fswt

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Contents lists available at ScienceDirect

Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 24 (2010) 491–507

0888-32

doi:10.1

� Cor

E-m

k006wc1 Te2 Te

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

An overall theoretical description of frequency slicewavelet transform

Zhonghong Yan a,b,�, Ayaho Miyamoto b,1, Zhongwei Jiang b,2, Xinglong Liu b,2

a Biomedical Department, ChongQing Institute of Technology, Chongqing 400050, Chinab Yamaguichi University, 2-16-1 Tokiwadai, Ube, Yamaguchi 755-8611, Japan

a r t i c l e i n f o

Article history:

Received 19 March 2009

Received in revised form

28 May 2009

Accepted 8 July 2009Available online 16 July 2009

Keywords:

Time–frequency analysis

Free style wavelet base

New method for signal processing

Wavelet transform

Guided-wave signals

70/$ - see front matter Crown Copyright & 2

016/j.ymssp.2009.07.002

responding author. Yamaguichi University, 2-

ail addresses: [email protected] (Z. Yan), miy

@yamaguchi-u.ac.jp (X. Liu).

l.:/fax: +81836 85 9530.

l./fax: +81836 85 91370.

a b s t r a c t

This paper presents an overall description of a new time–frequency signal analysis

method, called frequency slice wavelet transform (FSWT). Five new properties of the

FSWT are introduced as follows.

(1) The center of time–frequency window is the observing center in contrast to the

wavelet transform (WT).

(2) FSWT can be controlled by the frequency resolution ratio of the measured signal.

(3) The original signal can be decomposed on a frequency slice function (FSF);

theoretically, it can also be rebuilt by infinite correlation function (CF) with the FSF.

At the same time, FSWT has many reconstructive procedures that are not directly

related to the choice of the FSF. Furthermore, several useful transmutations for

FSWT are proposed.

(4) The FSF can be designed very freely and can be ensured to have perfect symmetry in

time and frequency domains.

(5) The FSWT has higher performance against noise than the WT.

009 P

16-1

a818

Meanwhile, this paper reveals the theoretical comparisons with the WT method.

Then some typical examples are used to demonstrate these facts.

Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Time–frequency analysis has been successfully used in dealing with rapidly varying transient signals, such as guided-wave signals and damping vibration signals [1]. For time–frequency representations (TFRs), the short-time Fouriertransform (STFT), the Wigner-Ville distribution (WVD) and the continuous wavelet transform (CWT) are commonly used.STFT and WVD have certain advantages over the CWT, but they also have some critical limitations in comparison with theCWT. The fixed time–frequency window of STFT can lead to undesirable time and frequency resolutions. In spite of itsexcellent time–frequency resolution, using WVD, it is often difficult to analyze a signal with composite-frequencycomponents because of the appearance of interference terms. In the case of the CWT, however, the window size changes

ublished by Elsevier Ltd. All rights reserved.

Tokiwadai, Ube, Yamaguchi 755-8611, Japan. Tel./fax: +81836 85 9530.

@yamaguchi-u.ac.jp (A. Miyamoto), [email protected] (Z. Jiang),

www.sciencedirect.com/science/journal/ymssp

www.elsevier.com/locate/jnlabr/ymssp

dx.doi.org/10.1016/j.ymssp.2009.07.002

mailto:[email protected]




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Z. Yan et al. / Mechanical Systems and Signal Processing 24 (2010) 491–507492

adaptively to the frequency component because of its constant bandwidth-to-frequency ratio property [2–4]. In otherwords, this analysis uses a shorter time window for higher frequency components and a longer window for lower-frequency components. In fact, CWT is only an extension of STFT in time domain with the constant bandwidth-to-frequency ratio. Unfortunately, even if under the same bandwidth-to-frequency ratio, the CWT may have different TFRfeatures, which will be verified in this paper. Since the wavelet window function must satisfy the strict admissiblecondition, the CWT often creates two problems, one is the center of time–frequency window, which usually is not theobserving center, and the other is the fact that most of wavelet functions are not symmetric. Therefore, the time–frequencywindow of the WT will lose the easy operation in application and theory analysis, especially in the frequency domain.Moreover, it is often not convenient in application that the WT uses the scale instead of the observing frequency.Comparing with STFT, the CWT can extract more accurately the instantaneous frequency information of signals, but themost important issue in the time–frequency analysis is how to achieve the best time–frequency energy localization forgiven signals. For instance, this localization is often employed to locate the arrival time and estimate the dispersedfrequency of guided-wave signal. Nevertheless, the characteristics of the mother wavelet function significantly affect theperformance of the time–frequency analysis of the CWT. For example, although the Gabor wavelet, which is one of the mostwidely used analytic wavelets, has the best time–frequency resolution, i.e. the smallest Heisenberg box, the centerfrequency and the time supporting width of the mother Gabor wavelet affect its time–frequency decompositioncharacteristics. This means that, depending on the signals to be analyzed, different Gabor wavelet shapes must be used.Since the characteristics of signals are unknown in general, the determination of the optimal shape is usually difficult [5].Based on the motivations, we aim at developing a new time–frequency transform with better properties than the WT toimprove the situations in application. The new transform is called the frequency slice wavelet transform (FSWT), which hasbeen reported in our paper [6] but not been analyzed systematically. Therefore, in this paper we attempt to present anoverall theoretical description for the FSWT. Five new properties of the FSWT are introduced as follows.

(1)
The center of the time–frequency window is the observing center in contrast to the WT. (2) The FSWT can be controlled by the frequency resolution ratio of the measured signal. (3) The original signal can be decomposed on a frequency slice function (FSF); theoretically, it can also be rebuilt by infinite
correlation function (CF) with the FSF. At the same time, FSWT has many reconstructive procedures that are not directlyrelated to the choice of the FSF. Furthermore, several useful transmutations for FSWT are proposed.

(4)
The FSF can be designed very freely and can be ensured to have perfect symmetry in time and frequency domains. (5) The FSWT has higher performance against noise than the WT.
Meanwhile, this paper reveals the theoretical comparisons with the WT method. Then some typical examples are used todemonstrate these facts.

1.1. Notation

R denotes the set of real numbers. L2(R) denotes the vectors space of measurable, square integrable one-dimensionalfunctions f(x).

Fourier transform (FT) for function f(x)AL2(R).

Fff g : f ðoÞ ¼Z 1�1

f ðtÞe�iot dt (1)

Fourier inverse transform:

F�1ff g : f ðtÞ ¼

1

2p

Z 1�1

f ðoÞeiot do (2)

The signal energy is recorded as:

f�� 2

2¼

Z 1�1

f ðtÞ�� 2 dt (3)

The J � J2 also denotes the classical norm in the space of square integrable functions.We define the following time–frequency localization features of the limited energy signals, which include the wavelet

functions and the STFT window functions, etc.The duration Dtf and bandwidth Dof are defined as

Dtf ¼1

f��

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 1�1

ðt � tf Þ2 f ðtÞ�� 2 dt

s; Dof ¼

1

f��

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZ 1�1

ðo�of Þ2 f ðoÞ�� 2 do

s(4)

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Z. Yan et al. / Mechanical Systems and Signal Processing 24 (2010) 491–507 493

where tf and of are the centers of f(t) and f ðoÞ, respectively,

tf ¼1

f�� 2

2

Z 1�1

t f ðtÞ�� 2 dt; of ¼

1

f�� 2

2

Z 1�1

o f ðoÞ�� 2 do (5)

2. Frequency slice wavelet transform analysis

2.1. Frequency slice wavelet transform

Suppose pðoÞ is the Fourier transformation of the function p(t). For any f(t)AL2(R), the frequency slice wavelet transform

(FSWT) is defined directly in the frequency domain as

Wf ðt; o; sÞ ¼1

2p

Z þ1�1

f ðuÞp� u�o

s

� �eiut du (6)

where the scale s is a constant or a function of o, t and u, and the star ‘*’ means the conjugate of a function. Here we call oand t the observed frequency and time, and u the assessed frequency. In fact, the FSWT is another extension of the STFT inthe frequency domain.

pðoÞ is also called a frequency slice function (FSF) in [6]. By using the Parseval equation, if s is not the function of theassessed frequency u, then Eq. (6) can be translated into its time domain [6].

Wf ðt; o; sÞ ¼ seiot

Z þ1�1

f ðtÞe�iotp�ðsðt� tÞÞdt (7)

Eq. (6) is a simplified version of [6], where we have finished a FSWT analysis flow and compared the different features ontransient signal with the fast Fourier transform (FFT), STFT, discrete wavelet transform (DWT), CWT and WVD, etc.Meanwhile, the FSWT discrete and the fast algorithm with aid of fast FFT can be found in [6]. Therefore, this paper will paymore attention to the theoretical features of the FSWT. Although a part of theoretical contents as the appendix of [6] hasbeen introduced, in this paper, we will further study on theory or give a clear explanation to application if necessary.

2.2. Time and frequency window property

Time and frequency window properties of the FSWT depend on the dilatation parameter s. In [6] we have given thelocalized domain of the FSWT as

t þ1

stp �

1

sDtp; t þ

1

stp þ

1

sDtp

� �oþ sop � sDop; oþ sop þ sDop

(8)

In fact, pðoÞ�� and pðtÞ

�� can be selected as even functions, respectively, so the window center is at the origin, i.e. op ¼ 0,and tp ¼ 0. For example, let the FSF be the Gaussian function pðoÞ ¼ e�1=2o2

, and pðtÞ ¼ e�1=2t2, it is easy to obtain op ¼ 0

and tp ¼ 0. Therefore, the localized window is further simplified as

t �1

sDtp; t þ

1

sDtp

� �o� sDop; oþ sDop

(9)

However, note that Eq. (8) in ordinary wavelet is Eq. (10) (e.g. [7,8])

bþ atc � aDtc; bþ atc þ aDtc oc

a�Doc

a;oc

aþDoc

a

� �(10)

where c is the mother wavelet function similar to the FSF function, a is the familiar wavelet scale, which is the same sensewith 1/s, b is a translation of time t.

From this, we can see that the center of the time–frequency window is always the observing center. This is a newproperty of the FSWT in contrast of the wavelet method. Because in the traditional wavelet Eq. (10) states that moving afrequency window must dilate the scale a, and this will lead to the decrease of either the frequency resolution or the timeresolution [6]. However, in the FSWT, the time–frequency window is adaptive to the observing center of the analyzedsignal, and the scale s is only a balance factor between the time resolution and the frequency resolution.

Due to the fact that the mother wavelet function must satisfy the strict admissible condition,

0oCc ¼

Z 1�1

cðoÞ�� 2

oj j doo1 (11)

the inequality implies that cð0Þ ¼ 0 and its energy center is not at o ¼ 0 . Therefore oc ¼ 0 is usually impossible, and somost of the mother wavelet functions in the frequency domain (i.e.cðoÞ) cannot be even symmetric functions. But tc ¼ 0 isusually allowable. Because of the asymmetry of cðoÞ and oca0, the WT is often biased for the signal frequency.

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Fortunately, the FSWT ensures that different frequency signals can be localized with the complete equality fortime–frequency representation. The latter example will reveal these facts.

2.3. Bandwidth-to-frequency ratio property

Consider the bandwidth-to-frequency ratio property of the FSF. From Eq. (9), we define frequency resolution ratio of anFSF as

Zp ¼half �width of frequency window

center frequency¼sDop

o ¼Dop

o=s (12)

where Dop is computed by Eq. (4). Thus, in FSWT Zp may not be constant.However, in traditional wavelet, from Eq. (10)

Zc ¼Doc

oc(13)

where Doc and oc can be determined by Eqs. (4) and (5). So Zc is constant.Furthermore, from Eq. (12), we therefore denote a new scale as

k ¼ os

(14)

where k40 is assumed. Zp is further rewritten again as

Zp ¼Dop

k (15)

Usually, we can assume that Zp51 in the FSWT. Notably, Zp can be controlled by the new scale k. When k is constant, Zp is aconstant similar to Zc in the CWT, otherwise it is not a constant. From Eq. (14), s ¼ o/k. As a result of the new scale k, theFSWT Eq. (6) naturally becomes into

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� ku�o

o

� �eiut du (16)

Due to the FSWT as all of CWT is also restrained by Heisenberg uncertain theorem, it is impossible that at the same timewe attempt to have high level of resolution in both time and frequency. Actually, any time–frequency transform mustanswer the same question: What is the relationship between the resolution and its scale? It is generally possible thatdifferent transforms have different strategies on their scales. So we introduce the frequency resolution ratio of a signal s(t)as a useful way to reach a compromise between the time and frequency resolution. Once the FSF has been selected, it isimportant to note that the scale k in Eq (16) is the unique parameter that should be chosen in application.

Naturally, the frequency resolution ratio of a signal s(t) is defined as

Zs ¼Dos

os(17)

where the Dos and os can be determined by Eqs. (4) and (5). Nevertheless, in a signal system, a measured signal usuallyincludes many frequency components, therefore Eqs. (4) and (5) can not be used to compute Dos and os for multiplecomponents directly. The determination of the maximum or minimum Zs is usually difficult in application, and themaximum or minimum Zs should be estimated according to the signal. However, in a damping vibration system, it can beproved that Zs is equivalent to the damping ratio of the measured signal. Therefore there are many methods can be used toestimate this parameter [1]. In fact, it is not necessary in FSWT that Zs is estimated accurately because [6] has shown thatthe FSWT is more steady than the CWT when scale changes. Therefore, we can use a certain experiential value instead ofthe estimation Zs of the measured signal. For example, we often use five percent as a default in application, i.e. Zs ¼ 0.05.

Since from Eq. (15), if we can decide parameter Zp, then we have

k ¼ Dop

Zp

(18)

Consequently, we need to discuss the relationship between Zp and Zs. A simplicity idea is to control Zp according to Zs of themeasured signal. Naturally, we assume

Zp ¼ Zs (19)

In fact, the CWT also employs the same assumption, but there are few papers to study this issue. Unfortunately, even ifunder the same bandwidth-frequency ratio with different center frequency and bandwidth parameter, the CWT might havedifferent TFR resolution. So the CWT does not support the assumption exactly. Moreover, the CWT method cannot provide astrategy to easily realize the control even if Zs is known. See the latter example in Section 3.

Since pðoÞ can be viewed as an implementation to sample the original signal in frequency domain, therefore, in order todistinguish two frequencies, we often use the following selection as a basic choice, but it is a biased selection for the

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frequency resolution

Zp �1

2Zs (20)

Obviously, if we choose

Zp � 2Zs (21)

then it is a biased selection for the time resolution.Consequently, according to Eqs. (19), (20) or (21), the scale k can be selected, respectively as

k ¼ Dop

Zs

(22)

k � 2Dop

Zs

(23)

k � Dop

2Zs

(24)

Finally, according to the different strategy about the time and frequency resolution in Eqs. (22)–(24), k can be chosen byZs. The prior knowledge or analysis about signal is naturally useful in FSWT. Therefore, once the frequencyresolution ratio Zs is estimated roughly in a signal system, the FSWT can be controlled. This is another new property ofthe FSWT.

Based on the above analysis of the scale and the resolution, the time–frequency localization of the FSWT can becontrolled by the frequency resolution ratio of the measured signal.

2.4. Reconstructive procedure independency property

The inner product of functions pðoÞ and qðoÞ is stated as

Cpq ¼

Z 1�1

qðoÞp�ðoÞdo (25)

If 0ojCpqjoN, we call that p and q are correlative, and q(t) is a correlation function of p(t). If Cpq ¼ 0, we call p and q non-correlative or orthogonal.

A signal’s time and frequency behaviors are not independent. So the information of the FSWT is redundant as thegeneral CWT. It is possible to construct various forms to rebuild the original signal.

Theorem 1 (Generalized reconstructing equation). If q is a correlative function of the FSF p, and s is not the function of oand t, then the original signal f(t) can be rebuilt by the following formula

f ðtÞ ¼1

Cpq

Z 1�1

Z 1�1

Wf ðt; o; sÞeioðt�tÞqðsðt � tÞÞdtdo (26)

The proof of Theorem 1 can be found in Appendix A. From Theorem 1, note that a signal f(t) can not only be decomposedby frequency slice function pðoÞ, theoretically, it can also be rebuilt by infinite correlation function (CF) with the FSF. And sothe FSWT is not limited by wavelet basic theory. Therefore, one can do that pðoÞ is selected to analyze easily in thefrequency domain, and q(t) is controlled and calculated easily in the time domain.

Theorem 2. If the pðoÞ satisfies pð0Þ ¼ 1, which is called the first class design condition, then the original signal f(t) can be

reconstructed by

f ðtÞ ¼1

2p

Z 1�1

Z 1�1

Wf ðt; o; sÞeioðt�tÞ dtdo (27)

Theorem 3. If the p(t) satisfies p(0) ¼ 1, which is called the second class design condition, and s is a constant, then the original

signal f(t) can also be rebuilt by

f ðtÞ ¼1

2ps

Z 1�1

Wf ðt; o; sÞdo (28)

Theorem 4. Traditional reconstructing equation: If p(t) is a limited energy function, which is called the third class design

condition, and s is not the function of o and t, then the original signal f(t) can be still rebuilt by

f ðtÞ ¼1

Cpp

Z 1�1

Z 1�1

Wf ðt; o; sÞeioðt�tÞpðsðt � tÞÞdtdo (29)

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Remark. The reconstruction transform Eq. (27) was proved in [6], and Eqs. (28) and (29) are only two special cases ofTheorem 1 with q(t) ¼ d(t) (Dirac delta function) and q(t) ¼ p(t), respectively. Here, we omit their proofs. Moreover, Eq. (29)is similar to the WT method [3,4].

In considering Theorem 2, it is significant to note that, for any non-constant scale s, Eq. (16) can be reconstructed only byEq. (27) whose computation process is not in relation with the FSF p(t) or pðoÞdirectly. Therefore, if the condition pð0Þ ¼ 1always remains unchanged in computation of the FSWT, we can conclude the following:

(1)
in Eq. (6) or k in Eq. (16) can be changed to fit a specific signal during computation if necessary. (2) The FSF pðoÞ can be modified during computation as needed. (3) The truncation of FSF pðoÞ does not create any error for reconstructing the original signal by Eq. (27). See [6].
In other words, dynamic scale is allowable in FSWT. For instance, k can be adaptively controlled by the signal spectrum as

k ¼12k0 f ðuÞ

�� f ðoÞ��

2k0 f ðuÞ�� 4 f ðoÞ

�� 8><>: (30)

where k040 is a constant that satisfies one of Eqs. (22), (23) and (24), f ðoÞ�� represents the energy of the signal at the

observing frequency in Eq. (16) and f ðuÞ�� for the assessing frequency. For the first controlling condition in Eq. (30), when

the energy of the current observing frequency is bigger, it usually means its duration is longer, especially for stationarycomponents, and so we expect that a decrease in k will increase the time resolution. Otherwise when the energy of thecurrent observing frequency is smaller, it usually means its duration is shorter, and it is interfered easily by other frequencycomponents, especially for transient signal; so an increase scale k will improve the frequency resolution. Under this case,Eq. (16) becomes a dynamic scale transformation, and the FSF or the FSWT may become very complex or nonlinear. Thereare many challenges left to study, such as, how to dynamically choose the FSF and the scale k similar to Eq. (30) and explorethe properties and applications. In Section 3, the dynamic scale method is used to distinguish two close frequency signals.

Eq. (27) is the unique form that ensures the original signal to be reconstructed from a TFR with dynamic scale. Notably,the general wavelet cannot allow this kind of dynamic scale similar to Eq. (30) because its reconstruction equation mustdepend on the selected wavelet base and scale, otherwise it is impossible to rebuild the original signal.

Moreover, based on the above important properties, the FSWT can provide many transmutations. Note that the factork((u�o)/o) in the window function pðoÞ in Eq. (16) is a relative ratio about the observing frequency o and the assessingfrequency u. The ratio relationship can be redefined similarly because the scale parameter s in Eq. (6) may be a constant ora function about o, t and u. For example, Eq. (6) can be changed significantly into

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� ku�o

u

� �eiut du (31)

or

Wf ðt; o;kÞ ¼1

2p

Z þ1�1

f ðuÞp� 2k

n

un �on

un þon

� �eiut du (32)

where n is a constant. Nevertheless, the observing centers in Eqs. (31) and (32) are always at the same frequency o. Eq. (31)is a manifest method and similar with Eq. (16). Eq. (32) is to compress (when no1) or expand (when n41) the observingfrequency range. At the same time, Eq. (32) is symmetric form about frequency o and u in contrast with Eq. (16) or Eq. (31).Usually n ¼ 1 and 2 are useful choices in Eq. (32).

Notably, it is not difficult to prove that Eqs. (31) and (32) still satisfy Eq. (15) (see Appendix A). Hence, the scaleselections with Eqs. (22)–(24) are still useable. Eq. (31) can be rebuilt by (26),(27),(29), but Eq. (32) can only be rebuilt byEq. (27). Nevertheless, Eqs. (31) and (32) cannot be computed in the time domain as Eq. (7).

On the contrast between the FSWT and the CWT, Eq. (6) can be rewritten as (see Appendix A)

Wf ðt; o; sÞ ¼1

2p eiot

Z þ1�1

f ðuþoÞp� u

s

� �eiut du (33)

The CWT has a relationship in the time–frequency domain as the following [3]:

1ffiffiffiap

Z þ1�1

f ðtÞct � b

a

� �dt ¼

1

2p

Z þ1�1

ffiffiffiap

f ðuÞc�

ðauÞeiub du (34)

So Eq. (33) changes the common views of the wavelet transform.

Remark. Ref. [6] has shown that the FSWT can be approximate to the CWT, STFT and WVD. Therefore, by comparing Eqs.(33) and (34), it is easy to know that the computational cost of FSWT is similar with the CWT,STFT,WVD in frequencydomain. Since the time–frequency information of FSWT is redundant, it is possible that we can reduce a great deal ofcalculation by reducing resample in time or frequency domain [6]. It is more important that the down-sampling technique

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does not create any error for reconstructing the original signal by Eq. (27). On the other hand, the truncation of an FSF alsodoes not create any error for signal reconstructing. Therefore, reducing the support length of FSF can also reduce thecomputation in application. Ref. [6] has shown more details. Therefore the computational cost of the FSWT algorithm isactually much less than the CWT.

2.5. FSF design property

More discussion of design conditions (DCs) to create an FSF can be found in [6]. In this paper, we will only give newexplanations and two extensions for FSF. These DCs are briefly listed here, but they are not necessarily simultaneous.

(1)
DC1: pð0Þa0 or pð0Þ ¼ 1 is always necessary to reconstruct the original signal by Eq. (27).� � � � (2) DC2: To ensure Eq. (12) the FSF pðoÞ� � and pðtÞ� � should be even functions. (3) DC3: To avoid the singularity of the insertion of the factor u�o/o or u�o/u in window function FSF in Eq. (16) or (31)
that requires

pð�1Þ ¼ 0 (35)

(4)
DC4: To distinguish two different frequency signals or two different time pulses, the following design conditions ofpðoÞ andpðtÞ are necessary:
pðoÞ�� pð0Þ

�� (36)

pðtÞ�� pð0Þ

�� (37)

The conditions (36) and (37) mean that their energy should be concentrated at the center of the time and frequencywindow. Therefore o ¼ 0 is the maximum point of pðoÞ

�� in FSWT, but it is the minimum point cðoÞ�� in WT. This means that

the FSWT is essentially different with any wavelet.The following gives several typically FSF functions.

FSF1: pðoÞ ¼ e�1=2o2, and pðtÞ ¼ e�1=2t2

FSF2: pðoÞ ¼ 11þo2 and pðtÞ ¼ e� tj jFSF3: pðoÞ ¼ e� oj j and pðtÞ ¼ 1

1þt2FSF4:

pðoÞ ¼ sinoo , and pðtÞ ¼

1 tj j � 1

0 tj j41

(

FSF5: pðoÞ ¼1 oj j � 1;

0 oj j41

(, and pðtÞ ¼ sin t

t .

Note that all above FSFs meet all design conditions, and they are exchangeable in time and frequency domains in contrastto the WT. Because the most important design condition pð0Þ ¼ 1 is very simple, FSF can be designed very freely and can beensured to have perfect symmetry in time and frequency domains. Furthermore, we easily know that the following FSFextensions are reasonable. If p1ðoÞ; p2ðoÞ; . . . pnðoÞ are the FSF functions

pðoÞ ¼ p1ðoÞp2ðoÞ � � � pnðoÞ (38)

The product is still an FSF. If a1,y, an are scalars, then the linear combination of those FSF functions with those scalars ascoefficients is still an FSF

pðoÞ ¼ a1p1ðoÞ þ a2p2ðoÞ þ � � � þ anpnðoÞ (39)

Under the condition:

a1 þ a2 þ � � � þ an ¼ 1 (40)

We can use the property to choose the ‘best’ scale to obtain the best TFR in application, this is much simpler than themethods based on the minimum Shannon entropy search [5]. For example, we can choose a group of scales: k1,k2,y,kn

and then chose scalars a1,y, an that satisfy the optimal solution

minða1 ; a2 ; ... anÞ

Xðt; oÞ2O

jðt; o; a1; a2; . . . anÞ�� 2

subject to a1 þ a2 þ � � � þ an ¼ 1 (41)

where jðt; o; a1; a2; . . . anÞ ¼ Sni¼1aiWf ðt; o; kiÞ, and O is an optimized area selected from the time–frequency plane. See

the later example in Section 3. Note that Eq. (41) can be changed into a linear problem to be solved. Moreover, the originalsignal can still be reconstructed from the optimal TFR j(t,o,a1,a2,y, an).

Remark. Since a number of functions can satisfy the FSF design conditions, an important problem is how to choose thebest FSF for the practical applications. However, it is very difficult to answer the question. Actually, the scale optimization is

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also an effort to answer the question because Eq. (39) can be viewed as a new FSF function. Therefore, the optimal linearcombination of many FSF functions or scales may be a good and simple approach to solve the question desirably.Unfortunately, its computational cost is very expensive. As we know, the Gaussian function has the best time–frequencylocalization window; therefore, we often use the Gaussian function as the FSF. Nevertheless, the later section will showmany functions can create approximate results as the Gaussian function. Notably, pursuing the best FSF function will be abig challenge in the future studies.

2.6. Filter property

From Eqs. (36) and (37), we require that the main energy of the FSF pðoÞ should be concentrated at the center of thetime–frequency window. Notice the insertion of factor k((u�o)/o) in the window function pðoÞ in Eq. (16). So the FSWT isa representation of the observing signal at frequency o in a short bandwidth that can be controlled by the scale k, both Eqs.(31) and (32) have similar properties with Eq. (16). Therefore, FSWT can be treated as an expanding band-pass filter to beimplemented in time and frequency domains simultaneously. The fact that FSWT has higher performance against noisethan the CWT will be demonstrated below.

3. Verification and application

In this section, we will verify the proposed properties of the FSWT and show the comparisons with the CWT. Meanwhile,some application problems are stated along the way. To compare with the CWT, we first introduce Gabor wavelet function

cðtÞ ¼ e�ð1=a2Þt2

ei2pbt (42)

Hence cðtÞ�� is a symmetric function with the origin of time domain, but cðoÞ

�� is not symmetric with the origin of thefrequency domain. It is easy to know that

Zc ¼1

2pab (43)

We then introduce a useful simulation signal to show the FSWT application. For the efficient damage inspection, it isimportant to use a wave pulse having good time and frequency localization, which is a foundation to pinpoint the arrivaltime and the main frequency of the received pulse. We consider the recommended methods in [9], the modulated Gaussianpulse (also known as Gabor pulse) is implemented to detect the damage in experiments by guided-wave. The common typeof the Gabor pulse is given as

sGpðtÞ ¼ e�t2=a2

cosð2pft þ fÞ (44)

and the real received signal can be approximated by

sðtÞ ¼Xn

i¼1

Aie�ðt�uiÞ

2=a2i cosð2pf iðt � uiÞ þ fiÞ (45)

where Ai is magnitude of the echo wave, ui is the arrival time, and fi is the dispersed frequency, fi is the phase shift. Now weinspect the properties of the FSWT with the Gabor pulses.

3.1. Resolution and symmetry

Example 1. A simulated signal with Eq. (45) is described in Table 1, where there are three Gaussian pulse signals with thesame damping features except the different frequency and the different arrival time. Note that by using Eq. (43) thebandwidth–frequency-ratios of the three components are also estimated in Table 1, i.e. Z1 ¼ 0.015, Z2 ¼ 0.009 andZ3 ¼ 0.0064. Therefore, as the first example, Zs is assumed to be known for the test. The three choices have no any essentialdifferent, and so we only take the maximum Zs ¼ 0.015 as one demonstration of them.

Table 1The parameters of simulated signal in Example 1.

s A f (104 Hz) u (10�4 s) a 10�4 f

s1 1 15 5 0.707 0

s2 1 25 4 0.707 0

s3 1 35 3 0.707 0

s ¼ s1+s2+s3, Ts ¼ 8�10�4 s, fs ¼ 2 MHz, Z1 ¼ 0.015, Z2 ¼ 0.009, Z3 ¼ 0.0064.

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Let the FSF be the Gaussian function pðoÞ ¼ e�1=2o2, henceDop ¼

ffiffiffi2p

=2. We then use Eq. (19) to keep the same ratio withthe maximum Zs ¼ 0.015. The chosen parameters in FSWT Eq. (16) or Eqs. (31) and (32) are summarized as follows:

pðoÞ ¼ e�1=2o2

; s ¼ ok; Zs ¼ 0:015; Zp ¼ 0:015; k ¼ 47:13 (46)

Fig. 1 (b) and (c) indicate the 2D and 3D maps of the FSWT coefficients, where a perfect symmetry and conformity imagefor the three Gaussian signals is shown. It is obvious that the three TFRs have the same time–frequency resolutions andamplitude responses, and this means that the FSWT is unbiased for the different frequency signals. The FSWT can becontrolled by the bandwidth–frequency-ratio.

Fig. 1(d)–(g) represent the results of the CWT Eq. (42) with two group of different bandwidth parameters and the centerfrequencies: a ¼ 21.22, b ¼ 0.5, and a ¼ 1, b ¼ 10.61. Since Eq. (43), their bandwidth–frequency-ratios are the sameZc ¼ 0.015 that still keeps the same value with the maximum Zs ¼ 0.015 of the signal. However, Fig. 1(d) and (f) obviouslyhave different time–frequency resolutions and amplitude responses, this drawback is obviously not an advantage forapplication. This fact shows that the CWT cannot be controlled only by the bandwidth–frequency-ratio. Therefore, the CWT

4

0

8

Tim

e

4

0

8

Tim

e

4

0

8

Tim

e

CWT

100

150

0

50

05

10

10 20 3040

FSWT

100

150

0

50

05

10

1020 30

40

100

150

0

50

05

10

1020 30

40

CWT

1.5

-1.5

0

0 2 (10−4s)

Am

plitu

de

10

(10−4s)

(10−4s)

(10−4s)

Time 10−4 s, Frequency 104Hz

1 3 4 5 6 7

(104Hz)30252015

10 (104Hz)30252015

10 (104Hz)30252015

� = 1 � = 10.61

� = 21.22 � = 0.5

� = 47.13

Fig. 1. Compare the FSWT with the CWT for the same parameters Z ¼ 0.015 and Zc ¼ 0.015; (a) Simulated signals, (b) and (c) the 2D and 3D maps of the

FSWT. (d)–(g) are the 2D and 3D maps of the of Gabor wavelet with a ¼ 21.22 b ¼ 0.5 and a ¼ 1 b ¼ 10.61.

ARTICLE IN PRESS


that includes the Gabor wavelet is a biased transformation for different frequency signals because cðoÞ�� is asymmetry with

the origin of frequency domain.

Remark. Since the differences among Eqs. (16), (31) and (32) for TFR features are very minor, we therefore omit theircomparisons in this paper. Nevertheless, Eq. (32) is verified to have the most perfect symmetry. Therefore, n ¼ 2 in Eq. (32)is used to test in this paper, and it is not difficult to analyze its time–frequency window property.

3.2. Energy leakage problem

Whether the DWT or the CWT has the frequency band energy leakage (FBEL) problem [10], it often gives bad influencesto the signal feature extraction. Fig. 1(d) and (f) show the FBEL phenomena, which can be clearly observed in eachfrequency. A lot of energy are leaked at low-frequency area or around the time–frequency center of signal. At the same time,the experiments show that as the bandwidth parameter a decreases, severe energy leakage increases. But Fig. 1(b) showsthat the FSWT with the Gaussian function pðoÞ ¼ e�1=2o2

is so conformable with the expectation; it seems that the FSWThas no any FBEL problem. However, we will found that the FBEL may exist in different FSF functions slightly. By using thesame example, Fig. 2(a)–(d) show four TFRs of the FSWT with four FSF functions, respectively. Fig. 2(a)–(b) exist the shorttime–frequency smearing; Fig. 2(c) has obvious FBEL since the support width of function pðoÞ ¼ sino=o is infinite. Theenergy leakage of first three functions nullifies the property of exact localization, but the last one has perfect property andvery similar with the Gaussian function. Therefore, the better localization property of the FSF is important to the FBEL.Nevertheless, even if in the FBEL cases, the FSWT can still keep the high consistency and the complete equality among thedifferent frequency signals.

3.3. Time and frequency domains filter

The Example 1 and the same FSWT parameters are still used in this section. Fig. 3 demonstrates the frequency slices ofthe FSWT and the CWT without noise or under high noise (+25%). Although the CWT looks like the FSWT in amplitudes (infact, their complex values are complete different), it is evident that the CWT is more sensitive to noise than the FSWT. TheFSWT shown in Fig. 3 presents higher performance against noise.

As a simple application of the FSWT, the FSWT provides a new approach that both filtering in time and frequencydomains can be processed simultaneously. The first task is to determine the needed parts in the TFR image, the left partsare set to be 0, whose procedure is called the image binarization, and then the original signal can be reconstructed byFSWT. Note that the general objective signal is a connected area in the TFR image. Therefore, many image binarizationmethods can be implemented to cut the background noise. A perfect denoising result is shown in Fig. 3(h).

3.4. Scale optimization

Example 2. Fig. 4(a) shows a simulated Gaussian signal by Eq. (45). It seems that this signal includes three or four differenttime components. In fact, the original signal is composed of two frequency components at the same time, which aredescribed in Table 2. Since Z4 ¼ 0.0067, Z5 ¼ 0.0065, thus they are low-damping signals. It is obvious that the experientialvalue Zs ¼ 0.05 (i.e. k ¼ 14.14) cannot fit this case. Nevertheless, generally, we do not know the real values Z4 and Z5, so wemay have to try some scales. Here we introduce a new usage of the FSWT. Eq. (41) is applied to find automatically the best

4

0

8(10−4s)

Tim

e

10

4

0

8(10−4s)

Tim

e

(104Hz)15 20 25 30

10 (104Hz)15 20 25 30

4

0

8(10−4s)

Tim

e

10

4

0

8(10−4s)

Tim

e

(104Hz)15 20 25 30

10 (104Hz)15 20 25 30

Fig. 2. Compare the TFRs of FSWT under four FSFs. (a) pðoÞ ¼ e� oj j; (b) pðoÞ ¼ 1=ð1þo2Þ, (c) pðoÞ ¼ sino=o; (d)_pðoÞ ¼ sin4o=o4.

ARTICLE IN PRESS

0 4

50

0

150

100

50

0

150

100

50

0

150

100

50

0

150

100

CWT Noise = 25%

FSWT Noise = 25%

FSWT Without noise

CWT Without noise

Mag

nitu

deM

agni

tude

1.5

-1.5

0

0 2 6 7

4

0

8

10

Tim

e

Tim

e

(10−4s) (10−4s)

0

4

8

1.5

-1.5

0

Am

plitu

deA

mpl

itude

(10−4s)1 3 4 5

0 2 6 7 (10−4s)1 3 4 5

(104Hz)15 20 25 30 10 (104Hz)15 20 25 30

(10−4s)2 6 0 4 (10−4s)2 6

Fig. 3. Compare the FSWT with the CWT on the influences of noise. (a) The noise (+25%) version of Fig. 1 (a); (b) 2D map of the FSWT; (c) 2D map of CWT;

(d)–(g) are two groups of slices of Gaussian signal s1 at 15�104 Hz without noise and under high noise, respectively. (h) The black curve is the

reconstructed signal from three marked rectangles in (b) and the blue curve is original signal.


time–frequency expression of this signal. The FSF function is still assumed in Eq. (46), we then take 25 scales fromk1 ¼14.14 to k25 ¼ 14.14� 4 with equal step for the optimal combination of scales. Fig. 4 shows the optimal procedure,where Fig. 4(b) and (c) represent two TFRs with the smallest and biggest scales. Obviously, Fig. 4(b) for k1 is biased to thetime resolution, and Fig. 4(c) for k25 is biased to the frequency resolution. It is not easy to judge which scale is right for thissignal. Fig. 4(d) is a binarized image from the following geometric mean image:

Wðt; oÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYk¼k25

k¼k1

Wf ðt; o; kÞ25

vuut (47)

ARTICLE IN PRESS

0 1 3 4 6 7 8 9 (10−4s)

Am

plitu

de

0

-2

2

14.0 (104Hz)

5

0

10

Tim

e

(10−4s)

5

0

10

Tim

e

(10−4s)

5

0

10

Tim

e

(10−4s)

5

0

10

Tim

e

(10−4s)

Ω

2 5

14.5 15.0 15.5

14.0 (104Hz)14.5 15.0 15.5 14.0 (104Hz)14.5 15.0 15.5

14.0 (104Hz)14.5 15.0 15.5

Fig. 4. Scale optimization. (a) A simulated Gaussian signal. (b) The smallest scale TFR with k ¼ 14.14/4; (c) The biggest scale TFR with k ¼ 14.14*4; (d) O is

the minimized energy area. (e) The best linear combination of 25 scales by Eq. (41).


S A f (104 Hz) U (10�4 s) a 10�4 f

s4 1 15 5 1.581 0

s5 1 15.6 5 1.581 � 14p

s ¼ s4+s5, Ts ¼ 10�10�4 s., fs ¼ 2 MHz, Z4 ¼ 0.0067, Z5 ¼ 0.0065.


S A f (104 Hz) U (10�4 s) a 10�4 f

s6 1 3.4 3 0.9676 14p

s7 1 4 3 0.8921 0

s ¼ s6+s7, Ts ¼ 6�10�4 s, fs ¼ 1 MHz, Z6 ¼ 0.0483, Z7 ¼ 0.0446.


The minimized energy area O in Eq. (41), which is marked in red color in Fig. 4(d), is defined as

O ¼ fðt;oÞj Wðt; oÞ�� o0:5cg (48)

where c is the average value of jW(t, o)j. Finally, by solving Eq. (41), we can get the best TFR with the linear combination ofscales. As a result, Fig. 4(e) clearly shows that the original signal has two frequency components with highertime–frequency resolution than Fig. 4(b) and (c). Therefore, the scale optimization is a useful idea.

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3.5. Dynamic scale

Example 3. We investigate the representation of the FSWT with dynamic scale. A simulated Gaussian signal with Eq. (45)is described in Table 3 and shown in Fig. 5(a). It is very difficult to separate accurately the couple of signals because theirresonant frequencies are very close and they have higher equivalent damping ratios (Z6 ¼ 0.0483, Z7 ¼ 0.0446). The Fouriertransform spectrum in Fig. 5(b) clarifies the fact. Actually, this case often happens in practical applications.

0

-1

1

Am

plitu

deA

mpl

itude

0

-1

1

0 3 4 (10−4s)

0 3 4 6 (10−4s)

0

-2

2

0 3 4 5 (104Hz)0

40

80

3

0

6

Tim

e

2.5 (104Hz)

(10−4s)

3

0

6

Tim

e

(10−4s)

Mag

nitu

de

1 2 5

1 2 6 7

3.0 3.5 4.0 2.5 (104Hz)3.0 3.5 4.0

21 5 6

Fig. 5. Dynamic scale and signal reconstruction. (a) Simulated Gaussian signal; (b) Fourier transform spectrum; (c) A TFR under static scale k ¼ 14.14. (d)

A TFR with dynamic scales k ¼ 14.14/2k ¼ 14.14*2; the black curves in (e) and (f) are the reconstructed signals from (d), the blue curves are original

signals s6 and s7 described in Table 3.

ARTICLE IN PRESS


The FSF is determined as same as Eq. (46), but we select Zs ¼ 0.05 firstly, hence k ¼ 14.14 is a constant. Fig. 5(c) thereforerepresents the time–frequency image of the FSWT coefficients with the static scale. However, it is not easy to observe thedifferent frequency components from that image. Unfortunately, we cannot find out another suitable scale k to distinguishthem clearly, even if we completely sacrifice the time resolution in Fig. 5(c). Similarly, in CWT, we cannot find a couple ofGabor wavelet parameters to recognize them easily.

Fortunately, Fig. 5(d) easily shows a result by using the dynamic scales in Eq. (30), where the same FSF function andk0 ¼ 14.14 are kept. Fig. 5(d) clearly represents the two components with much higher local time and frequency resolutionthan that of Fig. 5(c). Meanwhile a clear boundary between the two components can be seen. In fact, in order to obtain thesimilar result to Fig. 5(d), one can have many selections resembling to Eq. (30), but here we omit the further discussion. Thelater section will show the other example for dynamic scale.

3.6. Signal segmentation and reconstruction

Based on the above results of the FSWT, many image process methods [11] can be implemented to segment eachobjective signal presented in a FSWT image. Usually, we can pick up the interest regions (IRs) for the further aims, such assignal segmentation, pattern identification, signal reconstruction and parameter computation, etc.

Moreover, Example 3 is implemented a reconstructive procedure shown in Fig. 5(d)–(f). Two IRs are first markedin Fig. 5(d), and then two signals are reconstructed by Eq. (27) from them. Fig. 5(e) and (f) reveal that the two reconstructed

0 1.0 Sec.

0 2000

0 1.0 Sec.

0

2000

4000

0

2000

4000

0

2000

4000

0

1

-1

0

10

5

Freq

uenc

y H

z M

agni

tude

A

mpl

itude

CWT � = 6.37� = 0.5

FSWT � = 28.28

FSWT � = 28.28 / f (�)f (u)

0.5

Hz3000 40001000

0.5

Fig. 6. (a) A real life signal; (b) Fourier transform spectrum; (c) The TFR of CWT with a ¼ 6.37 b ¼ 0.5; (d) A TFR of FSWT with a scale k ¼ 28.28. (e) A TFR

with dynamic scales k ¼ 28.28jf(u)j/jf(o)j.

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Gaussian signals are desirable with their main energy and they are in good agreement with the original signal. The FSWTsignal analysis flow has been detailed in [6], here we omit the further discussions.

3.7. A real-life signal

A real-life signal is shown in Fig. 6(a), where there are two sounds, one is the train whistle from far away, and the otherone is the bird chirps in the nearby. The sample frequency is 8192 Hz. From the Fourier spectrum Fig. 6(b), the train whistlecan be viewed as stationary components with long duration and the bird chirps are transient components with very shortduration. Under the same bandwidth-to-frequency ratio, which is assumed as 0.025, we use the CWT and the FSWT tocompare their TFRs of this signal.

In CWT, the Gabor wavelet Eq. (42) is still used here, and let a ¼ 6.37, b ¼ 0.5, so Zc ¼ 0.025. Since the CWT has no fastalgorithm, the computational cost is very expensive. Fig. 6(c) shows the result of the CWT, where an obvioustime–frequency smearing can be seen.

In FSWT, the FSF is also taken as the Gaussian function shown in Eq. (46), so Dop ¼ffiffiffi2p

=2 and let Zp ¼ 0.025. By usingEq. (15), we have k ¼ 28.28. The FSWT computing with the down-sampling technique is much faster than the CWT. At thesame time, Fig. 6(d) shows that the FSWT is obviously better than the CWT both in time and frequency resolution. Fig. 6(e)provides another TFR with a different dynamic scale with Eq. (30): k ¼ 28:28

_

f ðuÞ�� = _

f ðoÞ�� .

Fig. 6(e) shows a significant fact: the frequency resolution to the stationary components is much higher than Fig. 6(c)and (d). However, in CWT, it is very difficult to find a couple of Gabor wavelet parameters to reach so higher frequencyresolution. On the other hand, Fig. 6(e) simultaneously keeps a high time resolution to the transients and is not less thanFig. 6(c) and (d). Therefore, it is possible that by using dynamic scale method one does not have to sacrifice the timeresolution for increasing the frequency resolution, conversely, the same reason is also possible for increasing the timeresolution.

Remark. Dynamic scale reveals the essential fact that redefining FSF in the computation of FSWT is a useful idea inpractical applications. If taking the spectrum as the controlling condition of dynamic scale as Eq. (30), since it is a switchfunction not a continuous condition, usually the experimental result is sensitive with noise spectrum. Therefore, it isnecessary that the chosen controlling condition should be stable. For example, k ¼ k0 f ðuÞ

�� = f ðoÞ�� is a seemly controlling

condition. The further discussions and applications for dynamic scale will be carried out in future research.

4. Concluding remarks

(1)
Some new properties of the FSWT are presented in this paper, such as symmetry, controllability, easy-to-design, andthe reconstruction independency, etc. Due to the new features, FSWT is more flexible to fit ever-changing signals thanthe classical methods. More comparisons based on application with FFT, CWT, DWT, STFT, and WVD can be found in [6].
(2)
Several significant comparisons based on time–frequency analysis theory with the CWT are shown in this paper. As anew time–frequency transform, FSWT has better performance than the CWT and several typical examples are used toverify these facts. The FSWT can be controlled by the frequency resolution ratio Zs of the measured signal. It issignificant to note that in a damping vibration system Zs is equivalent to the damping ratio of the signal. FSWTtherefore provides a new view for adaptive time–frequency analysis.
(3)
Some new usages of the FSWT, such as, the dynamic scale, the transmutations of the FSWT, the time–frequency filter,the scale optimization and the FBEL problem, etc., are also discussed in this paper.
(4)
However, more studies should be carried out on the analysis of FSFs, dynamic property of the FSWT, and the scaleoptimization, etc. There are still more discussions and applications to be done delicately.
Appendix A. Proof of Theorem 1

We can directly take the definition of the FSWT into the right side of Eq. (26) to verify the equation. Hence

1

Cpq

Z 1�1

Z 1�1

Wf ðt;o;sÞeioðt�tÞqðsðt � tÞÞdtdo

¼1

Cpq

Z 1�1

Z 1�1

1

2p

Z þ1�1

f ðuÞp� u�o

s

� �eiut du

� �eioðt�tÞqðsðt � tÞÞdtdo

¼1

2pCpq

Z 1�1

Z 1�1

f ðuÞp� u�o

s

� �eiut

Z þ1�1

qðsðt � tÞÞe�iðu�oÞðt�tÞ dt� �

dodu

¼1

2pCpq

Z 1�1

Z 1�1

f ðuÞp� u�o

s

� �eiut

Z �1þ1

qðsÞe�iðu�o=sÞs �1

s ds

� �� dodu

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Continuously,

¼1

2pCpq

Z 1�1

Z 1�1

f ðuÞp� u�o

s

� �eiutq

u�os

� � 1

s dodu

¼1

2pCpq

Z 1�1

f ðuÞeiut

Z 1�1

p� u�o

s

� �q

u�os

� � 1

sdo

� �du

¼1

2pCpq

Z 1�1

f ðuÞeiut

Z þ1�1

p�ðxÞqðxÞdx

� �du

¼1

2p

Z 1�1

f ðuÞeiut du

¼ f ðtÞ

Meanwhile, note that swapping the order of integral are used twice and finally we use the definition

Cpq ¼

Z 1�1

qðxÞp�ðxÞdx

Therefore, Theorem 1 is proved.Proof of variation replacements for Eqs. (31) and (32) from Eq. (6).Note that s in Eq. (6) may be a constant or a function of o, t and u. Thus, if we let s ¼ u/k, then Eq. (6) can be rewritten

into

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� ku�o

u

� �eiut du

Similarly, let s ¼ 2kn

ðun�onÞ

ðunþonÞðu�oÞ, then Eq. (6) can also be changed into

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� 2k

n

un �on

un þon

� �eiut du

Proof of Eqs. (31) and (32) satisfy Eq. (15).Since pð�Þ is an even function, so we only prove the positive case, i.e. u40, o40. From Eq. (31)

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� ku�o

u

� �eiut du

We have the energy window of the FSF as

�Dop � ku�ou� Dop

At the same time, usually we can assume that Dop/k51 in FSWT (see the above examples). And then

o= 1þDop

k

� �� u � o= 1�

Dop

k

� �

Then the center frequency and half-bandwidth are, respectively

1

2o= 1�

Dop

k

� �þo= 1þ

Dop

k

� �� 1

2o= 1�

Dop

k

� ��o= 1þ

Dop

k

� ��

Therefore, according to the definition of the bandwidth-to-frequency ratio in Eq. (12), we have

Zp ¼ o= 1�Dop

k

� ��o= 1þ

Dop

k

� �� = o= 1�

Dop

k

� �þo= 1þ

Dop

k

� ��

¼Dop

k

Using the above same method, we can prove Eq. (32) satisfies Eq. (15) approximately. From Eq. (32)

Wf ðt; o; kÞ ¼1

2p

Z þ1�1

f ðuÞp� 2k

n

un �on

un þon

� �eiut du

We have the main energy window of the FSF

�Dop �2kn

un �on

un þon� Dop

ARTICLE IN PRESS


Thus offiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k�nDop

2kþnDop

n

q� u � o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kþnDop

2k�nDop

n

qThen the center frequency and half-bandwidth are, respectively

1

2o

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kþ nDop

2k� nDop

n

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k� nDop

2kþ nDop

n

s !

1

2o


2k� nDop

n

s�


2kþ nDop

n

s !

Using Eq. (12), we thus have

Zp ¼


2k� nDop

n

s�


2kþ nDop

n

s !=


2k� nDop

n

sþ


2kþ nDop

n

s !

Since Dop/k51, usually we can also assume that nDop/2k51, and so it is not difficult to prove that

Zp Dop

kThus, the proof is finished.

Proof of Eq. (33)Since the definition of the FSWT Eq. (6)

Wf ðt; o; sÞ ¼1

2p

Z þ1�1

f ðuÞp� u�o

s

� �eiut du

¼1

2p

Z þ1�1

f ðxþoÞp� x

s

� �eiðxþoÞt dðxþoÞ

¼1

2p eiot

Z þ1�1

f ðxþoÞp� x

s

� �eixt dx

Therefore, Eq. (6) can be changed into Eq. (33).

References

[1] S.A. Neild, P.D. McFadden, M.S. Williams, A review of time–frequency methods for structural vibration analysis, Engineering Structures 25 (2003)713–728.

[2] S. Mallat, A Wavelet tour of Signal Processing, Academic Press, New York, 1998.[3] Ingrid Daubechies, Ten Lecture on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, SIAM, Philadelphia, 1992.[4] S. Qian, Introduction to time–frequency and wavelet transforms, Prentice-Hall PTR, Upper Saddle River, NJ, 2002.[5] J.-C. Hong, Y.Y. Kim, The determination of the optimal Gabor wavelet shape for the best time–frequency localization using the entropy concept,

Experimental Mechanics 44 (2004) 387–395.[6] Z. Yan, et al., Frequency slice wavelet transform for transient vibration response, Mechanical Systems and Signal Processing 23 (5) (2009) 1474–1489.[7] W. Christopher Lang, Kyle Forinash, Time–frequency analysis with the continuous wavelet transform, American Journal of Physics 66 (9) (1998)

794–797.[8] C.K. Chui, Jiang Zong Wang, A study of asymptotically optimal time–frequency localization by scaling function and wavelet, Annals of Numerical

Mathematics 4 (1997) 193–216.[9] J.-C. Hong, K.H. Sun, Y.Y. Kim, The matching pursuit approach based on the modulated-Gaussian pulse for efficient guided-wave damage inspection,

Smart Materials and Structures 14 (2005) 548–560.[10] P. Chethan, C. Mickey, Frequency Characteristics of Wavelets, IEEE Transactions on Power Delivery 17 (2002) 800–804.[11] P. Viola, M. Jones, Robust real time object detection, in: IEEE International Conference on Computer Vision Workshop on Statistical and

Computational Theories of Vision, Vancouver, Canada, July 13, 2001, pp. 905–910.

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