπφk(t)τ
2
}(η)|dη ≤ K3ε,˜ for some constant K3, then, provided ε is
sufficiently small,thefollowinghold:
(a) |Vfg(η,t)|≥ ˜ε onlywhen thereexistsk∈ {1,....,K} suchthat(η,t)∈ Zk.
(b) For allk∈ {1,...,K} andall(η,t)∈ Zk suchthat |Vfg(η,t)|> ˜ε and |∂t˜tf(η,t)|> ˜ε we have |ω(2)
f (η, t)− φk(t)| ≤ ˜ε. (45)
(c) Moreover, foreach k∈ {1,...,K},there existsaconstant D suchthat ⎛ ⎜ ⎝ limλ→0 Mk, ˜ε Tfλ,˜ε(ω, t)dω ⎞ ⎟ ⎠ − fk(t) ≤ D˜ε, (46)
whereMk,˜ε:={ω : |ω − φk(t)|< ˜ε},providedtheLebesguemeasureμ{η, s.t.(η,t) ∈ Zk,|∂t˜tf(η,t)|≤ ˜
ε}≤ γ˜ε forsome constant γ.
Theproof ofthisTheoremisavailableinSectionAppendix. Remark9.The assumption|η|> Δ⇒ |F{g(τ)eiπφk(t)τ2}(η)|≤ K
0ε issomewhat complex,so wetryhere
to explainwhatitreally means.Letus fixtheridgenumberk andsetc= φk(t), theassumption ensuresa sufficient decreaseoftheFouriertransformof:
l(τ ) = g(τ )eiπcτ2.
ForaGaussianwindow,g(t)= e−πt2,ˆl hasthefollowing closed-formexpression: |ˆl(η)| =%1 + c2&− 1 4 e1+c2−π η 2 .
It isclearthat,ifwetakeΔ largeenough,weachievetheassumption.Wheng isnotaGaussian function, we cannot compute ˆl, but can still perform an estimation of the decay of |ˆl| by means of the stationary phaseapproximation: thephasewithintheintegral
ˆ l(η) =
R
g(τ )eiπcτ2e−2iπητdτ
admits astationary pointatτc= ηc.Thestationaryphasetheoremfinally gives |ˆl(η)| ≈'1
|c|
g"ηc# .
Again, by assuming Δ large enough, one can satisfy the assumption. Now, if we consider all the ridges together, things are more complex, since choosing alarge Δ implies theridges need to be farther apart. But this issomethingthatis expected:the frequencymodulationmakes thefrequency supportofthe TF responseassociatedwithonemodelarger,thusitchangestheseparationcondition.Inpractice,thewindow’s choiceisdrivenbythistrade-off betweenseparationandlocalization.
4.3. Second order synchrosqueezingintheCWT framework
Itis stillpossible todefine asecondorder IFestimate, inthewaveletframework, byputtingt˜f(a,t):=
Rτ f (τ )1aψ(τ−ta )∗dτ Wfψ(a,t)
, q˜f(a,t)=∂t∂tωf˜tf˜(a,t)(a,t),andthendefining:
Definition9.
˜
ωf(2)(a, t) = $
˜
ωf(a, t) + ˜qf(a, t)(t− ˜tf(a, t)) if ∂t˜tf(a, t)= 0 ˜
ωf(a, t) otherwise, (47)
andthenitsreal partωˆf(2)(a,t)=R(˜ω(2)f (a,t)).
Then,todefine asecondorder synchrosqueezingoperatorinthewavelet caseisstraightforward:wejust replacethe IFestimate ωfˆ byωˆf(2). Unfortunately,itstheoretical analysis cannotbe directlyderivedfrom thejuststudiedSTFTframework,thereforeitisleftforfuturework.
5. Numericalresults
Inorder to illustrate the behavior of ournewsecond ordersynchrosqueezing transform, we apply it to thesametestsignalsasin[16].One(signal1)ismadeoflow-orderpolynomialchirps,thatbehavelocallyas linearchirps,andtheotherone(signal2)containsstronglynonlinearsinusoidalfrequencymodulations.We willcomparetheresultsoftheFSST(first-orderSTFT-basedSST)andthenewsecond-order synchrosqueez-ingtransform, designedbyVSST.Wechoosethis denominationafter theso-calledVSSTtechniquewhich istheothertransformbasedonthesecondorderapproximationofthephase,whichwasintroducedin[16].
Fig. 1shows theSTFT,FSST,and newVSSTforbothtest-signals1and2.Note thatinoursimulations, weused1024timesamplesover[0,1] andaGaussianwindowofsize400.Forreproducibilitypurposes,the Matlab codeforVSST, as well as thescripts generatingall theFigures of this paper,can be downloaded from http://oberlin.perso.enseeiht.fr/files/vsst.zip.
5.1. Quality ofrepresentation
In order to quantify the quality of the representation given bythe new VSST, we proposeto measure theamountofinformationcontainedinthecoefficientswiththelargestamplitude.Inthisregard,oneway to compare the different transformations is to compute the normalized energy associated with the first coefficients with the largest amplitude: the faster the growth of this energy towards 1, the sharper the representation. In Fig. 2A and B,these normalized energies are displayedwith respect to the number of coefficientskeptdividedbythelengthofthesignal,forboth test-signals,andforthethreerepresentations, namely VSSTproposedin[16], newVSSTand FSST. Inourcontext,thenormalized energyis computed as the cumulative sumof the squaredsorted coefficients over thesum of allthe squaredcoefficients. The firstremarkisthatbothVSSTandnewVSSTbehavesimilarlyfortest-signal1whileslightlybetterresults areobtainedwhen thelatteris usedontest-signal2,which containsstrongerfrequencymodulations.The second remark is thatoneneedsonly 3coefficientsper time instantto recoverthe signalenergy,which is consistentwiththethreemodesmakingupthetest-signal.Inordertoinvestigatetheinfluenceofnoiseon thesparsityoftherepresentation,wecarryoutthesameexperimentswhenthetest-signalsarecontaminated bywhiteGaussiannoise(noiselevel0dB).TheresultsdisplayedonFig. 2CandDexhibitaslowerincrease ofthenormalizedenergy,sincethecoefficientscorresponding tonoisearespreadoverthewholeTFplane. However,ourproposednewVSSTstillbehavesbetterthanbothVSSTandFSSTinthisnoisysituation.
Fig. 2. Normalized energyasafunctionofthenumberofsortedcoefficientsfortest-signals1(A)and2(B).Abscissagives the numberofcoefficientskeptoverthelengthN ofthesignal,i.e.,themeannumberofcoefficientskeptineachcolumnoftheTF plane.CandD:idemasAandBbutfornoisysignals(0 dB).
TobetterevaluatethequalityoftheTFrepresentation,wewillcomparetheobtainedrepresentationand theidealone,bymeansoftheEarthmover’sdistance(EMD)[17].EMDisasliced(fixedtime)Wasserstein distance aimed at comparing probabilitydistributions, thathas been already used in the time–frequency context by[10,24].Wewill compareour newVSSTtechniquewithFSST, andwith therecentlyproposed synchrosqueezed wave packettransform (SSWPT)[24].This last techniquehas provento be morerobust tonoisethanstandardsynchrosqueezingtransform,becauseitmakesuseofredundancyofthewavepacket decomposition.Wewillalsocomparedtheobtainedrepresentationwiththereassignmentmethod(RM)[1], whichbehaveswell withfrequencymodulation,butdoesnotallowforreconstruction.
ToclearlystatetheinterestofthenewVSSTweintroduced,weinvestigatethequalityofrepresentation associated withthemiddlemode oftest-signal2,whichexhibits relativelyhigh frequencymodulation(the sampling frequencyis 1024Hz).TheresultsdepictedinFig. 3show thebenefitsoftakingthemodulation intoaccount,sinceRMandnewVSSTbehavemuchbetterthantheothermethodsatlownoiselevels.Also, at veryhigh noiselevels,themethodbasedonwaveletpackettransformprovideswithamoreaccurateTF representation, butthenewVSSTremainsalwaysbetterthatFSSTevenintheseconfigurations.
An alternative technique,called ConceFT wasrecently proposedin [10] and consists inusing a multi-taperapproachintheWSSTframework,atechniquefirstintroducedin[23]inthecontextofspectrogram reassignment. Inanutshell,theapproachaimsto improvetheTFrepresentation associated withthe syn-chrosqueezingusingasinglewaveletbyaveragingWSSTsassociatedwithlinearcombinationsoforthogonal wavelets.Inmanysituations,thisapproachresultsinasignificantpartofthenoisebeingremoved.However, since it relies on transforms that do not explicitly take into account the modulation, the results are not satisfactory whenappliedtosignalswheretheformerisimportant.Inthisregard,webelievethatitwould
Fig. 3. ComparisonofthequalityoftheTFrepresentationseithergivenbySTFT,FSST,RM,VSST,SSWPT(forredundancy factorred=1or20),forthehighestfrequencymodeoftest-signal 2.
Fig. 4. Comparison of the quality of the TF representations given by FSST and VSST, with 4 different windows.
beofinteresttodesignanewversionofConceFTwhoseblockbasiswouldbeoursecond orderSSTrather thanawavelettransform,butthisisleftforfutureinvestigation.
5.2. Choiceof thewindow
Thechoiceofthewindow’slengthisacriticalstepforanytime–frequencyrepresentation.Acarefulstudy ofthe approximationtheorem offirst-order SSTshowsthatthewindowshouldbe takensmallenough, to minimize theestimation error;and large enoughto satisfy theseparation conditionbetween the different modes. Inpractice, oneneedsto tune thewindow’s lengthto achieveatrade-off between localization and separation.
In the VSST case, the window must still be chosen so as to satisfy the separation condition between modes.But,sinceitusesasecond-orderapproximationofthephase,VSSTdoesnotrequireasmallwindow to produce a concentrated representation. This phenomenon is illustrated in the following Fig. 4, which displaysthe EMD withrespect to theinput SNRfor thesamesignalas inFig. 3.It comparesFSST and VSST,computedwith aGaussian windowwith 4significant valuesof σ rankedincreasingly,meaningthat tochooseσ outside[σ1,σ4] leadstoworse results.Itisclearthatforagiven σ,VSSTalwaysconsiderably
improvestheresults. Moreover,theperformanceof FSSTfor theoptimalwindowσ3 issignificantly worse
than theresult provided byVSST. Finally, both σ2 and σ3 giveaccurate results forVSST, showing that
thewindow’schoicewithourmethodislesscriticalthanwith FSST.
To better illustrate the influenceof the window, we show inFig. 5 the resultsof FSST and VSST for window’s sizes σ3 and σ4. We remark that for σ3, which is the optimal parameter for FSST, the latter
does notgive avery sharp representation and, taking a largervalue, as for instance σ4, resultsinstrong
Fig. 5. Comparison between FSST and VSST, with two different window’s sizes (σ3< σ4).
5.3. Reconstructionofthemodes
The main advantageof synchrosqueezing techniquesover traditional reassignmenttechniques liesinits invertibility.Toimprovetheaccuracyofthereassignmentstepinsynchrosqueezingtechniquesbyusingnew VSSTnaturallyleadstobetterreconstructionresults.Thereconstructionformulausedto retrievethekth mode,assumingφˆk(t) isanestimateofφk(t),isas follows:
fk(t)≈ |ω− ˆφk(t)|<d
Tf(ω, t)dω, (48)
as explained inmore detailsin [16]. The parameter d ishere to compensate for the factthat φˆk(t) is an estimateofφk(t) andnotthetruevalue.Providedd isofthesameorderofmagnitudeastheestimationerror, thisreconstructionformulaensuresanasymptoticallyperfectreconstruction.Tocomputeanestimationof the ridges (t,φk(t))k, knowing the number K of modes, we use the algorithm introduced in [3], which computesalocalminimumofthefunctional:
Ef((ϕk)k=1,··· ,K) = K k=1 − R |Tf(t, ϕk(t))|2dt + R (λϕk(t)2+ βϕk(t)2)dt, (49)
λ and β beingtwo positivetuningparameters. Then ( ˆφk(t))k=1,··· ,K = argmin Ef((ϕk)k=1,··· ,K).In prac-tice, we notice thatsince the TF representation given by the second order synchrosqueezing is very well concentrated, thechoiceforregularizationparametersisnotessential.
To illustrate thebehavior ofour newVSSTtechnique,we firstdisplaythe reconstructionprocess asso-ciated withtest-signal2inFig. 6A,theridges used forreconstructionbeing depictedinFig. 6B.Then, to compare theproposed new VSST with the alternative technique proposed in[16], we display, inFig. 6C and D,thereconstructionerrorateachtimeinstantforthethreemodesandforbothVSSTandnewVSST. TheimprovementbroughtbyusingnewVSSTinsteadofoldVSSTappearstobethemostsignificantwhen the modulation is strong (i.e., in the mode associated with the highest frequency in the studied case). ComparedtothesamefigureplottedforFSST in[16], theerrorisreally decreased.
To further quantifythe accuracy of mode reconstructioninnoisy situations,we investigate theoutput SNRfortest-signal2asafunctionofinputSNR.TheproposedcomparisoninvolvesFSST andnewVSST and the computation is performed for two different values of parameter d, using the true instantaneous frequenciesto avoiddependencyontheridgedetectionstep.Itisworthmentioningherethatparameterd enablestheerrorassociatedwithIFestimationwiththedifferenttechniquestobecompensatedfor.Indeed, togetgoodreconstructionresults,d needstobeallthelargerthattheIFestimateisinaccurate.Also,alarger d stillcompatiblewiththeseparationconditiononthemodes, willalsomeanbetterreconstructionresults. These factsare numerically illustrated inFig. 7 where, for afixed d, the reconstructionresults is always betterwithnewVSSTthanwithFSST,andwhere,foragivenmethod,thequalityofthereconstructionis improvedbychoosingalargerd.Also,oneremarks thatwhatever thevalueofd,thereconstructiongiven bySSTreachesastepanddoesnotincreaseaftersomeinputSNR.Thisisbecauseevenwithnonoise,the SSTcannotreachaveryhigh reconstructiondue tothebadIFestimate.
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6. Conclusion
In this paper, we developed a novel synchrosqueezing transform for analyzing multicomponent signals made of strongly frequency-modulated modes, based on the short-time Fourier transform. It simply consists in a refinement of the instantaneous frequency estimate, computed using a second-order expansion of the phase. After having revisited the case of first-order synchrosqueezing, releasing the hypothesis of a window compactly supported in the frequency domain, we proved a novel approximation theorem involving the
proposed newsynchrosqueezingtransform appliedtomulticomponent signals madeofstrongly modulated modes. Numerical experimentsshowed thebenefits of taking into account frequency modulationfor both representation and reconstructionpurposes. Interestingly, thenew transformation has proven to be quite robust against noise and exhibits less sensitivity to the window’s choice. A limitation of the proposed theoretical study isthat itis restricted to modes with small amplitude modulations,and to improve this aspect isaworkcurrentlyunderway.
Acknowledgments
The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR) under grantANR-13-BS03-0002-01.
Appendix A
Theproofof Theorem 4involves anumberofestimates,whichareshownhereafter. Lemma 2.Forany (η,t),there isatmostone k∈ {1,· · · ,K} for which|η − φk(t)|≤ Δ.
Theproof isstraightforwardand islefttothereader.Forourpurpose,weneedtoanalyzethebehavior of theSTFTonalinearchirp,forwhichwehavethefollowingproposition.
Proposition 3.Let h(τ )= Ae2iπφ(τ ) be a linear chirp and consider Vtrg
h , for r∈ {0,1,2}, theSTFT of h obtained with awindowtrg, whereg satisfies thehypothesesof Theorem 4.If|η − φ(t)|> Δ then
|Vtrg
h (η, t)| ≤ εKrA.
Proof. WeknowthatVhtrg(η,t)= h(t)F{τrg(τ )eiπφ(t)τ2
}(η −φ(t)) sothat,if|η −φ(t)|> Δ,assumptions ong lead to|Vhtrg(η,t)|≤ εKrA. 2
Proposition4.Forany k∈ {1,· · · ,K},any r∈ {0,1,2},and any(η,t)∈ Zk/ ,wehave: |Vtrg
fk (η, t)| ≤ εEk,r(t), (50)
with Ek,r(t)= Ir+1+ (π3Ir+3+ Kr)Ak(t).Consequently, forany(η,t)∈ Zk: |Vtrg f (η, t)− V trg fk (η, t)| ≤ ε l=k El,r(t) := εΩk,r(t), (51)
Proof. First, wewrite
fk(τ ) = Ak(τ )e2iπφk(τ )= (Ak(τ )− Ak(t))e2iπφk(τ ) + Ak(t)e2iπ[φk(t)+φ k(t)(τ−t)+12φk(t)(τ−t) 2 ] + Ak(t)[e2iπ[φk(t)+φ k(t)(τ−t)+12φk(t)(τ−t) 2 +1 2 τ t φ k(x)(τ−x) 2 dx]− e2iπ[φk(t)+φk(t)(τ−t)+12φk(t)(τ−t) 2 ]] = fk,1(τ ) + fk,2(τ ) + fk,3(τ ). (52)
Then,forany (η,t), |Vtrg fk,1(η, t)| ≤ R |Ak(τ )− Ak(t)||τ − t|r|g(τ − t)|dτ, ≤ R ε|τ − t|r+1|g(τ − t)|dτ ≤ εIr+1, and |Vtrg fk,3(η, t)| ≤ πAk(t) R ⎛ ⎝ τ t |φ(x)||τ − x|2dx ⎞ ⎠ |τ − t|r|g(τ − t)| dτ ≤ π 3Ak(t) R ε|τ − t|r+3|g(τ − t)| dτ ≤ π 3Ak(t)εIr+3.
UsingProposition 3, weget(50), andinequality(51)follows. 2 Now,if(η,t)∈/ K l=1 Zl,weimmediatelyget: |Vg f(η, t)| ≤ ε K l=1 El,0(t)≤ ˜ε,
whenε is˜ sufficientlysmall, whichprovesitem(a)of Theorem 4.
Now,weintroduceseveralpropositionsthatareusefultoproveitem (b)ofTheorem 4. Proposition5.Forany (η,t)∈ Zk,assumingg satisfies thehypothesesof Theorem 4,wehave:
∂tVfg(η, t)− 2iπ"φk(t)Vfg(η, t) + φk(t)Vftg# ≤ εBk,1(t), (53) where Bk,1(t) = KI0+ K k=1 Ak∞πI2+ 2π l=k (φl(t)El,0(t) +|φl(t)|El,1(t)) + 2π(φk(t)Ωk,0(t) +|φk(t)|Ωk,1(t))
Proof. DifferentiatingtheSTFToff withrespectto t,wegetforany (η,t):
∂tVfg(η, t) = K k=1 R
Ak(τ )e2iπφk(τ )g(τ− t)e−2iπη(τ−t)dτ
+ K k=1 R
Ak(τ )2iπφk(τ )e2iπφk(τ )g(τ− t)e−2iπη(τ−t)dτ
= K k=1 R
+ K k=1 R Ak(τ )2iπ ⎡ ⎣φ k(t) + (τ− t)φk(t) + τ t (τ− u)φk(u)du ⎤
⎦ e2iπφk(τ )g(τ− t)e−2iπη(τ−t)dτ
= K k=1 R
Ak(τ )e2iπφk(τ )g(τ− t)e−2iπη(τ−t)dτ + K k=1 2iπφk(t)Vfkg(η, t) + K k=1 2iπφk(t)Vfktg(η, t) + K k=1 R Ak(τ )2iπ ⎛ ⎝ τ t (τ− u)φk(u)du ⎞ ⎠ e2iπφk(τ )e−2iπη(τ−t)g(τ− t)dτ. Wemaythenwrite:
∂tVfg(η, t)− 2iπ K k=1 " φk(t)Vfkg(η, t) + φk(t)Vfktg(η, t)# ≤ K k=1 R |A k(τ )||g(τ − t)|dτ + 2π K k=1 R Ak(τ ) ⎛ ⎝ τ t |τ − u||φ k(u)|du ⎞ ⎠ |g(τ − t)|dτ ≤ ε KI0+ K k=1 Ak∞πI2 .
From Proposition 4,wefirsthave,when(η,t)∈ Zk: ∂tVfg(η, t)− 2iπ " φk(t)Vfkg(η, t) + φk(t)Vfktg(η, t)# ≤ ε ⎛ ⎝KI0+ K k=1 Ak∞πI2+ 2π l=k (φl(t)El,0(t) +|φl(t)|El,1(t)) ⎞ ⎠ , and then, ∂tVfg(η, t)− 2iπ " φk(t)Vfg(η, t) + φk(t)Vftg(η, t)# ≤ ε ⎛ ⎝KI0+ K k=1 Ak∞πI2+ 2π l=k (φl(t)El,0(t) +|φl(t)|El,1(t)) + 2π(φk(t)Ωk,0(t) +|φk(t)|Ωk,1(t))) , hence(53). 2
Proposition 6.Forany (η,t)∈ Zk onehas: ∂2 ttV g f(η, t)− 2iπ " φk(t)Vfg(η, t) + φk(t)∂tVfg(η, t) + φk(t)∂tVftg(η, t)# ≤ εBk,2(t), (54) where
Bk,2(t) = Λ0+ 2π l=k (φl(t)El,0(t) + φl(t)Fl,0(t) + φ(t)Fl,1(t)) + 2π[φk(t)Ωk,0(t) + φk(t) l=k Fl,0(t) + φk(t) l=k Fl,1(t)
Proof. Since,forany(η,t),onehas:
∂tt2Vfg(η, t) = K l=1 R (Al(τ ) + 2iπφl(τ )Al(τ ) + 2iπφl(τ )Al(τ ) −4π2φ l(τ )2Al(τ ) &
e2iπφl(τ )g(τ− t)e−2iπητ−tdτ = K l=1 R
Al(τ )e2iπφl(τ )g(τ− t)e−2iπη(τ−t)dτ
+ 2iπ K l=1 R
φl(τ )Al(τ )e2iπφl(τ )g(τ− t)e−2iπη(τ−t)dτ
+ 2iπ K l=1 R φl(τ )fl(τ )g(τ− t)e−2iπη(τ−t)dτ,
wehave,forthefirstpartof∂tt2Vfg(η,t), K l=1 R
Al(τ )e2iπφl(τ )g(τ− t)e−2iπη(τ−t)dτ
≤KεI0. (55)
Then,forthesecond partof∂2
ttV g f(η,t),wemaywrite 2iπ K l=1 R
φl(τ )Al(τ )e2iπφl(τ )g(τ− t)e−2iπη(τ−t)dτ− 2iπ K l=1 φl(t)Vflg(η, t) ≤ 2π K l=1 R ⎛ ⎝ τ t |φ l (u)|du ⎞ ⎠ Al(τ )|g(τ − t)|dτ ≤ 2πε K l=1 Al∞I1.
Finally,wewriteforthethirdpartof∂tt2V g
f(η,t),usingsecond orderTaylorexpansionofφl(τ ): 2iπ K l=1 R
φl(τ )fl(τ )g(τ− t)e−2iπη(τ−t)dτ − 2iπ K l=1 φl(t)∂tV g fl(η, t) + φl(t)∂tV tg fl(η, t) = 2iπ K l=1 R τ t
(τ− u)φl (u)du (Al(τ ) + 2iπφl(τ )Al(τ )) e2iπφl(τ )g(τ− t)e−2iπη(τ−t)dτ ≤ πε K l=1 (Al∞+ 2πφl∞Al∞)I2.
From this,weget: ∂tt2V g f(η, t)− 2iπ K l=1 " φl(t)Vflg(η, t) + φl(t)∂tVflg(η, t) + φl(t)∂tVfltg(η, t)# ≤ εΛ0, (56) where Λ0= KI0+ 2π K l=1 Al(t)∞I1+ π K l=1 (Al∞+ 2πφl∞Al∞)I2.
Now, recallingthat:
|∂tVflg(η, t)− 2iπ"φl(t)Vflg(η, t) + φl(t)Vfltg(η, t)#| ≤ ε (I0+ πAl∞I2) ,
|∂tVtg
fl(η, t)− 2iπ "
φl(t)Vfltg(η, t) + φl(t)Vflt2g(η, t)#| ≤ ε(I1+ πAl∞I3),
we maywrite,if(η,t)∈ Zk,andifl= k,that |∂tVg
fl(η, t)| ≤ ε (I0+ πAl∞I2+ 2π(φl(t)El,0(t) + φl(t)El,1(t))) := εFl,0(t) |∂tVtg
fl(η, t)| ≤ ε (I1+ πAl∞I3+ 2π(φl(t)El,1(t) + φl(t)El,2(t))) := εFl,1(t), and, finally, |∂tVg f(η, t)− ∂tV g fk(η, t)| ≤ ε l=k Fl,0(t) |∂tVtg f (η, t)− ∂tV tg fk(η, t)| ≤ ε l=k Fl,1(t).
Using thesefourinequalities,wecanfinallywrite: ∂2 ttV g f(η, t)− 2iπ " φl(t)V g f(η, t) + φl(t)∂tV g f(η, t) + φl(t)∂tV tg f (η, t)# ≤ ε ⎛ ⎝Λ0+ 2π l=k (φl(t)El,0(t) + φl(t)Fl,0(t) + φ(t)Fl,1(t)) +2π[φk(t)Ωk,0(t) + φk(t) l=k Fl,0(t) + φk(t) l=k Fl,1(t) ⎞ ⎠ , hencetheresult. 2
Now wecanproveitem(b)ofTheorem 4.
Proposition 7.Forany (η,t)∈ Zk,suchthat |Vfg(η,t)|> ˜ε and |∂t˜tf(η,t)|> ˜ε one has:
|˜qf(η, t)− φk(t)| ≤ ˜ε. (57)
Proof. For thesakeofclarity,weomittowrite (η,t) intheproofthatfollows: |φ k(t)− ˜qf(η, t)| = φk(t)− Vfg∂2 ttV g f − (∂tV g f)2 2iπ(Vfg)2− Vg f∂ηt2V g f + ∂tVfg∂ηVfg
= ∂tVfg ( ∂tVfg+ φk(t)∂ηVfg ) 2iπ(Vfg)2− Vg f∂ηt2V g f + ∂tVfg∂ηVfg −V g f ( ∂2 ttV g f + φk(t)∂2ηtV g f − 2iπφk(t)V g f ) 2iπ(Vfg)2− Vg f∂2ηtV g f + ∂tVfg∂ηVfg = ∂tVfg[∂tVfg− 2iπφk(t)V tg f − 2iπφk(t)V g f] 2iπ(Vfg)2+ 2iπVg f∂tV tg f − 2iπ∂tV g fV tg f +
Vfg[∂tt2Vfg− 2iπφk(t)∂tVftg− 2iπφk(t)Vfg− 2iπφk(t)∂tVfg] 2iπ(Vfg)2+ 2iπVg f∂tVftg− 2iπ∂tV g fV tg f ≤ εBk,1(t) ∂tVfg 2iπ[(Vfg)2+ Vg f∂tVftg− ∂tV g fV tg f ] + εBk,2(t) Vfg 2iπ[(Vfg)2+ Vg f∂tV tg f − ∂tV g fV tg f ] ≤ ˜ε3"Bk,1(t)|∂tVg f| + Bk,2(t)|V g f| # ≤ ˜ε, ifε is˜ sufficientlysmall. 2
Lemma3.Forallk∈ {1,...,K} andany (η,t)∈ Zk suchthat |Vfg(η,t)|> ˜ε and |∂tt˜f(η,t)|> ˜ε,wehave |ω(2)
f (η, t)− φk(t)| ≤ ˜ε. (58)
Proof. Accordingto definitionof ω(2)f (η,t) in(40),wehave ω(2) f (η, t) = ˜ωf(η, t) + ˜qf(η, t)(t− ˜tf(η, t)). Itfollows that ω(2) f (η, t)− φk(t) = ∂tVfg(η, t) + ˜qf(η, t)∂ηVfg(η, t) 2iπVfg(η, t) − φ k(t) = ∂tVfg(η, t)− 2iπ˜qf(η, t)Vftg(η, t) 2iπVfg(η, t) − φ k(t) ≤ ε Bk,1(t) 2iπVg f(η, t) +|˜qf (η, t)− φk(t)|Vftg(η, t) Vg f(η, t) ≤ ε Bk,1(t) 2πVg f(η, t) + ˜ε3 " Bk,1(t)|∂tVfg(η, t)| + Bk,2(t)|Vfg(η, t)|# |Vg f(η, t)| ≤ ˜ε whenε is˜ sufficientlysmall. 2
Toprove(c)ofTheorem 4,isexactlythesameastheproofintheweakmodulationcase,exceptthatwe use,attheveryend oftheproof,thehypothesis:
1 g(0) |η|>Δ F{g(τ)eiπφl(t)τ 2 }(η)dη ≤ K4ε˜ |g(0)| for any l,
and 1 |g(0)| {|η−φ k(t)|<Δ} {{|Vg f(η,t)|≤˜ε} {|∂t˜tf(η,t)|≤˜ε}} Vfg(η, t)dη ≤ ( 2Δ |g(0)|+ γ|Ak(t)|)˜ε to conclude. References
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