Digital Signal Processing Reference
In-Depth Information
Fig. 13.1
Example of an expanded and translated wavelet
Fig. 13.2
The process of a
wavelet transform
Further, the wavelet is normalized
ψ
=
1, and is centered in the neighborhood
of
t
0. By expanding the wavelet
ψ
by a factor
s
and then translating it by
u
(see
the example in Fig.
13.1
), we obtain the family of wavelets
ψ
u,s
associated with
ψ
,
with the same standard unit (that is,
=
ψ
u,s
=
1):
√
s
ψ
t
.
1
−
u
ψ
u,s
(t)
=
(13.2)
s
L
2
(
The continuous wavelet transform of a signal
X
∈
R
)
at time
u
and scale
s
is
defined by:
−∞
√
s
ψ
∗
t
dt
X(t)
1
−
u
W
X
(u,s)
=
X,ψ
u,s
=
(13.3)
s
+∞
where
W
refers to the wavelet and
ψ
∗
denotes the complex conjugate of
ψ
.
The relationship (
13.3
) represents the scalar product of
X
and the set of wavelets
ψ
u,s
associated with
ψ
.
W
X
(u,s)
characterizes the “fluctuations” of the signal
X(t)
in the neighborhood of position
u
at scale
s
(see Fig.
13.2
; here
u
takes the specific
value
t
0
).
Examining expressions (
13.1
) and (
13.3
), it is clear that
W
X
(u,s)
will be insen-
sitive to the signal's most regular behaviors and more flexible than polynomials of
degree strictly less than
n
(the number of vanishing moments of
ψ
). Conversely,
W
X
(u,s)
takes into account the irregular behavior of polynomial trends. This im-
portant property plays a role in the detection of signal singularities.
W
X
(u,s)
can also be interpreted as a linear filter operation:
W
X
(u,s)
=
X
∗
ψ
s
(u)
(13.4)
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