Appl. Math. Inf. Sci. 6, No. 3, 689-695 (2012) 689

Applied Mathematics & Information SciencesAn International Journal

c⃝ 2012 NSPNatural Sciences Publishing Cor.

Nonlinear Distortion Identification based on Intra-waveFrequency ModulationYutian Wang1, Hui Wang1 and Qin Zhang1

Information Engineering School, Communication University of China, Beijing 100024, China

Received: Apr. 8, 2012; Revised May 4, 2012; Accepted May 26, 2012Published online: 1 September 2012

Abstract: Recently, most of the methods used to measure and analysis signal property are based on the linear transform theory, suchas FFT, STFT, wavelet, etc. Unfortunately, these methods usually cause meaningless results when it is used to analysis nonlinearsignal. In this paper, we use Hilbert-Huang transform (HHT) to review the nonlinear distortion and define a novel nonlinear parameternamed Nonlinear Distortion Degree (NDD) which is based on intra-wave frequency modulation measurement. The loudspeaker modelsimulations are used to illustrate the intra-wave frequency modulation caused by nonlinear distortion. The results agree that NDD canreveal more accurate and physical meaningful nonlinear distortion characteristic.

Keywords: Nonlinear distortion, HHT, intra-wave frequency modulation, loudspeaker.

1. Introduction

In the signal analysis field, Fourier based analysis meth-ods such as FFT and STFT take the dominated position.However, there are some crucial restrictions of the Fourierspectral analysis: the system must be linear and the datamust be strictly periodic or stationary; otherwise, the re-sulting spectrum will make little physical sense. By far,some time-frequency analysis methods, including waveletanalysis, Wigner-Ville distribution, etc. have been used fornon-stationary signal analysis and achieve excellent result-s, but they have their respective limitations [1]. Because oflimited length of the basic wavelet function, wavelet maycause energy leakage. Beside this, wavelet analysis has thedifficulty for its non-adaptive nature when it encountersnonlinear problem in which deformed wave-profile maycause spurious harmonics. The difficulty with WVD is thesevere cross terms which will make some frequency rangeshave negative power. On the other hand, limited by sig-nal analysis principle, researches on the nonlinear distor-tion phenomenon did not get significant progress. Recent-ly, nonlinear distortion is measured in term of total har-monic distortion (THD) which is the ratio of the harmon-ic wave energy to total signal energy. Obviously, it is toorough for the interesting details.

Huang et. al. raised a novel signal processing methodnamed Hilbert-Huang Transform (HHT) based on EMD(Empirical Mode Decomposition), which is suitable foranalyzing nonlinear and non-stationary signal [2]. Differ-ent from the traditional signal processing methods, the H-HT is an adaptive decomposition method and can yieldmore physical results. EMD is a complete, approximatelyorthogonal and self-adaptive method which has the abilityto decompose signal by time scale. Some numerical exper-iments show that EMD behaves as a dyadic filter bank [3,4].

As a strongly nonlinear system, loudspeaker is asso-ciated with several nonlinear effects such as electronic,magnetic, mechanical and sound. In the early stage, thedisplacement of the diaphragm in the low-frequency rangewas found can be described by Duffing equation [5]. Loud-speaker is considered as a mass/spring device driven by aLorentz force and loaded by acoustical impedance. Such amodel was presented in term of equivalent parameters byThiele and Small [6].

In this paper, the HHT method and its improvementswas studied firstly. Then, a novel nonlinear distortion pa-rameter called nonlinear distortion degree (NDD) is pre-sented based on intra-wave frequency modulation whichis a unique definition in HHT. Finally a Duffing-like loud-

∗ Corresponding author: e-mail: wangyutian@cuc.edu.cnc⃝ 2012 NSP

Natural Sciences Publishing Cor.

690 Yutian Wang et. al. : Nonlinear Distortion Identification ...

speaker model is learned and simulated under differentvoltages. The simulation results demonstrate that the waveprofile deformation caused by nonlinear distortion shouldbe represented as intra-wave frequency modulation. It alsoverifies NDD is more accuracy and physical for nonlinearmeasurement.

2. Hilbert-Huang Transform Method

To analysis the nonlinear and non-stationary signal, thedefinition of instantaneous frequency and energy is need-ed, which must be functions of time. The principle of in-stantaneous energy and envelop is accepted widely, but theopinion of instantaneous frequency was under argumentsfor a long time until Hilbert Transform was introduced todefine analytic signals by Gabor [7].

2.1. Instantaneous Frequency and EmpiricalMode Decomposition

Gabor’s theory is given as follow: for an arbitrary real-valued signal, X(t), the Hilbert transform is defined as

Y (t) =1

πP

∫ +∞

−∞

X(τ)

t− τdτ (1)

where P denotes the Cauchy principal value. X(t) andY (t) form the complex conjugate pair, then we can havethe analytic signal, Z(t).

Z(t) = X(t) + jY (t) = a(t)ejθ(t) (2)

in which

a(t) =√X2(t) + Y 2(t) (3)

θ = arctanY (t)

X(t)(4)

Here a is the instantaneous amplitude, and θ is the in-stantaneous phase function. The instantaneous frequencyis simply defined as

ω(t) =dθ(t)

dt(5)

With both amplitude and frequency being a function oftime, we can express the amplitude in terms of a functionof time and frequency. However, Cohen point out that onlythe ’monocomponent signal’ can be transformed to instan-taneous frequency by Hilbert transform[8]. The monocom-ponent signal means that at any given time, there is onlyone frequency value. The definition of ’monocomponent’is not clear before N. E. Huang proposed the definition ofintrinsic mode function (IMF). An IMF is a function thatsatisfies two conditions[2]: (1) in the whole data set, thenumber of extrema and the number of zero crossings must

either equal or differ at most by one; and (2) at any point,the mean value of the envelope defined by the local max-ima and the envelope defined by the local minima is ze-ro. Unfortunately, most of the practical data are not IMF-s. Therefore, a method called Empirical Mode Decompo-sition (EMD) was proposed to decompose a complicatedsignal into a group of IMFs. Given a signal x(t), the effec-tive algorithm of EMD can be summarized as follows:

1.Find out all local extrema of x(t), obtain the upper andlower envelope which are the cubic spline interpolatedof its local maxima and minima.

2.Calculate the mean function m1(t) of the envelopes.3.Compute the residue value h1(t) = x(t) − m1(t). Ifh1(t) satisfies the two conditions of IMF, it should bean IMF. Otherwise, treat h1(t) as the signal, repeat theshifting steps 1-3 on it until the shifting result is anIMF, we denoted it as c1(t).

4.Obtain the residue r1(t) = x(t) − c1(t). Apply theabove procedure on it to extract another IMFs.

5.The process is repeated until the last residue rn(t) is amonotonic function or has at most one local extremum.

The shift result is highly determined by the stop criterion.Different stop criterions make results vary. Although thereare several kinds of stop criterion, Wu et. al. suggest fixedsifting time criterion[9]. In separate research, Flandrin andWu point out that EMD is in fact a dyadic filter bank[3,4].In the research by Wu and Huang[10], the dyadic propertyis available only shift times is about 10. Too many or toofew iterator numbers would decrease the dyadic property.

2.2. Time-Frequency Distribution Spectrum

In order to obtain the time-frequency-energy distributionof given signal, a naturally step is to apply the Hilberttransform to each IMF component and calculate the in-stantaneous frequency by means of Equation (5). Unfortu-nately, some theoretic condition limit this presumable ap-plication. A major problem is caused by Bedrosian’s theo-ry[11].

From Equation (5) we can conclude that the instan-taneous frequency is only determinate by phase function.For any IMF x(t), we can express it in terms of

x(t) = a(t) cos θ(t) (6)

The physically meaningful instantaneous frequency requirethe signal to satisfy

H [a(t) cos θ(t)] = a(t)H [cos θ(t)] (7)

where H[·] denotes the Hilbert transform. The Bedrosiantheory point out that only if the Fourier spectrum of a(t)and cos θ(t) are absolutely disjoint in frequency space andthe the frequency range of the cos θ(t) is higher than a(t),the Equation (7) is valid. Unfortunately, practical data sel-dom satisfy this condition. To overcome this difficulty, a

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normalization scheme is proposed by Huang et. al.[12].The algorithm is given as follows: for arbitrary data x(t)given as Equation (6),

1.Find all of the maximum points of the absolute valueof x(t).

2.Get the upper envelop e1(t) by spline interpolation ofthese maximum points

3.Conduct the normalization scheme on the x(t):

f1(t) =x(t)

e1(t)(8)

Ideally, f1(t) should be identical to cos θ(t), but theenvelop e1(t) often cut x(t) when a(t) changes sharply.Under this circumstance, f1(t) may have values higherthan 1.

4.To obtain useful FM part, the normalization is used asan iterative process

fn(t) =fn−1(t)

en(t)n = 2, 3, . . . (9)

until all the maxium values are 1.5.Then the FM and AM components are defined as

F (t) = fn(t) (10)

A(t) =x(t)

F (t)(11)

The combination of the normalization scheme and the ap-plication of Hilbert transform to the FM component arecalled Normalized Hilbert transform (NHT). Then the time-frequency distribution can be defined as follows:

H(ω, t) = Re

[n∑

i=1

Ai(t)ej∫Fi(t)dt

](12)

3. Intra-wave Frequency Modulation andNonlinear Distortion Degree

In the HHT principle, the fluctuating of instantaneous fre-quency in one oscillation is regarded as intra-wave fre-quency modulation. The intra-wave frequency modulationis a unique definition of HHT and based on a commonphenomenon: if the frequency changes from time to timewithin a wave, its profile can no longer be a pure sine orcosine function. Therefore, any wave-profile deformationfrom the simple sinusoidal implies the intra-wave frequen-cy modulation [2]. However, this important nonlinear in-formation is totally lost in Fourier spectral analysis andwavelet analysis. In the past, the wave profile deformationwas treated as harmonic distortion. Intra-wave frequencymodulation offers new understanding on nonlinear oscilla-tion systems in more details.

With the power of HHT, we can discuss the definitionof nonlinear distortion. To quantify the nonlinear distor-tion, an index is needed to give a quantitative measure of

how far the final outputs of nonlinear system deviates fromoriginal signal.

To define the nonlinear distortion degree, NDD, thefirst step is to obtain the marginal spectrum M(ω) of sig-nal. Marginal spectrum is defined as the integral of HHTtime-frequency spectrum

M(ω) =

∫ ∞

0

H(ω, t)dt (13)

which offers a measure of total amplitude contribution fromeach frequency value.

Consequently, the degree of nonlinear distortion is de-fined as the max deviation from the carrier frequency.

NDD =max |M(ω)− fc|

fc(14)

Obviously, for a linear system, the output signal’s marginalspectrum will concentrate to the carrier signal frequency.Then, the NDD will be identically zero. Under this condi-tion, the Fourier spectrum can equal to the marginal spec-trum and make physical sense. If the marginal spectrumspreads into a frequency range, this index will no longerbe zero, then the physical sense of Fourier spectrum willdecrease. The higher the NDD value, the more nonlinearis the signal.

The degree of nonlinear can also be a function of timeimplicitly, because the definition depends on the time lengthof integration of marginal spectrum as shown in Equation(13). As a result, a signal can be piecewise linear. There-fore, the nonlinear distortion degree can be modified tovary with time. The NDD(t) is defined as

NDD(t) =max |M(ω)− fc|

fc(15)

in which the overline indicates average in a short time spanat time t. For based on intra-wave frequency modulation,the NDD have the potential to reveal more details in non-linear distortion.

4. Loudspeaker Experimental Results

4.1. The Loudspeaker Model

The large signal behavior in low frequency rang of loud-speaker can be modeled by a group of simplified nonlin-ear differential equations[13], such as Equation (16) andEquation (17).

E(t) = Rebic +d(L(x)ic)

dt+ Φ(x)x (16)

Φ(x)ic = Mmsx+Rmsx+ k(x)x− 1

2

dL(x)

dxi2c (17)

The Equation (16) is the electrical part of the mod-el, where E(t) is the voltage, ic is the current and x is

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692 Yutian Wang et. al. : Nonlinear Distortion Identification ...

the loudspeaker diaphragm displacement. x and x denotethe loudspeaker diaphragm velocity and acceleration re-spectively, where the overdots represent first and secondderivatives of x. The main electrical parameters are thevoice-coil electrical resistance Reb and the voice-coil non-linear self inductance L(x). The Equation (17) describesthe mechanical part, which is modeled as a damped mass-spring mechanical system, in which the Mms is movingmass (including air mass), Rms is mechanical dampingand k(x) is nonlinear stiffness. Beside them, there is anadded reluctance force − 1

2dL(x)dx i2c(t) caused by the non-

linear self-inductance. The two equations are connectedby the loudspeaker motor interdependence between them.Additionally, the nonlinear force factor Φ(x) of Equation(16) denotes the ratio between the force produced by themotor and the current.

The nonlinear differential equation that describes thenonlinear vibration can be derived if we approximate Φ(x),L(x) and k(x) by a truncated power series:

Φ(x) = ϕ0 + ϕ1x+ ϕ2x2 + · · ·+ ϕnx

n (18)

L(x) = l0 + l1x+ l2x2 + · · ·+ lnx

n (19)

k(x) = k0 + k1x+ k2x2 + · · ·+ knx

n (20)

The nonlinear coefficients ϕi, li and ki can be esti-mated from experimental results, using the procedure de-veloped by Park and Hong[14]. Using such a model, theharmonic distortion of the diaphragm velocity at low fre-quency range can be predicted with good agreement of realmeasurements.

The model simulation and real loudspeaker measure-ment are studied later. The loudspeaker to be analyzedis TianAi PW-11C, with the parameters:Reb = 5.87(Ω),Leb = 207.321(µH), Mms = 2.0144(Ns/m), k0 =

2282(N/m), Φ0 = 4.762(N/A) and E(t) =√2A sin(ωt).

The same parameters are also used in the model.

4.2. Model Simulation

The distortion analysis for the loudspeaker was carried outby means of simulation tests with 1v, 5v, 10v and 15vat 125Hz, 250Hz and 500Hz stimulate signal respective-ly.The instantaneous frequency is calculated from the di-aphragm displacement with different driven voltage andshown in terms of overlapping the diaphragm displace-ment diagram with the same time axis.

At the beginning of our simulation, we carried out thetest under small stimulate signal at low frequency. In theFigure (1), with the 125Hz sinusoidal driven signal at 1v,the vibration of the diaphragm is almost similar with puresinusoidal wave and the instantaneous frequency is fair-ly close to a horizontal line. We can learn form the figurethat deformation appears at both ends of the instantaneous

frequency curve. In the EMD algorithm, the cubic splinesinterpolation creates top and bottom envelopes which areimplemented in the first step of the shifting process. It isdifficult to interpolate data near the beginnings or ends,where the cubic splines can have swings. Estimation of topand bottom envelopes is difficult as there are not enoughdata [15]. From the diagram, we can reach the conclusion:with small stimulate voltage, loudspeaker behaves like lin-ear system.

0 0.05 0.1 0.15 0.2

−0.50

0.5

x 10−3

Time(s)

Dia

phra

gm d

ispl

acem

ent(

m)

HHT spectrum

0 0.05 0.1 0.15 0.20

100

200

Inst

anta

neou

s fr

eque

ncy(

Hz)

Diaphragm displacementInstantaneous frequency

Figure 1 loudspeaker model simulate result with 1v and 125Hzsine signal stimulate

With the increasing of the input signal voltage from1v to 15v, the nonlinear effect is enhanced notably. In thetraditional Fourier transform analysis, this nonlinear phe-nomenon is recognized as harmonics in infinite frequencyrange. Figure (2) demonstrate that the instantaneous fre-quency fluctuates in the whole time length. The degree ofthe oscillation reflects the nonlinear distortion.

0 0.02 0.04 0.06 0.08 0.150

100150

Instantaneous frequency spectrum

1v

0 0.02 0.04 0.06 0.08 0.150

100150

7v

0 0.02 0.04 0.06 0.08 0.150

100150

10v

0 0.02 0.04 0.06 0.08 0.10

100200

15v

time (s)

Figure 2 Instantaneous frequencies of model test results with 1v,7v, 10v, 15v stimulate at 125Hz

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Under different driven signal voltage, there is a signif-icant phenomenon: following the increasing of stimulatevoltage, the nonlinear effects grow correspondingly. It isdue to the nonlinear of electrodynamic coil-magnet systemand the large displacements of the rim[16]. In the processof oscillation, the nonlinear effects make the diaphragmdisplacement no longer a pure sinusoidal function, whichis a typical intra-wave frequency modulation phenomenon.By contrast, due to the nonlinear deformation of diaphrag-m vibration, in linear analysis methods, it leads to variousphenomena: harmonic, subharmonic and superharmonicfrequency entrainment and chaotic behavior in small rangeof control parameters[17–19]. These frequency structuresreveal that the output of loudspeaker have the nonlineardistortion which is expressed as intra-wave frequency mod-ulation in HHT principle uniquely. The intra-wave frequen-cy modulation means the frequency changes within a wave-length whose wave profile is no longer a pure sinusoidalfunction. Therefor, the the wave profile deformation whichis known as nonlinear distortion implies the intra-wave fre-quency modulation. The nonlinear distortion usually illus-trate in terms of harmonics in Fourier analysis. However,these harmonics generally do not have physical meaningbecause FFT is not suitable for nonlinear signal. The intra-wave frequency modulation is a better definition of thesenonlinear phenomena.Furthermore, based on the principleof HHT, the distortion components do not spread energyover whole frequency range. It just concentrate into a nar-row frequency band which match the practical perceptionbetter.

4.3. Experiment Results

The measurements for the loudspeaker was carried out with137.5Hz, 275Hz, 550Hz, 1100Hz, 2200Hz, 4400Hz and8800Hz respectively.

The test result of the stimulate signal from 137.5Hz to4400Hz is shown in Figure (3), in which the instantaneousfrequency of the loudspeaker output fluctuates 2 times inone wave cycle. Obviously, the structure of the loudspeak-er output instantaneous frequency is illustrated as intra-wave frequency modulation. Furthermore, the frequencydeviation varies under different stimulate frequency. From137.5Hz to 1100Hz, the IF fluctuate range decrease to al-most zero. On the other hand, from 1100Hz to 4400Hz, therange raise significantly. This means the smallest nonlin-ear distortion appears at 1100Hz and increases when stim-ulates become both higher and lower.

When stimulate frequency increases to 8800Hz, the in-stantaneous frequency of the loudspeaker output disuniteinto 3 IMF. The joint distribution is shown in Figure (4).From the diagram, we can see that the highest frequencyoscillation mode take the dominate percentage of the en-ergy. Nevertheless, there are two extra component whichindicate that the nonlinear distortion under 8800Hz stim-ulate is very high and have the crosstalk distortion withlower frequency components.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090

500

1000

1500

2000

2500

3000

3500

4000

4500

Time(s)

Inst

anta

neou

s fr

eque

ncy

Figure 3 Instantaneous frequency of loudspeaker output with137.5Hz, 275Hz, 550Hz, 1100Hz, 2200Hz and 4400Hz stimu-late.

Figure 4 The time-frequency-amplitude distribution of 8800Hzstimulate.

4.4. NDD and THD Comparison

Total Harmonic Distortion (THD) is a common parameterto characterize nonlinear distortion, which is defined as

THD =

√∞∑i=2

H2i

H1(21)

where H1 denotes the level of fundamental frequency andHi denotes the level of ith harmonic. The difficulty ofTHD is the Fourier spectrum lost some important nonlin-ear information which represents the characteristic of sys-tem. Therefore, the THD offers only an approximation forthe distortion.

The NDD and the THD are measurement for all theexperiment data. The results under different voltages areshown in Figure (5).

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694 Yutian Wang et. al. : Nonlinear Distortion Identification ...

0 5 10 150

0.1

0.2

0.3

0.4

stimulate voltage (v)

ND

D

125Hz250Hz500Hz

0 5 10 150

0.2

0.4

0.6

0.8

1

stimulate voltage (v)

TH

D (

%)

125Hz250Hz500Hz

Figure 5 NDD and THD measurement results for the simulationdata.

Both of NDD and THD reveal the nonlinear character-istic from different aspect. From the results of NDD andTHD, we can extract the general character: with the in-creasing of stimulate signal voltage, the distortion of loud-speaker output decrease notably. On the other hand, theNDD parameter clearly demonstrate the increasing trendof distortion with the raise of stimulate voltage. By con-trast, the THD results fail to show this important charac-teristic. The difference between the THD and NDD agreesthat NDD can represent more detail of nonlinear systemand have more physical sense.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.2

0.4

0.6

0.8

ND

D

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.2

0.4

0.6

0.8

1

TH

D(%

)

Frequency(HZ)

Figure 6 NDD and THD measurement results for the experi-ment data.

Figure (6) compares the NDD and THD on the real ex-periment measurements with different stimulate frequen-cy. Both of them agree that the distortion have low levelin middle frequency range. The NDD curve line is moresmooth than THD. The lowest point appears at 1100Hz in

both THD and NDD curve. However, in THD measure-ment, the distortion in high frequency range do not havesignificant increase. Part of the reason is the THD mea-surement do not consider the sub-harmonic energy. Onthe other hand, in NDD diagram, the distortion in highfrequency range increases notably. From Figure (3) weknow that in high frequency range, the intra-wave frequen-cy modulation degree increases with the rising of frequen-cy. Therefore, the NDD value render the real trend of loud-speaker distortion.

The Fourier spectrum uses linear superposition of trigono-metric functions, therefore, it needs additional harmonicsto simulate the deformed wave shape caused by nonlineareffects. As a result, the harmonic explanation for nonlin-ear phenomena just fulfil the mathematical requirementsfor fitting the data but not have physical mean. Therefore,the THD which is based on harmonics level loss some im-portant information about the nonlinear distortion. In thisway, the NDD is an better alternative to character nonlin-ear distortion.

5. Conclusion

In this paper a novel nonlinear distortion identification pa-rameter called nonlinear distortion degree (NDD) is pro-posed, which is based on Hilbert-Huang transform. TheHHT method absolutely lies on the character of the signaland have local-adaptive property. Consequently, HHT canavoid the shortcomings of linear transform methods andis very applicable to analyze nonlinear and non-stationarysignal. The loudspeaker model was simulated and the re-sults in terms of diaphragm displacement were obtained.By applying HHT on the output of the loudspeaker mod-el, the instantaneous frequency spectrum shows that theessential of loudspeaker distortion is intra-wave frequen-cy modulation which relates the deformation of diaphragmdisplacement from simple sinusoidal. By way of analyzingthe structure of time-frequency spectrum of model simu-lation output, NDD demonstrate better performance thanclassic THD parameter.

Acknowledgement

This work is supported by the National Natural ScienceFoundation of China (Grant No. 61071149) , the State Ad-ministration of Radio Film and Television of China (GrantNo. 2011-11) and Program for New Century Excellent Tal-ents of Ministry of Education of the People’s Republic ofChina (Grant No. NCET-10-0749).

References

[1] A. G. Espinosa, J. A. Rosero, J. Cusido, L. Romeral And J.Ortega, IEEE Trans. Energy Convers. 99, 1 (2010).

c⃝ 2012 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci. 6, No. 3, 689-695 (2012) / www.naturalspublishing.com/Journals.asp 695

[2] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih,R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.454, 903(1998).

[3] Z. H. Wu and N. E. Huang, R. Soc. Lond. Proc. Ser. A Math.Phys. Eng. Sci. 460, 1597 (2004).

[4] P. Flandrin, G. Rilling and P. Goncalves, IEEE Signal Pro-cess. Lett. 11, 112 (2004).

[5] H. F. Olson, J. Acoust. Soc. Am. 16, 100 (1944).[6] R. Small, IEEE Trans. Audio Electroacoust. 19, 269 (1971).[7] D. Gabor, J. El. Eng. 93, 429 (1946).[8] L. Cohen, Time-frequency analysis (Englewood Cliffs, NJ:

Prentice-Hall, 1995).[9] G. Wang, X. Y. Chen, F. L. Qiao, Z. Wu and N. E. Huang,

Adv. Adaptive Data Anal. 2, 277 (2010).[10] Z. Wu and N. E. Huang, Adv. Adaptive Data Anal. 2, 397

(2010).[11] E. Bedrosian and A. H. Nuttall, 54th Proc. IEEE, 1458

(1966).[12] N. E. Huang, Z. Wu, S. R. Long, K. C. Arnold, X. Chen and

K. Blank, Adv. Adaptive Data Anal. 1, 177 (2009).[13] W. Klippel, J. Audio Eng. Soc. 40, 483 (1992).[14] S. T. Park and S. Y. Hong, J. Audio Eng. Soc. 49, 99 (2001).[15] Z.D. Zhao and Y. Wang, Proceedings of the International

Conference on Communications, Circuits and Systems, 841(2007).

[16] W. Klippel, 109th Audio Engineering Society Convention,5261 (2000).

[17] A. Petosic, I. Djurek and D. Danijel, J Acoust Soc Am. 123,3696 (2008).

[18] R. Yamapi and S. Bowong, Communications in NonlinearScience and Numerical Simulation 11, 355 (2006).

[19] M. Siewe, F.M. Moukam Kakmeni, S. Bowong and C. Tcha-woua, Chaos Solitons Fractals 29, 431 (2006) .

Yutian Wang received theB. E. degree from Communi-cation University of China in2007. He is currently pursuingthe Ph. D. degree in the Com-munication University of Chi-na. His research interests lie pri-marily in the signal process-ing theory and nonlinear sys-tem toward acoustic application,

including timbre model, source separation and instrumentsrecognition, etc.

Hui Wang received the Ph.D.degree from Communication U-niversity of China, Beijing, Chi-na, in 2011. He is currently aprofessor of Information En-gineering School, Communica-tion University of China . Hisresearch interests include au-dio signal processing, Bayesianstatistical model, sound field re-

production and broadcasting and television technology.

Qin Zhang earned a doc-tor degree from the Universityof British Columbia, Vancou-ver, B.C. Canada 1990. Cur-rently, he is professor and di-rector for Advanced Media Labat the Communication Univer-sity of China, Beijing, China.He also served as a senior re-viewer for Nature Science Foun-

dation of China. His research interests are media technolo-gy and applications in image coding, theater music preser-vation and reproduction system, electromagnetic wave andacoustic wave propagation.

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