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Wavelets: Theoretical Concepts and VibrationsRelated Applications

ARTICLE in THE SHOCK AND VIBRATION DIGEST · SEPTEMBER 2005

DOI: 10.1177/0583102405055441

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The Shock and Vibration Digest

DOI: 10.1177/0583102405055441

2005; 37; 359The Shock and Vibration Digest Pol D. Spanos and Giuseppe Failla

Wavelets: Theoretical Concepts and Vibrations Related Applications

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Articles

Wavelets: Theoretical Concepts and Vibrations Related

Applications

Pol D. Spanos and Giuseppe Failla

ABSTRACT—In this paper we provide a review of waveletanalysis in the context of applications to vibrations problems.First, we give an introduction to the important concepts andmathematical properties of wavelets within the framework oftime–frequency analysis of signals. Next, wavelet analysis isdiscussed as applied to relevant themes in vibrations, such astime-varying spectra estimation, random field synthesis, sys-

tem identification, damage detection, and material characteri-zation. In view of the large number of related books and journalarticles published in recent decades, the list of selected refer-ences in the paper is not meant to be exhaustive. Neverthe-less, the cited references aim to point out the salient featuresof wavelet analysis, and to identify significant contributions ineach theme, with the goal of expediting additional researchand development efforts.

KEYWORDS: wavelets, localization, vibrations, applications

Nomenclature

E [] the operator of mathematical expectation

R(•, •) correlation function H [•] Hilbert transform operator

inner productinclusion

{•;•} set of all elements with a specified propertyabsolute valuenorm

1 the set of complex numbers2 the set of real numbers3 the set of integer numbers

complex conjugateω frequencyζ damping ratioa scaleb shiftψ ( x ) mother waveletφ( x ) scale functionx spatial variable

W wavelet transformiδmn Kronecker delta defined as

1. Introduction

Wavelet-based representations offer an important optionfor capturing localized effects in many signals. This isachieved by employing representations via double integrals(continuous transforms), or via double series (discrete trans-forms). Seminal to these representations are the processes of scaling and shifting of a generating (mother) function. Overa period of several decades, wavelet analysis has been set ona rigorous mathematical framework and has been applied toquite diverse fields. Wavelet families associated with spe-

cific mother functions have proven quite appropriate for avariety of problems. In this context, fast decomposition andreconstruction algorithms ensure computational efficiency,and rival classical spectral analysis algorithms such as thefast Fourier transform (FFT). The field of vibrations analy-sis has benefited from this remarkable mathematical devel-opment in conjunction with vibration monitoring, systemidentification, damage detection, and several other tasks.There is a voluminous body of literature focusing on wave-let analysis. However, this paper has the restricted objectiveof, on one hand, discussing concepts closely related tovibrations analysis, and on the other hand citing sourceswhich can be readily available to a potential reader. In this

sense, almost exclusively books and archival articles areincluded in the list of references. First, theoretical conceptsare briefly presented; for more mathematical details, thereader may consult the following references: Carmona et al.(1998), Chan (1995), Chui (1992a 1992b), Chui et al. (1994),Cohen (1995), Daubechies (1992), Hernández and Weiss(1996), Hubbard (1996), Jaffard et al. (2001), Kahane andLemarié-Rieusset (1995), Kaiser (1994), Mallat (1998), Meyer(1990), Misiti (1997), Newland (1993), Qian and Chen(1996), Qian (2001), Strang and Nguyen (1996), Vetterliand Kovacevic (1995), Walter (1994), Young (1993), andSpanos and Zeldin (1997). Next, the theoretical conceptsare supplemented by vibrations analysis related sections ontime-varying spectra estimation, random field synthesis,

• •,〈 〉⊂

••

•( )

1–

δmn0 for m n≠

1 for m n=

=

Pol D. Spanos, George R. Brown School of Engineering, L. B. Ryon Chair in Engineering, Rice University, P.O. Box 1892, Houston, Texas 77251,USA ([email protected]). Giuseppe Failla, Researcher, Dipartimento di Meccanica e Materiali, Università “Mediterranea” di Reggio Calabria, Località Graziella Feo di Vito, 89060 Reggio Calabria, Italy.

The Shock and Vibration Digest, Vol. 37, No. 5, September 2005 359–375

©2005 Sage Publications

DOI: 10.1177/0583102405055441


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360 The Shock and Vibration Digest / September 2005

structural identification, damage detection, and materialcharacterization. It is noted that most of the mathematical

developments pertain to the interval [0, 1] relating to dimen-sionless independent variables derived by normalizationwith respect to the spatial or temporal “lengths” of the entiresignals.

2. Time–frequency Analysis

A convenient approach to introduce the wavelet trans-form is through the concept of time–frequency representa-tion of signals. In the classical Fourier theory, a signal canbe represented either in the time domain or in the frequencydomain, and the Fourier coefficients define the averagespectral content over the entire duration of the signal. TheFourier representation is effective for signals, which are sta-

tionary, in terms of parameters that are deemed importantfor the problem in hand, but becomes inadequate for non-stationary signals in which these parameters may evolverapidly in time.

The need for a time–frequency representation is obviousin a broad range of physical problems, such as acoustics,image processing, earthquake and wind engineering, and aplethora of others. Among the time–frequency representa-tions available to date, the wavelet transform has uniquefeatures in terms of efficacy and versatility. In mathematicalterms, it involves the concept of scale as a counterpart to theconcept of frequency in the Fourier theory. Thus, it is alsoreferred to as time-scale representation. Its formulation

stems from a generalization of a previous time–frequencyrepresentation, known as the Gabor transform. For com-pleteness, and to underscore the significant advantagesachieved by the development of the wavelet transform, theGabor transform is briefly discussed in Section 2.1. Section2.2 is entirely devoted to the wavelet transform, and themost commonly used wavelet families are described in Sec-tion 2.3.

2.1. Gabor Transform

The first steps in time–frequency analysis trace back tothe work of Gabor (1946), who applied for signal analysisfundamental concepts developed in quantum mechanics by

Wigner (1932) a decade earlier. Given a function f (t ) belong-ing to the space of finite-energy one-dimensional functions,

denoted by L

2

(2

), Gabor introduced the transform

, (1)

where g(t ) is a window, and the bar denotes complex con- jugation. This transform, generally referred to as the contin-uous Gabor transform (CGT) or the short-time Fouriertransform of f (t ), is a complete representation of f (t ). That is,the original function f (t ) can be reconstructed as

, (2)

where . The Gabor transform (1) may beseen as the projection of f (t ) onto the family

of shifted and modulated copies (atoms) of g(t )expressed in the form

. (3)

These time–frequency atoms, also referred to as Gaborfunctions, are shown in Figure 1 for three different values of ω . Clearly, if g(t ) is an appropriate window function, Equa-tion (1) may be regarded as the standard Fourier transformof the function f (t ), localized at the time t 0. In this context,

t 0 is the time parameter, which gives the center of the win-dow, and ω is the frequency parameter, which is used tocompute the Fourier transform of the windowed signal.

As intuition suggests, the accuracy of the CGT represen-tation (2) of f (t ) depends on the window function g(t ), whichmust exhibit good localization properties in both the timeand frequency domains. As discussed in Cohen (1995), ameasure of the localization properties may be obtained bythe average and the standard deviation of the densityin the time domain, i.e.

(4a)

and

Figure 1. Plots of Gabor function g (ω , t 0) versus the independent variable t for three values of the frequency ω ; the effectivesupport is the same for the three values of the frequency.

G f ω t 0,( ) f t ( )g t t 0–( )eiω t t 0–( )–

∞–

∞

∫ = dt

f t ( ) 12π g 2---------------- G f ω t 0,( )g t t 0–( )e

iω t t 0–( )dω dt 0∞–

∞

∫ ∞–

∞

∫ =

g 2 g t ( ) 2∞–

∞

∫ dt =g ω t 0,( ) t ( ) ;{

ω t 0 2∈, }

g ω t 0,( )t ( ) e iω t t 0–( )g t t 0–( )=

g t ( ) 2

t 〈 〉 t g t ( ) 2dt ∞–

∞

∫ =


https://www.researchgate.net/publication/228109667_On_the_Quantum_Correction_For_Thermodynamic_Equilibrium?el=1_x_8&enrichId=rgreq-f797082a-b144-4f20-8b6f-5b797686fab8&enrichSource=Y292ZXJQYWdlOzIzODE5NDE2ODtBUzoxNDAzMDI5MTIzMzE3NzhAMTQxMDQ2MjIyOTQ5Mw==http://svd.sagepub.com/http://svd.sagepub.com/http://svd.sagepub.com/https://www.researchgate.net/publication/228109667_On_the_Quantum_Correction_For_Thermodynamic_Equilibrium?el=1_x_8&enrichId=rgreq-f797082a-b144-4f20-8b6f-5b797686fab8&enrichSource=Y292ZXJQYWdlOzIzODE5NDE2ODtBUzoxNDAzMDI5MTIzMzE3NzhAMTQxMDQ2MjIyOTQ5Mw==http://svd.sagepub.com/


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Spanos and Failla / THEORETICAL CONCEPTS AND VIBRATIONS RELATED APPLICATIONS 361

(4b)

The counterparts of Equations (4) in the frequency domainare

(5a)

and

, (5b)

where denotes the Fourier transform of g(t ) given bythe equation

. (6)

The well-known Heisenberg uncertainty principle is in actu-ality a mathematically proven property and states that thevalues σt and σω cannot be independently small (Cohen, 1995).Specifically, for an arbitrary window normalized so that

, it can be shown that

. (7)

Thus, high resolution in the time domain (small value of σt )may be generally achieved only at the expense of a poor res-olution (bigger than a minimum value σω ) in the frequencydomain, and vice versa. Note that the optimal time–frequencyresolution, i.e. σt σω = 1/2, may be attained when the Gaus-sian window

(8)

is selected.Clearly, as a time–frequency representation, the Gabor

transform exhibits considerable limitations. The time support,governed by the window function g(t ), is equal for all of theGabor functions (3) for all frequencies (see Figure 1). In orderto achieve good localization of high-frequency components,narrow windows are required; as a result, low-frequencycomponents are poorly represented. Thus, a more flexiblerepresentation with non-constant windowing is quite desira-

ble to enhance the time resolution for short-lived high-fre-quency phenomena and the frequency resolution for long-lasting low-frequency phenomena.

2.2. Wavelet Transform

The preceding shortcomings of the Gabor transform havebeen overcome with significant effectiveness and efficiencyby wavelet-based signal representation. Its two formula-tions, continuous and discrete, are described in the follow-ing sections. Due to the numerous applications of waveletsbeyond time–frequency analysis, the t -time domain will bereplaced by a generic x-space domain. For succinctness, theformulation will be developed for scalar functions only, but

generalizations for multidimensional spaces are well estab-lished in the literature (Carmona et al., 1998; Chan, 1995;Chui et al., 1994; Kaiser, 1994; Chui, 1992a 1992b; Cohen,1995; Daubechies, 1992; Hernández and Weiss, 1996; Hub-bard, 1996; Jaffard et al., 2001; Kahane and Lemarié-Rieusset,1995; Mallat, 1998; Meyer, 1990; Misiti, 1997; Newland,1993; Qian and Chen, 1996; Qian, 2001; Strang and Nguyen,1996; Vetterli and Kovacevic, 1995; Walter, 1994; Young,1993).

2.2.1 Continuous Wavelet Transform

The concept of wavelet transform was introduced first byGoupillaud et al. (1984) and Grossmann and Morlet (1984)for seismic records analysis. In analogy to the Gabor trans-form, the idea consists of decomposing a function f ( x ) into atwo-parameter family of elementary functions, each derivedfrom a basic or mother wavelet, ψ ( x ). The first parameter, a,corresponds to a dilation or compression of the mother wave-let, which is generally referred to as scale. The second param-eter, b, determines a shift of the mother wavelet along the x -domain. In mathematical terms,

, (9)

where a 2+, b 2. In the literature, Equation (9) is gener-

ally referred to as continuous wavelet transform (CWT).Note that the factor a

–1/2 is a normalization factor, included

to ensure that the mother wavelet and any dilated wavelet

a–1/2ψ ( x / a) have the same total energy (Goupillaud et al.,

1984). Clearly, alternative normalizations may also be cho-sen (Carmona et al., 1998).

An example of wavelet functions is shown in Figure 2, fordifferent values of the scale parameter a. Due to scaling, allthe wavelet functions exhibit the same number of cycleswithin the x-support of the mother wavelet. Obviously, thespatial and frequency localization properties of the wavelettransform depend on the value of the parameter a. As aapproaches zero, the dilated wavelet a

–1/2ψ ( x / a) is highlyconcentrated at the point x = 0; the wavelet transform,

W f (a, b), then gives increasingly sharper spatial resolutiondisplaying the small-scale/higher-frequency features of thefunction f ( x ), at various locations b. However, as a approaches+∞, the wavelet transform W f (a, b) gives increasinglycoarser spatial resolution, displaying the large-scale/low-fre-quency features of the function f ( x ).

For the function f ( x ) to be reconstructable from the set of coefficients (9) in the form

, (10)

where

,

the wavelet function ψ (•) must satisfy the admissibility con-dition

, (11)

σt 2 t t 〈 〉–( )2 g t ( ) 2dt .∞–

∞

∫ =

ω 〈 〉 ω Ĝ ω ( )2

dω ∞–

∞

∫ =

σω 2 ω ω 〈 〉–( )2 Ĝ ω ( )2dω

∞–

∞

∫ =

Ĝ ω ( )

Ĝ ω ( ) 12π

---------- g t ( )e iω t – dt ∞–

∞

∫ =

g 2 1=

σt σω 1 2 ⁄ ≥

g t ( ) 1

2πσ t 24-----------------exp

t 2

4σt 2---------–

=

W f a b,( )1

a------- f x ( )ψ x b–

a-----------

∞–

∞

∫ = d x

∈ ∈

f x ( ) 1πcψ --------- W f a b,( )ψ a b, x ( )

da

a2------db

∞–

∞

∫ 0

∞

∫ =

ψ a b, x ( ) a 1 2 / – ψ x b–

a-----------

=

cψ Ψ̂ ω ( ) 2

ω -------------------dω ∞<

∞–

∞

∫ =


https://www.researchgate.net/publication/216027353_Grossmann_A_Morlet_J_Decomposition_of_Hardy_functions_into_square_integrable_wavelets_of_constant_shape_SIAM_J_Math_Anal_15_723-736?el=1_x_8&enrichId=rgreq-f797082a-b144-4f20-8b6f-5b797686fab8&enrichSource=Y292ZXJQYWdlOzIzODE5NDE2ODtBUzoxNDAzMDI5MTIzMzE3NzhAMTQxMDQ2MjIyOTQ5Mw==http://svd.sagepub.com/http://svd.sagepub.com/http://svd.sagepub.com/https://www.researchgate.net/publication/216027353_Grossmann_A_Morlet_J_Decomposition_of_Hardy_functions_into_square_integrable_wavelets_of_constant_shape_SIAM_J_Math_Anal_15_723-736?el=1_x_8&enrichId=rgreq-f797082a-b144-4f20-8b6f-5b797686fab8&enrichSource=Y292ZXJQYWdlOzIzODE5NDE2ODtBUzoxNDAzMDI5MTIzMzE3NzhAMTQxMDQ2MjIyOTQ5Mw==https://www.researchgate.net/publication/222788746_Cycle-Octave_and_re-lated_transforms_in_Seismic_Signal_analysis?el=1_x_8&enrichId=rgreq-f797082a-b144-4f20-8b6f-5b797686fab8&enrichSource=Y292ZXJQYWdlOzIzODE5NDE2ODtBUzoxNDAzMDI5MTIzMzE3NzhAMTQxMDQ2MjIyOTQ5Mw==http://svd.sagepub.com/


6/19


where (ω ) denotes the Fourier transform of ψ ( x ). Aspointed out in Goupillaud et al. (1984), the condition (11)

includes a set of subconditions, such as

(i) the analyzing wavelet ψ (•) is absolutely integrable andsquare integrable, i.e.

(12a)

and

; (12b)

(ii) the Fourier transform (ω ) must be sufficiently small atthe vicinity of the origin ω = 0, or in mathematical terms

. (13)

So, subcondition (ii) implies that (0) = 0, i.e. d x = 0.From the reconstruction formula (10) it can be shown that

. (14)

Based on Equation (14), the square modulus of the wavelet

transform (9) is often taken as an energy density in a spatial-scale domain. Extensive use of this concept has been madefor spectra estimation purposes, as discussed in Section 3.

Note that the reconstruction wavelet in Equation (10) canbe different from the analyzing wavelet used in Equation(9). That is, under some admissibility conditions on 1 ( x )(Carmona et al., 1998), the original function f ( x ) may bereconstructed as

, (15)

where

and cψ 1 is a constant parameter depending on the Fouriertransforms of both ψ ( x ) and 1 ( x ). This property, referred toas redundancy in mathematical terms, may be advantageousin some applications for reducing the error due to noise in sig-nal reconstruction (Holschneider and Tchamitchian, 1991;Cohen and Kovacevic, 1996) but highly undesirable for sig-nal coding or compression purposes (Carmona et al., 1998).Further, under certain conditions (Carmona et al., 1998), thefollowing simplified reconstruction formula holds

, (16)

where k ψ is a constant parameter given by

. (17)

Use of this formula has been made, in a discrete version, inthe approximation theory of functional spaces (Carmona etal., 1998) and also in structural identification applications,as discussed in Section 5.

2.2.2 Discrete Wavelet Transform

For numerical applications, where fast decomposition orreconstruction algorithms are generally required, a discreteversion of the CWT is preferred. In this sense, a natural wayto define a discrete wavelet transform (DWT) is

. (18)

Equation (18) is derived from a straightforward discretizationof the CWT, Equation (9), by considering the discrete lat-tice a = , a0 > 1, b = k b0 , b0 0. In developing Equa-tion (18), however, the first mathematical concern is to ensurethat sampling the CWT on a discrete set of points does not

Figure 2. Plots versus the independent variable x of wavelet functions corresponding to three different values of a scale a ofthe same mother function; the effective time support increases with the magnitude of the scale.

Ψ̂

ψ x ( ) d x ∞<∞–

∞

∫

ψ x ( ) 2d x ∞<∞–

∞

∫

Ψ̂

Ψ̂ ω ( )ω

-----------------dω ∞<∞–

∞

∫

Ψ̂ ψ x ( )∞–

∞

∫

f 21

πcψ --------- W f a b,( ) 2

da

a2------db

∞–

∞

∫ 0

∞

∫ =

f x ( ) 1cψ 1 -------- W f a b,( )1 a b, x ( )

da

a2------db

∞–

∞

∫ 0

∞

∫ =

1 a b, x ( ) a 1 2 / – 1 x b–

a-----------

=

f x ( ) 1k ψ ----- W f a x ,( )

da

a3 2 / ---------

0

∞

∫ =

k ψ 2π ψ ˆ ω ( )

ω -------------dω

0

∞

∫ =

W f j k ,( )1

a0 j

--------- f x ( )ψ a 0 j– x kb0–( )d x , j k 3∈,

∞–

∞

∫ =

a0 j

a0 j ≠




7/19


lead to a loss of information about the wavelet-transformedfunction f ( x ). Specifically, the original function f ( x ) must befully recoverable from a discrete set of wavelet coefficients.That is,

, (19)

where ψ j,k ( x ) = . Another crucial aspectin Equation (18) involves selecting the wavelet functionsψ j, k ( x ) such that Equation (19) may be regarded as theexpansion of f ( x ) in a basis, thus eliminating the redundancyof the CWT.

These issues are addressed by using the theory of Hilbertspace frames, introduced by Duffin and Schaeffer (1952)in context with non-harmonic Fourier series. In general, if hλ ( x ) L

2(2) and Λ is a countable set, a family of functions

{hλ ( x ); λ Λ} constitutes a frame if for any f ( x ) L2(2)

, (20)

where and A > 0, B


8/19


tion is involved to compute wavelet and scale coefficients. Itrelies on the projection of f ( x ) onto a sufficiently fine scale jof the set (24). That is,

, (29)

where, for orthogonal wavelets,

. (30)

Based on Equations (22), the projection f j( x ) can be rewrit-ten in terms of the projection f j +1( x ) onto the coarser scale( j + 1) and the incremental detail δ j +1( x ), i.e. the pieces of information contained in the subspace V j and lost when“moving” to the subspace V j +1. Therefore,

(31)

As a fundamental result of multiresolution analysis, thedetails δ j( x ) can be decomposed in terms of the set of wave-let functions at the same scale. That is,

, (32)

where are the wavelet coefficients of f ( x ). Based on Equa-tion (28), both wavelet and scale coefficients can be com-puted recursively by the closed-form expressions

(33a)

and

. (33b)

Similarly, the reconstruction algorithm can be implementedby the formula

. (34)

The reconstruction algorithm described by Equation (34)lends itself to interpretation as a scale linear system (Basse-ville et al., 1992a, 1992b). Based on this concept, applica-tions have also been developed for random field simulation(Zeldin and Spanos, 1996).

2.3. Wavelet Families

A great number of wavelet families with various proper-ties are available. Selecting an optimal family for a specificproblem is not, in general, a trivial task, and there are proper-ties that prove more important to certain fields of application.For instance, symmetry may be of great help for preventing

dephasing in image processing, while regularity is critical forbuilding smooth reconstructed signals or accurate non-linearregression estimates. Compactly supported wavelets, in eitherthe time or frequency domain, may be preferable for enhancedtime or frequency resolution. The number of vanishingmoments M , i.e. the highest integer m for which the equation

, (35)

holds, is important in signal processing for compression, orin damage detection for enhancement of singularities in thevibration modes. Also, in some cases wavelets may be requiredto be progressive. In mathematical terms, this means thattheir Fourier transform is defined only for positive frequen-cies. That is,

. (36)

The progressive wavelet transform of a real-valued signal

f ( x ) and the associated analytic signal,

, (37)

are related by the equation

, (38)

where H [•] denotes the Hilbert transform operator (Carmonaet al., 1997). Equation (38) is quite useful for structural iden-tification. Note also that significant reduction of computa-tional costs is generally achieved if orthogonal wavelets inthe frequency domain or in the x -domain are used.

In the next subsection we give a brief description of themost used families. A distinction is made between real andcomplex wavelets, and the most relevant properties for appli-cation purposes are discussed. A more exhaustive reviewmay be in found in Misiti (1997).

2.3.1 Real Wavelets

(i) Daubechies orthonormal wavelets

A family of bases, each corresponding to a particularvalue of the parameter M in Equations (27) and (28) (Daub-echies, 1988). Closed-form expressions for φ( x ) in Equation(27) are available only for M = 1, to which the well-knownHaar basis corresponds. In this case, the scaling function and

the mother wavelet are

(39a,b)

Various algorithms, however, are available in the literaturefor determining φ( x ) and ψ ( x ) numerically for M > 1.

Daubechies wavelets support both CWT and DWT,although the latter is most generally performed due to thefast decomposition and reconstruction algorithm mentionedin Section 2.2.2. Both φ( x ) and ψ ( x ) are compactly supported

x ( ) f j x ( )≈ ck j φ j k , x ( )

k ∑=

ck j

f x ( )φ j k , x ( )d x ∞–

∞

∫ =

j x ( ) f j 1+ x ( ) δ j 1+ x ( )+= =

f j 1+ x ( ) δ j 1+ x ( ) K δ j l+ x ( )+ + += =

δ j 1+ x ( ) K δ j l+ x ( ).+ +≈

δ j x ( ) d k jψ j k , x ( )

k ∑=

d k j

ck j

hl 1+ c2k l 1–+ j 1–

l 0=

2 M 1–

∑=

d k j

gl 1+ c2k l 1–+ j 1–

l 0=

2 M 1–

∑=

ck j 1–

hk 2 l– 2+ cl j

l

∑ gk 2 l– 2+ d l j+=

x mψ x ( )∞–

∞

∫ d x 0,= m 0 1 … M 1–, , ,=

Ψ̂ ω ( ) 0= , for ω 0<

z f x ( ) f x ( ) iH f x ( )[ ]+=

W f a b,( )1

2---W z f a b,( )=

φ x ( )1 0 x 1


9/19


in the x -domain, and the support is equal to the segment[0; 2 M – 1]. Also, M is equal to the number of vanishingmoments of the wavelet function. Note that most Daub-echies wavelets are not symmetric; regularity and harmonic-like shape increases with M .

(ii) Meyer wavelets

Families of wavelets (Jaffard et al., 2001), each definedfor a particular choice of an auxiliary function v(ω ), whichappears in the following expression for the Fourier trans-form of the mother wavelet:

(40)

For v(ω ) to be an admissible auxiliary function it is required

that

(41a)

(41b)

The most common form of v(ω ) in the literature is

. (42)

The mother wavelet, for which only numerical expressionsare available, is then constructed by inverse Fourier trans-forming Equation (40).

Meyer wavelets are suitable for both CWT and DWT.Unlike Daubechies wavelets, they are compact in the fre-quency domain but not in the x -domain. Due to their fastdecay, however, an effective x -support [–8, 8] is generallyassumed. Appealing features of Meyer wavelets are orthog-onality, infinite regularity, and symmetry.

(iii) Mexican Hat wavelets

A family of wavelets in the x -domain (Misiti, 1997)related to a mother function which is proportional to thesecond derivative of the Gaussian probability density func-tion. That is,

. (43)

Mexican Hat wavelets allow CWT only. Unlike Daubechiesor Meyer wavelets, they are not compact in both the fre-quency domain and the x -domain, although an effective sup-port [–5, 5] may be considered for practical calculations.They are infinitely regular and symmetric.

(iv) Biorthogonal wavelets

Families of wavelets derived by generalizing the ordinaryconcept of wavelet bases, and creating a pair of dual wave-lets, say , satisfying the following properties(Kim et al., 2001; Spanos and Rao, 2001)

(44)

where δmn denotes the Kronecker delta. One wavelet, sayψ ( x ), may be used for reconstruction and the dual one,

, for decomposition. Therefore, Equations (18) and(19) can be rewritten as

(45)

and

. (46)

Biorthogonal wavelets support both CWT and DWT. Theproperties of a biorthogonal wavelet family are specified interms of a pair of integers ( N d , N r ) .These integers, in anal-ogy with the Daubechies wavelets, govern the regularity

and the number of vanishing moments N d of the decomposi-tion wavelet ψ ( x ), and the regularity and the number of van-ishing moments N r of the reconstruction wavelet .Obviously, this feature allows a greater number of choicesfor signal decomposition and reconstruction. Both waveletfunctions ψ ( x ) and are symmetric.

2.3.2 Complex Wavelets

(v) Harmonic wavelets

A family of bases defined in the frequency domain by theformula (Newland, 1993, 1994a; Spanos et al., 2005)

(47)

where m and n are positive numbers, but not necessarilyintegers. The pair of values m,n is referred to as level m,nand represents, for harmonic wavelets, the scale index j. Aharmonic wavelet basis thus corresponds to a complete setof adjacent levels m,n, spanning all the positive frequencyaxis. By inverse Fourier transforming Equation (47), thecorresponding wavelet function at a generic step k on the x -domain takes the complex form

(48)

A common choice for the pairs m, n is m,n = 0,1; 2,4;…;2 j,

2 j + 1

;…. In this case, all the wavelets have octave bandsexcept for the first one.

Harmonic wavelets have been devised in context with aDWT, for which extremely fast decomposition and recon-struction algorithms exist. They exhibit a compact supportin the frequency domain, see Equation (47), while in the x -

Ψ̂ ω ( )

1

2π----------eiω 2 ⁄ sin

π2---v

3

2π------ ω 1–

23---π ω 4

3---π≤ ≤,

1

2π----------eiω 2 ⁄

π2---v

3

4π------ ω 1–

43---π ω 8

3---π≤ ≤,cos

0 ω 23---π 8

3---π, .∉,

=

v ω ( )0 ω 0;≤,1 ω 1;≥,

=

v ω ( ) v 1 ω –( )+ 1;= 0 ω 1.≤ ≤

v ω ( ) ω 4 35 84ω – 70ω 2 20ω 3–+( ) 0 ω 1≤ ≤;=

ψ x ( ) 23

-------π 1 4 ⁄ – 1 x 2–( )e x 2 2 ⁄ –=

ψ x ( ) ψ ˜ x ( ),( )

ψ j k , x ( )ψ ˜ j′ k ′, x ( )d x ∞–

∞

∫ δ jj ′δkk ′= ,

ψ ˜ x ( )

W f j k ,( ) 2 j 2 ⁄ –

f x ( )ψ 2 j– x k –( )d x j k Z∈,,∞–

∞

∫ =

f x ( ) W f j k ,( )ψ ˜ j k , x ( ) j k 3∈,∑=

ψ ˜ x ( )

ψ ˜ x ( )

Ψ̂m n, ω ( )1

2π n m–( )------------------------- , mπ ω nπ≤ ≤

0, elsewhere

=

ψ m n k , , x ( )e

in2π x k n m–-------------–

eim2π x k

n m–-------------–

–

i2π n m–( ) x k n m–-------------–

-----------------------------------------------------------------.=




10/19


domain their rate of decay away from the wavelet’s center isrelatively low and proportional to x

–1. Further, they satisfy

relevant orthogonality properties (Newland, 1993).From Equation (48), it is seen that the real part of the

wavelet is even while the imaginary part is odd. For signalprocessing, this ensures that harmonic components in a sig-nal can be detected regardless of the phase. Note that thisfeature cannot be achieved by real wavelets such as theMeyer wavelets, which are all self-similar being derivedfrom a unique mother wavelet by scaling and shifting. Also,note that orthonormal basis of real wavelets can be gener-ated by considering either the real or the imaginary partsonly of Equation (48). For instance, the well-known Shan-non wavelets correspond to the imaginary parts of Equation(48), for m,n=1,2; 2,4; 4,8;…. Harmonic wavelets are usedin many mechanics applications such as acoustics, vibrationmonitoring, and damage detection (Newland, 1994b, 1994c,1999; Newland and Butler, 1998, 1999).

(vi) Complex Gaussian wavelets

Families of wavelets, each corresponding to a pth-order

derivative of a complex Gaussian function, i.e.

(49)

where C p is a normalization constant such that .Complex Gaussian wavelets support the CWT only. They haveno finite support in the x -domain, although the interval [–5,5] is generally taken as effective support. Despite their lack of orthogonality, they are quite popular in image processingapplications due to their regularity (Carmona et al., 1998).

(vii) Complex Morlet wavelets

Families of wavelets (Haase and Widjajakusuma, 2003),each obtained as the derivative of the classical Morlet wave-let , i.e.

(50)

Except for ψ 0( x ), which does not satisfy the admissibilitycondition (11) in a strict sense, all the other members of thefamily are proper wavelets. For practical purposes, however,

ψ 0( x ) is generally considered admissible for . Com-plex Morlet wavelets support the CWT only and are notorthogonal. However, they are all progressive, i.e. they sat-isfy the condition posed by Equation (36). Further, for theMorlet wavelet ψ 0( x ), there exists a relation between the

scale parameter a and the frequency ω at which its Fouriertransform focuses, i.e.

. (51)

Complex Morlet wavelets are then applied for structuralidentification purposes, as shown in Section 5.

3. Time-dependent Spectra Estimation ofStochastic Processes

Wavelet-based approaches are significant tools for jointtime–frequency analysis of problems related to vibrations of mechanical and structural systems. This applies to the char-

acterization of the system excitation, the system identifica-tion, and the system response determination. Several examplesexist in nature of stochastic phenomena with a time-depend-ent frequency content. The frequency content of earthquakerecords, for instance, evolves in time due to the dispersion of the propagating seismic waves (Trifunac, 1971; Spanos etal., 1987). Further, sudden changes in the wave frequency ata given location of the sea surface are often induced by fastmoving of meteorological fronts (Massel, 1996). Also, arapid change in the frequency content is generally associatedwith waves at the breaking stage. Similarly, turbulent gustsof time-varying frequency content are often embedded inwind fields.

Appropriate description of such phenomena is obviouslycrucial for design and reliability assessments. In an earlyattempt, concepts of traditional Fourier spectral theory weregeneralized to provide spectral estimates, such as the Wigner–Ville method (Wigner, 1932; Ville, 1948) or the CGT of Equa-tion (1). However, it soon became clear that the extension of the traditional concept of a spectrum is not unique, and pro-posed time-varying spectra could have contradictory proper-

ties (Loynes, 1968; Cohen, 1995).Wavelet analysis is readily applicable for estimating time-

varying spectral properties, and significant effort has beendevoted to formulating “wavelet energy principles” that work as alternatives to classical Fourier methods. Measures of atime-varying frequency content were first obtained by “sec-tioning”, at different time instants, the wavelet coefficientsmean square map (Gurley and Kareem, 1999; Newland andButler, 1999; Kareem and Kijewski, 2002; Gurley et al., 2003).Developing consistent spectral estimates from such sections,however, is not straightforward. From a theoretical point of view, either it requires an appropriate wavelet-based definitionof time-varying spectra or it must relate to well-establishednotions of time-varying spectra. From a numerical point of view, it involves certain difficulties in converting the scaleaxis to a frequency axis, especially when the frequency con-tent of wavelet functions at adjacent scales do overlap.

Investigations on wavelet-based spectral estimates maybe found in references such as Spanos et al. (2005), Basu andGupta (1997, 1998, 2000a, 2000b), Tratskas and Spanos(2003), where wavelet analysis has been applied in the con-text of earthquake engineering problems. In a particularapproach, a modified Littlewood–Paley (MLP) waveletbasis can be introduced, whose mother wavelet is defined inthe frequency domain by

(52)

In Equation (52) the symbol σ denotes a scalar factor, to beadjusted depending on the desired frequency resolution. TheMLP wavelets are orthogonal in the frequency domain; thatis, wavelets at adjacent scales span non-overlapping inter-vals. The MLP wavelets have been used in conjunction witha discretized version of the CWT proposed by Alkemade(1993) for a finite-energy process f (t )

, (53)

ψ p x ( ) C pd p

d x p-------- e i x – e x 2 2 ⁄ –( )= , p 1 2 …, , ,=

ψ x ( ) 2 1=

ψ 0 x ( ) e x 2– 2 ⁄ eiω 0 x =

ψ p x ( )d p

d x p-------- e x 2 2 ⁄ – e iω 0 x –( )= , p 1 2 K ., ,=

ω 0 5≥

aω 0ω ------=

Ψˆ

ω ( )

1

2 σ 1–( )π

---------------------------- , π ω σπ≤ ≤

0, elsewhere.

=

f t ( ) K ∆ba j

-----------W f a j bi,( )ψ a j bi, t ( )i j,∑=




11/19


where a j = σ j, ∆b is a time-step, and K is a constant parame-

ter depending on σ.In many instances Equation (53) can be construed as rep-

resenting realizations of a stochastic process and, in thiscase, the following estimate of the instantaneous mean-square value of f (t ) has been constructed

, (54)

where E [•] is the mathematical expectation operator overthe ensemble of realizations. From Equation (54), and basedon the orthogonality properties of the MLP wavelets, thequantity

, (55)

where the symbol denotes the Fourier transform of the wavelet function , can be taken as a measure of

the time-varying power spectral density (PSD) of the proc-ess f (t ). Based on Equation (55), closed-form expressionscan be derived between the input and the output PSDs (Trat-skas and Spanos, 2003). In this context, linear response sta-tistics such as the instantaneous rate of crossings of the zerolevel, or the instantaneous rate of occurrence of the peakshave been estimated with considerable accuracy. Analysisof non-linear systems has also been attempted by an equiva-lent statistical linearization procedure (Basu and Gupta,1999, 2000a).

Wavelet analysis for spectral estimation has also beenpursued by Kareem et al., who have used the squared wave-let coefficients of a DWT to estimate the PSD of stationaryprocesses (Gurley and Kareem, 1999). To improve the fre-quency resolution of the DWT, where only adjacent octavebands can be accounted for, a CWT can be implementedbased on a complex Morlet wavelet family. The latter ispreferable due to the one-to-one correspondence betweenthe scale a and the center frequency (51), which allows min-imizing the overlap between spectral estimates at adjacentscales. Further, the product of wavelet coefficients can beused as a measure of the cross-correlation between two non-stationary signals x (t ) and y(t ) (Gurley and Kareem, 1999).This concept can be refined by the introduction of a waveletcoherence measure (Kareem and Kijewski, 2002; Gurley etal., 2003) expressed by the equation

. (56)

In this equation, the local spectrum is defined as

, (57)

where the time integration window T , centered around b,depends on the desired time resolution. The local spectrum(57), due to the time average over T , allows smoothing of potential measurement noise effects. Measures of higher-ordercorrelation can be introduced (Gurley and Kareem, 1999; Kar-eem and Kijewski, 2002), such as the wavelet bicoherence

(58)

where , and

. (59)

Related remedies can be adopted to suppress spurious corre-lations induced by statistical noise, based on a referencenoise map created from artificially simulated signals (Kar-eem and Kijewski, 2002).

Signal energy representation concepts have been exam-ined in Iyama and Kuwamura (1999) by using quasi-orthog-onal Daubechies wavelets in the frequency domain tosimulate earthquake ground motion accelerations. Further,Massel (2001) has used wavelet analysis to capture time-varying frequency composition of sea surface records due to

fast moving atmospheric fronts in deep water, wave growth,and breaking or disintegration of mechanically generatedwave trains. In this regard, absolute value wavelet maps anda spectral measure called global wavelet energy spectrum,defined by

(60)

are used. The symbol E 1(τ,b) denotes a time-scale energydensity

. (61)

The scale in Equation (61) is readily translated into fre-quency by selecting a Morlet wavelet family.

Spanos and Failla (2004) have applied wavelet analysisto estimate the evolutionary power spectral density (EPSD)of non-stationary oscillatory processes defined as (Priestley,1981)

. (62)

The symbol A(ω ,t ) denotes a slowly varying time- and fre-quency-dependent modulating function, and Z (ω ) is a com-

plex random process with orthogonal increments such that, where is the two-sided

power spectral density of the zero-mean stationary process

. (63)

The two-sided EPSD of f (t ) is then taken as

. (64)

Due to its localization properties, the wavelet transform of f (t ), Equation (62), may be approximated as an oscillatorystochastic process. That is,

E f 2 t ( )[ ] t bi= K E W f a j bi,( )[ ]2

a j---------------------------------- j∑=

S f ω ( ) t bi= K E W f a j bi,( )[ ]2

a j---------------------------------- Ψ̂a j bi, ω ( )

2

j∑=

Ψ̂a j bi, ω ( )ψ a j bi, t ( )

cW a b,( )( )2 S xyW

a b,( )2

S xx W a b,( )S yyW a b,( )

-------------------------------------------=

S ijW a b,( )

S ijW a b,( ) W i a τ,( )W j a τ,( )dτ

T

∫ =

b xx yW a1 a2 b, ,( )( )2

B xx yW a1 a2 b, ,( ) 2

W x a1 τ,( )W x a2 τ,( ) 2T

∫ dτ W y ã τ,( ) 2dτT

∫ ------------------------------------------------------------------------------------------------= ,

1ã---

1a1-----

1a2-----+=

B xx yW a1 a2 b, ,( ) W x a1 τ,( )W x a2 τ,( )

T ∫ W y ã τ,( )dτ=

E 3 a( ) E 1 a b,( )db0

∞

∫ =

E 1 a b,( )W f a b,( ) 2

a-------------------------=

t ( ) A ω t ,( )e iω t ∞–

∞

∫ = d Z ω ( )

E d Z ω ( ) 2[ ] S f 0 f 0 ω ( )dω = S f 0 f 0 ω ( )

0 t ( ) e iω t ∞–

∞

∫ = d Z ω ( )

S ff ω t ,( ) A ω t ,( ) 2S f 0 f 0 ω ( )=




12/19


(65)

where . Based on Equation(65), the following integral relation is found between themean-squared wavelet coefficients at each scale a and theEPSD of f (t ), i.e.

. (66)

An adequate number of Equations (66), one for each scalea, can be solved by a standard solution algorithm applicablefor both orthogonal and non-orthogonal wavelet families inthe frequency domain. This procedure has proved quiteaccurate using both the Littlewood–Paley and the real Mor-let wavelets.

4. Random Field Simulation

The use of wavelets for random field synthesis can beexamined within the more general framework of scale-typemethods. The latter have been developed to improve thecomputational performances of Monte Carlo simulations.Classical methods such as the spectral approach (Shinozukaand Deodatis, 1988) or the autoregressive moving average(ARMA; Spanos and Mignolet, 1992) are not readily applica-ble for this purpose, especially when using non-uniform meshesor when enhancement of local resolution is desirable. Toaddress these shortcomings, Fournier et al. (1982) have pro-posed a “random mid-point method” to synthesize fractionalBrownian motion. That is, a scale-type method where valuesof the random field for points within a coarser scale are gen-erated first, and then the generated samples are used to deter-mine values for a finer scale. This approach has been extendedby Lewis (1987) into a “generalized stochastic subdivisionmethod”, suitable for a broad class of stationary processes, andby Fenton and Vanmarcke (1990) into a “local average sub-division method”, which includes a random field smoothingprocedure producing averages of the field for an increas-ingly finer scale.

An interpretation of scale-type approaches in the contextof random field synthesis has been given by Zeldin and Spanos(1996) using compactly supported Daubechies wavelets.Specifically, a synthesis algorithm has been developed,

which includes the previous methods proposed by Lewis(1987) and Fenton and Vanmarcke (1990) as a particularcase. To synthesize a sample of a given process, the closed-form expressions

; (67)

; (68)

, (69)

given in Walter (1994) and Zeldin and Spanos (1996) areconsidered to relate the autocorrelation function R f ( x 1, x 2) of the process to the coefficients of its wavelet transform, (seeSection 2.2.2) which in this case are random variables. Thesynthesis algorithm is based on the wavelet reconstructionalgorithm developed by Mallat (1989a, 1989b), which pro-ceeds from coarse to fine scales to determine the wavelet coef-ficients. Some relevant properties of wavelet analysis, whichhave emerged in other fields, ensure the computational effi-ciency of the algorithm. Specifically, using the quasi-differ-ential properties of wavelets shown by Belkin (1993), thecoefficients d are derived directly from c by the approxi-mate linear combination

, (70)

where uk are uncorrelated, zero mean, unit variance randomvariables, statistically independent of c . For a wide classof stochastic processes, wavelet coefficients prove weaklycorrelated as the difference k – l increases and, for this, the

summation in Equation (70) is generally restricted to adja-cent elements only. The algorithm is completed by an errorassessment procedure which allows refining of the trigger-ing scale j in order to fit the sought target statistical proper-ties of the synthesized field.

Further studies on the role of wavelet analysis in stochas-tic mechanics applications may be found in Walter (1994).Walter has shown how wavelet bases can be used in approx-imate Karhunen–Loève expansions. Any stationary processcan be then represented as

, (71)

where d are uncorrelated random variables, and ψ j,k (t ) area non-orthogonal wavelet-like basis.

5. System Identification

Wavelet analysis lends itself to system identification appli-cations. For instance, frequency localization properties allowdetection and decoupling of individual vibration modes of multi-degrees-of-freedom (MDOF) linear systems. The wave-let representation of the system response can be truncated toan appropriate scale parameter, in order to filter measure-ment noise. Also, the wavelet transform coefficients can berelated directly to the system parameters, as long as specific

wavelet families are used.Early investigations trace back to the work by Robertson

et al. (1995), who have used the DWT for the estimation of the impulse response function of MDOF systems. Comparedto alternative time-domain techniques, the DWT-based extrac-tion procedure offers significant advantages. It is robust,since singularities in the procedure related matrices can gen-erally be avoided by selecting orthonormal wavelet func-tions. Further, the reconstructed impulse response functioncaptures the low-frequency components, referred to as staticmodes and mode shape errors, which ordinarily are difficultto estimate.

An important application of wavelet analysis to structuralidentification is due to Staszewski (1997), who has used

W f a b,( ) A ω b,( )eiω bd Z ′ ω ( )∞–

∞

∫ ≈ ,

Z ′ ω ( ) 2πa= Ψ ω a( )d Z ω ( )ˆ

E W f a b,( )2[ ] 4πa Ψ̂ ω a( ) 2S ff ω b,( )dω 0

∞

∫ =

r k l, j i, E d k

j d li[ ] R f x 1 x 2,( )ψ j k , x 1( )ψ i l, x 2( )d x 1d x 2

∞–

∞

∫ ∞–

∞

∫ = =

bk l, j i, E ck

j d li[ ] R f x 1 x 2,( )φ j k , x 1( )ψ i l, x 2( )d x 1d x 2

∞–

∞

∫ ∞–

∞

∫ = =

ak l, j i, E ck

j cli[ ] R f x 1 x 2,( )φ j k , x 1( )φ i l, x 2( )d x 1d x 2

∞–

∞

∫ ∞–

∞

∫ = =

k j

k j

d k j αk l,

jc

l j βk

j uk +l

∑=

k j

t ( ) d k j ψ j k , t ( )

j k ,∑=

k j




13/19


complex Morlet wavelets for modal damping estimation.Specifically, Staszewski has interpreted in terms of thewavelet transform some concepts already used in well-established methods, where the Hilbert transform has beenapplied to a free-vibration linear response (Jones, 1988). Inthe case of light damping, the free response in each mode

x j(t ) may be approximated in the complex plane by an ana-lytical signal, given by Equation (37). The modulus of theMorlet wavelet transform of x j(t ) can be expressed as

, (72)

where A j is the residue magnitude, and ω j and ζ j are themode natural frequency and damping ratio, respectively. InEquation (72) the symbol a j denotes the specific scale valuerelated to the mode natural frequency ω j by the closed-formrelation (51), typical of Morlet wavelets. Assuming that thenatural frequency ω j has been previously computed, thedamping ratio ζ j can be then estimated as the slope of astraight line, representing the cross-section wavelet modu-

lus (72) plotted in a semilogarithmic scale, i.e.

.(73)

Staszewski (1997) has also proposed an alternative dampingestimation method based on the “ridge” and “skeletons” of the wavelet transform. A ridge is a curve of local maxima inthe mean-square wavelet map and the corresponding skele-ton is given by the values of the wavelet transform restrictedto the ridge. Due to the localization properties of the wave-let transform, the ridges and skeletons of the wavelet trans-form can be detected separately for each mode. Specifically,the real part of the skeleton of the wavelet transform givesthe impulse response function for each single mode, fromwhich a straightforward estimate of the damping ratio isobtained from a logarithmic equation analogous to Equation(73). A generalization of the method for non-linear systemscan also be formulated (Staszewski, 1998a).

Ruzzene et al. (1997) have also presented a damping esti-mation algorithm based on the same concepts and leading toanalogous results. Certain issues have been addressed in detail,concerning the frequency resolution of the adopted waveletfamilies, crucial for detecting coupled modes, and appropri-ate algorithms for ridge extraction (Haase and Widjajaku-suma, 2003). Lardies and Gouttebroze (2002) have estimatedmodal parameters via ambient records, without input meas-

urements. To this end, the random decrement method (seeSpanos and Zeldin, 1998, and references therein) has beenused to convert ambient vibration response into a free vibrationresponse. Also, a modified Morlet wavelet has been devel-oped with enhanced properties for modal parameter estima-tion. The method devised by Staszewski and Ruzzene et al.has also been implemented by Slavic et al. (2003), by replacingMorlet wavelets by Gabor wavelets, whose time and fre-quency resolutions may be adjusted by an appropriate param-eter. Explicit conditions have been given on the frequencybandwidths of the Gabor wavelet transform, in order to esti-mate the instantaneous frequencies of two adjacent modes.

Damping coefficients have been estimated using a loga-rithmic decrement formula, where the ratio of the wavelet

transform at two subsequent extremes of the pseudo-period

T j = 2π / ω j of the response in each mode is involved, for aselected wavelet transform scale (Hans et al., 2000; Lamarqueet al., 2000). For the procedure to estimate the damping coef-ficient associated with the fundamental mode, it is sufficientto adapt the analyzing scale so that the higher-frequencymodes are filtered. For an arbitrary mode j, low-pass filter-ing is used to cancel the fundamental and the first j – 1 modes.Ghanem and Romeo (2000) have formulated a wavelet–Galerkin method for time-varying systems, where bothdamping and stiffness parameters are computed by solving amatrix equation. The latter is built by a standard Galerkinmethod, by projecting the solution of the differential equa-tion of motion onto a subspace described by the waveletscaling functions of a compactly supported Daubechies wave-let basis. The method is accurate for both free and forcedvibration responses. A formulation for non-linear systemshas also been proposed (Ghanem and Romeo, 2001). Anotherapplication is due to Yu et al. (2000), who have used wave-let transform to identify the parameters of a Preisach modelof hysteresis; see Spanos et al. (2004) and Mayergoyz

(2003), and references therein. The output function of thePreisach model is expanded in terms of the scaling functionsof a given wavelet family. Then, the coefficients of such anexpansion are determined by fitting a number of experimen-tal data points with a minimum energy method. From theoutput function, the so-called Preisach function can be deter-mined in a closed form.

A comprehensive application of wavelet concepts to sys-tem identification problems has been given by Le and Argoul(2004). They have developed closed-form expressions tocompute the damping ratio, the natural frequency, and theshape of each mode based on ridges and skeletons of thewavelet transformed free vibration response. As an alterna-tive, Yin et al. (2004) have proposed to apply the wavelettransform to the frequency response function (FRF) of thesystem. Specifically, given the FRF of an N -degrees-of-free-dom system in the form

, (74)

where λ r is the r th complex pole and Ar the r th residue, acomplex fractional function

, (75)

is selected as a wavelet family. Based on Equation (75) aclosed-form expression can be established for the CWT of Equation (74) multiplied by . Specifically,

(76)

Natural frequencies and damping ratios can be estimated bylocating the maxima of Equation (76) in the (a, b) plane.

W x j a j b,( ) A je ζ jω jb–≈ Ψ̂ i± a jω j 1 ζ j2–( )

ln W x j a j b,( )( ) ζ jω jb– A j Ψ̂ i± a jω j 1 ζ j2–( )

ln+≈

H ω ( ) Ar

iω λ r –-----------------

Ar

iω λ r –-----------------+

r 1=

N

∑=

ψ y x ( )1

1 i x +( ) y 1+--------------------------- e y 1+( ) 1 i x +( )ln–= = y 2+∈,

a( ) y–

H y a b,( ) a y 1+( ) 2 ⁄ –= H ω ( )ψ y∞–

∞

∫ ω b–a------------- dω

2πa y 1+( ) 2 ⁄ Ar

a ib λ r –+( ) y 1+--------------------------------------

Ar

a ib λ r –+( ) y 1+--------------------------------------+

r 1=

N

∑= .




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Note that an interesting use of wavelet analysis for detect-ing non-linear behaviors in structures has been proposed byArgoul and Le (2003). From the Cauchy wavelet transformof the transient response, four instantaneous indicators havebeen singled out, based on which an appropriate analyticalmodel may be constructed for the structure. Specific applica-tions have involved a non-linear beam excited by an impacthammer. In this context, note that further applications of wavelet analysis to non-linear vibrating systems may befound in the extensive work by Lamarque and Pernot (2001)and Pernot and Lamarque (2001, 2003).

6. Damage Detection

Properties of the wavelet transform are also quite appeal-ing for damage detection purposes. Early investigations in thisfield (Staszewski and Tomlinson, 1994; Wang and McFad-den, 1995) used wavelet analysis to detect local faults inmachineries. Visual inspection of the modulus and phase of the wavelet transform has been used to localize the fault(Staszewski and Tomlinson, 1994). Further, it has been shown

that transient vibrations due to developing damage are dis-closed by the local maxima of the mean-square wavelet map(Wang and McFadden, 1995). Additional results have beenthen proposed by Yesilyurt and Ball (1997), Sung et al.(2000), Boulahbal et al. (1999). Specifically, the latter sug-gested a combined use of amplitude and phase map to distin-guish the nature of damage, such as a cracked tooth. Also,they have pointed out that a Morlet CWT amplitude mapperforms better if applied on an “overall residual” signal,obtained by filtering out the gear meshing frequency fromthe time synchronous averaged signal. A confirmation inthis sense has been given by Dalpiaz et al. (2000) and Wanget al. (2001), for a variety of types of damage. Applicationsin machinery fault diagnostics have been also proposed byAdewusi and Al-Bedoor (2001), who used Daubechieswavelets to monitor startup and steady-state vibrations of anoverhang rotor with a propagating crack. Results in terms of amplitude map have shown how the crack propagation mayreduce the critical speed of the rotor and determine continu-ous changes in the amplitude of the vibration harmonics,unlike imbalance or misalignment which generally show aconstant amplitude.

Most of the studies previously mentioned have outlined asomewhat qualitative approach to damage detection, whichis mainly based on intuitive features of amplitude and/orphase maps. An attempt to assess the damage severity maybe found in fact only in Wang and McFadden (1995). To

address this issue, a combined use of wavelet analysis andneural networks has been proposed by Essawy et al. (1997),Paya et al. (1997), Ye et al. (2000), and Chen and Wang(2002). Subsequently, an energy-based method has beenalso devised by Zheng et al. (2002), who has monitored faultadvancement in an automobile gear box via a time-averagedMorlet wavelet spectrum. The latter reveals that the energycontent of the vibration signal shifts to lower frequenciesand increases according to a conic law as the gear faultadvances.

Wavelet analysis for damage detection has also involvedstructural engineering components. Early studies have shownthe correlation existing between local maxima of the wavelettransform and damage in simply supported beams (Liew and

Wang, 1998). For this, a new family of wavelets reflecting theboundary conditions has been introduced. Then, Wang andDeng (1999) used Haar and Gabor wavelet transforms on thenumerical displacement response of statically loaded crackedbeams and plates. The method has proved robust for variousboundary conditions and damage characteristics, such ascrack length, embedment, orientation and width, with a rela-tively low spatial resolution of measurement data (Quek etal., 2001a). However, no investigation has been performedon the feasibility of the method in the presence of noise andno relation has been found between the characteristic valuesof the wavelet transform and the damage degree. A firstattempt to estimate the damage degree was made by Okaforand Dutta (2000). Specifically, Daubechies wavelets wereused to wavelet transform the mode shapes of a damagedcantilever beam, and a regression analysis by a least-squaresmethod was conducted to correlate the peaks of the waveletcoefficients with the corresponding damage degree.

A consistent mathematical framework for wavelet analy-sis of damaged beams is due to Hong et al. (2002). The focalconcept is that defects in structures, even if small, may sig-

nificantly affect the vibration mode shapes, depending onthe location and the type of damage. Such variations may notbe apparent in the measured data, but become detectable assingularities if wavelet analysis is used, due to its high-reso-lution properties. Specifically, Hong et al. have shown thatthe singularity of the vibration modes can be described interms of Lipschitz regularity, a concept also encountered inthe theory of differential equations and widely used in imageprocessing where object contours correspond to irregulari-ties in the intensity (Grossmann, 1986; Mallat and Hwang,1992). In mathematical terms, a function f ( x ) is Lipschitz

at x = x 0 if there exist K > 0, and a polynomial of orderm (m is the largest integer satisfying ), pm( x ), such that

, (77)

. (78)

The wavelet transform of Lipschitz α functions enjoys someproperties. Mallat and Hwang (1992) have shown that for awavelet family with a number of vanishing moments ,a local Lipschitz singularity at x 0 corresponds to maximalines of the wavelet transform modulus. That is, localmaxima with asymptotic decay across scales. Near the coneof influence x = x 0, such moduli satisfy the equation

, (79)

from which the Lipschitz exponent is computed as

. (80)

By plotting the wavelet coefficients on a logarithmic scale, A and α may be computed by setting the equality sign inEquation (80) and minimizing the error in the least-squaressense. Hong et al. have applied Equation (79) to the firstmode shape of a damaged free-free beam, via a MexicanHat wavelet transform. The first mode shape is preferablesince it is the most accurately determined by modal testing,it features the lowest curvature, and sets off the singularity

α 0≥m α≤

f x ( ) pm x ( ) ε x ( )+=

ε x ( ) K x x 0– α≤

α n≤

W f a x ,( ) Aa α 1 2 ⁄ +≤ A 0>,

log2 W f a x ,( ) log2≤ A α1

2---+

log2a+




15/19


better. A correlation between damage degree and the magni-tude of the Lipschitz exponent has been found, from anumber of beams with different damage parameters.

Some of the ideas presented by Hong et al. may be alsofound in the work by Douka et al. (2003, 2004), who havepursued crack identification in beams and plates usingDaubechies wavelets. The first mode vibration response hasbeen considered and the singularity induced by local defectshas been characterized in terms of Equation (79). The Lip-schitz exponent has been used to describe the type of singu-larity, and the parameter A has been taken as the factorrelating the depth of the crack to the amplitude of the wave-let transform. Specifically, a second-order polynomial lawhas been found for the intensity factor as a function of thecrack depth. The work by Douka et al. has pointed out theimportance of the number of vanishing moments M of thechosen wavelet family. It is intuitive that the capability of setting off singularities in a regular function increases with

M . However, wavelet functions with high M exhibit a longsupport and lack space resolution. A compromise must thenbe achieved depending on the application in hand. Further

insight into some mathematical details of both the methodsdeveloped by Hong et al. and Douka et al. may be found inHaase and Widjajakusuma (2003). Specifically, a fast algo-rithm to determine the maxima lines of the wavelet trans-form has been devised. Also, the performance of variouswavelet families, such as the Gaussian family of wavelets,has been assessed versus Daubechies wavelets used byDouka et al. Another interesting method has been devisedby Chang and Chen (2005), who estimated the positions anddepths of multiple cracks in a beam modeled as rotationalsprings. In a first step, the experimental vibration modeshave been wavelet transformed to determine the positionof the cracks. Then experimental natural frequencies, in anumber equal to the cracks, have been used to predict thecracks depth via the characteristic equation.

An alternative strategy for damage detection in beams hasbeen recently developed via wavelet analysis of flexuralwaves induced by an impact hammer at the free end andmeasured by conventional strain gages or piezoelectric sen-sors. In this context, work has been carried out by Zhang etal. (2001a), who applied Morlet wavelet transform to detectthe existence and location of multiple cracks in a cantileverbeam. The crack position has been derived by observingabnormal peaks in the wavelet transformed mid-frequencyflexural wave. Further investigations on the sensitivity of themethod to sensor position, damage height, and degree of damage have been carried out by Zhang et al. (2001b), while

practical issues involving the allowable range of waveletscale to process have been subsequently addressed by Quek et al. (2001b). Additional results on wavelet analysis of flex-ural waves in multicracked beams may be found in Tianet al. (2003).

Besides the above-mentioned applications on individualstructural components, wavelet analysis has also yieldedencouraging results for global structural health monitoring.In this regard, interesting results have been presented byHera and Hou (2004), who applied Daubechies wavelets toAmerican Society of Civil Engineers (ASCE) benchmark study data. Specifically, a four-story, two-bay by two-bayprototype steel building subjected to stochastic wind loadinghas been considered. It has been shown that the occurrence

of damage, due to a sudden breakage of interstory braces, isrevealed by a spike in the high-resolution wavelet details of the acceleration response data. Further, the location of thedamage region may be determined by the spatial distributionpattern of the spikes in the acceleration responses at somerepresentative points in the structure. An attempt to extendthe method to damage events of finite time duration has beenalso proposed by Hou et al. (2002) based on experimentaldata from a shaking table test of a full-size two-story woodenframe.

Wavelet analysis has been also used for damage detectionin composite structures. In a first attempt, Zhu et al. (1999)presented wavelet transformed experimental data of delami-nated carbon reinforced composite plates, but no quantita-tive determination of the location and amplitude of thedefect was given. Then, Staszewski et al. (1999) used across-wavelet analysis to improve the interpretation of Lambwave data related to defects in a carbon fiber compositeplate. The Lamb waves, in fact, prove quite effective since theycan propagate over long distances in the composite materialand can interfere with damage. Sung et al. (2002) applied the

Daubechies wavelet transform on the acoustic emissionwaves generated by low-velocity impact loads to determinedamage modes and size in composite laminates. Specifi-cally, they found a relation between levels of detail of thewavelet transform and damage modes such as matrix cracksand delamination. In order to detect small and incipient dam-age, Yam et al. (2003) have devised a method based on theenergy variation of the vibration response due to the occur-rence of damage. The method is implemented in two steps.The first involves the construction of damage feature proxyvectors using the energy at various scales of the wavelettransformed vibration response. Then, classification andidentification of the structural damage status is pursuedby using artificial neural networks (ANNs), which offer sig-nificant advantages compared to genetic algorithms (GAs),developed by Moslem and Nafaspour (2002) for damageidentification purposes. GA-based damage detection requiresrepeatedly searching among numerous damage parametersto find the optimal solution of the objective function. Yetanother approach for applications of wavelet analysis fordamage detection in composite plates has been discussedby Paget et al. (2003). It is based on Lamb waves generatedand received by embedded piezoceramic transducers. Tocharacterize the damage, the Lamb waves are wavelet trans-formed using an original wavelet family, devised from therecurrent waveforms of the Lamb waves. The changes inthe Lamb waves interacting due to the occurrence of damage

are captured by the amplitude change of the wavelet coef-ficients. From this effect, an estimate of the impact energyand the damage level is obtained based on experimentalresults.

It is worth mentioning that wavelet-based methods havebeen also devised to address denoising, which is a crucialaspect of damage diagnostics. In this regard, a significantcontribution has been given by Donoho (1995). Methodshave been also developed by Watson et al. to process non-destructive signals of piled foundations (1999), by Menon etal. (2000) to detect fatigue cracks in helicopter rotor headdynamic components, and by Lin and Qu (2000) to extractperiodic impulses in noisy gearbox vibration signals. Fur-ther, for the on-line health monitoring of machinery and




16/19


structures, an efficient wavelet-based data compression methodhas been implemented by Staszewski (1998b).

7. Material Characterization

The description of material properties is another applica-tion for wavelet analysis. Intuition suggests that multiscaleanalysis is a natural way for describing microstructure ormaterial heterogeneity. Various, in fact, are the examples of multiscale microstructures, such as porosity distributions inceramics, defects, dislocations, grain boundaries, and pores.It is important, however, to understand how information atdifferent scales is related, and whether large or small scalesaffect macroscopic material properties such as deformation,toughness, and electrical conductance. Additional interesttowards a multiscale description of material properties is moti-vated by the need for alternatives to the standard finite ele-ment method (FEM). The latter, although capable in principle,cannot simulate efficiently the actual behavior of materialssuch as aluminum alloys, where pores may attain a size up to500 µm, and inclusions may attain sizes up to 3–6 µm in

diameter. Further, in FEM-based methods the constitutiveresponse of the material at increasing scales is not the resultof microstructural analysis at smaller scales, but it is ratherassumed on the basis of macroscopic experiments.

Willam et al. (2001) have performed multiresolution homo-genization based on a recursive Schur reduction method, inconjunction with the Haar wavelet transform. The methodallows coarse grained parameters, such as Young’s modulusof elasticity, to be extracted from fine grained properties atthe mesoscale and microscale. Also, progressive elastic deg-radation can be modeled, which initiates at a quite fine scaleand evolves into a macroscopic zero stiffness at the contin-uum level.

Frantzikonis (2002a) has focused on stationary and iso-tropic porous media. The geometry of porous media is gen-erally described in terms of a fundamental function, definedas unity for spatial locations in the matrix and as zero forlocations in the pores or flaws. At a solid–flaw interface theporous medium is represented mathematically through alocal jump in the fundamental function. It has been foundthat such a jump can be captured by a wavelet transform, aslong as the finest scale is small enough relative to the size of the pores. From this fact, a relationship between the energyof the wavelet transform of the porous medium, and thevariance and the correlation distance of the solid phasecan be derived. In the presence of heterogeneous materials,with multiscale porosity, the role of porosity at each scale

has been identified through the variation of the energy of the wavelet transform as a function of scale. Peaks of theenergy reveal the dominant scale in determining macro-scopic properties of the materials, such as mechanical fail-ure. Specifically, a biorthogonal spline with four vanishingmoments has been employed as a wavelet family. Theresults obtained have been subsequently extended in a sec-ond study, addressing the crack formation in an aluminumalloy with distributed pores and inclusions (Frantziskonis,2002b). The problem, implemented for a one-dimensionalsolid, is tackled by wavelet transforming the flexibilityfunction, assumed to vary along the longitudinal axis of theone-dimensional solid. The relationship between the energyof the wavelet transform and the variance of the flexibility

is used to detect the dominant scale in the crack formationprocess.

Note that an application of a two-dimensional wavelettransform has been described in Ciliberto et al. (2002) forporosity classification on carbon fiber reinforced plastics.

8. Concluding Remarks

Concepts of wavelet-based continuous and discrete repre-sentations of signals have been reviewed. Further, we haveincluded an overview of vibration-related applications forevolutionary spectrum estimation, random field simulation,system identification, damage detection, and material char-acterization. The list of references cannot be exhaustive and,thus, other perhaps relevant applications of the wavelet trans-form have been omitted for succinctness; for instance, wemention the wavelet-based analysis of elastic waves in sol-ids for which interesting contributions can be found in refer-ences such as Kishimoto et al. (1995) and Park and Kim (2001).Nevertheless, it is believed that the references cited in thispaper can serve as readily available resources for canvassing

the multitude of concepts and applications of wavelet analy-sis, this remarkable tool for capturing and representing local-ization features of many physical phenomena. Wavelet-basedalgorithms and commercial codes are indeed an indispensa-ble family of tools for vibration analysis, and offer, in manycases, a potent improvement over the classical Fourier trans-form based approaches.

Acknowledgment

The support of this work by a grant from the NationalScience Foundation, USA, is gratefully acknowledged.

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