Digital Signal Processing Reference
In-Depth Information
√
s
ψ
∗
(
−
s
)
, in which
the Fourier transform,
ψ
s
(ω)
=
√
sψ
∗
(sω)
, is identified as a transfer function of
a bandpass filter [
2
].
The relationship (
13.4
) demonstrates that wavelet transforms can be calculated
by expanded bandpass filters (with variable
s
). The inverse of a continuous wavelet
transform in
L
2
denotes the product of convolution, with
ψ
s
(t)
1
where
∗
=
is provided by the wavelet admissibility condition:
−∞
0
|
ψ(ω)
|
ψ(ω)
2
2
|
|
K
ψ
=
dω
=
dω<
+∞
.
(13.5)
|
ω
|
|
ω
|
0
+∞
In order for this integral to be finite, it is necessary to ensure that
ψ(
0
)
=
0,
=
R
which is why wavelets must have a mean of zero (
ψ(
0
)
ψ(t)dt
=
0). This
condition is almost sufficient. If
ψ(
0
)
0 with continuously differentiable
ψ(ω)
,
=
the admissibility condition is satisfied.
In practice, choosing a wavelet with a zero mean (and highly localized in time
and in frequency) is sufficient. In this case, it is possibly to synthesize or reconstruct
signal
X(t)
by inverting the wavelet transform as follows:
0
−∞
√
s
ψ
t
du
ds
1
K
ψ
W
X
(u,s)
1
−
u
X(t)
=
s
2
,t
∈ R
.
(13.6)
s
+∞
+∞
This reconstruction uses all scales, and as such is highly redundant. Continuous
wavelet transform is calculated based on the scale factor
s
and the time
u
in the
set of real numbers (the time-scale plane is therefore continuously traversed), which
renders it extremely redundant. In reconstruction of a signal by a continuous in-
verse transform, this redundancy is extreme in the sense that all the expanded and
translated wavelets are employed such that they are linearly dependent, therefore
reflecting existing signal information without adding new information.
13.1.3 Discrete Wavelet Transforms
As previously noted, continuous wavelet transforms are highly redundant:
W
X
(u,s)
is the 2D (
(u,s)
-plane) representation of a signal
X(t)
in 1D! This redundancy can
be reduced using one of the countably infinite family of wavelets
{
ψ
j,k
}
, where
2
2
−
j/
2
ψ(
2
−
j
t
(j,k)
k)
. The time-scale
(u,s)
-plane is con-
verted to a “dyadic mesh” (or in base 2), as shown in Fig.
13.3
:
u
∈ Z
and
ψ
j,k
(t)
=
−
∈ Z × Z
.
2
j
k,s
2
j
,(j,k)
→
→
{
ψ
j,k
}
(j,k)
∈Z
2
must consti-
tute an orthonormal basis of
L
2
(
R
)
. The conditions under which this basis becomes
orthonormal and thus provides a “highly economical” wavelet transform are related
to the concept of multiresolution analysis (abbreviated MRA).
Clearly, to reduce or eliminate redundancy, the family
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