Applied Acoustics 73 (2012) 1045–1051

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Applied Acoustics

journal homepage: www.elsevier .com/locate /apacoust

AR model whitening and signal detection based on GLD algorithmin the non-Gaussian reverberation

Ye Liu ⇑, Zhi-ping Huang, Shao-jing Su, Chun-wu LiuCollege of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, PR China

a r t i c l e i n f o

Article history:Received 13 September 2011Received in revised form 25 February 2012Accepted 3 May 2012Available online 29 May 2012

Keywords:Non-Gaussian reverberationSymmetric alpha stable distributionAR modelGLD algorithmSignal detection

0003-682X/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.apacoust.2012.05.002

⇑ Corresponding author. Tel.: +86 15 873112119.E-mail addresses: liuyeforever@yahoo.cn (Y. Liu)

Huang), susj-5@163.com (S.-j. Su), liucw-888@163.co

a b s t r a c t

The paper analyzes the characteristics of shallow sea reverberation such as divergent variance and heavytails in probability density distribution and matches it to symmetric alpha stable (SaS) distribution. Afteranalyzing the AR model whitening method based on Gaussian distribution and Levinson–Durbin algo-rithm, the paper proposes and derives the generalized Levinson–Durbin (GLD) algorithm based on covari-ation theory of alpha stable distribution. The GLD algorithm can estimate the order and parameters of ARSaS model by iterations which are fast and effective in calculating, and then the performance tests bysimulations are given. The AR model whitening method using GLD algorithm has a faster speed and a bet-ter detection performance than AR Gaussian whitening method in impulsive reverberation. At last theMonte Carlo simulating tests are used to validate the validity of the new method.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Reverberation is a complex acoustic phenomenon which takesplace in the working cycles of active sonar systems. The formingprocess and power level of reverberation have much to do withthe factors such as the transmitted waveform, sea conditions,and transmission distance. As the reverberation is a stochastic pro-cess, there are kinds of methods in dealing with it in the field ofunderwater acoustic signal processing. Studies have shown thatreverberation can be regarded as a nonstationary process for itstime-varying and divergent variance; as a result, many methodstreat it as a local stationary process for simplification. The methodprewhitening the subsections of reverberation data and using thematched filter to detect the target signal always plays a good per-formance in Gaussian-like reverberations. However in some shal-low sea environments, the sea conditions become so badly thatthe scatters distributed randomly in the sea floor have a strongscattering effect driven by the transmitted waveform. The scatter-ing strength of scatters makes the variance of reverberation diver-gent and the tails of its probability distribution function (pdf)heavier. As a result the methods in Gaussian distributions cannotget correct results or good performances. However the family ofalpha stable distributions can describe a broad class of impulsiveinterference and they have heavier tails than those of Gaussiandistributions.

ll rights reserved.

, kdhuangzp@163.com (Z.-p.m (C.-w. Liu).

The paper is organized as follows. Section 2 gives the analysesof reverberation with impulsive characteristics and the traditionaldetecting method using AR whitening model, then discusses thefeasibility and virtues of GLD algorithm based on alpha stable the-ory. Section 3 gives the detailed analyses of the matching degree ofSaS model to impulsive reverberation and derivation of the GLDalgorithm, and then gives the simulation results. Signal detectionin reverberation by two AR whitening methods are comparedand their performances are given in Section 4.

2. Problem analyses

As we know, reverberation is treat as a nonstationary stochasticprocess as its time-varying mean and variance. But when the trans-mitted waveform is modulated by monochromatic wave, the meanis always approximate to zero. The reverberation which is drivenby monochromatic wave is the main research point in this paper.Fig. 1 shows a section of reverberation data and its statistical var-iance. We can see the variance is time-varying.

As we know dealing with nonstationary data is more difficultthan dealing with stationary data. As mentioned before, a simplemethod is to partition the nonstationary data into subsectionsaccording to some principles and then each subsection can be trea-ted as stationary data. Matched filter is the optimal detector in sta-tionary Gaussian white noise. Whereas reverberation is colorful asit has many similar characteristics with target echo that matchingthe signal directly cannot get good results. Prewhitening is neces-sary before matching. AR model is so useful that a finite order ARmodel can replace an infinite order MA model. More important,

Fig. 1. Statistical characteristics: (a) amplitude and (b) variance.

1046 Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051

in Gaussian theory autocorrelation coefficients of AR process andparameters of the model satisfy Yule–Walker (YW) equation,which makes the AR model parameters easy to be estimated. Asa Toeplitz matrix can be got from the YW equations, we can useLD algorithm to derive the parameters and order of AR model.The proposed algorithm cannot only less calculation amount butalso ensure the accuracy.

When dicky sea conditions make the reverberations moreimpulsive, the probability distributions of reverberations tend tobe non-Gaussian. The performances of detecting methods inGaussian theory will descend. However, alpha stable distributionwhich obeys the generalized central limited theory can match thisclass of impulsive interference. Alpha stable distributions are as ra-tional as Gaussian distribution, but more universal than Gaussiandistribution for their heavy tails. As we know that the perfor-mances of signal detection in white noises are always better thandetection in colorful noises, we need to whiten the reverberationfor preprocessing. As Gaussian linear process, SaS linear processalso can be described by AR models. Estimating the parametersof SaS AR model of reverberation and whitening the reverberationto be non-Gaussian white noise is the main job of this paper. Asmentioned before, using LD algorithm to estimate the order andparameters of AR model in Gaussian distribution is an effectivemethod; here in this paper we propose the GLD algorithm in SaSdistribution. As infinite variance is an important characteristic inalpha stable theory, we need not partition the data into subsec-tions any more. We need to estimate the parameters and orderof SaS AR model only once so that the AR whitening method basedon GLD in SaS theory has a higher efficiency and faster speed thanAR Gaussian whitening method.

3. Algorithm development

3.1. SaS distributions

The characteristic function [1] (CF) of the class of SaS distribu-tions is defined as

uðtÞ ¼ expðjat � cjtjaÞ ð1Þ

where a 2 ð0;2� is the characteristic exponent of SaS distributions, cis the disperse coefficient and a is the location parameter. Whena > 1, a is the mean of the distributions. Except some special distri-butions such as Gaussian (a = 2), Cauchy (a = 1) and Pearson(a = 0.5), there is no closed form expression for the pdf of alphastable distribution. But the pdf can be calculated numerically by

the inverse Fourier transform of CF or power series expansion.Many literatures like [2–4] have given detailed methods in param-eters estimation.

We use a section of impulsive reverberation data to make a sta-tistical distribution of amplitude, and then the normal and SaS dis-tributions are modeled to see their matching degrees to thestatistical distribution. The parameters of two distributions areestimated by the data series respectively. Fig. 2a shows the fittinglines of three pdfs and Fig. 2b is the partial detailed figure of Fig. 2a.From Fig. 2b we can see that the SaS distribution can match the‘impulsive’ data better than the Normal distribution. As the resultthe SaS distribution has a better matching degree than the normaldistribution.

3.2. GLD algorithm

As AR SaS process is similar to AR Gaussian process, it can bedepicted as

xðnÞ ¼ a1xðn� 1Þ þ � � � þ apxðn� pÞ þ uðnÞ ð2Þ

where u(n) is a independent and identically distributed (i.i.d.) SaSvariable with the characteristic exponent a and disperse coefficientc. When a = 2, the AR SaS process is just the AR Gaussian process. Asa strict AR SaS process, x(n) also can be expressed as a infinite MASaS process as

xðnÞ ¼X1i¼0

biuðn� iÞ ð3Þ

From (3) and the property of alpha stable distribution we cansee x(n) has the same characteristic exponent with white noiseu(n). As the parameters of AR Gaussian model can be estimatedby YW equation, we can get the matrix form as follow:

Ra ¼ r2

0

" #ð4Þ

where R is the (p + 1) � (p + 1) Toeplitz matrix composed of auto-correlation coefficients of AR process. a ¼ ½a1 . . . ap�T , r2 is the var-iance of Gaussian noise u(n). Due to inexistence of finite secondmoment in AR SaS process, we use the generalized Yule–Walker[5] (GYW) equation base on covariation theory proposed by Nikiasand Shao to derive the GLD algorithm, the processes are asfollows:

Step 1: Getting the conditional expectation of x(m) at both sidesof (2), we can get

Fig. 2. Model matching degree in pdf.

Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051 1047

EðxðnÞjxðmÞÞ ¼ a1Eðxðn� 1ÞjxðmÞÞ þ � � � þ apEðxðn� pÞjxðmÞÞ ð5Þ

For the stationary characteristic of alpha stable theory (5) canalso be expressed as

kðn�mÞxðmÞ ¼ a1kðn�m� 1ÞxðmÞ þ � � � þ a1kðn�m

� pÞxðmÞ ð6Þ

Step 2: From Step 1 we can get matrices L, a and p, where

Lp ¼

kð0Þ kð�1Þ � � � kð�pÞkð1Þ kð0Þ � � � kð1� pÞ� � � � � � � � � � � �

kðpÞ kðp� 1Þ � � � kð0Þ

26664

37775 ð7Þ

ap ¼ ½1 a1 � � � ap �T ð8Þ

lp ¼ ½ kð1Þ kð2Þ � � � kðpÞ �T ð9Þ

We can get the matrix form of GYW equations Lpap = lp.Step 3: Let

Lm�1 ¼

kð0Þ kð�1Þ � � � kð2�mÞkð1Þ kð0Þ � � � kð3�mÞ� � � � � � � � � � � �

kðm� 2Þ kðm� 3Þ � � � kð0Þ

26664

37775 ð10Þ

l0m�1 ¼ ½ kð1�mÞ kð2�mÞ � � � kð�1Þ � ð11Þ

lbm�1 ¼ ½ kðm� 1Þ kðm� 2Þ � � � kð1Þ � ð12Þ

where superscript b of lbm�1 means the reverse order of elements in

lm�1; and

lm�1 ¼ ½ kð1Þ kð2Þ � � � kðm� 1Þ � ð13Þ

So Lm, am can be expressed as follows:

Lm ¼Lm�1 l0Tm�1

lbm�1 kð0Þ

" #ð14Þ

am ¼

am;1

am;2

..

.

am;m

266664

377775 ¼

am�1

0

� �þ

dm�1

qm

� �ð15Þ

Lm�1 l0Tm�1

lbm�1 kð0Þ

" #am�1

0

� �þ

dm�1

qm

� �� �¼ lT

m�1

kðmÞ

" #ð16Þ

From (16) we can get two equations:

Lm�1am�1 þ Lm�1dm�1 þ qml0Tm�1 ¼ lTm�1 ð17Þ

lbm�1am�1 þ lb

m�1dm�1 þ qmkð0Þ ¼ kðmÞ ð18Þ

As Lm�1am�1 ¼ lTm�1; together with (15), we can get

dm�1 ¼ �qmL�1m�1l0Tm�1 ð19Þ

Due to the property of Toeplitz matrix, we change the matrix formto get

LTm�1ab

m�1 ¼ lbTm�1 ð20Þ

As we know that LTm�1 is a full-rank matrix, after changing the form

of (20) we can get the expression

abTm�1 ¼ lb

m�1L�1m�1 ð21Þ

Considering Lm�1am�1 ¼ lTm�1; we can get

Lm�1dm�1 þ qml0Tm�1 ¼ 0 ð22Þ

Multiplying abTm�1 at both sides of equation (20), we can get

qm ¼ �ðlbm�1dm�1Þ ðl0m�1ab

m�1ÞT

h i�1ð23Þ

Synthesizing (17) and (23), we can get

lbm�1am�1 þ Q lb

m�1dm�1 ¼ kðmÞ ð24Þ

where

Q ¼ 1� kð0Þ½ðl0m�1abm�1Þ

T ��1 ð25Þ

Then

lbm�1dm�1 ¼

1Q

kðmÞ � lbm�1am�1

h ið26Þ

Fig. 3. GLD algorithm testing: (a) u(n), (b) x(n), (c) order estimation and (d) parameters estimation.

Table 1Estimates means and variances of AR parameters by computer simulations.

a1 a2 a3 a4 a5 a6 a7 a8

Mean 0.8700 �0.3397 �0.1660 �0.4734 0.8313 �0.6432 0.3986 �0.2567Variance 0.0033 0.0084 0.0051 0.0011 0.0011 0.0054 0.0081 0.0032

1048 Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051

Synthesizing (22) and (26), we can get

qm ¼ �1Q

kðmÞ � lbm�1am�1

h iðl0m�1ab

m�1ÞT

h i�1

¼ �kðmÞ � lb

m�1am�1

h iðl0m�1ab

m�1ÞT

h i�1

1� kð0Þ ðl0m�1abm�1Þ

Th i�1

¼ kðmÞ � lbm�1am�1

kð0Þ � ðl0m�1abm�1Þ

T ¼kðmÞ � lb

m�1am�1

kð0Þ � l0m�1abm�1

ð27Þ

Step 4: In AR Gaussian model we use minimum mean squareerror criterion to keep the prediction error EN being minimum.As there is no finite second moment in AR SaS process, here wedefine

em ¼ EðxðnÞjxðnÞÞ � a1Eðxðn� 1ÞjxðnÞÞ � � � � � amEðxðn�mÞjxðnÞÞ

¼ ½kð0Þ � a1kð�1Þ � � � � � amkð�mÞ�xðnÞ ð28Þ

And then

qm ¼kðmÞ � lb

m�1am�1

kð0Þ � ðl0m�1abm�1Þ

T ¼½kðmÞ � lb

m�1am�1�xðnÞ½kð0Þ � l0m�1ab

m�1�xðnÞð29Þ

Then we define Em�1 ¼ kð0Þ � l0m�1abm�1; we can get

amðmÞ ¼ qm ¼kðmÞ � lb

m�1am�1

kð0Þ � l0m�1abm�1

¼ kðmÞ � lbm�1am�1

Em�1ð30Þ

Em ¼ kð0Þ � l0mabm ¼ kð0Þ � l0m

am�1

0

" #þ

dm�1

qm

" # !b

¼ kð0Þ � l0m�1abm�1 � l0m

dm�1

qm

" #b

¼ Em�1 � l0mdm�1

amðmÞ

" #b

ð31Þ

Step 5: Repeating Step 3 and 4 until the prediction error EN goesdown slowly as the order increases, and then we estimate theorder is the order of AR model.

Fig. 4. ROCs. (a) AR SaS whitened data. (b) ROC of unwhitened data in Gaussian detector. (c) ROC of Unwhitened data in Cauchy detector. (d) ROC of AR Gaussian whiteneddata in Gaussian detector. (e) ROC of AR SaS whitened data in Gaussian detector. (f) ROC of AR Gaussian whitened data in Cauchy detector.

Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051 1049

3.3. Computer simulation

We simulate a section of SaS data to test the GLD algorithm.Here is a sequence of 8 order AR SaS process with 6000 samples.

The AR coefficients are given as follows: 0.85, �0.33, �0.15,�0.48, 0.81, �0.62, 0.40, 0.26. u(n) is a i.i.d. SaS white noise withthe parameters a = 1.7 and c = 1. Fig. 3 gives the simulations ofnon-Gaussian white noise, AR SaS series and estimated results.

1050 Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051

Table 1 gives the estimation of means and variances of AR param-eters by 1000 Monte Carlo simulations.

3.4. Performance analysis

From Fig. 3c, we see the prediction error Em decreases as theorder increases. When the order gets to 8, Em reaches a very smallvalue and its declining rate becomes very slow. We can define anerror threshold in calculation and when Em overtakes it, we choosethe order. From Fig. 3d and Table 1, we can see there are someerrors between the estimated parameters and true’s. The errorsare formed by Em and the estimated errors of covariation coeffi-cients. In [5] Shao and Nikias give some methods such as leastsquare estimation, screened ratio (SCR) estimation and FLOMestimation to estimate the covariation coefficients. We chooseFLOM estimation here as it can get the consistent estimates ofthe covariation coefficients in AR SaS process. It is noteworthy thatif we set a very small error threshold for Em to get a higher accuracyfor AR parameters, it will not only increase the amount of compu-tation but also affect the model accuracy. As a result choosing a fiterror threshold is the key point.

The advantage of GLD algorithm is that, we can estimate the or-der in real time and need not use complicated order determinationprinciples. The parameters can be obtained by iterations. Theshortage is that, for the asymmetrical characteristic of covariationcoefficients ðkðmÞ–kð�mÞ; ðm–0ÞÞ, in the iterations l0m�1 cannotchange into lb

m�1. So we need calculate L�1m�1 each iteration. There-

fore GLD algorithm cost more in calculating than LD algorithm.

4. Signal detection

We use GLD algorithm to estimate the AR model of reverbera-tion data and whiten it. As mentioned before, we only need esti-mate AR SaS model once. We compare the AR SaS whiteningmethod with AR Gaussian whitening method [6]. In [6], a movingwindow determinated by resolving power of wave ambiguity func-tion is used to whiten the data. Kth subsection data is used to esti-mate the AR model and whiten this data, and a length of movingwindow is moved to (K + 1)th subsection for the next repeat. In thispaper, we simulate a segment data of target echo and mix it withreverberation data at a location of integral multiple length of trans-mitted signal for simplification.

Matched filter is the optimal detector in Gaussian theory. In ARSaS theory, though the optimal detector can be obtained due to thecharacteristic exponent theoretically, there is no closed formexpression of pdf except several special distributions. As a resultit is difficult to get the optimal detector. We use some suboptimaldetectors instead. In [7,8] Tsihrintzis and Nikias give Gaussian andCauchy two detectors for fast detecting purpose and test the per-formances of two suboptimal detectors in different stable noise.Here we also compare their performances in detecting signal inimpulsive reverberation data and find which detector has a betterperformance. The signal sample in detectors should also be whit-ened by corresponding AR model.

We simulate the reverberation data driven by monochromaticwave. The frequency of transmitted wave is 4 kHz, width is 4 ms,with 10 kHz sampling frequency. The data length is 6000 samples,and signal length is 40 samples. The simulating procedure of thereverberation data are as follows: we initialize the conditionparameters, including the depths of seafloor and sonar, the scatter-ing coefficients of sea surface, water and seafloor, attenuating coef-ficient of sound wave, the field angle and effective distance ofsonar, etc. The positions and number of the scatters are generalizedrandomly and the number should be a big positive random integer.The scattering intensity of each scatter and the delay the scattering

intensity reach sonar can be calculated by the above conditionparameters and relative position between sonar and the scatter.Then the reverberation data can be calculated by the contributionsof the scattering intensities and delays. Before AR SaS whiteningthe parameters of SaS distribution should be estimated. We useMa’s log jSaSj method in [9], for it is easy to estimate the parame-ters and it need not estimate the minus moment power exponent.This method is used with the location parameter a = 0 (X � Sa(c, 0,0)), when the mean of reverberation data is not equal to 0(X � Sa(c, 0, a)), we use the property

X � a � Saðc;0;0Þ ð32Þ

Fig. 4 gives the AR SaS whitened data and ROCs of unwhiteneddata, AR Gaussian whitened data and AR SaS whitened data. Themixed signal noise ratio (MSNR, here we define it as MSNR ¼10log10

1Nc

PNn¼1jsðnÞj

2� �

, where s(n) is the target signal and N is

its sample number) is set low. 1000 Monte Carlo tests are madeto get the statistical results. According to the analyses before, theAR Gaussian whitening method based on autocorrelation theorylose its predominance in detecting impulsive reverberation. It isnoticeable that as the impulsive reverberation is not a strict SaSprocess actually, after we whiten the reverberation data by ARSaS whitening method we find that the whitened data tend to beGaussian-like white noise (au > ax). As a result the Gaussian detec-tor has a better performance than Cauchy detector.

From the ROCs, we see the AR Gaussian whitening method per-form badly in impulsive reverberation with low MSNR. As theunwhitened data is colorful to signal, performance of detecting inunwhitened data is also of a sort. We find the AR SaS whiteningdata perform well especially in Gaussian detector. Though theCauchy detector performs not as well as the Gaussian detector indetecting AR SaS whitening data, its performance improves quicklyas the MSNR increases.

5. Conclusions

The paper has researched the impulsive reverberations in shal-low sea. After analyzing the statistical variance of reverberationdata and comparing the amplitude distribution with Gaussian dis-tribution and SaS distribution, we found that SaS distribution has abetter matching degree to this class of reverberations. According tothe LD algorithm and covariation theory in stable distribution, wepropose and derive the GLD algorithm, using it to whiten theimpulsive reverberation data. Simulations and tests proved thatthe AR SaS whitening method based on GLD algorithm has a betterperformance than AR Gaussian whitening method in detecting inthe impulsive reverberations. This method can be used for real-time calculating for its fast speed.

References

[1] Tsihrintzis George A, Nikias Chrysostomos L. Fast estimation of the parametersof alpha stable impulsive interference. IEEE Trans Signal Process1996;44(6):1492–503.

[2] Kuruoglu Ercan Engin. Density parameter estimation of skewed alpha stabledistributions. IEEE Trans Signal Process 2001;49(10):2192–201.

[3] Ma Xinyu, Nikias Chrysostomos L. Parameter estimation and blind channelidentification in impulsive signal environments. IEEE Trans Signal Process1995;43(12):2884–97.

[4] Bodenschatz John S, Nikias Chrysostomos L. Maximum-likelihood symmetricalpha-stable parameter estimation. IEEE Trans Signal Process 1999;47(5):1382–4.

[5] Shao Min, Nikias Chrysostomos L. Signal processing with fractional lower ordermoments stable processes and their applications. Proc IEEE 1993;81(7):986–1010.

[6] Ginolhac G, Jourdain G. Detection in presence of reverberation. In: Oceans 2000MTS/IEEE conference and exhibition, vol. 2; September 2000. p. 1043–46.

[7] Tsihrintzis George A, Nikias Chrysostomos L. Incoherent receivers in alpha-stable impulsive noise. IEEE Trans Signal Process 1995;43(9):1043–6.

Y. Liu et al. / Applied Acoustics 73 (2012) 1045–1051 1051

[8] Tsihrintzis George A, Nikias Chrysostomos L. Performance of optimum andsuboptimum receivers in the presence of impulsive noise modeled as an alphastable process. IEEE Trans Commun 1995;43(2/3/4):904–14.

[9] Ma Xinyu, Nikias Chrysostomos L. Joint estimation of time delay and frequencydelay in impulsive noise using fractional lower order statistics. EEE Trans SignalProcess 1996;44(11):2669–87.