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Correntropybased DOA estimation algorithm under impulsive noise environments
EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 154 (2020)
Abstract
In this paper, the direction of arrival (DOA) estimation of signals in the presence of impulsive noise environment is studied. Complex isotropic symmetric alphastable (SαS) random variables are modeled as impulsive noise, then a novel secondorder statistic method that correntropybased covariance matrix (CBCM) is defined, based on the combination of the CBCM of the array sensor outputs with the signal subspace technique (e.g., multiple signal classification (MUSIC)), which can be achieved source localization under impulsive noise environments. The MonteCarlo simulation results illustrate the improved performance of CBCMMUSIC for DOA estimation under a wide range of impulsive noise conditions.
Introduction
Array signal processing is an important branch in modern signal processing. It is widely used in the field of radar [1], sonar [2], 5G communication [3,4,5,6], and smart antenna [7]. DOA estimation algorithms under additive white noise have been extensively studied in the past several decades. However, literature [8,9,10,11] studies show that atmospheric environments, sea clutter, ground clutter, radar backscatter echoes, and urban mobile radio channels, sudden bursts, or sharp spikes are exhibited at the array outputs which can be characterized as impulsive; an impulsive environment can be well modeled by alphastable distribution [12], compared with the mixture Gaussian distribution, and the alphastable distribution has “thick tail” statistical characteristics of impulse noise, which makes it have strongly impulse characteristics in the time domain. Secondorder statistics are not finite [12] under alphastable distribution. In order to suppress these impulsive outliers, researchers have proposed a series of DOA estimation methods based on fractional lowerorder statistics (FLOS).
In [13], the authors proposed new subspace methods based on fractional lowerorder moment (FLOM) covariance estimates, and the robust covariationbased MUSIC (ROCMUSIC) used covariations under the assumption that the signals and the additive noise are jointly SαS, which does not hold always because signals of interest are generally of finite variances, and the ROCMUSIC is defined only for 1 < α < 2.
In [14], the authors proposed a new class of covariance matrices named FLOM matrices for impulsive noise. The FLOM outperforms the ROCMUSIC from the fact that it handles all types of signals. However, it is limited in the range of α (α > 1) for robust covariation.
In [15], the authors introduce a new subspace algorithm based on the phased fractional lowerorder moment (PFLOM), and the new subspace algorithm based on the PFLOM covariance estimation shows a higher resolution capability and lower estimation error.
In [16], the authors proposed a new operator referred to as the correntropybased correlation (CRCO), and it can be applied with MUSIC algorithm; despite the CRCOMUSIC shows robustness in highly impulsive noise environments or in low generalized signal to noise ratio (GSNR) situation, the formulation for the robust CRCO statistics needs quite a number of snapshots.
Professor Principe’s team first proposed the correntropy in [17]. The correntropy is a new statistic that can quantify the time structure as well as the statistical distribution of two stochastic random processes. The correntropy function conveys information about the quadratic Renyi’s entropy of the generating source. At the same time, correntropy can well suppress impulse noise and does not depend on the prior knowledge of alphastable noise. Therefore, it has been widely used in signal detection [18], time delay estimation [19], adaptive filtering [17], and image processing [20].
In this paper, we focused on the issue of DOA estimation algorithm in extremely high impulsive noise environments along with low generalized signal to noise ratio (GSNR) levels and fewer snapshots and introduced a new operator based on correntropy, namely, correntropybased covariance matrices (CBCM), which can be combined with subspace algorithm, such as MUSIC algorithm, that is CBCMMUSIC algorithm. The CBCMMUSIC algorithm exhibits an evident performance in low GSNR or in strong impulsive environments.
Our major contributions are listed as follows:

(1)
Consider the problem of DOA estimation in impulsive noise and proposes a new method to rebuild the covariance matrix based on correntropy.

(2)
Several experiments have been made for the selection of kernel size and suppression parameter.

(3)
The performances of different algorithms are compared, and the effectiveness of the proposed method is verified.
The paper is organized as follows: in Section 2, we define the problem of interest and briefly review some preliminaries on αstable distributions. In Section 3, we provide the CBCMMUSIC algorithm. Finally, some simulation examples are presented in Section 4, and the conclusion is given in Section 5.
Notation: Lowercase (capital) bold symbols denote vector (matrix). (·)^{T}, (·)^{*}, and (·)^{H} denote transpose, complex conjugation, and conjugate transposition, respectively; E{·} denotes expectation operator; and · stands for an absolute value of a random quantity.
Problem definition
Here, the array signal model has given in Section 2.1, and then the impulsive noise model (alphastable noise) is presented in Section 2.2.
Data model
Consider a uniform linear array (ULA) with M sensors and P narrowband farfield signal source impinging on the ULA from angular direction θ_{p}, which space half of the wavelength. The sensor received signal kth sample can be modeled as
where X(k) is the array received observation vector
A_{m} is an ideal steering matrix when setting the first sensor as the reference
S_{k}(θ) is the signal vector
N(k) is alphastable noise
Our objective is to estimate the DOA (θ_{1}, θ_{2}, …, θ_{p}) of the source from X(k), the secondorder statistical is the most commonly used method, that is
In practice, the covariance matrix R can be estimated with a finite number of snapshots via
By performing eigenvalue decomposition (EVD) on R^{^}, we can obtain
where U = [U_{s}, U_{n}], U_{s,} and U_{n} are the noise and signal subspace matrices and Λ_{s} and Λ_{n} are the corresponding eigenvalue matrices. According to the orthogonality between noise and signal subspaces [21], U_{s}⊥U_{n}, spatial power spectrum can be obtained from
Alphastable noise (SαS)
Recent studies show that alphastable distribution is well suited for describing impulsive noise [12], which is defined by a characteristic function
where
As can be seen from the above Eqs. (10, 11 and 12), alphastable distribution is determined by α, β, γ, and μ, and its characteristics are as follows.

(1)
α is the characteristic exponent which is characterized the strength of the impulsiveness, the smaller α, the heavier the tails of the alphastable probability density function, 0 < α ≤ 2;

(2)
β is the symmetry parameter which is characterized by the level of outofcenter for alphastable probability density function. Alphastable distribution is called symmetric alpha stable (SαS) when β = 0, − 1 ≤ β ≤ 1;

(3)
γ is the dispersion parameter which is characterized by the degree of outofmean for the random variable, γ > 0;

(4)
ξ is the location parameter which is determined by the central position of alphastable distribution. ξ denotes the mean of the random variable when the α satisfies with 1 < α ≤ 2; ξ denotes the median of the random variable when the characteristic exponent satisfies 0 < α < 1, − ∞ < ξ < + ∞.
Figure 1 presents the symmetric αstable probability density functions, and Figs. 2 and 3 are the SαS distributed signal in the time domain. There are some especial distributions as follows: Gaussian (α = 2, β = 0, γ = 1, ξ = 0), Cauchy (α = 1, β = 0, γ = 1, ξ = 0), Levy (α = 0.5, β = − 1, γ = 1, ξ = 0). The Matlab code was used to generate a complex isotropic symmetric alphastable distribution reference as ROCMUSIC [13].
Proposed solution
In this section, we proposed a new operator correntropybased covariance matrix (CBCM), and it applied with MUSIC to estimating DOA in the presence of an impulsive noise environment. We present the definition of correntropy in Section 3.1 and correntropyinduced metric (CIM) in Section 3.2. Correntropybased covariance matrix was proposed in Section 3.3, and then the implementation of CBCMMUSIC algorithm is given.
Correntropy
For two arbitrary random variables X and Y, the cross correntropy defined by [17]:
where κ_{σ}(•) is the kernel function that satisfies Mercer’s theory, σ is the kernel size, and E[•] represents the mathematical expectation. In practice, the joint probability density function (pdf) is unknown, and only a finite number of data {[x_{i}, y_{i}]^{N}_{i = 1}} can obtain to estimate the correntropy for random variables X and Y,
In general, we use the Gaussian kernel k_{σ}(•), using Taylor series expansion for Eq. (14), and the correntropy can be rewritten as [17]
Equation (16) indicates that correntropy involves all the evenorder moments of the (XY); note that if n = 1, we can get
Equation (17) reveals that correntropy includes the conventional relation.
Correntropyinduced metric
Correntropy can also induce a metric (CIM) [18]. There are two vector a = (a_{1}, a_{2},···,a_{N})^{T} and b = (b_{1},b_{2},···,b_{N})^{T}, and the function CIM defines as
Apparently, when the Gaussian kernel is used,
The properties of CIM can be listed as follows:

(1)
Nonnegativity:
CIM(a,b) ≥ 0, CIM(a,b) = 0 if and only if a = b;

(2)
Symmetric:
CIM(a,b) = CIM(b,a);

(3)
Triangle inequality:
CIM(a,c) ≤ CIM(a,b) + CIM(b,c).
Figure 4 shows the contours of distance from X to the origin in a 2D space, and Fig. 5 displays the surface. Compared with the conventional metric, CIM presents “mix norm” property. From Fig. 4, we can see that three zones have been divided. This metric divides space into three regions named the Euclidean region, transition region, and rectification region [18]. In the Euclidean region, CIM behaves as l_{2}norm, in the transition region, CIM behaves as l_{1}norm, and in rectification region, CIM behaves like l_{0}norm. The kernel size σ controls the bandwidth of the CIM “mix norm.”
Correntropybased covariance matrix
Theorem 1 If X and Y are jointly SαS and have a symmetric distribution, the correntropybased covariance matrix by using the Gaussian kernel of X and Y can be defined as
where μ is the suppression parameter and σ is the kernel size.
The proof of the boundedness of CBCM reference as Appendix B [16], here, inspired by the phase fractional lowerorder moment; the suppression parameter μ is introduced to exert different suppressed effects on random variables X and Y. Equation (21) can be expressed as
The implementation of CBCMMUSIC
Summarizing the existing algorithms, knowing that the key to implementing DOA is to modify the conventional covariance matrix to suit for impulsive noise environment, then the DOA estimation can be implemented in combination with the subspace technology. Inspired by correntropy and Gaussian kernel function to suppress impulse noise, and the prior parameters of noise do not need to know, this paper proposed a modified covariance matrix (CBCM) based on correntropy with Gaussian kernel function.
CBCM cannot only preserve the similarity measure in the sense for two random variables, but also it can suppress the “outliers” by using the rapid reduction of the exponential function; thus, CBCM achieves the purpose of adapting to the environment.
The main steps of the CBCMMUSIC are summarized as follows:

Step 1. Compute the M × M matrix R^{^}, whose (i,j)th entry is
Section 4.1 gives the discussion for the selection of the suppression parameter μ and kernel size σ.

Step 2. Perform eigenvalue decomposition (EVD) on the covariance matrix R^{^} to obtain the noise subspace matrix U^{ˆ}_{n}

Step 3. Compute the corresponding CBCMMUSIC spatial spectrum via

Step 4. Choose P local peaks of P_{CMCBMUSIC} as the estimates of DOAs.
Simulation and results
To assess the performance of the CBCMMUSIC, two performance criteria are used to evaluate the proposed algorithms: the probability of resolution and RMSE (rootmeansquareerror). The two sources are recognized to be successfully resolved if and only if [6]
where ƒ(·) stands for the spectral value. Rootmeansquareerror (RMSE) can be expressed as
where θ_{p} is the actual angle of the pth signal, and θ^{ˆ}_{i,p} is the estimated angle of θ_{p} in the ith MonteCarlo trial, where i = 1, 2, · · · , 200. All the numerical results were obtained with 200 independent trials.
In the following experiments, this paper considers two sources (Gaussian, QPSK, BPSK, QAM, PAM) that have the same variance imping on a uniform linear array (ULA), an M = 8 elements ULA with an interelement spacing equal to half a wavelength, supposing that the source number is known. A generalized signaltonoise (GSNR) ratio was used to describe signaltonoise ratio [13], that is
where N indicates the snapshots, γ indicates the dispersion parameter, and σ^{2}_{s} indicates the signal power.
Section 4.1 gives the recommendations for the selection of parameters; Section 4.2 compares the DOA spatial spectrum for ROCMUSIC, FLOMMUSIC, PFLOMMUSIC, CRCOMUSIC, and CBCMMUSIC; finally, the performance of analysis for CBCMMUSIC presents in Section 4.3.
Parameter selection
In this section, we have discussed the selection of the parameters of the CBCMMUSIC algorithm, including the kernel size σ and suppression parameter μ. According to the correntropy, the kernel bandwidth σ is full and controls the scale of the CIM norm, that is, a small kernel size will lead to a tight linear region (L_{2} norm) and to a large L_{0} region. The selection of the kernel size is described as conventional signals that fall into L_{2} norms and impulse signals that fall into L_{1} and L_{2} norms.
In order to make kernel size σ and suppression parameter μ widely applicable, different types of communication signals were used to DOA’s sources embedded in complex isotropic SαS noise, such as quaternary amplitude modulation (QAM), binary phaseshift keying (BPSK), quaternary phaseshift keying (QPSK), and Gaussian source, note that BPSK is noncircular signals.
Suppression parameter μ
Suppression parameter μ is limited in the range of [0, 1], and the covariance matrix degenerates into a traditional covariance matrix when μ = 1. Snapshots N = 256, a moderate impulsive noise condition with α = 1.4, GSNR = 8 dB, ULA with 8 sensors, we select the kernel size σ = 8 tentatively.
Figure 6 illustrates the influence of the suppression parameter, we can see that μ∈[0.5, 0.9] would be the optimal fields for CBCMMUSIC, and the suppression parameter μ is relatively small over a wide range of μ∈[0.4, 0.9] to MonteCarlo runs in terms of RMSE of CBCMMUSIC. According to Figs. 6 and 7, the desired results are achieved at μ = 0.7; hence, in the following experiment, we perform MonteCarlo runs with μ = 0.7.
Kernel size σ
Snapshots N = 256, μ = 0.7, and GSNR = 8 dB. α = 1.4, ULA with 8 sensors. From Figs. 8 and 9, it can be seen that the CBCMMUSIC algorithm obtains better DOA estimation performance in the range of σ ∈[6, 20], but the optimal value of the kernel size cannot be determined.
Since the selection of σ is closely related to GSNR [16], the rectification region is affected by the kernel size. Below, the RMSE and probability of resolution are analyzed at different GSNR σ ∈[2, 20], μ = 0.7, and α = 1.4. From Figs. 10 and 11, we can observe that kernel size σ = 10 would be the optimal value for CBCMMUSIC to reach its best performance. In particular, the probability of resolution in low SNR and high impulsive environments is shown in Fig. 10. Based on the above description and discussion, the simulations in Section 4.2 and Section 4.3 are implemented by setting μ = 0.7 and σ = 10.
Spatial spectrum estimation
In order to directly show the performance of the proposed CBCMMUSIC algorithm, we also compare spatial spectrum with that of the ROCMUSIC (p = 1.1) [13], FLOMMUSIC (p = 1.1) [14], PFLOMMUSIC (p = 0.2) [15], and CRCOMUSIC (μ = 0.5, σ = 1.4sqrt(σ_{s}^{^2})) [16], where σ_{s}^{^2} is the estimated variance of the noisefree signal S and p is the order of the moment. Consider a uniform linear array of 8 sensors with an interspacing of half a wave is used, two uncorrelated Gaussian sources (θ_{1} = − 5°, θ_{2} = 5°) impinge on this array, GSNR = 8 dB, snapshots N = 256, and α = 1.4.
The results of 10 runs of the normalized spatial spectrum are displayed in Fig. 12. Comparing the results of the above five algorithms, we can observe that CBCMMUSIC algorithm shows better focus ability for true DOA under impulsive noise environments, the performance of MUSIC is degraded seriously and 10 runs fail to distinguish two incident signals successfully. Seeing the performance degradation of the MUSIC, ROCMUSIC, FLOMMUSIC, and PFLOMMUSIC algorithms in the impulsive noise environment by no means, further analyses are made in the following experiments.
Performance analysis
In this section, the performance of the proposed CBCMMUSIC algorithm is compared with CRCOMUSIC (μ = 0.5, σ = 1.4sqrt(σ_{s}^{^2})) [16] and SCMMUSIC [22]. In terms of resolution probability and RMSE, the performance of the number of snapshots, GSNR, characteristic exponent α, and angular separation is investigated in this experiment.
Effect of the number of snapshots
In the first experiment, we study the effect of the number of snapshots and the results are exhibited in Figs. 13 and 14. An M = 8 element ULA with an interspacing of half a wave is used, two independent QAM sources are located at θ_{1} = − 5° and θ_{2} = 5°, the complex isotropic SαS is α = 1.4, and the GSNR is set as a constant at 8 dB. From Fig. 13, we can observe that CBCM–MUSIC gains a more evident decrease in RMSE than the other algorithms as the number of the snapshots increases. Figure 14 displays the probability of resolution against snapshots.
Effect of the GSNR
Figures 15 and 16 illustrate the performance of CBCMMUSIC, CRCOMUSIC [16], and SCMMUSIC [22] under a wide range of GSNRs from 0 to 10 dB, and the number of snapshots available to the algorithms is N = 256. A moderate impulsive noise α = 1.4 embedded in two QAM sources (θ_{1} = − 5°, θ_{2} = 5°). Figures 15 and 16 depict the improved performance of CBCMMUSIC over that conventional algorithm both in terms of resolution probability and RMSE.
Effect of the characteristic exponent α
In this experiment, we study the robustness of the CBCMMUSIC algorithm under a wide range of characteristic exponent α from 0.6 to 2. Consider two QAM sources (θ_{1} = − 5°, θ_{2} = 5°) impinge on the ULA with 8 sensors, the GSNR = 8 dB and the number of snapshots N = 256. Fig. 17 and Fig. 18 give the simulation results. We note that CBCMMUSIC showed better performance than SCMMUSIC and CRCOMUSIC as the characteristic exponent α is decreased.
Effect of the angular separation
In the final experiment, we study the variation of the algorithmic performance with respect to the angular separation of the two incoming QAM signals, for N = 256, GSNR = 10 dB, and α = 1.4. As expected, by contrast with the performance of the SCMMUSIC and CRCOMUSIC, the resolution capability of CBCMMUSIC algorithms is improved with increasing angleseparated value between the two sources based on Figs. 19 and 20.
Discussion
In this paper, we proposed a novel method that formulated the covariation matrix of the sensor outputs under the impulsive noises, which is based on correntropy. The improved performance of the proposed CBCMMUSIC algorithm in the presence of a wide range of impulsive noise environments was demonstrated via MonteCarlo experiments.
This paper assumed sources are independent that illuminated the array sensor. In many practical conditions, the sources are coherent signals due to multipath; hence, future research includes the development of methods for the DOA of the coherent signals in the presence of impulsive noise. Secondly, we will address the problem of localizing multiple wideband sources in impulsive noise
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Abbreviations
 BPSK:

Binary phaseshift keying
 CBCM:

Correntropybased covariance matrix
 CIM:

Correntropyinduced metric
 CRCO:

Correntropybased correlation
 DOA:

Direction of arrival
 EVD:

Eigenvalue decomposition
 FLOM:

Fractional lowerorder moments
 FLOS:

Fractional lowerorder statistic
 GSNR:

Generalization signal to noise ratio
 MUSIC:

Multiple signal classification
 pdf:

Probability density function
 PFLOM:

PhaseFLOM
 QAM:

Quaternary amplitude modulation
 QPSK:

Quaternary phaseshift keying
 ROCMUSIC:

Robust covarianceMUSIC
 SαS :

Symmetric alpha stable
 ULA:

Uniform linear array
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Acknowledgements
This work has been supported by the National Natural Science Foundation of China (Grant No: 61701344), Tianjin Higher Education Creative Team Funds Program in China, and Tianjin Normal University Doctoral Foundation (52XB1603, 52XB1713). I sincerely thank the anonymous referees for their technical suggestions and their advice on decorum.
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JinChen gives the overall research direction and ideas. ShengGuan read the relevant literature and books and drafts the article and makes the corresponding experimental simulation. The author(s) read and approved the final manuscript.
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Jin Chen was born in Wuhu, China, in 1976. He received the M.S. degree from Tianjin Normal University and the Ph.D. degree from College of Precision Instrument and Optoelectronics Engineering, Tianjin University, in 2002 and 2013, respectively. Since 2005, he has been working at Tianjin Normal University. He is an associate professor of the Tianjin Key Laboratory of Wireless Mobile Communications and Power Transmission. His research interests include acoustic signal acquisition and processing, broadband sensor array signal processing, and artificial intelligence.
Sheng Guan was born in JiLin, China, in 1993. He received the B.S. degree in textile engineering from the Inner Mongolia University of Technology in 2017. He is currently working toward the M.S. degree in information and communication engineering at Tianjin Normal University. His research interests include beamforming optimization and nonstationary signal processing.
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Chen, J., Guan, S. Correntropybased DOA estimation algorithm under impulsive noise environments. J Wireless Com Network 2020, 154 (2020). https://doi.org/10.1186/s13638020017666
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Keywords
 Direction of arrival
 Correntropy
 Complex isotropic symmetric alphastable