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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·dθi(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|ψt−b
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
√2j−k(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
1 1.05 1.1 1.15 1.2
Time [ms]
-1.5
-1
-0.5
0
0.5
1
1.5
Voltage [pu]
Intrinsic Mode Function 1
1 1.05 1.1 1.15 1.2
Time [ms]
-1.5
-1
-0.5
0
0.5
1
1.5
Voltage [pu]
Intrinsic Mode Function 2
0 0.01 0.02 0.03 0.04 0.05
frequency(Hz)
0
0.5
1
1.5
2
2.5
3
Power
×104Power Spectrum of Intrinsic Mode Function 1
0 0.01 0.02 0.03 0.04 0.05
frequency(Hz)
0
2
4
6
8
10
12
Power
×106Power Spectrum of Intrinsic Mode Function 2
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
0 500 1000 1500
Time [ms]
-5
0
5
10
Voltage
Intrinsic Mode Function 1
0 500 1000 1500
Time [ms]
-5
0
5
Voltage
Intrinsic Mode Function 2
0 500 1000 1500
Time [ms]
-10
0
10
Voltage
Intrinsic Mode Function 3
0 500 1000 1500
Time [ms]
-5
0
5
10
Voltage
Intrinsic Mode Function 4
0 500 1000 1500
Time [ms]
-5
0
5
Voltage
Intrinsic Mode Function 5
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
1
2
Power
×104Power Spectrum of Intrinsic Mode Function 1
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
5
Power
×104Power Spectrum of Intrinsic Mode Function 2
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
1
2
Power
×105Power Spectrum of Intrinsic Mode Function 3
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
5
Power
×105Power Spectrum of Intrinsic Mode Function 4
0 0.1 0.2 0.3 0.4 0.5
frequency(Hz)
0
2
4
Power
×105Power Spectrum of Intrinsic Mode Function 5
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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