Correntropy-based DOA estimation algorithm under impulsive noise environments

In this paper, the direction of arrival (DOA) estimation of signals in the presence of impulsive noise environment is studied. Complex isotropic symmetric alpha-stable (SαS) random variables are modeled as impulsive noise, then a novel second-order statistic method that correntropy-based 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 Monte-Carlo simulation results illustrate the improved performance of CBCM-MUSIC for DOA estimation under a wide range of impulsive noise conditions.

MUSIC (ROC-MUSIC) 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 ROC-MUSIC 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 ROC-MUSIC 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 lower-order 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 correntropy-based correlation (CRCO), and it can be applied with MUSIC algorithm; despite the CRCO-MUSIC 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 alpha-stable 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, correntropy-based covariance matrices (CBCM), which can be combined with subspace algorithm, such as MUSIC algorithm, that is CBCM-MUSIC algorithm. The CBCM-MUSIC 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. The paper is organized as follows: in Section 2, we define the problem of interest and briefly review some preliminaries on α-stable distributions. In Section 3, we provide the CBCM-MUSIC algorithm. Finally, some simulation examples are presented in Section 4, and the conclusion is given in Section 5.

Problem definition
Here, the array signal model has given in Section 2.1, and then the impulsive noise model (alpha-stable 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 Our objective is to estimate the DOA (θ 1 , θ 2 , …, θ p ) of the source from X(k), the second-order 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 Recent studies show that alpha-stable 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), alpha-stable distribution is determined by α, β, γ, and μ, and its characteristics are as follows.

Proposed solution
In this section, we proposed a new operator correntropy-based 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 correntropy-induced metric (CIM) in Section 3.2. Correntropy-based covariance matrix was proposed in Section 3.3, and then the implementation of CBCM-MUSIC 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 even-order moments of the (X-Y); note that if n = 1, we can get Equation (17) reveals that correntropy includes the conventional relation.

Correntropy-induced 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: (2) Symmetric: (3) Triangle inequality: 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."

Correntropy-based covariance matrix
Theorem 1 If X and Y are jointly SαS and have a symmetric distribution, the correntropy-based 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 lower-order 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 CBCM-MUSIC
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 CBCM-MUSIC are summarized as follows: Step 1. Compute the M × M matrix R^, whose (i,j)th entry is exp - 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 Uˆn Step 3. Compute the corresponding CBCM-MUSIC spatial spectrum via Step 4. Choose P local peaks of P CMCB-MUSIC as the estimates of DOAs.

Simulation and results
To assess the performance of the CBCM-MUSIC, two performance criteria are used to evaluate the proposed algorithms: the probability of resolution and RMSE (root-meansquare-error). The two sources are recognized to be successfully resolved if and only if where ƒ(·) stands for the spectral value. Root-mean-square-error (RMSE) can be expressed as where θ p is the actual angle of the pth signal, and θˆi ,p is the estimated angle of θ p in the ith Monte-Carlo 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 signal-to-noise (GSNR) ratio was used to describe signal-to-noise ratio [13], that is where N indicates the snapshots, γ indicates the dispersion parameter, and σ 2 s indicates the signal power.

Parameter selection
In this section, we have discussed the selection of the parameters of the CBCM-MUSIC 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 phase-shift 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 CBCM-MUSIC, and the suppression parameter μ is relatively small over a wide range of μ∈[0.4, 0.9] to Monte-Carlo runs in terms of RMSE of CBCM-MUSIC. According to Figs. 6 and 7, the desired results are achieved at μ = 0.7; hence, in the following experiment, we perform Monte-Carlo 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 CBCM-MUSIC 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 CBCM-MUSIC 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

Spatial spectrum estimation
In order to directly show the performance of the proposed CBCM-MUSIC algorithm, we also compare spatial spectrum with that of the ROC-MUSIC (p = 1.1) [13], FLOM-MUSIC (p = 1.1) [14], PFLOM-MUSIC (p = 0.2) [15], and CRCO-MUSIC (μ = 0.5, σ = 1.4sqrt(σ s^2 )) [16], where σ s^2 is the estimated variance of the noise-free 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 CBCM-MUSIC algorithm shows better focus ability for true DOA under impulsive noise environments, the performance of MUSIC is degraded seriously and 10 runs fail to

Performance analysis
In this section, the performance of the proposed CBCM-MUSIC algorithm is compared with CRCO-MUSIC (μ = 0.5, σ = 1.4sqrt(σ s^2 )) [16] and SCM-MUSIC [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 characteristic exponent α
In this experiment, we study the robustness of the CBCM-MUSIC 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 CBCM-MUSIC showed better performance than SCM-MUSIC and CRCO-MUSIC 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 SCM-MUSIC and CRCO-MUSIC, the resolution capability of CBCM-MUSIC algorithms is improved with increasing angle-separated value between the two sources based on Figs. 19 and 20. 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 CBCM-MUSIC algorithm in the presence of a wide range of impulsive noise environments was demonstrated via Monte-Carlo 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 wide-band sources in impulsive noise