基本信息

書名：Wavelets in Engineering Applications

定價：78.00元

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

編輯推薦

內容提要

《Wavelets in Engineering 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 in Engineering 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 Introduction
The Fourier trAnsform (FT) AnAlysis concept is widely used for signAl processing. The FT of A function x(t) is de.ned As

+∞
X.(ω)=x(t)e.iωtdt (1.1)
.∞
The FT is An excellent tool for deposing A signAl or function x(t)in terms of its frequency ponents, however, it is not locAlised in time. This is A disAdvAntAge of Fourier AnAlysis, in which frequency informAtion cAn only be extrActed for the plete durAtion of A signAl x(t). If At some point in the lifetime of x(t), there is A locAl oscillAtion representing A pArticulAr feAture, this will contribute to the
.
cAlculAted Fourier trAnsform X(ω), but its locAtion on the 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 locAlisAtion in time 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 cAn be expressed As A function of the frequency ω And the position b

1 +∞ X(ω, b)= x(t)w(t . b)e.iωtdt (1.2) 2π.∞ This is the FT of function x(t) windowed by w(t) for All b. Hence one cAn obtAin A time-frequency mAp of the entire signAl. The mAin drAwbAck, however, is thAt the windows hAve the sAme width of time slot. As A consequence, the resolution of
the WFT will be limited in thAt it will be di.cult to distinguish between successive events thAt Are sepArAted by A distAnce smAller thAn the window width. It will Also be di.cult for the WFT to cApture A lArge event whose signAl size is lArger thAn the window’s size.
WAvelet trAnsforms (WT) developed during the lAst decAde, overe these lim-itAtions And is known to be more suitAble for non-stAtionAry signAls, where the description of the signAl involves both time And frequency. The vAlues of the time-frequency representAtion of the signAl provide An indicAtion of the speci.c times At which certAin spectrAl ponents of the signAl cAn be observed. WT provides A mApping thAt hAs the Ability to trAde o. time resolution for frequency resolution And vice versA. It is e.ectively A mAthemAticAl microscope, which Allows the user to zoom in feAtures 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 function
(
. 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 An oscillAtory function, Ψdenotes the plex conjugAte
A of Ψ, b is the time delAy (trAnslAte pArAmeter) which gives the position of the wAvelet, A is the scAle fActor (dilAtion pArAmeter) which determines the frequency content.
The vAlue WT(b, A) meAsures the frequency content of x(t) in A certAin frequency bAnd within A certAin time intervAl. The time-frequency locAlisAtion property 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 Attention in the reseArch munity. And the technique of WT hAs been Applied in such 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
EquAtion (1.3) is the form of continuous wAvelet trAnsform (CWT). To AnAlyse Any .nite energy signAl, the CWT uses the dilAtion And trAnslAtion of A single wAvelet function Ψ(t) cAlled the mother wAvelet. Suppose thAt the wAvelet Ψ sAtis.es the Admissibility condition
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 on its rAnge, And An inverse trAnsform is given by the relAtion
1 +∞ dAdb
x(t)= WT(b, A)Ψb,A(t) (1.5)
A2
CΨ .∞
One would often require 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 cAn be seen thAt Ψ.(0) hAs to be 0, And, in pArticulAr, Ψ hAs to oscillAte. This hAs given Ψ the nAme wAvelet or “smAll wAve”. This shows the time-frequency locAlisAtion of the wAvelets, which is An importAnt 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 function of two independent vAriAbles A,b. It is cAlculAted by continuously shifting A continuously scAlAble function over A signAl And cAlculAting the correlAtion between the two. This provides A nAturAl tool for time-frequency signAl AnAlysis since eAch templAte Ψb,A is predominAntly locAlised in A certAin region of the time-frequency plAne with A centrAl frequency thAt is inversely proportionAl to A. The chAnge of the Amplitude Around A certAin frequency cAn then be observed. WhAt distinguishes it from the WFT is the multiresolution nAture 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 on A two-dimensionAl grid (Aj ,bj,k). This will not prevent the inversion of the discretised wAvelet trAnsform in generAl.
In equAtion (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). Often for 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 then to .nd the deposition of A function x(t) As A lineAr binAtion of the given bAsis 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 thAn orthogonAl bAses. There is A clAss of DWT thAt cAn be implemented using e.cient Algorithms. These types of wAvelet trAnsforms Are AssociAted with mAthemAticAl structures cAlled multi-resolution Ap-proximAtions. These fAst Algorithms use the property thAt the ApproximAtion spAces Are nested And thAt the putAtions At coArser resolutions cAn be bAsed entirely on the ApproximAtions At the previous .nest level.
In terms of the relAtionship between the wAvelet function Ψ(t) And the scAling function φ(t), nAmely
II ∞II
2 f
II II
I φ.(ω)I = I Ψ.(2j ω)I (1.7)
j=.∞
The discrete scAling function corresponding to the discrete wAvelet function is 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 deposition Algorithm 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-resolution wAvelet trAnsform
where the terms g And h Are high-pAss And low-pAss .lters derived from the wAvelet functionΨ(t) And the scAling function φ(t), the coe.cients dj+1(k)And cj+1(k)rep-re Wavelets in Engineering Applications 97870304 下載 mobi epub pdf txt 電子書