Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay

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20 Dynamical Properties of Intensity Fluctuation 247

-0.75 -0.5 -0.25 0 0.25 0.5 0.75

0.08

0.1

0.12

0.14

0 0.2 0.4 0.6

0.8

0.2

0.4

0.6

0.8

C(s)

————————— 0.9

——————————

0.5

——————————

0.5

——————————

0.9

Fig. 20.8 The laser system is operated at the threshold (a

D 0). (a) The relaxation time T

a function of the coupling strength . Parameters chosen are K D 120, A D 1,andQ D 2.

(b) The normalized correlation function C.s/ as a function of decay time s for different values of

the coupling strength . Parameters chosen are A D 1, K D 60, D D 1 and Q D 2

1. When the laser system is operated at the threshold and below the threshold,

.I / mainly concentrated on the region of I ! 0, so it cannot almost possess

the laser intensity output. When the laser system is operated above the thresh-

old, P

.I / possesses one-maximum structure, and the laser system possesses

the output of the steady laser intensity. The coupling strength  weakens the

peak of the SPD, and the position of the peak shifts to the smaller intensity vari-

able I as  increases; in contrast, the cross-correlation time  generates entirely

opposite effects.

2. The relaxation time T

reﬂects the evolution velocity of the system from arbitrary

initial state to the steady state, which can characterize the startup time of the laser

system from start to steady work. When the laser system is operated above the

threshold, the coupling strength  speeds up the startup; When <0, the cross-

correlation time  speeds up the startup; When >0, the cross-correlation time

 blows down the startup; The quantum noise intensity D and the pump noise

248 P. Z h u

intensity Q makes the relaxation time exhibit the extremum structures, that is,

the startup time possesses the least values at D D D

and Q D Q

, which are

as functions of other parameters, respectively.

3. The correlation function reﬂects the stability of laser intensity of laser output.

When the laser system is operated above the threshold, which the laser system

possesses the steady output of laser intensity,  enhances the stability of the in-

tensity output of the saturation laser system, however,  weakens the stability of

the intensity output.

4. When the laser system is operated at threshold, the coupling strength  expedites

the startup and enhances the stability of the startup.

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Chapter 21

Empirical Mode Decomposition Based

on Bistable Stochastic Resonance Denoising

Y.-J. Zhao, Y. Xu, H. Zhang, S.-B. Fan, and Y.-G. Leng

Abstract The empirical mode decomposition (EMD) of weak signals submerged

in a heavy noise was conducted and a method of stochastic resonance (SR) used

for noisy EMD was presented. This method used SR as pre-treatment of EMD

to remove noise and detect weak signals. The experiment result proves that this

method, compared with that using EMD directly, not only improve SNR, enhance

weak signals, but also improve the decomposition performance and reduce the

decomposition layers.

21.1 Introduction

In the quest of accurate time and frequency localization, Huang et al. [1, 2]pro-

posed the empirical mode decomposition (EMD) scheme which offers a different

approach in time-series processing. This method can decompose the signal into a set

of oscillatory modes by taking advantage of the characteristic time scales embedded

in the data. So there is no need for a basis function and no need for transforma-

tion [3]. But EMD method can not eliminate boundary problem because it uses

cubic splines method to obtain signal’s instantaneous average. If EMD is used in

strong noise condition it will affect the decomposition performance, especially in-

crease decomposition layers and lower efﬁciency of arithmetic, even may lead EMD

lose deﬁnitude signiﬁcance, so denoising process must be performed before EMD

decomposition.References [4]and[5] use wavelet and SVD to denoise and get good

effect. But these methods are not good at weak signals submerged in heavy noise.

At the meantime, stochastic resonance (SR), which is put forward by Benzi [6]

wherein he addresses the problem of the primary cycle of recurrent ice ages, has

a strong development for its advantages in weak signal’s enhancement and detec-

tion. With the cooperation effect of signal and noise in non-liner system, SR can put

Y. - J . Z h a o (



)

CSR Qingdao Sifang Locomotive and Rolling Stock Company, Chengyang, Qingdao 266111,

People’s Republic of China

e-mail: zhaoyanju215@yahoo.com.cn

A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,

DOI 10.1007/978-1-4419-5754-2

21,

 Springer Science+Business Media, LLC 2010

251

252 Y.-J. Zhao et al.

noise power into lower-frequency signal which can enhance weak signal and reduce

noise at the same time. In a word, this paper proposes a new method of weak sig-

nals detection based on SR theory and EMD decomposition. Firstly, do denoising

procession with SR and then do EMD decomposition. Experiments proved that this

method is better than direct EMD decomposition method.

21.2 Empirical Mode Decomposition

Empirical Mode Decomposition (EMD) is a novel method for adaptive of non-linear

and non-stationary signals. It can decompose any non-linear signal into several

Intrinsic Mode Functions (IMFs), and a residue.

21.2.1 IMFs

The components resulting from EMD, called Intrinsic Mode Functions (IMFs),

each admit an unambiguous deﬁnition of instantaneous frequency. By deﬁnition,

an Intrinsic Mode Function (IMF) satisﬁes two conditions

1. The number of extreme and the number of zero crossing may differ by no more

than one

2. The local average is zero

where the local average is deﬁned by the average of the maximum and minimum

envelopes discussed in the following section. These properties of IMFs allow for

instantaneous frequency and amplitude to be deﬁned unambiguously.

21.2.2 The Sifting Processing

In order to obtain the separate components called IMFs, we perform a sifting pro-

cess. The goal of sifting is to subtract away the large-scale features of the signal

repeatedly until only the ﬁne-scale features remain. A signal x.t/ is then divided

into the ﬁne-scale detail, h.t/ and a residual, m.t/ so x.t/ D m.t/ C h.t/.This

detail becomes the ﬁrst IMF and the sifting process is repeated on the residual

m.t/ D x.t/  d.t/.

The sifting process requires that a local average of the function be deﬁned. If

we knew the components before, we would naturally deﬁne the local average to be

the lowest frequency component. Since the goal of EMD is to discover these com-

ponents, we must approximate the local average of the signal. Huang’s solution to

ﬁnd a local average creates maximum and minimum envelopes around the signal by

21 Empirical Mode Decomposition Based on Bistable Stochastic Resonance 253

using natural cubic splines through the respective local extreme. The local average

is approximated as the mean of the two envelopes.

The ﬁrst IMF, c

.t/ of a signal, x.t/, is found by iterating through the follow-

ing loop.

1. Find the local extreme of x.t/.

2. Find the maximum envelope e

.t/ of x.t/ by passing a natural cubic spline

through the local maxima. Similarly ﬁnd the minimum envelope e



.t/ with the

local minima.

3. Compute an approximation to the local average, m.t/ D .e

.t/ C e



.t//=2.

4. Find the proto-mode function h

.t/ D x.t/  m.t/.

5. Check whether h

.t/ is an IMF. If h

.t/ is not an IMF, repeat the loop on h

.t/.

If h

.t/ is an IMF then set c

.t/ D h

.t/.

The sifting indicates the process of removing the lowest frequency information until

only the highest frequency remains. The sifting procedure performed x.t/ on can

then be performed on the residual r

.t/ D x.t/  c

.t/ to obtain r

.t/ and c

.t/,

repeat the process as described above for n time, then n IMFs of signal x.t/ could

be got. Then

.t/ D r

.t/  c

.t/

.t/ D r

.t/  c

.t/

.t/ D r

n1

.t/  c

.t/

(21.1)

At last, the signal x.t/ is decomposed into several Intrinsic Mode Functions (IMFs),

and a residue.

x.t/ D

iD1

.t/ C r

.t/ (21.2)

Residue r

.t/ is the mean trend of x.t/.TheIMFsc

;:::; c

include different

frequency bands ranging from high to low. The frequency components contained in

each frequency band are different and they change with the combination signal x.t/,

while r

.t/ represents the central tendency of signal x.t/.

21.2.3 Examples

In order to illustrate the performance of EMD, we use a combination of two pure

sine waves, which are added together as

x.t/ D a

sin.2f

t/ C a

sin.2f

t/ (21.3)

The values of f

and f

are respective 10 and 30 Hz, amplitudes are both 1.5.

254 Y.-J. Zhao et al.

Fig. 21.1 The EMD decomposed result of pure mixed signal

As shown in Fig. 21.1, mixed signal x.t/ is decomposed into two IMFs: c

and the residue r

c

is corresponding with sine wave of 30Hz, c

is the other sine

wave of 10 Hz. It can be seen from this example that, with the EMD method, signal

can be decomposed into some different time-scales IMFs by which the characteris-

tics of the signal can be presented in different resolution ratio.

However, signals are not always pure in practice. They usually mix much noise

which can affect EMD performance. In Fig. 21.2, the mixed signal is still composed

by two sine waves 30 and 10 Hz, but mixed noise of intensity D D 0:2. While the

amplitude of 30 Hz is still 1.5, but the one of 10 Hz changes into 0.1, belongs to

weak signals. As shown in Fig. 21.2, c

 c

are IMFs decomposed form EMD,

and r

is the residue. It can be seen from this ﬁgure that the ﬁrst IMF c

is mainly

composed of high frequency noise, and for the noise existence, the sine wave of

30 Hz is decomposed into c

and c

, the wave is serious distortion especially in

time domain. c

 c

are supposed to depict sine wave of 10 Hz, but due to small

amplitude, they cannot be distinguished in despite of in frequency spectrum. There-

fore, when signal is very weak and mixed with noise, direct EMD performance is

bad. It will result distorted IMFs and cannot detect the weak signal. So before EMD

operation denoising process is necessary.

21.3 Stochastive Resonance

Stochastic resonance (SR) is a phenomenon in which a nonlinear system heightens

the sensitivity to a weak signal input and when noise with an optimal intensity is

presented simultaneously. So SR doesn’t remove but make use of noise to gain the

optimal and desirable signal-to-noise ration (SNR) of output signals.

21 Empirical Mode Decomposition Based on Bistable Stochastic Resonance 255

Fig. 21.2 The EMD decomposed result of noisy and weak signal

21.3.1 Bistable System

SR is most often considered by the example of the motion equation of a light particle

in a bistable potential ﬁeld. It is disturbed by a small periodic signal and additive

white noise:

Px C f.t/ D A sin.!t / C n.t/; (21.4)

where x is the particle displacement, A sin !t is a weak periodic signal at frequency

!; f.x/ D dU.x/=dx; U.x/ is a symmetric double-well potential We will consider

the simplest case of such a potential, namely, U.x/ Dax

=2Cbx

=4n.t/ is white

noise of intensity D,i.e.n.t/n.t C / D Dı./. As it follows from [7–9], under

the condition of adiabatic elimination and small parameters (frequency, amplitude

and intensity of noise are less than 1), the power spectrum of (21.4) is described that

high frequency noise is weakened, and spectra energy is center around in the low fre-

quency area, especially, there is a peak at the frequency of !. However, the adiabatic

elimination SR theory in small parameters cannot meet larger signals in practice.

256 Y.-J. Zhao et al.

So the method of twice sampling stochastic resonance is recommended. According

to this method, we ﬁrst compress the measurement signals to low frequency ones in

a linear mode to meet the requirement of small parameters. Secondly, analyze the

compressed data spectrum to acquire characteristics of signals. At last we recon-

struct the compressed data to origin according to the linear mode mentioned earlier.

21.3.2 Examples

Figure 21.3 is a given example to illustrate the implementation of stochastic reso-

nance under larger parameters signal condition. As shown in Fig. 21.3,thef

value

is much more than 1, belongs to large parameter signals, so the twice sampling

Fig. 21.3 Implement of SR of large parameters single, where f

D 10 Hz;f

D 2;000 Hz, and

data length is 2,048. Parameter values are a D 0:1; b D 1; A D 0:5,andD D 0:6

21 Empirical Mode Decomposition Based on Bistable Stochastic Resonance 257

stochastic resonance method is adopted to compress it to small parameter signal.

The twice sampling frequency is selected 8 Hz, then the compressed frequency is

calculated 0.04 Hz. In the ﬁgure, (c) and (d) graphs display waveform and spectrum

of compressed signal, (c) and (d) is the SR output of compressed signal. Compared

with graph (c) and (e), because high-frequency noise in original signal is weakened,

the waveform of SR output becomes smooth, at the same time the amplitude of

signal is heightened, as shown in graph there is a peak at the frequency f

21.4 Test of EMD Based on SR Denoising

We still use the 10 and 30 Hz sine wave signals. Sampling frequency is 2 kHz and

data length is 2,048. Amplitude of 30 and 10 Hz signal is 1.5 and 0.1. The noise

of intensity is 0.2. Parameters of SR are set as: a D 0:1; b D 1, twice sampling

frequency is 8 Hz.

Figure 21.4 shows signal’s waveform and spectrum before and after SR process.

The 10 Hz weak signal nearly submergence in noise is seen in graph (b), but after

SR system the weak signal is enhanced as shown in graph (d). Figure 21.5 gives the

EMD decomposition results of signal which is processed by SR. It is clear that the

high frequency noise is ﬁltered by SR system and 10-Hz weak signal is enhanced. It

can be seen from frequency domain, the ﬁrst IMF c

is 30 Hz frequency component,

it is smooth and obvious. IMF c

is correspondence with 10Hz signal. IMF c

 c

for their small amplitudes, consider as residue.

Fig. 21.4 Waveform and spectrum of original signal before and after SR