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Journal ArticleDOI

Complex Empirical Mode Decomposition

15 Jan 2007-IEEE Signal Processing Letters (IEEE)-Vol. 14, Iss: 2, pp 101-104
TL;DR: A method for the empirical mode decomposition (EMD) of complex-valued data is proposed based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum.
Abstract: A method for the empirical mode decomposition (EMD) of complex-valued data is proposed. This is achieved based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum. The so-generated intrinsic mode functions (IMFs) are complex-valued, which facilitates the extension of the standard EMD to the complex domain. The analysis is supported by simulations on both synthetic and real-world complex-valued signals

Summary (2 min read)

I. INTRODUCTION

  • T HE empirical mode decomposition (EMD) is a novel signal analysis tool, whereby the underlying notion of instantaneous frequency provides insight into the time-frequency signal features.
  • Unlike other signal decomposition techniques, which map the signal space onto a space spanned by a predefined basis, the idea behind this method is to decompose a general data set into a number of "basis functions" termed intrinsic mode functions (IMFs), which are derived directly from the data, in a natural way.
  • Notice that an extension of EMD to the complex domain is not trivial; this is due to the mutual dependence between the real Manuscript received May 12, 2006; revised June 19, 2006.
  • The problem they introduce is that in this way, a complex, bivariate quantity with some existing mutual information between the real and imaginary part is mapped onto two independent real-valued univariate quantities where this mutual information is lost.

II. EMPIRICAL MODE DECOMPOSITION AND THE CONCEPT OF INSTANTANEOUS FREQUENCY

  • The EMD aims at representing an arbitrary signal via a number of IMFs and the residual .
  • 3) Find an "envelope," (resp. ) that interpolates all local minima (resp. maxima).
  • Once the first IMF is obtained, to obtain the next IMF, the above procedure is applied to the residual .
  • The set of so-obtained IMFs, in fact, represents a unique "time-frequency" analyzer that allows for the analysis of the instantaneous frequency, defined via the Hilbert transform [6] .
  • Unfortunately, this method is defined only for real-valued data.

III. EMPIRICAL MODE DECOMPOSITION FOR COMPLEX-VALUED DATA

  • To derive complex EMD, the authors first decompose a complexvalued data set into its positive and negative frequency components, whereby either component becomes an analytic signal.
  • There are two possibilities to obtain a desired real time sequence from .
  • Let be an ideal band-pass filter specified by ( 4) From ( 4), the authors can generate two analytic signals where symbol denotes the operator that extracts the real part of a complex function.
  • Such a procedure can be expressed as EQUATION where and denote sets of IMFs corresponding, respectively, to and , and and denote the associated residuals.
  • The authors next define the th complex IMF of a complex process (that is a complex-valued counterpart of a real IMF) as ( 12) By using complex-valued IMFs from (11), the proposed complex EMD can be described as (13) where residual , for instance, may represent a trend within the data set.

A. Complex EMD as a Complex Filter Bank

  • Following the approach from [5] , the authors first analyze the power spectra of IMFs generated from random complex-valued time series.
  • A set of 1000 independent Gaussian complex-valued time series of 1024 samples each were generated, and the resulting spectra were averaged.
  • Fig. 1 depicts these averaged spectra for the first seven IMFs in both the positive and negative frequency range.
  • Observe from Fig. 1 that, on the average, the complex EMD exhibits the desired behavior of a filter bank.
  • This result conforms with their proposed complex EMD being a natural extension of the standard EMD.

B. Decomposition of Complex-Valued Wind Signal

  • To further support the analysis, experiments were conducted by applying the complex EMD to real-world wind measurements 1 consisting of wind speed and wind direction , which can be represented by a single complex variable: (for more detail of this representation, see [7] ).
  • Again, the major difference between the complex EMD and its real counterpart is the existence of positive and negative frequencies.
  • Fig. 3 shows the IMFs from the signal components with positive/negative frequency, obtained by (11)-(13), and the corresponding Hilbert-Huang spectra of the wind data plotted in Fig. 2 .
  • Observe from Fig. 3 that IMFs generated using the proposed complex EMD do indeed have a physical meaning and represent the instantaneous frequency and power of the analyzed data.

V. CONCLUSION

  • The authors have introduced the complex EMD method, which represents an extension of the real-valued EMD to the complex domain.
  • This has been achieved based on some inherent properties of complex signals, such as the relationship between their positive and negative frequency components.
  • This way, the authors have been able to apply standard EMD to the corresponding analytic components of complex-valued data.
  • Examples illustrating the operation of the proposed complex EMD as a filter bank, and also the potential of complex EMD when processing real-world complex-valued data, support the analysis.

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IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 2, FEBRUARY 2007 101
Complex Empirical Mode Decomposition
Toshihisa Tanaka and Danilo P. Mandic
Abstract—A method for the empirical mode decomposition
(EMD) of complex-valued data is proposed. This is achieved based
on the filter bank interpretation of the EMD mapping and by
making use of the relationship between the positive and negative
frequency component of the Fourier spectrum. The so-generated
intrinsic mode functions (IMFs) are complex-valued, which facili-
tates the extension of the standard EMD to the complex domain.
The analysis is supported by simulations on both synthetic and
real-world complex-valued signals.
Index Terms—Complex-valued signals, empirical mode de-
composition (EMD), nonlinear signal analysis, time-frequency
analysis.
I. INTRODUCTION
T
HE empirical mode decomposition (EMD) is a novel
signal analysis tool, whereby the underlying notion of in-
stantaneous frequency provides insight into the time-frequency
signal features. This technique has been first introduced in
ocean research [1] and has since become an established tool
for the analysis of nonstationary and nonlinear data [1] with a
number of important applications in signal processing [2]–[4].
Unlike other signal decomposition techniques, which map
the signal space onto a space spanned by a predefined basis,
the idea behind this method is to decompose a general data
set into a number of “basis functions” termed intrinsic mode
functions (IMFs), which are derived directly from the data, in
a natural way. In spite of the well-established and understood
EMD-based analysis of real-valued processes, a major issue
that prevents a wider application of EMD in signal processing
is that this concept has been developed strictly for real-valued
data. On the other hand, several important signal processing
areas (telecommunications, sonar, radar, to mention but a few)
use complex-valued data structures. To analyze these within
the EMD framework, it is necessary to develop an extension
of the standard EMD suitable for dealing with complex-valued
data. In addition, a strong motivation for the development of
the EMD for complex-valued data comes from the concept of
so-called
instantaneous frequency [1], which gives EMD an
edge over other established time-frequency analyzers.
Notice that an extension of EMD to the complex domain is
not trivial; this is due to the mutual dependence between the real
Manuscript received May 12, 2006; revised June 19, 2006. This work was
supported in part by the a Royal Society U.K. International Exchange grant.
The associate editor coordinating the review of this manuscript and approving
it for publication was Dr. Xiang-Gen Xia.
T. Tanaka is with the Department of Electrical and Electronic Engineering,
Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan and
also with the Laboratory for Advanced Brain Signal Processing, RIKEN Brain
Science Institute, Saitama 351-0198, Japan (e-mail: tanakat@cc.tuat.ac.jp).
D. P. Mandic is with the Department of Electrical and Electronic Engineering,
Imperial College London, London SW7 2AZ, U.K. (E-mail: d.mandic@impe-
rial.ac.uk).
Digital Object Identifier 10.1109/LSP.2006.882107
and imaginary component of complex data and the fact that arith-
meticoperationson
formanalgebra.Asimplewaytoextendthe
EMD to the field of complex numbers
would be to apply EMD
separately to the real and imaginary part of a complex-valued
signal. Alternatively, consider the amplitude-phase representa-
tion of complex quantities, and apply EMD separately to the am-
plitude and the phase function. Although these approaches may
at first seem appealing, the problem they introduce is that in this
way, a complex, bivariate quantity with some existing mutual in-
formationbetweentherealandimaginarypart ismappedontotwo
independent real-valued univariate quantities where this mutual
information is lost. Moreover, the power of EMD comes from the
possibility of physical interpretation of the IMFs within EMD,
which in the above case is not applicable.
To develop our proposed complex EMD, we first introduce
the concept of complex-valued IMFs, for which we make use
of the fact that EMD behaves stochastically as a filter bank
[5]. More precisely, when the nature of IMFs within the EMD
framework is stochastic, the EMD behaves as a dyadic sub-
band decomposition structure. We embark upon this result and
derive a complex EMD method, for which the only require-
ment is that the complex EMD preserves the desired property
of a filter bank “on the average.” This is achieved by oper-
ating directly in
, which opens up the possibility to divide a
complex signal in hand into its positive and negative frequency
parts. This provides the basis for a subsequent application of the
standard EMD. To illustrate the proposed complex EMD oper-
ating as a filter bank, examples on randomly generated com-
plex signals support the analysis. In addition, simulations com-
bining the complex EMD and the instantaneous frequency anal-
ysis method termed Hilbert–Huang spectrum [1] are conducted
on real-world complex-valued signals.
II. E
MPIRICAL MODE DECOMPOSITION AND THE
CONCEPT OF INSTANTANEOUS FREQUENCY
The EMD aims at representing an arbitrary signal via a
number of IMFs
and the residual . By design, an IMF
is a function for which the number of extrema and the number
of zero crossings are either equal or they differ at most by one,
together with the mean value of two envelopes associated with
the local maxima and minima being zero. More precisely, for a
real-valued signal
, the EMD performs the mapping
(1)
where
is a set of IMFs, and is the residual. The
first IMF can be obtained as follows [1].
1) Let
.
2) Identify all local minima and maxima of
.
1070-9908/$25.00 © 2006 IEEE

102 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 2, FEBRUARY 2007
3) Find an envelope, (resp. ) that interpo-
lates all local minima (resp. maxima).
4) Extract the detail,
.
5) Let
and go to step 2); repeat until
be-
comes an IMF.
Once the rst IMF is obtained, to obtain the next IMF, the above
procedure is applied to the residual
.In
the same spirit, by applying this procedure recursively, the re-
maining IMFs are calculated. The set of so-obtained IMFs, in
fact, represents a unique time-frequency analyzer that allows
for the analysis of the instantaneous frequency, dened via the
Hilbert transform [6].
For illustration, consider a real-valued signal
, and apply
the Hilbert transform to generate the corresponding analytic
signal (for more information, see [1])
(2)
where
is the Hilbert transform pair,
is the amplitude of , and is its phase given by
. The instantaneous frequency
is derived from as [1]
(3)
It is this combination of the concept of instantaneous fre-
quency and EMD that makes the EMD framework so powerful
for time-frequency signal analysis. To illustrate this, further
consider signal
, which by means of EMD is described
as
, where is the set
of IMFs representing modes, and
is a residual of the
decomposition. This way, we obtain a spectgram [1], given
by
, that provides a new insight into the
time-frequency characteristics of a signal.
Unfortunately, this method is dened only for real-valued
data. Given the importance of the EMD framework, and
the rapidly increasing number of the applications of com-
plex-valued signal processing [7], it would be highly benecial
to nd a natural way to extend the conventional EMD to the
complex domain
.
III. E
MPIRICAL MODE DECOMPOSITION
FOR
COMPLEX-VALUED DATA
To derive complex EMD, we rst decompose a complex-
valued data set into its positive and negative frequency compo-
nents, whereby either component becomes an analytic signal.
Owing to the well-known properties of signal representations in
the complex domain (Fourier), this provides us with an oppor-
tunity to deal with only the real part of such signal and without
loss of information.
Let
be a complex-valued time sequence and
the discrete-time Fourier transform of . There are two
possibilities to obtain a desired real time sequence from
.
First, if
is already an analytic signal, say, for
, we may opt to analyze only the real part of ,
Fig. 1. Complex-valued EMD and the equivalent lters within a lter bank
deduced by complex-valued IMFs
y
[
n
]
.
Fig. 2. Time sequences of wind speed and direction.
since it can be converted back into by using the Hilbert
transform. Unfortunately,
is generally not guaranteed to be
analytic. We therefore propose to extract positive and negative
frequency components from
, as follows. Let be an
ideal band-pass lter specied by
(4)
From (4), we can generate two analytic signals
(5)
(6)
where
denotes the complex conjugate of . Ac-
cording to the Hilbert transform functional relationship, by em-
ploying the inverse Fourier transform, denoted by
,we
obtain
(7)
(8)
where symbol
denotes the operator that extracts the real part
of a complex function. This way, since
and are

TANAKA AND MANDIC: COMPLEX EMPIRICAL MODE DECOMPOSITION 103
Fig. 3. (Left) HilbertHuang spectrum and (right) the corresponding IMFs of complex-valued wind data. (a) Positive frequency. (b) Negative frequency.
now real valued, we can obtain the corresponding IMF using
standard EMD. Such a procedure can be expressed as
(9)
(10)
where
and denote sets of IMFs corre-
sponding, respectively, to
and , and and
denote the associated residuals. The reconstruction of so-de-
composed complex-valued signal is accomplished by
(11)
where
denotes the Hilbert transform operator.
We next dene the
th complex IMF of a complex process
(that is a complex-valued counterpart of a real IMF) as
(12)
By using complex-valued IMFs from (11), the proposed com-
plex EMD can be described as
(13)
where residual
, for instance, may represent a trend within
the data set. This completes the derivation of complex EMD,
which retains the generic form of standard EMD.
In practice, the ideal band-pass lter from (4) for which the
transfer function is
may need to be approximated, using
standard methods from the theory of digital lters [6]. Another

104 IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 2, FEBRUARY 2007
practical issue to consider is the behavior of residual , for
the different approximations of
from (3).
IV. E
XAMPLES
We provide two sets of simulations, an illustration of the be-
havior of complex EMD as a lter bank, followed by a time-fre-
quency analysis of a real-world complex-valued data set.
A. Complex EMD as a Complex Filter Bank
Following the approach from [5], we rst analyze the power
spectra of IMFs generated from random complex-valued time
series. In this experiment, a set of 1000 independent Gaussian
complex-valued time series of 1024 samples each were gen-
erated, and the resulting spectra were averaged. Fig. 1 depicts
these averaged spectra for the rst seven IMFs in both the posi-
tive and negative frequency range. Observe from Fig. 1 that, on
the average, the complex EMD exhibits the desired behavior of
a lter bank. This result conforms with our proposed complex
EMD being a natural extension of the standard EMD.
B. Decomposition of Complex-Valued Wind Signal
To further support the analysis, experiments were conducted
by applying the complex EMD to real-world wind measure-
ments
1
consisting of wind speed
and wind direction ,
which can be represented by a single complex variable:
(for more detail of this representation, see [7]). Again,
the major difference between the complex EMD and its real
counterpart is the existence of positive and negative frequen-
cies. We applied the complex EMD to the complex-valued wind
signal with sampling period of 20 min, which is depicted in
Fig. 2.
Fig. 3 shows the IMFs from the signal components with pos-
itive/negative frequency, obtained by (11)(13), and the cor-
responding HilbertHuang spectra of the wind data plotted in
1
The data examined here are publicly available from [8] and are recorded
by the Automated Weather Observing System (AWOS), Iowa, managed by the
Iowa Department of Transportation.
Fig. 2. Observe from Fig. 3 that IMFs generated using the pro-
posed complex EMD do indeed have a physical meaning and
represent the instantaneous frequency and power of the analyzed
data.
V. C
ONCLUSION
We have introduced the complex EMD method, which repre-
sents an extension of the real-valued EMD to the complex do-
main. This has been achieved based on some inherent proper-
ties of complex signals, such as the relationship between their
positive and negative frequency components. This way, we have
been able to apply standard EMD to the corresponding analytic
components of complex-valued data. Examples illustrating the
operation of the proposed complex EMD as a lter bank, and
also the potential of complex EMD when processing real-world
complex-valued data, support the analysis.
R
EFERENCES
[1] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng,
N.-C. Yen, C. C. Tung, and H. H. Liu, The empirical mode decompo-
sition and the Hilbert spectrum from nonlinear and non-stationary time
series analysis,
Proc. R. Soc. Lond. A, vol. 454, pp. 903995, 1998.
[2] T. M. Rutkowski, F. Vialatte, A. Cichocki, D. P. Mandic, and A. K.
Barros, Auditory feedback for brain computer interface manage-
mentan EEG data sonication approach, in Knowledge-Based
Intelligent Information Engineering Systems, B. Gabrys, R. J. Howlett,
and L. C. Jain, Eds. New York: Springer, 2006, vol. 4253, pp.
12321239.
[3] A. Linderhed, Compression by image empirical mode decomposi-
tion, in Proc. IEEE Int. Conf. Image Processing, Genova, Italy, Sep.
2006, vol. I, pp. 553556.
[4] H. Hariharan, A. Gribok, M. A. Abidi, and A. Koschan, Image fu-
sion and enhancement via empirical mode decomposition, J. Pattern
Recognit. Res., vol. 1, pp. 1632, Jan. 2006.
[5] P. Flandrin, G. Rilling, and P. Gonçalvès, Empirical mode decompo-
sition as a lter bank, IEEE Signal Process. Lett., vol. 11, no. 2, pp.
112114, Feb. 2004.
[6] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Pro-
cessing. Englewood Cliffs, NJ: Prentice-Hall, 1989.
[7] S. L. Goh, M. Chen, D. Popovic, D. Obradovic, K. Aihara, and D.
P. Mandic, Complex-valued forecasting of wind prole, Renew. En-
ergy, 2006, available online.
[8] IEMAWOS 1 minute Data Download. [Online]. Available:
http://mesonet.agron.iastate.edu/request/awos/1min.php. 20012006,
Iowa State Univ., Dept. Agronomy
Citations
More filters
Journal ArticleDOI
TL;DR: The proposed algorithm to use real-valued projections along multiple directions on hyperspheres in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD.
Abstract: Despite empirical mode decomposition (EMD) becoming a de facto standard for time-frequency analysis of nonlinear and non-stationary signals, its multivariate extensions are only emerging; yet, they are a prerequisite for direct multichannel data analysis. An important step in this direction is the computation of the local mean, as the concept of local extrema is not well defined for multivariate signals. To this end, we propose to use real-valued projections along multiple directions on hyperspheres ( n -spheres) in order to calculate the envelopes and the local mean of multivariate signals, leading to multivariate extension of EMD. To generate a suitable set of direction vectors, unit hyperspheres ( n -spheres) are sampled based on both uniform angular sampling methods and quasi-Monte Carlo-based low-discrepancy sequences. The potential of the proposed algorithm to find common oscillatory modes within multivariate data is demonstrated by simulations performed on both hexavariate synthetic and real-world human motion signals.

800 citations


Cites methods from "Complex Empirical Mode Decompositio..."

  • ...(a) Bivariate/complex extensions of EMD The first complex extension of EMD was proposed by Tanaka & Mandic (2006); it employed the concept of analytical signal and subsequently applied standard EMD to analyse complex/bivariate data; however, this method cannot guarantee an equal number of real and…...

    [...]

  • ...Recent multivariate extensions of EMD include those suitable for the processing of bivariate (e.g. Tanaka & Mandic 2006; Altaf et al. 2007; Rilling et al. 2007) and trivariate (Rehman & Mandic in press) signals; however, general original n-variate extensions of EMD are still lacking, and are…...

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Journal ArticleDOI
TL;DR: The empirical mode decomposition is extended to bivariate time series that generalizes the rationale underlying the EMD to the bivariate framework and is designed to extract zero-mean rotating components.
Abstract: The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal, and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin.

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Cites background from "Complex Empirical Mode Decompositio..."

  • ...It is worth noticing that two other bivariate extensions have been introduced very recently [2], [3]....

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Journal ArticleDOI
TL;DR: It is found that, similarly to EMD, MEMD also essentially acts as a dyadic filter bank on each channel of the multivariate input signal, but better aligns the corresponding intrinsic mode functions from different channels across the same frequency range which is crucial for real world applications.
Abstract: The multivariate empirical mode decomposition (MEMD) algorithm has been recently proposed in order to make empirical mode decomposition (EMD) suitable for processing of multichannel signals. To shed further light on its performance, we analyze the behavior of MEMD in the presence of white Gaussian noise. It is found that, similarly to EMD, MEMD also essentially acts as a dyadic filter bank on each channel of the multivariate input signal. However, unlike EMD, MEMD better aligns the corresponding intrinsic mode functions (IMFs) from different channels across the same frequency range which is crucial for real world applications. A noise-assisted MEMD (N-A MEMD) method is next proposed to help resolve the mode mixing problem in the existing EMD algorithms. Simulations on both synthetic signals and on artifact removal from real world electroencephalogram (EEG) support the analysis.

399 citations


Cites methods from "Complex Empirical Mode Decompositio..."

  • ...Algorithm 1: Multivariate Extension of EMD 1: Generate the pointset based on the Hammersley sequence for sampling on an -sphere [9]....

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Journal ArticleDOI
TL;DR: Simulations using real-world case studies illuminate several practical aspects, such as the role of noise in T-F localization, dealing with unbalanced multichannel data, and nonuniform sampling for computational efficiency.
Abstract: This article addresses data-driven time-frequency (T-F) analysis of multivariate signals, which is achieved through the empirical mode decomposition (EMD) algorithm and its noise assisted and multivariate extensions, the ensemble EMD (EEMD) and multivariate EMD (MEMD). Unlike standard approaches that project data onto predefined basis functions (harmonic, wavelet) thus coloring the representation and blurring the interpretation, the bases for EMD are derived from the data and can be nonlinear and nonstationary. For multivariate data, we show how the MEMD aligns intrinsic joint rotational modes across the intermittent, drifting, and noisy data channels, facilitating advanced synchrony and data fusion analyses. Simulations using real-world case studies illuminate several practical aspects, such as the role of noise in T-F localization, dealing with unbalanced multichannel data, and nonuniform sampling for computational efficiency.

359 citations


Cites methods from "Complex Empirical Mode Decompositio..."

  • ...Direct MEMD algorithms were first developed for the bivariate (complex) case and include the complex EMD [25], which exploits univariate analyticity of data channels but does not guarantee coherent bivariate IMFs, and the bivariate EMD (BEMD) [26], which applies standard EMD to multiple data projections and averages the so-obtained local means to yield the true bivariate local mean....

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Journal ArticleDOI
TL;DR: An extension of empirical mode decomposition (EMD) is proposed, which extracts rotating components embedded within the signal and performs accurate time-frequency analysis, via the Hilbert-Huang transform.
Abstract: An extension of empirical mode decomposition (EMD) is proposed in order to make it suitable for operation on trivariate signals. Estimation of local mean envelope of the input signal, a critical step in EMD, is performed by taking projections along multiple directions in three-dimensional spaces using the rotation property of quaternions. The proposed algorithm thus extracts rotating components embedded within the signal and performs accurate time-frequency analysis, via the Hilbert-Huang transform. Simulations on synthetic trivariate point processes and real-world three-dimensional signals support the analysis.

236 citations


Cites background from "Complex Empirical Mode Decompositio..."

  • ...Real-valued EMD [1] aims to adaptively decompose a signal into a finite set of oscillatory components called “intrinsic mode 1053-587X/$26.00 © 2010 IEEE functions” (IMFs)....

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References
More filters
Journal ArticleDOI
TL;DR: In this paper, a new method for analysing nonlinear and nonstationary data has been developed, which is the key part of the method is the empirical mode decomposition method with which any complicated data set can be decoded.
Abstract: A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

18,956 citations

Book
01 Jan 1989
TL;DR: In this paper, the authors provide a thorough treatment of the fundamental theorems and properties of discrete-time linear systems, filtering, sampling, and discrete time Fourier analysis.
Abstract: For senior/graduate-level courses in Discrete-Time Signal Processing. THE definitive, authoritative text on DSP -- ideal for those with an introductory-level knowledge of signals and systems. Written by prominent, DSP pioneers, it provides thorough treatment of the fundamental theorems and properties of discrete-time linear systems, filtering, sampling, and discrete-time Fourier Analysis. By focusing on the general and universal concepts in discrete-time signal processing, it remains vital and relevant to the new challenges arising in the field --without limiting itself to specific technologies with relatively short life spans.

10,388 citations

Journal ArticleDOI
TL;DR: It turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions, and the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.
Abstract: Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.

2,304 citations


"Complex Empirical Mode Decompositio..." refers background in this paper

  • ...Given the importance of the EMD framework, and the rapidly increasing number of the applications of complex-valued signal processing [7], it would be highly beneficial to find a natural way to extend the conventional EMD to the complex domain ....

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TL;DR: This paper presents a novel approach for the simultaneous modelling and forecasting of wind signal components by using novel neural network algorithms and architectures based on modular complex-valued recurrent neural networks (RNNs).

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Journal ArticleDOI
01 Jan 2006
TL;DR: A novel technique for image fusion and enhancement, using Empirical Mode Decomposition, which has devised weighting schemes which emphasize features from both modalities by decreasing the mutual information between IMFs, thereby increasing the information and visual content of the fused image.
Abstract: In this paper, we describe a novel technique for image fusion and enhancement, using Empirical Mode Decomposition (EMD). EMD is a non-parametric data-driven analysis tool that decomposes non-linear non-stationary signals into Intrinsic Mode Functions (IMFs). In this method, we decompose images, rather than signals, from different imaging modalities into their IMFs. Fusion is performed at the decomposition level and the fused IMFs are reconstructed to realize the fused image. We have devised weighting schemes which emphasize features from both modalities by decreasing the mutual information between IMFs, thereby increasing the information and visual content of the fused image. We demonstrate how the proposed method improves the interpretive information of the input images, by comparing it with widely used fusion schemes. Apart from comparing our method with some advanced techniques, we have also evaluated our method against pixelby-pixel averaging, a comparison, which incidentally, is not common in the literature.

91 citations


Additional excerpts

  • ...More precisely, for a real-valued signal , the EMD performs the mapping (1) where is a set of IMFs, and is the residual....

    [...]

Frequently Asked Questions (3)
Q1. What are the contributions in "Complex empirical mode decomposition" ?

In this paper, a method for the empirical mode decomposition ( EMD ) of complex-valued data is proposed based on the filter bank interpretation of the EMD mapping and by making use of the relationship between the positive and negative frequency component of the Fourier spectrum. 

In practice, the ideal band-pass filter from (4) for which the transfer function is may need to be approximated, using standard methods from the theory of digital filters [6]. 

By design, an IMF is a function for which the number of extrema and the number of zero crossings are either equal or they differ at most by one, together with the mean value of two envelopes associated with the local maxima and minima being zero.