Biomedical Engineering Reference
In-Depth Information
Morlet wavelet
1
0.5
0
−
0.5
−
1
−
5
0
5
(a)
Mexican hat wavelet
1
0.5
0
−
0.5
−
5
0
5
(b)
Figure 3.15
Two popular wavelets: (a) the Morlet and (b) the Mexican hat.
∞
∫
1
tb
a
−
⎛
⎜
⎞
⎟
{
}
()
()
*
WT
xt ab
;,
=
xt
ψ
dt
(3.30)
ab
,
a
−∞
where
a
,
b
0 represent the scale and translation parameters, respectively;
t
is the time; and the asterisk stands for complex conjugation. If
a
∈ℜ
,
a
≠
>
1, then
ψ
is
stretched along the time axis and if 0
1,
then the wavelet is termed the mother wavelet. The wavelet coefficients describe the
correlation or similarity between the wavelet at different dilations and translations
and the signal
x
. As an example of a CWT, Figure 3.16 shows the continuous wave-
let transform using the Morlet wavelet of the EEG signal depicted earlier in Figure
3.12(a).
<
a
<
1, then
ψ
is contracted. If
b
=
0 and
a
=
3.1.3.3 Discrete Wavelet Transform
If we are dealing with digitized signals, then to reduce the number of redundant
wavelet coefficients,
a
and
b
must be discretized. The discrete wavelet transform
(DWT) attains this by sampling
a
and
b
along the dyadic sequence:
a
=
2
j
and
b
=
k
2
j
, where
j
,
k
Z
and represent the discrete dilation and translation numbers,
respectively. The discrete wavelet family becomes
∈
{
}
(
)
()
ψ
t
=
2
−
j
2
ψ
2
−
1
t
−
k
,,
j k
∈
Z
(3.31)
jk
,
The scale 2
-
j
/2
normalizes
ψ
j,k
so that
||ψ
j,k
||
=
||ψ||
.
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