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11.2 Time – Frequency Analysis

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A convenient way to introduce the wavelet transform is through the concept of time – frequency

representation of signals. In the classical Fourier theory, a signal can be represented either in the time or

in the frequency domain, and the Fourier coefficients define the average spectral content over the entire

duration of the signal. The Fourier representation is appropriate for signals that are stationary, in terms

of parameters which are deemed important for the problem in hand, but becomes inadequate for

nonstationary signals, in which important parameters may evolve rapidly in time.

The need for a time – frequency representation is obvious in a broad range of physical problems, such as

acoustics, image processing, earthquake and wind engineering, and a plethora of others. Among the time –

frequency representations available to date, the wavelet transform has unique features in terms of efficacy

and versatility. In mathematical terms, it involves the concept of scale as a counterpart to the concept of

frequency in the Fourier theory. Thus, it is also referred to as time-scale representation. Its formulation

stems from a generalization of a previous time – frequency representation, known as the Gabor transform.

For completeness, and to underscore the significant advantages achieved by the development of the wavelet

transform, the Gabor transform is briefly discussed in Section 11.2.1. Section 11.2.2 is entirely devoted to

the wavelet transform, and the most commonly used wavelet families are described in Section 11.2.3.

11.2.1 Gabor Transform

The first steps in time – frequency analysis trace back to the work of Gabor [24], who applied in signal

analysis fundamental concepts developed in quantum mechanics by Wigner a decade earlier [25]. Given

a function f ðtÞ belonging to the space of finite-energy one-dimensional functions, denoted by L 2(R),

Gabor introduced the transform

Gf ðv; t0Þ ¼

ð1

f ðtÞgðt 2 t0Þ e2iv ðt2t0 Þ dt ð11:1Þ

where gðtÞis a window and the bar ð􀀊Þdenotes complex conjugation. This transform, generally referred to as

the continuous Gabor transform (CGT) or the short-time Fourier transform of f ðtÞ; is a complete

representation of f ðtÞ: That is, the original function f ðtÞ can be reconstructed as

f ðtÞ ¼

􀀉 􀀉

ð1

Gf ðv; t0Þgðt 2 t0Þeivðt2t0 Þ dv dt0 ð11:2Þ

where

􀀉 􀀉

2 ¼

121

􀀈 􀀈gðtÞ

􀀈 􀀈

2 dt: The Gabor transform (Equation 11.1) may be seen as the projection of f ðtÞonto

the family {gðv;t0 ÞðtÞ; v; t0 [ R} of shifted and modulated copies (atoms) of gðtÞ expressed in the form

gðv;t0 ÞðtÞ ¼ eivðt2t0 Þgðt 2 t0Þ ð11:3Þ

These time – frequency atoms, also referred to as Gabor functions, are shown in Figure 11.1 for three

different values of v: Clearly, if gðtÞ is an appropriate window function, then Equation 11.1 may be regarded

as the standard Fourier transform of the function f ðtÞ; localized at the time t0: In this context, t0 is the

11-2 Vibration and Shock Handbook

time parameter which gives the center of the window, and v is the frequency parameter which is used to

compute the Fourier transform of the windowed signal.

As intuition suggests, the accuracy of the CGT representation (Equation 11.2) of f ðtÞ depends on the

window function gðtÞ; which must exhibit good localization properties both in the time and the

frequency domains. As discussed in Ref. [6], a measure of the localization properties may be obtained by

the average and the standard deviation of the density

􀀈 􀀈

gðtÞ

􀀈 􀀈

2 in the time domain. That is,

ktl ¼

ð1

􀀈 􀀈

gðtÞ

􀀈 􀀈

2 dt ð11:4aÞ

s2t

ð1

21 ðt 2 ktlÞ2􀀈 􀀈

gðtÞ

􀀈 􀀈

2 dt ð11:4bÞ

The counterparts of Equation 11.4a and Equation 11.4b in the frequency domain are

kvl ¼

ð1

􀀈 􀀈

G^ ðvÞ

􀀈 􀀈

2 dv ð11:5aÞ

s2v

ð1

21 ðv 2 kvlÞ2􀀈 􀀈

G^ ðvÞ

􀀈 􀀈

2 dv ð11:5bÞ

where G^ ðvÞ denotes the Fourier transform of gðtÞ given by the equation

G^ ðvÞ ¼

1ffiffiffiffi

p2p

ð1

gðtÞe2ivt dt ð11:6Þ

The well-known Heisenberg uncertainty principle is in actuality a mathematically proven property and

states that the values st and sv cannot be independently small [6]. Specifically, for an arbitrary

window normalized so that

􀀉 􀀉

2 ¼ 1; it can be shown that

stsv $

2 ð11:7Þ

Thus, high resolution in the time domain (small value of st ) may be generally achieved only at the

expense of a poor resolution (bigger than a minimum value sv ) in the frequency domain and vice versa.

Note that the optimal time – frequency resolution, that is stsv ¼ 1=2; may be attained when the Gaussian

window

gðtÞ ¼

ffiffiffiffiffiffiffi

2ps2t

4p exp 2

4s 2

􀁻 !

ð11:8Þ

is selected.

Clearly, as a time – frequency representation, the Gabor transform exhibits considerable limitations.

The time support, governed by the window function gðtÞ; is equal for all of the Gabor functions

(Equation 11.3) for all frequencies (see Figure 11.1). In order to achieve good localization of highfrequency

components, narrow windows are required; as a result of that, low-frequency components are

poorly represented. Thus, a more flexible representation with nonconstant windowing is quite desirable,

w1 w2 w3

FIGURE 11.1 Plots of Gabor function gðv; t0 Þ versus the independent variable x for three values of the frequency v;

the effective support is the same for the three values of the frequency.

Wavelets — Concepts and Applications 11-3

to enhance the time resolution for short-lived high-frequency phenomena and frequency resolution for

long-lasting low-frequency phenomena.

11.2.2 Wavelet Transform

The preceding shortcomings of the Gabor transform have been overcome with significant effectiveness

and efficiency by wavelets-based signal representation. Its two formulations, continuous and discrete, are

described in the ensuing sections. Because of the numerous applications of wavelets beyond time –

frequency analysis, the t-time domain will be replaced by a generic x-space domain. For succinctness, the

formulation will be developed for scalar functions only, but generalization for multidimensional spaces is

well established in the literature [1 – 22].

11.2.2.1 Continuous Wavelet Transform

The concept of wavelet transform was introduced first by Goupillaud et al. for seismic records analysis

[26,27]. In analogy to the Gabor transform, the idea consists of decomposing a function f ðxÞ into a twoparameter

family of elementary functions, each derived from a basic or mother wavelet, c ðxÞ: The first

parameter, a; corresponds to a dilation or compression of the mother wavelet that is generally referred to

as scale. The second parameter, b; determines a shift of the mother wavelet along the x-domain. In

mathematical terms

Wf ða; bÞ ¼

1ffiffi

ð1

f ðxÞc

x 2 b

􀀏 􀀐

dx ð11:9Þ

where a [ Rþ; b [ R: In the literature, Equation 11.9 is generally referred to as continuous wavelet

transform (CWT). Note that the factor a21=2 is a normalization factor, included to insure that the mother

wavelet and any dilated wavelet a21=2c ðx=aÞ have the same total energy [26]. Clearly, alternative

normalizations may also be chosen [1].

An example of wavelet functions is shown in Figure 11.2 for different values of the scale parameter a:

As a result of scaling, all the wavelet functions exhibit the same number of cycles within the x-support of

the mother wavelet. Obviously, the spatial and frequency localization properties of the wavelet transform

depend on the value of the parameter a: As a approaches zero, the dilated wavelet a21=2cðx=aÞ is highly

concentrated at the point x ¼ 0; the wavelet transform, Wf ða; bÞ; then gives increasingly sharper spatial

resolution displaying the small-scale/higher-frequency features of the function f ðxÞ; at various locations

b: However, as a approaches þ1, the wavelet transform Wf ða; bÞ gives increasingly coarser spatial

resolution, displaying the large-scale/low-frequency features of the function f ðxÞ:

For the function f ðxÞ to be reconstructable from the set of coefficients (Equation 11.9), in the form

f ðxÞ ¼

pcc

ð1

Wf ða; bÞca;bðxÞ

a2 db ð11:10Þ

a1 a2 a3

FIGURE 11.2 Plots versus time of wavelet functions corresponding to three different values of a scale a of the same

mother function; the effective time support increases with the magnitude of the scale.

11-4 Vibration and Shock Handbook

where ca;bðxÞ ¼ a21=2c½ðx 2 bÞ=a􀀉; the wavelet function cð·Þ must satisfy the admissibility condition

cc ¼

ð1

􀀈 􀀈

C^ ðvÞ

􀀈 􀀈

dv , 1 ð11:11Þ

where C^ ðvÞ denotes the Fourier transform of cðxÞ: As pointed out in Ref. [26], the condition 11.11

includes a set of subconditions, such as:

1. The analyzing wavelet cð·Þ is absolutely integrable and square integrable. That is,

ð1

􀀈 􀀈

cðxÞ

􀀈 􀀈

dx , 1 ð11:12aÞ

ð1

􀀈 􀀈

cðxÞ

􀀈 􀀈

2 dx , 1 ð11:12bÞ

2. The Fourier transform C^ ðvÞ must be sufficiently small at the vicinity of the origin v ¼ 0; or in

mathematical terms

ð1

􀀈 􀀈

C^ ðvÞ

􀀈 􀀈

dv , 1 ð11:13Þ

Subcondition 2, then, implies that C^ ð0Þ ¼ 0; that is,

1 cðxÞdx ¼ 0: Therefore, for an analyzing

wavelet to be admissible, its real and imaginary parts must both be symmetric with respect to the x-axis.

From the reconstruction formula (Equation 11.10), it can be shown that

􀀉 􀀉

2 ¼

pcc

ð1

􀀈 􀀈

Wf ða; bÞ

􀀈 􀀈

2 da

a2 db ð11:14Þ

Based on Equation 11.14, the square modulus of the wavelet transform (Equation 11.9) is often taken as

an energy density in a spatial-scale domain. Extensive use of this concept has been made for spectra

estimation purposes, as discussed in Section 11.3.

Note that the reconstruction wavelet in Equation 11.10 can be different from the analyzing wavelet

used in Equation 11.9. That is, under some admissibility conditions on xðxÞ [1], the original function

f ðxÞ may be reconstructed as

f ðxÞ ¼

ccx

ð1

Wf ða; bÞxa;bðxÞ

a2 db ð11:15Þ

where xa;bðxÞ ¼ a21=2x½ðx 2 bÞ=a􀀉 and ccx is a constant parameter depending on the Fourier transforms

of both cðxÞ and xðxÞ: This property, referred to as redundancy in mathematical terms, may be

advantageous in some applications for reducing the error due to noise in signal reconstruction [28,29],

but highly undesirable for signal coding or compression purposes [1]. Further, under certain conditions

[1], the following simplified reconstruction formula holds

f ðxÞ ¼

ð1

Wf ða; xÞ

a3=2 ð11:16Þ

where kc is a constant parameter given by the equation

kc ¼

ffiffiffiffi

2p p ð1

C^ ðvÞ

dv ð11:17Þ

Use of this formula has been made, in a discrete version, in the approximation theory of functional spaces

[1] and also in structural identification applications, as discussed in Section 11.5.

Wavelets — Concepts and Applications 11-5

11.2.2.2 Discrete Wavelet Transform

For numerical applications, where fast decomposition or reconstruction algorithms are generally

required, a discrete version of the CWT is to prefer. In this sense, a natural way to define a discrete wavelet

transform (DWT) is

Wf ð j; kÞ ¼

1ffiffiffi

a j

ð1

f ðxÞcða2j

0 x 2 kb0Þdx; j; k [ Z ð11:18Þ

Equation 11.18 is derived from a straightforward discretization of the CWT (Equation 11.9) by

considering the discrete lattice a ¼ a j

0; a0 . 1; b ¼ kb0a j

0; b0 – 0: In developing Equation 11.18,

however, the main mathematical concern is to ensure that sampling the CWT on a discrete set of points

does not lead to a loss of information about the wavelet-transformed function f ðxÞ: Specifically, the

original function f ðxÞ must be fully recoverable from a discrete set of wavelet coefficients. That is,

f ðxÞ ¼

j;k[Z

Wf ðj; kÞcj;kðxÞ ð11:19Þ

where cj;kðxÞ ¼ a2j=2

0 cða2j

0 x 2 kb0Þ. Another crucial aspect in Equation 11.18 involves selecting the

wavelet functions cj;kðxÞ such that Equation 11.19 may be regarded as the expansion of f ðxÞ in a basis,

thus eliminating the redundancy of the CWT.

This issues are addressed by using the theory of Hilbert space frames, introduced in 1952 by Duffin and

Schaeffer in context with non-harmonic Fourier series [30]. In general, if hl ðxÞ [ L2ðRÞ and L is a

countable set, a family of functions {hl ðxÞ; l [ L} constitutes a frame, if for any f ðxÞ [ L2ðRÞ

􀀉 􀀉

2 #

l[L

􀀈 􀀈

kf ; hl l

􀀈 􀀈

2 # B

􀀉 􀀉

2 ð11:20Þ

where k f ; hl l ¼

1 21 f ðxÞhl ðxÞ dx and A . 0; B , 1; the so-called frame bounds, are independent of f ðxÞ

[1]. The concept of frame may be interpreted as an extension of the concept of basis, in the sense that the

reconstruction of the original function is possible via stable numerical expressions in terms of the set

{hl ðxÞ; l [ L}: For instance, if the frame is tight, that is A ¼ B; the simple formula

f ðxÞ ¼

l[L

k f ; hl lhl ðxÞ ð11:21Þ

holds [29].

In contrast to a basis, however, the vectors of a frame may be linearly dependent and, for this, a certain

degree of redundancy is still retained in the reconstruction formula (Equation 11.21) [29,31].

The concept of frame has played a crucial role in the formulation of the DWT. The first wavelet frames

were constructed by Daubechies et al. [32]. Later, Battle [33] constructed orthonormal bases with an

exponential decay. The ensemble of these results has demonstrated the advantages of the wavelet

transform over the Gabor transform. In fact, it has been shown that discrete versions of Gabor transform

are not capable of generating orthonormal bases [32] due to the so-called Balian – Low phenomenon [1].

Mallat [34] has shown that the orthonormal wavelet bases proposed by Battle can all be derived by a

multiresolution analysis. The latter involves representing an arbitrary f ðxÞ [ L2ðRÞ as a limit of successive

approximations, at different resolutions. That is, if {Vj}j[Z is a sequence of subspaces of L2ðRÞ; and fj is the

orthogonal projection of f ðxÞ on Vj; in a multiresolution analysis the following conditions hold

lim

j !21

fj ¼ f ð11:22aÞ

lim

j !1

fj ¼ 0 ð11:22bÞ

Each approximation fj; then, represents a smoothed version of f ðxÞ and, in the limit, more and more

localized smoothing functions lead to the function f ðxÞ: From a mathematical point of view [29,31], a

11-6 Vibration and Shock Handbook

multiresolution analysis requires that

1. The subspaces Vj’s are closed and embedded, that is

· · · , V2 , V1 , V0 , V21 , V22 , · · · ð11:23Þ

where V2m ! L2ðRÞ for m ! 1 and f [ Vm , f ð2·Þ [ Vm21:

2. A scaling function fðxÞ [ L2ðRÞ exists, such that, for each j; the family of functions

fj;kðxÞ ¼ 22j=2fð22jx 2 kÞ; k [ Z ð11:24Þ

spans the subspace Vj and constitutes a Riesz basis for Vj; that is, there exists 0 , C0 # C00 , 1

such that

C 0

􀀈 􀀈

2 #

ð1

ckfj;k ðxÞ

􀀈 􀀈 􀀈 􀀈 􀀈

dx # C 00

􀀈 􀀈

2 ð11:25Þ

for all sequences of numbers ðck Þk[Z: Equation 11.25 is a more stringent condition of

Equation 11.20 and includes the latter as a special case.

The concept of multiresolution analysis offers a straightforward and mathematically coherent

approach to discrete wavelet analysis. Given a scaling function fðxÞ as in 2, in fact, families of

orthonormal wavelet bases

cj;kðxÞ ¼ 22j=2cð22jx 2 kÞ; j; k [ Z ð11:26Þ

can be developed by appropriate algorithms. For this, Mallat has used the frequency response of a highpass

filter [35], while Daubechies has devised a systematic approach to build orthonormal wavelet bases

with compact support in the x-domain [36]. Specifically, for each even integer 2M; the Daubechies

scaling function fðxÞ can be computed as

fðxÞ ¼

ffiffi

2 p 2MX21

k¼0

hkþ1fð2x 2 kÞ ð11:27Þ

where hk’s are 2M coefficients obtained by imposing M orthogonality conditions and M accuracy

conditions to enhance the rate of convergence of the approximation to the original function f ðxÞ: In turn,

the mother wavelet is related to the scaling function fðxÞ by the equation

cðxÞ ¼

ffiffi

2 p 2MX21

k¼0

gkþ1fð2x 2 kÞ ð11:28Þ

where gk’s are the same as hk’s but reversed in order and with alternate signs. Numerical values of both

series hk’s and gk’s are readily available in the literature [16].

Also based on multiresolution analysis concepts, a wavelet decomposition algorithm for image analysis

has been developed [34,35]. If associated to Daubechies wavelets, the algorithm becomes quite efficient

from a computational point of view, since no numerical integration is involved to compute wavelet

and scale coefficients. It relies on the projection of f ðxÞ onto a sufficiently fine scale j of the set 11.24.

That is,

f ðxÞ < fjðxÞ ¼

c j

kfj;kðxÞ ð11:29Þ

where, for orthogonal wavelets,

c j

k ¼

ð1

f ðxÞfj;kðxÞdx ð11:30Þ

Based on Equation 11.22a and Equation 11.22b, the projection fjðxÞ can be rewritten in terms of the

projection fjþ1ðxÞ onto the coarser scale ðj þ 1Þ and the incremental detail djþ1ðxÞ; that is the pieces of

Wavelets — Concepts and Applications 11-7

information contained in the subspace Vj and lost when “moving” to the subspace Vjþ1: Therefore,

fjðxÞ ¼ fjþ1ðxÞ þ djþ1ðxÞ ¼ fjþlðxÞ þ djþ1ðxÞ þ · · · þ djþlðxÞ < djþ1ðxÞ þ · · · þ djþl ðxÞ ð11:31Þ

As a fundamental result of multiresolution analysis, the details djðxÞ ____________can be decomposed in terms of the

set of wavelet functions at the same scale. That is,

djðxÞ ¼

d j

kcj;kðxÞ ð11:32Þ

where d j

k ’s are the wavelet coefficients of f ðxÞ: Based on Equation 11.28, both wavelet and scale coefficients

can be computed recursively by the closed-form expressions

c j

k ¼

2MX21

l¼0

hlþ1c j21

2kþl21 ð11:33aÞ

d j

k ¼

2MX21

l¼0

glþ1c j21

2kþl21 ð11:33bÞ

Similarly, the reconstruction algorithm can be implemented by the formula

c j21

k ¼

hk22lþ2c j

l þ gk22lþ2d j

l ð11:34Þ

The reconstruction algorithm described by Equation 11.34 lends itself to interpretation as a scale

linear system [37,38]. Based on this concept, applications have also been developed for random field

simulation [39].

11.2.3 Wavelet Families

A great number of wavelet families with various properties are available. Selecting an optimal family for a

specific problem is not, in general, an easy task and there are properties 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 for building smooth reconstructed signals or accurate

nonlinear regression estimates. Compactly supported wavelets, either in the time or in the frequency

domain, may be preferable for enhanced time or frequency resolution. The number of vanishing

moments, M; that is the highest integer m for which the equation

ð1

xmc ðxÞdx ¼ 0; m ¼ 0; 1; …; M 2 1 ð11:35Þ

holds is important in signal processing for compression, or in damage detection for enhancement of

singularities in the vibration modes. Also, in some cases, wavelets may be required to be progressive.

In mathematical terms, this means that their Fourier transform is defined only for positive frequencies.

That is,

C^ ðvÞ ¼ 0; for v , 0 ð11:36Þ

The progressive wavelet transform of a real-valued signal f ðtÞ and the associated analytic signal

zf ðtÞ ¼ f ðtÞ þ iH½f ðtÞ􀀉 ð11:37Þ

are related by the equation

Wf ða; bÞ ¼

Wz f ða; bÞ ð11:38Þ

where H½·􀀉 denotes the Hilbert transform operator [40]. Equation 11.38 is quite useful for structural

identification. Note also that significant reduction of computational costs is generally achieved if

orthogonal wavelets in the frequency or in the x-domain are used.

11-8 Vibration and Shock Handbook

A brief description of the most-used families is given below. A distinction is made between real and

complex wavelets, and the most relevant properties for application purposes are discussed. A more

exhaustive review may be in found in Ref. [15].

11.2.3.1 Real Wavelets

1. Daubechies orthonormal wavelets — A family of bases, each corresponding to a particular value of

the parameter M in Equation 11.27 and Equation 11.28 [36]. Closed-form expressions for fðxÞ in

Equation 11.27 are available only for M ¼ 1; to which the well-known Haar basis corresponds. In

this case, the scaling function and the mother wavelet are

fðxÞ ¼

1; 0 # x , 1;

0; elsewhere;

(

cðxÞ ¼

1; 0 # x ,

;

21;

# x , 1;

0; elsewhere:

8>>>>><

>>>>>:

ð11:39Þ

Various algorithms, however, are available in the literature for determining fðxÞ and cðxÞ

numerically for M . 1:

Daubechies wavelets support both CWT and DWT, although the latter is most generally

performed due to the fast decomposition and reconstruction algorithm mentioned in

Section 11.2.2.2. Both fðxÞ and cðxÞ are compactly supported in the x-domain, and the support

is equal to the segment ½0; 2M 2 1􀀉: Also, M is equal to the number of vanishing moments of the

wavelet function. Note that most Daubechies wavelets are not symmetric; regularity and

harmonic-like shape increases with M:

2. Meyer wavelets — Families of wavelets [10], each defined for a particular choice of an auxiliary

function vðvÞ which appears in the following expression for the Fourier transform of the mother

wavelet:

C^ ðvÞ ¼

1ffiffiffiffi

p2p eiv=2 sin

􀀈 􀀈

2 1

􀀒 􀀏 􀀐􀀓

;

p #

􀀈 􀀈

1ffiffiffiffi

p2p eiv=2 cos

􀀈 􀀈

2 1

􀀒 􀀏 􀀐􀀓

;

p #

􀀈 􀀈

􀀊

􀀒 􀀓

;

8>>>>>>><

>>>>>>>:

ð11:40Þ

for vðvÞ to be an admissible auxiliary function it is required that

vðvÞ ¼

0; v # 0;

1; v $ 1;

(

ð11:41aÞ

vðvÞ þ vð1 2 vÞ ¼ 1; 0 # v # 1 ð11:41bÞ

The most common form of vðvÞ in the literature is

vðvÞ ¼ v4ð35 2 84v þ 70v2 2 20v3Þ; 0 # v # 1 ð11:42Þ

The mother wavelet, for which only numerical expressions are available, is then constructed by

inverse Fourier-transforming Equation 11.40.

Meyer wavelets are suitable for both CWT and DWT. Unlike Daubechies wavelets, they are

compact in the frequency domain but not in the x-domain. Because of their fast decay, however,

an effective x-support [2 8,8] is generally assumed. Appealing features of Meyer wavelets are

orthogonality, infinite regularity, and symmetry.

3. Mexican Hat wavelets — A family of wavelets in the x-domain [15] related to a mother function

that is proportional to the second derivative of the Gaussian probability density function.

Wavelets — Concepts and Applications 11-9

That is,

cðxÞ ¼

2ffiffi

3 p p21=4ð1 2 x2Þe2x2 =2 ð11:43Þ

The Mexican Hat wavelets allow CWT only. Unlike Daubechies or Meyer wavelets, Mexican Hat

wavelets are not compact both in the frequency and in the x-domain, although an effective

support [2 5,5] may be considered for practical calculations. They are infinitely regular and

symmetric.

4. Biorthogonal wavelets — Families of wavelets derived by generalizing the ordinary concept of

wavelet bases, and creating a pair of dual wavelets, say ðcðxÞ;c~ðxÞÞ; satisfying the following

properties [41,42]:

ð1

cj;kðxÞ c~j0;k0 ðxÞdx ¼ djj0dkk 0 ð11:44Þ

where the symbol dmn denotes the Kronecker delta. One wavelet, say cðxÞ; may be used for

reconstruction and the dual one, c~ðxÞ; for decomposition. Therefore, Equation 11.18 and

Equation 11.19 can be rewritten as

Wf ð j; kÞ ¼ 22j=2

ð1

f ðxÞcj;kð22jx 2 kÞdx; j; k [ Z ð11:45Þ

f ðxÞ ¼

j;k[Z

Wf ð j; kÞ c~j;kðxÞ ð11:46Þ

Biorthogonal wavelets support both CWT and DWT. The properties of a biorthogonal basis

are specified in terms of a pair of integers ðNd; NrÞ: These integers, in analogy with the

Daubechies wavelets, govern the regularity and the number of vanishing moments Nd of

the decomposition wavelet c ðxÞ; and the regularity and the number of vanishing moments Nr of

the reconstruction wavelet c~ðxÞ: Obviously, this feature allows a greater number of choices for

signal decomposition and reconstruction. Both wavelet functions cðxÞ and c~ðxÞ are symmetric.

11.2.3.2 Complex Wavelets

5. Harmonic wavelets — A Family of bases defined in the frequency domain by the formula

[16,43,44]:

C^ m;nðvÞ ¼

2p ðn 2 mÞ

; mp # v # np;

0; elsewhere;

8><

ð11:47Þ

where m and n are positive numbers but not necessarily integers. The pair of values m, n is

referred to as level m, n and represents, for harmonic wavelets, the scale index j: A harmonic

wavelet basis thus corresponds to a complete set of adjacent levels m, n, spanning all the

positive frequency axis. By inverse-Fourier transforming Equation 11.47, the corresponding

wavelet functions at a generic step k on the x-domain take the complex form:

cm;n;kðxÞ ¼

exp in2p x 2

n 2 m

􀀒 􀀏 􀀐􀀓

2 exp im2p x 2

n 2 m

􀀒 􀀏 􀀐􀀓

i2p ðn 2 mÞ x 2

n 2 m

􀀏 􀀐 ð11:48Þ

A common choice for the pairs m; n is m; n ¼ 0; 1; 2; 4; …; 2j; 2jþ1; … . In this case, all the

wavelets have octave bands, except for the first one.

Harmonic wavelets have been devised in context with a DWT, for which extremely fast

decomposition and reconstruction algorithms exist. They exhibit a compact support in the

frequency domain (see Equation 11.47), while in the x-domain their rate of decay away from

11-10 Vibration and Shock Handbook

the wavelet’s center is relatively low and proportional to x21: Further, they satisfy relevant

orthogonality properties [16].

From Equation 11.48, it is seen that the real part of the wavelet is even, while the imaginary part

is odd. For signal processing, this ensures that harmonic components in a signal can be detected

regardless of the phase. Note that this feature cannot be achieved by real wavelets such as the Meyer

wavelets, which are all self-similar, being derived from a unique mother wavelet by scaling and

shifting. Also, note that orthonormal basis of real wavelets can be generated by considering either

the real or the imaginary parts only of Equation 11.48. For instance, the well-known Shannon

wavelets correspond to the imaginary parts of Equation 11.48, for m; n ¼ 1; 2; 2; 4; 4; 8; … :

Harmonic wavelets are used in many mechanics applications such as acoustics, vibration

monitoring, and damage detection [45 – 49].

6. Complex Gaussian wavelets — Families of wavelets, each corresponding to a pth order derivative of

a complex Gaussian function. That is,

cpðxÞ ¼ Cp

dx p ðe2ix e2x2 =2Þ; p ¼ 1; 2; … ð11:49Þ

where Cp is a normalization constant such that

􀀉 􀀉

cðxÞ

􀀉 􀀉

2 ¼ 1: Complex Gaussian wavelets support

the CWT only. They have no finite support in the x-domain, although the interval [2 5,5] is

generally taken as effective support. Despite their lack of orthogonality, they are quite popular in

image-processing applications due to their regularity [1].

7. Complex Morlet wavelets — Families of [50], each obtained as the derivative of the classical Morlet

wavelet c0ðxÞ ¼ e2x2 =2 eiv0 x : That is,

cpðxÞ ¼

dxp ðe2x2 =2 eiv0 x Þ; p ¼ 1; 2; …: ð11:50Þ

Except for c0ðxÞ; which does not satisfy the admissibility condition (Equation 11.11) in a strict

sense, all the other members of the family are proper wavelets. For practical purposes, however,

c0ðxÞ is generally considered admissible for v0 $ 5: Complex Morlet wavelets support the CWT

only and are not orthogonal. However, they are all progressive, that is, they satisfy the condition

posed by Equation 11.36. Further, for the Morlet wavelet c0ðxÞ; there exists a relation between the

scale parameter a and the frequency v at which its Fourier transform focuses. That is,

a ¼

v ð11:51Þ

Complex Morlet wavelets are then applied for structural identification purposes, as shown in

Section 11.5.

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