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Research ArticleMedian-Difference Correntropy for DOA under the ImpulsiveNoise Environment

Fuqiang Ma,1,2,3 Jie He ,1,2,3 and Xiaotong Zhang 1,2,3

1Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and Technology Beijing,Beijing 100083, China2Department of Computer Science and Technology, University of Science and Technology Beijing, Beijing 100083, China3Beijing Key Laboratory of Knowledge Engineering for Materials Science, Beijing 100083, China

Correspondence should be addressed to Xiaotong Zhang; [email protected]

Received 25 May 2019; Revised 21 August 2019; Accepted 16 September 2019; Published 3 October 2019

Guest Editor: Peio Lopez-Iturri

Copyright © 2019 Fuqiang Ma et al. is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

e source localization using direction of arrival (DOA) of target is an important research in the eld of Internet ofings (IoTs).However, correntropy suers the performance degradation for direction of arrival when the two signals contain the similarimpulsive noise, which cannot be detected by the dierence between two signals.is paper proposes a new correntropy, called themedian-dierence correntropy, which combines the generalized correntropy and the median dierence. e median dierence isdened as the deviation between the sampling value and the median of the signal, and it intuitively reects the abnormality ofimpulsive noise. en, the median dierence is combined with the generalized correntropy to form a new weighting factor thatcan eectively suppress the amplitude level of impulsive noise. To improve the robustness of the algorithm, an adaptive kernel sizeis also integrated into the weighting factor to obtain the optimal local feature. e inuence of adaptive kernel sizes on theproposed algorithm is simulated, and the comparison between three typical direction-of-arrival estimation algorithms isconducted. e results show that the accuracy of the median-dierence correntropy is signicantly superior to the correntropy-based correlation and the phased fractional lower-order moment for a wide range of alpha-stable distribution noise environments.

1. Introduction

e source localization using direction of arrival (DOA) oftarget is an important research in the eld of Internet ofings (IoTs). e direction-of-arrival (DOA) approachesbased on acoustics have many applications including radar,sonar, seismic exploration, navigation, and sound sourcetracking [1–3]. DOA estimation is usually regarded as aproblem of signal matching, and the performance is sig-nicantly inuenced by noise. A majority of existing DOAestimations are based on the concept that noise followsGaussian distribution [4, 5]. Since the Gaussian process hassecond-order and higher-order statistics, the traditionalDOA algorithms can easily evaluate the signal characteristicsaccording to second-order statistics [6]. e multiple signalclassication (MUSIC) algorithm [7, 8] and estimationmethod of signal parameters via rotational invariance

techniques (ESPRIT) are the basic subspace algorithmswhich have good performance [9, 10]. MUSIC is the rep-resentative of the noise subspace algorithm, and ESPRIT isthe representative of the signal subspace algorithm.

Because of atmospheric noise, electromagnetic in-terference, sea clutter, car ignitions, and o¦ce equipment,the signal is corrupted by the extremely impulsive noise thatexhibits irregularity in time domain. In addition, theprobability density functions of impulsive noise decay withheavy tails and do not follow a common Gaussian distri-bution [11]. erefore, alpha-stable distribution is usuallyused to dene impulsive noise [12]. e conventional co-variance matrix is calculated from the second-order statisticsof the signal, which may be innite when the data arecorrupted by the extremely impulsive noise [13, 14]. Inaddition, the conventional DOA algorithms cannot bedecomposed into the signal subspace and the noise subspace

HindawiWireless Communications and Mobile ComputingVolume 2019, Article ID 8107176, 12 pageshttps://doi.org/10.1155/2019/8107176

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with covariance matrix. 0us, it has become increasinglyimportant to study DOA under the impulsive noiseenvironment.

0e relevant statistical algorithms mainly focus on theexistence of statistics. 0e fractional lower-order statistics(FLOS) algorithms [15] exhibit a more desirable perfor-mance than the second-order statistics for alpha-stabledistribution [16]. When the signal contains impulsive noise,the signal bears the fractional lower-order statistics. 0eFLOS algorithms employ the minimum dispersion (MD)criterion to suppress impulsive noise, such as the robustcovariation in ROC-MUSIC [17], the fractional lower-orderstatistics- (FLOM-) based MUSIC, and the phased fractionallower-order moment (PFLOM). 0e FLOM algorithm-based MUSIC obtains a finite covariance by suppressing oneof two cross-correlation signals when the characteristicexponent of alpha-stable distribution ranges from 1 to 2 [18].0e PFLOM algorithm gets the accurate DOA estimationwith circular symmetrical signals, embedded in the additiveimpulsive noise [19]. However, investigators demonstratethat the performance of FLOM and PFLOM algorithmsdepends on the relationship between the parameter offractional lower-order moment and the characteristic ex-ponent of alpha-stable distribution. If the characteristicexponent is unknown, the performance of FLOM andPFLOM algorithms seriously decreases.

0e correntropy criterion [20] is a relatively simplemethod that can measure the local similarity between twosignals [21, 22]. Because it has the properties of M-esti-mation, the correntropy has been widely used in the im-pulsive noise environment [23]. Zhang et al. [24]investigated a narrowband model based on the generalizedcorrentropy which is called the correntropy-based correla-tion (CRCO) in impulsive noise environment. 0e CRCOalgorithm imposes a correntropy operator on the covariancematrix to depress impulsive noise. 0e generalized cor-rentropy is suitably for dealing with the template matchingbetween the received signal and the template signal [25].Since DOA estimation is a matching problem between twosignals, the generalized correntropy is incapable of mea-suring the difference between the outliers and fails tosuppress impulsive noise. 0us, when data contain thesimilar impulsive noise, the cross-correlation of the gen-eralized correntropy is infinite.

In order to solve the problem that correntropy cannotdistinguish the similar impulsive noise, we propose a me-dian-difference correntropy (MDCO) algorithm. 0eMDCO depending on the inner product of vectors measuresthe similarity of multidimensional properties from inputspace. A weighting factor of a median difference is definedand evaluates the similarity between the sample value andthe median of the signal. 0e median difference intuitivelyreflects the abnormality of impulsive noise to guarantee thatthe autocorrelation is finite. 0en, MDCO derives theweighting factor of the generalized correntropy from thecorrentropy criterion which suppresses the larger impulsivenoise. Hence, the weighted covariance function of signalemploys the generalized correntropy instead of second-or-der statistics. 0ese two criteria map the signal from the

low-dimensional space to an infinite-dimensional repro-ducing kernel Hilbert space (RKHS) with impulsive noise,thus including higher-order signal statistics. 0e adaptivekernel size is applied to the weighting factor forMDCOwhichcan obtain the optimal local feature. At convergence, MDCOis unbiased and it is applicable to achieve CRLB under variousparameters. 0e main work is summarized as follows:

(1) We propose a median-difference correntropy(MDCO) algorithm, which can effectively combinethe generalized correntropy and the median differ-ence to suppress impulsive noise

(2) To improve the robustness of MDCO, we also in-troduce a novel adaptive kernel size into theweighting factor of the generalized correntropy andthe median difference

0e rest of this paper is organized as follows: 0eproblem model is defined with DOA in Section 2. In Section3, we present the median-difference correntropy (MDCO)for DOA estimation under impulsive noise environment.0e performance evaluation is presented in Section 4. Fi-nally, conclusions are drawn in Section 5.

2. Problem Formulation

It is considered that a uniform linear array of M isotropicacoustic sensors receives the far-field signal generated byP(M>P) narrowband sources. 0en, Figure 1 illustrates thelinear array model.

0e first array sensor is a reference sensor, and the re-ceived signal of the mth sensor can be expressed as

xm � P

p�1sp(t)e

− j2πfτmp + nm(t), (1)

where sp(t) is the pth signal at time t, nm(t) is the noise of themth sensor, τmp denotes the time delay of the mth acousticssensor relative to the reference sensor for the pth signal, and fis the center frequency of signal.

0e received signal of acoustics can be expressed as

X(t) � A(θ)S(t) + N(t), (2)

where

X(t) � x1(t), x2(t), . . . , xM(t) T,

A(θ) � a θ1( , a θ2( , . . . , a θP( ,

S(t) � s1(t), s2(t), . . . , sP(t) T,

N(t) � n1(t), n2(t), . . . , nM(t) T,

(3)

in which the superscript [•]T represents transpose, X(t) isthe M × 1 vector of the signal received by the acoustic arraysensors, S(t) is the P × 1 vector of the acoustic source, N(t)is the M × 1 vector of impulsive noise that follows alpha-stable distribution, andA(θ) is the M × P array manifolds ofthe array sensors. 0ere is an assumption that the numberMof the array elements is greater than the number P of theacoustic sources. a(θp) is the steering vector that can beexpressed as

2 Wireless Communications and Mobile Computing

a θp � 1, e− j2πfτ2p , . . . , e

− j2πfτMp T. (4)

Impulsive noise N(t) in (2) is usually defined as sym-metric alpha-stable distribution. However, the probabilitydensity function of symmetric alpha-stable distributioncannot be expressed by a general expression [24]. 0erefore,it is generally introduced by its characteristic function whichcan be expressed as

ϕ(t) � exp − c|t|α , (5)

where α is the characteristic exponent of symmetric alpha-stable distribution whose range is 0< α≤ 2. Moreover, thesmaller the α becomes, the more impulsive the non-Gaussian noise will be. c is the dispersion.

Usually, the received data contain very little impulsivenoise and can also be represented as

xm � P

p�1sp(t)e

− j2πfτmp + wm(t) + em(t), (6)

where em(t) denotes the noise without outliers, wm(t) de-notes outliers with sparse characteristics, W(t) � [w1(t),w2(t), . . . , wM(t)]

T, and E(t) � [e1(t), e2(t), . . . , eM(t)]T.

Meanwhile, x′ � s + e denotes the received data withoutoutliers. We can assume that the noise em(t) followsGaussian distribution. 0en, the noise nm(t) � em(t) +wm(t) follows symmetric alpha-stable distribution. 0epurpose of our algorithm is to eliminate outliers wm(t).

Many DOA estimation algorithms usually use the co-variance to represent the second-order statistics of the signal.0e conventional covariance matrix can be given as

R � E X(t)X(t)H � A(θ)RsAH

(θ) + Rn, (7)

where the superscript [•]H represents transpose, the nota-tion E[•] represents the expectation operation, and Rs andRn are the covariance matrix of signal and noise, re-spectively. If the noise contains impulsive noise, the signaland noise cannot be completely orthogonal. 0e conven-tional DOA algorithms that describe signal with impulsivenoise cannot be decomposed into the signal subspace and

the noise subspace by the covariance matrix. 0erefore, thecovariance matrix (7) can also be represented as

R � E X(t)X(t)H � A(θ)RsAH

(θ) + Re + Rw,

Rw � EWH

+ WEH + WWH,(8)

where Rw denotes the covariance term generated by out-liers and the covariance R is dominated by impulsive noisewhen some elements of covariance Rw is much larger thanthe corresponding elements of A(θ)RsAH(θ) + Re. Becausenatural noise and man-made noise often include outliers,some elements of matrix Rw may have a large value whichgives rise to false DOA estimation. From Figure 1, we cansee that the received data of the acoustic sensors sufferimpulsive noise with large values a1, a2, a3, b1, b2,b3, c1, c2, c3}. For example, because data b1 and c2 which arelocated on the different sensors are both impulsive noise atthe same time, the cross-correlation with a relatively largevalue is infinite:

RcomM t2( � YZ � Xm t2( XM t2(

� XM t1 + Δt( XM t2( ej2πfΔt

� b1c2,(9)

where Y and Z are two sampling points of total snapshotsand RcomM(t2) represents the cross-correlation of data b1 andc2 at time t2. Assuming that only one acoustic source im-pinges on the array, the signal Xm(t2) is a delayed signal ofXM(t1) with time Δt when the signal XM(t1) is received bythe array sensor M at time t1. Furthermore, array sensors mandM both contain impulsive noise at time t2. At this point,the noise covariance Rw(t2) is dominant so that the noisesubspace would spread to the signal subspace, causing thecharacteristics of the signal to be covered. In addition, theautocorrelation of data a1, b1, and c1 is also nonexistent.

0e goal is to search for an efficient strategy of sup-pressing impulsive noise that makes it possible to

limwm(t)⟶∞

ψ Rw( ≈ 0, (10)

where ψ(Rw) indicates the operator of suppressing impul-sive noise. In principle, as long as Rw provides a small

Outliers

RmM(t2)ge

RmM(t2)co

RmM(t1)coOutliers

Outliers Cross-correlationof sensor m and M

|c2 – b1| < ε

a3

b3

c3a2a1

b1 b2

c2∆t

c1

t2t2 t3t1

d

90° 1

m

M

CorrelationGeneralized correntropyMDCO

θ

Figure 1: Model of the uniform linear array.

Wireless Communications and Mobile Computing 3

contribution, we can accurately estimate DOA from thecovariance.

3. Median-Difference Correntropy (MDCO)

In order to solve impulsive noise, the structure of this sectionis as follows: firstly, the median-difference correntropy(MDCO) is proposed. Next, we summarize some propertiesfor MDCO. Finally, the application of MDCO for DOAestimation is designed.

Normally, if the signal follows Gaussian distribution,there is a 99.73% probability that results in X′ ∈ (μ′ − 3σ′,μ′ + 3σ′), where X′ is the collection of Gaussian distributionsignal, μ′ is the mean value, and σ′ is the standard deviation(μ′ and σ′ have no relationship to the similar parametersbehind). If X1′, X2′ ∈ (μ′, σ′

2), the X1′ − X2′ is a zero-mean

Gaussian process with variance 2σ′2. 0erefore, there is a

99.73% probability of X1′ − X2′ ∈ (− 6σ′, 6σ′). At this point,we can simply set a threshold ε to measure the similaritybetween the signals and set ε � 6σ′.

Definition 1. For two variables Y and Z of total snapshots X,where Y, Z ∈ X, if |Y − Z|< ε, we say that the variable Y issimilar to Z.

As shown in Table 1, the similarity of the variables Y andZ is divided into four cases. If Y, Z ∈ X′, (X′ ∈ X), thevariables Y and Z are considered normal values. But, ifY, Z ∈ X′ + W, where X′ + W ∈W, we consider the vari-ables Y and Z to be the outliers containing impulsive noise.One of our goals is to suppress the three abnormal cases thatcontain impulsive noise.

3.1. Median-Difference Correntropy (MDCO). In this sec-tion, we present a median-difference correntropy (MDCO)algorithm for DOA with alpha-stable distribution. 0ecorrentropy is a newmethod for nonlinear and local optimalmeasurement of two random variables Y and Z. It is definedas follows:

V(Y, Z) � EYZ[κ(Y, Z)] � Bκ(y, z)pY,Z(y, z)dy dz,

(11)

where κ(., .) is a translation function of the shift-invariantMercer kernel [26] and pY,Z(., .) is the joint probabilitydensity function of Y and Z. In this paper, the Gaussiandensity function is used as kernel function. 0e Gaussiankernel creates a RKHS [26] with universal approximatingcapability. 0e Gaussian kernel is numerically stable andusually gets reasonable results. In this paper, we use aGaussian kernel to suppress impulsive noise.

Assuming that the random variables Y and Z followsymmetric alpha-stable distribution whose characteristicexponent is 1< α< 2, the median-difference correntropy(MDCO) can be considered as

RMDCO � Eexp −||Y| − B|2

2σ2 exp−

||Z| − B|2

2σ2

· exp −|Y − Z|2

2σ2 YZ,

(12)

where σ is the kernel size of Gaussian kernel. 0e variable Bis the average of l1 − norm of the preprocessing data whichcan approximately represent themedian of the received data.

Because impulsive noise contained in the signal is sparse,the variable B can effectively weigh the amplitude of thesignal. 0e variables |Y| − B and |Z| − B reflect the mediandifference to which the received signal deviates from themedian B. 0erefore, the variables |Y| − B and |Z| − B arecalled the weighting factors of the median difference. 0ecorrentropy Y − Z measures the similarity between thesignals and is called the weighting factor of the generalizedcorrentropy. 0e median difference evaluates the deviationbetween the sampling value and the median of the signal,and it intuitively reflects the abnormality of impulsive noise.0en, the median difference is combined with the gener-alized correntropy to form a newweighting factor, which caneffectively suppress the amplitude level of impulsive noise.0en, the median difference and the generalized correntropyare combined with the traditional covariance, and a novelgeneralized weighted covariance is obtained.

To simplify (12), the median differences |Y| − B and |Z| −B can be combined. 0e variable |Y| + |Z| − 2B reflects thedeviation degree of |Y| + |Z|. 0e relaxation of (12) can beexpressed as

RMDCO � E exp −||Y| +|Z| − 2B|2

2σ21 exp −

|Y − Z|2

2σ22 YZ ,

(13)

where |Y| and |Z| are the absolute value of the signal and σ1and σ2 are the kernel sizes that control the scale of themetric.

0e short-time energy method is a common method todetect impulsive noise in time domain [27]. We use a short-time average energy method to preprocess the received data.However, the data processed by the short-time averageenergy method does not directly be used as the input ofMDCO. 0e short-time average energy indicates that theenergy of the signal is average in a short segment. 0e short-time average energy in the kth segment can be expressed asEk:

Ek �

k+I− 1i�k (X(i))

2

I, (14)

Table 1: Comparison of similarity between two variables.

Variables Y, Z ∈ X′

|Y − Z|< ε

Y ∈ X′Z ∈ X′ + W|Y − Z|≫ ε

Y ∈ X′ + WZ ∈ X′ + W

|Y − Z|≫ ε |Y − Z|< εCondition C1 C2 C3 C4Similarity Yes No No Yes


where X(i) is the sampling sequence of the original signal.By the formula above, we can obtain the average energy withI data points.

If the signal energy is much larger than the short-timeaverage energy, the preprocessing signal Xpre can be rep-resented as

Xpre(i) �0, E (X(i))2 >Ek,

X(i), others.

⎧⎨

⎩ (15)

Assuming that N0 is the sum of E[(X(i))2]>Ek in

total snapshots. 0erefore, in this study, B can beexpressed as

B �Xpre

��

��1N − N0

, (16)

where N is the total sampling size of snapshots. 0e signalwith impulsive noise does not participate in calculating themedian of the signal. 0erefore, for a majority of data, themedian B is less than ε/2 due to preprocessing.

0e key to the median-difference correntropy is that thekernel size can work in the confidence interval to eliminateimpulsive noise. 0e kernel size is usually related to thedispersion coefficient c by considering the local feature ofthe signal. We take a modified Sigmoid function to make thekernel size adaptive [24, 28–30]. 0en, kernel sizes σ1 and σ2can be expressed as

σ21 �g1

1 + exp − h1(||Y| +|Z| − 2B|)( · c,

σ22 �g2

1 + exp − h2(|Y − Z|)( · c,

(17)

where g1 and g2 indicate the scale of the modifiedSigmoid function and h1 and h2 control the monotonicityof the modified Sigmoid function [28]. 0e shrinkagedirection and rate are determined by h1 and h2,respectively.

0e adaptive median-difference correntropy provides anew metric criterion for impulsive noise. And the M-esti-mators of the autocorrelation and cross-correlation ofrandom variables Y and Z are in existence associated withthe generalized correntropy and median difference.

In order to prove the effectiveness of MDCO in theory,five crucial properties of MDCO are listed as follows:

Property 1 (C1). When the noise follows temporally sta-tionary zero-mean white Gaussian processes, MDCO isreduced to traditional second-order statistics, and theperformance of MDCO is equal to traditional signalsubspace algorithms (see Appendix A for this proof ).

Property 2 (C2). If the variable Y is an outlier rather than Z,the MDCO can eliminate impulsive noise. Rw(t) of MDCOapproaches zero; that is to say, RMDCO is close to zero at thispoint (see Appendix B for this proof).

Property 3 (C3, C4). When the variables Y and Z bothcontain impulsive noise, Rw(t) and RMDCO both approachzero (see Appendix C for this proof).

Property 4. When the variables Y and Z both contain im-pulsive noise, the MDCO is more effective than the CRCOfor noise suppression (see Appendix D for this proof).

Property 5. Assuming that the random variables Y and Zfollow the symmetric alpha-stable distribution, RMDCO isbounded and the MDCO has the generalized correntropystatistics (see Appendix E for this proof).

Because of the abovementioned properties, the gener-alized weighted covariance RMDCO has finite autocorrelationand cross-correlation at all moments. When the samplingdata contain impulsive noise, Rw(t) of MDCO approacheszero; that is to say, RMDCO is close to zero at this point. If twosignals contain the different degrees of impulsive noise at thesame time, MDCO can induce a very small weighted co-variance by means of the median difference operator andthen obtain a convergent covariance matrix. As shown inTable 2, the performances of the MDCO is listed for differentnoise cases. In summary, the MDCO has reliably goodperformance in all cases.

4. Performance Simulations

In this experiment, we compare the MDCO-MUSIC toFLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC andtest the performance of the proposed MDCO-MUSIC underdifferent parameter conditions. In our experiment, weevaluate the performance of the algorithm from five aspects,namely, various kernel sizes of MDCO-MUSIC, GSNR,snapshots of sampling, different characteristic exponents ofalpha-stable distribution, and angular separation of twodirection angles.

0e resolution probability can well define the perfor-mance of the four algorithms. We use a popular resolutioncriterion to measure the spatial spectrum which can be givenas

E P θm(

the spatial spectrum. 0e two direction angles are resolvableif the result on the left of the equation is smaller than that onthe right; otherwise, the two direction angles are notresolvable.

RMSE is the deviation criterion between the observationand the true value and can be given by

RMSE �

��

Ll�1

θ1(l) − θ1 2

+ Ll�1

θ2(l) − θ2 2

2L,

(19)

where L represents the total number of MC run and θ1(l)and θ2(l) are estimations of θ1 and θ2 in the lth MC run,respectively.

Kozick and Sadler has come to a closed-form expressionof CRLB for the impulsive noise [31], which can be expressedas

CRLB �1Ic

L

1Re SHd (t)D

H I − A AHA − 1AH DSd(t) ,

(20)

4 5 6 7 8 9 10 11 12 13Kernel parameter (g1)

0.24

0.26

0.28

0.3

0.32

0.34

0.36

RMSE

α = 1.4, h1 = –0.1α = 1.8, h1 = –0.1α = 1.4, h1 = –0.4

α = 1.8, h1 = –0.4α = 1.4, h1 = –0.8α = 1.8, h1 = –0.8

Figure 2: Performance comparison of the proposed algorithmversus kernel parameter g1.

–1.8 –1.6 –1.4 –1.2 –1 –0.8 –0.6 –0.4 –0.2 0Kernel parameter (h1)

0.25

0.3

0.35

0.4

0.45

0.5

0.55

RMSE

α = 1.4, g1 = 5α = 1.8, g1 = 5α = 1.4, g1 = 10

α = 1.8, g1 = 10α = 1.4, g1 = 15α = 1.8, g1 = 15

Figure 3: Performance comparison of the proposed algorithmversus kernel parameter h1.

4 6 8 10 12 14 16 18Kernel parameter (g2)

0.28

0.3

0.32

0.34

0.36

0.38

0.4

RMSE

α = 1.4, h2 = –0.4α = 1.8, h2 = –0.4α = 1.4, h2 = –1.2

α = 1.8, h2 = –1.2α = 1.4, h2 = –2.0α = 1.8, h2 = –2.0

Figure 4: Performance comparison of the proposed algorithmversus kernel parameter g2.

–4 –3.5 –3 –2.5 –2 –1.5 –1 –0.5 0Kernel parameter (h2)

0.2

0.25

0.3

0.35

0.4

0.45

0.5

RMSE

α = 1.4, g2 = 5α = 1.8, g2 = 5α = 1.4, g2 = 10

α = 1.8, g2 = 10α = 1.4, g2 = 15α = 1.8, g2 = 15

Figure 5: Performance comparison of the proposed algorithmversus kernel parameter h2.


where Sd(t) � diag s1(t), s2(t), . . . , sP(t) is a diagonalmatrix of the received signal, the subscript θ of A is omittedfrom A(θ), D � [za(θ1)/zθ1, . . . , za(θP)/zθP] is the differ-ential of the array manifolds A, Re ·{ } is the real part, andthe coefficient Ic can be expressed as Ic �

∞0 ((f′(x))2/

f(x))x dx, in which x � |e| is the modulus of the impulsivenoise and f(x) is probability density function of x. Note thatI1 � 1/2c with α � 1 and I2 � 3/5c2 with α � 2. For

simplicity, the coefficient Ic for 1< α< 2 can be approxi-mated by first-order linear interpolation with I1 and I2 [32].

4.1. Experimental Setup. 0e linear array is set to M � 6omnidirectional acoustic sensors which are placed in a halfof the wavelength at the center frequency of the signal. Weconsider the case that the number of narrowband sources is

–10 –8 –6 –4 –2 0 2 4 6 8GSNR

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Pr

obab

ility

of r

esol

utio

n

FLOM_MUSICPFLOM_MUSIC

CRCO_MUSICMDCO_MUSIC

(a)

–10 –8 –6 –4 –2 0 2 4 6 8GSNR

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

RMSE

FLOM_MUSICPFLOM_MUSICCRCO_MUSIC

MDCO_MUSICCRLB

(b)

Figure 6: Performance comparison of four algorithms versus GSNR: (a) probability resolution of four algorithms; (b) RMSE of fouralgorithms.

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9Characteristic exponent (α)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abili

ty o

f res

olut

ion



(a)

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9Characteristic exponent (α)

0

0.5

1

1.5

2

2.5

3

3.5

RMSE


MDCO_MUSICCRLB

(b)

Figure 7: Performance comparison of four algorithms versus α values: (a) probability resolution of four algorithms; (b) RMSE of fouralgorithms.


two which follow temporally stationary zero-mean whiteGaussian processes. 0e central frequency of signal is set tof � 100Hz. In order to distinguish the signal, the power ofeach signal is different. 0e symmetric alpha-stable distri-bution is used to model impulsive noise with the charac-teristic exponent α � 1.4 and dispersion c � 1. 0e GSNR isset to GSNR � 5 dB. 0e sampling frequency of each arrayelement is Fs � 240Hz. All experiments are carried outthrough two directions of arrival associated with θ1 � − 10∘and θ2 � 15∘. 0e snapshots are snaps � 512. Assume thatthe parameter p of the fractional lower-order statistics is

equal to 1.1 in FLOM-MUSIC and PFLOM-MUSIC algo-rithms. In every experiment, 500Monte Carlo (MC) runs areperformed. Furthermore, we compare the resolutionprobability and the root-mean-square error (RMSE) be-tween the different algorithms.

4.2. Kernel Parameters of MDCO. In this experiment, wetest the performance of MDCO-MUSIC with respect to thedifferent kernel parameters of the weighting factors for theseparate characteristic exponents α � 1.4 and α � 1.8. In

0 100 200 300 400 500 600 700 800 900 1000Snapshot

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Pr

obab

ility

of r

esol

utio

n



(a)

0 100 200 300 400 500 600 700 800 900 1000Snapshot

0

0.5

1

1.5

2

2.5

RMSE


MDCO_MUSICCRLB

(b)

Figure 8: Performance comparison of four algorithms versus snapshot values: (a) probability resolution of four algorithms; (b) RMSE offour algorithms.

0 2 4 6 8 10 12 14 16 18 20Angular separation (degree)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abili

ty o

f res

olut

ion



(a)

0 2 4 6 8 10 12 14 16 18 20Angular separation (degree)



0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

RMSE

(b)

Figure 9: Performance comparison of four algorithms versus angular separation: (a) probability resolution of four algorithms; (b) RMSE offour algorithms.


order to eliminate impulsive noise, it is necessary to satisfythe requirement that the large local weighting factor of themedian difference and generalized correntropy associatewith the small kernel sizes σ21 and σ

22; that is, h1 and h2 must

be less than zero. Because the weighting factor of themedian difference and generalized correntropy are in-dependent parameters, the performance can be analyzedseparately. Figures 2 and 3 show that the appropriate g1 andh1 for the median difference can effectively suppress im-pulsive noise and evidently decrease RMSE. From Figure 2,we can see that g1 ∈ [8, 11] would get the best result withthree different values of h1. 0e convergence of g1 is notaffected by h1. Empirically, it is revealed that impulsivenoise can be effectively alleviated in Figure 3, whenh1 ∈ [− 0.5, − 0.3].

Figures 4 and 5 illustrate the robustness of MDCO-MUSIC in terms of the appropriate g2 and h2 for thegeneralized correntropy. MDCO-MUSIC gains the optimalperformance when the kernel parameter g2 varies from 9 to12. Figure 5 shows that RMSE gains a more significantdecrease if h2 ∈ [− 1.4, − 1]. Usually, it is a common methodthat the optimal kernel parameters are obtained by exper-iments. 0erefore, we set g1 � 10, h1 � − 0.4, g2 � 10, andh2 � − 1.2 in other experiments.

4.3. Performance of Different Algorithms

4.3.1. Performance Comparison versus GSNR. Figure 6 an-alyzes the performance of MDCO-MUSIC, FLOM-MUSIC,PFLOM-MUSIC, and CRCO-MUSIC under the averageGSNR from − 10 dB to 8 dB.We can derive the conclusion thatMDCO-MUSIC outperforms FLOM-MUSIC, PFLOM-MU-SIC, and CRCO-MUSIC algorithms both in resolutionprobability and in RMSE. 0e MDCO-MUSIC algorithmyields particularly robust estimation to impulsive noise andcan accurately estimate the DOA when the GSNR is 2 dB. 0eRMSE ofMDCO-MUSIC is much smaller than that of FLOM-MUSIC and PFLOM-MUSIC. It is clear from Figure 6 thatthat the performance of MDCO coincides with the CRLB athigh SNR.

4.3.2. Performance Comparison versus Characteristic Expo-nent of Alpha-Stable Distribution. In this simulation, we aimat evaluating the performance of the characteristic exponenton the basis of probability of resolution and RMSE. 0echaracteristic exponent of the symmetric alpha-stable dis-tribution ranges from 1 to 2. Figure 7 shows that theperformance of MDCO-MUSIC outperforms the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC both inresolution probability and in RMSE. Moreover, the smallerthe α becomes, the worse the performance of FLOM-MUSICand PFLOM-MUSIC will be. Another observation is that it isbeneficial to employ MDCO-MUSIC rather than PFLOM-MUSIC if the source signal is not circularly symmetrical.Furthermore, when α approaches 1, FLOM-MUSIC andPFLOM-MUSIC gain a large RMSE if the parameters of thefractional lower-order statistics cannot automatically adapt.0e result shows that the accuracy of the PFLOM-MUSIC

algorithm is influenced by the relationship of the FLOMparameter p and the characteristic exponent α. Because ofthe contribution of two weighting factors, MDCO-MUSIC isnot affected by the variety of characteristic exponent and caneffectively estimate the DOA in any case.

4.3.3. Performance Comparison versus Snapshots ofSampling. In this simulation, the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC about snapshots is tested under symmetricalpha-stable distribution. Considering that the snapshotsstart at 50 and end at about 1000. Figure 8 illustrates thatMDCO-MUSIC gains a more significant enhancement inresolution probability with the increasing number ofsnapshots and a more evident decrease in RMSE comparedto the other three algorithms. Consequently, MDCO-MUSIC can work effectively for a small number ofsnapshots.

4.3.4. Performance Comparison versus Angular Separation.Figure 9 depicts the performance of four algorithms whenthe direction angle of the second signal is gradually awayfrom that of the first angle. 0e first direction of arrival isθ1 � − 10∘. As expected, the MDCO-MUSIC can effectivelyestimate the two directions of arrival with 7∘ angle sepa-ration. Furthermore, MDCO-MUSIC requires a smallerangle separation threshold than FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC associated with a fixedprobability of resolution, and then MDCO-MUSIC gains alower RMSE than other algorithms associated with a fixedangular separation threshold.

0rough the RMSE curve in the above experiment,MDCO can effectively converge to a stable trend underdifferent parameters, which ensure the boundedness andimprove the robustness of the algorithm.

5. Conclusion

0e issue of DOA in the presence of the abnormal similarityis solved through the adaptive median-difference corren-tropy. 0e MDCO combines the weighting factor of themedian difference and generalized correntropy to suppressimpulsive noise. 0e median difference evaluates the de-viation between the sampling value and the median of thesignal, and it intuitively reflects the abnormality of im-pulsive noise. 0e generalized correntropy measures thesimilarity between the two signals. 0ese two weightingfactors map the signal to an infinite-dimensional space withimpulsive noise, thus including much statistical in-formation. By controlling the adaptive kernel size, MDCOcan effectively deal with DOA. We also prove that theMDCO satisfies some robust properties and applies theMDCO to DOA estimation combined with MUSIC. Ex-perimental results illustrate that the MDCO-MUSIC ismore robust than the FLOM-MUSIC, PFLOM-MUSIC,and CRCO-MUSIC at a much lower GSNR and in theextremely impulsive noise environment.


Appendix

A. Proof of Property 1

In (12), the weighting factors can be represented as

C � exp −||Y| − B|2

2σ2 exp −

||Z| − B|2

2σ2 exp −

|Y − Z|2

2σ2 .

(A.1)

When the noise does not contain impulsive noise, it canbe obtained:

0 < ||Y| − B|< ε,

0 < ||Z| − B|< ε,

0 < |Y − Z|< ε.

(A.2)

0e constraint of the weight factor C can be translatedinto

exp −ε2

2σ2

3

ε. Let us discuss the two cases together,and RMDCO can be expressed as

RMDCO � Eexp −WY

2

2σ2⎛⎝ ⎞⎠exp −

WZ

2

2σ2

· exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠YZ

≈ E 0 × 0 × exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠ × YZ⎡⎢⎢⎣ ⎤⎥⎥⎦ � 0.

(C.1)

Although the variables Y and Z involve impulsive noise,the median difference of |Y| − B and |Z| − B works well on it.

D. Proof of Property 4

To prove Property 4, it is only necessary to prove that|Rmdcow |< |R

crcow | when the sampling data contains impulsive

noise. According to Property 3, 0e authors only need toprove |RMDCO|< |RCRCO|.

For the MDCO and CRCO algorithms, the authors have

RMDCO

− RCRCO

�

Eexp −

||Y| − B|2

2σ2 exp −

||Z| − B|2

2σ2

· exp −|Y − Z|2

2σ2 YZ

− E exp −

|Y − Z|2

2σ2 YZ

≈Eexp −

WY

2

2σ2⎛⎝ ⎞⎠exp −

WZ

2

2σ2

· exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠YZ

− E exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠YZ⎡⎢⎢⎣ ⎤⎥⎥⎦

.

(D.1)

For the generalized covariance of the two variables Y andZ at a given moment, (D.1) can be rewritten as

RMDCO

− RCRCO

≈ Eexp −WY

2

2σ2exp −

WZ

2

2σ2⎛⎝ ⎞⎠ − 1

· exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠|YZ|.

(D.2)

When the variables Y and Z contain impulsive noise,the authors can get exp(− |WY|

2/2σ2)< 1 andexp(− |WZ|

2/2σ2)< 1. 0en,


RMDCO

− RCRCO

≈ Eexp −WY

2

2σ2⎛⎝ ⎞⎠exp −

WZ

2

2σ2⎛⎝ ⎞⎠ − 1

· exp −WY − WZ

2

2σ2⎛⎝ ⎞⎠|YZ|< 0.

(D.3)

In summary, the MDCO algorithm performs better thanthe CRCO algorithm under impulsive noise environments,and results in RMDCO⟶0.

E. Proof of Property 5

Under the premise of Property 4, there exists 0< η< 1 whichmakes

RMDCO � ηRCRCO. (E.1)

Meanwhile, RCRCO is bounded with [24] and can beexpressed as

−σ3��2π

√c≤RCRCO ≤ 4

Γ(1)Γ(− 1/α)α

��π

√Γ(− 1/2)

c1/α

2

. (E.2)

Combining (E.3) with (E.2), the authors can easily get

−σ3��2π

√c≤

RMDCO

η≤ 4Γ(1)Γ(− 1/α)α

��π

√Γ(− 1/2)

c1/α

2

. (E.3)

Transforming (E.4), the authors have

−σ3��2π

√cη≤RMDCO ≤ 4

Γ(1)Γ(− 1/α)α

��π

√Γ(− 1/2)

c1/α

2

η, (E.4)

which means that RMDCO is bounded.

Data Availability

All the data generated or analyzed in this study are availablefrom the corresponding author on reasonable request.

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

0e paper was sponsored by National Key R&D Program ofChina (no. 2017YFB0702300) and National Natural ScienceFoundation of China (no. 61971031).

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