Gao R.X., Yan R. Wavelets: Theory and Applications for Manufacturing

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bearing imbalance at 10 Hz frequency and the structural defect at 59 Hz frequency,

as they can be calculated as 59  10 Hz.

6.4 Summary

A wavelet-based multiscale enveloping technique is introduced in this chapter. This

multidomain signal processing technique combines band-pass ﬁltering (implemen-

ted through variation of the scale parameter s of the base wavelet) and enveloping

120

150

0.05

0.1

0.15

0.2

Frequency (Hz)

Scale

Power (Watt)

105 Hz

20 Hz

Fig. 6.12 MuSEnS of vibration signals measured on a roller bearing with a structural defect on the

outer raceway (speed: 1,200 rpm; radial load: 3,665 N)

0 100 200 300 400 500

−1

Time (ms)

Signal (V)

0 500 1000 1500 2000 2500

0.2

0.4

Frequency (Hz)

Power (Watt)

0 20

40 60 80 100

0.2

0.4

Frequency (Hz)

Power (Watt)

rpm

2Xf

rpm

BPFO2

BPFO1

2Xf

BPFO1

Fig. 6.13 Signal measured from a ball bearing and its power spectrum

6.4 Summary 99

(obtained through modulus of the wavelet coefﬁcients) into a single-step operation.

The effectiveness of the multiscale enveloping technique is demonstrated through

studies in ultrasonic pulse differentiation for pressure measurement in injection

molding and bearing defect diagnosis in rotary machines, both numerically and

experimentally.

When the multiscale enveloping techniq ue is used to identify the ultrasonic

pulses generated from injection molding process, not only spectrally identical and

timely adjacent but also timely overlapped and spectrally adjacent ultrasonic pulses

could be detected and differentiated. This allows for comprehensive spatial cover-

age of the cavity pressure proﬁle by placing multiple sensors with different working

frequency ranges at different locations in the injection mold. When the wavelet-

based enveloping is combined with spectral domain postprocessing, a new algo-

rithm termed MuSEnS was developed, which has been shown to be more accurate

and illustrative in depicting critical features related to structural defects embedded

in the bearings than the traditional enveloping spectral analysis. As many of the

applications in manufacturing equipment and system monitoring involve rotary

machine components (e.g., bearings, spindles, gearboxes, etc.), it is possible that

the MuSEnS technique may contribute to improving solutions to a wide variety of

machine monitoring problems.

6.5 References

Choy FK, Zhou J, Braun MJ, Wang L (2005) Vibration monitoring and damage quantiﬁcation of

faulty ball bearings. ASME J Tribol 127(3):776 783

100

Frequency (Hz)

Scale

Power (mW)

10 Hz

30 Hz

41 Hz

49 Hz

59 Hz

69 Hz

20 Hz

Fig. 6.14 MuSEnS of vibration signals measured on a ball bearing with a structural defect on the

inner raceway (speed: 600 rpm; radial load: 10,000 N)

100 6 Wavelet Based Multiscale Enveloping

Gao R, Kazmer D, Theurer C, Zhang L (2001) Fundamental aspects for the design of a

self energized sensor for injection molding process monitoring. In: Proceedings of NSF design

and manufacturing research conference, Tempa, FL

Greguss P (1980) Ultrasonic imaging: seeing by sound. The principles and widespread applica

tions of image formation by sonic, ultrasonic, and other mechanical waves. Focal Press, New

York

Hahn SL (1996) Hilbert transform in signal processing. Artech House Inc., Norwood, MA

Harris TA (1991) Rolling bearing analysis, 3rd edn. Wiley, New York

Ho D, Randall RB (2000) Optimization of bearing diagnostic techniques using simulated and

actual bearing fault signals. Mech Syst Signal Process, 14(5):763 788

Kiral Z, Karag

€

ulle H (2003) Simulation and analysis of vibration signals generated by rolling

element bearing with defects. Tribol Int 36:667 678

Lawrence MS (1999) Computing the discrete time analytic signal via FFT. IEEE Trans Signal

Process 47(9):2600 2603

Lee BY, Tang YS (1999) Application of the discrete wavelet transform to the monitoring of tool

failure in end milling using the spindle motor current. Int J Adv Manuf Technol 15(4):238 243

Liang W, Que PW, Yang G (2006) Ultrasonic ﬂaw detection during NDE of oil pipelines via a

resonance ﬁlter. Russ J Nondestruct Test 42(6):398 403

McFadden P, Smith J (1984) Vibration monitoring of rolling element bearings by the high

frequency resonance technique a review. Tribol Int 17(1):3 10

McGonnagle WJ (1966) Nondestructive testing, 2nd edn. Gordon and Breach Science Publisher,

New York

Oppenheim AV, Schafer RW, Buck JR (1999) Discrete time signal processing, 2nd edn. Prentice

Hall, Upper Saddle River, NJ

Prabhakar S, Mohanty AR, Sekhar, AS (2002) Application of discrete wavelet transform for

detection of ball bearing race faults. Tribol Int 35:793 800

Rawabdeh IA, Petersen PF (1999) In line monitoring of injection molding operations: a literature

review. Injection Molding Technol 3:47 53

Theurer C, Zhang L, Gao R, Kazmer D (2001) Acoustic telemetry in injection molding. In:

Proceedings of society of plastics engineers annual technical conference, process monitoring

and control division, vol 51. Dallas, TX, pp 208 213

Tse PT, Peng YH, Yam R (2001) Wavelet analysis and envelope detection for rolling element

bearing fault diagnosis their effectiveness and ﬂexibilities. ASME J Vib Acoust 123

(4):303 310

Wang W (2001) Early detection of gear tooth cracking using the resonance demodulation

technique. Mech Syst Signal Process 15(5):887 903

Wu JD, Huang CW, Huang R (2004) An application of a recursive Kalman ﬁltering algorithm in

rotating machinery fault diagnosis. NDT&E Int 37:411 419

Yan R, Gao R (2005a) An efﬁcient approach to machine health evaluation based on harmonic

wavelet packet transform. Robot Comput Integrated Manuf 21:291 301

Yan R, Gao R (2005b) Multi scale enveloping spectrogram for bearing defect detection. World

tribology congress III, Washington, DC, pp 855 856

Yen G, Lin K (2000) Wavelet packet feature extraction for vibration monitoring. IEEE Trans Ind

Electron 47(3):650 667

Yoshida A, Ohue Y, Ishikawa H (2000) Diagnosis of tooth surface failure by wavelet transform of

dynamic characteristics. Tribol Int 33:273 279

Zhang L, Theurer C, Gao R, Kazmer D (2004) A self energized sensor for wireless injection mold

cavity pressure measurement: design and evaluation. ASME J Dyn Syst Meas Control 126

(2):309 318

6.5 References 101

Chapter 7

Wavelet Integrated with Fourier Transform:

A Uniﬁed Technique

Fourier transform-ba sed spectral analysis has been widely applied to processing

signals, such as vibration and acoustic signals (Mori et al. 1996; Tandon and

Choudhury 1999; Cavacece and Introini 2002), acquired from manufacturing

systems. Because of noise contamination and signal interference, the constituent

components of interest may be submerged in the signal and difﬁcult to be revealed

through a spectral analysis (Ho and Randall 2000). Furthermore, events occurred in

the manufacturing system may be transient in nature, for example, the initiation and

propagation of surface spalling in a ball bearing (Gao and Yan 2006; Orhan et al.

2006). As another example, the process of metal removal can be viewed as

consisting of multipl e, indivi dual transi ent events in which a single chip of metal

is removed (Ge et al. 2004; Obikawa and Shinozuka 2004; Byrne and O’Donnell

2007; Malekian et al. 2009). On the one hand, given the global analysis nature, it is

difﬁcult to apply the Fourier transform for localizing these transient events. On the

other hand, the Fourier transform can identify a signal’s frequency components,

from which a speciﬁc event (e.g., localized bearing defects, which have distinct

characteristic frequencies at inner raceway, outer raceway, or rolling element itself)

can be detected. Leveraging the capability of wavelet transform in transient signal

analysis, this chapter introduces a uniﬁed time scale frequency analysis technique

through spectral postprocessing on the data set extracted by wavelet transforms to

enhance the effectiveness of signal representation and identiﬁcation.

7.1 Generalized Signal Transformation Frame

Fourier transform and wavelet transform were originated from different theoretical

platforms, and each technique analyzes a signal from a different perspective.

Speciﬁcally, the Fourier transform depicts the energy concentration of constituent

frequency components within the signal, whereas the wavelet transform presents

the similarity between the signal being analyzed and the base wavelet, in the

time scale domain. To enable cross-domain uniﬁcation of the two techniques for

signal analysis, a common signal transform ation platform needs to be established

ﬁrst, which is the focus of this chapter.

R.X. Gao and R. Yan, Wavelets: Theory and Applications for Manufacturing,

DOI 10.1007/978 1 4419 1545 0 7,

Springer Science+Business Media, LLC 2011

103

Let us ﬁrst deﬁne a function W

1,0

(t) within a certain time interval, or support,

expressed as [0, L), where the symbol L represents the width of support. The

function W

1,0

(t) is called the base template function for signal analysi s. Next, we

deﬁne W

s, u

(t), which is a derived version of W

1,0

(t). Comparing to W

1,0

(t), the

magnitude of W

s,u

(t)isscaled by scale s, where s  0 is an integer number, and its

location along the time axis is shifted by time u, with u ∈ R, and R represents the set

of all real numbers. The function W

s,u

(t) is called the derived template functi on, at

scale s and time u, and is supported within the time interval [u, u þ sL]. Generally,

s,u

(t) can be expressed in terms of the base template function W

1,0

(t) as:

s;u

ðtÞ¼

1;0

t  u



(7.1)

where 1

is a factor for purpose of normalization. Speciﬁcally, it ensures that the

following relationship between the derived template function W

s,u

(t) and the base

template function W

1,0

(t) is always satisﬁed:

s;u

ðtÞdt ¼

1;0

ðtÞdt jjW

1;0

ðtÞjj

(7.2)

The physical signiﬁcance of (7.2) is that all the derived template functions

preserve the same amount of energy as the base template function does.

In a linear signal space, the set of all derived template functions {W

s,u

(t): s  0,

u ∈ R} forms a continuous frame G

, spanned by scale s and time u. In accordance

with the discrete data sampling process, where data points are taken at time

instances u ¼ mkL from a derived template function with a scale factor of s ¼ k,

a discrete expression o f the derived template function is obtained as:

k;m

1;0

t  mkL



(7.3)

where W

k, m

(t) is a simpliﬁed expression for W

k, mkL

(t). In the above notation, k or

-1

∈ N, m ∈ Z, and k and m represent a discrete version of the continuous

parameters s and u. The notation k

-1

∈ N corresponds to s< 1. The set {W

k, m

(t):

k or k

-1

∈ N, m ∈ Z}, with N being the set of all nonnegative integers and Z being

the set of all integers, forms a discrete frame G

, spanned by the parameters k and m.

The continuous frame G

(or discrete fra me G

) provides a gener alized frame for

signal transformation, and is deﬁned as comple te in the linear signal space, if any

signal function x(t) can be expressed in it as (Kaiser 1994 ):

xðtÞ¼

Cðs; uÞW

s;u

ðtÞds du (7.4)

or, in the case of discrete frame G

104 7 Wavelet Integrated with Fourier Transform: A Uniﬁed Technique

xðtÞ¼

k¼1

m¼ 1

Cðk; mÞW

k;m

ðtÞ (7.5)

In (7.4) and (7.5), the coefﬁcient of the functions, C(s, u)orC(k, m), can be

considered as a measure function, which expresses the extent to which the signal

function x(t) is correlated to the derived template function {W

s,u

(t): s  0, u ∈ R}of

scale s, at the speciﬁc time u.

Under the discrete frame G

, the signiﬁcance of the measure function expressed

in (7.5) can be further illustrated, considering that there exists a complete orthogo-

nal set {W

k,m

(t)} within such a frame. The orthogonal identity states that:

ðtÞW

ðtÞdt ¼

ðtÞdt; for k

¼ k

; m

¼ m

0; otherwise

(7.6)

where k

and k

∈ {k}, and m

and m

∈ {m}. Using the iden tity of (7.6), multi-

plying the two sides of (7.5)byW

ðtÞ, and integrating over the time interval

(1, 1 ), we have:

xðtÞW

ðtÞdt ¼

Cðk; mÞW

k;m

ðtÞW

ðtÞdt

¼Cðk

; m

ðtÞdt

(7.7)

With k ¼ k

and m ¼ m

, rearranging (7.7) yields,

Cðk; mÞ¼

xðtÞW

k;m

ðtÞdt

k;m

ðtÞdt

xðtÞW

1;0

t mkL



1;0

ðtÞdt

(7.8)

The equation indicates that the measure function C(k, m) expresses the correlation

(or similarity) between the signal function x (t) and the derived template function

k,m

(t) of scale k and at time mkL. This concept can be expanded to view a signal

transformation operation, such as Fourier or wavelet transform, as a correlation

operation between a signal and a template function, and the result expresses the

measures of correlation between the two functions. When a template function

derived from a base template function has a large value of C(k, m ), or correlation,

with a signal feature at certain scale k and time mkL, the template function is said to

have a good match with that corresponding feature. As a result, the feature will be

effectively extracted by this particular template function. Signal components that

are of little correlation to the template function will show small or no correlation

measures, and thus be suppressed in the analysis. A signal may show different

correlation measures with different base templa te functions. In Table 7.1, several

7.1 Generalized Signal Transformation Frame 105

basic template functions are expressed in the generalized signal transformation

frame, and they are discussed in the following sections.

7.1.1 Fourier Transform in the Generalized Frame

A signal x(t) of period T can be expresse d through its Fourier transform as

(Bracewell 1999):

xðtÞ¼

n¼0

j2pnt=T

; 1<t<1; n 2 N (7.9)

where c

is the nth-order transformation coefﬁcient. If a single-period complex

exponential function is deﬁned as the base template function:

1;0

ðtÞ¼e

j2pt=L

; t 2½0; LÞ (7.10)

then the corresponding derived template function can be expressed, per deﬁnition in

(7.1), as:

k;m

ðtÞ¼

j2p

ðÞ

; t 2½mkL; mkL þ kLÞ (7.11)

Using the derived template function, the signal x(t) shown in (7.9) can be

expressed, in the generalized transformation frame, as:

xðtÞ¼

Cðk; mÞW

k;m

ðtÞ¼

Cðk; mÞe

j2p

ðÞ

Cðk; mÞ

j2p

ðtÞ

(7.12)

Table 7.1 Basic template functions expressed in the generalized transformation frame

Frame {W

k,m

(t)} Properties

Fourier

function

1;0

ðtÞ e

j2pt=L

; t 2½0; LÞ

Exponential function forms a

complete orthogonal base

Haar

function

1;0

ðtÞ

þ10 t<L=2

1 L=2  t<L

(

; t 2½0; LÞ

Rectangular waveform forms a

complete orthogonal base

Daubechies

function

1;0

ðtÞ c

ðnÞ

1;0

ðtÞ; t 2½0; LÞ

Fractal shape forms a complete

orthogonal base

There are different Daubechies functions c

ðnÞ

1;0

ðtÞ

depending on different order of n

106 7 Wavelet Integrated with Fourier Transform: A Uniﬁed Technique

where the term

Cðk ; mÞ (7.13)

represents the sum of the normalized individual measure functions corresponding to

the discrete scale k in the Fourier transform, and the term

ðtÞ¼e

j2p

(7.14)

represents a periodic exponential function with a period of kL over the time

interval (1, 1), for a given scale k. Comparing (7.9) with (7.12), it can be

seen that k ¼ 1/n when L ¼ T, and c

¼ C

, The expression k¼ 1/n indicates that a

higher scale (k) corresponds to a lower frequency (n), and c

¼ C

is deﬁned only

by the frequency n or scale k. From this discussion, the Fourier transform can be

viewed in the generalized frame as a 1D function, deﬁned by the scale k with the

orthogonal base {V

(t)}. There is no time information of the extracted signal

features contained in this function. This explains why the Fourier transform does

not provide time information of the extracted frequency components.

7.1.2 Wavelet Transform in the Generalized Frame

Wavelet transform decomposes a signal using ﬁnite time intervals (or support)at

different scales, thus maint ains the time location information of the signal features.

Through variation of the scales of the template function used, nonstationary or

transient features within a signal can be extracted more effectively than by using

the exponential (sine and cosine) functions in the Fourier transform. The wavelet

transform can be expressed as:

Cðs; uÞ¼

xðtÞW

s;u

ðtÞdt (7.15)

where the term C(s, u) represents the wavelet coefﬁcients (or measure function, in the

generalized signal transformation frame). The wavelet function W

s,u

(t)isdeﬁned

by (7.1). To reduce computational load and avoid redundancy in signal representa-

tion, discrete instead of continuous wavelet transform is often employed for analyz-

ing signals consisting of discrete data points acquired through a data acquisition

process. Generally, a discrete wavelet transform discretizes a signal by using a scale

of the power of 2 (Daubechies 1992;Kaiser1994). At the scale k ¼ 2

(n ∈ N), the

discrete wavelet function W

k,m

(t)(m ∈ Z)hasthesupportofT

¼ 2

L. Physically, the

term 2

L represents the time resolution, which increases linearly with the logarithm of

the scale, enabling signal feature extractions at different resolutions. The frequency

7.1 Generalized Signal Transformation Frame 107

resolution of wavelet transform is the inverse of the time resolution, or 1/T

It increases as the scales become higher (i.e., when n increases), and is therefore

well suited for analyzing slow-changing signals. At lower scales, the frequency

resolution decreases, enabling analysis of fast-changing signals. Such is in contrast

to the Fourier transform, which maintains a constant frequency resolution over the

entire spectrum, and have a time resolution that is deﬁned by the signal duration.

To illustrate the ability of wavelet transform in signal feature extraction, we

analyze an frequency shifted keying (FSK) signal, which is com monly used for data

modulation and wireless data transmission (Gibson 1999 ). An FSK signal is

expressed as:

xðtÞ¼

square ð2pf

tÞ for message ‘‘1’’

square ð2pf

tÞ for message ‘‘0’’

(

(7.16)

where square (2pf

t) represents a periodic square wave with a unit amplitud e and

frequency f

. Figure 7.1 illustrates an example of such an FSK signal, where the

frequency f

¼ 30 Hz is used to transmit the digit “1” and f

¼ 125 Hz is used to

transmit the digit “0.” The message to be transmitted is [1 0 0 1101000].Sucha

signal x(t) is nonsinusoidal and nonstationary.

To analyze this signal, we choose the Haar wavelet as the base wavelet, since its

square-shaped wave form best matches the shape of the FSK signal. Given that the

wavelet has a support of L ¼ 1 s (refer to Table 7.1), the measure function C(s

, m)

can be calculated by using (7.8). For the message “1,” C(s

, m) is calculated to

be s

, at scale s

¼ 1/f

and time t ¼ ms

. The measure C(s

, m) is zero for the

message “0.” Similarly, at scale s

¼ 1/f

, C(s

, m)is s

at time t ¼ ms

for

the message “0,” and zero for a message “1.” The result of such a wavelet transform

operation indicates that the Haar wav elet was able to locate the time instant of the

message “1” or “0” at scales s

and s

, respectively, and expresses the FSK signal

x(t) in the generalized signal transform frame as a single-form expression:

xðtÞ¼

m¼ 1

Cðs

; mÞc

ð1Þ

ðtÞþCðs

; mÞc

ð1Þ

ðtÞ (7.17)

0 1 2 3 4 5 6 7 8 9 10

-1

Fig. 7.1 A FSK signal x(t)

108 7 Wavelet Integrated with Fourier Transform: A Uniﬁed Technique

The results are shown in Fig. 7.2, where the messages “1” and “0” can be clearly

separated into two different scales. In comparison, it is not feasible to use the

Fourier transform to specify at which time which message (1 or 0) is transmitted.

The sine and cosine template functions in the Fourier transform do not match the

square waveform of the FSK signal, and consequently, the frequency component of

the FSK signal will be spread out in a broad spectrum, especially when the message

“0” and “1” are randomly transmitted.

7.2 Wavelet Transform with Spectral Postprocessing

As a time scale domain technique, the wavelet transform utilizes template func-

tions of different time resolutions at different scales to extract “transient” features

embedded in a signal. Such transient features can be generated by the interactions

between the rolling elements in bearing and a localized defect (e.g., surface spal-

ling) on the surface of the raceway. As the rolling elements periodically roll over

0 1 2 3 4 5 6 7 8 9 10

-8

-6

-4

-2

0 1 2 3 4 5 6 7 8 9 10

-4

-3

-2

-1

Extracted FSK si

nal for messa

e “0”

Extracted FSK signal for message “1”

Fig. 7.2 Extracted FSK signals. (a) Extracted FSK signal for message “1” and (b) extracted FSK

signal for message “0”

7.2 Wavelet Transform with Spectral Postprocessing 109