# Median-Difference Correntropy for DOA under the Impulsive Noise Environment

Authors: Fuqiang Ma; Jie He; Xiaotong Zhang
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

The source localization using direction of arrival (DOA) of target is an important research in the field of Internet of Things (IoTs). However, correntropy suffers the performance degradation for direction of arrival when the two signals contain the similar impulsive noise, which cannot be detected by the difference between two signals. This paper proposes a new correntropy, called the median-difference correntropy, which combines the generalized correntropy and the median difference. The median difference is defined as the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise. Then, the median difference is combined with the generalized correntropy to form a new weighting factor that can effectively suppress the amplitude level of impulsive noise. To improve the robustness of the algorithm, an adaptive kernel size is also integrated into the weighting factor to obtain the optimal local feature. The influence of adaptive kernel sizes on the proposed algorithm is simulated, and the comparison between three typical direction-of-arrival estimation algorithms is conducted. The results show that the accuracy of the median-difference correntropy is significantly superior to the correntropy-based correlation and the phased fractional lower-order moment for a wide range of alpha-stable distribution noise environments.


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## Body

## 1. Introduction

The source localization using direction of arrival (DOA) of target is an important research in the field of Internet of Things (IoTs). The direction-of-arrival (DOA) approaches based on acoustics have many applications including radar, sonar, seismic exploration, navigation, and sound source tracking [1–3]. DOA estimation is usually regarded as a problem of signal matching, and the performance is significantly influenced by noise. A majority of existing DOA estimations are based on the concept that noise follows Gaussian distribution [4, 5]. Since the Gaussian process has second-order and higher-order statistics, the traditional DOA algorithms can easily evaluate the signal characteristics according to second-order statistics [6]. The multiple signal classification (MUSIC) algorithm [7, 8] and estimation method of signal parameters via rotational invariance techniques (ESPRIT) are the basic subspace algorithms which have good performance [9, 10]. MUSIC is the representative of the noise subspace algorithm, and ESPRIT is the representative of the signal subspace algorithm.Because of atmospheric noise, electromagnetic interference, sea clutter, car ignitions, and office equipment, the signal is corrupted by the extremely impulsive noise that exhibits irregularity in time domain. In addition, the probability density functions of impulsive noise decay with heavy tails and do not follow a common Gaussian distribution [11]. Therefore, alpha-stable distribution is usually used to define impulsive noise [12]. The conventional covariance matrix is calculated from the second-order statistics of the signal, which may be infinite when the data are corrupted by the extremely impulsive noise [13, 14]. In addition, the conventional DOA algorithms cannot be decomposed into the signal subspace and the noise subspace with covariance matrix. Thus, it has become increasingly important to study DOA under the impulsive noise environment.The relevant statistical algorithms mainly focus on the existence of statistics. The fractional lower-order statistics (FLOS) algorithms [15] exhibit a more desirable performance than the second-order statistics for alpha-stable distribution [16]. When the signal contains impulsive noise, the signal bears the fractional lower-order statistics. The FLOS algorithms employ the minimum dispersion (MD) criterion to suppress impulsive noise, such as the robust covariation in ROC-MUSIC [17], the fractional lower-order statistics- (FLOM-) based MUSIC, and the phased fractional lower-order moment (PFLOM). The FLOM algorithm-based MUSIC obtains a finite covariance by suppressing one of two cross-correlation signals when the characteristic exponent of alpha-stable distribution ranges from 1 to 2 [18]. The PFLOM algorithm gets the accurate DOA estimation with circular symmetrical signals, embedded in the additive impulsive noise [19]. However, investigators demonstrate that the performance of FLOM and PFLOM algorithms depends on the relationship between the parameter of fractional lower-order moment and the characteristic exponent of alpha-stable distribution. If the characteristic exponent is unknown, the performance of FLOM and PFLOM algorithms seriously decreases.The correntropy criterion [20] is a relatively simple method that can measure the local similarity between two signals [21, 22]. Because it has the properties of M-estimation, the correntropy has been widely used in the impulsive noise environment [23]. Zhang et al. [24] investigated a narrowband model based on the generalized correntropy which is called the correntropy-based correlation (CRCO) in impulsive noise environment. The CRCO algorithm imposes a correntropy operator on the covariance matrix to depress impulsive noise. The generalized correntropy is suitably for dealing with the template matching between the received signal and the template signal [25]. Since DOA estimation is a matching problem between two signals, the generalized correntropy is incapable of measuring the difference between the outliers and fails to suppress impulsive noise. Thus, when data contain the similar impulsive noise, the cross-correlation of the generalized correntropy is infinite.In order to solve the problem that correntropy cannot distinguish the similar impulsive noise, we propose a median-difference correntropy (MDCO) algorithm. The MDCO depending on the inner product of vectors measures the similarity of multidimensional properties from input space. A weighting factor of a median difference is defined and evaluates the similarity between the sample value and the median of the signal. The median difference intuitively reflects the abnormality of impulsive noise to guarantee that the autocorrelation is finite. Then, MDCO derives the weighting factor of the generalized correntropy from the correntropy criterion which suppresses the larger impulsive noise. Hence, the weighted covariance function of signal employs the generalized correntropy instead of second-order statistics. These two criteria map the signal from the low-dimensional space to an infinite-dimensional reproducing kernel Hilbert space (RKHS) with impulsive noise, thus including higher-order signal statistics. The adaptive kernel size is applied to the weighting factor for MDCO which can obtain the optimal local feature. At convergence, MDCO is unbiased and it is applicable to achieve CRLB under various parameters. The main work is summarized as follows:(1)
We propose a median-difference correntropy (MDCO) algorithm, which can effectively combine the generalized correntropy and the median difference to suppress impulsive noise(2)
To improve the robustness of MDCO, we also introduce a novel adaptive kernel size into the weighting factor of the generalized correntropy and the median differenceThe rest of this paper is organized as follows: The problem model is defined with DOA in Section2. In Section 3, we present the median-difference correntropy (MDCO) for DOA estimation under impulsive noise environment. The performance evaluation is presented in Section 4. Finally, conclusions are drawn in Section 5.


## 2. Problem Formulation

It is considered that a uniform linear array ofM isotropic acoustic sensors receives the far-field signal generated by PM>P narrowband sources. Then, Figure 1 illustrates the linear array model.Figure 1
Model of the uniform linear array.The first array sensor is a reference sensor, and the received signal of themth sensor can be expressed as(1)xm=∑p=1Pspte−j2πfτmp+nmt,where spt is the pth signal at time t, nmt is the noise of the mth sensor, τmp denotes the time delay of the mth acoustics sensor relative to the reference sensor for the pth signal, and f is the center frequency of signal.The received signal of acoustics can be expressed as(2)Xt=AθSt+Nt,where(3)Xt=x1t,x2t,…,xMtT,Aθ=aθ1,aθ2,…,aθP,St=s1t,s2t,…,sPtT,Nt=n1t,n2t,…,nMtT,in which the superscript •T represents transpose, Xt is the M×1 vector of the signal received by the acoustic array sensors, St is the P×1 vector of the acoustic source, Nt is the M×1 vector of impulsive noise that follows alpha-stable distribution, and Aθ is the M×P array manifolds of the array sensors. There is an assumption that the number M of the array elements is greater than the number P of the acoustic sources. aθp is the steering vector that can be expressed as(4)aθp=1,e−j2πfτ2p,…,e−j2πfτMpT.Impulsive noiseNt in (2) is usually defined as symmetric alpha-stable distribution. However, the probability density function of symmetric alpha-stable distribution cannot be expressed by a general expression [24]. Therefore, it is generally introduced by its characteristic function which can be expressed as(5)ϕt=exp−γtα,where α is the characteristic exponent of symmetric alpha-stable distribution whose range is 0<α≤2. Moreover, the smaller the α becomes, the more impulsive the non-Gaussian noise will be. γ is the dispersion.Usually, the received data contain very little impulsive noise and can also be represented as(6)xm=∑p=1Pspte−j2πfτmp+wmt+emt,where emt denotes the noise without outliers, wmt denotes outliers with sparse characteristics, Wt=w1t,w2t,…,wMtT, and Et=e1t,e2t,…,eMtT. Meanwhile, x′=s+e denotes the received data without outliers. We can assume that the noise emt follows Gaussian distribution. Then, the noise nmt=emt+wmt follows symmetric alpha-stable distribution. The purpose of our algorithm is to eliminate outliers wmt.Many DOA estimation algorithms usually use the covariance to represent the second-order statistics of the signal. The conventional covariance matrix can be given as(7)R=EXtXtH=AθRsAHθ+Rn,where the superscript •H represents transpose, the notation E• represents the expectation operation, and Rs and Rn are the covariance matrix of signal and noise, respectively. If the noise contains impulsive noise, the signal and noise cannot be completely orthogonal. The conventional DOA algorithms that describe signal with impulsive noise cannot be decomposed into the signal subspace and the noise subspace by the covariance matrix. Therefore, the covariance matrix (7) can also be represented as(8)R=EXtXtH=AθRsAHθ+Re+Rw,Rw=EWH+WEH+WWH,where Rw denotes the covariance term generated by outliers and the covariance R is dominated by impulsive noise when some elements of covariance Rw is much larger than the corresponding elements of AθRsAHθ+Re. Because natural noise and man-made noise often include outliers, some elements of matrix Rw may have a large value which gives rise to false DOA estimation. From Figure 1, we can see that the received data of the acoustic sensors suffer impulsive noise with large values a1,a2,a3,b1,b2,b3,c1,c2,c3. For example, because data b1 and c2 which are located on the different sensors are both impulsive noise at the same time, the cross-correlation with a relatively large value is infinite:(9)RmMcot2=YZ=Xmt2XMt2=XMt1+ΔtXMt2ej2πfΔt=b1c2,where Y and Z are two sampling points of total snapshots and RmMcot2 represents the cross-correlation of data b1 and c2 at time t2. Assuming that only one acoustic source impinges on the array, the signal Xmt2 is a delayed signal of XMt1 with time Δt when the signal XMt1 is received by the array sensor M at time t1. Furthermore, array sensors m and M both contain impulsive noise at time t2. At this point, the noise covariance Rwt2 is dominant so that the noise subspace would spread to the signal subspace, causing the characteristics of the signal to be covered. In addition, the autocorrelation of data a1, b1, and c1 is also nonexistent.The goal is to search for an efficient strategy of suppressing impulsive noise that makes it possible to(10)limwmt⟶∞ψRw≈0,where ψRw indicates the operator of suppressing impulsive noise. In principle, as long as Rw provides a small contribution, we can accurately estimate DOA from the covariance.


## 3. Median-Difference Correntropy (MDCO)

In order to solve impulsive noise, the structure of this section is as follows: firstly, the median-difference correntropy (MDCO) is proposed. Next, we summarize some properties for MDCO. Finally, the application of MDCO for DOA estimation is designed.Normally, if the signal follows Gaussian distribution, there is a99.73% probability that results in X′∈μ′−3σ′,μ′+3σ′, where X′ is the collection of Gaussian distribution signal, μ′ is the mean value, and σ′ is the standard deviation (μ′ and σ′ have no relationship to the similar parameters behind). If X1′,X2′∈μ′,σ′2, the X1′−X2′ is a zero-mean Gaussian process with variance 2σ′2. Therefore, there is a 99.73% probability of X1′−X2′∈−6σ′,6σ′. At this point, we can simply set a threshold ε to measure the similarity between the signals and set ε=6σ′.Definition 1.
For two variablesY and Z of total snapshots X, where Y,Z∈X, if Y−Z<ε, we say that the variable Y is similar to Z.
As shown in Table1, the similarity of the variables Y and Z is divided into four cases. If Y,Z∈X′, X′∈X, the variables Y and Z are considered normal values. But, if Y,Z∈X′+W, where X′+W∈W, we consider the variables Y and Z to be the outliers containing impulsive noise. One of our goals is to suppress the three abnormal cases that contain impulsive noise.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
′
+
W
Z
∈
X
′
+
W
Y
−
Z
≫
ε
Y
−
Z
<
ε
Condition
C1
C2
C3
C4
Similarity
Yes
No
No
Yes


### 3.1. Median-Difference Correntropy (MDCO)

In this section, we present a median-difference correntropy (MDCO) algorithm for DOA with alpha-stable distribution. The correntropy is a new method for nonlinear and local optimal measurement of two random variablesY and Z. It is defined as follows:(11)VY,Z=EYZκY,Z=∬κy,zpY,Zy,zdydz,where κ.,. is a translation function of the shift-invariant Mercer kernel [26] and pY,Z.,. is the joint probability density function of Y and Z. In this paper, the Gaussian density function is used as kernel function. The Gaussian kernel creates a RKHS [26] with universal approximating capability. The Gaussian kernel is numerically stable and usually gets reasonable results. In this paper, we use a Gaussian kernel to suppress impulsive noise.Assuming that the random variablesY and Z follow symmetric alpha-stable distribution whose characteristic exponent is 1<α<2, the median-difference correntropy (MDCO) can be considered as(12)RMDCO=Eexp−Y−B22σ2exp−Z−B22σ2exp−Y−Z22σ2YZ,where σ is the kernel size of Gaussian kernel. The variable B is the average of l1−norm of the preprocessing data which can approximately represent the median of the received data.Because impulsive noise contained in the signal is sparse, the variableB can effectively weigh the amplitude of the signal. The variables Y−B and Z−B reflect the median difference to which the received signal deviates from the median B. Therefore, the variables Y−B and Z−B are called the weighting factors of the median difference. The correntropy Y−Z measures the similarity between the signals and is called the weighting factor of the generalized correntropy. The median difference evaluates the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise. Then, the median difference is combined with the generalized correntropy to form a new weighting factor, which can effectively suppress the amplitude level of impulsive noise. Then, the median difference and the generalized correntropy are combined with the traditional covariance, and a novel generalized weighted covariance is obtained.To simplify (12), the median differences Y−B and Z−B can be combined. The variable Y+Z−2B reflects the deviation degree of Y+Z. The relaxation of (12) can be expressed as(13)RMDCO=Eexp−Y+Z−2B22σ12exp−Y−Z22σ22YZ,where Y and Z are the absolute value of the signal and σ1 and σ2 are the kernel sizes that control the scale of the metric.The short-time energy method is a common method to detect 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 average energy method does not directly be used as the input of MDCO. The short-time average energy indicates that the energy of the signal is average in a short segment. The short-time average energy in the kth segment can be expressed as Ek:(14)Ek=∑i=kk+I−1Xi2I,where Xi is the sampling sequence of the original signal. By the formula above, we can obtain the average energy with I data points.If the signal energy is much larger than the short-time average energy, the preprocessing signalXpre can be represented as(15)Xprei=0,EXi2>Ek,Xi,others.Assuming thatN0 is the sum of EXi2>Ek in total snapshots. Therefore, in this study, B can be expressed as(16)B=Xpre1N−N0,where N is the total sampling size of snapshots. The signal with impulsive noise does not participate in calculating the median of the signal. Therefore, for a majority of data, the median B is less than ε/2 due to preprocessing.The key to the median-difference correntropy is that the kernel size can work in the confidence interval to eliminate impulsive noise. The kernel size is usually related to the dispersion coefficientγ by considering the local feature of the signal. We take a modified Sigmoid function to make the kernel size adaptive [24, 28–30]. Then, kernel sizes σ1 and σ2 can be expressed as(17)σ12=g11+exp−h1Y+Z−2B⋅γ,σ22=g21+exp−h2Y−Z⋅γ,where g1 and g2 indicate the scale of the modified Sigmoid function and h1 and h2 control the monotonicity of the modified Sigmoid function [28]. The shrinkage direction and rate are determined by h1 and h2, respectively.The adaptive median-difference correntropy provides a new metric criterion for impulsive noise. And the M-estimators of the autocorrelation and cross-correlation of random variablesY and Z are in existence associated with the 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 stationary zero-mean white Gaussian processes, MDCO is reduced to traditional second-order statistics, and the performance of MDCO is equal to traditional signal subspace algorithms (see Appendix A for this proof).Property 2 (C2).
If the variableY is an outlier rather than Z, the MDCO can eliminate impulsive noise. Rwt of MDCO approaches zero; that is to say, RMDCO is close to zero at this point (see Appendix B for this proof).Property 3.
(C3, C4). When the variablesY and Z both contain impulsive noise, Rwt and RMDCO both approach zero (see Appendix C for this proof).Property 4.
When the variablesY and Z both contain impulsive noise, the MDCO is more effective than the CRCO for noise suppression (see Appendix D for this proof).Property 5.
Assuming that the random variablesY and Z follow the symmetric alpha-stable distribution, RMDCO is bounded and the MDCO has the generalized correntropy statistics (see Appendix E for this proof).Because of the abovementioned properties, the generalized weighted covarianceRMDCO has finite autocorrelation and cross-correlation at all moments. When the sampling data contain impulsive noise, Rwt of MDCO approaches zero; that is to say, RMDCO is close to zero at this point. If two signals contain the different degrees of impulsive noise at the same time, MDCO can induce a very small weighted covariance by means of the median difference operator and then obtain a convergent covariance matrix. As shown in Table 2, the performances of the MDCO is listed for different noise cases. In summary, the MDCO has reliably good performance in all cases.Table 2
Comparison of performance between CRCO and MDCO.
Variables
Y
,
Z
∈
X
′, Y−Z<ε
Y
∈
X
′, Z∈X′+W, Y−Z≫ε
Y
∈
X
′
+
W, Z∈X′+W
Y
−
Z
≫
ε
Y
−
Z
<
ε
Condition
C1
C2
C3
C4
MDCO
Success
Success
Success
Success


## 3.1. Median-Difference Correntropy (MDCO)

In this section, we present a median-difference correntropy (MDCO) algorithm for DOA with alpha-stable distribution. The correntropy is a new method for nonlinear and local optimal measurement of two random variablesY and Z. It is defined as follows:(11)VY,Z=EYZκY,Z=∬κy,zpY,Zy,zdydz,where κ.,. is a translation function of the shift-invariant Mercer kernel [26] and pY,Z.,. is the joint probability density function of Y and Z. In this paper, the Gaussian density function is used as kernel function. The Gaussian kernel creates a RKHS [26] with universal approximating capability. The Gaussian kernel is numerically stable and usually gets reasonable results. In this paper, we use a Gaussian kernel to suppress impulsive noise.Assuming that the random variablesY and Z follow symmetric alpha-stable distribution whose characteristic exponent is 1<α<2, the median-difference correntropy (MDCO) can be considered as(12)RMDCO=Eexp−Y−B22σ2exp−Z−B22σ2exp−Y−Z22σ2YZ,where σ is the kernel size of Gaussian kernel. The variable B is the average of l1−norm of the preprocessing data which can approximately represent the median of the received data.Because impulsive noise contained in the signal is sparse, the variableB can effectively weigh the amplitude of the signal. The variables Y−B and Z−B reflect the median difference to which the received signal deviates from the median B. Therefore, the variables Y−B and Z−B are called the weighting factors of the median difference. The correntropy Y−Z measures the similarity between the signals and is called the weighting factor of the generalized correntropy. The median difference evaluates the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise. Then, the median difference is combined with the generalized correntropy to form a new weighting factor, which can effectively suppress the amplitude level of impulsive noise. Then, the median difference and the generalized correntropy are combined with the traditional covariance, and a novel generalized weighted covariance is obtained.To simplify (12), the median differences Y−B and Z−B can be combined. The variable Y+Z−2B reflects the deviation degree of Y+Z. The relaxation of (12) can be expressed as(13)RMDCO=Eexp−Y+Z−2B22σ12exp−Y−Z22σ22YZ,where Y and Z are the absolute value of the signal and σ1 and σ2 are the kernel sizes that control the scale of the metric.The short-time energy method is a common method to detect 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 average energy method does not directly be used as the input of MDCO. The short-time average energy indicates that the energy of the signal is average in a short segment. The short-time average energy in the kth segment can be expressed as Ek:(14)Ek=∑i=kk+I−1Xi2I,where Xi is the sampling sequence of the original signal. By the formula above, we can obtain the average energy with I data points.If the signal energy is much larger than the short-time average energy, the preprocessing signalXpre can be represented as(15)Xprei=0,EXi2>Ek,Xi,others.Assuming thatN0 is the sum of EXi2>Ek in total snapshots. Therefore, in this study, B can be expressed as(16)B=Xpre1N−N0,where N is the total sampling size of snapshots. The signal with impulsive noise does not participate in calculating the median of the signal. Therefore, for a majority of data, the median B is less than ε/2 due to preprocessing.The key to the median-difference correntropy is that the kernel size can work in the confidence interval to eliminate impulsive noise. The kernel size is usually related to the dispersion coefficientγ by considering the local feature of the signal. We take a modified Sigmoid function to make the kernel size adaptive [24, 28–30]. Then, kernel sizes σ1 and σ2 can be expressed as(17)σ12=g11+exp−h1Y+Z−2B⋅γ,σ22=g21+exp−h2Y−Z⋅γ,where g1 and g2 indicate the scale of the modified Sigmoid function and h1 and h2 control the monotonicity of the modified Sigmoid function [28]. The shrinkage direction and rate are determined by h1 and h2, respectively.The adaptive median-difference correntropy provides a new metric criterion for impulsive noise. And the M-estimators of the autocorrelation and cross-correlation of random variablesY and Z are in existence associated with the 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 stationary zero-mean white Gaussian processes, MDCO is reduced to traditional second-order statistics, and the performance of MDCO is equal to traditional signal subspace algorithms (see Appendix A for this proof).Property 2 (C2).
If the variableY is an outlier rather than Z, the MDCO can eliminate impulsive noise. Rwt of MDCO approaches zero; that is to say, RMDCO is close to zero at this point (see Appendix B for this proof).Property 3.
(C3, C4). When the variablesY and Z both contain impulsive noise, Rwt and RMDCO both approach zero (see Appendix C for this proof).Property 4.
When the variablesY and Z both contain impulsive noise, the MDCO is more effective than the CRCO for noise suppression (see Appendix D for this proof).Property 5.
Assuming that the random variablesY and Z follow the symmetric alpha-stable distribution, RMDCO is bounded and the MDCO has the generalized correntropy statistics (see Appendix E for this proof).Because of the abovementioned properties, the generalized weighted covarianceRMDCO has finite autocorrelation and cross-correlation at all moments. When the sampling data contain impulsive noise, Rwt of MDCO approaches zero; that is to say, RMDCO is close to zero at this point. If two signals contain the different degrees of impulsive noise at the same time, MDCO can induce a very small weighted covariance by means of the median difference operator and then obtain a convergent covariance matrix. As shown in Table 2, the performances of the MDCO is listed for different noise cases. In summary, the MDCO has reliably good performance in all cases.Table 2
Comparison of performance between CRCO and MDCO.
Variables
Y
,
Z
∈
X
′, Y−Z<ε
Y
∈
X
′, Z∈X′+W, Y−Z≫ε
Y
∈
X
′
+
W, Z∈X′+W
Y
−
Z
≫
ε
Y
−
Z
<
ε
Condition
C1
C2
C3
C4
MDCO
Success
Success
Success
Success


## 4. Performance Simulations

In this experiment, we compare the MDCO-MUSIC to FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC and test the performance of the proposed MDCO-MUSIC under different parameter conditions. In our experiment, we evaluate the performance of the algorithm from five aspects, namely, various kernel sizes of MDCO-MUSIC, GSNR, snapshots of sampling, different characteristic exponents of alpha-stable distribution, and angular separation of two direction angles.The resolution probability can well define the performance of the four algorithms. We use a popular resolution criterion to measure the spatial spectrum which can be given as(18)EPθm<EPθ1+Pθ22,where θ1 and θ2 are the two independent direction angles, θm=θ1+θ2/2 is the midangle between them, and Pθ is the spatial spectrum. The two direction angles are resolvable if the result on the left of the equation is smaller than that on the right; otherwise, the two direction angles are not resolvable.RMSE is the deviation criterion between the observation and the true value and can be given by(19)RMSE=∑l=1Lθ^1l−θ12+∑l=1Lθ^2l−θ222L,where L represents the total number of MC run and θ^1l and θ^2l are estimations of θ1 and θ2 in the lth MC run, respectively.Kozick and Sadler has come to a closed-form expression of CRLB for the impulsive noise [31], which can be expressed as(20)CRLB=1Ic∑1LReSdHtDHI−AAHA−1AHDSdt,where Sdt=diags1t,s2t,…,sPt is a diagonal matrix of the received signal, the subscript θ of A is omitted from Aθ, D=∂aθ1/∂θ1,…,∂aθP/∂θP is the differential of the array manifolds A, Re⋅ is the real part, and the coefficient Ic can be expressed as Ic=∫0∞f′x2/fxxdx, in which x=e is the modulus of the impulsive noise and fx is probability density function of x. Note that I1=1/2γ with α=1 and I2=3/5γ2 with α=2. For simplicity, the coefficient Ic for 1<α<2 can be approximated by first-order linear interpolation with I1 and I2 [32].


### 4.1. Experimental Setup

The linear array is set toM=6 omnidirectional acoustic sensors which are placed in a half of the wavelength at the center frequency of the signal. We consider the case that the number of narrowband sources is two which follow temporally stationary zero-mean white Gaussian processes. The central frequency of signal is set to f=100Hz. In order to distinguish the signal, the power of each signal is different. The symmetric alpha-stable distribution is used to model impulsive noise with the characteristic exponent α=1.4 and dispersion γ=1. The GSNR is set to GSNR=5dB. The sampling frequency of each array element is Fs=240Hz. All experiments are carried out through two directions of arrival associated with θ1=−10∘ and θ2=15∘. The snapshots are snaps=512. Assume that the parameter p of the fractional lower-order statistics is equal to 1.1 in FLOM-MUSIC and PFLOM-MUSIC algorithms. In every experiment, 500 Monte Carlo (MC) runs are performed. Furthermore, we compare the resolution probability and the root-mean-square error (RMSE) between the different algorithms.


### 4.2. Kernel Parameters of MDCO

In this experiment, we test the performance of MDCO-MUSIC with respect to the different kernel parameters of the weighting factors for the separate characteristic exponentsα=1.4 and α=1.8. In order to eliminate impulsive noise, it is necessary to satisfy the requirement that the large local weighting factor of the median difference and generalized correntropy associate with the small kernel sizes σ12 and σ22; that is, h1 and h2 must be less than zero. Because the weighting factor of the median difference and generalized correntropy are independent parameters, the performance can be analyzed separately. Figures 2 and 3 show that the appropriate g1 and h1 for the median difference can effectively suppress impulsive noise and evidently decrease RMSE. From Figure 2, we can see that g1∈8,11 would get the best result with three different values of h1. The convergence of g1 is not affected by h1. Empirically, it is revealed that impulsive noise can be effectively alleviated in Figure 3, when h1∈−0.5,−0.3.Figure 2
Performance comparison of the proposed algorithm versus kernel parameterg1.Figure 3
Performance comparison of the proposed algorithm versus kernel parameterh1.Figures4 and 5 illustrate the robustness of MDCO-MUSIC in terms of the appropriate g2 and h2 for the generalized correntropy. MDCO-MUSIC gains the optimal performance when the kernel parameter g2 varies from 9 to 12. Figure 5 shows that RMSE gains a more significant decrease if h2∈−1.4,−1. Usually, it is a common method that the optimal kernel parameters are obtained by experiments. Therefore, we set g1=10, h1=−0.4, g2=10, and h2=−1.2 in other experiments.Figure 4
Performance comparison of the proposed algorithm versus kernel parameterg2.Figure 5
Performance comparison of the proposed algorithm versus kernel parameterh2.


### 4.3. Performance of Different Algorithms

#### 4.3.1. Performance Comparison versus GSNR

Figure6 analyzes the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC under the average GSNR from −10 dB to 8 dB. We can derive the conclusion that MDCO-MUSIC outperforms FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC algorithms both in resolution probability and in RMSE. The MDCO-MUSIC algorithm yields particularly robust estimation to impulsive noise and can accurately estimate the DOA when the GSNR is 2 dB. The RMSE of MDCO-MUSIC is much smaller than that of FLOM-MUSIC and PFLOM-MUSIC. It is clear from Figure 6 that that the performance of MDCO coincides with the CRLB at high SNR.Figure 6
Performance comparison of four algorithms versus GSNR: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


#### 4.3.2. Performance Comparison versus Characteristic Exponent of Alpha-Stable Distribution

In this simulation, we aim at evaluating the performance of the characteristic exponent on the basis of probability of resolution and RMSE. The characteristic exponent of the symmetric alpha-stable distribution ranges from 1 to 2. Figure7 shows that the performance of MDCO-MUSIC outperforms the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC both in resolution probability and in RMSE. Moreover, the smaller the α becomes, the worse the performance of FLOM-MUSIC and PFLOM-MUSIC will be. Another observation is that it is beneficial to employ MDCO-MUSIC rather than PFLOM-MUSIC if the source signal is not circularly symmetrical. Furthermore, when α approaches 1, FLOM-MUSIC and PFLOM-MUSIC gain a large RMSE if the parameters of the fractional lower-order statistics cannot automatically adapt. The result shows that the accuracy of the PFLOM-MUSIC algorithm is influenced by the relationship of the FLOM parameter p and the characteristic exponent α. Because of the contribution of two weighting factors, MDCO-MUSIC is not affected by the variety of characteristic exponent and can effectively estimate the DOA in any case.Figure 7
Performance comparison of four algorithms versusα values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


#### 4.3.3. Performance Comparison versus Snapshots of Sampling

In this simulation, the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC about snapshots is tested under symmetric alpha-stable distribution. Considering that the snapshots start at 50 and end at about 1000. Figure8 illustrates that MDCO-MUSIC gains a more significant enhancement in resolution probability with the increasing number of snapshots and a more evident decrease in RMSE compared to the other three algorithms. Consequently, MDCO-MUSIC can work effectively for a small number of snapshots.Figure 8
Performance comparison of four algorithms versus snapshot values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


#### 4.3.4. Performance Comparison versus Angular Separation

Figure9 depicts the performance of four algorithms when the direction angle of the second signal is gradually away from that of the first angle. The first direction of arrival is θ1=−10∘. As expected, the MDCO-MUSIC can effectively estimate the two directions of arrival with 7∘ angle separation. Furthermore, MDCO-MUSIC requires a smaller angle separation threshold than FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC associated with a fixed probability of resolution, and then MDCO-MUSIC gains a lower RMSE than other algorithms associated with a fixed angular separation threshold.Figure 9
Performance comparison of four algorithms versus angular separation: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)Through the RMSE curve in the above experiment, MDCO can effectively converge to a stable trend under different parameters, which ensure the boundedness and improve the robustness of the algorithm.


## 4.1. Experimental Setup

The linear array is set toM=6 omnidirectional acoustic sensors which are placed in a half of the wavelength at the center frequency of the signal. We consider the case that the number of narrowband sources is two which follow temporally stationary zero-mean white Gaussian processes. The central frequency of signal is set to f=100Hz. In order to distinguish the signal, the power of each signal is different. The symmetric alpha-stable distribution is used to model impulsive noise with the characteristic exponent α=1.4 and dispersion γ=1. The GSNR is set to GSNR=5dB. The sampling frequency of each array element is Fs=240Hz. All experiments are carried out through two directions of arrival associated with θ1=−10∘ and θ2=15∘. The snapshots are snaps=512. Assume that the parameter p of the fractional lower-order statistics is equal to 1.1 in FLOM-MUSIC and PFLOM-MUSIC algorithms. In every experiment, 500 Monte Carlo (MC) runs are performed. Furthermore, we compare the resolution probability and the root-mean-square error (RMSE) between the different algorithms.


## 4.2. Kernel Parameters of MDCO

In this experiment, we test the performance of MDCO-MUSIC with respect to the different kernel parameters of the weighting factors for the separate characteristic exponentsα=1.4 and α=1.8. In order to eliminate impulsive noise, it is necessary to satisfy the requirement that the large local weighting factor of the median difference and generalized correntropy associate with the small kernel sizes σ12 and σ22; that is, h1 and h2 must be less than zero. Because the weighting factor of the median difference and generalized correntropy are independent parameters, the performance can be analyzed separately. Figures 2 and 3 show that the appropriate g1 and h1 for the median difference can effectively suppress impulsive noise and evidently decrease RMSE. From Figure 2, we can see that g1∈8,11 would get the best result with three different values of h1. The convergence of g1 is not affected by h1. Empirically, it is revealed that impulsive noise can be effectively alleviated in Figure 3, when h1∈−0.5,−0.3.Figure 2
Performance comparison of the proposed algorithm versus kernel parameterg1.Figure 3
Performance comparison of the proposed algorithm versus kernel parameterh1.Figures4 and 5 illustrate the robustness of MDCO-MUSIC in terms of the appropriate g2 and h2 for the generalized correntropy. MDCO-MUSIC gains the optimal performance when the kernel parameter g2 varies from 9 to 12. Figure 5 shows that RMSE gains a more significant decrease if h2∈−1.4,−1. Usually, it is a common method that the optimal kernel parameters are obtained by experiments. Therefore, we set g1=10, h1=−0.4, g2=10, and h2=−1.2 in other experiments.Figure 4
Performance comparison of the proposed algorithm versus kernel parameterg2.Figure 5
Performance comparison of the proposed algorithm versus kernel parameterh2.


## 4.3. Performance of Different Algorithms

### 4.3.1. Performance Comparison versus GSNR

Figure6 analyzes the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC under the average GSNR from −10 dB to 8 dB. We can derive the conclusion that MDCO-MUSIC outperforms FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC algorithms both in resolution probability and in RMSE. The MDCO-MUSIC algorithm yields particularly robust estimation to impulsive noise and can accurately estimate the DOA when the GSNR is 2 dB. The RMSE of MDCO-MUSIC is much smaller than that of FLOM-MUSIC and PFLOM-MUSIC. It is clear from Figure 6 that that the performance of MDCO coincides with the CRLB at high SNR.Figure 6
Performance comparison of four algorithms versus GSNR: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


### 4.3.2. Performance Comparison versus Characteristic Exponent of Alpha-Stable Distribution

In this simulation, we aim at evaluating the performance of the characteristic exponent on the basis of probability of resolution and RMSE. The characteristic exponent of the symmetric alpha-stable distribution ranges from 1 to 2. Figure7 shows that the performance of MDCO-MUSIC outperforms the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC both in resolution probability and in RMSE. Moreover, the smaller the α becomes, the worse the performance of FLOM-MUSIC and PFLOM-MUSIC will be. Another observation is that it is beneficial to employ MDCO-MUSIC rather than PFLOM-MUSIC if the source signal is not circularly symmetrical. Furthermore, when α approaches 1, FLOM-MUSIC and PFLOM-MUSIC gain a large RMSE if the parameters of the fractional lower-order statistics cannot automatically adapt. The result shows that the accuracy of the PFLOM-MUSIC algorithm is influenced by the relationship of the FLOM parameter p and the characteristic exponent α. Because of the contribution of two weighting factors, MDCO-MUSIC is not affected by the variety of characteristic exponent and can effectively estimate the DOA in any case.Figure 7
Performance comparison of four algorithms versusα values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


### 4.3.3. Performance Comparison versus Snapshots of Sampling

In this simulation, the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC about snapshots is tested under symmetric alpha-stable distribution. Considering that the snapshots start at 50 and end at about 1000. Figure8 illustrates that MDCO-MUSIC gains a more significant enhancement in resolution probability with the increasing number of snapshots and a more evident decrease in RMSE compared to the other three algorithms. Consequently, MDCO-MUSIC can work effectively for a small number of snapshots.Figure 8
Performance comparison of four algorithms versus snapshot values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


### 4.3.4. Performance Comparison versus Angular Separation

Figure9 depicts the performance of four algorithms when the direction angle of the second signal is gradually away from that of the first angle. The first direction of arrival is θ1=−10∘. As expected, the MDCO-MUSIC can effectively estimate the two directions of arrival with 7∘ angle separation. Furthermore, MDCO-MUSIC requires a smaller angle separation threshold than FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC associated with a fixed probability of resolution, and then MDCO-MUSIC gains a lower RMSE than other algorithms associated with a fixed angular separation threshold.Figure 9
Performance comparison of four algorithms versus angular separation: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)Through the RMSE curve in the above experiment, MDCO can effectively converge to a stable trend under different parameters, which ensure the boundedness and improve the robustness of the algorithm.


## 4.3.1. Performance Comparison versus GSNR

Figure6 analyzes the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC under the average GSNR from −10 dB to 8 dB. We can derive the conclusion that MDCO-MUSIC outperforms FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC algorithms both in resolution probability and in RMSE. The MDCO-MUSIC algorithm yields particularly robust estimation to impulsive noise and can accurately estimate the DOA when the GSNR is 2 dB. The RMSE of MDCO-MUSIC is much smaller than that of FLOM-MUSIC and PFLOM-MUSIC. It is clear from Figure 6 that that the performance of MDCO coincides with the CRLB at high SNR.Figure 6
Performance comparison of four algorithms versus GSNR: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


## 4.3.2. Performance Comparison versus Characteristic Exponent of Alpha-Stable Distribution

In this simulation, we aim at evaluating the performance of the characteristic exponent on the basis of probability of resolution and RMSE. The characteristic exponent of the symmetric alpha-stable distribution ranges from 1 to 2. Figure7 shows that the performance of MDCO-MUSIC outperforms the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC both in resolution probability and in RMSE. Moreover, the smaller the α becomes, the worse the performance of FLOM-MUSIC and PFLOM-MUSIC will be. Another observation is that it is beneficial to employ MDCO-MUSIC rather than PFLOM-MUSIC if the source signal is not circularly symmetrical. Furthermore, when α approaches 1, FLOM-MUSIC and PFLOM-MUSIC gain a large RMSE if the parameters of the fractional lower-order statistics cannot automatically adapt. The result shows that the accuracy of the PFLOM-MUSIC algorithm is influenced by the relationship of the FLOM parameter p and the characteristic exponent α. Because of the contribution of two weighting factors, MDCO-MUSIC is not affected by the variety of characteristic exponent and can effectively estimate the DOA in any case.Figure 7
Performance comparison of four algorithms versusα values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


## 4.3.3. Performance Comparison versus Snapshots of Sampling

In this simulation, the performance of MDCO-MUSIC, FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC about snapshots is tested under symmetric alpha-stable distribution. Considering that the snapshots start at 50 and end at about 1000. Figure8 illustrates that MDCO-MUSIC gains a more significant enhancement in resolution probability with the increasing number of snapshots and a more evident decrease in RMSE compared to the other three algorithms. Consequently, MDCO-MUSIC can work effectively for a small number of snapshots.Figure 8
Performance comparison of four algorithms versus snapshot values: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)


## 4.3.4. Performance Comparison versus Angular Separation

Figure9 depicts the performance of four algorithms when the direction angle of the second signal is gradually away from that of the first angle. The first direction of arrival is θ1=−10∘. As expected, the MDCO-MUSIC can effectively estimate the two directions of arrival with 7∘ angle separation. Furthermore, MDCO-MUSIC requires a smaller angle separation threshold than FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC associated with a fixed probability of resolution, and then MDCO-MUSIC gains a lower RMSE than other algorithms associated with a fixed angular separation threshold.Figure 9
Performance comparison of four algorithms versus angular separation: (a) probability resolution of four algorithms; (b) RMSE of four algorithms.
(a)
(b)Through the RMSE curve in the above experiment, MDCO can effectively converge to a stable trend under different parameters, which ensure the boundedness and improve the robustness of the algorithm.


## 5. Conclusion

The issue of DOA in the presence of the abnormal similarity is solved through the adaptive median-difference correntropy. The MDCO combines the weighting factor of the median difference and generalized correntropy to suppress impulsive noise. The median difference evaluates the deviation between the sampling value and the median of the signal, and it intuitively reflects the abnormality of impulsive noise. The generalized correntropy measures the similarity between the two signals. These two weighting factors map the signal to an infinite-dimensional space with impulsive noise, thus including much statistical information. By controlling the adaptive kernel size, MDCO can effectively deal with DOA. We also prove that the MDCO satisfies some robust properties and applies the MDCO to DOA estimation combined with MUSIC. Experimental results illustrate that the MDCO-MUSIC is more robust than the FLOM-MUSIC, PFLOM-MUSIC, and CRCO-MUSIC at a much lower GSNR and in the extremely impulsive noise environment.



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*Source: 8107176-2019-10-03.xml*