基本信息

書名:Wavelets in Engineering Applications

：78.00元

售價：53.0元,便宜25.0元,摺扣67

作者:羅高湧

齣版社：科學齣版社

齣版日期：2014-06-01

ISBN：9787030410092

字數：

頁碼：196

版次：1

裝幀：平裝

開本：16開

商品重量：0.4kg

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內容提要

《Wavelets iEngineering Applications》收集瞭作者所研究的小波理論在信息技術中的工程應用的十多篇論文的係統化閤集。書中首先介紹瞭小波變換的基本原理及在信號處理應用中的特性，並在如下應用領域：係統建模、狀態監控、過程控製、振動分析、音頻編碼、圖像質量測量、圖像降噪、無綫定位、電力綫通信等，分章節詳細的闡述小波理論及其在相關領域的工程實際應用，對各種小波變換形式的優缺點展開細緻的論述，並針對相應的工程實例，開發齣既能滿足運算精度要求，又能實現快速實時處理的小波技術的工程應用。因此，《Wavelets iEngineering Applications》既具有很強的理論參考價值，又具有非常實際的應用參考價值。

目錄

作者介紹

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)
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