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Understanding Instantaneous frequency detection: A discussion of Hilbert-Huang Transform versus Wavelet Transform

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Abstract

Nonlinear and/or nonstationary properties have been observed in measurements coming from microgrids in modern power systems and biological systems. Generally, signals from these two domains are analyzed separately although they may share many features and can bene�t from the use of the same methodology. This paper explores the use of Hilbert-Huang transform (HHT) and Wavelet transform (WT) for instantaneous frequency detection in these two di�erent domains, in the search for a new adaptive algorithm that can be used to analyze signals from these domains without the need to make many a-priory adjustments. Two signals are selected for the investigation: a synthetic signal containing a time varying component and a real EEG signal obtained from The Ecole Polytechnique Federale de Lausanne. The two signals are analyzed with HHT and a discrete WT (DWT). When interpreting the results obtained with the synthetic signal, it is clear that the HHT reproduces the true components, while the DWT does not, making a meaningful interpretation of the modes more challenging. The results obtained when applying HHT to the EEG signal shows 5 modes of oscillations that appear to be well behaved Intrinsic Mode Functions (IMFs), while the results with DWT are harder to interpret in terms of modes. The DWT requires a higher level of decomposition to get closer to the results of the HHT, however multi-frequency bands may be useful depending on the application. The reconstruction of the signal from the approximation and detail coe�cients shows a good behavior and this is one application for DWT especially for removing the unwanted noise of a signal.
Understanding Instantaneous frequency
detection: A discussion of Hilbert-Huang
Transform versus Wavelet Transform
Maximiliano Bueno-L´opez1,2, Marta Molinas2and Geir Kulia3
1Facultad de Ingenier´ıa
Universidad de la Salle, Bogot´a, Colombia
2Department of Engineering Cybernetics
Norwegian University of Science and Technology
Trondheim, Norway
3Signal Analysis Lab, Norway
Abstract. Nonlinear and/or nonstationary properties have been observed
in measurements coming from microgrids in modern power systems and
biological systems. Generally, signals from these two domains are ana-
lyzed separately although they may share many features and can benefit
from the use of the same methodology. This paper explores the use of
Hilbert-Huang transform (HHT) and Wavelet transform (WT) for in-
stantaneous frequency detection in these two different domains, in the
search for a new adaptive algorithm that can be used to analyze signals
from these domains without the need to make many a-priory adjustments.
Two signals are selected for the investigation: a synthetic signal contain-
ing a time varying component and a real EEG signal obtained from The
Ecole Polytechnique Federale de Lausanne. The two signals are analyzed
with HHT and a discrete WT (DWT). When interpreting the results ob-
tained with the synthetic signal, it is clear that the HHT reproduces the
true components, while the DWT does not, making a meaningful inter-
pretation of the modes more challenging. The results obtained when ap-
plying HHT to the EEG signal shows 5 modes of oscillations that appear
to be well behaved Intrinsic Mode Functions (IMFs), while the results
with DWT are harder to interpret in terms of modes. The DWT requires
a higher level of decomposition to get closer to the results of the HHT,
however multi-frequency bands may be useful depending on the applica-
tion. The reconstruction of the signal from the approximation and detail
coefficients shows a good behavior and this is one application for DWT
especially for removing the unwanted noise of a signal.
Keywords: Electrical signals, EEG signals, Empirical mode decompo-
sition, Hilbert-Huang transform, Hilbert spectral analysis, Wavelets
1 Introduction
1.1 Analysis of modern power systems signals
For many years, Fourier-based analyses was enough to study signals in power
systems and the notion of Instantaneous Frequency (IF) has not been previously
2 Understanding Instantaneous frequency detection
explored in such systems. The arrival of new technologies such as distributed
generation, nonlinear loads and power electronic devices has increased the net-
work complexity and caused new problems in power quality. This has called for
new methodologies for analyzing signals which have different characteristics to
previously studied ones [1]. One of the most common techniques used in power
systems for signal analysis is the Fast Fourier Transform (FFT) with its intrinsic
limitation of frequency resolution. Therefore, new methods that can accurately
detect the presence of instantaneous frequency are now necessary. Hilbert-Huang
transform (HHT) and Wavelet Transform (WT) have emerged as options to help
solve this problem [2], [3], [4]. The disturbance detection method for power sys-
tems application has to be capable of dealing with not only harmonic signals but
also nonlinear and non-stationary signals. HHT is an adaptive time-frequency
analysis method which can deal with this type of signals [2], [5]. Compared
with Fourier transform, HHT can analyze and interpret non-stationary and non-
periodic signals [6], [7]. WT is a powerful signal-processing tool that has also
shown the ability to deal with nonstationary signals and in recent years has
taken greater strength in power systems [8]. Usually these techniques are used
independently and sometimes combined to provide better performance. In this
paper, HHT and WT are applied to two different signals: a synthetic signal that
mimics a power system signal and a EEG signal of eye blinks. The observed
results and the interpretation from the application of these two methods are
discussed in the paper.
1.2 Analysis of Electroencephalography (EEG) signals
The human body has been studied using multiple tools and devices. Nowadays,
it is possible to detect diseases, learn about problems and different behaviors
using information taken from the body with different types of sensors. Some
researchers have focused on analyzing information from the heart [9], [10], [11],
but the brain remains one of the organs of greatest interest [12], [13], [14], [15].
Richard Caton discovered electrical currents in the brain in 1875 and Hans Berger
recorded these currents and published the first human Electroencephalogram
(EEG) in 1924 [16]. The EEG is a measure of neural activity and is used to
study cognitive processes, physiology, and complex brain dynamics [17], [18],
[19]. EEG signals are nonstationary. In [15] a method is proposed to quantify
interaction between nonstationary cerebral blood flow velocity (BFV) and blood
pressure (BP) for the assessment of dynamic cerebral autoregulation (CA) using
HHT. In [18] an application was presented to detect the emotions of video viewers
emotions from electroencephalogram (EEG) signals and facial expressions, while
[20] shows the classification of a three-class mental task-based brain-computer
interface (BCI) that uses the Hilbert-Huang transform for feature extraction
and fuzzy particle swarm optimization with cross-mutated-based artificial neural
network (FPSOCM-ANN) for the classifier. In [21] a discrete wavelet transform-
based feature extraction scheme for the classification of EEG signals is described.
The relative wavelet energy is calculated in terms of detailed coefficients and
the approximation coefficients of the last decomposition level. Discrete Wavelet
Understanding Instantaneous frequency detection 3
Transform (DWT) has been widely used to analyze EEG signals. One example of
application in [22], refers to providing information related to epilepsy diagnosis.
2 Instantaneous Frequency detection
The notion of Instantaneous Frequency (IF) has been widely studied [23], be-
cause in practice, signals are not truly sinusoidal and the concept of frequency
must be analyzed in greater depth. Much research has been done on this sub-
ject, however it remains an open debate, since some streams of research deny its
existence. Generally, signals coming from the physical world have been analyzed
using Fourier transform, which gives time invariant amplitude and frequency val-
ues. The inherited uncertainty principle associated with the Fourier transform
makes the concept of an Instantaneous Frequency hard to define. However, this
can be clarified because the uncertainty principle is a consequence of the Fourier
transform (or any other type of integral transform). Thus, if we don’t apply
an integral transform in the frequency computation, we would not be bounded
by the uncertainty principle [24]. Historically, IF was computed from analytic
signals (AS) through the Hilbert transform. HHT estimates the instantaneous
frequency and amplitude of a given signal. To do so, it decomposes any signal
down to monocomponents called intrinsic mode function (IMF) by using the
Empirical mode decomposition (EMD). With the function v(t) defined as
x(t) = r(t) + Xci(t) = r(t) + Xai(t) cos(θi(t)),(1)
where ci(t) is the IMF number i,ai(t) and (θi(t)) are ci(t)0sinstantaneous am-
plitude and phase respectively. The residual r(t) is a monotone function. The
instantaneous frequency fi(t) for each IMF ci(t) is defined by
fi(t),1
2π·i(t)
dt .(2)
3 Hilbert-Huang transform
The HHT was created initially to study ocean waves, which are non-stationary
and non-linear in nature but over time its application has been spread to other
fields. The HHT consits of empirical mode decomposition (EMD) and Hilbert
spectral analysis [5]. In this section, we will show some components and proper-
ties of HHT. The basic structure of HHT is show in Fig. 1.
The main goal of the development of HHT is to have a tool to manage the
time-frequency-energy paradigm of data. The nonlinearity and nonstationarity
can be dealt with better than by using the traditional paradigm of constant
frequency and amplitude. One way to express the nonstationarity is to find
instantaneous frequency and instantaneous amplitude. This was the reason why
Hilbert spectrum analysis was included as a part of HHT. Spectral analysis is
a powerful tool to analyze the statistical characteristics of stochastic data [23].
4 Understanding Instantaneous frequency detection
Signal Input 
Find the local maxima and minima  
Compute the interpolating signals (envelopes
of the signal)  and 
Compute the mean value
e     
Subtract   from the signal  
     
Satisfies the
stopping
criterion
IMF 
 
 
Hilbert Transform
Hilbert Spectrum
NO
YES
YES
NO
Fig. 1. Flowchart for the Hilbert-Huang transform.
A Hilbert Spectrum (HS) is a 3D representation of the instantaneous amplitude
and frequency as a function of time for each Intrinsic Mode Function (IMF).
The HS is defined as
Hi(f, t),(ai(t) for f=fi(t)
0 otherwise (3)
For a general multicomponent signal, the Hilbert Spectrum is defined as the sum
of Hilbert Spectra of all the IMFs, as given in
H(f, t),
N
X
i=1
Hi(f, t) (4)
where Nis the total number of IMFs.
4 Wavelet Transform
The concept of wavelets started to appear in the early 1980s. This new concept
can be viewed as a synthesis of various ideas originating from different disci-
plines. In 1982, the idea of wavelet was discovered the by Jean Morlet. The first
application was focused on seismic wave analysis [25].
Understanding Instantaneous frequency detection 5
The Wavelet Transform (WT) is a widely used tool in signal processing, it is
a quick and efficient way of analyzing transients in voltage and current signals
and characterize EEG signals. WT decomposes a signal into frequency bands,
which are generated by means of mother wavelet and operations of translation
in the time domain. It also responds to the needs of an optimal time-frequency
resolution in all frequency ranges.
In order to achieve good time resolution for the high-frequency transients and
good frequency resolution for the low-frequency components, the idea of mother
wavelet was introduced . They are defined by
ψa,b(t) = 1
p|a|ψtb
a, a, b R, a 6= 0,(5)
where ais called a scaling parameter which measures the degree of compres-
sion or scale, and ba translation parameter which determines the time location
of the wavelet [22].
A wavelet is a function ψ(t) =L2(R) with a zero average
Z
−∞
ψ(t)dt = 0 (6)
The discrete wavelet transform (DWT) is obtained by discretizing the param-
eters aand b. In its most common form, the DWT employs a dyadic sampling
with parameters aand bbased on powers of two a= 2jand b=k2j, with
j, k Z. By substituting the discretized parameters into (5), we obtain the
dyadic wavelets
ψj,k(t) = 1
2jψt
2jk(7)
Therefore, the DWT of a signal s(t) can be written as
dj,k =Z
−∞
s(t)ψ
j,k(t)dt =hs(t), ψj,k (t)i(8)
where dj,k are known as wavelet (or detail) coefficients at level jand location
k.s(t) is the continuous wavelet transform of a signal s(t).
The DWT uses an analysis filter bank to decompose a signal into wavelet
coefficients at various frequency bands. The basic structure of DWT for 3 de-
composition levels is show in Fig. 2.
5 Illustrative Examples
We have carried out some simulations using HHT and the DWT to analyze two
signals whose behavior is easily found in modern power systems and biological
signals. In Fig. 3, we show a signal with two frequency components whose model
is represented by the equation (9)
y(t) = sin(2π50t)+0.5cos(2π70t2),(9)
6 Understanding Instantaneous frequency detection
s(t)% cD1%
cA1%cD2%
cA2%
cD3%
cA3%
s(t)=cD
1+cD2+cD3+cA3
Original%Signal%
cA%%%%%%%%%%%approxima8on%coefficients%%%%%
cD%%%%%%%%%%%detail%coefficients%
Fig. 2. Discrete wavelet transform
and Fig. 4 shows an EEG signal obtained from The Ecole Polytechnique Federale
de Lausanne, Signal Processing Institute [26]. The data matrix used contains
the information of 34 electrodes, for our study we used information from one of
these. The data were recorded using Biosemi Active Two. The sampling rate is
2048Hz. EEG signals were recorded from 10 healthy subjects who participated
in 2 sessions. Using a speller, subjects can spell characters by focusing their
attention on a given character displayed on a computer screen.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Time [ms]
-1
-0.5
0
0.5
1
Voltage [pu]
Original Signal
1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2
Time [ms]
-1
-0.5
0
0.5
1
Voltage [pu]
Zoom in Signal
ZOOM
Fig. 3. Synthetic signal with similar behaviour to modern power systems
Understanding Instantaneous frequency detection 7
0 500 1000 1500
Time [ms]
1.519
1.52
1.521
1.522
1.523
1.524
1.525
Voltage [pu]
×104Original Signal
500 520 540 560 580 600 620 640 660 680 700
Time [ms]
1.519
1.52
1.521
1.522
1.523
1.524
1.525
Voltage [pu]
×104Zoom in Signal
Zoom
Fig. 4. EEG Signal taken under real-world conditions. This means that the data might
contain artifacts coming from eye-blinks, eye-movements, muscle-activity, etc
5.1 Example 1- Modern Power System
In this section, we present the results obtained for the signal in Fig. 3. In Fig. 5
the IMFs and their respective power spectrum are shown, 2 IMFs have been
obtained. The Multilevel Wavelet Decomposition using DWT is shown in Fig. 6.
For this case we used the Morlet function as mother wavelet. Fig. 7 show the HS.
In Fig. 8 the scalogram obtained using Continuous Wavelet Transform (CWT) is
shown. The vertical axis is the scale factor, the higher scale factor corresponds to
lower frequencies. The time-frequency analysis in wavelet transform, it’s a good
tool to know the distribution of signal energy of wavelet details coefficient with
the change of time. In WT exists a trade-off between the order of the wavelet
function and the computation time. Higher order wavelets are better able to
distinguish between the various frequencies, but require more computation time.
5.2 Example 2- EEG Signal
In this section, we present the results obtained for the signal in Fig. 4. In Fig. 9
the IMFs and their respective Fast Fourier Transform (FFT) are shown. The
Multilevel Wavelet Decomposition using DWT is shown in Fig. 10. For this case
we used the Daubechies function as mother wavelet. Finally, in Fig. 11, we show
the original signal and the reconstructed signal using the wavelet coefficients.
6 Discussion
We have used HHT and DWT to a synthetic signal and to a real signal, in
an attempt to better understand the notion of instantaneous frequency and for
8 Understanding Instantaneous frequency detection
Fig. 5. Intrinsic Mode functions of signal y(t)
0 2000 4000 6000 8000 10000
Time(s)
-2
-1
0
1
Amplitud
Approximation coefficient 3
0 2000 4000 6000 8000 10000
Time(s)
-0.1
0
0.1
Amplitud
Detail coefficient 1
0 2000 4000 6000 8000 10000
Time(s)
-0.01
0
0.01
0.02
Amplitud
Detail coefficient 2
0 2000 4000 6000 8000 10000
Time(s)
-5000
0
5000
Amplitud
Detail coefficient 3
0 0.01 0.02 0.03 0.04 0.05
frequency(Hz)
0
5
10
Power
×106Power Spectrum A3
0 0.05 0.1 0.15 0.2
frequency(Hz)
0
10
Power
Power Spectrum D1
0 0.1 0.2 0.3 0.4
frequency(Hz)
0
5
Power
×10-3 Power Spectrum D2
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
0.5
1
Power
×10-4 Power Spectrum D3
Fig. 6. Multilevel Wavelet Decomposition using DWT for signal y(t)
exploring the applicability of these two methods to different applications, that
share nonstationary nature. In the case of the synthetic signal, HHT is clearly
more suitable for decomposition and meaningful interpretation of the monocom-
ponents. The fast oscillations can be separated from the slow oscillations present
in the signal, and this can allow to obtain more information. The HHT has better
Understanding Instantaneous frequency detection 9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Time [s]
0
50
100
150
200
250
300
Frequency [Hz]
Hilbert Spectrum
-9
-8
-7
-6
-5
-4
-3
-2
Power [dB]
c1(t)
c2(t)
Fig. 7. Hilbert Spectrum for signal y(t)
Scalogram
1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Time b
1
14
27
40
53
66
79
92
105
118
131
144
157
170
183
196
209
222
235
248
Scales a
0.5
1
1.5
2
2.5
3
3.5
×10-4
Fig. 8. Scalogram using CWT
resolution in time domain and in frequency domain than the DWT. This makes
the HHT more powerful for detecting the impacts of continuous variations in
signals. The HHT has better computing efficiency, which means that the HHT
is more suitable for large size signal analysis. Fig. 5 shows the different IMFs
obtained for a very common signal in modern power systems, in this case we
only need 2 IMFs to describe the behavior of the phenomenon. When we com-
pare the IMFs and their respective power spectrum, the IMF1 contains many
frequency components and in IMF2 the frequency components are reduced, and
the same happens with oscillations. The DWT applied to the same signal has 3
decomposition levels, which is computationally more demanding. The HS often
10 Understanding Instantaneous frequency detection
Fig. 9. Intrinsic Mode functions for EEG signal
0 2000 4000 6000 8000 10000 12000
Time(s)
1.52
1.525
1.53
Amplitud
×104Approximation coefficient 3
0 2000 4000 6000 8000 10000 12000
Time(s)
-20
0
20
Amplitud
Detail coefficient 1
0 2000 4000 6000 8000 10000 12000
Time(s)
-20
0
20
Amplitud
Detail coefficient 2
0 2000 4000 6000 8000 10000 12000
Time(s)
-10
0
10
Amplitud
Detail coefficient 3
0 2000 4000 6000 8000 10000 12000
Time(s)
-5
0
5
Amplitud
Detail coefficient 4
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
2
4
Power
×1016 Power Spectrum A4
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
5
10
Power
×106Power Spectrum D1
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
2
4
Power
×106Power Spectrum D2
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
1
2
Power
×106Power Spectrum D3
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
5
10
Power
×105Power Spectrum D4
Fig. 10. Multilevel Wavelet Decomposition using DWT
has better resolution than WS, and WS shows a rich distribution of harmon-
ics but does not offer many details about the exact frequency values at which
some disturbances may appear. The frequency content of the signal is descom-
posed into frequency bands (Fig. 6), Approximation coefficient 3 corresponds to
the low frequency component and Detail coefficient 1 corresponds to the high
frequency component. In Detail coefficient 1, it is possible to observe the two
Understanding Instantaneous frequency detection 11
0 2000 4000 6000 8000 10000 12000
Time [ms]
1.518
1.52
1.522
1.524
1.526
1.528
1.53
Voltage [uV]
×104Original Signal
0 2000 4000 6000 8000 10000 12000
Time [ms]
1.518
1.52
1.522
1.524
1.526
1.528
1.53
Voltage [uV]
×104 Signal Reconstructed
Fig. 11. Signal reconstructed using DWT
frequency components of the signal. In the other frequency bands (Detail coef-
ficient 2 and Detail coefficient 3) also appear frequency components but with
lower amplitudes. This behaviour also appears in Fig. 10.
The HS (Fig. 7) reflects the signals instantaneous frequency pattern and
allows to observe with greater clarity frequency trends. With the EEG signal,
we need more IMFs, but they all appear to be well behaved IMFs. The frequency
components are decreasing in each IMF and this is a good approximation for
the detection and classification of patterns. When we compare the behavior of
HHT in each case, it is possible to affirm that this method works well in both
applications. The scalogram (Fig. 8) shows the two frequencies present in the
original signal(red rectangle), but the resolution is lower compared wit HS. The
frequency component of higher value is difficult to identify in the scalogram.
In Fig. 11, we show the reconstructed signal using the wavelet coefficients, this
is one of the great advantages of the DWT. This suggests that a preprocessing
signal using DWT and then apply HHT can offer better results than the two
tools working separately.
The IMF eliminate the high frequency oscillation (Fig. 5 and 9) and this
allows to observe in each component different behaviors. This would allow focus
on a specific component and analyze behaviors in power systems and EEG signals
according to the application or diagnosis.
In EEG signal analysis one application of HHT and WT can be to detect
and characterize sleep spindles (SSs) [12] or Epileptic Focus [22]. The firs step in
this process is to mark by an expert in an EEG recording behavior patterns and
then decompose the signal to obtain a better resolution. In Fig. 9 is possible to
detect changes in the oscillations, for example in Intrinsic Mode Function 2 we
have a strong variation in the envelope. In the DWT is necessary to have more
decomposition levels to observe changes in the signal behaviour.
12 Understanding Instantaneous frequency detection
7 Conclusion and further work
In this paper a comparison between the use of HHT and DWT to two different
nonstationary signals was presented. The results indicate that for signal decom-
position into meaningful modes, the HHT will be a preferable method. One of
the advantages of HHT versus DWT is that by being a data driven method, no
reference functions are required (mother wavelets) and the adaptability of the
method is greater. One of the main challenges of HHT is to improve the calcu-
lation process of the IMFs. The best way for this is to optimize the selection
method. The reconstruction of the signal from the approximation and detail co-
efficients shows a good behavior for DWT especially for removing the unwanted
noise of a signal. Further investigation is currently being focused on the im-
provement of the IMFs extraction algorithm in the EMD. The use of black-box
optimization is going to be explored as an alternative to the cubic splines used
in this paper.
Acknowledgments. This work was carried out during the tenure of an ERCIM
‘Alain Bensoussan’ Fellowship Programme.
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... The concept of wavelets started to appear in the early 1980s and today is a widely used tool in signal processing, it is a quick and efficient way of analyzing transients in voltage and current signals. WT decomposes a signal into frequency bands, which are generated by means of mother wavelet and operations of translation in the time domain [11]: ...
... N 2017, Maximiliano Bueno-Lopez, Marta Molinas and Geir Kulia analyzed the use of HHT and WT for instantaneous frequency detection in two different domains: a synthetic signal and a real EEG signal [11]. The results indicate that for signal decomposition into meaningful modes, the HHT will be a preferable method. ...
... 11: FFTs spectrums of the original signal minus the 50 Hz component's IMFs ...
Thesis
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Modern electrical systems have introduced distributed generations and power converters. This brings out different issues such as low inertia of the grids and an increasing number of harmonics and non linear distortions injected. However, the fluctuations of instantaneous frequency are arguably the most characteristic feature of microgrids. Frequency drift during step load change as well all quasi-periodic fluctuation has an impact on various conditions as generators’ prime movers and governor’s characteristic. Quasi periodic frequency fluctuations can have besides detrimental effect on electrical receivers’ work, especially electric motors will be affected. The thesis aims at providing a deep insight into the phenomena of non-constant instantaneous frequency. Origin of instantaneous frequency fluctuations are analyzed, tools for the phenomenon identification are proposed. These considerations are based on careful analysis of data from the Norwegian power system and behavior of three real marine systems onboard of a ro-ro ship with shaft generator, research-training ship during work of auxiliary generating sets as well as integrated power system of ship with electrical propulsion. For each ship, the cases of rough and calm sea are compared. To analyze data, the empirical mode decomposition algorithm and the Hilbert transform implemented in Matlab will be used.
... For example, Fourier transform is a very easy and quick to use tool, but it only obtains the frequency spectrum and it does not have the capacity to analyze non-stationary signals in time [5], [6]. In order to overcome this delimitation, the STFT is introduced, which allows the satisfactory analysis of stationary signals. ...
... Both amplitude and frequency are functions of time, so it is possible to express the amplitude in terms of a time and frequency function H(ω,t) [6]. However, the frequency obtained from the Hilbert transform does not necessarily have a physical meaning. ...
Conference Paper
Full-text available
Disturbances in power quality have increased due to the use of electronic equipment, causing deviations in current and voltage waveforms, which can cause many failures and damage to equipment used in different demand points. Therefore, an efficient disturbance detection method is required in order to provide relevant information regarding its ocurrence. However, there are many difficulties detecting disturbances throughout traditional data extraction methods. These methods have not been able to perform the detection process with the efficiency, speed and accuracy required for this type of work, due to the non-stationary and non-linear behavior of these disturbances. In this study, the Hilbert-Huang Transform and the Multilayer Perceptron Neural Network model are implemented in order to detect and classify disturbances in power quality. Eight common types of disturbances were analyzed based on the parameters stated in the IEEE 1159 standard. By means of instantaneous frequencies and intrinsic mode functions of each disturbance, the neural network is trained for the classification of these disturbances. The implemented method reached a precision percentage of 94.6, demonstrating the versatility and great potential that this method provides when detecting disturbances in power quality.
... The HHT was created initially to study ocean waves, which are non-stationary and non-linear in nature, but over time its application has been spread to other fields. The HHT consists of the empirical mode decomposition (EMD) and Hilbert spectral analysis [35]. ...
Article
Full-text available
Smart meter (SM) deployment in the residential context provides a vast amount of data that allows diagnose the behavior of household inhabitants. However, the conventional methods to analyze household-consumption load profiles based on time-series techniques, such as Fourier and Wavelet transform, have problems with nonlinear and nonstationary processes. This paper presents a methodology to perform a comprehensive analysis of consumption load profile features based on the detection of oscillation modes in the time-frequency domain for off-line systems. The methodology is based on the Hilbert-Huang transform (HHT) that evaluates the instantaneous frequency (IF) using the empirical mode decomposition (EMD) and some of the variants of this standard technique such as ensemble empirical mode decomposition (EEMD) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). The study presented in this paper compares the accuracy of different techniques by conducting a case study of two households in Spain. Nonetheless, there is a trade-off between the computational burden and accuracy. Furthermore, the methodology reveals the importance of the data sampling frequency (temporal data granularity) for accurate characterization of the load profile. Some household loads produced oscillation modes that became increasingly ill-suited for resolutions of 1 min or higher.
... Al comparar la HHT con la transformada de Fourier, es posible afirmar que la primera tiene la habilidad de detectar patrones de comportamiento con mayor facilidad en señales con fuertes oscilaciones. El análisis mediante WT es bastante poderoso en señales no estacionarias [13], mientras la WVD ha demostrado mejor resolución que la FFT en diferentes aplicaciones. Usualmente cada una de estas estrategias ha sido utilizada de manera independiente y en algunas ocasiones se han visto aplicaciones donde se combinan buscando aprovechar los beneficios individuales de cada herramienta [14], [15]. ...
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Full-text available
The smart grid concept is being applied more and more frequently and this is due to the need to integrate all the components that are part of power systems today, starting from generation units, storage systems, communications and connected loads. Non-linear and non-stationary signals have been obtained in this type of systems, which have high penetration of non-conventional energy sources (NCSRE) and non-linear loads. The power quality criterion has had to be adapted to the new conditions of the electrical systems and this has led to the need to search for new analysis methodologies for the acquired signals. In this article we present a review on non-linear and non-stationary signal analysis methods in electrical systems with high NCSRE penetration. To this end we explore the application of the Hilbert-Huang Transform (HHT), Wavelet Transform (WT) and Wigner-Ville Distribution (WVD), exposing each of the advantages and disadvantages of these methods. To validate the methodology, we have selected some synthetic signals that adequately describe the typical behaviors in these systems.
... En este trabajo se emplea una estrategia de análisis que pueda ofrecer respuestas más adecuadas a las exigencias de sistemas de alta complejidad. La Transformada de Hilbert-Huang es una técnica de análisis en tiempo-frecuencia que utiliza el concepto de frecuencia instantánea lo cual permite observar con una adecuada resolución como la frecuencia va cambiando con el tiempo, y que ofrece una mayor resolución y mejor precisión en eventos transitorios, no lineales y no estacionarios de señales que las técnicas convencionales como las transformadas de Fourier y Wavelet (Bueno-Lopez et al. (2017b), Gasca Segura et al. (2018)). ...
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In communication links between satellites and ground stations, the transmitted signal is exposed to disturbances such as noise of different types that hinder reception or retrieval of information. For that reason, different methods are used to process the received signal and to remove these perturbations, and thus recover the information. Different techniques of signal analysis have been used to achieve the above described but the results are not completely satisfactory. The Hilbert Huang Transform has been applied increasingly in different areas and satellite signals are not the exception. In this work, a study where the Hilbert-Huang transform is used to analyze the received signal and to remove these perturbations, and thus recover the transmitted signal is presented
... The MEMD is a method that reduces the mode mixing problem and it is a good alternative for multichannel data analysis, as is the case of EEG signals. Some previous papers have reported the use of EMD for neural activity reconstruction and other applications in bioengineering Bueno-Lopez et al. (2017a), Bueno-Lopez et al. (2017b), Okcana (2016), Men-Tzung et al. (2008), but the use of MEMD for the same application is less. In Yin et al. (2012) the authors presented a method for data analysis based on MEMD, applying a pre-processing step with Independent Component Analysis (ICA) to calculate and evaluate the energy presented in an EEG record from the quasi brain deaths and to evaluate the brain activity. ...
Preprint
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Several approaches can be used for estimating neural activity. The main differences between them are in the apriori information used and their sensibility to high noise levels. Empirical Mode Decomposition (EMD) has been recently applied to Electroencephalography EEG-based neural activity reconstruction to provide apriori time-frequency information to improve the neural activity estimation. EMD has the specific ability to identify independent oscillatory modes in non-stationary signals with multiple oscillatory components. The various attempts to use EMD in EEG analysis, however, did not provide yet the best reconstructions due to the intrinsic mode mixing problem of EMD. Some previous works have used a single-channel analysis and in other cases, multiple-channel have been used for other applications. In this paper, we present a study about multiple-channel analysis using Multivariate Empirical Mode Decomposition (MEMD) as a method to attenuate the mode mixing problem and to provide apriori useful time-frequency information to the reconstruction of neuronal activity using several low-density EEG electrode montages. The methods were evaluated over real and synthetic EEG data, in which the reconstructions were performed using multiple sparse priors (MSP) method with several electrode numbers of 32, 16, and 8, and the source reconstruction quality was measured using the Wasserstein Metric. Comparing the solutions when no pre-processing was made and when MEMD was applied, the source reconstructions were improved using MEMD as apriori information in the low-density montage of 8 and 16 electrodes. The mean source reconstruction error on a real EEG dataset was reduced a 59.42% and 66.04% for the 8 and 16 electrodes montages respectively, and on a simulated EEG with three active sources, the mean error was reduced an 87.31% and 31.45% for the same electrodes montages.
... EMD does not require any restrictive assumptions on the underlying model of the process/system under analysis and can accommodate both non-linear and non-stationary signals. However, the algorithm has been shown to have limitations in identifying closely-spaced spectral tones and intermittently appearing components in the signal (Bueno-Lopez et al. (2017b)). The aim of the EMD method is to decompose the nonlinear and non-stationary signal y(t k ) into a sum of IMFs that satisfies two conditions (Mandic et al. (2013)): ...
Article
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The localization of active brain sources from Electroencephalogram (EEG) is a useful method in clinical applications, such as the study of localized epilepsy, evoked-related-potentials, and attention deficit/hyperactivity disorder. The distributed-source model is a common method to estimate neural activity in the brain. The location and amplitude of each active source are estimated by solving the inverse problem by regularization or using Bayesian methods with spatio-temporal constraints. Frequency and spatio-temporal constraints improve the quality of the reconstructed neural activity. However, separation into frequency bands is beneficial when the relevant information is in specific sub-bands. We improved frequency-band identification and preserved good temporal resolution using EEG pre-processing techniques with good frequency band separation and temporal resolution properties. The identified frequency bands were included as constraints in the solution of the inverse problem by decomposing the EEG signals into frequency bands through various methods that offer good frequency and temporal resolution, such as empirical mode decomposition (EMD) and wavelet transform (WT). We present comparative analysis of the accuracy of brain-source reconstruction using these techniques. The accuracy of the spatial reconstruction was assessed using the Wasserstein metric for real and simulated signals. We approached the mode-mixing problem, inherent to EMD, by exploring three variants of EMD: masking EMD, Ensemble-EMD (EEMD), and multivariate EMD (MEMD). The results of the spatio-temporal brain source reconstruction using these techniques show that masking EMD and MEMD can largely mitigate the mode-mixing problem and achieve a good spatio-temporal reconstruction of the active sources. Masking EMD and EEMD achieved better reconstruction than standard EMD, Multiple Sparse Priors, or wavelet packet decomposition when EMD was used as a pre-processing tool for the spatial reconstruction (averaged over time) of the brain sources. The spatial resolution obtained using all three EMD variants was substantially better than the use of EMD alone, as the mode-mixing problem was mitigated, particularly with masking EMD and EEMD. These findings encourage further exploration into the use of EMD-based pre-processing, the mode-mixing problem, and its impact on the accuracy of brain source activity reconstruction.
... Some methodologies such as the masking signal [16] or Ensamble Empirical Mode Decomposition (EEMD) have been proposed to avoid this problem [17]. The mode mixing does not disappear completely, however the technique is very interesting when it is compared with other strategies quite common for this type of application, for example with Discrete Wavelet Transform (DWT) is necessary to consider four factors affecting the performance in epileptic focus localization: the mother wavelet, the level of decomposition, frequency bands, and features [18], [19]. Based on the above, we have proposed a new and simple methodology based on an entropy function that allows to select the IMFs regardless of the mode mixing problem. ...
Conference Paper
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Empirical Mode Decomposition (EMD) is an adaptive time-frequency analysis method, which is very useful for extracting information from noisy nonlinear or nonstationary data. The applications of this technique in Biomedical Signal analysis has increased and is now common to find publications that use EMD to identify behaviors in the brain or heart. In this paper, a novel identification method of relevant Intrinsic Mode Functions (IMFs), obtained from EEG signals, using an entropy analysis is proposed. The idea is to reduce the number of IMFs that are necessary for the reconstruction of neural activity. The entropy cost function is applied on the IMFs generated by the EMD in order to automatically select the IMFs with relevant information. A relative error measure has been used to validate our proposal.
Conference Paper
Recent studies have attempted to recognize emotions by extracting features from electroencephalographic (EEG) signals using either linear and stationary, or linear and non-stationary transformations. However, as EEG signals are non-linear and non-stationary, it seems that a non-linear and non-stationary transformation may be more suitable. Despite the attractiveness of this hypothesis, until now, little studies have used such transformation. The current work presents a comparison between an approach to recognize positive and negative emotions using a non-linear and non-stationary transformation (Hilbert-Huang Transformation) with an approach using linear and non-stationary transformation (Discrete Wavelet Transform). The two approaches were compared using 200 EEG signals recorded from 10 subjects. The comparison indicated that an approach using the Hilbert-Huang Transformation statistically significantly classified emotions more accurately than a Wavelet-based approach (P < 0.02). This result implies that Hilbert-Huang Transformation is a promising tool to increase the prediction of emotional states, thereby helping to designing and developing more robust emotion recognition approaches.Clinical relevance- This remarks the potential of the Hilbert-Huang transform to enhance EEG-based emotion recognition systems, which can potentially help to diagnose and treat mental diseases, such as autism and depression.
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Several approaches can be used to estimate neural activity. The main differences between them concern the a priori information used and its sensitivity to high noise levels. Empirical mode decomposition (EMD) has been recently applied to electroencephalography EEG-based neural activity reconstruction to provide a priori time-frequency information to improve the estimation of neural activity. EMD has the specific ability to identify independent oscillatory modes in non-stationary signals with multiple oscillatory components. However, attempts to use EMD in EEG analysis have not yet provided optimal reconstructions, due to the intrinsic mode-mixing problem of EMD. Several studies have used single-channel analysis, whereas others have used multiple-channel analysis for other applications. Here, we present the results of multiple-channel analysis using multivariate empirical mode decomposition (MEMD) to reduce the mode-mixing problem and provide useful a priori time-frequency information for the reconstruction of neuronal activity using several low-density EEG electrode montages. The methods were evaluated using real and synthetic EEG data, in which the reconstructions were performed using the multiple sparse priors (MSP) algorithm with EEG electrode montages of 32, 16, and 8 electrodes. The quality of the source reconstruction was assessed using the Wasserstein metric. A comparison of the solutions without pre-processing and those after applying MEMD showed the source reconstructions to be improved using MEMD as a priori information for the low-density montages of 8 and 16 electrodes. The mean source reconstruction error on a real EEG dataset was reduced by 59.42% and 66.04% for the 8 and 16 electrode montages respectively, and that on a simulated EEG with three active sources, by 87.31% and 31.45% for the same electrode montages.
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This paper describes a platform for obtaining and analyzing real-time measurements in Microgrids. A key building block in this platform is the Empirical Mode Decomposition (EMD) used to analyze the electrical voltage and current waveforms to identify the instantaneous frequency and amplitude of the monocomponents of the original signal. The method was used to analyse the frequency fluctuation and obtain information about the linearity of electrical current and voltage waveforms measured in the field. Comparison between grid-connected and stand-alone microgrid voltage and currents’ monocomponents were conducted. Fluctuations in the grid frequency occurred in both the grid-connected and stand-alone microgrid, but the degree of the observed fluctuations were different, revealing more apparent nonlinear distortions in the latter. The observed instantaneous frequency from the collected data indicates potential nonstationary electrical signals when compared to synthetic data containing periodic signals coming from nonlinear loads. This observation leads us to expect the next generation of real-time measuring devices for the micro power grids to be designed on the principle of instantaneous frequency detection. Further efforts will be directed to a more rigorous characterization of the nonstationary nature of the signals by analyzing more and longer set of data.
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Over the past decade, with the development of machine learning, Discrete Wavelet Transform (DWT) has been widely used in computer-aided epileptic EEG signal analysis as a powerful timefrequency tool. But some important problems have not yet been benefitted from DWT, including epileptic focus localization, a key task in epilepsy diagnosis and treatment. Additionally, the parameters and settings for DWT are chosen empirically or arbitrarily in previous work. In this work, we propose a framework to use DWT and Support Vector Machine (SVM) for epileptic focus localization problem based on EEG. To provide a guideline in selecting the best settings for DWT, we decompose the EEG segments in 7 commonly used wavelet families to their maximum theoretical levels. The wavelet and its level of decomposition providing the highest accuracy in each wavelet family are then used in a grid search for obtaining the optimal frequency bands and wavelet coefficient features. Our approach achieves promising performance on two widely-recognized intrancranial EEG datasets that are also seizurefree, with an accuracy of 83.07% on the Bern-Barcelona dataset and an accuracy of 88.00% on the UBonn dataset. Compared with existing DWT-based approaches in epileptic EEG analysis, the proposed approach leads to more accurate and robust results. A guideline for DWT parameter setting is provided at the end of the paper.
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The Holo-Hilbert spectral analysis (HHSA) method is introduced to cure the deficiencies of traditional spectral analysis and to give a full informational representation of nonlinear and non-stationary data. It uses a nested empirical mode decomposition and Hilbert–Huang transform (HHT) approach to identify intrinsic amplitude and frequency modulations often present in nonlinear systems. Comparisons are first made with traditional spectrum analysis, which usually achieved its results through convolutional integral transforms based on additive expansions of an a priori determined basis, mostly under linear and stationary assumptions. Thus, for non-stationary processes, the best one could do historically was to use the time–frequency representations, in which the amplitude (or energy density) variation is still represented in terms of time. For nonlinear processes, the data can have both amplitude and frequency modulations (intra-mode and inter-mode) generated by two different mechanisms: linear additive or nonlinear multiplicative processes. As all existing spectral analysis methods are based on additive expansions, either a priori or adaptive, none of them could possibly represent the multiplicative processes. While the earlier adaptive HHT spectral analysis approach could accommodate the intra-wave nonlinearity quite remarkably, it remained that any inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase-lock modulations were left untreated. To resolve the multiplicative processes issue, additional dimensions in the spectrum result are needed to account for the variations in both the amplitude and frequency modulations simultaneously. HHSA accommodates all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions. The Holo prefix in HHSA denotes a multiple dimensional representation with both additive and multiplicative capabilities.
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This chapter presents an introductory overview and a tutorial of signal-processing techniques that can be used to recognize mental states from electroencephalographic (EEG) signals in brain–computer interfaces. More particularly, this chapter presents how to extract relevant and robust spectral, spatial, and temporal information from noisy EEG signals (e.g., band-power features, spatial filters such as common spatial patterns or xDAWN, etc.), as well as a few classification algorithms (e.g., linear discriminant analysis) used to classify this information into a class of mental state. It also briefly touches on alternative, but currently less used approaches. The overall objective of this chapter is to provide the reader with practical knowledge about how to analyze EEG signals as well as to stress the key points to understand when performing such an analysis.
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Overview Historically, the concept of "ondelettes" or "wavelets" originated from the study of time-frequency signal analysis, wave propagation, and sampling theory. One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform analysis does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean MorIet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary signals. However, this new concept can be viewed as the synthesis of various ideas originating from different disciplines including mathematics (Calder6n-Zygmund operators and Littlewood-Paley theory), physics (coherent states in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing). Wavelet analysis is an exciting new method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines, and improvement in CAT scans and other medical image technology. Wavelets allow complex information such as music, speech, images, and patterns to be decomposed into elementary forms, called the fundamental building blocks, at different positions and scales and subsequently reconstructed with high precision.
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