# DOA Estimation with Nested Arrays in Impulsive Noise Scenario: An Adaptive Order Moment Strategy

Xudong Dong, Jun Zhao, Jingjing Pan, Meng Sun, Xiaofei Zhang, Peihao

Dong, Yide Wang

► To cite this version:

Xudong Dong, Jun Zhao, Jingjing Pan, Meng Sun, Xiaofei Zhang, et al.. DOA Estimation with Nested Arrays in Impulsive Noise Scenario: An Adaptive Order Moment Strategy. IEEE Open Journal of Signal Processing, 2024, 5, pp.493 - 502. 10.1109/OJSP.2024.3360896. hal-04434609

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# DOA Estimation with Nested Arrays in Impulsive Noise Scenario: An Adaptive Order Moment Strategy

Xudong Dong, Jun Zhao, Jingjing Pan, Member, IEEE, Meng Sun, Member, IEEE, Xiaofei Zhang, Peihao Dong, and Yide Wang, Senior Member, IEEE

**Abstract**—Most of the existing direction of arrival (DOA) estimation methods in impulsive noise scenario are based on the fractional low-order moment statistics (FLOSs), such as the robust covariation-based (ROC), fractional low-order moment (FLOM), and phased fractional low-order moment (PFLOM). However, an unknown order moment parameter *p* needs to be selected in these approaches, which inevitably increases the computational load if the optimal value of the parameter *p* is determined by a large number of Monte Carlo experiments. To address this issue, we propose the adaptive order moment function (AOMF) and improved AOMF (IAOMF), which are applicable to the existing FLOSs-based methods and can also be extended to the case of sparse arrays. Moreover, we analyze the performance of AOMF and IAOMF, and simulation experiments verify the effectiveness of proposed methods.

**Index Terms**—Impulsive noise, Direction of arrival (DOA) estimation, sparse arrays, adaptive order moment.

## I. INTRODUCTION

DIRECTION of arrival (DOA) estimation has been a research focus in array signal processing, and a series of super-resolution parameters estimation methods, such as MUSIC, [1], ESPRIT [2], and maximum likelihood estimation [3], have been widely applied to wireless communications, sonar, radar, and electronic reconnaissance. However, the above mentioned methods only consider the scenario of Gaussian noise. In practical communication systems, noise generally exhibits impulsive characteristics [4], [5], such as astronomical noise, sparking noise, electrical and industrial equipment operation noise, etc. In addition, the work in [5] shows that the $\alpha$-stable distribution is the most appropriate model for impulsive noise.

To combat the impulsive noise, the robust covariation-based MUSIC (ROC-MUSIC, [6]) method for DOA estimation utilizes the covariation property and assumes that the signal and additive noise obey a joint symmetric $\alpha$ stable ($S\alpha S$, [4]) distribution. However, the hypothesis that the signal and additive noise jointly obey $S\alpha S$ distribution is unrealistic because the signal usually has a finite variance, while the signal obeying $S\alpha S$ distribution has an infinite variance.

This work is supported by the open research fund of National Mobile Communications Research Laboratory, Southeast University, under Grant 2022D08. (Corresponding author: Meng Sun. e-mail: mengsun@nuaa.edu.cn)

X. Dong, J. Pan, M. Sun, X. Zhang and P. Dong are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210000, China, and also with the Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum Space.

P. Dong is also with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 211111.

Yide Wang is with the IETR, University of Nantes, 44306 Nantes, France
J. Zhao is with the College of Electronic and Information Engineering,
Tongji University, Shanghai 201800, China.

Although the fractional low-order moment MUSIC (FLOM-MUSIC, [7]) outperform the ROC-MUSIC method, these two methods on work with $1 < \alpha \le 2$. In [8], the sign covariance matrix MUSIC (SCM-MUSIC) is shown to obtain convergent estimates of the signal and noise subspaces, solving the problem that FLOM method are only applicable to $1 < \alpha < 2$. A new subspace algorithm based on the phased fractional low-order moment MUSIC (PFLOM-MUSIC) is presented in [9], which can also obtain good direction finding performance at $0 < \alpha < 1$. The infinity-norm-normalization MUSIC (IN-MUSIC, [10]) method is proposed to limit the influence of impulsive noise by pruning the amplitude of array received signal, and IN-MUSIC method yields a more accurate DOA estimation than FLOM-MUSIC, PFLOM-MUSIC, and SCM-MUSIC methods. However, its performance may degrade with the increase of signal subspace rank. In addition, the correlation entropy [11]–[14], sparse Bayesian learning (SBL, [15]–[17]), sparse representation [18], [19], $l_p$-MUSIC [20] and the bounded nonlinear covariance (BNC)-based method [21] are used for DOA estimation in impulsive noise. Nevertheless, these methods are only applicable to conventional uniform linear arrays (ULAs), and there are still gaps in sparse array scenarios.

Unlike the traditional ULA, the structure of sparse arrays is different in that antenna positions are not continuous and may have vacancies. This discontinuity will cause the traditional ULA-based algorithms (such as ROC, FLOM and PFLOM, etc.) cannot be directly applied to sparse arrays. Recently, the impulsive noise DOA estimation technique has also been extended to the sparse array scenario [22]–[24]. In these works, the ULA-based FLOM, PFLOM, and SCM methods are extended to the coprime array scenario [25]–[28] by virtualization technique, which generates virtual ULAs with longer apertures than the original ULAs, achieving better estimation performance. The simultaneous presence of impulsive noise and Gaussian noise is considered in [22], and simulation results show that the PFLOM method is valid. On this basis, [24] proposes an augmented phased fractional low-order moment (APFLOM) method with better performance than the work in [23] by utilizing the non-circular property of signal. Additionally, [29] extends the previous work in [21] to the nested array scenario [30]–[33] and proposes an enhanced bounded nonlinear covariance (EBNC) method, which presents better estimation performance than the BNC method [21]. However, the above mentioned methods rely on an unknown parameter that needs to be adjusted artificially (the choice of the order moment parameter is involved in the ROC, FLOM, PFLOM, and EBNC methods). The uncertainty of the
parameters leads to a limited resistance of their equivalent covariance matrix to impulsive interference.

In this paper, we propose two adaptive order moment functions allowing to determine the specific values of the unknown parameters. Our contributions are as follows:

* We apply the signal local entropy theory to the array signal processing scenario and propose the adaptive order moment function (AOMF), which can be applied to the existing FLOS-based methods. In this paper, we propose to use the AOMF value as the order moment parameter of the PFLOM method and discuss the dependence of PFLOM method on the AOMF value.

* Based on the AOMF, we propose its improved version, namely IAOMF. The AOMF value is further adjusted by introducing a scale parameter $\delta$, whose value does not require any prior knowledge and depends on the array noise environment. Theoretical analysis verified that the IAOMF value is more suitable as the order moment parameter of the PFLOM method.

* We extend the proposed method to the sparse array scenario (nested arrays as an example), and simulation experiments verify its effectiveness.

This paper is organized as follows: Section II provides the signal and noise model. The FLOS technologies for S$\alpha$S process are described in section III. Section IV presents the proposed adaptive low-order moment function and its improved version. The numerical result and conclusion are included in Section V and VI, respectively.

## II. SIGNAL AND NOISE MODEL

### A. Signal model

Considering the nested arrays of P sensors with positions

$$ \mathcal{L}_{\text{nested}} = \{(m-1)d_0 | m = 1, \dots, N_1\} \\ \cup \{N_1 d_0 + (n-1)(N_1+1)d_0 | n = 1, \dots, N_2\}, \quad (1) $$

where $N_1$ and $N_2$ are positive integers. Let $\mathcal{L}_{\text{nested}} = \{l_1, l_2, \dots, l_P, l_1 = 0\}$, each array sensor location $l_p, p = 1, \dots, P$ is selected as an integer multiple of the minimum distance $d_0 = \lambda/2$ between any two sensors, where $\lambda$ is the carrier frequency wavelength. Given that there are K far field narrowband independent sources with DOAs $\theta_1, \dots, \theta_K$ impacting into the array, the direction vector of the k-th source can be expressed as

$$ \mathbf{a}(\theta_k) = [1, e^{-j2\pi l_2 \sin \theta_k / \lambda}, \dots, e^{-j2\pi l_P \sin \theta_k / \lambda}]^T, \quad (2) $$

then the received signal vectors can be expressed as:

$$ \mathbf{x}(t) = \mathbf{A}\mathbf{s}(t) + \mathbf{n}(t), \quad (3) $$

where $\mathbf{A} = [\mathbf{a}(\theta_1), \dots, \mathbf{a}(\theta_k), \dots, \mathbf{a}(\theta_K)]$ is the directional matrix and $\mathbf{s}(t) = [s_1(t), \dots, s_k(t), \dots, s_K(t)]^T$ is the signal vector, $(\cdot)^T$ denotes the transpose and $\mathbf{n}(t) = [n_1(t), \dots, n_P(t)]^T$ is the impulsive noise term that follows symmetric $\alpha$ stable (S$\alpha$S) [4] distribution.

### B. Complex symmetric $\alpha$ stable distribution

The $\alpha$ stable distribution appears naturally in the study of heavy-tailed noise and has applications in economics

and physics as models of rare but extreme events (such as earthquakes or stock market collapses). One of the most discouraging problems of $\alpha$ stable distribution is that their parameterization is inconsistent. The parameterization we have selected here is consistent with the book [34]. The $\alpha$ stable distribution is a four-parameter family of distributions, usually denoted by $S(\alpha, \beta, \gamma, \delta)$. The first parameter $\alpha \in (0, 2]$ is called the characteristic exponent and describes the tail of the distribution. The second $\beta \in [-1, 1]$ is the skewness, as the name specifies, whether the distribution is right-skewed ($\beta < 0$) or left-skewed ($\beta > 0$). The last two parameters are the scale $\gamma > 0$, and the position $\delta \in \mathbb{R}$. We can think of these two as similar to the variance and mean in the normal distribution. In fact, if $X \sim S(\alpha, \beta, 1, 0)$, then if $\alpha \neq 1$, $Z = \gamma X + \delta \sim S(\alpha, \beta, \gamma, \delta)$, where the variable X is generally referred to as a standard $\alpha$-stable random variable.

The $\alpha$ stable distribution family is an extensive class that includes the following distributions as subclasses:

**A1:** The Gaussian distribution $N(\mu, \sigma^2)$ can be expressed as $S(2, \beta, \frac{\sigma}{\sqrt{2}}, \mu)$. Notice that $\beta$ does not matter in this example.

**A2:** The Cauchy distribution of scale $\gamma$ and position $\delta$ can be expressed as $S(1, 0, \gamma, \delta)$.

**A3:** The Levy distribution (also known as the inverse-Gaussian or Pearson V), with scale $\gamma$ and location $\delta$ are given by $S(1/2, 1, \gamma, \delta)$.

A complex random variable $X = X_1 + jX_2$ obeys S$\alpha$S distribution if $X_1$ and $X_2$ are jointly S$\alpha$S, and it is isotropic if $(X_1, X_2)$ has a uniform spectral measure. The characteristic function of an isotropic complex S$\alpha$S distribution can be expressed as

$$ \phi(u) = \exp(-\gamma|u|^{\alpha}), \quad (4) $$

where $u = u_x + j u_y$, $0 < \alpha \le 2$ is the characteristic exponent and $\gamma$ denotes the dispersion parameter. When $\alpha = 2$, it becomes a Gaussian distribution. And the variance of a Gaussian process and the dispersion parameter of S$\alpha$S process have the relationship $\sigma^2 = 2\gamma$, i.e., $CS(\cdot|2, \gamma) \sim CN(\cdot|0, 2\gamma)$.

The 1000 random samples are generated from the symmetric $\alpha$-stable distribution with $\alpha = 0.5$ or $\alpha = 1.5$, and the histograms are produced in Fig. 1. Fig. 2 gives the amplitude values corresponding to the 1000 samples, and it can be seen that the noise emerges at irregular intervals with impulsive characteristics, and the smaller the value of $\alpha$, the more obvious the impulsive characteristic is.

## III. FLOS TECHNOLOGIES FOR S$\alpha$S PROCESS

When the noise vector $\mathbf{n}(t)$ in (3) is an impulsive noise, the second-order or higher-order statistics do not exist. In recent years, a large amount of researches have been dedicated to propose replaceable equivlent covariance matrices that make traditional subspace techniques (e.g. MUSIC) still applicable, such as robust covariation-based (ROC, [6]), fractional low-order moment (FLOM, [7]), phased fractional low-order moment (PFLOM, [9]), sign covariance matrix (SCM, [8]) and infinity-norm-normalization (IN, [10]). In this section, we briefly describe these methods and their respective advantages and disadvantages.
<figure><img src="image_2.png" /><figcaption>Fig. 1: Histogram of 1000 samples, $\beta = 0, \gamma = 1, \delta = 0$.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: Amplitude of 1000 samples, $\beta = 0, \gamma = 1, \delta = 0$.</figcaption></figure>

### A. Some replaceable covariance matrices

1) ROC-based matrix: From [6], the (i, j)th element of ROC matrix $\mathbf{R}_{\text{ROC}}$ can be denoted as

$$ \mathbf{R}_{\text{ROC}}(i, j) = \frac{\sum_{t=1}^{T} x_i(t) |x_j(t)|^{p-2} x_j^*(t)}{\sum_{t=1}^{T} |x_j(t)|^p}, \quad (5) $$

where $i, j = 1, \dots, P$, $x_i(t)$ is the i-th row of $\mathbf{x}(t)$ and $1/2 < p < \alpha/2$ is the order moment parameter, T denotes the number of snapshots and the ROC matrix $\mathbf{R}_{\text{ROC}}$ can be rewritten as

$$ \mathbf{R}_{\text{ROC}} = \mathbf{A} \mathbf{\Lambda}_{\text{ROC}} \mathbf{A}^H + \gamma_{\text{roc}} \mathbf{I}_p, \quad (6) $$

where $\mathbf{\Lambda}_{\text{ROC}}$ is a robust covariance-based matrix of the signals. For uncorrelated signals, $\mathbf{\Lambda}_{\text{ROC}}$ is a diagonal matrix. $\gamma_{\text{roc}}$ is a scalar, and its specific expression can be found in [6].

2) FLOM-based matrix: According to [7], with conditions A1-A2, the (i, j)th element of FLOM matrix $\mathbf{R}_{\text{FLOM}}$ can be expressed as

$$ \mathbf{R}_{\text{FLOM}}(i, j) = \mathbb{E} \left\{ x_i(t) |x_j(t)|^{p-2} x_j^*(t) \right\}, \quad (7) $$

where $i, j = 1, \dots, P$ and $1 < p < \alpha \le 2$ is the order moment parameter, and the FLOM matrix $\mathbf{R}_{\text{FLOM}}$ can be rewritten as

$$ \mathbf{R}_{\text{FLOM}} = \mathbf{A} \mathbf{\Lambda}_{\text{F}} \mathbf{A}^H + \gamma_f \mathbf{I}_p, \quad (8) $$

where $\mathbf{\Lambda}_{\text{F}}$ is a fractional lower-order correlation matrix of the signals. For uncorrelated signals, $\mathbf{\Lambda}_{\text{F}}$ is a diagonal matrix. $\gamma_f$ is a scalar, and its specific expression can be found in [7].

3) PFLOM-based matrix: From [9], the PFLOM matrix $\mathbf{R}_{\text{PFLOM}}$, is applicable for $0 < \alpha \le 2$, and its (i, j) element can be expressed as:

$$ \mathbf{R}_{\text{PFLOM}}(i, j) = \mathbb{E} \left\{ (x_i(t))^{\langle b \rangle} \cdot (x_j(t))^{\langle -b \rangle} \right\}, \quad (9) $$

$$ c^{\langle b \rangle} = \begin{cases} |c|^{b-1} c, & c \neq 0 \\ 0, & c = 0 \end{cases}, c = x_i(t) \text{ or } x_j(t), \quad (10) $$

where $c^{\langle -b \rangle} = (c^*)^{\langle b \rangle} = (c^{\langle b \rangle})^*$ and $0 < b < \frac{\alpha}{2}$ is the order moment. Substituting (10) into (9), we can obtain

$$ \mathbf{R}_{\text{PFLOM}}(i, j) = \mathbb{E} \left\{ |x_i(t)|^{b-1} |x_j(t)|^{b-1} x_i(t) x_j^*(t) \right\}. \quad (11) $$

The covariance matrix in (11) also can be rewritten as

$$ \mathbf{R}_{\text{PFLOM}} = \mathbf{A} \mathbf{\Lambda}_{\text{PF}} \mathbf{A}^H + \gamma_{\text{pf}} \mathbf{I}_p \quad (12) $$

where $\mathbf{\Lambda}_{\text{PF}}$ is the phased fractional lower-order correlation matrix of signals. For uncorrelated signals, $\mathbf{\Lambda}_{\text{PF}}$ is a diagonal matrix. $\gamma_{\text{pf}}$ is a scalar, and its specific expression can be found in [9].

4) Sign covariance matrix (SCM): The sample SCM or normalized covariance matrix exhibits [8] good robust estimation performance in a heavy trailed noise environment. It is a normalization operation for each array received signal vector as follows

$$ f(\mathbf{x}) = \begin{cases} \frac{\mathbf{x}}{\|\mathbf{x}\|}, & \mathbf{x} \neq \mathbf{0} \\ \mathbf{0}, & \mathbf{x} = \mathbf{0} \end{cases}, \quad (13) $$

where $\|\mathbf{x}\| = \sqrt{\mathbf{x}^H \mathbf{x}}$. The SCM can be estimated by

$$ \hat{\mathbf{R}}_{\text{SCM}} = \frac{1}{T} \sum_{t=1}^{T} f(\mathbf{x}(t)) f^{H}(\mathbf{x}(t)). \quad (14) $$

5) Infinity-norm-normalization (IN) preprocessing: The IN vector of array received data [10] can be obtained by

$$ \mathbf{y}(t) = \omega(t) \mathbf{x}(t), \quad (15) $$

$$ \omega(t) = \frac{1}{\|\mathbf{x}(t)\|_{\infty}} = \frac{1}{\max\{x_i(t), \dots, x_P(t)\}}, \quad (16) $$

and the IN covariance matrix can be estimated by

$$ \hat{\mathbf{R}}_{\text{IN}} = \frac{1}{T} \sum_{t=1}^{T} \mathbf{y}(t) \mathbf{y}^{H}(t). \quad (17) $$

Remark 1: In fact, Eqs. (14) and (17) can also be simplified into an expression similar to Eqs. (6), (8) and (12), which will not be specified in this paper. If you want to obtain further results, you can refer to the works in [8] and [10].

### B. FLOS-based methods with nested array

In the following descriptions, the subscript ROC in $\mathbf{R}_{\text{ROC}}$. $\mathbf{\Lambda}_{\text{ROC}}$, the subscript FLOM in $\mathbf{R}_{\text{FLOM}}$, $\mathbf{\Lambda}_{\text{FLOM}}$, the subscript PFLOM in $\mathbf{R}_{\text{PFLOM}}$, $\mathbf{\Lambda}_{\text{PFLOM}}$ and the index $\text{roc}/f/pf$ in $\gamma_f/\gamma_{pf}/\gamma_{roc}$ are dropped for notational convenience.
By vectorizing the equivalent covariance matrix **R**, then
$$
\mathbf{z} = \text{vec}(\mathbf{R}) = (\mathbf{A}^* \odot \mathbf{A})\mathbf{p} + \gamma\text{vec}(\mathbf{I}_P) = \tilde{\mathbf{A}}\mathbf{p} + \gamma\mathbf{i}, \quad (18)
$$

where **p** is a *K* × 1 vector, whose elements are the diagonal elements of matrix **Λ**. Through the nested array technique [30], **z** can be considered as a single snapshot vector from a virtual difference co-array, which contains many repeated informations. By introducing a selection matrix to remove these redundant terms, we can obtain

$$
\tilde{\mathbf{z}} = \mathbf{F}\mathbf{z} = \tilde{\mathbf{A}}_{\mathbb{D}}\mathbf{p} + \gamma\tilde{\mathbf{i}}, \mathbf{F} \in \mathbb{R}^{(2G+1) \times P^2}, \quad (19)
$$

with

$$
\mathbf{F}_{m+G+1,i+(j-1)P} = \begin{cases} \frac{1}{\omega(m)}, & m = l_i - l_j, (l_i, l_j) \in \mathbb{D}(m) \\ 0 & \end{cases} \\
m = -G, \dots, 0, 1, G; G = N_1 N_2 + N_2 - 1; i, j = 1, \dots, P \tag{20}
$$

where $\tilde{\mathbf{A}}_{\mathbb{D}} = [\tilde{\mathbf{a}}_{\mathbb{D}}(\theta_1), \dots, \tilde{\mathbf{a}}_{\mathbb{D}}(\theta_K)]$ is the difference coarray directional matrix, which corresponds to the directional matrix of a consecutive virtual ULA with $2G+1$ antenna elements located between $-Gd_0$ and $Gd_0$ (only for nested array, whose difference coarray has no holes). $\tilde{\mathbf{i}}$ represents a vector with all elements equal to zero except that the $G+1$-th element equals to 1, $\omega(m)$ stands for the weight function and can be defined as

$$
\omega(m) = |\mathbb{D}(m)|, \quad (21)
$$

$$
\mathbb{D}(m) = \{ (l_i, l_j) | l_i - l_j = m, l_i, l_j \in \mathcal{L}_{\text{nested}}/d_0 \}, \quad (22)
$$

$$
\mathbb{D} = \{ l_i - l_j \mid l_i, l_j \in \mathcal{L}_{\text{nested}}/d_0 \}, \quad (23)
$$

where $|X|$ denotes the element number of the set $X$ and $G$ indicates the maximum value of the difference operation $|l_i - l_j|$.

For the directional matrix $\tilde{\mathbf{A}}_{\mathbb{D}}$, the DOA can be estimated using either the subspace-based or the compressive sensing-based approaches [26]. However, the signal vector **p** received from the nested virtual arrays only has one snapshot. In a single snapshot scenario, the rank of data covariance matrix is always equal to 1, which leads to performance degradation in DOA estimation. Therefore, common decoherence methods (e.g., spatial smoothing-based methods, Toeplitz reconstruction methods, etc.) should be applied for restoring the rank of the “data covariance matrix” with the cost of sacrificing some array apertures.

**Remark 2:** The FLOM and ROC based methods are only suitable for $\alpha > 1$, while the PFLOM, SCM and IN methods are able to handle the DOA estimation problem for $0 < \alpha \le 2$. The literature [9] shows that the PFLOM method exhibits better estimation performance than the SCM method for $0 < \alpha < 1$. The value of the order moment parameter $b$ determines the performance of the PFLOM method, but it needs to be set manually, and its uncertainty brings a performance degradation to the PFLOM method. In contrast, the SCM and IN based methods do not use the order moment parameter and are simple to calculate, but the performance needs to be further improved.

IV. THE PROPOSED ADAPTIVE LOW-ORDER MOMENT FUNCTION AND ITS IMPROVED VERSION

Unfortunately, the order moment parameters *p* or *b* for methods such as ROC, FLOM and PFLOM are uncertain. One way is to determine the order moment parameter value by multiple Monte Carlo experiments [9]. However, when performing various experiments (e.g., signal-to-noise ratio (SNR), snapshots, characteristic exponent $\alpha$, and other experimental parameters), it is necessary to find the optimal order moment parameter value for each experiment, which unavoidably increases the computational complexity. Therefore, we propose an adaptive order moment function, which can be determined based on the change of entropy values associated with the array received signal. Therefore, it is not necessary to manually adjust the size of the order moment parameter.

A. Adaptive entropy function

**Definition 1:** Local entropy for array signal processing

Based on information theory, for an event set $X = \{x_1, x_2, ..., x_n\}$, in which event $x_i, i = 1, ..., n$ occurs with probability $p_i$, the amount of self-information is defined as

$$
I(x_i) = - \log_{10} p_i. \tag{24}
$$

Then its average self-information (i.e., entropy) can be expressed as

$$
H(X) = \mathbb{E}(I(x_i)) = - \sum_{i=1}^{n} p_i \log_{10} p_i. \quad (25)
$$

For the array output (3), the local entropy is defined as

$$
H = - \sum_{i=1}^{P} \sum_{t=1}^{T} h_{it} \log_{10} (h_{it}), \quad (26)
$$

with

$$
h_{it} = \frac{|x_i(t)|}{\sum_{i=1}^{P} \sum_{t=1}^{T} |x_i(t)|} = \frac{|x_i(t)|}{\mathbb{L}_1(\mathbf{x}(t))}, \quad (27)
$$

where $x_i(t)$ is the $i$-th row element $\mathbf{x}(t)$, $\mathbb{L}_1(\cdot)$ denotes the $l_1$ norm.

**Definition 2:** Adaptive entropy function

Actually, Eq. (26)-(27) depend only on the array output data and do not require any priori knowledges. Using the non-negativity of entropy (i.e., $H \ge 0$), we define the adaptive entropy function (AEF) as follows

$$
p(H) = \frac{1}{1 + e^{-H}}. \qquad (28)
$$

The AEF (28) has the following important properties:

**Property 1:** The derivative of the AEF can be expressed as

$$
\frac{\partial p}{\partial H} = \frac{e^{-H}}{(1 + e^{-H})^2} > 0. \quad (29)
$$

Therefore, the AEF is monotonic.

**Property 2:** Since the AEF satisfies

$$
\frac{1}{2} \le \frac{1}{1 + e^{-H}} < 1. \qquad (30)
$$
Then the AEF is positive and bounded, which obtains the minimum value if and only if $H = 0$.

**Property 3:** From (26) and (29), it is known that the AEF is a differentiable real-valued function and has a non-negative derivative at each point.

## B. The proposed adaptive order moment function (AOMF)

In fact, it is clear from Property 2 that the AEF $p(H)$ is no longer applicable to ROC-MUSIC and FLOM-MUSIC methods because they require $1 < p \le \alpha$ and cannot solve the DOA estimation problem in highly impulsive scenarios ($\alpha < 1$). Inspired by the PFLOM order moment parameter $0 < b < \frac{\alpha}{2}$, we propose an adaptive order moment function (AOMF), defined as

$$p_{\text{AOMF}}(H, \alpha) = \frac{\alpha/2}{1 + e^{-H}}. \qquad (31)$$

In comparison with (28), the $p_{\text{AOMF}}$ is a two-dimensional function with properties 1-3, and the property 1 can be extended as follows:

**Property 1:** The AOMF (31) satisfies

$$\frac{\partial p_{\text{AOMF}}}{\partial H} = \frac{\alpha e^{-H}}{2(1 + e^{-H})^2} > 0, \qquad (32)$$

$$\frac{\partial p_{\text{AOMF}}}{\partial \alpha} = \frac{1}{2(1 + e^{-H})} > 0. \qquad (33)$$

Thus the AOMF (31) is monotonic with $H$ or $\alpha$, with Property 2, we have $\frac{\alpha}{4} \le p < \frac{\alpha}{2}$. Obviously, the $p_{\text{AOMF}}$ can be applied for the PFLOM method. Note that $p_{\text{AOMF}}$ satisfies $\frac{\alpha}{2} \le p < \alpha$ when the numerator of (31) is $\alpha$. In this case, AOMF is no longer applicable to the PFLOM method. In this paper, we mainly address the problem of direction finding in highly impulsive scenario, so we focus on how to choose the parameter $p$ for better performance.

According to some recent researches ([9], [22]–[24]), the order moment parameter $b$ in the PFLOM method generally yields a better algorithm performance when its value is small (e.g., $b = 0.1$). Therefore, the algorithm performance corresponding to the AOMF degrades significantly when $\alpha > 1$. Figs. 3-5 give the comparison results between the PFLOM method with AOMF and the existing SCM method in different $\alpha$ scenarios with conditions of $N_1 = 4, N_2 = 5$, the snapshots $T = 500$, and DOAs are $[10^\circ, 20^\circ, 30^\circ]$.

<figure><img src="image_4.png" /><figcaption>Fig. 3: RMSE with different GSNR, $\alpha = 0.5$, MC=500.</figcaption></figure>

Figs. 3-4 show the root mean square error (RMSE, defined in section IV) performance of the proposed method with the NA-SCM method (SCM with nested array) in impulsive noise

<figure><img src="image_5.png" /><figcaption>Fig. 4: RMSE with different GSNR, $\alpha = 1.5$, MC=500.</figcaption></figure>

environment when $\alpha = 0.5$ (or $\alpha = 1.5$). From Figs. 3-4, it can be seen that the AEF $p(H)$ corresponding to equation (28) is no longer applicable to the highly impulsive noise scenario, and it can only work with $\alpha > 1$. In comparison, the PFLOM method with AOMF outperforms the NA-SCM method at $\alpha < 1$, while the performance at $\alpha > 1$ is close to that of the NA-SCM method only in the case of highly generalized signal-to-noise ratios (GSNR).

<figure><img src="image_6.png" /><figcaption>Fig. 5: RMSE versus different characteristic exponent $\alpha$, GSNR = 10 dB, MC=500.</figcaption></figure>

Fig. 5 depicts the algorithm performance versus the characteristic exponent $\alpha$ for different GSNR. It can be seen that the PFLOM algorithm with AOMF outperforms the NA-SCM algorithm when GSNR = 10 dB. According to the property 2 of the AOMF $p_{\text{AOMF}}$, which increases with increasing $\alpha$, and it is known from literature [9] that the performance of the PFLOM algorithm decreases when the order moment parameter is large, i.e., the performance of the AOMF-PFLOM algorithm degrades when $\alpha$ is large (see Fig. 4).

## C. The proposed improved AOMF

To solve the performance degradation problem of the above AOMF-PFLOM algorithm when $\alpha > 1$, we introduce the scaling parameter $\delta$ in the AOMF $p_{\text{AOMF}}$ and further propose the improved AOMF (IAOMF), defined as follows:

$$p_{\text{IAOMF}}(\alpha, H, \delta) = \frac{\alpha/2}{1 + \delta \cdot e^{-H}}, \delta \ge 1. \qquad (34)$$

Similar to **Definition 2**, it has the following properties:

**Property 1:**

$$\frac{\partial p_{\text{IAOMF}}}{\partial H} = \frac{\alpha \delta e^{-H}}{2(1 + \delta e^{-H})^2} > 0, \qquad (35)$$

$$\frac{\partial p_{\text{IAOMF}}}{\partial \alpha} = \frac{1}{2(1 + \delta e^{-H})} > 0, \qquad (36)$$

$$\frac{\partial p_{\text{IAOMF}}}{\partial \delta} = \frac{-\alpha e^{-H}}{2(1 + \delta e^{-H})^2} < 0. \qquad (37)$$
**Property 2:** It takes values in the range $\frac{\alpha}{2(1+\delta)} \le p_{IAOMF} < \frac{\alpha}{2}$, where $\delta$ is the scale parameter, which serves to make $p_{IAOMF}$ infinitely close to 0.

Therefore, we can conclude that

* When $\delta = 1$, the $p_{IAOMF}$ degenerates to the $p_{AOMF}$, and still retains properties 1-3.

* When $\delta = +\infty, 0 < \frac{\alpha}{2(1+\delta)} \le p < \frac{\alpha}{2}$. Therefore, the order moment parameter $b$ in the PFLOM method can be seen as a special case of the $p_{IAOMF}$ version.

**Remark 3:** According to (35)-(37), $p_{IAOMF}(\alpha, H, \delta)$ is strictly monotonically decreasing with respect to $\delta$. Theoretically, $\delta$ should be fixed to a constant such that $p_{IAOMF}$ takes a small value. The optimal value of the PFLOM order moment parameter $b$ in [9] can only be determined by many Monte Carlo experiments, but $b$ still cannot be specified to a certain value because of the influence of the characteristic exponent $\alpha$ and the GSNR. We consider that $\delta$ should be correlated with the array data $\mathbf{x}(t)$ and defined as

$$ \delta = \max_t \{\text{mean} |\mathbf{x}(t)|\}, t = 1, \dots, T, \quad (38) $$

where $\text{mean}\{\cdot\}$ is the mean value operation.

## D. The main steps of our proposed AOMF and IAOMF based methods

**Step 1:** Calculate the AOMF $p_{AOMF}$ and IAOMF $p_{IAOMF}$ according to (31) and (34), respectively.

**Step 2:** Construct the PFLOM matrix (9) with $b = p_{AOMF}$ or $b = p_{IAOMF}$.

**Step 3:** Perform nested array technology (in section III-B) of the PFLOM matrix (11) to obtain the DOA estimation, including vectorization operation, spatial smoothing technology (or Toeplitz reconstruction and compressive sensing methods).

## V. NUMERICAL RESULT

In the following simulations, several Monte Carlo simulations illustrate the estimated performance of our proposed method. The proposed method is compared with several existing methods as follows:

* **ULA-ROC:** This method is proposed in [6], where the array configuration is a ULA, and the ROC matrix is employed to replace the original received signal covariance matrix.

* **ULA-FLOM:** This method is presented in [7], where the array configuration is a ULA, and the FLOM equivalent covariance matrix is employed to replace the original received signal covariance matrix.

* **ULA-PFLOM:** This method can be found in [9], where the array configuration is a ULA, and the PFLOM equivalent covariance matrix is applied to replace the original received signal covariance matrix.

* **NA-SCM:** Based on SCM [8], we extend it to the nested array scenario and develop the NA-SCM method.

* **NA-IN:** Based on IN [10] method, we extend it to the nested array scenario and develop the NA-IN method.

* **NA-PFLOM:** We replace the co-prime array employed in the PFLOM based method of [23] with the nested array for comparison.

<table><caption>TABLE I: Parameter for different methods</caption><thead><tr><th>Algorithms</th><th>Correlation theories</th><th>Parameter</th></tr></thead><tbody><tr><td>ULA-PFLOM [9]</td><td>FLOS</td><td>b = { 0.3, if α ≥ 1 <br> 0.05, if α < 1 </td></tr><tr><td>ULA-ROC [6]</td><td>FLOS</td><td>p = 0.4</td></tr><tr><td>ULA-FLOM [7]</td><td>FLOS</td><td>p = 1.2</td></tr><tr><td>NA-SCM [8]</td><td>Amplitude suppression</td><td>no</td></tr><tr><td>NA-IN [10]</td><td>Amplitude suppression</td><td>no</td></tr><tr><td>NA-PFLOM [23]</td><td>FLOS</td><td>b = { 0.3, if α ≥ 1 <br> 0.05, if α < 1 </td></tr><tr><td>The proposed AOMF</td><td>FLOS</td><td>p = <sup>α/2</sup>&frasl;<sub>1+α/2H</sub></td></tr><tr><td>The proposed IAOMF</td><td>FLOS</td><td>p = <sup>α/2</sup>&frasl;<sub>1+δ·e<sup>-H</sup></sub></td></tr></tbody></table>

In the aforementioned methods, after the equivalent covariance matrix is obtained, the MUSIC subspace estimation method is used for DOA estimation with a spectral peak search interval of 0.01°. Moreover, the ROC and FLOM based methods only applicable to the scenario of $1 < \alpha \le 2$, the remaining methods can work in $0 < \alpha \le 2$. Some parameters are provided in TABLE I.

Also the Cramer-Rao bound (CRB, [3]) in the presence of impulsive noise is added for comparison. Nevertheless, the CRB of the impulsive noise is not available for $\alpha \ne 1$ and $\alpha \ne 2$. In this case, [3] indicates that the CRB between $\alpha = 1$ and $\alpha = 2$ can be determined by linear interpolation.

## A. Performance analysis of AOMF and IAOMF

Fig. 6 gives the variation of AEF, AOMF and IAOMF with the value of the characteristic exponent $\alpha$. Following conclusions can be obtained: a) The values of the AEF always remain at the highest level; b) The proposed AOMF values increase with increasing $\alpha$ from the range [$\alpha/4, \alpha/2$). c) Compared to the AEF and AOMF, the value of the $p_{IAOMF}$ is always the smallest, as shown by the expression of the PFLOM (10), the smaller the $b$, the more effective the suppression of the impulsive noise outliers. d) When the $\alpha$ is greater than 1.4, the values of AOMF and IAOMF are close to $\alpha/2$. In summary, the proposed $p_{IAOMF}$ addresses the shortcomings of the AEF and AOMF, and its further performance will be discussed in the simulation.

<figure><img src="image_7.png" /><figcaption>Fig. 6: AEF, AOMF and IAOMF values versus characteristic exponent $\alpha$ when GSNR = 5 dB and $T = 500$, the DOA of three signals is [10°, 20°, 30°] and $N_1 = 4, N_2 = 5$.</figcaption></figure>

Figs. 7(a)-7(d) give the performance comparison experiments of the proposed $p_{AOMF}$ and $p_{IAOMF}$ (combined with PFLOM) methods with NA-SCM and NA-IN algorithms when $\alpha = 0.2, 0.5, 1.0, 1.5$, respectively. For the proposed $p_{AOMF}$
method, $p_{IAOMF}$ version method, NA-SCM method and NA-IN method, the RMSE of DOA decreases continuously with increasing GSNR. As GSNR increases, the performance of the proposed $p_{IAOMF}$ method is similar to that of NA-IN. For each value of $\alpha$, the $p_{IAOMF}$ version of the proposed method gives more accurate estimation results than the other methods, especially when the GSNR is low and $\alpha$ is small. With the

<figure><img src="image_8.png" /><figcaption>Fig. 7: RMSE performance versus GSNR, the DOA of three signals is [10°, 20°, 30°] and $N_1 = 4$, $N_2 = 5$, $T = 500$, MC = 500. (a) $\alpha = 0.2$; (b) $\alpha = 0.5$; (c) $\alpha = 1.0$; (d) $\alpha = 1.5$.</figcaption></figure>

<table><caption>TABLE II: Computational complexity of the equivalent covariance matrix</caption><tr><td>ROC [6]</td><td>O{2P<sup>2</sup>T}</td></tr><tr><td>FLOM [7]</td><td>O{(P<sup>2</sup> + P)T}</td></tr><tr><td>SCM [8]</td><td>O{(P<sup>2</sup> + 2P)T}</td></tr><tr><td>PFLOM [9]</td><td>O{3P<sup>2</sup>T}</td></tr><tr><td>IN [10]</td><td>O{(P<sup>2</sup> + P)T}</td></tr><tr><td>AOMF</td><td>O{(3P<sup>2</sup> + 2P)T}</td></tr><tr><td>IAOMF</td><td>O{(3P<sup>2</sup> + 2P)T}</td></tr></table>

introduction of the scaling parameter $\delta$, not only the $p_{IAOMF}$ version is more suitable for the PFLOM scheme, but also possesses a better performance than the NA-SCM method and NA-IN method, especially in the highly impulsive noise environment.

In the traditional PFLOM method, the order moment parameter is generally determined by a large number of experiments, which undoubtedly increases the computational burden. In contrast, the proposed AOMF and IAOMF are only related to the array received signal $x(t)$, which does not require any prior knowledges. Therefore, the AOMF and IAOMF are particularly suitable for the scenario of practical applications.

## B. Computational complexity analysis

For convenience, we only use the number of complex or real number multiplications as the computational complexity metric. In this case, we only compare the computational complexity of the proposed AOMF-PFLOM and IAOMF-PFLOM algorithms with the equivalent covariance matrices of ROC, FLOM, PFLOM, SCM and IN. The computational complexity of the subsequent nested array DOA estimation algorithms is equivalent, as can be seen in [29]. The computational complexity of all the aforementioned algorithms is shown in TABLE II, and it should be noted that the computation of the proposed AOMF and IAOMF is only distinguished from that of the scaling parameter $\delta$, and thus their computational complexity is equivalent. Compared to the FLOS-based algorithms, the computational complexity of the proposed algorithms comes from the computation of AOMF and IAOMF, but all the FLOS-based algorithms need to set the unknown order moment paramete in advance. Moreover, although the computational complexity of the proposed algorithm is higher than that of the SCM and IN algorithms, the computational complexity is affordable as it results in a great improvement in the DOA estimation performance.

## C. Simulation results

The root mean square error (RMSE) can be defined as

$$ \text{RMSE} = \sqrt{\frac{1}{\text{MC}} \sum_{j=1}^{\text{MC}} \frac{1}{K} \sum_{k=1}^{K} (\tilde{\theta}_{kj} - \theta_k)^2}, \quad (39) $$

$\tilde{\theta}_{kj}$ is the estimated DOA of the $k$th source $\theta_k$ at the $j$th Monte Carlo. In addition, the GSNR [35] in the presence of impulsive noise is as follows

$$ \text{GSNR} = 10 \log \left( \frac{\mathbb{E}\{|s(t)|^2\}}{\gamma} \right), \quad (40) $$

where $\gamma$ denotes the dispersion parameter.
**Case 1:** RMSE performance comparison versus GSNR.

Consider three statistically incoherent phase-modulated (PM) signals with DOA $[-10°, 20°, 30°]$. The number of snapshots $T = 500$, array elements $N_1 = 4, N_2 = 5$, i.e., the number of sensors of ULA $P = 9$, and the number of Monte Carlo experiments $MC = 500$.

Figs. 8 and 9 demonstrate the RMSE performance of the proposed IAOMF method with different GSNR ranging from -6 dB to 10 dB, where $\alpha = 0.5$ in Fig. 8 and $\alpha = 1.5$ in Fig. 9, respectively. In Fig. 8, all the algorithms' performance improves with the increasing of GSNR. With the same sparse array scenario, as the same method to suppress the received signal amplitude, the NA-IN method still outperforms the NA-SCM algorithm because the $||\cdot||_\infty$ norm of the NA-IN algorithm is more sensitive to impulsive noise than the Frobenius-norm of NA-SCM. For the NA-PFLOM method, we set its order-moment parameter $b$ to 0.05 and find that it has stronger suppression ability for the impulsive noise than the NA-IN and NA-SCM methods. In comparison, the IAOMF proposed in this paper achieves the best performance when applied to PFLOM, as seen in the previous Fig. 6, where the IAOMF value is very small when $\alpha = 0.5$, and thus the reduction for highly impulsive noise outliers is more significant.

<figure><img src="image_9.png" /><figcaption>Fig. 8: RMSE versus different GSNRs, $\alpha = 0.5$, $T = 500$, MC = 500.</figcaption></figure>

In Fig. 9, $\alpha = 1.5$ is adjusted to the general impulsive scene. It can be seen that the performance of the nested array-based method is generally better than that of the ULA-based method. The performance of all algorithms is improved compared to Fig. 8, and the RMSE performance of the NA-PFLOM method is slightly weaker than that of the NA-SCM and NA-IN methods when the GSNR < 0 dB. Moreover, the performance of the proposed IAOMF method consistently outperforms other methods.

**Case 2:** RMSE performance comparison versus characteristic exponent $\alpha$.

Fig. 10 demonstrates the RMSE curve as a function of the characteristic exponent $\alpha$ from a highly impulsive scenario $\alpha = 0.2$ to a moderately impulsive environment $\alpha = 1.8$. For the ULA-ROC and ULA-FLOM methods, they cannot work at $0 < \alpha < 1$ and therefore are not available for comparison in Fig. 10(a). To demonstrate a straightforward comparison of the estimation performance of the various algorithms, GSNR is set to 0 dB, the number of snapshots $T = 500$, and the number of Monte Carlo experiments $MC = 500$. The

remaining parameters (e.g., source information, number of array elements) are consistent with Case 1.

In general, the performance of all algorithms improves with increasing $\alpha$ in the highly impulsive scenario. From Fig. 10(a), it can be seen that the changes in $\alpha$ have a small performance impact on the proposed algorithm, while the other algorithms converge only at $\alpha > 0.7$ (can be seen in Fig. 10(b)). Additionally, it can be found that the nested array based

<figure><img src="image_10.png" /><figcaption>Fig. 9: RMSE versus different GSNRs, $\alpha = 1.5$, $T = 500$, MC = 500.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 10: RMSE performance versus different characteristic exponents $\alpha$, where GSNR = 0 dB, $T = 500$, MC = 500 (a) $\alpha \in [0.2, 1]$, (b) $\alpha \in [1.1, 1.8]$.</figcaption></figure>
method outperforms the ULA based method for $\alpha > 1$. For the NA-PFLOM method, we take the order moment parameters $b = 0.05$ ($0 < \alpha < 1$) and $b = 0.3$ ($\alpha \ge 1$), respectively, thus it performs better than the NA-IN and NA-SCM algorithms. However, the overall results show that the performance of the proposed method is always better than the other algorithms, considering all possible impulsive noise conditions.

## VI. CONCLUSION

This paper proposes an adaptive order moment function (AOMF) scheme and its improved version (IAOMF) for DOA estimation in the impulsive noise scenario. Additionally, our proposed AOMF and IAOMF are combined with PFLOM theory, which have the advantage that the order moment parameter $p$ does not need to be predetermined. Moreover, the parameter $\delta$ in the proposed IAOMF strategy changes with the complicated noise environment and is more adaptable in the highly impulsive scenes. Compared with the existing methods, the performance of the proposed algorithm is significantly improved. Performance analysis and simulation results verify the superiority of the proposed IAOMF method, and it can be adapted to arbitrary impulsive scenarios.

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