The Hilbert–Huang transform (HHT) is a way to decompose a
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
into so-called intrinsic mode functions (IMF) along with a trend, and obtain
instantaneous frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
data. It is designed to work well for data that is
nonstationary and
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
. In contrast to other common transforms like the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
, the HHT is an algorithm that can be applied to a data set, rather than a theoretical tool.
The Hilbert–Huang transform (HHT), a
NASA
The National Aeronautics and Space Administration (NASA ) is an independent agency of the US federal government responsible for the civil space program, aeronautics research, and space research.
NASA was established in 1958, succeedi ...
designated name, was proposed by
Norden E. 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
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
into so-called intrinsic mode functions (IMF) with a trend, and applies the HSA method to the IMFs to obtain
instantaneous frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
data. Since the signal is decomposed in time domain and the length of the IMFs is the same as the original signal, HHT preserves the characteristics of the varying frequency. This is an important advantage of HHT since real-world signal usually has multiple causes happening in different time intervals. The HHT provides a new method of analyzing
nonstationary and
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
time series data.
Definition
Empirical mode decomposition
The fundamental part of the HHT is the empirical mode decomposition (EMD) method. Breaking down signals into various components, EMD can be compared with other analysis methods such as
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
and
Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components. These components form a complete and nearly orthogonal basis for the original signal. In addition, they can be described as intrinsic mode functions (IMF).
Because the first IMF usually carries the most oscillating (high-frequency) components, it can be rejected to remove high-frequency components (e.g., random noise).
EMD based smoothing algorithms have been widely used in seismic data processing, where high-quality seismic records are highly demanded.
Without leaving the time domain, EMD is
adaptive
Adaptation, in biology, is the process or trait by which organisms or population better match their environment
Adaptation may also refer to:
Arts
* Adaptation (arts), a transfer of a work of art from one medium to another
** Film adaptation, ...
and highly efficient.
Since the decomposition is based on the local characteristic time scale of the data, it can be applied to
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
and
nonstationary processes.
Intrinsic mode functions
An intrinsic mode function (IMF) is defined as a function that satisfies the following requirements:
# In the whole data set, the number of
extrema and the number of zero-crossings must either be equal or differ at most by one.
# At any point, the mean value of the envelope defined by the local
maxima and the envelope defined by the local
minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ...
is zero.
It represents a generally simple
oscillatory
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
mode as a counterpart to the simple
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
function. By definition, an IMF is any function with the same number of
extrema and zero crossings, whose envelopes are symmetric with respect to zero.
This definition guarantees a well-behaved
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...
of the IMF.
Hilbert spectral analysis
Hilbert spectral analysis (HSA) is a method for examining each IMF's
instantaneous frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
as functions of time. The final result is a frequency-time distribution of signal amplitude (or energy), designated as the
Hilbert spectrum, which permits the identification of localized features.
Techniques
The Intrinsic Mode Function (IMF) amplitude and frequency can vary with time and it must satisfy the rule below:
# The number of extremes(local maximums & local minimums) and the number of zero-crossings must either equal or differ at most by one.
# At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is near zero.
Empirical mode decomposition
The empirical mode decomposition (EMD) method is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the
Hilbert spectral analysis can be applied.
IMF represents a
simple oscillatory mode as a counterpart to the simple
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
function, but it is much more general: instead of constant amplitude and frequency in a simple
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the '' fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', ...
component, an IMF can have variable amplitude and frequency along the time axis.
The procedure of extracting an IMF is called sifting. The sifting process is as follows:
# Identify all the local
extrema in the test data.
# Connect all the local
maxima by a
cubic spline line as the upper envelope.
# Repeat the procedure for the local
minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ...
to produce the lower envelope.
The upper and lower envelopes should cover all the data between them. Their
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
is ''m''
1. The difference between the data and ''m''
1 is the first component ''h''
1:
:
Ideally, ''h''
1 should satisfy the definition of an IMF, since the construction of h
1 described above should have made it
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
and having all
maxima positive and all
minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ...
negative. After the first round of sifting, a crest may become a local
maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
. New
extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, h
1 can only be treated as a proto-IMF. In the next step, ''h''
1 is treated as data:
:
After repeated sifting up to ''k'' times, h
1 becomes an IMF, that is
:
Then, ''h''
1k is designated as the first IMF component of the data:
:
Stoppage criteria of the sifting process
The stoppage criterion determines the number of sifting steps to produce an IMF. Following are the four existing stoppage criterion:
Standard deviation
This criterion is proposed by Huang et al. (1998). It is similar to the
Cauchy convergence test, and we define a sum of the difference, SD, as
:
:Then the sifting process stops when SD is smaller than a pre-given value.
S Number criterion
This criterion is based on the so-called S-number, which is defined as the number of consecutive siftings for which the number of zero-crossings and
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 siftings, the numbers of zero-crossings and extrema stay the same, and are equal or at most differ by one.
Threshold method
Proposed by Rilling,
Flandrin Flandrin is a French surname. Notable people with the surname include a family of 19th and 20th-century painters:
*Jean-Hippolyte Flandrin (1809–1864), French painter, the best-known of the family
* Paul Flandrin (1811–1902), painter, brother o ...
and Gonçalvés, threshold method set two threshold values to guaranteeing globally small fluctuations in the meanwhile taking in account locally large excursions.
Energy difference tracking
Proposed by Cheng, Yu and Yang, energy different tracking method utilized the assumption that the original signal is a composition of orthogonal signals, and calculate the energy based on the assumption. If the result of EMD is not an orthogonal basis of the original signal, the amount of energy will be different from the original energy.
Once a stoppage criterion is selected, the first IMF, c
1, can be obtained. Overall, c
1 should contain the finest scale or the shortest period component of the
signal
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
. We can, then, separate c
1 from the rest of the data by
Since the residue, r
1, 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.
This procedure can be repeated for all the subsequent r
j's, and the result is
:
The sifting process finally stops when the
residue, r
n, becomes a
monotonic function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
from which no more IMF can be extracted. From the above equations, we can induce that
:
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.
Hilbert spectral analysis
Having obtained the intrinsic mode function components, the
instantaneous frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
can be computed using the
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...
. After performing the Hilbert transform on each IMF component, the original data can be expressed as the real part, Real, in the following form:
:
Current applications
*Improved EMD on ECG signals: Ahmadi et al.
019
Nineteen or 19 may refer to:
* 19 (number), the natural number following 18 and preceding 20
* one of the years 19 BC, AD 19, 1919, 2019
Films
* ''19'' (film), a 2001 Japanese film
* ''Nineteen'' (film), a 1987 science fiction film
Music ...
presented an Improved EMD and compared with other types of EMD. Results show the proposed algorithm provides no spurious IMF for these functions and is not placed in an infinite loop. EMD types comparison on ECG(Electrocardiography) signals reveal the improved EMD was an appropriate algorithm to be used for analyzing biological signals.
*Biomedical applications: Huang et al.
999banalyzed the
pulmonary arterial pressure on conscious and unrestrained
rat
Rats are various medium-sized, long-tailed rodents. Species of rats are found throughout the order Rodentia, but stereotypical rats are found in the genus ''Rattus''. Other rat genera include '' Neotoma'' ( pack rats), '' Bandicota'' (bandico ...
s.
*Neuroscience: Pigorini et al.
011
The following is a list of different international call prefixes that need to be dialled when placing an international telephone call from different countries.
Countries by international prefix
Countries using optional carrier selection code ...
analyzed Human EEG response to Transcranial Magnetic Stimulation; Liang et al.
005
''005'' is a 1981 arcade game by Sega. They advertised it as the first of their RasterScan Convert-a-Game series, designed so that it could be changed into another game in minutes "at a substantial savings". It is one of the first examples of ...
analyzed the visual evoked potentials of macaque performing visual spatial attention task.
*Epidemiology: Cummings et al.
004 004, 0O4, O04, OO4 may refer to:
* 004, fictional British 00 Agent
* 0O4, Corning Municipal Airport (California)
* O04, the Oversea-Chinese Banking Corporation
* Abdul Haq Wasiq, Guantanamo detainee 004
* Junkers Jumo 004 turbojet engine
* Lauda ...
applied the EMD method to extract a 3-year-periodic mode embedded in Dengue Fever outbreak time series recorded in Thailand and assessed the travelling speed of Dengue Fever outbreaks. Yang et al.
010 010 may refer to:
* 10 (number)
* 8 (number) in octal numeral notation
* Motorola 68010, a microprocessor released by Motorola in 1982
* 010, the telephone area code of Beijing
* 010, the Rotterdam
Rotterdam ( , , , lit. ''The Dam on the ...
applied the EMD method to delineate sub-components of a variety of neuropsychiatric epidemiological time series, including the association between seasonal effect of Google search for depression
010 010 may refer to:
* 10 (number)
* 8 (number) in octal numeral notation
* Motorola 68010, a microprocessor released by Motorola in 1982
* 010, the telephone area code of Beijing
* 010, the Rotterdam
Rotterdam ( , , , lit. ''The Dam on the ...
association between suicide and air pollution in Taipei City
011
The following is a list of different international call prefixes that need to be dialled when placing an international telephone call from different countries.
Countries by international prefix
Countries using optional carrier selection code ...
and association between cold front and incidence of migraine in Taipei city
011
The following is a list of different international call prefixes that need to be dialled when placing an international telephone call from different countries.
Countries by international prefix
Countries using optional carrier selection code ...
*Chemistry and chemical engineering: Phillips et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
investigated a conformational change in
Brownian dynamics
Brownian dynamics (BD) can be used to describe the motion of molecules for example in molecular simulations or in reality. It is a simplified version of Langevin dynamics and corresponds to the limit where no average acceleration takes place. Thi ...
and
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of th ...
simulations using a
comparative analysis of HHT and
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
methods. Wiley et al.
004 004, 0O4, O04, OO4 may refer to:
* 004, fictional British 00 Agent
* 0O4, Corning Municipal Airport (California)
* O04, the Oversea-Chinese Banking Corporation
* Abdul Haq Wasiq, Guantanamo detainee 004
* Junkers Jumo 004 turbojet engine
* Lauda ...
used HHT to investigate the effect of reversible digitally filtered molecular dynamics which can enhance or suppress specific frequencies of motion. Montesinos et al.
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
applied HHT to signals obtained from BWR
neuron
A neuron, neurone, or nerve cell is an membrane potential#Cell excitability, electrically excitable cell (biology), cell that communicates with other cells via specialized connections called synapses. The neuron is the main component of nervous ...
stability.
*Financial applications: Huang et al.
003bapplied HHT to nonstationary financial time series and used a weekly mortgage rate data.
*Image processing: Hariharan et al.
006
Alec Trevelyan (006) is a fictional character and the main antagonist in the 1995 James Bond film ''GoldenEye'', the first film to feature actor Pierce Brosnan as Bond. Trevelyan is portrayed by actor Sean Bean. The likeness of Bean as Alec T ...
applied EMD to image fusion and enhancement. Chang et al.
009 009 may refer to:
* OO9, gauge model railways
* O09, FAA identifier for Round Valley Airport
* 0O9, FAA identifier for Ward Field, see List of airports in California
* British secret agent 009, see 00 Agent
* BA 009, see British Airways Flight 9 ...
applied an improved EMD to iris recognition, which reported a 100% faster in computational speed without losing accuracy than the original EMD.
*Atmospheric turbulence: Hong et al.
010 010 may refer to:
* 10 (number)
* 8 (number) in octal numeral notation
* Motorola 68010, a microprocessor released by Motorola in 1982
* 010, the telephone area code of Beijing
* 010, the Rotterdam
Rotterdam ( , , , lit. ''The Dam on the ...
applied HHT to turbulence data observed in the stable boundary layer to separate turbulent and non-turbulent motions.
*Scaling processes with intermittency correction: Huang et al.
008 008, OO8, O08, or 0O8 may refer to:
* The Streetwear Brand @008us , inspired by Ian Fleming & Virgil Abloh
*"030", the fictional 030 Agent of MI6
* '' 038: Operation Exterminate'', a 1965 Italian action film
* '' Explosivo 030'' a 1940 Argentine c ...
has generalized the HHT into arbitrary order to take the intermittency correction of scaling processes into account, and applied this HHT-based method to hydrodynamic turbulence data collected in laboratory experiment,; daily river discharge,; Lagrangian single particle statistics from direct numerical simulation,; Tan et al.,
014 014 may refer to:
* Argus As 014
* BIND-014
* 014 Construction Unit
* Divi Divi Air Flight 014
* Pirna 014
* Tyrrell 014
See also
* 14 (disambiguation)
Fourteen or 14 may refer to:
* 14 (number), the natural number following 13 and preceding 1 ...
vorticity field of two dimensional turbulence,; Qiu et al.
016
HV-016 is a former military unit of Norway, that was a part of the Home Guard. It was established after 1985 to "stop terror- or sabotage actions that could weaken or paralyze Norway's ability to mobilize its military and its ability to resist".
...
two dimensional bacterial turbulence,; Li & Huang
014 014 may refer to:
* Argus As 014
* BIND-014
* 014 Construction Unit
* Divi Divi Air Flight 014
* Pirna 014
* Tyrrell 014
See also
* 14 (disambiguation)
Fourteen or 14 may refer to:
* 14 (number), the natural number following 13 and preceding 1 ...
China stock market,; Calif et al.
013
013 is a music venue in Tilburg, the Netherlands. The venue opened in 1998 and replaced the ''Noorderligt'', the ''Bat Cave'' and the ''MuziekKantenWinkel''. 013 is the largest popular music venue in the southern Netherlands.
There are two concer ...
solar radiation,. A source code to realize the arbitrary order Hilbert spectral analysis can be found at .
*Meteorological and atmospheric applications: Salisbury and Wimbush
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
using Southern Oscillation Index data, applied the HHT technique to determine whether the
Sphere of influence
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military or political exclusivity.
While there may be a formal al ...
data are sufficiently noise free that useful predictions can be made and whether future
El Nino southern oscillation
EL, El or el may refer to:
Religion
* El (deity), a Semitic word for "God"
People
* EL (rapper) (born 1983), stage name of Elorm Adablah, a Ghanaian rapper and sound engineer
* El DeBarge, music artist
* El Franco Lee (1949–2016), American ...
events can be predicted from SOI data. Pan et al.
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
used HHT to analyze
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioiso ...
scatterometer
A scatterometer or diffusionmeter is a scientific instrument to measure the return of a beam of light or radar waves scattered by diffusion in a medium such as air. Diffusionmeters using visible light are found in airports or along roads to measure ...
wind data over the northwestern Pacific and compared the results to vector
empirical orthogonal function
In statistics and signal processing, the method of empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. The term is also interchangeabl ...
results.
*Ocean engineering: Schlurmann
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
introduced the application of HHT to characterize
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
water waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction of ...
from two different perspectives, using laboratory experiments. Veltcheva
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
applied HHT to wave data from nearshore sea. Larsen et al.
004 004, 0O4, O04, OO4 may refer to:
* 004, fictional British 00 Agent
* 0O4, Corning Municipal Airport (California)
* O04, the Oversea-Chinese Banking Corporation
* Abdul Haq Wasiq, Guantanamo detainee 004
* Junkers Jumo 004 turbojet engine
* Lauda ...
used HHT to characterize the
underwater electromagnetic environment In telecommunication, the term electromagnetic environment (EME) has the following meanings:
#For a telecommunications system, the spatial distribution of electromagnetic fields surrounding a given site. The electromagnetic environment may be expr ...
and identify transient manmade electromagnetic disturbances.
*Seismic studies: Huang et al.
001 001, O01, or OO1 may refer to:
*1 (number), a number, a numeral
*001, fictional British agent, see 00 Agent
*001, former emergency telephone number for the Norwegian fire brigade (until 1986)
*AM-RB 001, the code-name for the Aston Martin Valkyrie ...
used HHT to develop a spectral representation of
earthquake
An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves. Earthquakes can range in intensity, from ...
data. Chen et al.
002aused HHT to determine the
dispersion
Dispersion may refer to:
Economics and finance
* Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
* Wage dispersion, the amount of variat ...
curves of
seismic surface waves and compared their results to
Fourier-based time-frequency analysis. Shen et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
applied HHT to
ground motion
Ground motion is the movement of the earth's surface from earthquakes or explosions. Ground motion is produced by seismic waves that are generated by sudden slip on a fault or sudden pressure at the explosive source and travel through the earth ...
and compared the HHT result with the
Fourier spectrum.
*Solar physics: Nakariakov et al.
010 010 may refer to:
* 10 (number)
* 8 (number) in octal numeral notation
* Motorola 68010, a microprocessor released by Motorola in 1982
* 010, the telephone area code of Beijing
* 010, the Rotterdam
Rotterdam ( , , , lit. ''The Dam on the ...
used EMD to demonstrate the triangular shape of quasi-periodic pulsations detected in the hard X-ray and microwave emission generated in
solar flares
A solar flare is an intense localized eruption of electromagnetic radiation in the Sun's atmosphere. Flares occur in active regions and are often, but not always, accompanied by coronal mass ejections, solar particle events, and other sola ...
. Barnhart and Eichinger
010 010 may refer to:
* 10 (number)
* 8 (number) in octal numeral notation
* Motorola 68010, a microprocessor released by Motorola in 1982
* 010, the telephone area code of Beijing
* 010, the Rotterdam
Rotterdam ( , , , lit. ''The Dam on the ...
used HHT to extract the periodic components within
sunspot data, including the 11-year Schwabe, 22-year Hale, and ~100-year Gleissberg cycles. They compared their results with traditional
Fourier analysis.
*Structural applications: Quek et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
illustrate the feasibility of the HHT as a signal processing tool for locating an anomaly in the form of a
crack,
delamination
Delamination is a mode of failure where a material fractures into layers. A variety of materials including laminate composites and concrete can fail by delamination. Processing can create layers in materials such as steel formed by rolling an ...
, or stiffness loss in beams and plates based on physically acquired propagating wave signals. Using HHT, Li et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
analyzed the results of a pseudodynamic test of two rectangular reinforced
concrete
Concrete is a composite material composed of fine and coarse aggregate bonded together with a fluid cement (cement paste) that hardens (cures) over time. Concrete is the second-most-used substance in the world after water, and is the most ...
bridge columns.
*Health monitoring: Pines and Salvino
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
applied HHT in structural health monitoring. Yang et al.
004 004, 0O4, O04, OO4 may refer to:
* 004, fictional British 00 Agent
* 0O4, Corning Municipal Airport (California)
* O04, the Oversea-Chinese Banking Corporation
* Abdul Haq Wasiq, Guantanamo detainee 004
* Junkers Jumo 004 turbojet engine
* Lauda ...
used HHT for damage detection, applying EMD to extract damage spikes due to sudden changes in
structural stiffness. Yu et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
used HHT for fault diagnosis of roller bearings.
*System identification: Chen and Xu
002 002, 0O2, O02, OO2, or 002 may refer to:
Fiction
*002, fictional British 00 Agent
*''002 Operazione Luna'',
*1965 Italian film
*Zero Two, a '' Darling in the Franxx'' character
Airports
*0O2, Baker Airport
*O02, Nervino Airport
Astronomy
*1996 ...
explored the possibility of using HHT to identify the
modal damping ratio
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
s of a structure with closely spaced modal frequencies and compared their results to
FFT
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in th ...
. Xu et al.
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
compared the modal frequencies and
damping ratio
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
s in various time increments and different winds for one of the tallest composite buildings in the world.
*Speech recognition: Huang and Pan
006
Alec Trevelyan (006) is a fictional character and the main antagonist in the 1995 James Bond film ''GoldenEye'', the first film to feature actor Pierce Brosnan as Bond. Trevelyan is portrayed by actor Sean Bean. The likeness of Bean as Alec T ...
have used the HHT for speech pitch determination.
* Astroparticle physics : Bellini et al.
014 014 may refer to:
* Argus As 014
* BIND-014
* 014 Construction Unit
* Divi Divi Air Flight 014
* Pirna 014
* Tyrrell 014
See also
* 14 (disambiguation)
Fourteen or 14 may refer to:
* 14 (number), the natural number following 13 and preceding 1 ...
(Borexino collaboration), Measurement of the seasonal modulation of the solar neutrino fluxes with Borexino experiment, Phys. Rev. D 89, 112007 2014
Limitations
Chen and Feng
003 003, O03, 0O3, OO3 may refer to:
*003, fictional British 00 Agent
*003, former emergency telephone number for the Norwegian ambulance service (until 1986)
*1990 OO3, the asteroid 6131 Towen
* OO3 gauge model railway
*''O03 (O2)'' and other related ...
proposed a technique to improve the HHT procedure. The authors noted that the EMD is limited in distinguishing different components in
narrow-band
Narrowband signals are signals that occupy a narrow range of frequencies or that have a small fractional bandwidth. In the audio spectrum, narrowband sounds are sounds that occupy a narrow range of frequencies. In telephony, narrowband is usu ...
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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
intensity
Intensity may refer to:
In colloquial use
* Strength (disambiguation)
*Amplitude
*Level (disambiguation)
*Magnitude (disambiguation)
In physical sciences
Physics
*Intensity (physics), power per unit area (W/m2)
* Field strength of electric, ma ...
than the other components. The improved technique is based on beating-phenomenon waves.
Datig and Schlurmann
004 004, 0O4, O04, OO4 may refer to:
* 004, fictional British 00 Agent
* 0O4, Corning Municipal Airport (California)
* O04, the Oversea-Chinese Banking Corporation
* Abdul Haq Wasiq, Guantanamo detainee 004
* Junkers Jumo 004 turbojet engine
* Lauda ...
conducted a comprehensive study on the performance and limitations of HHT with particular applications to
irregular water waves. The authors did extensive investigation into the
spline interpolation
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all ...
. The authors discussed using additional points, both forward and backward, to determine better envelopes. They also performed a
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.
Huang and Wu
008 008, OO8, O08, or 0O8 may refer to:
* The Streetwear Brand @008us , inspired by Ian Fleming & Virgil Abloh
*"030", the fictional 030 Agent of MI6
* '' 038: Operation Exterminate'', a 1965 Italian action film
* '' Explosivo 030'' a 1940 Argentine c ...
reviewed applications of the Hilbert–Huang transformation emphasizing that the HHT theoretical basis is purely empirical, and noting that "one of the main drawbacks of EMD is mode mixing". They also outline outstanding open problems with HHT, which include: End effects of the EMD, Spline problems, Best IMF selection and uniqueness. Although the ensemble EMD (EEMD) may help mitigate the latter.
End effect
End effect occurs at the beginning and end of the signal because there is no point before the first data point and after the last data point to be considered together. In most cases, these end points are not the extreme value of the signal. While doing the EMD process of the HHT, the extreme envelope will diverge at the end points and cause significant error. This error distorts the IMF waveform at its endpoints. Furthermore, the error in the decomposition result accumulates through each repetition of the sifting process. Various methods are proposed to solve the end effect in HHT:
*Characteristic wave extending method
*Mirror extending method
*Data extending method
*Similarity searching method
Mode mixing problem
Mode mixing problem happens during the EMD process. Straightforward implementation of sifting procedure produces mode mixing due to IMF mode rectification. Specific signal may not be separated into the same IMFs every time. This problem makes it hard to implement feature extraction, model training and pattern recognition since the feature is no longer fixed in one labeling index. Mode mixing problem can be avoided by including an intermittence test during the HHT process.
[Hilbert-Huang Transform and Its Applications]
*Masking Method
*Ensemble Empirical Mode Decomposition
Ensemble empirical mode decomposition (EEMD)
The proposed Ensemble Empirical Mode Decomposition is developed as follows:
# add a white noise series to the targeted data;
# decompose the data with added white noise into IMFs;
# repeat step 1 and step 2 again and again, but with different white noise series each time;and
# obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result.
The effects of the decomposition using the EEMD are that the added white noise series cancel each other, and the mean IMFs stays within the natural dyadic filter windows, significantly reducing the chance of mode mixing and preserving the dyadic property.
Comparison with other transforms
See also
*
Hilbert transform
In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, of a real variable and produces another function of a real variable . This linear operator is given by convolution with the func ...
*
Hilbert spectral analysis
*
Hilbert spectrum
*
Instantaneous frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also known as local phase or simply phase) of a ''comple ...
*
Multidimensional empirical mode decomposition
In signal processing, multidimensional empirical mode decomposition (multidimensional EMD) is an extension of the one-dimensional (1-D) EMD algorithm to a signal encompassing multiple dimensions. The Hilbert–Huang empirical mode decomposition ( ...
*
Nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
*
Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
*
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
*
Signal envelope
References
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{{DEFAULTSORT:Hilbert-Huang transform
Signal processing
Telecommunication theory