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Abstract and Figures

The Empirical Mode Decomposition (EMD) is a signal analysis method that separates multi-component signals into single oscillatory modes called intrinsic mode functions (IMFs), each of which can generally be associated to a physical meaning of the process from which the signal is obtained. When the phenomena of mode mixing occur, as a result of the EMD sifting process, the IMFs can lose their physical meaning hindering the interpretation of the results of the analysis. In the paper, "One or Two frequencies? The Empirical Mode Decomposition Answers", Gabriel Rilling and Patrick Flandrin [3] presented a rigorous mathematical analysis that explains how EMD behaves in the case of a composite two-tones signal and the amplitude and frequency ratios by which EMD will perform a good separation of tones. However, the authors did not propose a solution for separating the neighboring tones that will naturally remain mixed after an EMD. In this paper, based on the findings by Rilling and Flandrin, a method that can separate neighbouring spectral components, that will naturally remain within a single IMF, is presented. This method is based on reversing the conditions by which mode mixing occurs and that were presented in the map by Rilling and Flandrin in the above mentioned paper. Numerical experiments with signals containing closely spaced spectral components shows the effective separation of modes that EMD can perform after this principle is applied. The results verify also the regimes presented in the theoretical analysis by Rilling and Flandrin.
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1
Method for Mode Mixing Separation in Empirical
Mode Decomposition
Olav B. Fosso*, Senior Member, IEEE, Marta Molinas*, Member, IEEE,
Abstract—The Empirical Mode Decomposition (EMD) is a sig-
nal analysis method that separates multi-component signals into
single oscillatory modes called intrinsic mode functions (IMFs),
each of which can generally be associated to a physical meaning
of the process from which the signal is obtained. When the
phenomena of mode mixing occur, as a result of the EMD sifting
process, the IMFs can lose their physical meaning hindering the
interpretation of the results of the analysis. In the paper, One
or Two frequencies? The Empirical Mode Decomposition Answers,
Gabriel Rilling and Patrick Flandrin [3] presented a rigorous
mathematical analysis that explains how EMD behaves in the
case of a composite two-tones signal and the amplitude and
frequency ratios by which EMD will perform a good separation
of tones. However, the authors did not propose a solution for
separating the neighboring tones that will naturally remain mixed
after an EMD. In this paper, based on the findings by Rilling
and Flandrin, a method that can separate neighbouring spectral
components, that will naturally remain within a single IMF, is
presented. This method is based on reversing the conditions
by which mode mixing occurs and that were presented in the
map by Rilling and Flandrin in the above mentioned paper.
Numerical experiments with signals containing closely spaced
spectral components shows the effective separation of modes that
EMD can perform after this principle is applied. The results
verify also the regimes presented in the theoretical analysis by
Rilling and Flandrin.
Index Terms—EMD, Mode Mixing, Sifting process, Intermit-
tency, Masking signal, Closely spaced spectral tones.
I. INTRODUCTION
Empirical Mode Decomposition (EMD) has since it was first
proposed in [1], demonstrated its capabilities within many ap-
plication areas. EMD’s data driven approach does not require
any specific assumptions behind the underlying model and
is able to handle both non-linear and non-stationary signals.
However, the algorithm has some limitations in identifying
two closely spaced spectral tones and components appearing
intermittently in the signal. In such cases the EMD will
have trouble to separate the closely spaced tones and the
intermittency. This paper addresses the problem of separation
of closely spaced spectral tones and proposes a new method for
separation. The structure of the paper is: the concept of mode
mixing is discussed first, then some of the existing solutions
are highlighted before the new method is presented. A case
study based on a synthetic signal demonstrates the method’s
capabilities.
O.B. Fosso is with the Department of Electric Power Engineering, Nor-
wegian University of Science and Technology (NTNU), 7491 Trondheim,
Norway, e-mail: olav.fosso@ntnu.no
M. Molinas is with Department of Engineering Cybernetics, Norwegian
University of Science and Technology (NTNU), 7491 Trondheim Norway,e-
mail: marta.molinas@ntnu.no
* These two authors contributed equally to the work.
Manuscript received September 14, 2017
A. What is Mode Mixing?
Although Mode Mixing has not been strictly defined in the
literature, it is known to happen as a result of the way in which
the Empirical Mode Decomposition is designed to extract
monocomponents from a multi-component signal. Only modes
that clearly contribute with their own maxima and minima can
be identified by the sifting process of the EMD. When a mode
cannot clearly contribute with extremas, the EMD will not
be able to separate the mode in a single IMF and the mode
will remain mixed in another IMF. The paper in [3], provides
restrictions on when it is possible to extract a tone from a
composite two tones signals. The ratio of the amplitudes and
of the frequencies of the individual components of the signal,
will determine whether the EMD will be able to separate them
in two different IMFs or whether they will be interpreted as
one single IMF.
In this paper, Mode Mixing is broadly categorized in two
groups. Depending on the source at the origin, they can be
originated by:
presence of closely spaced spectral components
presence if intermittence
In general, the reported literature offers solutions to the mode
mixing caused by the presence of intermittence, while sepa-
ration of closely spaced spectral tones has remained unsolved
[8].
This paper proposes a principle that when implemented
improves the ability of the EMD to separate closely spaced
spectral tones. In the following section, the main contributions
to the solution of the mode mixing problem are discussed in
light of their evolution in time. The new method is introduced
within this perspective.
B. Existing Mode Mixing Solutions
Mode mixing, observed in the context of the Empirical
Mode Decomposition, and caused by either intermittency of
a signal component or by closely spaced spectral tones, is a
well recognized limitation of the EMD method [2][3][4]. The
principle behind the EMD is to always extract the highest
frequency component present is the residue of the signal.
In the presence of intermittence, the next lower frequency
components may appear in the same IMF where the inter-
mittent signal is identified, even though they are in different
octaves and should appear as individual tones. In the case
of closely spaced spectral tones, the signals will appear in
the same IMF unless successfully separated. In 2005, in the
paper: The Use of Masking Signal to Improve Empirical Mode
Decomposition, Ryan Deering and James F. Kaiser [2] discuss
the phenomena of mode mixing and presented for the first
arXiv:1709.05547v1 [stat.ME] 16 Sep 2017
2
time the idea of using a masking signal to separate mixed
tones caused by the presence of intermittency in the signal. A
formula for choosing the masking signal was suggested based
on the observations of empirical trials with several signals
with mode mixing. The demonstration was essentially made
for mode mixing caused by the presence of intermittency,
since the frequency ratio between the signals was 0.57 which
from the perspective of closely spaced spectral tones, was only
moderately mixed. Rilling and Flandrin presented in 2008 a
theoretical analysis that explains the behavior of the EMD
in the presence of two closely spaced spectral components.
Their work demonstrates which spectral components could
be expected to be separated by the EMD based on their
frequency and amplitude ratios. A Boundary Map prepared
by the authors, provides a visual indication of the efficiency
of separation of two tones depending on their amplitude and
frequency ratios. This work did not present a solution for the
mode mixing problem caused by closely spaced spectral tones.
It rather establishes restrictions on when it is possible, using
EMD, to extract a tone from a composite two tones signals.
In 2009, Wu and Huang presented the Ensemble Empirical
Mode Decomposition (EEMD) as a solution to cope with the
mode mixing phenomena [4][5]. Again, this approach was
intended to solve the mode mixing caused by the presence
of intermittent components in the signal. The principle behind
EEMD is to average the modes obtained by EMD after several
realizations of Gaussian white noise that are added to the
original signal. After this work by Huang, several versions
of the Ensemble EMD were proposed in the literature and
the method is widely used [6] but it is considered to be
computationally expensive. Most recently, in April 2017, a
patent application, System and Method of Conjugate Adaptive
Conjugate Masking Empirical Mode Decomposition filed by
Norden Huang et. al [7] discloses a method for directly pro-
cessing an original signal into a plurality of mode functions.
The invention claims to exclude the problem of mode mixing
caused by an intermittent disturbance but does not apparently
address the mode mixing caused by closely spaced spectral
tones.
Although Deering and Kaiser introduced the idea of a
masking signal, an open question remains on how to choose
the frequency and the amplitude of the masking signal, to
separate closely spaced spectral tones.
The following section presents and discusses the principle
that can enable EMD to separate these class of mode mixing.
Based on the idea of the masking signal presented by Deering
and Kaiser and the knowledge of the restrictions for closely
spaced spectral tones presented by Rilling and Flandrin, a
masking signal can be designed in order to reverse an existing
mode mixing condition.
II. TH E PROP OS ED ME TH OD
The method is based on the original idea of Deering and
Kaiser, of injecting a masking signal. The extension to this
original idea is the way in which the masking signal is defined.
In this work, the frequency and amplitude of the masking
signal are chosen according to the boundary conditions pre-
sented in [3]. The principle is explained in the following.
Based on the boundary conditions presented in[3], the existing
mode mixing conditions of a given signal can be reversed by
adding a masking signal that can enforce a controlled artificial
mode mixing with one of the signal’s components, leaving the
other free of mode mixing. If a masking signal of appropriate
frequency and amplitude is added to the original signal, this
masking signal will be able to attract one of the mixed signals
but not the other. This principle of attraction between closely
spaced spectral tones, will create a new controlled artificial
mode mixing, where one of the mixed signals is combined
with the known masking signal. In this way, one of the
originally mixed signals becomes free of mode mixing and
comes as a pure mode (IMF) out of the EMD process. The
controlled artificial mode mixing can be easily removed, since
the masking signal is known.
The step by step procedure for extracting IMFs when modes
are mixed, based on the proposed method in this paper is the
following:
1) Construct masking signal xmbased on the new principle
defined in this paper,
2) Perform EMD on x+=x+xmand obtain the IMF y+.
Similarly obtain yfrom x=x-xm
3) Define IMF as y= (y++y)/2
Step 1 will require to obtain the frequency information from
the original data. This information is obtained as explained in
section II.B.
A. Mode Mixing of Closely Spaced Spectral Components
In [3], it is pointed out that the amplitude and frequency
ratios of the signal components are crucial for the under-
standing of the basic principles behind mode mixing. In the
same work, a Boundary Condition Map portrays the different
regions where mode mixing is likely to occur as a function
of the relative frequencies and amplitudes involved in the two
signals. It can be observed from the map that in the region
where the ratio between the frequencies involved are between
0 and 0.5, mode-mixing will not be observed for a range of
amplitude ratios. Mode mixing is observed as the frequency
ratio is higher than 0.67 and approaches 1.0, where mode
mixing will always occur for all amplitude ratios. For the sake
of clarity and to aid the discussions, the same Boundary Map
is reconstructed in this paper and shown in Figure 1.
The map illustrates well how closely spaced spectral tones
attract each other in a mode mixing. This same property
is exploited in this paper for constructing effective masking
signals to separate closely spaced spectral tones.
B. Principle for separation of closely spaced spectral compo-
nents
The principle applied in this paper is based on the combi-
nation of the technique presented by Kaiser and the Boundary
Conditions Map presented by Flandrin[2][3]. The map of
boundary conditions guides the choice of the masking signal’s
frequency and amplitude.
To be able to extract a frequency by applying this principle,
the ratio of that frequency to the frequency of the masking
3
Fig. 1. Mode mixing boundary conditions map reproduced from [3] used for
defining masking signal
signal, must be located in the red area of the Boundary Map
(mode mixing area), while the ratio of the next frequency
should be located in the blue area of the map. The amplitude
ratios need to be adopted to ensure that condition as well. It
is therefore necessary to operate with a frequency sufficiently
close to the first mode and sufficiently distant from the next
mode, to be successful. This criteria will be discussed in more
detail in the case study.
Assume a signal with the two frequencies f1and f2(f1>
f2), where the ratio between them will cause mode mixing
due to close spectral tones. A masking signal of frequency fm
larger than f1will attract f1if the ratio f1/fmfalls into the
attraction region of the map (red color). If the ratio between
f2/fmfalls in the region where there is no attraction (blue
color) , adding a positive masking signal of frequency fm
will separate the two signal f1and f2and the first IMF will
have a controlled mode mixing of the signals f1and fm. To
separate f1and fma negative masking signal may be added,
and by averaging the two first IMFs the new IMF will be a
signal of frequency f1. However, depending on how close the
two frequencies f1and f2are, some amplitude modulation
will be observed between the signals f1and f2. To identify
the frequencies involved in the original signal, a Fast Fourier
Transform (FFT) can be used as first screening tool. In this
paper, a technique has been developed to identify the involved
instantaneous frequencies and amplitudes, to assist in choosing
the right masking signal. Assume a signal xdefined by:
x=Asin(2πf1t) + Bsin(2πf2t)(1)
After a standard EMD, these two signals will be mixed into
one IMF. After a Hilbert-transform of the mode mixed IMF
(s=x+jy) followed by an amplitude and an instantaneous
frequency calculation, the required information is available for
identifying the amplitudes and frequencies of the two signals
involved. The instantaneous frequency used here is defined by:
f=1
2π
∂φ
∂t (2)
where:
tan φ=y
x(3)
φ= arctan y
x(4)
From the amplitudes of the Hilbert-transformed signal, the
following expressions can be derived:
Kmin =pA2+B22AB = (AB)(5)
Kmax =pA2+B2+ 2AB = (A+B)(6)
Similarly, the expressions for the extreme values of the
instantaneous frequency plots are:
F min =Af
(A+B)+f2(7)
F max =Af
(AB)+f2(8)
Kmin Minimum value of the amplitude plot
Kmax Maximum value of the amplitude plot
Fmin Minimum value of the instantaneous frequency plot
Fmax Maximum value of the instantaneous frequency plot
fDifference between the two frequencies (f1f2)
It is also demonstrated that fis equivalent to the number
of peaks/second in the instantaneous frequency and the ampli-
tude plots. The frequencies f1and f2can now be calculated
and may be further validated with FFT calculation.
In the case of synthetic signals, these calculations are
accurate and in principle the signal components could have
been calculated directly by applying the above presented tech-
nique. However, for real signals the instantaneous amplitude
and frequency functions are less smooth. Still they reveal
information about the amplitudes and frequencies involved in
the different periods of a mode mixed signal and from this,
an optimal mask signal based on the map can be constructed.
III. CAS E STU DY
To demonstrate the method presented in this paper, a syn-
thetic signal consisting of the following components is chosen:
x= 0.7 sin(2π8t)+0.7 sin(2π24t)+1.4 sin(2π30t)(9)
The individual components and the final signal are shown
in Figure 2.
After separation with a standard EMD, the IMFs are shown
in Figure 3. The first plot corresponds to the signal. The first
IMF is the mix of the signals 24Hz and 30 Hz while the second
IMF is the 8Hz signal well separated. The other IMFs reported
are not relevant and only a result of the endpoint conditions,
sifting process and applied tolerances.
To identify the frequencies and amplitudes involved in the
first IMF, a Hilbert transform is performed and the instanta-
neous amplitudes and frequencies are calculated. The instanta-
neous amplitudes are shown in Figure 4 and the instantaneous
frequencies in Figure 5. Both figures portray the instantaneous
values from the two IMFs.
To estimate the amplitudes and frequencies of the tones
involved, we need to find the maximum and minimum of
4
Fig. 2. Synthetic signal and its components for case study
Fig. 3. Original signal, two IMFs and residuals after separation with a
standard EMD
the amplitude function and of the instantaneous frequency
function. Additionally we have to find the fwhich is the
number of peaks/second in either the instantaneous amplitude
or the instantaneous frequency plots. The maximum values
extracted from the plots are indicated in Table 1.
TABLE I
MAX IMU M VALUE S OB TAIN ED FR OM INS T. AMPLITUDE AND INS T.
FRE QUE NCY P LOT S
Frequency (Peaks/s) f6
Extreme values Minimum Maximum
Amplitudes 0.74 2.08
Instantaneous Frequencies 27.9 35.5
Using the obtained values in Table I into Equations 5-8,
the estimated amplitudes and frequencies involved in the IMF
with mode mixing are shown in Table II. These values are in
good agreement with the original signal components.
TABLE II
EST IMATE D (AMPLITUDES / FRE QU ENC IES )FRO M MOD EL
Estimated parameters Mode 1 (A and f1) Mode2 (B and f2)
Amplitudes 1.41 0.67
Frequencies 30.06 24.06
Using the estimated values of A and B in Equation 8, the
estimated frequency is: f2= 24.06. The process of extracting
Fig. 4. Instantaneous amplitudes of IMF1and IMF2
Fig. 5. Instantaneous frequencies of the IMF1and IMF2
the IMFs depends on the number of iterations in the sifting
process and may introduce some inaccuracies. This is verified
by applying the technique directly on the signal, which gives
exact values. For a synthetic signal, the separation could
have been done directly. When applied in conjunction with
EMD, the purpose is to extract amplitudes and frequencies
for different parts of the signal to assist in constructing an
optimal masking signal.
Based on the values obtained in Table II and following
the recommendation of amplitude and frequency ratios
observed in the Boundary Map for mode mixing f1/fm>
0.67 (red/attraction area) and separation f2/fm<< 0.67
(blue/separation area), an appropriate masking signal is
chosen. Generally a good compromise is to keep f1/fm>
0.7 and f2/fm<0.6 if possible. According to the map,
a value of 2.5 for the amplitude will be appropriate as
log10(f1/fm) = 0.25 and log10(f2/fm) = 0.4. With
these values, IMF1after masking, will be located in the
attraction area while IMF2in the separation areas (blue color).
Now, using the chosen masking signal, a new separation
with EMD is conducted:
xm= 2.5cos(2πfmt)(10)
Following the step by step procedure described in section II,
the IMFs obtained with this procedures are shown in Figure
6.
5
Fig. 6. IMFs after applying the masking signal defined by the proposed
method
Fig. 7. Instantaneous amplitudes of IMF1and IMF2after proposed masking
signal
The three spectral components are now well separated.
However, some amplitude modulation is observed on the 24
Hz and 30 Hz signals. When increasing the masking signal
frequency, the amplitude modulation moves from the 30 Hz
signal to the 24 Hz signal, while reducing the frequency will
move the amplitude modulation to the 30Hz signal as the
attraction is stronger according to the Boundary Map.
A Hilbert-transform is now performed on the two first
IMFs followed by instantaneous amplitude and instantaneous
frequency calculations. The results are shown in Figures 7 and
8.
Fig. 8. Instantaneous frequencies of IMF1and IMF2after proposed masking
signal
The instantaneous amplitude plots clearly shows the am-
plitude modulation and the presence of the other frequency
component in both, IMF1and IMF2. As the instantaneous
frequency plots are less smooth, it is more difficult to identify
the real extreme values needed in Equations 7 and 8. However,
in this case we extract values from both IMF1and IMF2. The
extracted values are shown in Table III.
TABLE III
EXT REM E VALUE S FRO M INS T. AMPLITUDE AND INS T. FRE QUE NC Y
PL OTS
Frequency (Peaks/s) f6
Extreme values Minimum Maximum
IMF1: Amplitudes 1.0 1.6
IMF1: Instantaneous Frequencies 28.8 31.79
IMF2: Amplitudes 0.32 0.55
IMF2: Instantaneous Frequencies 21.4 26.9
Using the extracted values from Table III into Equations
5-8, the estimated amplitudes and frequencies involved in the
IMF1and IMF2are shown in Table IV.
TABLE IV
EST IMATE ( AMPLITUDES / FREQ UEN CI ES)F ROM MO DE L
Estimated parameters Mode 1 (A and f1) Mode2 (B and f2)
IMF1: Amplitudes 1.3 0.3
IMF1: Frequencies 29.96 23.96
IMF2: Amplitudes 0.12 0.43
IMF2: Frequencies 29.7 23.7
With reference to Equation 1, and the estimated values from
Table IV, the IMF1and IMF2are estimated by:
x1= 1.3 sin(2π29.96t)+0.3 sin(2π23.96t)(11)
x2= 0.12 sin(2π29.96t)+0.43 sin(2π23.96t)(12)
The frequency components should be exactly the same (in
Table IV) but since the extreme values of the instantaneous
frequency were identified with low reliability, this can explain
the difference. The plot of the estimates of the IMF1(red
curve) together with the IMF1from the EMD (blue curve) are
shown in Figure 9 for a period of 0.5 second. A very good
agreement is also observed for the period of 2 seconds.
A similar plot is shown in Figure 10 for the IMF2.
For IMF2, the agreement between the model and the IMF
was also very good. In principle, the two corresponding
components could have been added to get two pure sinusoidal
functions of frequency 30 Hz and 24 Hz without any ampli-
tude modulation. However, the phase shift of the sinusoidal
functions are not accurately accounted for and therefore the
conclusion could be misleading.
A. Summary of Findings
This research aimed at enhancing the ability of the Empir-
ical Mode Decomposition to separate closely spaced spectral
tones. When two tones are in the same octave (the ratio
between the f2/f1>0.5), the standard EMD will generate
IMFs that are not mono-components. The main findings prove
6
Fig. 9. First IMF and estimated model - plot for 0.5 s
Fig. 10. Second IMF and estimated model - plot for 0.5 seconds
that by using the Boundary Map presented in [3], it is possible
to construct a masking signal which separates closely spectral
tones. Even cases where the ratio f2/f1exceeds 0.8, are
separated with limited amplitude modulation. Further, it is
found that using the instantaneous amplitudes and instan-
taneous frequencies obtained from the Hilbert-transform of
the signal, it is possible to identify the frequencies and the
amplitudes of the tones included in IMF with mode mixing.
This information can be used to propose an optimal masking
signal for separation of the closely spaced tones. In the case
where the the signal cannot be separated without amplitude
modulation, it is demonstrated how the same frequency and
amplitude identification process can be used to find the am-
plitude and the frequencies of the amplitude modulated IMFs.
This information may in specific cases be used to create mono-
component IMFs.
IV. DISCUSSION
The work has essentially proposed a new method for sepa-
rating closely spaced spectral tones using EMD. The principle
was demonstrated and validated using synthetic signals. The
model estimation process uses essentially local information
obtained as a result of the EMD and Hilbert Transform
process (number of peaks/s, extreme values of instantaneous
amplitudes and instantaneous frequencies). The information is
used to estimate the properties of the signal after a standard
EMD has produced IMFs that may be mono-component or
mixed mode IMFs.
Further research in this topic is being dedicated to the
definition of an optimal masking signal obtained from an
optimization formulation aimed at minimizing the amplitude
modulation.
ACKNOWLEDGMENT
The authors would like to thank NTNU summer interns
Francois Beline and Clara Renard for their contribution with
the EMD study and the reconstruction of the Boundary Map
in [3] used for defining the right value of the masking signal.
REFERENCES
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analysis, Proc. of the Royal Society of London A: Math., Physical and
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[2] R. Deering and J. F. Kaiser, The use of a masking signal to improve
empirical mode decomposition, in Proc. IEEE Int. Conf. Acoust., Speech,
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[3] G. Rilling and P. Flandrin, One or two frequencies? The empirical mode
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[7] Norden E. Huang, Zhao-Hua Wu and Jia-Rong Yeh, System and Method
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[8] Maans Klingspor: Hilbert transform: Mathematical theory
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portal.org/smash/record.jsf?pid=diva2
Olav Bjarte Fosso is Professor at the Department
of Electric Power Engineering of the Norwegian
University of Science and Technology (NTNU). He
has previously held positions as Scientific Advisor
and Senior Research Scientist at SINTEF Energy
Research, Head of the Department of Electric Power
Engineering 2009 2013 and Director of NTNUs
Strategic Thematic Area Energy from September
2014 - September 2016. He has been Chairman of
CIGRE SC C5 Electricity Markets and Regulation
and Member of CIGRE Technical Committee (2008
2014), Chairman of the board of NOWITECH (Norwegian Research Centre
for Offshore Wind Technology) 2015-2017 and currently Board member of
Energy21 (Norwegian National Strategy for Research, Development, Demon-
stration and Commercialization of New Energy Technology). He has been
expert evaluator in Horizon2020 and in a number of science foundations,
internationally. His research activities involve hydro scheduling, market in-
tegration of intermittent generation and signal analysis for study of power
systems dynamics and stability.
7
Marta Molinas (M97) received the Diploma degree
in electromechanical engineering from the National
University of Asuncin, San Lorenzo, Paraguay, in
1992, the M.E. degree from the University of the
Ryukyus, Nishihara, Japan, in 1997, and the D.Eng.
degree from the Tokyo Institute of Technology,
Tokyo, Japan, in 2000. She was a Guest Researcher
with the University of Padova, Padua, Italy, in 1998.
From 2004 to 2007, she was a Post-Doctoral Re-
searcher with the Norwegian University of Science
and Technology (NTNU), Trondheim, Norway. She
was a JSPS Fellow at AIST in Tsukuba from 2008 to 2009. From 2008
to 2014, she was Professor with the NTNU Department of Electric Power
Engineering. Since August 2014, she is a Professor with the NTNU De-
partment of Engineering Cybernetics. Her current research interests include
stability of power electronics systems, harmonics, instantaneous frequency,
and non-stationary signals from the human and the machine. Dr. Molinas
is a Member of the IEEE Power Electronics Society, where she currently
serves as an Associate Editor of the IEEE TRANSACTIONS ON POWER
ELECTRONICS, the IEEE POWER ELECTRONICS LETTERS, and the
IEEE JOURNAL OF EMERGING AND SELECTED TOPICS IN POWER
ELECTRONICS. She has, in 2016, become Editor of the JOURNAL ON
ADVANCES IN DATA SCIENCE AND ADAPTIVE ANALYSIS, World
Scientific. She is a member of the IEEE Industrial Electronics Society and
Power Engineering Society.
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A multi-dimensional ensemble empirical mode decomposition (MEEMD) for multi-dimensional data (such as images or solid with variable density) is proposed here. The decomposition is based on the applications of ensemble empirical mode decomposition (EEMD) to slices of data in each and every dimension involved. The final reconstruction of the corresponding intrinsic mode function (IMF) is based on a comparable minimal scale combination principle. For two-dimensional spatial data or images, f(x,y), we consider the data (or image) as a collection of one-dimensional series in both x-direction and y-direction. Each of the one-dimensional slices is decomposed through EEMD with the slice of the similar scale reconstructed in resulting two-dimensional pseudo-IMF-like components. This new two-dimensional data is further decomposed, but the data is considered as a collection of one-dimensional series in y-direction along locations in x-direction. In this way, we obtain a collection of two-dimensional components. These directly resulted components are further combined into a reduced set of final components based on a minimal-scale combination strategy. The approach for two-dimensional spatial data can be extended to multi-dimensional data. EEMD is applied in the first dimension, then in the second direction, and then in the third direction, etc., using the almost identical procedure as for the two-dimensional spatial data. A similar comparable minimal-scale combination strategy can be applied to combine all the directly resulted components into a small set of multi-dimensional final components. For multi-dimensional temporal-spatial data, EEMD is applied to time series of each spatial location to obtain IMF-like components of different time scales. All the ith IMF-like components of all the time series of all spatial locations are arranged to obtain ith temporal-spatial multi-dimensional IMF-like component. The same approach to the one used in temporal-spatial data decomposition is used to obtain the resulting two-dimensional IMF-like components. This approach could be extended to any higher dimensional temporal-spatial data.
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Empirical mode decomposition (EMD) provides a new method for analyzing signals from a nonlinear viewpoint. EMD is defined by an algorithm requiring experimental investigation instead of rigorous mathematical analysis. We show that EMD yields its own interpretation of combinations of pure tones. We present the problem of mode mixing and give a solution involving a masking signal. The masking signal method also allows EMD to be used to separate components that are similar in frequency that would be inseparable with standard EMD techniques.
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This paper investigates how the empirical mode decomposition (EMD), a fully data-driven technique recently introduced for decomposing any oscillatory waveform into zero-mean components, behaves in the case of a composite two-tones signal. Essentially two regimes are shown to exist, depending on whether the amplitude ratio of the tones is greater or smaller than unity, and the corresponding resolution properties of the EMD turn out to be in good agreement with intuition and physical interpretation. A refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments. The analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the EMD are assessed.
System and Method of Conjugate Adaptive Conjugate Masking Empirical Mode Decomposition
  • E Norden
  • Zhao-Hua Huang
  • Jia-Rong Wu
  • Yeh
Norden E. Huang, Zhao-Hua Wu and Jia-Rong Yeh, System and Method of Conjugate Adaptive Conjugate Masking Empirical Mode Decomposition, U.S. Patent 2017/0116155 A1, April 27, 2017