Implementation of Hilbert-Huang Transform Using C Language

生 物 医 学 信 号 检 测 与 处 理
Implementation of Hilbert-Huang Transform Using C Language
Wang Lei, Vai Mang I, Mak Peng Un
BME Lab, EEE, FST, University of Macau, Macau, China
Abstract: Hilbert-Huang Transform, proposed by N. E. Huang in 1998, is a novel algorithm for nonlinear and
non-stationary signal processing. Up to now, most of the researchers focus on the applications of HHT in different
areas. However, in this paper we try to focus on the implementation of HHT in C and the possibility in porting to
FPGA. With our implementation, IMFs with acceptable accuracy were achieved, which can be used in other
applications. Since the program is written in ANSI C, it also offers a good portability in different platforms
including hardware implementation (FPGA or ASIC).
Keywords: HHT; EMD; IMF; C; Instantaneous Frequency
1. Introduction
Digital Signal Processing technology is widely used
in Biomedical Engineering. As it is known, most of
the biomedical signals, such as ECG and EMG, are
nonlinear and non-stationary signals. However, most
of the methods, e.g. Fourier Integral Transform, Fast
Fourier Transform (FFT) and Wavelet Transform have
a strong priori assumption that the signals being
processed should be linear and/or stationary. They are
actually not suitable for biomedical signals which are
nonlinear and non-stationary.
Hilbert-Huang Transform (HHT), a novel algorithm
proposed by N. E. Huang in 1998, provides a possible
solution to those nonlinear and non-stationary signals.
The algorithm is composed of two main parts –
Empirical Mode Decomposition (EMD) and Hilbert
Transform. The first part will decompose the signal
into Intrinsic Mode Functions (IMF) which will meet
the requirement of the Hilbert Transform and then can
be further analyzed to get the instantaneous
frequencies of the original signals.
HHT have been widely used in a variety of
research areas, including researches in Biomedical
Engineering, such as EEG, EMG, ECG, pulse wave
and so on. Some fruitful results have been achieved.
However, the HHT is a very computation oriented
algorithm and will consume much time.
Currently, most of the researches were based on the
software (mainly in Matlab) implementation which
may be not running fast enough for real time
application, especially when running on an embedded
platform such as home health care devices. What we
are planning to do is trying to develop a hardware
based real-time HHT analysis system. As the first step,
we try to implement the HHT in ANSI C that is
possible to migrate to other hardware description
language to make the hardware implementation
become possible, for example, migrate to FPGA or
ASIC chip.
2. Hilbert-Huang Transform
The general accepted definition of Instantaneous
Frequency (IF) is the derivative of an analytic signal.
IF can be express with Hilbert Transform. The
Instantaneous Frequency of a signal x(t) is
'
'
1 H [ xt ()] xt () H[ xt ()] x()
t
IF
(1)
2 2
2 x ( t) H [ x( t)]
Here H[x(t)] is the Hilbert Transform of the signal
x(t). However, not all kinds of signals have a
meaningful IF according to equation (1). It’s only
works on narrow band signal.
2.1 IMF
To make clear the limitation of Hilbert Transform
The work presented in this paper is supported by the Research Committee of the University of Macau under Grants RG061/06-07S/VMI/FST,
RG075/07-08S/VMI/FST, UL012/09-Y1/EEE/VMI01/FST and by The Science and Technology Development Fund of Macau under grant
014/2007/A1.
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生 物 医 学 信 号 检 测 与 处 理
that most of the nonlinear and non-stationary signals
can not have a meaningful Instantaneous Frequency,
N. E. Huang proposed some restrictions to the signals
which can have meaningful IF. The restrictive
conditions called Intrinsic Mode Function (IMF). An
IMF must satisfy the following two conditions [2]:
1. In the whole data set, the number of extrema and
the number of zero crossings must either equal or
differ by one.
2. At any data point, the mean value of the envelope
defined by the local maxima and envelop defined
by the local minima is zero.
n
x()
t imf r
i1
n
(2)
2.2 EMD
As discussed above, an IMF has a meaningful IF
using Hilbert Transform. However, most of the signals
are not IMF. To solve this problem, N. E. Huang
proposed the Empirical Mode Decomposition (EMD).
This method can decompose the nonlinear and nonstationary
signals into finite IMFs, so we can have the
Instantaneous Frequencies from those IMFs using
Hilbert Transform. And add the IFs from IMFs
together we can have the IFs of the original signal. To
implement this method, a sifting process has been
designed. The steps are:
First, we extract the maxima and minima data
points from the original signal x(t). Then, interpolate
all the maxima and minima data points separately with
cubic spline to get the upper and lower envelopes,
followed by the calculation of the mean of the upper
and lower envelopes (m). After finding the m, the
difference between the original signal and the mean
can be calculated:
h x()
t m
If h is not an IMF, we set it as the original signal to
repeat the above steps until get an IMF. If h is an IMF,
it will be regarded as the IMF (imf) and the residua
should be found as: r=x(t)-imf. Then the residua will
be treated as the original signal x(t) and the above
steps will be repeated to get another IMF. This process
will be stopped when the residua r or imf satisfy the
stop criteria. Finally the original signal can be express
as the sum of all the IMFs and residua:
Figure 1
2.3 Hilbert Transform
For signal x(t), the definition of Hilbert Transform
is:
1
H[
x(
t)]
P
'
x(
t )
dt
'
t t
'
1
* x(
t)
t
(3)
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生 物 医 学 信 号 检 测 与 处 理
P is Cauchy principal value here. And for every
IMF after EMD sifting, we do the Hilbert transform
for them and calculate the corresponding analytic
signal. After further processing we can expressed x(t)
in a three-dimensional figure which is energy-timefrequency
distribution.
3. Implementation
3.1 Structure of The Program
There are two main loops in the program. One is to
extract IMFs (small loop). The other is to check if the
residua meet the EMD stop criteria (big loop). Figure
1 shows the flow chat of the program.
The first loop is to identify extremum data points
and the mean of upper and lower envelopes for the ‘h’.
If ‘h’ is an IMF, the iteration times of decomposition
should be saved and the first loop should be stopped.
If ‘h’ is not an IMF, set ‘h’ as the original signal and
start the decomposition again. The mean between
upper and lower envelopes of ‘h’ will be used to get
another difference of the signals.
The second loop is to verify if the EMD sifting
should be stopped or not. About the EMD sifting, N. E.
Huang gave such a description: the sifting process can
be stopped either when the component, imf n or the
residua, r n , becomes so small that it is less than
predetermined value of substantial consequence, or
when the residua, r n , becomes a monotonic function
from which no more IMF can be extracted[2].
According to this description, stop criteria for EMD
sifting can be set.
Because the program needs to save both upper and
lower envelopes, and the mean and difference between
mean and originals, as well as IMFs, the program
needs a large memory space.
3.2 Experiments
For testing our implementation, we took the ECG
signal ‘sig100a’ from MIT-BIH Arrhythmia Database
as the source signal. By decomposing the ECG signal,
totally 11 IMFs were extracted from the original
signal. The original ECG signal and the first to fifth
IMFs are shown in Figure 2, other IMFs and residua
are shown in Figure 3.
E CG signal
im f1
im f2
im f3
im f4
im f5
1
0
-1
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.5
0
-0.5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.5
0
-0.5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time(normalized)
Figure 2
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.5
0
-0.5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.2
0
-0.2
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.1
0
-0.1
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0.05
0
-0.05
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-0.3
residua-0.2
-0.4
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time(normalized)
im f6
im f7
im f8
im f9
im f1 0
im f1 1
Figure 3
We also tried to apply EMD sifting to other ECG
signals and found that for different signals the execute
time of program are different. That may because of the
number of IMFs and iteration time of every IMF is
depends on the complexity of signals.
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生 物 医 学 信 号 检 测 与 处 理
To test the speed of the implementation, we
compare the time consumed by our implementation
and an implementation in Matlab running on the same
PC. It was found that the C implementation is slightly
faster than the Matlab implementation.
4. Conclusions and Discussions
We have done an effective EMD sifting system
with pure ANSI C. With the program we can extract
IMFs from nonlinear non-stationary signals. But the
program needs a lot of memory, and for different
signals it will take different execution time. So it still
needs effort to improve the algorithm to make it uses
as less memory as possible, and make the program
running faster. At the other hand, it is also needed to
modify the algorithm to make it be more “parallel” so
suitable to be implemented in hardware (FPGA).
5. References
[1] N. E. Huang , Z. Shen and S.R. Long, “The empirical
Mode decomposition and Hilbert spectrum for nonlinear
and non-station time series analysis,” Proc Roy, Soc.
London A, vol. 454, pp, 903~995, 1998
[2] N. E. Huang, S. P. Shen, “Hilbert-Huang Transform and
Its Application.” World Scientific Pub Co Inc, 2005
[3] Semion Kizhner, Thomas P. Flatley, Dr. Norden E.
Huang, “On the Hilbert-Huang Transform Data
Processing System Development” IEEE Aerospace
Conference Proceedings, 2004
[4] Xiaoli Yang, Jing0tian Tang, “Hilbert-Huang Transform
and Wavelet Transform for ECG Detection.” Wireless
Communications, Networking and Mobile Computing,
2008. WiCOM’08. 5 th International Comference
[5] Li Hong sheng, Li Zhengqin, Chen Wenwu, Zhao Zhao,
“Implementation of Hilbert-Huang Transform(HHT)
Based on DSP.” Signal Processing, 2004. Proceedings.
ICSP '04. 2004 7th International Conference
通 讯 作 者 : 王 磊
单 位 : 澳 門 大 學 科 技 學 院 生 物 醫 學 工 程 實 驗 室
邮 箱 :ma76559@umac.mo
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