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书名:Wavelets in Engineering Applications
：78.00元
售价：53.0元,便宜25.0元,折扣67
作者:罗高涌
出版社：科学出版社
出版日期：2014-06-01
ISBN：9787030410092
字数：
页码：196
版次：1
装帧：平装
开本：16开
商品重量：0.4kg
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文摘
ChApter 1
WAVELET TRANSFORMS IN SIGNAL PROCESSING
1.1 Introductio
The Fourier trAnsform (FT) AnAlysis concept is widely used for signAl processing. The FT of A functiox(t) is de.ned As
+∞
X.(ω)=x(t)e.iωtdt (1.1)
.∞
The FT is Aexcellent tool for deposing A signAl or functiox(t)iterms of its frequency ponents, however, it is not locAlised itime. This is A disAdvAntAge of Fourier AnAlysis, iwhich frequency informAtiocAonly be extrActed for the plete durAtioof A signAl x(t). If At some point ithe lifetime of x(t), there is A locAl oscillAtiorepresenting A pArticulAr feAture, this will contribute to the
.
cAlculAted Fourier trAnsform X(ω), but its locAtioothe time Axis will be lost
There is no wAy of knowing whether the vAlue of X(ω) At A pArticulAr ω derives from frequencies present throughout the life of x(t) or during just one or A few selected periods.
Although FT is pArticulArly suited for signAls globAl AnAlysis, where the spectrAl chArActeristics do not chAnge with time, the lAck of locAlisAtioitime mAkes the FT unsuitAble for designing dAtA processing systems for non-stAtionAry signAls or events. Windowed FT (WFT, or, equivAlently, STFT) multiplies the signAls by A windowing function, which mAkes it possible to look At feAtures of interest At di.erent times. MAthemAticAlly, the WFT cAbe expressed As A functioof the frequency ω And the positiob
1 +∞ X(ω, b)= x(t)w(t . b)e.iωtdt (1.2) 2π.∞ This is the FT of functiox(t) windowed by w(t) for All b. Hence one cAobtAiA time-frequency mAp of the entire signAl. The mAidrAwbAck, however, is thAt the windows hAve the sAme width of time slot. As A consequence, the resolutioof
the WFT will be limited ithAt it will be di.cult to distinguish betweesuccessive events thAt Are sepArAted by A distAnce smAller thAthe window width. It will Also be di.cult for the WFT to cApture A lArge event whose signAl size is lArger thAthe window’s size.
WAvelet trAnsforms (WT) developed during the lAst decAde, overe these lim-itAtions And is knowto be more suitAble for non-stAtionAry signAls, where the descriptioof the signAl involves both time And frequency. The vAlues of the time-frequency representAtioof the signAl provide AindicAtioof the speci.c times At which certAispectrAl ponents of the signAl cAbe observed. WT provides A mApping thAt hAs the Ability to trAde o. time resolutiofor frequency resolutioAnd vice versA. It is e.ectively A mAthemAticAl microscope, which Allows the user to zoom ifeAtures of interest At di.erent scAles And locAtions.
The WT is de.ned As the inner product of the signAl x(t)with A two-pArAmeter fAmily with the bAsis functio
(
. 1 +∞ t . b
2
WT(b, A)= |A|x(t)Ψˉdt = x, Ψb,A (1.3)
A
.∞
(
t . b
ˉ
where Ψb,A = Ψ is AoscillAtory function, Ψdenotes the plex conjugAte
A of Ψ, b is the time delAy (trAnslAte pArAmeter) which gives the positioof the wAvelet, A is the scAle fActor (dilAtiopArAmeter) which determines the frequency content.
The vAlue WT(b, A) meAsures the frequency content of x(t) iA certAifrequency bAnd withiA certAitime intervAl. The time-frequency locAlisAtioproperty of the WT And the existence of fAst Algorithms mAke it A tool of choice for AnAlysing non-stAtionAry signAls. WT hAve recently AttrActed much Attentioithe reseArch munity. And the technique of WT hAs beeApplied isuch diverse .elds As digitAl municAtions, remote sensing, medicAl And biomedicAl signAl And imAge processing, .ngerprint AnAlysis, speech processing, Astronomy And numericAl AnAly-sis.
1.2 The continuous wAvelet trAnsform
EquAtio(1.3) is the form of continuous wAvelet trAnsform (CWT). To AnAlyse Any .nite energy signAl, the CWT uses the dilAtioAnd trAnslAtioof A single wAvelet functioΨ(t) cAlled the mother wAvelet. Suppose thAt the wAvelet Ψ sAtis.es the Admissibility conditio
II
.2
II
+∞ I Ψ(ω)I CΨ =dω< ∞ (1.4)
ω
.∞
where Ψ.(ω) is the Fourier trAnsform of Ψ(t). Then, the continuous wAvelet trAnsform WT(b, A) is invertible oits rAnge, And Ainverse trAnsform is giveby the relAtion
1 +∞ dAdb
x(t)= WT(b, A)Ψb,A(t) (1.5)
A2
CΨ .∞
One would ofterequire wAvelet Ψ(t) to hAve pAct support, or At leAst to hAve fAst decAy As t goes to in.nity, And thAt Ψ.(ω) hAs su.cient decAy As ω goes to in.nity. From the Admissibility condition, it cAbe seethAt Ψ.(0) hAs to be 0, And, ipArticulAr, Ψ hAs to oscillAte. This hAs giveΨ the nAme wAvelet or “smAll wAve”. This shows the time-frequency locAlisAtioof the wAvelets, which is AimportAnt feAture thAt is required for All the wAvelet trAnsforms to mAke them useful for AnAlysing non-stAtionAry signAls.
The CWT mAps A signAl of one independent vAriAble t into A functioof two independent vAriAbles A,b. It is cAlculAted by continuously shifting A continuously scAlAble functioover A signAl And cAlculAting the correlAtiobetweethe two. This provides A nAturAl tool for time-frequency signAl AnAlysis since eAch templAte Ψb,A is predominAntly locAlised iA certAiregioof the time-frequency plAne with A centrAl frequency thAt is inversely proportionAl to A. The chAnge of the Amplitude Around A certAifrequency cAthebe observed. WhAt distinguishes it from the WFT is the multiresolutionAture of the AnAlysis.
1.3 The discrete wAvelet trAnsform
From A putAtionAl point of view, CWT is not e.cient. One wAy to solve this problem is to sAmple the continuous wAvelet trAnsform oA two-dimensionAl grid (Aj ,bj,k). This will not prevent the inversioof the discretised wAvelet trAnsform igenerAl.
IequAtio(1.3), if the dyAdic scAles Aj =2j Are chosen, And if one chooses bj,k = k2j to AdApt to the scAle fActor Aj , it follows thAt
( II. 1 ∞ t . k2j
2
dj,k =WT(k2j , 2j)= I2jI x(t)Ψˉdt = x(t), Ψj,k(t) (1.6) .∞ 2j
where Ψj,k(t)=2.j/2Ψ(2.j t . k).
The trAnsform thAt only uses the dyAdic vAlues of A And b wAs originAlly cAlled the discrete wAvelet trAnsform (DWT). The wAvelet coe.cients dj,k Are considered As A time-frequency mAp of the originAl signAl x(t). Oftefor the DWT, A set of
{}
bAsis functions Ψj,k(t), (j, k) ∈ Z2(where Z denotes the set of integers) is .rst chosen, And the goAl is theto .nd the depositioof A functiox(t) As A lineAr binAtioof the givebAsis functions. It should Also be noted thAt Although
{}
Ψj,k(t), (j, k) ∈ Z2is A bAsis, it is not necessArily orthogonAl. Non-orthogonAl bAses give greAter .exibility And more choice thAorthogonAl bAses. There is A clAss of DWT thAt cAbe implemented using e.cient Algorithms. These types of wAvelet trAnsforms Are AssociAted with mAthemAticAl structures cAlled multi-resolutioAp-proximAtions. These fAst Algorithms use the property thAt the ApproximAtiospAces Are nested And thAt the putAtions At coArser resolutions cAbe bAsed entirely othe ApproximAtions At the previous .nest level.
Iterms of the relAtionship betweethe wAvelet functioΨ(t) And the scAling functioφ(t), nAmely
II ∞II
2 f
II II
I φ.(ω)I = I Ψ.(2j ω)I (1.7)
j=.∞
The discrete scAling functiocorresponding to the discrete wAvelet functiois As follows
(
1 t . 2j k
φj,k(t)= √ φ (1.8)
2j 2j
It is used to discretise the signAl; the sAmpled vAlues Are de.ned As the scAling coe.cients cj,k
∞
cj,k = x(t)φˉ j,k(t)dt (1.9)
.∞
Thus, the wAvelet depositioAlgorithm is obtAined
f
cj+1(k)= h(l)cj (2k . l)
l∈Z
f
dj+1(k)= g(l)cj (2k . l) (1.10)
l∈Z
Fig.1.1 Algorithm of fAst multi-resolutiowAvelet trAnsform
where the terms g And h Are high-pAss And low-pAss .lters derived from the wAvelet functionΨ(t) And the scAling functioφ(t), the coe.cients dj+1(k)And cj+1(k)rep-resent A depositioof the (j .1) th scAling coe.cient into high frequency (detAil informAtion) And low frequency (ApproximAtioinformAtion) terms. Thus, the Al-gorithm deposes the originAl signAl x(t) into di.erent frequency bAnds ithe time domAin. WheApplied recursively, the formulA (1.10) de.nes the fAst wAvelet trAnsform. Fig.1.1 shows the corresponding multi-resolutiofAst Algorithm, where 2 denotes down-sAmpling.
1.4 The heisenberg uncertAinty principle And time-frequency depositions
WAvelet AnAlysis is essentiAlly time-frequency deposition. The underlying prop-erty of wAvelets is thAt they Are well locAlised iboth time And frequency. This mAkes it possible to AnAlyse A signAl iboth time And frequency with unprecedented eAse And AccurAcy, zooming iovery brief intervAls of A signAl without losing too much informAtioAbout frequency. It is emphAsised thAt the wAvelets cAonly be well or optimAlly locAlised. This is becAuse the Heisenberg uncertAinty principle still holds, which cAbe expressed As the product of the two “uncertAinties”, or spreAds of possible vAlues Δt(time intervAl) And Δf(frequency intervAl)thAtis AlwAys AtleAst A certAiminimum number. The expressiois Also cAlled Heisenberg inequAlity.
WAvelets cAnnot overe this limitAtion, Although they AdApt AutomAticAlly to A signAl’s ponents, ithAt they bee wider to AnAlyse low frequencies And thinner to AnAlyse high frequencies.
1.5 Multi-resolutioAnAlysis
As discussed ithe previous section, multi-resolutioAnAlysis links wAvelets with the .lters used isignAl processing. Ithis ApproAch, the wAvelet is upstAged by A new function, the scAling function, which gives A series of pictures of the signAl, eAch At A resolutiodi.ering by A fActor of two from the previous resolution. Multi-resolutioAnAlysis is A powerful tool for studying signAls with feAtures At vArious scAles. IApplicAtions, the prActicAl implementAtioof this trAnsformAtiois performed by using A bAsic .lter bAnk, iwhich wAvelets Are incorporAted into A system thAt uses A cAscAde of .lters to depose A signAl. EAch resolutiohAs its owpAir of .lters: A low-pAss .lter AssociAted with the scAling function, giving AoverAll picture of the signAl, And A high-pAss .lter AssociAted with the wAvelet, letting through only the high frequencies AssociAted with the vAriAtions, or detAils.
By judiciously choosing the scAling function, which is Also referred to As the fAther wAvelet, one cAmAke customised wAvelets with the desired properties.
And the wAvelets generAted for multi-resolutioAnAlysis cAbe orthogonAl or non-orthogonAl. ImAny cAses no explicit expressiofor the scAling functiois AvAilAble. However, there Are fAst Algorithms thAt use the re.nement or dilAtioequAtioAs expressed iequAtio(1.10) to evAluAte the scAling functioAt dyAdic points.ImAny ApplicAtions, it mAy not be necessAry to construct the scAling functioitself, but to work directly with the AssociAted .lters.
1.6 Some importAnt properties of wAvelets
So fAr, there is no consensus As to how hArd one should work to choose the best wAvelet for A giveApplicAtion, And there Are no .rm guidelines ohow to mAke such A choice. IgenerAl, there Are two kinds of choices to mAke: the system of rep-resentAtio(continuous or discrete, orthogonAl or nonorthogonAl) And the properties of the wAvelets themselves.
1.6.1 CompAct support
If the scAling functioAnd wAvelet Are pActly supported, the .lters h And g Are .nite impulse response (FIR) .lters, so thAt the summAtions ithe fAst wAvelet trAnsform Are .nite. This obviously is of use iimplementAtion. If they Are not pActly supported, A fAst decAy is desirAble so thAt the .lters cAbe ApproximAted reAsonAbly by .nite impulse response .lters.
1.6.2 RAtionAl coe.cients
For puter implementAtions, it is of use if the coe.cients of the .lters h And g Are rAtionAls.
1.6.3 Symmetry
If the scAling functioAnd wAvelet Are symmetric, thethe .lters hAve generAlised lineAr phAse. The Absence of this property cAleAd to phAse distortion. This is importAnt isignAl processing ApplicAtions.
1.6.4 Smoothness
The smoothness of wAvelets is very importAnt iApplicAtions. A higher degree of smoothness corresponds to better frequency locAlisAtioof the .lters. Smooth bA-sis functions Are desired inumericAl AnAlysis ApplicAtions where derivAtives Are involved. The order of regulArity of A wAvelet is the number of its continuous derivA-tives.
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