# Enhanced Near-Field Coherent Source Localization via Alternating Oblique Projection and Polynomial Root-Finding

Jingjing Pan, Shaohui Lv, Yuehe Ning, Meng Sun, Yide Wang

► To cite this version:

Jingjing Pan, Shaohui Lv, Yuehe Ning, Meng Sun, Yide Wang. Enhanced Near-Field Coherent Source Localization via Alternating Oblique Projection and Polynomial Root-Finding. IEEE Sensors Journal, 2026, pp.1-1. 10.1109/JSEN.2026.3660254. hal-05520168

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Submitted on 20 Feb 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Enhanced Near-Field Coherent Source Localization
via Alternating Oblique Projection and Polynomial
Root-Finding

Jingjing Pan, Shaohui Lv, Yuehe Ning, Meng Sun and Yide Wang

***Abstract***—Near field (NF) source localization plays an essential role in the field of array signal processing. However, for coherent sources, conventional SSP and Topelitz matrix reconstruction algorithms cannot be applied directly, because of the nonlinear phase variations in near-field source. Although probability model-based methods like enhanced maximum likelihood method (EML) can deal with coherent near-field sources, they are fundamentally constrained by computational complexity or resolution performance. To address these issues, this paper proposes an enhanced method for near-field coherent source localization using a two-step alternating oblique projection (AOP) and polynomial root-finding, comprising a coarse localization stage followed by a fine estimation. Firstly, the coarse estimates of DOA and range parameters are derived by applying the anti-diagonal components of the covariance matrix through root-MUSIC and root-Capon, respectively, followed by an iterative refinement process that alternately applies the AOP method and root-finding to achieve fine estimation. The performance of the proposed algorithm is validated through simulations. Results demonstrate that the algorithm achieves accurate localization of NF coherent sources with reduced computational complexity and improved resolution. The accuracy of the proposed method is also validated by a filed experiment.

***Index Terms***—Direction of arrival (DOA), near-field (NF), alternating oblique projection (AOP), polynomial root-finding, coherent source localization.

## I. INTRODUCTION

NEAR-FIELD (NF) source localization is a fundamental research focus in modern communication systems, with important applications in radar, medical imaging, sonar, geophysics, and oceanography [1]-[7]. Unlike far-field (FF) source localization, where the wavefront can be approximated as planar and only the direction of arrival (DOA) needs to be estimated, NF localization involves spherical wavefronts. As a result, both DOA and range must be jointly estimated [8], [9].

Recent advancements in near-field source localization have been largely inspired by far-field methodologies. Early approaches, such as near-field beamforming [10], which are adapted from far-field localization techniques, utilize phase differences across sensor arrays for parameter estimation but

suffer from resolution limitations due to beamwidth constraints. The near-field multiple signal classification (MUSIC) algorithm is introduced in [11], which achieves high resolution by exploiting the subspace orthogonality principle. However, its reliance on a two-dimensional (2D) search over both DOA and range leads to high computational complexity. To mitigate this issue, techniques based on parameter decoupling are developed, transforming the joint estimation of range and DOA into sequential subproblems. Representative approaches include the parameter-separated polynomial rooting method [12], which decouples DOA and range into independent polynomials for closed-form solutions; focus-based techniques [13], which map the received data into a virtual far-field model for simplified DOA estimation; and second-order statistics (SOS) frameworks [14], which extract separable components from covariance matrices to achieve independent DOA and range optimization. Recent research expand towards addressing the fundamental limitations of classic near-field estimators, for example, the under-determined estimation problem [15], [16] and the model mismatch problem with approximation-based signal model [17], [18]. The authors in [15], [16] propose sparse array design in near-field source localization, such as coprime and nested arrays, which enlarge the array aperture and achieve a higher number of degrees of freedom (DOF) than the physical sensors. In contrast to the commonly used second-order Taylor series, the work in [17], [18] develops NF source localization techniques based on the exact source-sensor spatial propagation geometry, thereby achieving precise parameter estimation.

Nevertheless, aforementioned NF source localization methods typically assume independent signals, and their performance degrades significantly when applied to coherent sources [19]. To this end, several decorrelation techniques have been proposed, including spatial smoothing processing (SSP) [20], [21] and ESPRIT-like algorithms based on Toeplitz matrix structures [22]. Nevertheless, as these techniques are designed for far-field scenarios and rely on the assumption of linear phase progression across the array, they fail to accommodate the nonlinear phase behavior inherent in near-field wavefronts, and thus are unsuitable for coherent near-field source localization [20].

In coherent scenarios, the authors in [23] propose a NF coherent source localization method based on planar array design. By employing a specific sensor arrangement, the method constructs a full-rank covariance matrix to mitigate the impact of signal coherence. Despite this, this approach

Jingjing Pan, Shaohui Lv, Yuehe Ning and Meng Sun are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: jingjingpan @ nuaa.edu.cn; SZ2304120 @ nuaa.edu.cn; SZ2404116 @ nuaa.edu.cn; mengsun @ nuaa.edu.cn).

Yide Wang is with Nantes Université, CNRS, IETR, UMR 6164, F-44000 Nantes, France (e-mail: yide.wang @ univ-nantes.fr).
is limited by specific array configurations and has a reduced effective aperture. As an alternative, the enhanced maximum likelihood method (EML) [24] is proposed, in which, the alternating oblique projection (AOP) method is introduced to approximate the maximum likelihood estimator (MLE) [25], [26] via iterative subspace projections. Although AOP reduces computational complexity by avoiding exhaustive multi-dimensional searches, its resolution is constrained due to the use of beamforming-based formulations. The focusing technique is employed to address the limited resolution issue in [27]. Nevertheless, this technique requires closely spaced signal sources, and its performance degrades significantly when the targets are far apart. Besides, sparse representation [28], [29] and deep learning [30] methods provide effective alternatives for localizing coherent sources in near-field scenarios. As in [28], [29], sparse representation techniques are successfully applied in near-field coherent source localization by exploiting the inherent sparsity of source signals, rather than the covariance matrix structure, thereby achieving robustness against signal coherence. Unfortunately, sparse representation in [28], [29] often faces prohibitive computational complexity in its parameter searching process and limited resolution, while deep learning method in NF coherent source localization [30] is fundamentally limited by its requirement for massive labeled datasets.

In this context, we propose an enhanced near-field coherent source localization by integrating AOP and polynomial root-finding techniques. The proposed method is executed in a two-stage process. The first stage, the coarse estimation, involves obtaining initial coarse estimates for the DOA and range parameters. This is achieved through the use of the anti-diagonal components of the covariance matrix in conjunction with root-MUSIC and root-Capon algorithms. In the second stage, the fine estimation, these preliminary estimates are iteratively refined by alternately applying the AOP method and root-finding, ultimately yielding a high-resolution and low-complexity estimation.

The main contributions of this paper are detailed below:

1) The resolution of near-field coherent source localization is improved by using the anti-diagonal elements of the covariance matrix through MUSIC, which can not only deal with sources that are closely distributed but also far apart.

2) The complexity is significantly reduced through a series of polynomial root-finding procedures both in the coarse and fine estimations, compared to spectral searching methods [24]-[29].

The rest of this paper is structured in the following way. In Section II, the signal model is presented. In Section III, the proposed method is described in detail. Section IV and V provide analysis and comparison of the proposed method with other algorithms. In Section VI, practical experiments are conducted to validate the performance of the proposed method. The conclusions of this paper are given in Section VII.

## II. SIGNAL MODEL

The structure of the array of sensors and the schematic of the array receiving near-field incident signals, as presented

in [31], are shown in Fig. 1. We suppose that there are K narrowband sources located within the Fresnel region $r_F = [0.62(D^3/\lambda)^{1/2}, 2D^2/\lambda]$ [5], with wavelength $\lambda$ and aperture D. The incident signals are supposed to be coherent. It is assumed that the uniform linear array (ULA) is composed of $N = 2M+1$ sensors, with the elements numbered from left to right as $(-M, ..., 0, ..., M)$, where the center element is taken as the reference. The distance between two adjacent sensors is denoted by $d$ ($d \le 1/4\lambda$). The signal received by the mth element, emitted by the kth source, is given by:

$$z_{m,k}(t) = s_k(t)e^{j\varphi_{m,k}} + n_m(t), t = 1, 2, \dots, T_s, \quad (1)$$

where $s_k(t) = \alpha_k e^{j\beta_k} s_0(t)$, represents the signal received from the kth source; $\alpha_k$ and $\beta_k$ denote the amplitude and phase offset of the kth source, respectively; $s_0(t)$ is the normalized narrow-band signal, satisfying $E[s_0(t)s_0^*(t)] = 1$; $T_s$ stands for the number of snapshots; $n_m(t)$ denotes the Gaussian white noise added to the signal; and $\varphi_{m,k}$ represents the phase difference between the signal received by the mth sensor and that by the reference sensor from the kth source, which can be written as:

$$\varphi_{m,k} = \frac{2\pi}{\lambda} \left( \sqrt{r_k^2 + (md)^2 + 2r_k md \sin \theta_k} - r_k \right), \quad (2)$$

where $r_k$ denotes the range between the kth source and the reference element; $\theta_k$ indicates the direction of the kth source; and $\varphi_{m,k}$ is the phase shift, which is commonly approximated using a Taylor expansion:

$$\varphi_{m,k} = \left(-\frac{2\pi d}{\lambda} \sin \theta_k\right) m + \left(\frac{\pi d^2}{\lambda r_k} \cos^2 \theta_k\right) m^2 + o\left(\frac{d^2}{r_k^2}\right), \\ \approx \omega_k m + \phi_k m^2 \quad (3)$$

where $\omega_k = -\frac{2\pi d}{\lambda} \sin \theta_k$, $\phi_k = \frac{\pi d^2}{\lambda r_k} \cos^2 \theta_k$.

The vector form of the array output in (1) is:

$$\mathbf{z}(t) = \mathbf{B}(\theta, r)\mathbf{s}(t) + \mathbf{n}(t), \quad (4)$$

where $\mathbf{z}(t) = [z_{-M}(t), \dots, z_0(t), \dots, z_M(t)]^T$, is the received signal vector; $\{\bullet\}^T$ denotes the transpose operator; $\mathbf{s}(t) = [s_1(t), \dots, s_K(t)]^T$, represents the source vector; $\mathbf{B}(\theta, r)$ indicates the array manifold:

$$\mathbf{B}(\theta, r) = [\mathbf{b}(\theta_1, r_1), \mathbf{b}(\theta_2, r_2), \dots, \mathbf{b}(\theta_K, r_K)], \quad (5)$$

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic of ULA sensing near-field signals.</figcaption></figure>
whose kth column is:

$$ \mathbf{b}(\theta_k, r_k) = [e^{j[(-M)\omega_k + (-M)^2\phi_k]}, \dots, e^{j[M\omega_k + M^2\phi_k]}]^T. \quad (6) $$

The covariance matrix $\mathbf{R}$ can be obtained as:

$$ \mathbf{R} = E\{\mathbf{z}(t)\mathbf{z}^H(t)\}, \quad (7) $$

where $\{\bullet\}^H$ is the conjugate transpose operation, and $E\{\bullet\}$ denotes the mathematical expectation.

The following constraints are applied to the signal model discussed in this paper:

1) The number of array elements $N$ exceeds the number of signal sources $K$.

2) The incident signals are assumed to be coherent.

3) The noise received by the array elements is zero-mean, statistically independent of the source signals, and follows white Gaussian distribution.

### III. METHODOLOGY

This section introduces an improved approach for high-resolution and low-complexity NF coherent source localization, which employs a two-stage AOP and polynomial root-finding framework. The method first obtains coarse estimates of the DOA and range by exploiting the anti-diagonal elements in the covariance matrix combined with the root-MUSIC and root-Capon algorithms, respectively. These initial estimates are then refined through an iterative process that alternately employs the AOP method and polynomial rooting, enhancing overall localization performance. The complete procedure is illustrated in Fig. 2.

#### A. Coarse Estimation

Based on the previous assumptions, we propose to reconstruct a decoupled covariance matrix to roughly obtain the DOA and range parameters.

Denote $r_{i,\ell}$ as the element in the $\ell$th row and $\ell$th column of $\mathbf{R}$, and we can derive the expression of $r_{i,\ell}$ as:

$$ \begin{aligned} r_{i,\ell} &= E[\mathbf{z}_i(t)\mathbf{z}_\ell^H(t)] \\ &= \sum_{k_1=1}^{K} \sum_{k_2=1}^{K} \left( \alpha_{k_1}\alpha_{k_2}e^{j(\beta_{k_1}-\beta_{k_2})} \times \right. , \\ & \qquad \left. e^{j(\omega_{k_1}i+\phi_{k_1}i^2-\omega_{k_2}\ell-\phi_{k_2}\ell^2)} \right) + \sigma^2 \delta_{i,\ell} \end{aligned} \quad (8) $$

where $\mathbf{z}_i(t), \mathbf{z}_\ell(t)$ denote the received signals of the $i$th and $\ell$th sensor elements at time $t$, respectively; $\sigma^2$ represents the variance of the noise; $\delta$ is the Dirac delta function.

The components of $\mathbf{R}$ on its anti-diagonal are expressed as:

$$ \begin{aligned} r_{i,-i} &= \underbrace{\sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k i} + \sigma^2 \delta_{i,-i}}_{\text{primary term}} \underbrace{\text{noise term}}_{\text{cross term}} \\ &+ \underbrace{\sum_{k_1=1}^{K} \sum_{k_2=k_1+1}^{K} \zeta(k_1, k_2) e^{j(\omega_{k_1} + \omega_{k_2})i}_{,} }_{\text{cross term}} \end{aligned} \quad (9) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Main process of the proposed algorithm.</figcaption></figure>

where $\zeta(k_1, k_2) = 2\alpha_{k_1}\alpha_{k_2}\mathfrak{Re}\{e^{j(\beta_{k_1}-\beta_{k_2})}e^{j(\phi_{k_1}-\phi_{k_2})i^2}\}$, and $\mathfrak{Re}\{\bullet\}$ stands for the real part.

Ignoring the cross term, (9) can be approximated as:

$$ r_{i,-i} \approx \sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k i} + \sigma^2 \delta_{i,-i}. \quad (10) $$

Neglecting the influence of the noise term, the vector formed by the elements on the anti-diagonal of $\mathbf{R}$ can be approximated by:

$$ \mathbf{p} \approx \left[ \sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k (-M)}, \sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k (-M+1)}, \dots, \sum_{k=1}^{K} \alpha_k^2 e^{-j2\omega_k}, \sum_{k=1}^{K} \alpha_k^2, \sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k}, \dots, \sum_{k=1}^{K} \alpha_k^2 e^{j2\omega_k (M)} \right]^T. \quad (11) $$

The above equation can be estimated easily by Prony, which supports diverse array structures, including sparse arrays. Notably, the approximation of equation (11) introduces certain errors in the coarse estimation. In the fine estimation stage, however, these errors can be eliminated iteratively by applying the alternating projection method. Under this quasi-linear phase structure, certain methods originally designed for far-field signal processing, such as SSP, can be effectively applied. Based on the principle of spatial smoothing, $\mathbf{p}$ is partitioned into $Q$ subarrays, where each subarray contains $2M+2-Q$ elements. The $(2M+2-Q) \times (2M+2-Q)$ covariance matrix associated with the $Q$ subvectors can be obtained:
$$ \mathcal{D} = \frac{1}{Q} \sum_{q=1}^{Q} \mathbf{p}_q \mathbf{p}_q^H. \quad (12) $$

As can be seen from the previous process, $\mathcal{D}$ depends only on the DOAs of the signal. Therefore, a polynomial can be constructed from $\mathcal{D}$ to solve for the DOAs.

Construct the following vector:

$$ \mathbf{v}(y) = [1, y, \dots, y^{2M+1-Q}]^T, \quad (13) $$

where $y = e^{-\frac{j4\pi d}{\lambda} \sin \theta_k}$. Then, we can obtain the root-finding polynomial for the DOA:

$$ J(y) = \mathbf{v}^T (y^{-1}) \mathbf{\Omega}_n \mathbf{\Omega}_n^H \mathbf{v}(y), \quad (14) $$

where, $\mathbf{\Omega}_n$ is the matrix composed of the eigenvectors associated with the $2M+2-Q$ smallest eigenvalues of $\mathcal{D}$.

Perform root-finding on (14), the roots of $J(y)$ exhibit symmetry with respect to the unit circle. The $K$ roots with the largest magnitudes located inside the unit circle, denoted as $\hat{y}_1, \hat{y}_2, \dots, \hat{y}_K$, provide the phase information for estimating the DOA. Specifically, the DOA estimates can be obtained:

$$ \tilde{\theta}_k = - \arcsin \left( \frac{\lambda}{4\pi d} \angle \hat{y}_k \right), \quad k = 1, 2, \dots, K, \quad (15) $$

where $\angle$ denotes the phase of a complex variable.

Following the DOA estimation, the corresponding range parameters are obtained via root-Capon method. This method decouples the multidimensional parameter search into multiple one-dimensional (1D) polynomial equation solving steps, thereby significantly reducing the complexity.

The steering vector $\mathbf{b}(\theta, r)$ can be expressed in the following way:

$$ \mathbf{b}(\theta, r) = \underbrace{\begin{bmatrix} e^{j(-M)\omega} & & & \\ & \ddots & & \\ & & 1 & \\ e^{jM\omega} & & & \end{bmatrix}}_{\eta(\omega)} \underbrace{\begin{bmatrix} e^{j(-M)^2\phi} \\ e^{j(-M+1)^2\phi} \\ \vdots \\ e^{j(-1)^2\phi} \\ 1 \end{bmatrix}}_{\Upsilon(\phi)} \quad (16) \\ = \eta(\omega)\Upsilon(\phi), $$

where $\eta(\omega)$ depends only on the DOA of the signal source, while $\Upsilon(\phi)$ is influenced by both the range and the DOA information of the signal source.

The spectral peak search function of the Capon algorithm under near-field conditions is expressed as:

$$ \begin{aligned} P_{\text{Capon}} (\theta, r) &= \min_{\theta,r} \mathbf{b}^H (\theta, r) \mathbf{R}^{-1} \mathbf{b}(\theta, r) \\ &= \min_{\omega, \phi} \mathbf{\Upsilon}^H (\phi) \mathbf{\eta}^H (\omega) \mathbf{R}^{-1} \mathbf{\eta}(\omega) \mathbf{\Upsilon}(\phi). \end{aligned} \quad (17) $$

By successively substituting the estimated DOA values into equation (17), we obtain the following root-finding polynomial, which contains only the range as the unknown parameter:

$$ \mathcal{F}(v) = \mathbf{\Upsilon}^T (v^{-1}) \mathbf{\eta}^H (\tilde{\omega}_k) \mathbf{R}^{-1} \mathbf{\eta} (\tilde{\omega}_k) \mathbf{\Upsilon}(v), \quad (18) $$

where $v = e^{j \frac{\pi d^2}{\lambda r}} \cos^2 \tilde{\theta}_k$, $\tilde{\omega}_k = e^{-j \frac{2\pi d}{\lambda} \sin \tilde{\theta}_k}$.

For each DOA, the root with the minimum deviation from the unit circle, denoted as $\hat{v}_1, \hat{v}_2, \dots, \hat{v}_K$ is substituted into (19), the corresponding range estimates are derived as follows:

$$ \tilde{r}_k = \frac{\pi d^2}{\lambda \angle \hat{v}_k} \cos^2 \tilde{\theta}_k, \quad k = 1, 2, \dots, K. \quad (19) $$

## B. Fine Estimation

The coarse estimation yields roughly estimated DOA and range parameters with errors. Then the fine estimation is followed to iteratively improve the estimation results with AOP and polynomial root-finding.

In each iteration, the alternating projection method is applied to the covariance matrix to extract information about a single source. Then, a polynomial rooting procedure is used to estimate the DOA information. Next, the obtained DOA is substituted into a root-finding-based range estimator to determine its corresponding range. Unlike the traditional MLE multi-dimensional search, this algorithm establishes a closed-form solution that bypasses the searching process entirely, which significantly reduces the computational complexity while maintains good estimation accuracy.

Assume $\tilde{r}_k$ and $\tilde{\theta}_k$ as the input values of the first iteration, that is:

$$ \{\hat{\theta}_k^{(0)}, \hat{r}_k^{(0)}\} = \{\tilde{\theta}_k, \tilde{r}_k\}, \quad k = 1, 2, \dots, K, \quad (20) $$

During the $l$th iteration, the oblique projector is given by:

$$ \begin{aligned} \hat{\mathbf{P}}_k^{(l)} &\triangleq \mathcal{B} (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)}) \left[ \mathcal{B}^H (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)})^H \right. \\ &\qquad \left. \times \Gamma (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)}) \mathcal{B} (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)}) \right]^{-1}, \quad (21) \\ \mathcal{B}^H (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)}) \Gamma (\hat{\theta}_k^{(l-1)}, \hat{r}_k^{(l-1)}) \end{aligned} $$

where $\mathcal{B} (\hat{\theta}_k, \hat{r}_k) \in \mathbb{C}^{N \times (K-1)}$ is obtained by removing the $k$th column from $\mathbf{B}$, that is $\mathcal{B} (\hat{\theta}_k, \hat{r}_k) = [\mathbf{b}(\theta_1, r_1), \dots, \mathbf{b}(\theta_{k-1}, r_{k-1}), \mathbf{b}(\theta_{k+1}, r_{k+1}), \dots, \mathbf{b}(\theta_K, r_K)]$. $\{\bullet\}^{(l)}$ represents the iteration number, $l < L$, and $L$ is the total number of iterations. $\Gamma_k$ represents the orthogonal projection of the source onto the null space:

$$ \begin{aligned} \Gamma_k (\hat{\theta}_k, \hat{r}_k) &= \mathbf{I} - \mathbf{b} (\hat{\theta}_k, \hat{r}_k) \\ &\quad \times [\mathbf{b}^H (\hat{\theta}_k, \hat{r}_k) \mathbf{b} (\hat{\theta}_k, \hat{r}_k)]^{-1} \mathbf{b}^H (\hat{\theta}_k, \hat{r}_k). \end{aligned} \quad (22) $$

Denote $\bar{\mathbf{R}}$ as the noiseless covariance matrix:

$$ \bar{\mathbf{R}} = \mathbf{R} - \sigma^2 \mathbf{I} = \mathbf{B} \mathbf{R}_s \mathbf{B}^H, \quad (23) $$

where $\mathbf{R}_s$ denotes the source covariance matrix

The projection-based covariance matrix of the $k$th source in the $l$th iteration is

$$ \hat{\mathcal{R}}_k^{(l)} = (\mathbf{I} - \hat{\mathbf{P}}_k^{(l)}) \bar{\mathbf{R}} (\mathbf{I} - \hat{\mathbf{P}}_k^{(l)})^H. \quad (24) $$

According to (14), the following polynomial is constructed:

$$ \mathcal{G}(c) = \mathbf{g}^T (c^{-1})^T \hat{\mathbf{U}}_n^{(l)} (\hat{\mathbf{U}}_n^{(l)})^H \mathbf{g}(c), \quad (25) $$
where $\hat{\mathbf{U}}_n^{(l)}$ is the matrix formed by the eigenvectors corresponding to the 2M smallest eigenvalues of $\check{\mathfrak{R}}_k^{(l)}$; $\mathbf{g}(c) = [1, c, \dots, c^{2M}]^T$; $c = e^{-j \frac{2\pi d}{\lambda} \sin \hat{\theta}_k}$.

In each iteration, K alternating projections are performed. During each projection, the root closest to the unit circle obtained from (25) is preserved. After completing all K projections, the K retained roots $\hat{c}_1, \dots, \hat{c}_K$ are then substituted into (26), and the DOA estimation in the lth iteration can be computed:

$$ \hat{\theta}_k^{(l)} = -\arcsin\left(\frac{\lambda}{2\pi d} \angle(\hat{c}_k)\right), k = 1, 2, \dots, K. \quad (26) $$

The obtained $\hat{\theta}_k^{(l)}$ is used to update other parameters to solve for the range estimation in the lth iteration. The updated oblique projectors are given by:

$$ \check{\mathbf{P}}_k^{(l)} \triangleq \mathcal{B} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \left[ \left( \mathcal{B} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \right)^H \mathbf{\Gamma} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \mathcal{B} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \right]^{-1} \left( \mathcal{B} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \right)^H \mathbf{\Gamma} \left( \hat{\theta}_k^l, \hat{r}_k^{(l-1)} \right) \quad (27) $$

Replace $\check{\mathbf{P}}_k^{(l)}$ with $\check{\mathbf{P}}_k^{(l)}$ to obtain the updated covariance matrix in (24):

$$ \check{\mathfrak{R}}_k^{(l)} = (\mathbf{I} - \check{\mathbf{P}}_k^{(l)}) \bar{\mathbf{R}} (\mathbf{I} - \check{\mathbf{P}}_k^{(l)})^H . \quad (28) $$

Similarly, based on (16)–(18), the following polynomial is constructed to obtain the corresponding range:

$$ \mathcal{I}(\mu) = \mathbf{\Upsilon}^T ((\mu^{(l)})^{-1}) \eta^H (\hat{\omega}_k^{(l)}) \check{\mathbf{U}}_n^{(l)} (\check{\mathbf{U}}_n^{(l)})^H \eta (\hat{\omega}_k^{(l)}) \mathbf{\Upsilon}(\mu^{(l)}), \quad (29) $$

where, $\mu^{(l)} = e^{j \frac{2\pi d}{\lambda r}} \cos^2 \hat{\theta}_k^{(l)}$, $\hat{\omega}_k^{(l)} = e^{-j \frac{2\pi d}{\lambda} \sin \hat{\theta}_k^{(l)}}$, and $\check{\mathbf{U}}_n^{(l)}$ is the matrix constructed from the eigenvectors pertaining to the 2M smallest eigenvalues off $\check{\mathfrak{R}}_k^{(l)}$.

Preserve the root nearest to the unit circle obtained from (29) during each of the K alternating projections, denote these roots as $\hat{\mu}_1, \dots, \hat{\mu}_K$ and substitute them into the following formula to compute the corresponding range estimation at the lth iteration.

$$ \hat{r}_k^{(l)} = \frac{\pi d^2}{\lambda \angle \hat{\mu}_k} \cos^2 \hat{\theta}_k^{(l)}, k = 1, 2, \dots, K. \quad (30) $$

The iterative process in fine estimation will proceed until any of the following conditions are satisfied: either the maximum number of iterations L is reached or convergence is achieved. Convergence is defined as the difference between the estimates obtained in two consecutive iterations falling below a predefined threshold:

$$ \sum_{k=1}^{K} \left| \hat{\theta}_{k}^{(l)} - \hat{\theta}_{k}^{(l-1)} \right| + \sum_{k=1}^{K} \left| \hat{r}_{k}^{(l)} - \hat{r}_{k}^{(l-1)} \right| < \varepsilon, \quad (31) $$

where $\varepsilon$ represents the error threshold.

TABLE I presents the pseudo-code for the proposed algorithm.

<table><caption>TABLE I
PSEUDO-CODE OF THE PROPOSED ALGORITHM</caption><tr><td colspan="2"><b>Input:</b></td></tr><tr><td colspan="2">M, N, d, K, T<sub>s</sub>, z(t)</td></tr><tr><td colspan="2"><b>Output:</b></td></tr><tr><td colspan="2">{θ&#x0302;<sub>k</sub>, r&#x0302;<sub>k</sub>}<sub>K=1</sub><sup>K</sup> as final estimates</td></tr><tr><td colspan="2"><b>Coarse Estimation:</b></td></tr><tr><td colspan="2">1. Construct covariance matrix R = E{z(t)z<sup>H</sup>(t)}</td></tr><tr><td colspan="2">2. Extract anti-diagonal elements of R to form p using (11)</td></tr><tr><td colspan="2">3. Apply SSP to build D and perform EVD on it</td></tr><tr><td colspan="2">4. Apply root-MUSIC for DOA estimation using:</td></tr><tr><td colspan="2">θ&#x0302;<sub>k</sub> = -arcsin(λ/(4πd)∠y&#x0302;<sub>k</sub>), k = 1, 2, ..., K</td></tr><tr><td colspan="2">5. Apply root-Capon for range estimation using:</td></tr><tr><td colspan="2">r&#x0302;<sub>k</sub> = πd<sup>2</sup> cos<sup>2</sup>θ&#x0302;<sub>k</sub>, k = 1, 2, ..., K</td></tr><tr><td colspan="2"><b>Fine Estimation:</b></td></tr><tr><td colspan="2"><b>Initialization:</b></td></tr><tr><td colspan="2">l = 0, {θ&#x0302;<sub>k</sub><sup>(0)</sup>, r&#x0302;<sub>k</sub><sup>(0)</sup>}<sub>K=1</sub><sup>K</sup> = {θ&#x0302;<sub>k</sub>, r&#x0302;<sub>k</sub>}<sub>K=1</sub><sup>K</sup></td></tr><tr><td colspan="2"><b>while</b> not converged: l &lt; L or</td></tr><tr><td colspan="2">&nbsp;&nbsp;&nbsp;&nbsp;∑<sub>K=1</sub><sup>K</sup> |θ&#x0302;<sub>k</sub><sup>(l)</sup> - θ&#x0302;<sub>k</sub><sup>(l-1)</sup>| + ∑<sub>K=1</sub><sup>K</sup> |r&#x0302;<sub>k</sub><sup>(l)</sup> - r&#x0302;<sub>k</sub><sup>(l-1)</sup>| &lt; ε</td></tr><tr><td colspan="2"><b>do</b></td></tr><tr><td colspan="2">1. Calculate denoised covariance matrix:</td></tr><tr><td colspan="2">&nbsp;&nbsp;&nbsp;&nbsp;R&#x0302; = R - σ<sup>2</sup>I</td></tr><tr><td colspan="2">2. Compute oblique projector P&#x0302;<sub>k</sub><sup>(l)</sup> using (21)</td></tr><tr><td colspan="2">3. Update projected covariance matrix:</td></tr><tr><td colspan="2">&nbsp;&nbsp;&nbsp;&nbsp;R&#x0302;<sub>k</sub><sup>(l)</sup> = (I - P&#x0302;<sub>k</sub><sup>(l)</sup>) R&#x0302; (I - P&#x0302;<sub>k</sub><sup>(l)</sup>)<sup>H</sup></td></tr><tr><td colspan="2">4. Estimate DOA via:</td></tr><tr><td colspan="2">&nbsp;&nbsp;&nbsp;&nbsp;θ&#x0302;<sub>k</sub><sup>(l)</sup> = -arcsin(λ/(2πd)∠(c&#x0302;<sub>k</sub>)), k = 1, 2, ..., K</td></tr><tr><td colspan="2">5. Recompute projector R&#x0302;<sub>k</sub><sup>(l)</sup> with updated θ&#x0302;<sub>k</sub><sup>(l)</sup> using (28)</td></tr><tr><td colspan="2">6. Update range estimation via (30):</td></tr><tr><td colspan="2">&nbsp;&nbsp;&nbsp;&nbsp;r&#x0302;<sub>k</sub><sup>(l)</sup> = πd<sup>2</sup> cos<sup>2</sup>θ&#x0302;<sub>k</sub><sup>(l)</sup>, k = 1, 2, ..., K</td></tr><tr><td colspan="2">7. l &lt; l + 1</td></tr><tr><td colspan="2"><b>end while</b></td></tr></table>

## IV. THEORETICAL ANALYSIS

### A. Algorithm Complexity Analysis

The computational complexity of the proposed algorithm, the MLE algorithm, SS-MUSIC, EML [24], the method in [27] and the sparse reconstruction method in [29] is compared in this section. The MLE method requires multidimensional search in range and DOA estimation, and its computational complexity increases exponentially, significantly higher than other methods. The SS-MUSIC and the method in [27] require two-dimensional spectral peak searching, which leads to high computational complexity. The EML [24] and the sparse reconstruction method in [29] employ parameter decoupling to reduce the computational complexity, but 1D spectral peak searching and dictionary construction still introduce considerable complexity. In comparison, the proposed method achieves parameter estimation solely through polynomial root-solving without search procedure, which effectively reduces the complexity. The complexity comparison of the algorithms is illustrated in Table II and Fig. 3. In Table II, $n_{\theta}$ and $n_r$ denote the number of sampling points for DOA search and the number of sampling points for range search, respectively.

### B. Cramer-Rao bound (CRB)

The CRB represents the minimum possible mean squared error (MSE) of any unbiased estimator, providing a theoretical
<figure><img src="image_4.png" /><figcaption>Fig. 3. Complexity comparison.</figcaption></figure>

<table><caption>TABLE II
COMPLEXITY ANALYSIS</caption><thead><tr><th>Method</th><th>Complexity</th></tr></thead><tbody><tr><td>MLE</td><td>O(N<sup>2</sup>T<sub>s</sub> + n<sub>θ</sub><sup>2</sup>n<sub>r</sub><sup>2</sup>N<sup>3</sup>)</td></tr><tr><td>EML</td><td>O(N<sup>2</sup>T<sub>s</sub> + n<sub>θ</sub>N<sup>2</sup> + Kn<sub>r</sub>N<sup>2</sup> + KL(2N<sup>3</sup> + n<sub>θ</sub>N<sup>2</sup> + n<sub>r</sub>N<sup>2</sup>))</td></tr><tr><td>Method in [27]</td><td>O(N<sup>2</sup>T<sub>s</sub> + n<sub>θ</sub>n<sub>r</sub>N<sup>2</sup> + (2M + 2 − Q)<sup>2</sup>Q + N<sup>3</sup> + K<sup>3</sup> + K<sup>2</sup>N + KL(2N<sup>3</sup> + n<sub>θ</sub>N<sup>2</sup> + n<sub>r</sub>N<sup>2</sup>))</td></tr><tr><td>SS-MUSIC</td><td>O(N<sup>2</sup>T<sub>s</sub> + N<sup>3</sup> + (2M + 2 − Q)<sup>2</sup>Q + n<sub>θ</sub>n<sub>r</sub>N<sup>2</sup>)</td></tr><tr><td>Sparse reconstruction method [29]</td><td>O(2N<sup>2</sup>T<sub>s</sub> + N<sup>3</sup> + n<sub>θ</sub>N + Kn<sub>r</sub>N + (2M + 2 − Q)<sup>2</sup>Q + (2M + 2 − Q)<sup>3</sup>)</td></tr><tr><td>Proposed</td><td>O(N<sup>2</sup>T<sub>s</sub> + N<sup>3</sup>(KL + 3L + K + 1) + 2(2M + 2 − Q)<sup>3</sup> + (2M + 2 − Q)<sup>2</sup>Q)</td></tr></tbody></table>

lower bound for the estimator's performance. The expression for the CRB in NF is given in [32] as follows:

$$ \text{CRB} = \frac{\sigma^2}{2T_s} \left\{ \Re \left( (\mathbf{D}^H \Pi_{\mathbf{B}}^{\perp} \mathbf{D}) \otimes (\mathbf{J}_I \otimes (\mathbf{R}_s \mathbf{B}^H \mathbf{R}^{-1} \mathbf{B} \mathbf{R}_s)^T) \right) \right\}^{-1}, \quad (32) $$

where $\odot$ and $\otimes$ represents the Hadamard–Schur and the Kronecker product, respectively, $\Pi_{\mathbf{B}}^{\perp} = \mathbf{I} - \mathbf{B}(\mathbf{B}^H \mathbf{B})^{-1} \mathbf{B}^H$, and

$$ \mathbf{J}_I = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, $$

$$ \mathbf{D} = \begin{bmatrix} \frac{\partial \mathbf{b}}{\partial \theta}(\theta_1, r_1), \dots, \frac{\partial \mathbf{b}}{\partial \theta}(\theta_K, r_K), \\ \frac{\partial \mathbf{b}}{\partial r}(\theta_1, r_1), \dots, \frac{\partial \mathbf{b}}{\partial r}(\theta_K, r_K) \end{bmatrix}. $$

## V. SIMULATIONS AND ANALYSIS

In this section, the performance of the proposed near-field coherent source localization algorithm is evaluated under various conditions. First, its estimation performance is discussed. Then, the algorithm's estimation accuracy across various scenarios is examined and compared with EML [24], SS-MUSIC, the method in [27], the sparse reconstruction

<figure><img src="image_5.png" /><figcaption>Fig. 4. Near-field coherent source localization of three sources with 30 iterations.</figcaption></figure>

method in [29] and CRB [32]. The receiving array in this section is configured as a ULA with $N = 11$ elements, where the inter-element spacing is $d = 1/4$ λ, and it is assumed that SSP divides the array into 6 subarrays, and the maximum iteration number in the iteration of the proposed method is chosen as $L = 30$, and $\varepsilon$ is fixed at 0.01. Therefore, the angle resolution limit would be: $\Lambda \le 101^{\circ}/N \approx 9.2^{\circ}$.

In the statistical analysis, the root mean square error (RMSE) is employed and formulated as:

$$ \text{RMSE} = \sqrt{\frac{1}{K C'} \sum_{c'=1}^{C'} \sum_{k=1}^{K} (\hat{h}_{kc'} - \hat{h}_k)^2}, \quad (33) $$

where, $C'$ represents the number of Monte Carlo trials. It is assumed that the RMSE statistics in this section are based on the results of 200 experiments, i.e. $C' = 200$. $\hat{h}_{kc'}$ denotes the estimation result of the algorithm for the $k$th source in the $c'$th experiment.

### A. Efficiency validation

The first simulation shows the spatial distribution of the three estimated near-field sources. The settings are $[\theta_1, r_1] = [-56^{\circ}, 2.7\lambda]$, $[\theta_2, r_2] = [20^{\circ}, 4.4\lambda]$, $[\theta_3, r_3] = [35^{\circ}, 6.6\lambda]$, $\beta_1 = -\pi/8$, $\beta_2 = -3\pi/2$, $\beta_1 = \pi/4$, $\alpha_1 = \alpha_2 = \alpha_3 = 1$, SNR is set to 15 dB; the number of snapshots is 100. Fig. 4 displays the estimation results obtained by the algorithm for the incident signals. As shown in Fig. 4, the coarse estimates exhibit noticeable deviations from the true values, but the estimates progressively converge toward the true values with each iteration. The final estimation of the algorithm is the closest to the true values, which indicates that the proposed method can accurately estimate coherent sources.

### B. Performance versus SNR

The second simulation validates the statistical performance of the proposed method under different SNR conditions with
<figure><img src="image_6.png" /><figcaption>Fig. 5. RMSE of DOA estimation versus SNR in Case a.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. RMSE of range estimation versus SNR in Case a.</figcaption></figure>

two coherent near-field sources. Since methods like EML [24] and the sparse reconstruction method in [29] have resolution problems, two scenarios are utilized for the comparison. SNR varies from 0 to 30 dB. The other settings are the same with Section V-A. Two cases are as follows:

* Case a: $[0°, 3.8λ]$ and $[9°, 4λ]$;

* Case b: $[0°, 4λ]$ and $[35°, 6λ]$.

Figs. 5 and 6 show the performance of the proposed method in Case a with closely located sources, where the angle separation is smaller than 9.2°. Considering the resolution, the estimates of SS-MUSIC, the method in [27], and EML [24] are included for comparison. When the sources are close, the proposed method yields the lowest RMSE under varying SNR conditions, not only for the estimation of DOA but also for range.

Figs. 7 and 8 show the performance of the proposed method in Case b with far-apart sources with the angle separation = 35°. Here, the results of SS-MUSIC, EML [24], and the sparse

<figure><img src="image_8.png" /><figcaption>Fig. 7. RMSE of DOA estimation versus SNR in Case b.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. RMSE of range estimation versus SNR in Case b.</figcaption></figure>

reconstruction method in [29] are also analyzed. Since the method in [27] only deals with close sources, its estimation results are not included in Case b. In this scenario, the proposed algorithm achieves estimation accuracy approaching the CRB, demonstrating extremely high estimation accuracy. As is evident from the figures, under such conditions, SS-MUSIC struggles to achieve effective signal estimation. Even though the EML [24] and the sparse reconstruction method in [29] can correctly identify the two sources under this condition, its RMSE is still higher than that of the proposed algorithm.

### C. Performance versus the number of snapshots

The third simulation analyzes the statistical performance of the proposed algorithm under different number of snapshots, together with SS-MUSIC, EML [24] and CRB. The method in [27] and the sparse reconstruction method in [29] are involved in the comparisons under Case a and Case b, respectively. The
<figure><img src="image_10.png" /><figcaption>Fig. 9. RMSE of DOA estimation versus the number of snapshot in Case a.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 11. RMSE of DOA estimation versus the number of snapshot in Case b.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. RMSE of range estimation versus the number of snapshot in Case a.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. RMSE of range estimation versus the number of snapshot in Case b.</figcaption></figure>

snapshots vary from 50 to 550. The other settings are the same with section V-A.

The simulation results are depicted in Figs. 9 - 12. As the number of snapshots increases, the RMSE of the algorithm proposed remains at a low level and continues to decline. In Case a, where the sources are close, the proposed method achieves better estimation accuracy compared to the other algorithms. The EML [24], due to the limited resolution of its algorithm, fails to accurately estimate the target, and its RMSE is high throughout different numbers of snapshots. When the sources are far apart, the proposed method outperforms EML [24] and the sparse reconstruction method in [29]. Additionally, it can be observed that the SS-MUSIC algorithm exhibits poor performance in signal estimation under such conditions. In comparison, the proposed method maintains favorable estimation performance for both close and far apart sources across varying numbers of snapshots.

## D. Performance versus DOA and range separations

The fourth simulation presents the RMSE of the proposed algorithm under different DOA and range separations. In the simulation with different DOA separations, two coherent sources locate at $[\theta_1, r_1] = [0^\circ, 4\lambda]$, $[\theta_2, r_2] = [1^\circ + \Delta\theta, 4\lambda]$, with $\Delta\theta \in [1^\circ, 31^\circ]$. The number of snapshots is 100 and SNR = 15 dB. Figs. 13 and 14 show the RMSEs of DOA and range estimates with different DOA separations. With the increase of DOA separation, the RMSEs of all the algorithms decrease continuously. But the proposed method exhibits superior estimation accuracy in both DOA and range estimation.

Similarly, in the simulation with different range separations, two coherent sources locate at $[\theta_1, r_1] = [0^\circ, 4\lambda]$, $[\theta_2, r_2] = [9^\circ, 4\lambda + \Delta\lambda]$, with $\Delta\lambda \in [0\lambda, 2.1\lambda]$. The other settings are the same with the DOA separation case. Figs. 15 and 16 plot the RMSEs of DOA and range estimates with different
<figure><img src="image_14.png" /><figcaption>Fig. 13. RMSE of DOA estimation versus DOA separation.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14. RMSE of range estimation versus DOA separation.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15. RMSE of DOA estimation versus range separation.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 16. RMSE of range estimation versus range separation.</figcaption></figure>

range separations. As shown in the figures, the RMSE of the proposed algorithm continuously decreases as the range separation increases. The proposed algorithm outperforms the other compared methods in both DOA and range estimation, as confirmed in Figs. 13 and 14.

## VI. EXPERIMENTAL VERIFICATION

To further validate the performance of the proposed method in near-field coherent scenarios, we conduct 3 outdoor field experiments using a Zynq-based FPGA platform (xczu4ev-sfvc784-1-i), like in [33]. The experimental configuration is illustrated in Fig. 17.

The experimental system comprises signal generators, transmitting and receiving antennas, a signal receiver, and a host computer. Two signal generators are synchronized to transmit single-tone signals to ensure coherency. Each generator operates at a center frequency of 1.2 GHz ($\lambda$ = 25 cm) with an output power of 10 dBm. The receiving array is

a 7-element ULA with an inter-element spacing of 6.25 cm (1/4$\lambda$). Accordingly, the Fresnel region is $r_F$ = [28.5, 112.5] cm. The transmitting antennas are precisely positioned within the Fresnel region (Table III), with their exact locations measured using a laser rangefinder and protractor to define the ground truth. The signals are first downconverted and then synchronously sampled at 100 MHz (16-bit resolution) for a total of 1024 snapshots. To address channel inconsistencies, an active calibration mechanism is conducted to compensate for amplitude and phase errors.

The proposed method along with MLE and EML [24] are tested within 3 experiments, as shown in Table III. As an efficient estimator, MLE demonstrates the highest estimation accuracy but at the expense of complexity. While EML [24] and the proposed method deliver comparable performance, the latter yields a tighter fit to the ground truth, which is consistent with Figs. 13 and 14. The bias in the experiment might be attributed to non-ideal factors such as drift in sensor character-
istics, differences in ADC properties, mutual coupling between
array elements, and array position calibration deviations.

<figure><img src="image_18.png" /><figcaption>Fig. 17. Schematic diagram of the experimental scenario</figcaption></figure>

<table><caption>TABLE III
EXPERIMENTAL RESULTS</caption><thead><tr><th rowspan="2">Group</th><th rowspan="2">Method</th><th rowspan="2">Source</th><th colspan="2">DOA (°)</th><th colspan="2">Range (cm)</th></tr><tr><th>Ground Truth</th><th>Estimated</th><th>Ground Truth</th><th>Estimated</th></tr></thead><tbody><tr><td rowspan="6">1</td><td rowspan="2">Proposed method</td><td>1</td><td>-10.00</td><td>-11.29</td><td>80.00</td><td>84.48</td></tr><tr><td>2</td><td>5.00</td><td>5.77</td><td>75.00</td><td>76.54</td></tr><tr><td rowspan="2">MLE</td><td>1</td><td>-10.00</td><td>-10.34</td><td>80.00</td><td>80.00</td></tr><tr><td>2</td><td>5.00</td><td>5.17</td><td>75.00</td><td>76.00</td></tr><tr><td rowspan="2">EML [24]</td><td>1</td><td>-10.00</td><td>-17.33</td><td>80.00</td><td>86.01</td></tr><tr><td>2</td><td>5.00</td><td>9.01</td><td>75.00</td><td>82.09</td></tr><tr><td rowspan="6">2</td><td rowspan="2">Proposed method</td><td>1</td><td>0.00</td><td>-0.59</td><td>80.00</td><td>82.00</td></tr><tr><td>2</td><td>-20.00</td><td>-22.26</td><td>75.00</td><td>80.79</td></tr><tr><td rowspan="2">MLE</td><td>1</td><td>0.00</td><td>0.00</td><td>80.00</td><td>80.00</td></tr><tr><td>2</td><td>-20.00</td><td>-21.20</td><td>75.00</td><td>75.00</td></tr><tr><td rowspan="2">EML [24]</td><td>1</td><td>0.00</td><td>-0.67</td><td>80.00</td><td>85.50</td></tr><tr><td>2</td><td>-20.00</td><td>-22.39</td><td>75.00</td><td>81.22</td></tr><tr><td rowspan="6">3</td><td rowspan="2">Proposed method</td><td>1</td><td>-15.00</td><td>-15.81</td><td>80.00</td><td>81.26</td></tr><tr><td>2</td><td>20.00</td><td>19.64</td><td>64.00</td><td>61.41</td></tr><tr><td rowspan="2">MLE</td><td>1</td><td>-15.00</td><td>-14.89</td><td>80.00</td><td>79.00</td></tr><tr><td>2</td><td>20.00</td><td>19.82</td><td>64.00</td><td>64.00</td></tr><tr><td rowspan="2">EML [24]</td><td>1</td><td>-15.00</td><td>-15.81</td><td>80.00</td><td>83.26</td></tr><tr><td>2</td><td>20.00</td><td>19.60</td><td>64.00</td><td>57.19</td></tr></tbody></table>

VII. CONCLUSION

In this paper, a near-field coherent source localization
method with high resolution and low complexity is proposed,
in which coarse and fine estimations are carried out through
a two-step process combining AOP and polynomial root-
findings. The coarse estimates of DOA and range param-
eters are first obtained by applying root-MUSIC and root-
Capon respectively, to the elements on the anti-diagonal of
the covariance matrix. Then, an iterative refinement process
is performed, which alternately uses the AOP method and
polynomial rooting to achieve the fine estimation. Simulations
and experiments confirm that the algorithm proposed in this
paper enables accurate localization of NF coherent sources
with improved resolution and reduced complexity. Further-
more, field experiments validate its effectiveness in practical
physical environments.

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<figure><img src="image_19.png" /></figure>

**Yuehe Ning** was born in 2002. She received her B.S. degree from Nanchang Hangkong University in 2024. She is currently pursuing the M.S. degree in electronic engineering at Nanjing University of Aeronautics and Astronautics, Nanjing, China.

Her research interests include signal source localization.

<figure><img src="image_20.png" /></figure>

**MENG SUN** (Senior Member, IEEE) received the B.S. degree from Northwest University, in 2010, and the M.S. degree from the South China University of Technology, China, in 2013, and the Ph.D. degree from the IETR Laboratory, Polytech Nantes (Nantes Université), France, in 2016. He is currently with the Department of Electronic Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His main research interests include signal processing, deep learning.

<figure><img src="image_21.png" /></figure>

**Jingjing Pan** Jingjing Pan (Member, IEEE) received the B.S. and M.S. degrees in cartography and geographic information science from East China Normal University, Shanghai, China, in 2012, and Beijing Normal University, Beijing, China, in 2015, and the Ph.D. degree from Polytech Nantes, Université de Nantes, Nantes, France, in 2019. She is an associate professor with the Electronic Engineering Department, Nanjing University of Aeronautics and Astronautics, Nanjing, China. Her research interests include array signal processing, ground penetrating radar, and nondestructive testing.

<figure><img src="image_22.png" /></figure>

**Yide Wang** received the B.S. degree in electrical engineering from the Beijing University of Post and Telecommunication, Beijing, China, in 1985, and the M.S. and Ph.D. degrees in signal processing and telecommunications from the University of Rennes, France, in 1986 and 1989, respectively.

He is currently a Professor with the Ecole Polytechnique de l'Université de Nantes. His research interests include array signal processing, spectral analysis, and mobile wireless communication systems.

<figure><img src="image_23.png" /></figure>

**Shaohui Lv** was born in 2000. He received the B.S. degree in information engineering from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2019, where he is currently pursuing the M.S. degree with the College of Electronic and Information Engineering.

His research interests include array signal processing and near-field source localization.