Hilbert-Huang transf..

Flow Measurement and Instrumentation 18 (2007) 37–46
Flow Measurement
and Instrumentation
www.elsevier.com/locate/flowmeasinst
Hilbert–Huang transform based signal analysis for the characterization of
gas–liquid two-phase flow
Abstract
Hao Ding a , Zhiyao Huang a,∗ , Zhihuan Song a , Yong Yan b,∗∗
a National Laboratory of Industrial Control Technology, Department of Control Science and Engineering, Zhejiang University, Hangzhou 310027,
Zhejiang, PR China
b Department of Electronics, University of Kent, Canterbury, Kent CT2 7NT, UK
Received 25 August 2006; received in revised form 3 December 2006; accepted 19 December 2006
This paper reports the application of the Hilbert–Huang Transform (HHT) to the dynamic characterization of gas–liquid two-phase flow
in a horizontal pipeline. A differential pressure fluctuation signal of gas–liquid two-phase flow is adaptively decomposed into Intrinsic Mode
Functions (IMFs) through the use of Empirical Mode Decomposition (EMD) methods. Based on the EMD, the associated time–frequency–energy
distribution, i.e., the Hilbert spectrum, is obtained for the analysis of the differential pressure fluctuation signal and subsequent identification of its
corresponding energy characteristics. The relationship between the energy distribution of the signal and the flow pattern is established. In order
to assess the effectiveness of the approach, the results obtained using the HHT are compared with those from Fourier analysis and wavelet based
methods. It is found that the extracted energy characteristics give a good indication of the dynamic state of the gas–liquid two-phase flow and
thus can be used for flow pattern recognition. The proposed method is a useful tool for the in-depth understanding and subsequent quantitative
characterization of gas–liquid two-phase flow.
c○ 2007 Elsevier Ltd. All rights reserved.
Keywords: Gas–liquid two-phase flow; Differential pressure fluctuation; Flow pattern; Hilbert–Huang transform; Empirical mode decomposition
1. Introduction
Gas–liquid two-phase flow is a complex process that exists
widely in industry. With the advent of modern signal processing
technology, signal analysis has become an important
method in two-phase flow studies. In view of its common occurrence
in industry and its usefulness in representing information
from a two-phase flow, the differential pressure fluctuation
signal has been used in the study of two-phase flow by
many researchers. In recent years, the Fourier Transform (FT)
and Wavelet Transform (WT) have been applied to analyze the
differential pressure fluctuation signal of two-phase flow in a
horizontal pipeline. However, it is difficult to localize an event
precisely in both time and frequency domains concurrently because
the differential pressure fluctuation signal is nonlinear
∗ Corresponding author. Tel.: +86 571 8795 2145; fax: +86 571 8795 1219.
∗∗ Corresponding author. Tel.: +44 1227 823015; fax: +44 1227 456084.
E-mail addresses: zyhuang@iipc.zju.edu.cn (Z. Huang),
y.yan@kent.ac.uk (Y. Yan).
0955-5986/$ - see front matter c○ 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.flowmeasinst.2006.12.004
and non-stationary in nature [1]. The FT is strictly restricted to
linear systems and stationary data series. Since the data section
in a wavelet window can be regarded as approximately stationary,
WT can be used to analyze a non-stationary signal. However,
WT is basically suited to linear signals and cannot classify
time or frequency dependent information with the same accuracy
[2,3].
The Hilbert–Huang Transform (HHT) is a new time–
frequency analysis method [4]. The main difference between
the HHT and all other methods is that the elementary wavelet
function is derived from the signal itself and is adaptive. The
main feature of the HHT is the Empirical Mode Decomposition
(EMD), which is capable of extracting all the oscillatory modes
present in a signal. Each extracted mode is referred to as an Intrinsic
Mode Function (IMF), which has a unique local characteristic
[5,6]. After the Hilbert transform on each IMF has been
performed, the time–frequency distribution of the signal energy
is obtained, which is referred to as the Hilbert spectrum. The
primary advantages of the HHT over other methods are that it

38 H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46
can deal with the nonlinear problem objectively and the resolutions
of the Hilbert spectrum are satisfactory in both time and
frequency domains. The HHT is therefore suited to the exploration
of the local properties of a signal [7].
The HHT has been widely applied in recent years in the
fields of meteorology, ocean engineering, earthquake studies,
etc. Useful research results have been achieved to date, but
very limited work has been reported on the method for the
monitoring and characterization of two-phase flow. This paper
describes how the HHT is applied to analyze a differential
pressure fluctuation signal for the dynamic characterization
of gas–liquid two-phase flow. The local characteristics of
the signal are extracted and quantified under typical flow
conditions. A direct comparison between the HHT and FT
methods is also presented.
2. Fourier transform versus wavelet transform
The conventional FT uses linear superposition of trigonometric
functions to approach the original signal [4]. The FT of
the signal f (t) ∈ L2 (R) is defined as
F(ω) =
∞
−∞
f (t)e − jωt dt (1)
where ω is the angular frequency.
Although the FT is valid under general conditions, there
are some critical restrictions. As the FT is used to analyze a
signal, many additional components are introduced, giving rise
to misleading results. In order to overcome this shortcoming,
windowed FT is proposed, in which a time–frequency
distribution is obtained by sliding the window along the time
axis. Since this method depends on the conventional FT, the
analysis data are assumed to be piecewise-stationary. However,
there are practical difficulties in applying the method: in
order to localize an event in the time domain, the window
width must be narrow enough; on the other hand, the
frequency resolution requires longer time series [2]. These
contradictory requirements have posed significant limitations
on the applicability of the FT.
Unlike the FT, WT is a time–frequency analysis tool suitable
for non-stationary signal analysis. WT can be interpreted as a
decomposition of a signal into a set of frequency bands with the
same bandwidth on a logarithmic scale. The elementary wavelet
function is expressed as
ψa,b(t) = |a| − 1
t − b
2 ψ
a, b ∈ R a = 0 (2)
a
where a is the dilation factor and b is the translation factor. On
the other hand, 1/a denotes the frequency scale and b defines
the location of an event. The corresponding continuous WT can
be defined by
W f (a, b) = 〈 f, ψa,b〉 = |a| − 1
t − b
2 f (t)ψ dt (3)
a
where f (t) represents the signal to be analyzed.
WT can be viewed as an extended version of Fourier analysis
but with an adjustable window using short windows at high
R
frequency and long windows at low frequency [2,3]. Since
WT is fundamentally still based on Fourier analysis, all the
possible complications associated with the FT still exist. WT
satisfies the Heisenberg inequality, thus the resolutions in the
time and frequency domains cannot have the same accuracy
simultaneously. For practical applications, the elementary
wavelet function, ψ(·), can be adaptively modified by changing
the dilation factor and the translation factor. However, the
elementary wavelet function has to be given beforehand and
is difficult to change during analysis. The main drawback of
WT is that the fixed elementary wavelet function does not
necessarily match the varying nature of a signal. In practice,
it is difficult for WT to obtain a quantitative definition of the
energy–frequency–time distribution by the limited size of the
elementary wavelet function [4,8].
3. Hilbert–Huang transform
The HHT process comprises two main steps. Firstly, IMFs
are extracted from the signal itself based on EMD. Secondly,
the Hilbert transform is applied to each IMF component.
Consequently, the instantaneous frequency can be calculated
and the energy distribution of the signal is obtained in the
time–frequency plane [9–11].
3.1. EMD method
EMD is an adaptive technique that decomposes the original
signal into a family of IMF components. Each component
emphasizes the local embedded characteristics of the signal. An
IMF must satisfy two constraints: (i) the number of extrema and
the number of zero crossings must either equal each other or
differ at most by one; and (ii) at any time, the mean value of the
envelope defined by the local maxima and the envelope defined
by the local minima is zero.
The first constraint is similar to the traditional narrow-band
requirements for a stationary Gaussian process. The second
constraint modifies the global requirement to a local one and is
necessary so that the instantaneous frequency will not include
unwanted fluctuations.
The decomposition of data series x(t) follows a number of
logical steps [12,13]. These are described below.
1. Identify all of the local maxima and local minima of the
signal x(t). Then use cubic spline interpolation to define the
upper envelope xmax(t) and the lower envelope xmin(t) of the
original data series. The mean m1(t) of the upper and lower
envelopes is calculated as follows:
m1(t) = xmax(t) + xmin(t)
. (4)
2
2. Subtracting m1(t) from the data series x(t) gives:
h1(t) = x(t) − m1(t). (5)
3. In general, h1(t) is just an IMF candidate that is unlikely
to satisfy all the requirements of IMF. Iterate the above
procedure k times until the mean envelope is close to zero,

so the first IMF component I1(t) that contains the highest
frequency component of the signal can be designated as
h1(k−1)(t) − m1k(t) = h1k(t), I1(t) = h1k(t). (6)
The above procedure is a sifting process that has two effects:
(i) to eliminate riding waves; and (ii) to smooth uneven
amplitude. In order to avoid obliterating the physically
meaningful amplitude fluctuations, a stop criterion for the
sifting process is defined [10].
4. Separate I1(t) from the rest of the data series. The residue,
r1(t), is treated as the new data series. Since the residue
r1(t) still contains much information from lower frequency
components, it is subjected to the same sifting process.
Repeat the above procedure on the subsequent residual r j(t)
until the range of the residue is below a predetermined value
or the residue contains the lowest frequency component, and
the result is
r1(t) − I2(t) = r2(t), . . . , rn−1(t) − In(t) = rn(t) (7)
where rn(t) denotes the trend of the signal from which no
more IMF can be extracted and In is the nth IMF. Each IMF
contains lower frequency components than the one extracted
just before. At the end of the EMD procedure, the signal x(t)
can be exactly reconstructed using a linear combination:
n
x(t) = Ii(t) + rn(t). (8)
i=1
3.2. Hilbert spectrum
The Hilbert transform is applied to every IMF component
Ii(t) to obtain a data series Li(t):
Li(t) = 1
π P
∞
−∞
H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46 39
Ii(τ)
dτ (9)
t − τ
where P is Cauchy principal value. An analytical signal Zi(t)
can be constructed by
Zi(t) = Ii(t) + j Li(t) = ai(t)e jθ j (t)
(10)
where the amplitude ai(t) and the phase θi(t) are determined
from
ai(t) = [Ii(t) + Li(t)] 1 2 , θi(t) = arctan Li(t)
. (11)
Ii(t)
The instantaneous frequency ωi(t) of the analytical signal is
defined as the first derivative of the phase θi(t) [14–16]:
ωi(t) = 1
2π
dθi(t)
. (12)
dt
¿From Eqs. (11) and (12), the instantaneous frequency
and amplitude (energy) are all functions of time. The
time–frequency distribution of the amplitude is referred to as
the Hilbert spectrum:
H(ω, t) = Re
n
ai(t)e
i=1
j ωi (t)dt
where n is the number of IMFs.
(13)
Fig. 1. Experimental system: (1) water tank, (2) water pump, (3) water storage,
(4) water flowmeter, (5) compressor, (6) air storage, (7) gas flowmeter, (8)
mixer, (9) differential pressure transducer, (10) A/D converter, (11) computer.
Then the marginal spectrum can be defined as
h(ω) =
T
0
H(ω, t)dt (14)
where T is the total data length. The contribution of the total
amplitude from each frequency is measured by the marginal
spectrum.
4. Experimental test rig
The experimental set-up used in this study is shown in Fig. 1.
The differential pressure transducer used was a PD-23 type
differential pressure sensor. Fig. 1 indicates that two pressure
probes were installed on each pipe. A long transparent section
was installed upstream of the transducer so that the flow pattern
could be observed. The internal diameters of the three pipelines
at the test section are 15 mm, 25 mm and 40 mm, respectively.
The spacing between the two pressure probes is 0.25 m. The
gas flowrate mg was varied in the range of 0–10 m 3 /h and
the water flowrate mw in the range of 0–3.5 m 3 /h. In each
pipeline, the water flowrate was set to a constant value and
the gas flowrate was increased gradually. With the change in
gas flowrate, various flow patterns were created. A series of
experiments was conducted for different flow patterns such as
bubble flow, plug flow, wavy flow and slug flow. Through the
A/D converter, the differential pressure fluctuation signal was
fed into the computer for processing. The sampling frequency
and the sampling duration were 200 Hz and 15 s, respectively.
5. Results and discussion
5.1. Energy characteristics extracted
Fig. 2 shows side by side the IMF components and the
wavelet components of the differential pressure fluctuation
signal from the 25 mm-diameter pipeline.
It can be seen from Fig. 2 that the IMFs have different
local frequencies. The EMD method picks out the highestfrequency
component present in the signal. Thus, each IMF
component that contains a lower frequency than the preceding
one extracted is presented in turn. Furthermore, the quantitative
time–frequency representation is also obtained. EMD is a full
data driven method. However, WT disintegrates the signal into

40 H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46
Fig. 2. Comparison between the IMF components and wavelet components of the differential pressure fluctuation signals from the 25 mm-diameter pipeline: (a)–(d)
the IMF components; (e)–(h) the wavelet components.
a series of scales that cannot reveal precisely the relationship
between time and instantaneous frequency [4]. The smaller
scales represent the high-frequency information in the signal
and the larger scales represent the low-frequency information
in the signal.
In this study, the energy distribution H j of different
frequency bands is defined as
H j = E j
E
( j = 1, 2, 3) (15)

Table 1
Distribution of energy characteristics H j of a 15 mm-diameter pipeline
mg (m 3 /h) Flow pattern Energy characteristics H j (%)
H1 H2 H3
0.3 Bubble flow 87.69 10.17 2.14
1.0 Plug flow 33.23 62.29 4.48
1.5 Wavy flow 35.49 31.16 33.35
2.0 Slug flow 10.60 22.88 66.52
where E j stands for the local energy of the different frequency
bands ( j = 1 denotes the high-frequency band, j = 2 the
middle-frequency band and j = 3 the low-frequency band) and
E is the total energy. The energy distribution H j is selected as
the characteristics of the differential pressure fluctuation signal.
In this study a number of logical steps were followed:
(1) acquire the differential pressure fluctuation signal;
(2) apply HHT to the signal;
(3) compare the analysis results between HHT and WT;
(4) calculate the total signal energy, each frequency band of
local energy and the energy distribution;
(5) determine the energy distribution of the signal.
As per the extensive experimental work undertaken and
literature published, the IMF components are divided into three
frequency bands in terms of their inherent frequencies for
quantitative analysis of the energy distribution: high-frequency
band H1 (>25 Hz) with IMF components of I1 and I2, middlefrequency
band H2 (5–25 Hz) with IMF components of I3,
I4 and I5 and low-frequency band H3 (

42 H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46
Fig. 3. Comparison between Hilbert spectra and wavelet spectra of the differential pressure fluctuation signals for the 15 mm-diameter pipeline: (a)–(d) Hilbert
spectra; (e)–(h) wavelet spectra. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The above results imply that the energy characteristic H j
changes with the flow pattern and can therefore be regarded
as an indicator for the identification of the flow pattern of
gas–liquid two-phase flow. The results also demonstrate that
the HHT is suitable for analyzing the differential pressure
fluctuation signal. Moreover, the influence of the pipe diameter
on the energy characteristic is not significant. Thus, the
extracted energy characteristics can be used as an objective
indicator for flow pattern monitoring of gas–liquid twophase
flow. Though WT method can also be used to study
gas–liquid two-phase flow, it is difficult to demonstrate the
local characteristics of the signal in both time and frequency

domains because of the non-adaptive feature of the form of
elementary wavelet function in a practical application [4]. In
the next section, detailed analysis results for both HHT and WT
will be given.
5.2. Comparison between the HHT and WT methods
The experimental results presented above indicate that the
differential pressure fluctuation signals from different sized
pipelines have similar spectral characteristics. So we take the
experimental results for the 15 mm-diameter pipeline as an
example to analyze the Hilbert spectrum and the WT spectrum
of the signal. The Hilbert spectra and wavelet spectra of the
typical flow patterns in the 15 mm-diameter pipeline are plotted
in Fig. 3. The Hilbert spectra as the greyscale coded maps are
shown on the left, while the wavelet spectra of the signals are
illustrated on the right. The colour bars beside each Hilbert
spectrum denote the time and frequency dependent energy. The
energy is normalized and represented on a logarithmic scale.
The colour bars next to each wavelet spectrum represent the
absolute wavelet coefficient, i.e., the amplitude of the signal [3].
Fig. 3(a) shows the Hilbert spectrum of bubble flow. The
fluctuations of differential pressure are stable and the amplitude
of the signals is small. The main fluctuation locates at the higher
frequency, consequently the energy concentrates mainly on the
high-frequency components and the total energy is low. The
energy is evenly distributed in the high-frequency plane over
25 Hz in Fig. 3(a) and there is very little energy below 25 Hz.
The wavelet spectrum shows a similar energy distribution as the
Hilbert spectrum, as shown in Fig. 3(e). It can be seen that the
energy concentrates mostly in the smaller scale range.
Fig. 3(b) shows the Hilbert spectrum of the plug flow.
Compared to the bubble flow, the ratio of energy embedded
in different frequency bands is changed. The ratio of energy
embedded in the high-frequency range has reduced rapidly
whereas the ratio of energy embedded in the middle–lowfrequency
range has increased. As there still exist many tiny
bubbles between two successive plugs, there is low energy in
H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46 43
Fig. 3. (continued)
the high-frequency range. Fig. 3(f) shows the results derived
from WT. It is clear that the peak amplitude is greater than that
of the bubble flow.
The Hilbert spectrum of the wavy flow (Fig. 3(c)) indicates
that the ratio of energy embedded in the higher-frequency range
is greater than that in other frequency ranges. Compared to
the plug flow, the ratio of energy embedded in the middlefrequency
range begins to decrease and that in the lowerfrequency
range begins to increase slowly. As shown in
Fig. 3(g), the ratio of energy embedded in smaller scales begins
increasing.
Fig. 3(d) depicts the Hilbert spectrum of the slug flow. The
size of gas slug is bigger than the bubble and the fluctuation of
the signal is more drastic than other three flow patterns. As a result,
the ratio of energy embedded in the lower-frequency range
is the greatest, which is consistent with the analysis result from
the wavelet spectrum shown in Fig. 3(h). As shown in Fig. 3, the
variation in the ratio of energy embedded in different frequency
bands is distinctive in the Hilbert spectrum and is in agreement
with the wavelet spectrum. However, the frequency and time
localization of the Hilbert spectrum is superior to that of the
wavelet spectrum. The localizations of WT are accomplished
by modifying the scale of the elementary wavelet function. If a
local event occurs in the low-frequency range, it is necessary to
modify the scale for characterizing the signal continuously [4].
WT indeed improves the time resolution as compared to the FT,
but HHT gives much finer local properties than WT.
Although Fig. 3 gives a comparison between the Hilbert and
the wavelet spectra for the same set of data, the differences
between the two methods are not so obvious according
to the counter-intuitive figures. From a theoretical point of
view, the choice of the elementary wavelet function will
have a significant impact on the analysis results because the
elementary wavelet function is fixed during analysis. Once
the elementary function is determined, it will be used to
analyze all the data [2,4]. Since WT is Fourier based, the
wavelet spectrum possesses some fundamental characteristics
of a Fourier spectrum. WT can analyze a non-stationary signal

44 H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46
Fig. 4. Comparison between marginal spectra and Fourier spectra of the differential pressure fluctuation signals for the 40 mm-diameter pipeline: (a)–(d) the
marginal spectra; (e)–(h) the Fourier spectra.
under the assumption that the signal inside each sliding window
is stationary [17]. However, HHT is an efficient and adaptive
method which requires no assumption about signal stationarity.
HHT is capable of analyzing a nonlinear and non-stationary
signal not limited by the Heisenberg inequality. A more precise
time–frequency–energy representation may be obtained by
using an adaptive basis to preserve intrinsic properties of a
signal [8].
The marginal spectrum provides a measure of total
amplitude (or energy) contribution from each frequency value.
It presents the cumulative amplitude over the entire data span.
The Fourier spectrum depicts the global frequency–energy
distribution [2]. Because the pipe diameter is not very
significant for the signal analysis, only the experimental results
for the 40 mm-diameter pipeline are presented here. The
same variation trends exist for 15 mm- and 25 mm-diameter

pipelines. The marginal spectra and Fourier spectra are plotted
in Fig. 4.
It is evident from Fig. 4(a)–(d) that the ratio of energy
embedded in different frequency bands varies with the change
in flow pattern. Meanwhile, the energy amplitude of the signal
increases with the variation of flow pattern. These results are
in agreement with those of Hilbert spectra. Because stationarity
and linearity are the crucial restrictions of the FT, the adoption
of the stationary and linear assumptions may give misleading
results [2]. When the FT is used to analyze the differential
pressure fluctuation signal, many frequency components are
introduced to simulate the inherent features of the signal, but
their existence disperses the energy to a much wider frequency
range [4]. Thus, these additional frequency components and
the wide frequency spectra cannot accurately represent the true
frequency–energy distribution.
6. Conclusions
The HHT has been applied in the analysis of a differential
pressure fluctuation signal derived from gas–liquid two-phase
flow in a horizontal pipeline. Results presented in this paper
have demonstrated that the extracted energy characteristics
reflect the energy shift with the variation of flow pattern. The
energy characteristics can therefore be used as an indication of
the flow pattern of gas–liquid two-phase flow. The results have
also indicated that the HHT method is suitable for analyzing
nonlinear and non-stationary signals of gas–liquid two-phase
flow. Additionally, the HHT method has the potential to identify
local characteristics of the flow signal.
In this study, the EMD has been used to produce a series
of IMFs embedded in the differential pressure fluctuation
signal. The subsequent Hilbert spectrum presents a quantitative
analysis of variation of energy and frequency with time.
Analysis results obtained using the HHT, WT and FT have
been compared and appraised. The results have demonstrated
that WT improves the time resolution compared to the Fourier
method but the HHT shows no obvious constraints compared to
the WT. The HHT offers a different perspective for the analysis
of two-phase flow signals.
H. Ding et al. / Flow Measurement and Instrumentation 18 (2007) 37–46 45
Fig. 4. (continued)
The HHT method is potentially applicable to the measurement
of two-phase flow parameters, such as bubble frequency,
bubble velocity and bubble coalescence ratio. There is no doubt
that there is still a long way to go before effective practical application
of the method.
Acknowledgements
This research is supported by National Natural Science
Foundation of China under Grant Nos. 50576084 and
60532020.
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