<figure><img src="image_1.png" /></figure>

HAL
open science

# Real-Valued DOA Estimation of Coprime Array via Toeplitz Construction-Based Spline Interpolation

Jingjing Pan, Qixuan Lu, Yide Wang, Meng Sun

► To cite this version:

Jingjing Pan, Qixuan Lu, Yide Wang, Meng Sun. Real-Valued DOA Estimation of Coprime Array via Toeplitz Construction-Based Spline Interpolation. IEEE Signal Processing Letters, 2026, pp.1-5. 10.1109/LSP.2026.3667810. hal-05529283

HAL Id: hal-05529283

https://hal.science/hal-05529283v1

Submitted on 27 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Real-Valued DOA Estimation of Coprime Array via Toeplitz Construction-Based Spline Interpolation

Jingjing Pan, Qixuan Lu, Yide Wang and Meng Sun

**Abstract**—Coprime array enables derivation of an extended array with the number of virtual elements beyond the number of physical sensors, resolving the underdetermined direction-of-arrival (DOA) estimation problem while mitigating mutual coupling effects. However, the virtual array is discontinuous and contains some holes. To fully exploit the information of the virtual array, conventional methods typically employ complex matrix completion (MC) to iteratively recover the data of the covariance matrix, resulting in high complexity, particularly with large-scale coprime arrays. In this letter, a real-valued DOA estimation via the construction of Toeplitz covariance matrix and spline interpolation is proposed. The algorithm complexity is reduced by real-valued Toeplitz matrix, and then the holes are filled by spline interpolation. The recovered matrix can be used for DOA estimation by conventional subspace approaches. The superior computational performance of our approach is validated through comparative simulations with established methods.

**Index Terms**—Direction of arrival (DOA), Toeplitz matrix, spline interpolation, coprime array, real-valued.

## I. INTRODUCTION

DIRECTION of arrival (DOA) estimation, a critical task in array signal processing, is widely employed in engineering practices such as microphone arrays, wireless communications, radar, and sonar [1], [2]. Leveraging their advantage in degrees of freedom (DOFs), sparse arrays [3], [4], [5], [6] enable underdetermined DOA estimation, i.e., scenarios with more sources than physical sensors, thereby outperforming a uniform linear array (ULA) of comparable size. Among various sparse array geometries, diverse configurations of coprime arrays [7], [8], [9], [10] offer distinct advantages, such as reduced mutual coupling and explicitly defined element placement.

The coprime array design is based on two interleaved ULAs, operating under a dual condition of coprimality between their respective number of sensors and inter-sensor spacings. Current DOA estimation techniques for such arrays primarily involve two methods: subarray decomposition method [11] and difference co-array method [12]. The subarray decomposition method operates by applying subspace algorithms independently to each subarray for angle estimation, and the common

angles estimated by the two subarrays are the desired DOAs. While this method achieves superior estimation accuracy, it fails to maximally utilize the high DOFs capabilities provided by the coprime array. The alternative approach utilizes the vectorization of the covariance matrix to derive the equivalent single-snapshot data that represents the signal received by the virtual array. It should be noted in [13] that the virtual array exhibits many holes, indicating its discontinuity in the formation of a complete ULA. Conventional method uses the continuous positions of virtual array to estimate DOA [14]. However, the information of the virtual array in non-consecutive positions will be lost, resulting in loss of DOFs and performance.

In an effort to resolve the incomplete utilization of virtual array information, there are usually two kinds of methods. One is to use coprime array motion [15] or coprime configuration design [16] to extend the maximum continuous positions of the virtual array. The other is to use matrix completion (MC) to fill the vacant positions. Among them, augmented Lagrange multiplier (ALM) [17], singular value thresholding (SVT) [18] and alternating direction method of multipliers (ADMM) [19] use singular value decomposition and iteration to recover holes in the virtual arrays. Additionally, nuclear norm minimization (NNM) [20] recovers the matrix by minimizing the sum of all singular values. Later, a gridless algorithm based on rank minimization atomic norm minimization (Rank-Min ANM) [21] is proposed to fill the matrix. Nevertheless, these algorithms are characterized by high computational complexity owing to their inherent reliance on iterative computations and convex optimization procedures.

This letter introduces an efficient DOA estimation method utilizing covariance matrix reconstruction, where holes are addressed through real-valued spline interpolation [22]. Unlike existing virtual array interpolation methods for coprime arrays which rely on computationally intensive processes like iterative matrix completion (e.g., SVT, ALM, ADMM) or convex optimization (e.g., ANM, NNM), our approach reformulates the matrix completion problem as a real-valued curve interpolation task, which dramatically reduces complexity while maintaining accuracy.

## II. SIGNAL MODEL

A general configuration of coprime array is considered with the sensor positions defined by

$$ S = \{nMd|n = 0, 1, \dots, N-1\} \\ \cup \{mNd|m = 0, 1, \dots, M-1\}, \qquad (1) $$

This work is supported in part by the National Natural Science Foundation of China under Grant 62427801, the Fundamental Research Funds for the Central Universities under Grant NS2024024, and the Jiangsu Provincial Young Science and Technology Talent Support Program under grant JSTJ-2024-530.

Jingjing Pan, Qixuan Lu, and Meng Sun are with the College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: mengsun@nuaa.edu.cn)

Yide Wang is with Nantes Universite, CNRS, IETR, UMR 6164, F-44000 Nantes, France.
where $d = \lambda/2$ indicates the standard space between two adjacent elements; $\lambda$ is the wavelength, and $M$ and $N$ denote a pair of prime integers ($M < N$). The physical array configuration consists of $M + N - 1$ sensors. The model analyzes $K$ uncorrelated narrowband signals originating from far-field sources arriving at different angles of $\theta_k$, $k = 1, 2, \dots, K$. The received signal can be formulated as

$$ \mathbf{x}(t) = \mathbf{A}\mathbf{s}(t) + \mathbf{n}(t), \quad (2) $$

with:

* $\mathbf{x}(t) = [x_1(t), x_2(t), \dots, x_{M+N-1}(t)]^T$ is the $((M + N - 1) \times 1)$ received signal vector; $(\cdot)^T$ represents the transpose operation;

* $\mathbf{A} = [\mathbf{a}(\theta_1), \mathbf{a}(\theta_2), \dots, \mathbf{a}(\theta_K)]$ is the $((M + N - 1) \times K)$ directional matrix;

* $\mathbf{a}(\theta_k) = [\exp(-j2\pi d_1 \sin \theta_k / \lambda), \exp(-j2\pi d_2 \sin \theta_k / \lambda), \dots, \exp(-j2\pi d_{M+N-1} \sin \theta_k / \lambda)]^T$;

* $\mathbf{s}(t) = [s_1(t), s_2(t), \dots, s_K(t)]^T$ is the $(K \times 1)$ source vector;

* $\mathbf{n}(t) = [n_1(t), n_2(t), \dots, n_{M+N-1}(t)]^T$ is the $((M + N - 1) \times 1)$ independent identically distributed additive Gaussian white noise vector, with zero mean and covariance matrix $\sigma_n^2 \mathbf{I}$.

The covariance matrix of the received signal is then given by

$$ \mathbf{R} = E(\mathbf{x}(t)\mathbf{x}^H(t)) = \mathbf{A}\mathbf{R}_s\mathbf{A}^H + \sigma_n^2\mathbf{I}, \quad (3) $$

where $E(\cdot)$ represents the mathematical expectation; $\mathbf{R}_s$ is the $(K \times K)$ covariance matrix of sources; $\mathbf{I}$ is the $((M+N-1) \times (M+N-1))$ identity matrix; $(\cdot)^H$ represents the conjugate transpose.

### III. TOEPLITZ CONSTRUCTION OF VIRTUAL ARRAY

By transforming $\mathbf{R}$ into a vector, the virtual array can be derived as

$$ \mathbf{v} = \mathrm{vec}(\mathbf{R}) = \mathbf{B}\mathbf{p} + \sigma_n^2\mathbf{i}, \quad (4) $$

where $\mathbf{B} = [\mathbf{a}^*(\theta_1) \otimes \mathbf{a}(\theta_1), \dots, \mathbf{a}^*(\theta_K) \otimes \mathbf{a}(\theta_K)]$ is the directional matrix of virtual array. Symbols $(\cdot)^*$ and $\otimes$ represent the complex conjugate and Kronecker product, respectively; $\mathrm{vec}(\cdot)$ represents matrix vectorization; $\mathbf{p} = [\sigma_1^2, \sigma_2^2, \dots, \sigma_K^2]^T$ is the power of $K$ signal sources and $\mathbf{i} = \mathrm{vec}(\mathbf{I})$. The spatial distribution of equivalent virtual sensors can be written as

$$ \mathbb{V} = \{\pm(Nm - Mn)d, 0 \le m \le M-1, 0 \le n \le N-1\}. \quad (5) $$

It is worth noting that some virtual sensors appear repeatedly, and the corresponding positions of each virtual sensor are disordered. The weight function $w(p)$ denotes the number of virtual sensor elements at position $p$, where $p$ represents the position of the virtual sensor, $p = -M(N-1), \dots, M(N-1)$. The redundant element integration matrix $\mathbf{D}$ is characterized by row vectors formulated as

$$ \mathbf{D}(p + M(N - 1) + 1, :) = \mathrm{vec}(\mathbf{H}^p), \quad (6) $$

where each element of $\mathbf{H}^p$ can be expressed as

$$ \mathbf{H}_{ij}^p = \begin{cases} \frac{1}{w(p)}, & S_i - S_j = pd, \\ 0, & \text{else,} \end{cases} \quad (7) $$

where $S_i$ represents the $i$-th element of the set $S$, $i, j = 1, 2, \dots, M+N-1$. The de-redundant signal can be expressed as

$$ \bar{\mathbf{z}} = \mathbf{D}\mathbf{v}. \quad (8) $$

To fully exploit the complete virtual array aperture, we perform hole-filling interpolation to construct an equivalent virtual ULA. Since the information of the holes is unknown, it is initialized to zero, from which the virtual ULA signal can be obtained as

$$ [\mathbf{z}]_i = \begin{cases} [\bar{\mathbf{z}}]_i, & i \in \mathbb{V}, \\ 0, & i \in \mathbb{I} \setminus \mathbb{V}, \end{cases} \quad (9) $$

where $\mathbb{I} = \{-Q+1, -Q+2, \dots, Q-1\}$ and $[\cdot]_i$ denotes the receiving data at position $id$. $\mathbb{I} \setminus \mathbb{V}$ represents the complement of $\mathbb{V}$ in $\mathbb{I}$. We divide $\mathbb{I}$ into $Q = M(N-1)+1$ intersecting virtual sub-arrays, each of which contains $Q$ sensors. $\mathbf{r}_q = [[\mathbf{z}]_{1-q}, [\mathbf{z}]_{2-q}, \dots, [\mathbf{z}]_{Q-q}]^T$, $q = 1, 2, \dots, Q$ is the received signal vector of each virtual subarray.

Construct a virtual matrix $\mathbf{Z} = [\mathbf{r}_1, \mathbf{r}_2, \dots, \mathbf{r}_Q]$. Considering the structure of matrix $\mathbf{Z}$ is centrosymmetric conjugate, it can be directly constructed as

$$ \mathbf{Z} = \mathbf{T}(\mathbf{r}_1) = \begin{bmatrix} [\mathbf{z}]_0 & [\mathbf{z}]_{-1} & \cdots & [\mathbf{z}]_{-Q+1} \\ [\mathbf{z}]_1 & [\mathbf{z}]_0 & \cdots & [\mathbf{z}]_{-Q+2} \\ \vdots & \vdots & \ddots & \vdots \\ [\mathbf{z}]_{Q-1} & [\mathbf{z}]_{Q-2} & \cdots & [\mathbf{z}]_0 \end{bmatrix}, \quad (10) $$

where $\mathbf{T}(\mathbf{r}_1)$ is a Toeplitz matrix with $\mathbf{r}_1$ as its first column. The matrix $\mathbf{Z}$ contains no rows consisting entirely of zeros. Therefore, MC algorithm or interpolation method can be used to estimate the data.

### IV. REAL-VALUED SPLINE INTERPOLATION FOR DOA ESTIMATION

This part develops a novel DOA estimator that employs real-valued spline interpolation techniques on the matrix $\mathbf{Z}$ which is Toeplitz and Hermitian. In order to reduce the complexity and facilitate subsequent interpolation operations, the complex matrix $\mathbf{Z}$ is transformed into the real matrix $\mathbf{Y}$. Following the framework established in [23], the unitary transformation matrix $\mathbf{U}_Q$ is characterized under two scenarios:

1) When $Q$ is even, we have

$$ \mathbf{U}_Q = \frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I}_1 & \mathbf{J} \\ j\mathbf{J} & -j\mathbf{I}_1 \end{bmatrix}, \quad (11) $$

where $\mathbf{I}_1$ is the $(Q/2 \times Q/2)$ identity matrix and $\mathbf{J}$ is the $(Q/2 \times Q/2)$ exchange matrix characterized by ones along the secondary diagonal and zeros otherwise.

2) When $Q$ is odd, we have

$$ \mathbf{U}_Q = \frac{1}{\sqrt{2}} \begin{bmatrix} \mathbf{I}_1 & \mathbf{O} & \mathbf{J} \\ \mathbf{O} & \sqrt{2} & \mathbf{O} \\ j\mathbf{J} & \mathbf{O} & -j\mathbf{I}_1 \end{bmatrix}, \quad (12) $$

where $\mathbf{I}_1$ is the $((Q-1)/2 \times (Q-1)/2)$ identity matrix and $\mathbf{J}$ is the $((Q-1)/2 \times (Q-1)/2)$ exchange matrix.

Through algebraic manipulation, the matrix $\mathbf{Y} = \mathbf{U}_Q \mathbf{Z} \mathbf{U}_Q^H$ can be proved to be a real matrix. We define $\Gamma_i$ and $\bar{\Gamma}_i$,
respectively, as the set of positions of non-zero elements and zero elements in the *i*-th row of the matrix $\mathbf{Y}$, $i = 1, 2, \dots, Q$. Therefore, we can use classical spline interpolation [16] to estimate the zero element of each row. In addition, considering the symmetric property of the matrix $\mathbf{Y}$, the computational efficiency is enhanced by limiting interpolation operations to the first $\lceil Q/2 \rceil$ rows, where $\lceil \cdot \rceil$ means round up. The details are as follows: for each row, in order to estimate $\mathbf{Y}(\bar{\Gamma}_i)$, the spline function $S_i(x)$ is constructed with $|\Gamma_i|$ elements and $|\Gamma_i| - 1$ intervals. $S_i(x)$ should obey the following conditions:

1) $S_{ij}(x)$ is a cubic polynomial in the $j$-th interval, $j = 1, 2, \dots, |\Gamma_i| - 1$.

2) $S_i(x)$ traverses all known data points.

3) The first and second derivatives at the junction are the same, $S'_{ij}(\Gamma_i(j+1)) = S'_{i(j+1)}(\Gamma_i(j+1))$, $S''_{ij}(\Gamma_i(j+1)) = S''_{i(j+1)}(\Gamma_i(j+1))$, $j = 1, 2, \dots, |\Gamma_i| - 2$.

4) $S'''_{i1}(\Gamma_i(2)) = S''_{i2}(\Gamma_i(2))$, $S'''_{i(|\Gamma_i|-2)}(\Gamma_i(|\Gamma_i| - 1)) = S'''_{i(|\Gamma_i|-1)}(\Gamma_i(|\Gamma_i| - 1))$.

General spline interpolation establishes cubic equations. Therefore, for each row, $S_{ij}(x)$ has the form

$$S_{ij}(x) = a_{ij}x^3 + b_{ij}x^2 + c_{ij}x + d_{ij}, \quad (13)$$

where $a_{ij}, b_{ij}, c_{ij}, d_{ij}$ are unknowns. By using constraints, $4|\Gamma_i|$ unknowns in $S_i(x)$ admit a unique solution. Subsequently, we can reconstruct the spline function $S_i(x)$. Accordingly, by using the non-zero values $\mathbf{Y}(\bar{\Gamma}_i)$ of each row, the zero values $\mathbf{Y}(\bar{\Gamma}_i)$ of each row are interpolated with cubic spline. The interpolated real matrix can be expressed as $\mathbf{Y}_{nz}$. Through eigenvalue decomposition (EVD), the eigenvectors corresponding to the $K$ largest eigenvalues span the signal subspace $\mathbf{E}_s$, whereas those associated with the subsequent $Q - K$ eigenvalues generate the noise subspace $\mathbf{E}_n$. The relationship between the real matrices $\mathbf{E}_s, \mathbf{E}_n$ and the complex matrices $\mathbf{E}'_s, \mathbf{E}'_n$ is

$$\mathbf{E}'_s = \mathbf{U}_Q^H \mathbf{E}_s, \quad (14)$$

$$\mathbf{E}'_n = \mathbf{U}_Q^H \mathbf{E}_n. \quad (15)$$

Consequently, we can use traditional subspace algorithms such as MUSIC for DOA estimation. The overarching framework of our algorithm consists of the following essential steps:

1) Estimate the covariance matrix $\hat{\mathbf{R}}$;

2) Vectorize the matrix $\hat{\mathbf{R}}$ and eliminate redundant elements to derive the vector $\bar{\mathbf{z}}$;

3) Make the vector $\mathbf{z}$ Toeplitz and Hermitian, ded by the matrix $\mathbf{Z}$;

4) Use the unitary matrix to build the real matrix and spline interpolation for each row of the matrix to obtain $\mathbf{Y}_{nz}$;

5) Apply EVD to obtain $\mathbf{E}'_s, \mathbf{E}'_n$ and use MUSIC to estimate DOA.

**Remark 1:** The proposed method recovers holes in Hermitian matrix through real-valued transformation and spline interpolation, demonstrating strong potential for applications such as DOA estimation in wideband scenarios, hole filling in sparse array structures.

Table I presents a comparative analysis of computational complexity among various algorithms. In each iteration, SVD

<table><caption>TABLE I
ALGORITHM COMPLEXITY COMPARISON</caption><thead><tr><th>Method</th><th>Algorithm complexity</th></tr></thead><tbody><tr><td>SVT, ALM, ADMM</td><td>O(2b(M(N-1)+1)<sup>3</sup>)</td></tr><tr><td>Rank-Min ANM</td><td>O(2(M(N-1)+1)<sup>3</sup> + (M(N-1)+1)<sup>4.5</sup>)</td></tr><tr><td>NNM</td><td>O((M(N-1)+1)<sup>4.5</sup>)</td></tr><tr><td>Proposed</td><td>O((M(N-1)+1)<sup>2</sup>/8 + 2(M(N-1)+1)<sup>3</sup>)</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Fig. 1. Algorithm complexity comparison vs. the number of physical sensors.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. PSD of the proposed method, $M = 4$, $N = 5$, $K = 15$.</figcaption></figure>

and the related matrix multiplication operation in SVT, ALM, and ANMM are with complexity $O(2(M(N-1)+1)^3)$. Thus, their overall complexity is $O(2b(M(N-1)+1)^3)$, where $b$ is the number of iterations. Moreover, Rank-Min ANM adopts CVX toolbox with complexity $O((M(N-1)+1)^{4.5})$, and its overall complexity along with SVD and matrix multiplication operation would be $O(2(M(N-1)+1)^3 + (M(N-1)+1)^{4.5})$. In addition, NNM only uses the CVX toolbox with complexity $O((M(N-1)+1)^{4.5})$. Implementation of our method necessitates applying spline interpolation to every individual matrix row. Since the matrix is symmetric, only the first half of the rows need to be calculated. At the same time, it is a real matrix, the algorithm complexity is 4 times less than complex operation. In addition, the operation of converting the complex matrix and the real matrix into each other twice requires complexity with $O(2(M(N-1)+1)^3)$. Hence, the algorithm complexity of our method is $O((M(N-1)+1)^2/8 + 2(M(N-1)+1)^3)$. As illustrated in Fig. 1, the algorithm complexity of the proposed approach is significantly lower than that of compared algorithms.

## V. SIMULATION RESULTS

We consider a coprime array with 8 physical sensors, that is, $M = 4$, $N = 5$. Three numerical results are analyzed to
<figure><img src="image_5.png" /><figcaption>Fig. 3. RMSE comparison. (a) RMSE vs. SNR, K = 2, 500 snapshots; (b) RMSE vs. the number of snapshots, K = 2, SNR = 5 dB; (c) RMSE vs. SNR, K = 15, 500 snapshots; (d) RMSE vs. the number of snapshots, K = 15, SNR = 5 dB.</figcaption></figure>

assess the effectiveness of the developed approach.

In the first simulation, the spatial spectrum of the proposed method is evaluated with 15 incoherent unit power sources uniformly distributed over [−50°, 50°] under a fixed condition of 500 snapshots and SNR = 20 dB. As illustrated in Fig. 2, the proposed method shows enlarged DOF and high accuracy in underdetermined scenario.

In the second simulation, the statistical performance of our algorithm is analyzed in relation to both the number of snapshots and the SNR in scenarios of 2 sources. Here, $\theta_1 = -6°$, $\theta_2 = 7°$. The CRB [24] is also included as a benchmark in the comparison. The assessment metric employs root mean square error (RMSE) for DOA estimation accuracy, mathematically represented as

$$ \text{RMSE} = \sqrt{\frac{1}{KL} \sum_{k=1}^{K} \sum_{l=1}^{L} (\hat{\theta}_{kl} - \theta_k)^2}, \quad (16) $$

where L is the total trial count; during the l-th trial, $\hat{\theta}_{kl}$ represents the estimated DOA for the k-th source, while $\theta_k$ is its true direction. With 500 snapshots considered and SNR ranging from -15 dB to 15 dB, Fig. 3(a) illustrates that the RMSEs of all employed methods exhibit a decreasing trend as the SNR increases. When the SNR exceeds 10 dB, all methods tend to have similar performance and our algorithm has the best estimation accuracy. Fig. 3(b) presents the results obtained at fixed SNR of 5 dB but with different numbers of snapshots. As the number of snapshots increases, the accuracy of all methods is improving while the proposed method has small RMSE and outperforms the others.

The final simulation validates the enhanced effectiveness of our method in scenarios where the number of sources surpasses that of the physical sensors. The other settings are the same as second simulation except that 15 sources are uniformly distributed over [−50°, 50°]. As illustrated in Fig. 3(c), NNM fails at different SNRs, which might be attributed to its inherent inability to exploit the full non-uniform virtual array for covariance matrix reconstruction. Besides, the other methods show decreased RMSE as SNR increases while our method outperforms the other competitors, especially when SNR exceeds -5 dB. Then, the performance of our method is further analyzed by varying the number of snapshots. Fig. 3(d) indicates that our method demonstrates superior performance with lower error metrics compared to the others, which is consistent with Fig. 3(c). This improved precision is likely attributable to the integration of both real and imaginary components in the real-valued approach.

## VI. CONCLUSION

This letter proposes an efficient DOA estimation algorithm for coprime array based on real-valued Toeplitz matrix construction and spline interpolation. By converting the covariance matrix into real-valued form and using cubic spline interpolation to recover the holes in the virtual array, it achieves high-precision direction of arrival estimation while reducing computational complexity. Simulation results demonstrate that the proposed method maintains superior performance even in underdetermined scenarios, with computational efficiency significantly outperforming existing iterative and convex optimization-based matrix completion algorithms.
REFERENCES

[1] W. Manamperi, T. D. Abhayapala, J. Zhang, and P. N. Samarasinghe, "Drone audition: Sound source localization using on-board microphones," *IEEE/ACM Transactions on Audio, Speech, and Language Processing*, vol. 30, pp. 508-519, 2022.

[2] A. Gaber and A. Omar, "A study of wireless indoor positioning based on joint TDOA and DOA estimation using 2-D matrix pencil algorithms and ieee 802.11 ac," *IEEE Transactions on Wireless Communications*, vol. 14, no. 5, pp. 2440–2454, 2014.

[3] A. Moffet, "Minimum-redundancy linear arrays," *IEEE Transactions on antennas and propagation*, vol. 16, no. 2, pp. 172–175, 1968.

[4] P. Pal and P. P. Vaidyanathan, "Nested arrays: A novel approach to array processing with enhanced degrees of freedom," *IEEE Transactions on Signal Processing*, vol. 58, no. 8, pp. 4167–4181, 2010.

[5] P. P. Vaidyanathan and P. Pal, "Sparse sensing with co-prime samplers and arrays," *IEEE Transactions on Signal Processing*, vol. 59, no. 2, pp. 573–586, 2010.

[6] P. Vaidyanathan and P. Pal, "Theory of sparse coprime sensing in multiple dimensions," *IEEE Transactions on Signal Processing*, vol. 59, no. 8, pp. 3592–3608, 2011.

[7] Y. Zhang, J. Shi, G. Zheng, H. Zhou, Y. Song, and G. Hu, "Augmented coprime array design via hole analysis for direction of arrival estimation," *IEEE Transactions on Aerospace and Electronic Systems*, 2024.

[8] W. Zheng, X. Zhang, Y. Wang, J. Shen, and B. Champagne, "Padded coprime arrays for improved DOA estimation: Exploiting hole representation and filling strategies," *IEEE Transactions on Signal Processing*, vol. 68, pp. 4597–4611, 2020.

[9] X. Wang and X. Wang, "DOA estimation with k-times extended co-prime arrays," in *2017 51st Asilomar Conference on Signals, Systems, and Computers*, pp. 1183–1187, IEEE, 2017.

[10] X. Lai, X. Zhang, W. Zheng, J. Li, and F. Zhou, "Fragmented coprime arrays with optimal inter subarray spacing for DOA estimation: Increased DOF and reduced mutual coupling," *Signal Processing*, vol. 215, p. 109273, 2024.

[11] C. Zhou, Z. Shi, Y. Gu, and X. Shen, "Decom: DOA estimation with combined MUSIC for coprime array," in *2013 International Conference on Wireless Communications and Signal Processing*, pp. 1–5, IEEE, 2013.

[12] P. Pal and P. P. Vaidyanathan, "Coprime sampling and the MUSIC algorithm," in *2011 Digital signal processing and signal processing education meeting (DSP/SPE)*, pp. 289–294, IEEE, 2011.

[13] E. BouDaher, Y. Jia, F. Ahmad, and M. G. Amin, "Multi-frequency co-prime arrays for high-resolution direction-of-arrival estimation," *IEEE Transactions on Signal Processing*, vol. 63, no. 14, pp. 3797–3808, 2015.

[14] J. Pan, M. Sun, X. Dong, Y. Wang, and X. Zhang, "Enhanced DOA estimation with co-prime array in the scenario of impulsive noise: A pseudo snapshot augmentation perspective," *IEEE Transactions on Vehicular Technology*, vol. 72, no. 9, pp. 11603–11616, 2023.

[15] G. Qin, M. G. Amin, and Y. D. Zhang, "DOA estimation exploiting sparse array motions," *IEEE Transactions on Signal Processing*, vol. 67, no. 11, pp. 3013–3027, 2019.

[16] W. Si, F. Zeng, Z. Qu, and Z. Peng, "Two-dimensional DOA estimation via a novel sparse array consisting of coprime and nested subarrays," *IEEE Communications Letters*, vol. 24, no. 6, pp. 1266–1270, 2020.

[17] Z. Lin, M. Chen, and Y. Ma, "The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices," *arXiv preprint arXiv:1009.5055*, 2010.

[18] J.-F. Cai, E. J. Candès, and Z. Shen, "A singular value thresholding algorithm for matrix completion," *SIAM Journal on optimization*, vol. 20, no. 4, pp. 1956–1982, 2010.

[19] S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, et al., "Distributed optimization and statistical learning via the alternating direction method of multipliers," *Foundations and Trends® in Machine learning*, vol. 3, no. 1, pp. 1–122, 2011.

[20] C.-L. Liu, P. Vaidyanathan, and P. Pal, "Coprime coarray interpolation for DOA estimation via nuclear norm minimization," in *2016 IEEE International Symposium on Circuits and Systems (ISCAS)*, pp. 2639–2642, IEEE, 2016.

[21] S. Liu, Z. Mao, Y. D. Zhang, and Y. Huang, "Rank minimization-based toeplitz reconstruction for DOA estimation using coprime array," *IEEE Communications Letters*, vol. 25, no. 7, pp. 2265–2269, 2021.

[22] M. Unser, "Splines: A perfect fit for signal and image processing," *IEEE Signal processing magazine*, vol. 16, no. 6, pp. 22–38, 2002.

[23] K.-C. Huarng and C.-C. Yeh, "A unitary transformation method for angle-of-arrival estimation," *IEEE Transactions on Signal Processing*, vol. 39, no. 4, pp. 975–977, 1991.

[24] P. Stoica, E. G. Larsson, and A. B. Gershman, "The stochastic CRB for array processing: A textbook derivation," *IEEE Signal Processing Letters*, vol. 8, no. 5, pp. 148–150, 2001.