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2.2 Wavelet Transform
In 1980s, the wavelet transform (WT) was developed by Morlet and Grossmam, et al.
In recent years, the WT has become a very popular method and is introduced into the
analysis of sound and vibration signals in engineering. Due to the high efficiency and
superiority in multiresolution analysis of the WT, the analyzed signals can be observed
from coarse to fine [3]. The WT of the signal
x(t)
can be described as:
1
tb
−
∫
∫
WT
(,)
a b
=
+∞
−∞
x t
()
ϕ
*
()
t dt
=
x t
() (
ϕ
*
)
dt
(4)
x
a b
,
a
a
.
2.3 Wigner-Ville Distribution
The Wigner-Ville distribution (WVD) was presented by Wigner in the research of
quantum mechanics in 1932 and applied to signal processing by Ville later. The WVD
of the signal
x(t)
can be described as:
∫
+∞
*
−
j
2
πτ
f
WVD
( ,
t
f
)
=
x t
(
+
τ
/2)
x
(
t
−
τ
/2)
e
d
τ
(5)
.
x
−∞
If the signal has two components
x
1
and
x
2
, the WVD can be expressed as:
WVD
(,)
t
f
=
WVD
(,)
t
f
+
WVD
(,)
t
f
+
WVD
(,)
t
f
+
WVD
(,)
t
f
(6)
.
x
11
22
21
12
Here
WVD
11
(t, f)
and
WVD
22
(t, f)
are self terms of the WVD, while
WVD
12
(t, f)
and
WVD
21
(t, f)
are cross terms. The cross terms are involved due to that the WVD belongs
to quadratic analysis. To suppress influences of the cross terms, the pseudo-WVD
(PWVD) as an equivalent smoothed WVD was developed. The PWVD of
x(t)
can be
expressed as:
∫
+∞
*
−
j
2
πτ
f
PWVD
( ,
t
f
)
=
h
( ) (
τ
x t
+
τ
/2)
x
(
t
+
τ
/2)
e
d
τ
(7)
.
x
−∞
2.4 Hilbert-Huang Transform
A new time-frequency analysis method called Hilbert-Huang transform (HHT) was
presented by Norden E. Huang et al in 1998. The HHT contains two important parts:
empirical mode decomposition (EMD) and Hilbert transform. First, the EMD is used to
obtain the intrinsic mode function (IMF). Then Hilbert transform may be applied to
obtain a three-dimensional (time-frequency-amplitude) distribution, i.e., the Hilbert
spectrum of the signals. A great contribution of HHT is that it can obtain the
instantaneous frequency feature of the nonstationary signals from the Hilbert spectra
[4, 5]. Essentially, the EMD is used for smoothing the nonstationary signals and
decomposing them into a set of data sequences with different scale characteristics, i.e.,
IMFs. The target signal
x(t)
can be expressed as the sum of IMF
c
i
(i=1,2,3,…,n) and
the residue component
r
n
after the EMD. i.e.,
n
∑
xt
()
=
=+
c
r
i
n
(8)
.
i
1
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