Complex Empirical Mode Decomposition
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.
Did you find this useful? Give us your feedback
Citations
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…...
[...]
504 citations
Cites background from "Complex Empirical Mode Decompositio..."
...It is worth noticing that two other bivariate extensions have been introduced very recently [2], [3]....
[...]
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]....
[...]
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....
[...]
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)....
[...]
References
18,956 citations
10,388 citations
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 ....
[...]
125 citations
"Complex Empirical Mode Decompositio..." refers methods in this paper
...We therefore propose to extract positive and negative frequency components from , as follows....
[...]
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....
[...]
Related Papers (5)
Frequently Asked Questions (3)
Q2. What is the way to describe the IMF?
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].
Q3. What is the definition of an IMF?
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.