Civil Engineering Reference
In-Depth Information
The scale decomposition is accomplished by wavelet
analysis. Let us use the notation
(
)
to represent the
Ω
T
W
,
t
wavelet coefficient of
ut
()
1
τ
−
t
⎡
⎤
∞
(
)
()
∫
[4.73]
Ω=
Tt
,
u
τ
g
d
τ
⎢
⎣ ⎦
W
2
T
a
−∞
W
where
a
2
T
W
and
t
are, respectively, the scale and shift
parameters, and
g
is the mother wavelet function. The
frequency scale is defined by
k
=
T
W
. Any signal, and
=
2
π
particularly the coefficients
()
, can be expressed by
Ω
k
,
t
t
⎡
⎤
(
)
(
)
(
)
∫
Ω=
kt
,
r kt
,
cos
ω
kt
,
[4.74]
⎢
⎥
i
⎢
⎥
⎣
⎦
0
are, respectively,
the amplitude and the instantaneous frequency. We can
represent
In the above relation,
rk
,
t
()
and
()
ω
i
k
,
t
in different forms depending on the dual
process in question. This is the Hilbert transform in light of
the Rice representation [PAP 84] which minimizes the
variation of the signal envelope. The instantaneous local
amplitude therefore is
()
Ω
k
,
t
(
)
(
)
ˆ
(
)
rkt
2
,
=Ω
2
kt
,
+Ω
2
kt
,
is the Hilbert transform of
where
ˆ
()
.
()
Ω
k
,
t
Ω
k
,
t
With the Rice representation, the instantaneous
frequency is expressed by
ˆ
ˆ
(
)
(
)
(
)
(
)
ΩΩ −Ω Ω
kt
,
'
kt
,
'
kt
,
kt
,
ω
(,)
kt
=
(
)
i
2
rkt
,
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