具體描述
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
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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-resent A deposition of the (j .1) th scAling coe.cient into high frequency (detAil informAtion) And low frequency (ApproximAtion informAtion) terms. Thus, the Al-gorithm deposes the originAl signAl x(t) into di.erent frequency bAnds in the time domAin. When Applied recursively, the formulA (1.10) de.nes the fAst wAvelet trAnsform. Fig.1.1 shows the corresponding multi-resolution fAst 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 in both time And frequency. This mAkes it possible to AnAlyse A signAl in both time And frequency with unprecedented eAse And AccurAcy, zooming in on very brief intervAls of A signAl without losing too much informAtion About frequency. It is emphAsised thAt the wAvelets cAn only be well or optimAlly locAlised. This is becAuse the Heisenberg uncertAinty principle still holds, which cAn be expressed As the product of the two “uncertAinties”, or spreAds of possible vAlues Δt(time intervAl) And Δf(frequency intervAl)thAtis AlwAys AtleAst A certAin minimum number. The expression is Also cAlled Heisenberg inequAlity.
WAvelets cAnnot overe this limitAtion, Although they AdApt AutomAticAlly to A signAl’s ponents, in thAt they bee wider to AnAlyse low frequencies And thinner to AnAlyse high frequencies.
1.5 Multi-resolution AnAlysis
As discussed in the previous section, multi-resolution AnAlysis links wAvelets with the .lters used in signAl processing. In this ApproAch, the wAvelet is upstAged by A new function, the scAling function, which gives A series of pictures of the signAl, eAch At A resolution di.ering by A fActor of two from the previous resolution. Multi-resolution AnAlysis is A powerful tool for studying signAls with feAtures At vArious scAles. In ApplicAtions, the prActicAl implementAtion of this trAnsformAtion is performed by using A bAsic .lter bAnk, in which wAvelets Are incorporAted into A system thAt uses A cAscAde of .lters to depose A signAl. EAch resolution hAs its own pAir of .lters: A low-pAss .lter AssociAted with the scAling function, giving An overAll 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 cAn mAke customised wAvelets with the desired properties.
And the wAvelets generAted for multi-resolution AnAlysis cAn be orthogonAl or non-orthogonAl. In mAny cAses no explicit expression for the scAling function is AvAilAble. However, there Are fAst Algorithms thAt use the re.nement or dilAtion equAtion As expressed in equAtion (1.10) to evAluAte the scAling function At dyAdic points.In mAny ApplicAtions, it mAy not be necessAry to construct the scAling function itself, 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 given ApplicAtion, And there Are no .rm guidelines on how to mAke such A choice. In generAl, there Are two kinds of choices to mAke: the system of rep-resentAtion (continuous or discrete, orthogonAl or nonorthogonAl) And the properties of the wAvelets themselves.
1.6.1 CompAct support
If the scAling function And wAvelet Are pActly supported, the .lters h And g Are .nite impulse response (FIR) .lters, so thAt the summAtions in the fAst wAvelet trAnsform Are .nite. This obviously is of use in implementAtion. If they Are not pActly supported, A fAst decAy is desirAble so thAt the .lters cAn be 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 function And wAvelet Are symmetric, then the .lters hAve generAlised lineAr phAse. The Absence of this property cAn leAd to phAse distortion. This is importAnt in signAl processing ApplicAtions.
1.6.4 Smoothness
The smoothness of wAvelets is very importAnt in ApplicAtions. A higher degree of smoothness corresponds to better frequency locAlisAtion of the .lters. Smooth bA-sis functions Are desired in numericAl AnAlysis ApplicAtions where derivAtives Are involved. The order of regulArity of A wAvelet is the number of its continuous derivA-tives.
序言
用戶評價
當我看到“Wavelets in Engineering Applications”這個書名時，我的思緒立刻被帶入瞭一個充滿可能性和挑戰的領域。小波變換，這個詞語給我一種“分解與重構”的聯想，仿佛它能夠將一個龐雜的係統拆解成更易於理解的組成部分，然後又能夠根據這些部分精確地重建齣完整的圖像。工程領域，在我看來，就是一個不斷追求效率、精度和創新的舞颱。因此，我非常好奇這本書會如何闡釋小波變換在其中扮演的角色。我想象著，它或許能幫助工程師們更有效地分析結構振動，預測潛在的危險；或許能輔助醫療診斷，通過分析醫學影像中的微小異常來早期發現疾病；甚至可能在環境保護領域，用於分析復雜的環境數據，揭示汙染的來源和傳播規律。這本書，對我而言，不僅是對一種數學工具的介紹，更是一次對工程實踐中智慧運用的一次深刻洞察。我期待書中能夠提供紮實的理論基礎，但更希望它能通過豐富的實際案例，讓我親身感受到小波變換在解決工程難題時的強大威力，以及它如何推動著工程技術的不斷進步，為人類社會的發展貢獻力量。
評分當我瞥見這本書名時，腦海中湧現的畫麵並非那些抽象的數學公式，而是那些在現實世界中默默發揮著巨大作用的工程奇跡。小波變換，這個詞語對我來說，或許不像“結構力學”或“流體力學”那樣直觀，但它卻暗示著一種更深層次的洞察力。我常常驚嘆於工程師們如何能夠化繁為簡，在錯綜復雜的係統中找到關鍵的節點。而“Wavelets in Engineering Applications”這個組閤，仿佛就指嚮瞭那樣一種“化繁為簡”的藝術。我好奇書中會如何闡釋小波變換在解決實際工程難題中的具體方法，比如它如何幫助監測橋梁的健康狀況，如何優化通信信號的傳輸質量，又或者如何在生物力學研究中捕捉微小的肌肉運動。我期待書中能夠提供一係列引人入勝的案例研究，讓我能夠清晰地看到這些理論工具如何在實際工程場景中大放異彩。這不僅僅是關於理論的知識，更是關於智慧的實踐。它讓我思考，究竟是什麼樣的工程挑戰，需要如此精妙的數學工具來應對？而這些工具，又如何塑造瞭我們今天所見證的工程技術進步？這本書，對我來說，是一次瞭解工程界“幕後英雄”的機會，是一次學習如何用更具穿透力的方式去審視和解決工程問題的寶貴經曆。
評分“Wavelets in Engineering Applications”這個書名，瞬間在我的腦海中勾勒齣一幅充滿活力的畫麵：數據如同海浪般湧動，而小波變換則像是能夠捕捉並分析這些海浪的關鍵技術。我對工程領域一直懷有濃厚的興趣，尤其是那些能夠解決實際問題的創新方法。小波變換，這個詞匯本身就帶著一種數學的優雅和力量，我曾隱約聽說過它在信號處理和圖像分析中的強大能力。但“Engineering Applications”的後綴，則將這種理論上的魅力延伸到瞭更廣闊的應用場景。我迫不及待地想知道，這本書會展示哪些令人驚嘆的工程案例？例如，它是否能幫助我們更精確地監測航空發動機的運行狀態，從而預測潛在的故障？它是否能在通信領域實現更可靠的數據傳輸，即使在信號乾擾嚴重的環境下？或者，它能否在材料科學中幫助我們理解材料在受力過程中的微觀形變？這本書，對我來說，不僅僅是一本關於數學工具的書，更是一次深入瞭解工程領域是如何運用尖端技術解決復雜問題的絕佳機會。我期望它能給我帶來新的視角，讓我看到工程技術背後更深層的智慧和創造力。
評分這本書名讓我立刻聯想到瞭一係列充滿想象的圖像，比如如海浪般湧動的數據流，又或是巧妙地嵌入瞭微妙細節的工程設計。雖然我尚未翻開這本書，單是“Wavelets in Engineering Applications”這個標題就足以勾起我強烈的閱讀欲望。我一直對那些能夠揭示隱藏模式、分解復雜信號的數學工具充滿好奇。小波變換，這個詞本身就帶著一種優雅和力量，仿佛能夠穿透事物的錶象，觸及到其最核心的構成。想象一下，在復雜的振動分析中，它如何捕捉那些轉瞬即逝的異常；在圖像處理領域，它又如何實現精妙的去噪和特徵提取。工程應用，這個詞匯更是為我打開瞭一個充滿可能性的世界。從航空航天的精密控製到醫學影像的精準診斷，再到材料科學的微觀結構分析，我都能預見到小波變換在其中扮演的關鍵角色。我非常期待書中能夠展現那些前沿的研究成果，以及那些將理論轉化為實際應用的創新案例。這本書，對我而言，不僅僅是一本技術書籍，更像是一扇通往更深層次工程理解的大門，一扇能夠幫助我看到事物更細微之處，更全麵地把握問題本質的窗戶。它似乎預示著一次智識上的旅程，一次對工程世界奧秘的探索，我已迫不及待想要開始這段發現之旅。
評分這本書的名字，簡潔而有力，直接點明瞭其核心主題——小波變換在工程領域的應用。這個組閤讓我聯想到瞭那些在浩瀚數據海洋中精準定位關鍵信息的“偵探”工具。工程，在我的認知裏，是一個充滿挑戰和創新的領域，而小波變換，則像是為工程師們量身定製的“顯微鏡”和“透鏡”，能夠幫助他們揭示肉眼難以察覺的細節，分析那些瞬息萬變的現象。我非常好奇書中會如何詳盡地介紹小波變換的數學原理，但更讓我感興趣的是它如何被巧妙地應用於各種工程學科。比如，在信號處理領域，它或許能實現更高效的濾波和壓縮；在圖像分析方麵，它或許能帶來更清晰的診斷圖像；甚至在地球物理勘探中，它也可能幫助科學傢們更好地理解地下的結構。我期待書中能夠提供清晰的圖示和生動的解釋，讓我能夠理解那些復雜的概念。這本書，對我來說，是一次深入瞭解工程領域前沿技術的機會，一次學習如何運用先進數學工具解決實際問題的絕佳途徑。它仿佛是一個寶庫，裏麵藏著能夠幫助工程師們突破技術瓶頸、實現更高精度和效率的“秘籍”。
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