Biomedical Engineering Reference
In-Depth Information
2.4.2.2.6 Wavelets in MATLAB
In MATLAB wavelets analysis can be conve-
niently performed with the Wavelet Toolbox distributed by MathWorks. The col-
lection of functions delivered by this toolbox can be obtained by command
help
wavelet
.
Another MATLAB toolbox for wavelet analysis is WaveLab
http://www-stat.
a collection of MATLAB functions that implement a variety of algorithms related
to wavelet analysis. The techniques made available are: orthogonal and biorthogonal
wavelet transforms, translation-invariant wavelets, interpolating wavelet transforms,
cosine packets, wavelet packets.
2.4.2.2.7 Matching pursuit—MP
The atomic decompositions of multicompo-
nent signals have the desired property of explaining the signal in terms of time-
frequency localized structures. The two methods presented above: spectrogram and
scalogram are working well but they are restricted by the a priori set trade-off be-
tween the time and frequency resolution in different regions of the time-frequency
space. This trade-off does not follow the structures of the signal. In fact interpretation
of the spectrogram or scalogram requires understanding which aspects of the repre-
sentation are due to the signal and which are due to the properties of the methods.
A time-frequency signal representation that adjusts to the local signal properties
is possible in the framework of matching pursuit (MP) [Mallat and Zhang, 1993].
In its basic form MP is an iterative algorithm that in each step finds an element
g
γ
n
(atom) from a set of functions
D
(dictionary) that best matches the current residue of
the decomposition
R
n
x
of signal
x
; the null residue being the signal:
⎧
⎨
R
0
x
=
x
R
n
x
R
n
x
R
n
+
1
x
=
,
g
γ
n
g
γ
n
+
(2.109)
⎩
R
n
x
g
γ
n
=
arg max
g
γ
i
∈
D
|
,
g
γ
i
|
where: arg max
g
γ
i
∈
D
means the atom
g
γ
i
which gives the highest inner product with
the current residue:
R
n
x
. Note, that the second equation in (2.109) leads to orthogo-
nality of
g
γ
n
and
R
n
+
1
x
,so:
R
n
+
1
x
2
2
2
R
n
x
R
n
x
=
,
g
γ
n
+
(2.110)
The signal can be expressed as:
n
=
0
R
n
x
,
g
γ
n
g
γ
n
+
R
k
+
1
x
k
x
=
(2.111)
It was proved [Davis, 1994] that the algorithm is convergent, i.e., lim
k
→
∞
R
k
s
2
=
0.
Thus in this limit we have:
n
=
0
R
n
x
,
g
γ
n
g
γ
n
∞
x
=
(2.112)
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