Biomedical Engineering Reference
In-Depth Information
expense of poor time resolution, and frequency resolution gradually deteriorates for
high frequency bands but the time resolution for these bands improves.
The CWT can be used to form a time-frequency representation of the energy
distribution—scalogram, in a way analogous to the way in which a spectrogram is
obtained from STFT. The time-scale representation is:
2
S
x
(
t
,
a
;Ψ
)=
|
T
x
(
t
,
a
;Ψ
)
|
(2.94)
Due to equation (2.93) it can be understood as time-frequency representation:
2
S
x
(
t
,
f
;Ψ
)=
|
T
x
(
t
,
f
0
/
f
;Ψ
)
|
(2.95)
Analogously to the spectrogram, the scalogram is disturbed by cross-terms of
multi-component signals in the regions of time-frequency space where the CWT of
individual components overlap.
2.4.2.2.3 Discrete wavelet transform
The CWT in its theoretical form (2.91)
operates on continuous time-scale (time-frequency) space. In any practical appli-
cation the time and scale dimensions have to be sampled. The sampling, in gen-
eral, can be performed in any way resulting in different approximations of CWT.
The common way is to select
t
0
>
0and
a
0
>
0 and then generate a grid of points
t
a
0
for
m
nt
0
a
0
m
∈
Z
The discrete wavelet transform can be defined then as:
=
,
=
,
a
n
Z
∞
Ψ
n
,
m
(
T
x
(
n
,
m
;Ψ
)=
x
(
s
)
s
)
ds
;
m
,
n
∈
Z
(2.96)
−
∞
where Ψ
n
,
m
is a scaled and translated version of Ψ:
Ψ
s
nt
0
a
0
a
0
1
a
0
−
Ψ
n
,
m
(
s
)=
(2.97)
2.4.2.2.4 Dyadic wavelet transform—multiresolution signal decomposition
However, there is one special manner of sampling which, for some forms of wavelets,
creates the orthonormal basis in the time-scale space. This is the
dyadic sampling
ob-
tained for
t
0
2. The dyadic sampled wavelets are related to multiresolu-
tion analysis and the transform can be efficiently computed using filter banks [Mallat,
1989].
To better understand the multiresolution signal decomposition we should start with
considering the so-called
scaling function
. The scaling function is a unique measur-
able function
2
φ
=
1and
a
0
=
L
2
by 2
j
(
t
)
∈
(R)
such that, if the dilation of φ
(
t
)
is expressed by:
2
j
φ
2
j
t
φ
2
j
(
t
)=
(
)
(2.98)
2
i.e., the integral of squared function is finite.
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