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Mechanical Systems
and
Signal Processing
Mechanical Systems and Signal Processing 22 (2008) 1374–1394
On the HHT, its problems, and some solutions
R.T. Rato, M.D. Ortigueira
,1
, A.G. Batista
Campus da FCT da UNL, Quinta da Torre, 2825 - 114 Monte da Caparica, Portugal
Received 15 December 2006; accepted 30 November 2007
Available online 18 January 2008
Abstract
The empirical mode decomposition (EMD) is reviewed and some questions related to its effective performance are
discussed. Its interpretation in terms of AM/FM modulation is done. Solutions for its drawbacks are proposed. Numerical
simulations are carried out to empirically evaluate the proposed modified EMD.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Empirical mode decomposition; Intrinsic mode function; Parabolic interpolation; Extrema detection; Amplitude–frequency
modulation
1. Introduction
The empirical mode decomposition (EMD) [1] is a technique to decompose a given signal into a set of
elemental signals called ‘‘intrinsic mode functions’’ (IMFs). The EMD is the base of the so-called
‘‘Hilbert–Huang transform (HHT)’’ [1] that comprises the EMD and the Hilbert spectral analysis that
performs a spectral analysis using the Hilbert transform (HT) followed by an instantaneous frequency
computation.
The algorithm is simple and gives good results in situations where other methods fail. However, it has some
drawbacks, tied with some of the assumptions needed to implement the algorithm, leading to unexpected
results. There have been several attempts to solve such problems. For example, Rilling et al. [2,3] made some
algorithmic variations and proposed a new stopping criterion. Besides, they gave an interpretation in terms of
filter banks. They also studied the influence of sampling [4]. On the other hand, Junsheng et al. [5] studied the
behaviour of the decomposition algorithm and proposed an energy difference tracking method to defin e a
coherent stopping criterion. In another attempt [6] they use the Teager–Kaiser [7] energy operator to extract
the amplitude and instantaneous frequency of a multi-component amplitude-modulated and frequency-
modulated (AM/FM) signals. The problem of envelope estimation is considered by Qin and Zhong who
proposed the segment power function method [8].
In this paper, we make a global appreciation of the different aspects of the algorithm, consider ing its
drawbacks and suggesting modifications to alleviate such problems. The most important is the extrema
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doi:10.1016/j.ymssp.2007.11.028
Corresponding author. Tel.: +351 1 2948520; fax: +351 1 2957786.
E-mail addresses: rtr@uninova.pt (R.T. Rato), mdo@fct.unl.pt (M.D. Ortigueira), agb@fct.unl.pt (A.G. Batista).
1
Also with INESC_ID, R. Alves Redol, 9,21, Lisbon, Portugal.
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determination by using a parabolic approximation and the instantaneous amplitude and frequency
computation that we propose to be done through an amplitude demodulation and first-order autoregressive
(AR) approximation [9,10].
The paper outline is as follows. In Section 2 we describe the EMD algorithm as proposed in [1] and look
inside it to understand the main difficulties and ways of avoiding them. In Secti on 3 we present the proposed
solutions to such problems. In Section 4, we present some examples to illustrate the behaviour of the algorithm
and to understand its features. Finall y, we present some conclusions.
2. The EMD
2.1. Outline of the EMD algor ithm
The EMD as proposed by Huang et al. [1] is a signal decomposition algorithm based on a successive
removal of elemental signals: the IMFs. Given any signal, xðtÞ, the IMFs are found by an iterative procedure
called sifting algorithm, which is composed of the following steps:
(a) Find all the local maxima, M
i
; i ¼ 1; 2; ...; and minima, m
k
, k ¼ 1, 2; ...; in xðtÞ.
(b) Compute the corresponding interpolating signals MðtÞ:¼f
M
ðM
i
; tÞ, and mðtÞ :¼f
m
ðm
k
; tÞ. These signals are
the upper and lower envelopes of the signal.
(c) Let eðtÞ:¼ðMðtÞþmðtÞÞ=2.
(d) Subtract eðtÞ from the signal: xðtÞ:¼xðtÞeðtÞ.
(e) Return to step (a)—stop when xðtÞ remains nearly unchanged.
(f) Once we obtain an IMF, jðtÞ, remove it from the signal xðtÞ:¼xðtÞjðtÞ and return to (a) if xðtÞ has more
than one extremum (neither a constant nor a trend).
The interpolating function is a cubic spline. By construction, the number of extrema should decrease when
going from one IMF to the next, and the whole decomposition is expected to be completed with a finite
number of IMFs (see [2]). We must remark that, at least co nceptually, the algorithm:
is simple;
appears naturally;
does not assume anything about the signal, mainly stationarity;
can be applied to a wide class of signals.
2.2. The EMD as an AM/FM decomposition
Consider the sifting process. The first step finds two sets of points that constitute samples of two discrete-
time signals. The interpolated signals give estimates of the upper and lower envelopes. If the envelopes were
symmetric we would say that xðtÞ is an AM signal [11]. The sifting procedure is an iterative way of removing
the dissymmetry between the upper and lower envelopes in order to transform the original signal into an AM
signal. At least conceptually, this goal would be achieved in a few steps if:
The extrema were correctly determined.
We had no problems with the interpolation at the extremities.
We had no computational errors.
These difficulties will be considered later.
The above procedure allows us to conclude that each IMF is an AM signal. Moreover, as the instantaneous
frequency can change from instant to instant, we can say that each IMF is an amplitude/frequency
modulated (AM/FM) signal. So, the EMD is nothing else than a decomposition into a set of AM/FM
modulated signals. As it is not hard to understand, the envelopes cannot vary as fast as the signal , xðtÞ.
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In spectral terms, we can say that the bandwidth of the envelopes must be a fraction of the central frequency of
xðtÞ (normally called carrier). This means that when performing the sifting we are removing the low frequency
components. So, we are leaving a high frequency signal. This explains why the IMFs appear in a high to low
frequency order and why the EMD is essentially a time-frequency decomposition. This also explains why the
EMD behaves like a bank of filters [3]. This also explains an interesting phenomenon: if we add two IMFs with
non-intercepting bands, they will be decomposed without great distortion, but if the bands intercept, they will
be decomposed into a set of several IMFs. We will show this later.
2.3. On the IMF
For the continuous case, an abstract IMF is defined as a signal that satisfies two conditions [1]:
In the whole signal segment, the number of extrema and the number of zero crossings must be either equal
or differ at most by one.
At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the
local minima is zero.
This is the original definition of IMF presented in [1]. However, the first condition is redundant unless the
function at hand is discontinuous. In fact, if a function verifies the second condition, it verifies the first also,
because between a peak and a valley there is always a zero, since the envelopes are symmetric.
It is not difficult to see that sinusoidal signal s sinð 2 pftÞ or co sð2pftÞ for any real f and t are IMFs. It is not
difficult to recognize that if FðtÞ is a continuous function, the same happens with sin½Fð tÞ and cos½FðtÞ.
These are well-known functions in telecommunications where they are studied under the name of ‘‘angle
modulation’’ [11]. The instantaneous frequency is, aside a constant, the derivative of FðtÞ. Now, let us
consider a function gðtÞ¼AðtÞ sin½FðtÞ. This is what is called ‘‘double side band’’ modulated sinusoid. In
general, it is not an IMF, even if AðtÞ is also a sinusoid . But, if AðtÞ changes slowly when compared
with the changes in sin½jðtÞ, we have really an IMF. So, we can say that a function of the type A ðtÞ sin½FðtÞ
represents an IMF, provided that AðtÞ is a slowly varying function. This is the so-called envelope. But we must
remark that the above definition of envelope may not be a true envelope. A simple example shows this. The
function xðtÞ¼e
jtj
sinð2pftÞ has e
jtj
as envelope, but almost all the extrema points of xðtÞ do not belong to
e
jtj
. In some signals this fact has as consequence that the function may have segments above (below ) the
envelope defined by the maxima (minima). As a conjecture, we can say that the extrema envelope coincides
with the ‘‘true’’ envelope when this has extrema locations equal to those of the signal. However, there is no
simple way of defining the ‘‘true’’ envelope and the extrema defining envelope seems to be suitable to our
objectives.
We can conclude that the definition of IMF is tied with the definition of envelope that depends also on the
interpolating function used to estimate it. So, the Huang et al. [1] IMFs correspond to the cubic spline
interpolator. With other interpolator we may obtain a different set of IMFs. In our simulations we also used
the Akima interpolator.
2
The results were very similar.
2.4. Main drawbacks of the EMD
Let us take a look into the above described algorithm an d consider the most important steps:
extrema locations;
extrema interpolation;
end effects;
sifting stopping criteri on;
IMF removal.
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We used the routine available at the MatLab site.
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As we have seen in the earlier last sections, the algorithm has some implicit difficulties and the procedures used
in previous papers [1–3] for the above steps create several drawbacks that originate ‘‘strange’’ decompositions.
For example, when trying to decompose a sinusoidal segment, it is exp ected to obtain only one IMF and no
residual. This may not happen with the available algorithm implementations [1–3].
We are going to look into each step and see why it creates difficulties. We begin by the
extrema computation. This does not have an obvious solution. Although most data are originated by
continuous-time processes, in practice, the algorithm operates on quantized discrete-time signals. When
working with this kind of signals, some special attention is necessary because the extrema may not be correctly
identified. Most, if not all, of the continuous waveform actual extrema will fall in between sampling instants
and will not be correctly localized. To avoid this difficulty, Rilling et al. [2] proposed the use of a fair amount
of oversampling.
We have considered before the interpolation problem and the interpolation function choice. However, even
if we made a reasonable choice, the effectiveness of the approximation is highly dependent on the extrema
computation and may lead to some undesirable results. One of them is the ‘‘overshoots’’: the signal crosses the
envelope. This affects the IMF estimation.
The end effects appear when we have to decide what to do with the first and last sampl es. The solution will
affect the final decomposition:
To consider them as maxima and minima simultaneously (this forces all the IMFs to be zero at those
points).
To consider them as maxima or minima according to the nearest extremum in order to guarantee the
alternation between maxima and minima.
To leave them free [1].
The stopping criterion is another source of problems due to its degree of arbitrariness, since it may not
guarantee a total signal removal to obtain a ‘‘true’’ IMF.
According to the above description, the final step is the removal of the IMF from the signal. However,
if the IMF is not well computed, we may be ‘‘adding’’ to the remaining signal a component that will
appear in the following IMF. This explains partially why we do not obtain only one IMF in the case of a
pure sinusoid.
3. Some attempts to obtain a better algorithm
3.1. A framework for modifying the algor ithm
The EMD does not have an analytical formulation: it is based on a co mputational algorithm. So it performs
according to its implementation details. When the same signal is fed into different EMD implementations,
different results are obtaine d, as we will see in Section 5.1. This is confusing and inadequate. To achieve
consistency and to be able to compare results, different implementations should be equivalent. Our goal is to
establish a framework for EMD implementation to avoid the referred drawbacks (Section 2.4).
Based on this rationale, we state the following guiding principles to be followed in the design,
implementation and checking of EMD programs:
The IMF set obtained by multiplying a constant value to all the samples in the signal should be the IMF set
of the original signal multiplied by the same constant.
Changing the mean of the signal, it should only change the trend related IMF (the last one), leaving all the
others unchanged.
The EMD of an IMF should be the IMF itself.
The IMF set obtained from a time reversed signal should be the time reversed IMF set of the original
signal.
3
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This condition is not fundamental and can be released.
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