Biomedical Engineering Reference
In-Depth Information
In a general case the first candidate
h
11
doesn't satisfy the IMF conditions. In such
case the sifting is repeated taking
h
11
as the signal. The sifting is repeated iteratively:
h
1
k
(
t
)=
h
1
(
k
−
1
)
(
t
)
−
m
1
k
(
t
)
(2.120)
until the assumed threshold for standard deviation
SD
computed for the two consec-
utive siftings is achieved. The
SD
is defined as:
2
t
=
0
|
h
1
(
k
−
1
)
(
t
)
−
h
1
k
(
t
)
|
T
SD
=
(2.121)
h
1
(
k
−
1
)
(
t
)
Authors of the method suggest the
SD
of 0.2-0.3 [Huang et al., 1998]. At the end of
the sifting process after
k
iterations the first IMF is obtained:
c
1
=
h
1
k
(2.122)
The
c
1
mode should contain the shortest period component of the signal. Subtracting
it from the signal gives the first residue:
r
1
=
x
(
t
)
−
c
1
(2.123)
The procedure of finding consecutive IMFs can be iteratively continued until the
variance of the residue is below a predefined threshold, or the residue becomes a
monotonic function—the trend (the next IMF cannot be obtained). The signal can be
expressed as a sum of the
n
-empirical modes and a residue:
n
i
=
1
c
i
−
r
n
x
(
t
)=
(2.124)
Each of the components can be expressed by means of a Hilbert transform as a
product of instantaneous amplitude
a
j
(
t
)
and an oscillation with instantaneous fre-
e
i
R
ω
j
(
t
)
dt
. Substituting this to (2.124) gives
representation of the signal in the form:
quency ω
j
(
t
)
(
Sect. 2.4.1)
:
c
j
=
a
j
(
t
)
n
i
=
1
a
j
(
t
)
e
i
R
ω
j
(
t
)
dt
x
(
t
)=
(2.125)
Equation (2.125) makes possible construction of time-frequency representation—the
so-called Hilbert spectrum. The weight assigned to each time-frequency coordinate
is the local amplitude.
2.5 Non-linear methods of signal analysis
Non-linear methods of signal analysis were inspired by the theory of non-linear
dynamics— indeed the biomedical signal may be generated by the non-linear pro-
cess. Dynamical systems are usually definedbyasetoffirst-order ordinary differ-
ential equations acting on a phase space. The phase space is a finite-dimensional
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