A robust method for parameter estimation of AR systems using empirical mode decomposition

SIViP (2010) 4:451–461DOI 10.1007/s11760-009-0134-3

ORIGINAL PAPER

A robust method for parameter estimation of AR systemsusing empirical mode decomposition

Md. Kamrul Hasan · M. Shakib Apu ·Md. Khademul Islam Molla

Received: 27 October 2008 / Revised: 20 August 2009 / Accepted: 22 August 2009 / Published online: 10 September 2009© Springer-Verlag London Limited 2009

Abstract This paper presents a robust algorithm for param-eter estimation of autoregressive (AR) systems in noise usingempirical mode decomposition (EMD) method. The basicidea is to represent the autocorrelation function of the noise-free AR signal as the summation of damped sinusoidalfunctions and use EMD for extracting these component func-tions as intrinsic mode functions (IMFs). Unlike conven-tional correlation-based techniques, the proposed schemefirst estimates the damped sinusoidal model parameters fromthe IMFs of autocorrelation function using a least-squaresbased method. The AR parameters are then directly obtainedfrom the extracted sinusoidal model parameters. Simulationresults show that EMD is a very promising tool for AR systemidentification at a very low signal-to-noise ratio (SNR).

Keywords AR system · Parameter estimation · Empiricalmode decomposition · Intrinsic mode functions

1 Introduction

Parameter estimation of stochastic signal model is an impor-tant issue in various fields of science and engineering, e.g.,econometrics, geophysics, speech processing, image process-ing, biomedical signal processing, and communication [1].The most popular stochastic signal model is the Gaussian,

Md. K. Hasan (B) · M. S. ApuDepartment of Electrical and Electronic Engineering,Bangladesh University of Engineering and Technology,Dhaka 1000, Bangladeshe-mail: [email protected]

Md. K. I. MollaDepartment of Computer Science and Technology,Rajshahi University, Rajshahi, Bangladeshe-mail: [email protected]

minimum phase, AR model. In time series analysis and signalmodeling, both noise-free and noisy autoregressive (AR)systems have been extensively studied by many researchers[1,4].

Various methods have been reported in the literature forthe identification of AR systems, e.g., Yule–Walker method,Least-square (LS) method, improved least-square (ILS)method, lattice filter, Burg algorithm [5], and maximum like-lihood method [4,6]. Several noise compensation techniqueshave also been reported as an extension of these basic algo-rithms to identify AR systems from noise-corrupted observa-tions [7]. Identification of all the modes of the AR system in asingle pass is a common feature to all these algorithms. Noneof these algorithms look for the intrinsic modes of the system.A damped sinusoidal model based method has been reportedin [2] to identify AR systems in low-noise environments. Thismethod has demonstrated improved robustness by extractingthe system modes one by one sequentially. It is, however,computationally expensive as compared to the one pass tech-niques. The increased computational cost is due to iterativesearch of the system mode parameters from their full domain.A partial knowledge of the system modes will obviouslyreduce computational cost of the algorithm significantly.

Empirical mode decomposition (EMD), proposed byHuang et al. [8], is a technique developed specifically fordecomposing signals from non-stationary and nonlinear pro-cesses into intrinsic mode functions (IMFs). The method isfully data-driven and requires no prior assumption on thebasis functions for performing the decomposition. This inher-ent adaptive nature of the algorithm is particularly suitable fordecomposition and analysis of signals of diverse applicationareas. The basis of EMD is the IMFs constructed from thegiven signal using a sifting process. As this process is fullydependent on extrema finding and an interpolation method,the EMD is highly sensitive to data uncertainty due to noise.

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In this paper, we introduce a new AR parameter estima-tion scheme via modeling of the autocorrelation function ofthe noise-free AR signal by the IMFs. As the EMD methodwas found sensitive to data uncertainty and closely spacedpole positions, we modify the procedure of selecting IMFsusing the concept of the sifting algorithm reported in [2]to improve its robustness. The AR parameters are obtainedfrom the IMFs of the modified EMD. Unlike [2], a dampedsinusoidal is fitted to each IMF rather than to the total auto-correlation function or to its successive residues. This makesthe proposed EMD-based scheme more noise robust than thetechnique in [2].

The rest of the paper is organized as follows. In Sect. 2,we formulate the problem. In Sect. 3, we derive the dampedsinusoidal model of the autocorrelation function of an ARsignal. Section 4 introduces the EMD method for AR param-eter estimation. Section 5 presents numerical results to dem-onstrate the strength of the proposed method compared toother well-known methods, followed by Sect. 6 where thepaper is concluded with some remarks.

2 Problem formulation

The input–output relationship of a pth order AR process canbe expressed by the difference equation as

A(z)x(n) = u(n) (1)

where the unknown input u(n) is a sequence of zero-meanwhite Gaussian noise with unknown variance σ 2

u , x(n)

denotes the output signal, and A(z) = 1 + a1z−1 + a2z−2 +· · · + apz−p. Here, z−1 is the unit delay operator, i.e., z−1

x(n) = x(n − 1) and ak (k = 1, 2, . . . , p) are the unknownAR parameters. The order p of the AR system is assumed tobe known.

In many practical situations, observation noise corruptsthe data samples. Then the observed process y(n) can beexpressed as

y(n) = x(n) + v(n) (2)

The additive noise v(n) is assumed to be a zero-mean whiteGaussian noise with unknown variance σ 2

v .The objective of this paper is to propose a novel method

for parameter estimation of AR systems using EMD andthe damped sinusoidal model (DampSine) of autocorrelationfunction of the noise-free AR signal. The IMFs of noise-com-pensated autocorrelation function, which are also theoreti-cally the damped sinusoidal components, are first extractedusing the EMD method. The decay rate and frequency ofoscillation of each IMF are then computed using the least-squares (LS) method. The desired AR parameters {ak} arethen directly obtained from these model parameters.

3 Damped sinusoidal model representationof autocorrelation of AR signals

As reported in [2], the damped sinusoidal model of auto-correlation function of clean AR signals can be obtained asfollows. The transfer function of a pth order AR system inthe z-domain can be expressed as

H(z) = 1

A(z)=

p∑

k=1

Ck

1 − zk z−1 (3)

where zk denotes the kth pole of the AR system and Ck isthe partial fraction coefficient corresponding to the kth pole.The unit impulse response h(n) of the causal AR systemdescribed in (3) can be expressed as

h(n) =p∑

k=1

Ck(zk)n (4)

If this AR system is excited by a white noise sequence u(n),the response x M (n) is given by

x M (n) = u(n) × h(n) =n∑

l=0

u(l)h(n − l) (5)

Using (4), (5) can be written as

x M (n) =p∑

k=1

n∑

l=0

Cku(l)(zk)n−l (6)

Clearly, x(n) and x M (n) are the same except for a delaybecause (1) is the difference equation implementation ofinput–output using the system parameters and (6) is the con-volution sum implementation of the same using the systemroots. Using (6), the autocorrelation of the noise-free signalx M (n) can be obtained as

RMxx (m) = Rxx (m) =

p∑

k=1

βk(zk)m (7)

where

βk = σ 2u

⎡

⎣ C2k

1 − z2k

+p∑

q=1,q �=k

CkCq

1 − zk zq

⎤

⎦ (8)

The coefficient βk may be real or complex depending onwhether the pole is real or complex. Since x(n) is real, in thelatter case, a complex pole will always be accompanied by itscomplex conjugate pole. Considering the effect of complexand real poles, (7) can be simplified as

Rxx (m) =g∑

j=1

(r j

)m [Pj cos(ω j m) + Q j sin(ω j m)

](9)

where g = {number of complex conjugate pair of poles +number of real poles}, r j is the magnitude of the j th pole

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and Pj and Q j are constants. In general, r j governs the decayrate of the AR system response and ω j determines the angularposition of the pole of the AR system in the z-plane.

4 EMD method for AR parameter estimation

4.1 EMD of autocorrelation of AR signals

The key benefit of EMD method lies in its power of automaticdecomposition of signals into a sum of oscillatory functions,called the IMFs. An IMF satisfies two definitive require-ments: (i) in the whole data set, the numbers of extrema andzero-crossings are the same or they differ at most by one,and (ii) it is symmetric with respect to the local mean. Thereexist many approaches of computing EMD [9,10].

The EMD method will be used in this paper to decomposethe estimated autocorrelation function from noisy AR signalsinto IMFs. As the damped sinusoidal functions in (9) satisfythe two aforesaid conditions of an IMF, the autocorrelationfunction of an AR signal has a close relation with the IMFsof the EMD method. The sifting process for extracting theIMFs from a given signal Rxx (m) is summarized below asfollows:

1. Detect the extrema (maxima and minima) of Rxx (m),−M ≤ m ≤ M .

2. Generate the upper and lower envelopes U (m) and L(m),respectively, by connecting the maxima and minima sep-arately with cubic spline interpolation.

3. Determine the local mean as M(m) = [U (m)+L(m)]/2.4. IMF should have zero local mean. Subtract M(m) from

the original signal Rxx (m) to get the first componentR1(m) = Rxx (m) − M(m).

5. Decide if R1(m) is an IMF or not by checking the twobasic conditions as described above.

6. Repeat steps 1 to 5 assuming R1(m) be the new dataset until R1(m) is an IMF. The first IMF is denoted byR1(m), which is the smallest temporal scale in Rxx (m).To find the rest of the IMF components, generate the res-idue Res1(m) of the data by subtracting R1(m) from thesignal Rxx (m) as Res1(m) = Rxx (m)− R1(m). The sift-ing process is continued until Resg(m) is smaller than apredefined value or becomes a monotonic function fromwhich no more IMF can be extracted. At the end of thesifting process, the signal Rxx (m) can be represented as

Rxx (m) =g∑

i=1

Ri (m) + Resg(m) (10)

where g denotes the number of IMFs and Resg(m) is thefinal residue.

In practical applications, the signal model in (9) isunknown and therefore an estimate of the autocorrelationfunction can only be obtained as

Rxx (m) = 1

N

N−|m|+1∑

n=0

x(n + m)x(n) (11)

Because of finite data length, Rxx (m) differs from Rxx (m) =E[x(n + m)x(n)], where E[·] denotes the expectation oper-ator. When the input to the AR system is a random whitenoise, Rxx (m) is a stochastic process. Now, the question ishow EMD behaves when the estimate of the autocorrelationfunction given by (11) is used instead of true autocorrelationfunction in (9). It is observed that when angular separation ofthe poles is less than a threshold, the EMD cannot accuratelyrecover the true intrinsic mode functions from the estimatedautocorrelation function, and thus fails to estimate the ARparameters even if there is no noise in the system.

When the observations are corrupted by noise, the cleanAR signal autocorrelation function is to be estimated fromthe noisy observations y(n) for EMD analysis. Otherwise,the impulse present in the autocorrelation function due tonoise at zero lag will corrupt the desired IMFs significantly.Conventionally, the autocorrelation sequence, Rxx (m), of thenoise-free AR signal is related to the observed noisy signalautocorrelation, Ryy(m), as given below:

Rxx (m) ={

Ryy(m) − σ 2v , m = 0

Ryy(m), otherwise(12)

Equation 12 is obtained using the strict assumption that thenoise and AR signal are uncorrelated. This assumption ismainly responsible for poor performance of the conventionalapproaches for AR system identification at low SNRs [2]. Inthis work, we do not strictly employ this assumption. Butto reduce the strong noise effect at zero lag, we subtract thenoise power as in (12). The effect of noise in other lags isreduced by using a curve fitting technique to the IMFs asexplained in the next section.

In practical applications the effect of noise cannot beavoided and, therefore, to obtain an estimate of Rxx (m) using(12), the noise power, σ 2

v , is to be estimated first from thegiven noisy observations only. It can be estimated from thehigh-frequency components of the discrete cosine transform(DCT) coefficients of y(n) [11]:

σ 2v =

[MAD

0.6745

]2

(13)

where MAD refers to the median absolute deviation of the last512 coefficients of 4,096-point DCT of y(n). If there is a poleat ‘π ’, this estimate will be biased because of the presence ofsome signal components in these coefficients. To overcomethis problem we use the following strategy. The DCT coef-ficients have larger values at the pole locations; elsewhere

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they have lower values representing the DCT coefficients ofwhite noise. We, therefore, divide the whole range of 4,096points into 32 slots, each containing 128 coefficients. Thenthe root-mean-square (RMS) value of the 128 coefficients ineach slot is calculated and the average of 32 such RMS valuesis determined. The slots containing the effect of a pole willobviously have higher RMS value than this average value.The noise power is estimated from the MAD of 512 high-frequency DCT coefficients, but excluding the slots contain-ing effect of poles. From Fig. 1a, it is evident that little or noinformation can be obtained about the pole locations fromthe direct plot of the DCT coefficients of the noisy observa-tions. But as can be seen from Fig. 1b, when the average ofthe slot-wise RMS values is compared with the individualslot RMS value, the pole locations become visible.

4.2 Modified EMD for AR systems in noise

We now propose a modification in the EMD algorithm toimprove its robustness to the identification of AR systems innoise. The proposed modification is based on exploiting theprior knowledge of the analytical model of the autocorrela-tion function of AR systems. We replace an IMF extractedin any stage of the sifting process by an analytic functionthat best fits the IMF in the least-square sense. An obviouschoice for the analytic function is a damped sinusoidal func-tion as revealed by the model in (9). The cost function forthe minimization process is defined as

J j =M∑

m=1

∣∣∣R j (m) − d j (m)

∣∣∣2, j = 1, 2, . . . , g (14)

where R j (m) is the normalized function of the j th IMFR j (m) and d j (m) = rm

j cos(ω j m). The normalization isdone to match its peak value with that of d j (m) which is1. Note that the phase term in d j (m) is ignored as we areonly interested of its frequency of oscillation and dampingratio. As the total autocorrelation function is an even func-tion and has its peak value at m = 0, the component IMFfunctions R j (m) usually also have their peaks at m = 0 andare symmetric with respect to the vertical axis. The functiond j (m) that minimizes J j is termed as the normalized mod-ified IMF. It is scaled by the maximum value of R j (m) tobe used in the subsequent sifting process. At the end of thesifting process, we obtain

Rxx (m) =g∑

i=1

R′i (m) + Resg(m) (15)

where R′i (m) = max(|(Ri (m)|)d j (m) is the modified IMF

and Resg(m) is the final residue of the modified siftingprocess.

5 Results

In this section, we present AR system identification resultsto demonstrate the effectiveness of the proposed modifiedEMD method. The results obtained are also comparedwith the high-order Yule–Walker (HOYW) method, noise

Fig. 1 Determination of thepole locations in the DCTcoefficients

0 512 1024 1536 2048 2560 3072 3584 4096−40

−20

0

20

40

DCT points

(a)

Am

plitu

de

0 512 1024 1536 2048 2560 3072 3584 40964

6

8

10

12

DCT points

(b)

Am

plitu

de

Slotwise RMS valueAverage value

DCT coefficientsAverage value

slots including poles

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SIViP (2010) 4:451–461 455

Fig. 2 Showing the effect ofnoise on the poles using theBurg algorithm. Unfilled circleTrue, plus sign 40 dB, multiplesign 20 dB, asterisks 10 dB,square 0 dB, inverted triangle−5 dB, star −10 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(a) System 2

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(b) System 5

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(c) System 4

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(d) System 3

Fig. 3 EMD of noisy anddenoised autocorrelationfunctions at SNR = −5 dB forAR System 1. Two IMFs areshown corresponding to twosystem modes

20 40 60 80 100 120 140 160 180 200

−0.5

0

0.5

1

20 40 60 80 100 120 140 160 180 200

−0.5

0

0.5

Am

plitu

de

20 40 60 80 100 120 140 160 180 200−0.1

0

0.1

Autocorrelation lag, m

noisy ACFnoise−free ACF

denoised IMF2DS2noisy IMF2

denoised IMF1DS1noisy IMF1

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Fig. 4 Estimate of the poles ofAR System 2 at different SNRsby different methods. Unfilledcircle True, plus sign HOYW,multiple symbol LOYW,asterisks ILSNP, squareDampSine, inverted trianglemodified EMD

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(a) SNR=20 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(b) SNR=10 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(c) SNR=0 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(d) SNR= −5 dB

compensated low-order Yule–Walker (LOYW) method, iter-ative least-squares method with no prefiltering (ILSNP) [3],and the damped sinusoidal (DampSine) method [2] to showthe strength of the proposed EMD based method.

The following AR systems are used in the simulation:

System 1:: aT = [−2.3559 3.2457 − 2.2356 0.9036]System 2:: aT = [−0.5500 −0.1550 0.5495 −0.6241]System 3:: aT = [−0.8600 1.0494 − 0.6680 0.9592

−0.7563 0.5656]System 4:: aT = [−2.7607 3.8106 − 2.6535 0.9238]System 5:: aT = [0.2080 − 0.6749 0.1991 0.9238]Data were generated using the difference equation of the

AR system. Computer-generated white noise with differentpower was added to the AR signal. In all the simulationsN = 4,000 data samples were used. For determining the de-noised IMFs we have used Rxx (m) for −M ≤ m ≤ M . Inthe simulations, M = 100 was used. In order to estimate ω

and r , a domain of ω from 0 to π was scanned at a resolutionof 5 × 10−2 for different values of r . Scanning interval of rwas taken to be 0 to 1 and scanning resolutions were chosento be 1×10−2. Alternatively, a course estimate of the natural

frequency of oscillation and decay rate can be obtained fromthe IMFs of the EMD analysis of the data. The search domaincan be greatly reduced using this approach.

First, we show the effect of increase in noise power on thepoles of all four AR systems (System 2–System 5) in Fig. 2.The Burg algorithm [5] was used to estimate the AR systemsassuming as if there were no noise. It is evident that as thenoise power increases or equivalently, the SNR decreases (40to −10 dB), the poles move toward inside the unit circle fromtheir original position. For SNR≤0 dB, the impact of noiseis very significant on all the four systems. At 10 dB SNR,the noise has noticeable effect on Systems 3 and 4. Here, wepresent comparative results using different methods for theSNR range of 20 to −5 dB. We expect that for 20 and 10 dBall the methods will perform satisfactorily for the reason asexplained above.

Now, we present results of the EMD analysis of theestimated noisy autocorrelation function of the AR System 1with two pairs of complex conjugate poles at 0.6928 ±0.6930i and 0.4851 ± 0.8400i . As noted in Sect. 3, eachpair of complex conjugate poles will constitute one dampedsinusoidal function (DSk). Figure 3 shows the IMFs extracted

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Fig. 5 Estimate of the poles ofAR System 3 at different SNRsby different methods. Unfilledcircle True, plus HOYW,multiple symbol LOYW,asterisks ILSNP, squareDampSine, inverted trianglemodified EMD

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(a) SNR=20 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(b) SNR=10 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(c) SNR=0 dB

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Real Part

Imag

inar

y P

art

(d) SNR= −5 dB

Table 1 Comparative results ofdifferent methods for ARSystem 4 at SNR = 20 dB

Method a1 a2 a3 a4 RMSE (%)

HOYW −2.7671 3.8374 −2.6860 0.9419 35.24

±0.5455 ±1.3108 ±1.2834 ±0.5376

±0.5387 ±1.2946 ±1.2677 ±0.5311

LOYW −3.5890 5.7872 −4.6278 1.7439 119.41

±1.6372 ±3.9164 ±3.9133 ±1.6301

±1.8164 ±4.3430 ±4.3392 ±1.8064

ILSNP −1.0983 0.4380 0.4523 −0.1420 89.61

±0.0209 ±0.0250 ±0.0170 ±0.0096

±1.6625 ±3.3727 ±3.1059 ±1.0659

DampSine −2.7480 3.7891 −2.6369 0.9201 0.55

±0.0014 ±0.0031 ±0.0029 ±0.0013

±0.0128 ±0.0218 ±0.0168 ±0.0039

EMD −2.6986 3.6697 −2.5148 0.8631 3.90

±0.0070 ±0.0163 ±0.0162 ±0.0073

±0.0625 ±0.1418 ±0.1396 ±0.0611

True value −2.7607 3.8106 −2.6535 0.9238

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HOYW −1.0123 −0.9818 2.2789 −1.3501 1137.89

±15.5852 ±41.2368 ±41.9589 ±18.9944

±15.4881 ±40.9992 ±41.7237 ±18.8928

LOYW −1.9274 1.8287 −0.6793 0.1054 196.77

±2.8969 ±6.9600 ±6.9658 ±2.9104

±2.9794 ±7.1525 ±7.1559 ±2.9880

ILSNP −0.5619 0.0954 0.3728 0.2449 95.42

±0.0268 ±0.0187 ±0.0201 ±0.0249

±2.1989 ±3.7153 ±3.0264 ±0.6793

DampSine −2.7569 3.8009 −2.6440 0.9199 0.30

±0.0025 ±0.0056 ±0.0055 ±0.0025

±0.0045 ±0.0112 ±0.0110 ±0.0046

EMD −2.6997 3.6743 −2.5199 0.8658 3.80

±0.0105 ±0.0240 ±0.0238 ±0.0099

±0.0619 ±0.1384 ±0.1357 ±0.0588

True value −2.7607 3.8106 −2.6535 0.9238



HOYW −4.2794 −1.1199 5.7082 −5.6745 847.84

±20.3030 ±8.3148 ±27.3435 ±30.2580

±20.1050 ±9.5769 ±28.2647 ±30.5973

LOYW −1.8543 0.6496 0.9597 −0.9882 260.85

±6.5609 ±7.0376 ±7.1564 ±6.6770

±6.5415 ±7.6342 ±7.9366 ±6.8646

ILSNP −0.5115 0.0826 0.3501 0.3003 95.70

±0.1269 ±0.1062 ±0.1218 ±0.1238

±2.2527 ±3.7295 ±3.0060 ±0.6354

DampSine −2.6178 3.4172 −2.2526 0.7488 23.35

±0.2647 ±0.7562 ±0.7796 ±0.3325

±0.2978 ±0.8439 ±0.8680 ±0.3721

EMD −2.6957 3.6688 −2.5154 0.8643 4.03

±0.0174 ±0.0395 ±0.0391 ±0.0164

±0.0672 ±0.1470 ±0.1434 ±0.0617

True value −2.7607 3.8106 −2.6535 0.9238

from noisy autocorrelation function at SNR = −5 dB alongwith the damped sinusoids (DS1 and DS2) using the conven-tional EMD and modified EMD methods. The conventionalEMD is applied on the noise uncompensated autocorrela-tion function, while the proposed modified EMD operateson the estimated autocorrelation function obtained using theproposed noise compensation technique described in the pre-

ceding section. It is clear that the conventional EMD methodfails to extract the true second IMF (IMF2). It fails evenif the noise-compensated autocorrelation function is used.The reason is obviously the distortions in extrema mainlydue to noise. In contrast, the modified EMD method accu-rately extracts both the first and second IMFs (IMFk, k =1, 2). These results imply that the conventional EMD method

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Table 4 Comparative results ofdifferent methods for ARSystem 4 at SNR = −5 dB


HOYW −1.1215 0.0449 0.9831 −0.6105 183.55

±4.0272 ±2.2651 ±4.6083 ±5.6015

±4.3012 ±4.3799 ±5.8250 ±5.7399

LOYW 2.1743 −6.3051 6.8758 −2.6845 26.50

±14.7894 ±32.1618 ±32.0836 ±14.6156

±15.4147 ±33.3294 ±33.0822 ±14.8760

ILSNP −0.3746 0.0998 0.2368 0.4659 96.23

±0.5194 ±0.4026 ±0.3281 ±0.4360

±2.4406 ±3.7320 ±2.9084 ±0.6285

DampSine −1.4275 0.1291 1.0261 −0.6192 100.36

±0.0582 ±0.0458 ±0.0538 ±0.0255

±1.3344 ±3.6818 ±3.6800 ±1.5432

EMD −2.6799 3.6303 −2.4762 0.8482 5.68

±0.0514 ±0.1036 ±0.0979 ±0.0389

±0.0954 ±0.2073 ±0.2020 ±0.0848

True value −2.7607 3.8106 −2.6535 0.9238



HOYW 0.2108 −0.6765 0.1894 0.9141 0.92

±0.0010 ±0.0012 ±0.0015 ±0.0015

±0.0030 ±0.0020 ±0.0098 ±0.0098

LOYW 0.2134 −0.6730 0.1918 0.9154 0.82

±0.0009 ±0.0019 ±0.0010 ±0.0011

±0.0055 ±0.0027 ±0.0074 ±0.0085

ILSNP 0.2641 −0.7177 0.1696 0.9594 5.57

±0.0023 ±0.0115 ±0.0025 ±0.0135

±0.0561 ±0.0442 ±0.0296 ±0.0380

DampSine 0.2053 −0.6687 0.1967 0.9154 0.73

±0.0012 ±0.0014 ±0.0014 ±0.0011

±0.0029 ±0.0063 ±0.0027 ±0.0085

EMD 0.2069 −0.6771 0.1987 0.9224 0.18

0 0 0 0

±0.0011 ±0.0022 ±0.0004 ±0.0014

True value 0.2080 −0.6749 0.1991 0.9238

applied to the autocorrelation function corrupted heavily bynoise is not suitable for AR parameter estimation, becauseaccurate IMF extraction is the basic step for AR parameterestimation using the EMD. On the contrary, the effectivenessof the noise-compensation technique and the modified EMDis vividly shown in Fig. 3.

Next, parameter estimation results for two different ARsystems (System 2 and System 3) corrupted by different noise

levels are presented in Figs. 4 and 5, respectively. A compar-ison of the results with HOYW, LOYW, ILSNP, and Damp-Sine method is also given. Notice that for the LOYW methodtrue value of the noise power was used to compensate forthe noise effect, whereas, the proposed method estimatedthe noise power from the observed signal using the methoddescribed in the preceding section. As can be seen from Fig. 4,the complex poles as well as the real poles on either side

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460 SIViP (2010) 4:451–461



HOYW 0.2102 −0.6774 0.1894 0.9145 1.12

±0.0036 ±0.0043 ±0.0061 ±0.0064

±0.0042 ±0.0049 ±0.0114 ±0.0112

LOYW 0.2134 −0.6726 0.1920 0.9154 1.03

±0.0033 ±0.0074 ±0.0036 ±0.0046

±0.0063 ±0.0076 ±0.008 ±0.0095

ILSNP 0.2624 −0.7187 0.1693 0.9595 6.64

±0.0078 ±0.0078 ±0.0065 ±0.0445

±0.0549 ±0.0583 ±0.0304 ±0.0567

DampSine 0.2069 −0.6733 0.1942 0.9127 0.86

±0.0030 ±0.0025 ±0.0025 ±0.0023

±0.0031 ±0.0030 ±0.0055 ±0.0113

EMD 0.2069 −0.6771 0.1987 0.9224 0.18

0 0 0 0

±0.0011 ±0.0022 ±0.0004 ±0.0014

True value 0.2080 −0.6749 0.1991 0.9238



HOYW 0.2050 −0.6848 0.1919 0.9178 4.74

±0.0228 ±0.0247 ±0.0469 ±0.0445

±0.0227 ±0.0263 ±0.0469 ±0.0443

LOYW 0.2148 −0.6672 0.1936 0.9159 4.27

±0.0188 ±0.0497 ±0.0191 ±0.0330

±0.0198 ±0.0497 ±0.0196 ±0.0335

ILSNP 0.2602 −0.7293 0.1849 0.9894 2.95

±0.0479 ±0.2674 ±0.0901 ±0.3480

±0.0705 ±0.2696 ±0.0901 ±0.3499

DampSine 0.2236 −0.6626 0.1680 0.8630 4.70

±0.0080 ±0.0081 ±0.0083 ±0.0063

±0.0175 ±0.0147 ±0.0322 ±0.0611

EMD 0.2152 −0.6748 0.1895 0.9163 1.46

±0.0080 ±0.0046 ±0.0091 ±0.0123

±0.0107 ±0.0045 ±0.0131 ±0.0143

True value 0.2080 −0.6749 0.1991 0.9238

of the imaginary axis are almost correctly identified by theproposed modified EMD and other methods up to SNR =−5 dB. This is an example where pole locations with respectto the unit circle as well as pole separation are suited to allthe algorithms. Next, from Fig. 5 it is clear that almost allthe methods are capable of identifying the sixth order ARsystem up to SNR = 0 dB. But at SNR = −5 dB only themodified EMD method can successfully identify the system.

Further next, the parameter estimation results of AR Sys-tem 4 and System 5 are presented in Tables 1, 2, 3, 4 andTables 5, 6, 7, 8, respectively, for different SNRs. Under

each estimated parameter, first standard deviation withrespect to mean (SDM) and then standard deviation withrespect to true value (SDT) are shown. In order to illustratethe accuracy of the identification methods, the root meansquared error (RMSE) of the estimates is also shown in thetables. RMSE is defined by

RMSE =√√√√ 1

L

L∑

t=1

||at − a||2||a||2 (16)

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SIViP (2010) 4:451–461 461

Table 8 Comparative results ofdifferent methods for ARSystem 5 at SNR = −5 dB


HOYW 0.1910 −0.7012 0.2035 0.9249 14.31

±0.0710 ±0.0690 ±0.1462 ±0.1343

±0.0722 ±0.0730 ±0.1444 ±0.1326

LOYW 0.2211 −0.6558 0.2002 0.9248 12.87

±0.0541 ±0.1528 ±0.0553 ±0.1038

±0.0550 ±0.1521 ±0.0546 ±0.1025

ILSNP 0.2738 −0.5664 0.1557 0.8257 45.26

±0.1194 ±0.3884 ±0.1342 ±0.5394

±0.1350 ±0.3986 ±0.1395 ±0.5416

DampSine 0.2392 −0.6271 0.1375 0.7727 11.39

±0.0185 ±0.0183 ±0.0177 ±0.0135

±0.0362 ±0.0511 ±0.0640 ±0.1517

EMD 0.2338 −0.6635 0.1676 0.8925 4.82

±0.0219 ±0.0171 ±0.0264 ±0.0374

±0.0337 ±0.0204 ±0.0409 ±0.0484

True value 0.2080 −0.6749 0.1991 0.9238

where at stands for an estimator of a in the t th run, andL denotes the total number of independent runs. In all thesimulations, L = 40 was used. The estimated parametersas shown in the tables denote the average of 40 independentruns. It can be observed from the tables that, for both the sys-tems, the proposed method outperforms, in terms of RMSE,all other methods at all SNRs shown except for the Damp-Sine which shows somewhat better performance at 20 and10 dB SNRs for the System 4. The values of SDM and SDTas shown in the tables demonstrate that the proposed EMD-based method consistently gives accurate results. Thus, theresults of extensive simulation tests suggest that the proposedmethod is a powerful technique for identifying AR systemsin strong noise.

6 Conclusions

In this paper, a new method has been presented for estimat-ing the parameters of autoregressive signals using the EMDmethod. The AR parameters are computed from the IMFsof autocorrelation sequence of the AR signal. The least-squares algorithm is used for estimating improved IMFs iter-atively from data which contain some sort of uncertainty. Thenumerical results presented in the paper have demonstratedthe superiority of the proposed modified EMD method oversome other very well-known and powerful techniques forAR system identification with noise. The improvement inthe results stemmed from the concept of separating the ARsystem modes using the IMFs of EMD.

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A robust method for parameter estimation of AR systems using empirical mode decomposition