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| {{Refimprove|date=July 2010}}
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| The '''Hilbert–Huang transform''' ('''HHT''') is a way to decompose a [[Signal processing|signal]] into so-called intrinsic mode functions (IMF), and obtain [[instantaneous frequency]] data. It is designed to work well for data that is [[Stationary process|nonstationary]] and [[nonlinear]]. In contrast to other common transforms like the [[Fourier transform]], the HHT is more like an algorithm (an empirical approach) that can be applied to a data set, rather than a theoretical tool.
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| == Introduction ==
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| The '''Hilbert–Huang transform''' ('''HHT'''), a [[NASA]] designated name, was proposed by Huang et al. (1996, 1998, 1999, 2003, 2012). It is the result of the empirical mode decomposition (EMD) and the [[Hilbert spectral analysis]] (HSA). The HHT uses the EMD method to decompose a [[Signal processing|signal]] into so-called intrinsic mode function, and uses the HSA method to obtain [[instantaneous frequency]] data. The HHT provides a new method of analyzing [[Stationary process|nonstationary]] and [[nonlinear]] time series data.
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| === Introduction to EMD and IMF ===
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| The fundamental part of the HHT is the '''empirical mode decomposition''' ('''EMD''') method. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components, which is a collection of '''intrinsic mode functions''' ('''IMF'''). An IMF represents a generally simple [[Oscillation|oscillatory]] mode as a counterpart to the simple [[harmonic]] function. By definition, an IMF is any function with the same number of [[Maxima and minima|extrema]] and zero crossings, with its envelopes being symmetric with respect to zero. The definition of an IMF guarantees a well-behaved [[Hilbert transform]] of the IMF. This decomposition method operating in the time domain is [[adaptive behavior|adaptive]] and highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it can be applied to [[nonlinear]] and [[Stationary process|nonstationary]] processes.
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| === Introduction to HSA ===
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| The [[Hilbert spectral analysis]] (HSA) provides a method for examining the IMF's [[instantaneous frequency]] data as functions of time that give sharp identifications of embedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the [[Hilbert spectrum]].
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| == Techniques ==
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| === The empirical mode decomposition (EMD)===
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| The EMD method is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the [[Hilbert spectrum|Hilbert spectral]] analysis can be applied. An IMF is defined as a function that satisfies the following requirements:
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| # In the whole data set, the number of [[maxima and minima|extrema]] and the number of zero-crossings must either be equal or differ at most by one.
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| # At any point, the mean value of the envelope defined by the local [[maxima and minima|maxima]] and the envelope defined by the local [[maxima and minima|minima]] is zero.
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| Therefore, an IMF represents a [[Simple harmonic motion|simple oscillatory mode]] as a counterpart to the simple [[harmonic]] function, but it is much more general: instead of constant amplitude and frequency in a simple [[harmonic]] component, an IMF can have variable amplitude and frequency along the time axis.
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| The procedure of extracting an IMF is called sifting. The sifting process is as follows:
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| # Identify all the local [[maximum and minimum|extrema]] in the test data.
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| # Connect all the local [[maxima and minima|maxima]] by a [[Spline (mathematics)|cubic spline line]] as the upper envelope.
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| # Repeat the procedure for the local [[maxima and minima|minima]] to produce the lower envelope.
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| The upper and lower envelopes should cover all the data between them. Their [[mean]] is m<sub>1</sub>. The difference between the data and m<sub>1</sub> is the first component h<sub>1</sub>: <br />
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| :<math>X(t)-m_1=h_1.\,</math>
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| Ideally, h<sub>1</sub> should satisfy the definition of an IMF, for the construction of h<sub>1</sub> described above should have made it [[symmetry|symmetric]] and having all [[maxima and minima|maxima]] positive and all [[maxima and minima|minima]] negative. After the first round of sifting, a crest may become a local [[maxima and minima|maximum]]. New [[maxima and minima|extrema]] generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, h<sub>1</sub> can only be treated as a proto-IMF. In the next step, it is treated as the data, then<br />
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| :<math>h_{1}-m_{11}=h_{11}.\,</math>
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| After repeated sifting up to k times, h<sub>1</sub> becomes an IMF, that is<br />
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| :<math>h_{1(k-1)}-m_{1k}=h_{1k}.\,</math>
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| Then, it is designated as the first IMF component from the data:
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| :<math>c_1=h_{1k}.\,</math>
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| === The stoppage criteria of the sifting process===
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| The stoppage criterion determines the number of sifting steps to produce an IMF. Two different stoppage criteria have been used traditionally:
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| *1. The first criterion is proposed by Huang et al. (1998). It similar to the [[Cauchy convergence test]], and we define a sum of the difference, SD, as
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| :<math>SD_k=\frac{\sum_{t=0}^{T}|h_{k-1}(t)-h_k(t)|^2}{\sum_{t=0}^{T} h_{k-1}^2 (t)}.\,</math>
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| :Then the sifting process is stop when SD is smaller than a pre-given value.
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| *2. A second criterion is based on the number called the S-number, which is defined as the number of consecutive siftings when the numbers of zero-crossings and [[minima and maxima|extrema]] are equal or at most differing by one. Specifically, an S-number is pre-selected. The sifting process will stop only if for S consecutive times the numbers of zero-crossings and extrema stay the same, and are equal or at most differ by one.
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| Once a stoppage criterion is selected, the first IMF, c<sub>1</sub>, can be obtained. Overall, c<sub>1</sub> should contain the finest scale or the shortest period component of the [[Signal (electronics)|signal]]. We can, then, separate c<sub>1</sub> from the rest of the data by <math>X(t)-c_1=r_1.\,</math> Since the residue, r<sub>1</sub>, still contains longer period variations in the data, it is treated as the new data and subjected to the same sifting process as described above.
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| This procedure can be repeated to all the subsequent r<sub>j</sub>'s, and the result is
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| :<math>r_{n-1}-c_n=r_n.\,</math>
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| The sifting process stops finally when the [[residue (complex analysis)|residue]], r<sub>n</sub>, becomes a [[monotonic function]] from which no more IMF can be extracted. From the above equations, we can induce that
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| :<math>X(t)=\sum_{j=1}^n c_j+r_n.\,</math>
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| Thus, a decomposition of the data into n-empirical modes is achieved. The components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data. Flandrin et al. (2003) and Wu and Huang (2004) have shown that the EMD is equivalent to a dyadic filter bank.
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| === Hilbert spectral analysis ===
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| Having obtained the intrinsic mode function components, the [[instantaneous frequency]] can be computed using the [[Hilbert Transform]]. After performing the [[Hilbert transform]] on each IMF component, the original data can be expressed as the real part, Real, in the following form:
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| :<math>X(t)=\text{Real}{\sum_{j=1}^n a_j(t)e^{i\int\omega_j(t)dt}}.\,</math>
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| == Current applications ==
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| *'''Biomedical applications''': Huang et al. [1999b] analyzed the [[blood pressure|pulmonary arterial pressure]] on conscious and unrestrained [[rat]]s. Bajaj and Pachori (2012) have used EMD for classification of seizure and nonseizure EEG signals.
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| *'''Neuroscience''': Pigorini et al. [2011] analyzed Human EEG response to Transcranial Magnetic Stimulation; Liang et al. [2005] analyzed the visual evoked potentials of macaque performing visual spatial attention task.
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| *'''Chemistry and chemical engineering''': Phillips et al. [2003] investigated a conformational change in [[Brownian dynamics]](BD) and [[molecular dynamics]](MD) simulations using a [[Comparative bullet-lead analysis|comparative analysis]] of HHT and [[wavelet]] methods. Wiley et al. [2004] used HHT to investigate the effect of reversible digitally filtered molecular dynamics(RDFMD) which can enhance or suppress specific frequencies of motion. Montesinos et al. [2002] applied HHT to signals obtained from BWR [[neuron]] stability.
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| *'''Financial applications''': Huang et al. [2003b] applied HHT to nonstationary financial time series and used a weekly mortgage rate data.
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| *'''Image processing''': Hariharan et al. [2006] applied EMD to image fusion and enhancement. Chang et al. [2009] applied an improved EMD to iris recognition, which reported a 100% faster in computational speed without losing accuracy than the original EMD.
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| *'''Meteorological and atmospheric applications''': Salisbury and Wimbush [2002], using Southern Oscillation Index(SOI) data, applied the HHT technique to determine whether the [[Sphere of influence|SOI]] data are sufficiently noise free that useful predictions can be made and whether future [[El Niño-Southern Oscillation|El Nino southern oscillation]](ENSO) events can be predicted from SOI data. Pan et al. [2002] used HHT to analyze [[satellite]] [[scatterometer]] wind data over the northwestern Pacific and compared the results to vector [[empirical orthogonal function]](VEOF) results.
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| *'''Ocean engineering''':Schlurmann [2002] introduced the application of HHT to characterize [[nonlinear]] [[water waves]] from two different perspectives, using laboratory experiments. Veltcheva [2002] applied HHT to wave data from nearshore sea. Larsen et al. [2004] used HHT to characterize the [[underwater]] [[electromagnetic environment]] and identify transient manmade electromagnetic disturbances.
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| *'''Seismic studies''': Huang et al. [2001] used HHT to develop a spectral representation of [[earthquake]] data. Chen et al. [2002a] used HHT to determined the [[Dispersion relation|dispersion]] curves of [[Earthquake engineering|seismic surface]] waves and compared their results to [[Fourier analysis|Fourier-based]] [[time-frequency analysis]]. Shen et al. [2003] applied HHT to ground motion and compared the HHT result with the [[Fourier analysis|Fourier spectrum]].
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| *'''Solar physics''': Barnhart and Eichinger [2010] used HHT to extract the periodic components within [[Sunspots|sunspot]] data, including the 11-year Schwabe, 22-year Hale, and ~100-year Gleissberg cycles. They compared their results with traditional [[Fourier analysis]].
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| *'''Structural applications''': Quek et al. [2003] illustrate the feasibility of the HHT as a signal processing tool for locating an anomaly in the form of a [[Fracture|crack]], [[delamination]], or stiffness loss in beams and plates based on physically acquired propagating wave signals. Using HHT, Li et al. [2003] analyzed the results of a pseudodynamic test of two rectangular reinforced [[concrete]] bridge columns.
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| *'''Health monitoring''': Pines and Salvino [2002] applied HHT in structural health monitoring. Yang et al. [2004] used HHT for damage detection, applying EMD to extract damage spikes due to sudden changes in [[Structural engineering|structural stiffness]]. Yu et al. [2003] used HHT for fault diagnosis of roller bearings. Parey and Pachori (2012) have applied EMD for gear fault diagnosis.
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| *'''System identification''': Chen and Xu [2002] explored the possibility of using HHT to identify the [[Modal analysis|modal]] [[damping ratio]]s of a structure with closely spaced modal frequencies and compared their results to [[Fast Fourier transform|FFT]]. Xu et al. [2003] compared the modal frequencies and [[damping ratio]]s in various time increments and different winds for one of the tallest composite buildings in the world.
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| *'''Speech recognition''': Huang and Pan [2006] have used the HHT for speech pitch determination.<ref>{{ cite journal | author = Huang, H.; Pan, J. | title = Speech pitch determination based on Hilbert-Huang transform | journal = Signal Processing | year = 2006 | volume = 86 | issue = 4 | pages = 792–803 | doi = 10.1016/j.sigpro.2005.06.011 | url = http://spl.mt.ntnu.edu.tw/courses_data/%E7%A2%A9%E5%A3%AB%E7%8F%AD/99/%E6%9E%97%E5%AE%B6%E9%BD%8A/EMD%E8%AB%96%E6%96%87/EMD/%E5%85%B6%E4%BB%96/Speech%20pitch%20determination%20based%20on%20Hilbert-Huang%20transform.pdf | format = pdf }}</ref>
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| == Limitations ==
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| Chen and Feng [2003] proposed a technique to improve the HHT procedure.<ref>{{ cite journal | author = Chen, Y.; Feng M.Q. | title = A technique to improve the empirical mode decomposition in the Hilbert-Huang transform | journal = Earthquake Engineering and Engineering Vibration | year = 2003 | volume = 2 | issue = 1 | pages = 75–85 | doi = 10.1007/BF02857540 | url = http://academiccommons.columbia.edu/download/fedora_content/download/ac:158504/CONTENT/BF02857540.pdf }}</ref> The authors noted that the EMD is limited in distinguishing different components in [[narrowband|narrow-band]] signals. The narrow band may contain either (a) components that have adjacent frequencies or (b) components that are not adjacent in frequency but for which one of the components has a much higher [[energy]] [[intensity (physics)|intensity]] than the other components. The improved technique is based on beating-phenomenon waves.
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| Datig and Schlurmann [2004] did the most comprehensive studies on the performance and limitations of HHT with particular applications to [[irregular]] waves. The authors did extensive investigation into the [[spline interpolation]]. The authors discussed using additional points, both forward and backward, to determine better envelopes. They also performed a [[parametric model|parametric study]] on the proposed improvement and showed significant improvement in the overall EMD computations. The authors noted that HHT is capable of differentiating between time-variant components from any given data. Their study also showed that HHT was able to distinguish between riding and carrier waves.
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| In the US, where patents on algorithms are permitted, the HHT is heavily encumbered by patents in almost all of its domains of possible application {{Citation needed|date=August 2012}}.
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| == See also ==
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| *[[Hilbert transform]]
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| *[[Hilbert spectral analysis]]
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| *[[Hilbert spectrum]]
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| *[[Instantaneous frequency]]
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| *[[Nonlinear]]
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| *[[Wavelet transform]]
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| *[[Fourier transform]]
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| == References ==
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| {{reflist}}
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| * {{ cite journal | author = Ditommaso, R.; Mucciarelli, M.; Parolai, S.; Picozzi, M. | title = Monitoring the Structural Dynamic Response of a Masonry Tower: Comparing Classical and Time-Frequency Analyses | journal = Bulletin of Earthquake Engineering | year = 2012 | volume = 10 | issue = 4 | pages = 1221–1235 | doi = 10.1007/s10518-012-9347-x | url = http://roccoditommaso.xoom.it/index_file/Empirical%20Mode%20Decomposition%20Ditommaso.pdf | format = pdf }}
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| * {{ cite journal | author = Barnhart, B. L.; Eichinger, W. E. | title = Analysis of Sunspot Variability Using the Hilbert-Huang Transform | journal = Solar Physics | year = 2011 | volume = 269 | issue = 2 | pages = 439–449 | doi = 10.1007/s11207-010-9701-6 }}
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| * {{ cite journal | author = Boudraa, A.O.; Cexus, J.C. | title = EMD-Based Signal Filtering | journal = IEEE Transactions on Instrumentation and Measurement | year = 2007 | volume = 56 | issue = 6 | pages = 2196–2202 | doi = 10.1109/TIM.2007.907967 | url = http://ao.boudra.free.fr/publications/R25.pdf | format = pdf }}
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| * {{ cite journal | author = Huang, N. E.; Shen, Z.; Long, S. R.; Wu, M. C.; Shih, H. H.; Zheng, Q.; Yen, N. C.; Tung, C. C.; Liu, H. H. | title = The Empirical Mode Decomposition and the Hilbert Spectrum for Nonlinear and Nonstationary Time Series Analysis | journal = Proceedings of the Royal Society of London A | year = 1998 | volume = 454 | issue = 1971 | pages = 903–995 | doi = 10.1098/rspa.1998.0193 | url = http://keck.ucsf.edu/~schenk/Huang_etal98.pdf | format = pdf }}
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| * {{ cite book | author = Huang, N. E.; Attoh-Okine, N. O. | title = The Hilbert-Huang Transform in Engineering | publisher = CRC Taylor & Francis | year = 2005 | isbn = 978-0849334221 }}
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| * {{ cite book | author = Huang, N. E.; Shen, S. S. P. | title = Hilbert-Huang Transform and its Applications | location = London | publisher = World Scientific | year = 2005 | isbn = 978-9812563767 }}
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| * {{ cite journal | author = Flandrin, P.; Rilling, G.; Gonçalves, P. | title = Empirical Mode Decomposition as a Filterbank | journal = IEEE Signal Processing Letters | year = 2003 | volume = 11 | issue = 2 | pages = 112–114 | doi = 10.1109/LSP.2003.821662 | url = http://perso.ens-lyon.fr/patrick.flandrin/IEEE_SPL2004.pdf | format = pdf }}
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| * {{ cite journal | author = Huang, N. E.; Long, S. R.; Shen, Z. | title = The Mechanism for Frequency Downshift in Nonlinear Wave Evolution | journal = Advances in Applied Mechanics | year = 1996 | volume = 32 | pages = 59–111 | doi = 10.1016/S0065-2156(08)70076-0 }}
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| * {{ cite journal | author = Huang, N. E.; Shen, Z.; Long, R. S. | title = A New View of Nonlinear Water Waves — The Hilbert Spectrum | journal = Annual Review of Fluid Mechanics | year = 1999 | volume = 31 | pages = 417–457 | doi = 10.1146/annurev.fluid.31.1.417 | url = http://authors.library.caltech.edu/8905/1/HUAarfm99.pdf | format = pdf }}
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| * {{ cite journal | author = Huang, N. E.; Wu, M. L.; Long, S. R.; Shen, S. S.; W. D. Qu, W. D.; Gloersen, P.; Fan, K. L. | title = A Confidence Limit for the Empirical Mode Decomposition and Hilbert Spectral Analysis | journal = Proceedings of the Royal Society of London A | year = 2003 | volume = 459 | issue = 2037 | pages = 2317–2345 | doi = 10.1098/rspa.2003.1123 | url = http://rspa.royalsocietypublishing.org/content/459/2037/2317.full.pdf | format = pdf }}
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| * {{ cite journal | author = Wu, Z.; Huang, N. E. | title = A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method | journal = Proceedings of the Royal Society of London A | year = 2004 | volume = 460 | issue = 2046 | pages = 1597–1611 | doi = 10.1098/rspa.2003.1221 | url = http://rspa.royalsocietypublishing.org/content/460/2046/1597.full.pdf | format = pdf }}
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| *{{ cite journal | author = Hariharan H.; Gribok, A.; Abidi, M. A.; Koschan, A. | title = Image Fusion and Enhancement via Empirical Mode Decomposition | journal = Journal of Pattern Recognition Research | year = 2006 | volume = 1 | issue = 1 | pages = 16–31 | url = http://www.jprr.org/index.php/jprr/article/viewFile/6/3 | format = pdf }}
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| * {{ cite journal | author = Chang, J. C.; Huang, M. Y.; Lee, J. C.; Chang, C. P.; Tu, T. M. | title = Iris Recognition with an Improved Empirical Mode Decomposition Method | journal = Optical Engineering | year = 2009 | volume = 48 | issue = 4 | pages = 047007–047007–15 | doi = 10.1117/1.3122322 }}
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| * {{ cite journal | author = Parey, A.; Pachori, R.B. | title = Variable cosine windowing of intrinsic mode functions:
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| Application to gear fault diagnosis | journal = Measurement | year = 2012 | volume = 45 | issue = 3 | pages = 415–426 | doi =10.1016/j.measurement.2011.11.001 }}
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| * {{ cite journal | author = Pigorini, A.; Casali, A.G.; Casarotto, S.; Ferrarelli, F.; Baselli, G.; Mariotti, M.; Massimini, M.; Rosanova, M.C.E. | title = Time-frequency spectral analysis of TMS-evoked EEG oscillations by means of Hilbert-Huang transform | journal = J Neurosci Methods | year = 2011 | volume = 198 | issue = 2 | pages = 236–245 | doi = 10.1016/j.jneumeth.2011.04.013 }}
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| * {{ cite journal | author = Bajaj, V.; Pachori, R.B. | title = Classification of Seizure and Nonseizure EEG Signals Using Empirical
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| Mode Decomposition", | journal = IEEE Transactions on Information Technology in Biomedicine | year = 2012 | volume = 16 | issue = 6 | pages = 1135–1142 | doi = 10.1109/TITB.2011.2181403 }}
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| {{DEFAULTSORT:Hilbert-Huang transform}}
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| [[Category:Signal processing]]
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| [[Category:Telecommunication theory]]
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