Biomedical Engineering Reference
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on its location inside the considered window.
The decomposition based on the matching pursuit algorithm offers the step-wise
adaptive compromise between the time and frequency resolution. The resulting de-
composition is time and frequency invariant. The time-frequency energy density esti-
mator derived from the MP decomposition has explicitly no cross-term, which leads
to clean and easy-to-interpret time-frequency maps of energy density. The price for
the excellent properties of the MP decomposition is the higher computational com-
plexity.
The sparsity of the DWT and MP decompositions has a different character which
has an effect on their applicability. DWT is especially well suited to describing time
locked phenomena since it provides the common bases. MP is especially useful for
structures appearing in the time series at random. The sparsity of MP stems from the
very redundant set of functions, which allows to represent the signal structures as a
limited number of atoms. The MP decomposition gives the parameterization of the
signal structures in terms of the amplitude, frequency, time of occurrence, time, and
frequency span which are close to the intuition of practitioners.
2.4.2.2.9 Empirical mode decomposition and Hilbert-Huang transform
The
Hilbert-Huang transform (HHT) was proposed by Huang et al. [Huang et al., 1998].
It consists of two general steps:
•
The empirical mode decomposition (EMD) method to decompose a signal into
the so-called intrinsic mode function (IMF)
•
The Hilbert spectral analysis (HSA) method to obtain instantaneous frequency
data
The HHT is a non-parametric method and may be applied for analyzing non-
stationary and non-linear time series data.
Empirical mode decomposition (EMD) is a procedure for decomposition of a sig-
nal into so called intrinsic mode functions (IMF). An IMF is any function with the
same number of extrema and zero crossings, with its envelopes being symmetric with
respect to zero. The definition of an IMF guarantees a well-behaved Hilbert trans-
form of the IMF. The procedure of extracting an IMF is called sifting. The sifting
processisasfollows:
1. Between each successive pair of zero crossings, identify a local extremum in
the signal.
2. Connect all the local maxima by a cubic spline line as the upper envelope
E
u
(
t
)
.
3. Repeat the procedure for the local minima to produce the lower envelope
E
l
(
t
)
.
1
2
(
)=
(
(
)+
(
))
4. Compute the mean of the upper and lower envelope:
m
11
t
E
u
t
E
l
t
.
5. A candidate
h
11
for the first IMF component is obtained as the difference be-
tween the signal
x
(
t
)
and
m
11
(
t
)
:
h
11
(
t
)=
x
(
t
)
−
m
11
(
t
)
.
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