An Effective Off-Grid DOA Estimation Algorithm Using Difference Coarrays with Limited Snapshots

Abstract

A significant advantage of off-grid direction-of-arrival (DOA) estimation algorithms using difference coarrays is their ability to resolve more sources than the number of physical sensors. Current coarray-based off-grid DOA estimation algorithms experience a significant decline in estimation accuracy with limited snapshots. Moreover, most existing DOA estimation techniques exhibit a high computational complexity, limiting their practical implementation in real-time systems. To address these limitations, in this work, we propose a novel coarray-based off-grid DOA estimation algorithm that achieves a computationally efficient performance while maintaining a high estimation accuracy under snapshot-constrained conditions. The proposed algorithm first performs DOA estimation through coarray-augmented spatial smoothing multiple signal classification (SS-MUSIC), followed by noise suppression via multiplication with a constructed selection matrix. The off-grid angular deviations are sequentially refined based on the iterative correction mechanism. The disadvantage of a large number of snapshots requirement is overcome thanks to the combination of noise elimination and sequential angle refinement. Theoretical performance bounds are established through Cramér–Rao bound (CRB) analysis, while comprehensive simulations validate the estimation accuracy of the proposed algorithm and the robustness in off-grid scenarios.

1. Introduction

Direction-of-arrival (DOA) estimation has emerged as a critical signal processing technique with extensive applications across multiple domains, such as wireless communication, radar target detections, and satellite navigation, [1,2,3,4,5,6]. Conventional subspace-based approaches, like multiple signal classification (MUSIC) and estimation of the signal parameters via rotational invariance techniques (ESPRIT), perform DOA estimation by applying eigendecomposition to the array covariance matrix, leveraging the orthogonality between the signal and noise subspaces [7,8]. Recently, phase interferometry-based digital synchronous architectures and adaptive Direct Digital Synthesizer Phase-Locked Loop beam steering techniques have been proposed for DOA estimation [9,10]. Although these methods can accuratly estimate the DOAs of sources with low computational complexity, they are limited to resolving fewer sources than the number of sensors.Alternately, various methods have been developed to estimate the DOAs of more sources than the number of sensors by utilizing non-uniform linear arrays (NULAs) [11]. The difference coarray provides expanded degrees-of-freedom (DOF), enabling an N element array to resolve more than N sources [12,13].

One representative coarray-based approach is the SS-MUSIC algorithm, which employs an augmented sample covariance matrix and requires a large number of snapshots [14,15]. The SS-MUSIC algorithm relies on an on-grid assumption, where true DOAs are presumed to lie precisely on predefined grid points. This basis mismatch inherently limits estimation accuracy in practical off-grid scenarios. An algorithm combining OptSpace and coarray ESPRIT was developed to perform DOA estimation effectively [16]. The OptSpace algorithm has been utilized to fill the midding elements in the coarray domain, effectively mapping the steering matrix of NLAs onto that of ULAs. This enables the application of ULA-based methods, like ESPRIT, within the coarray domain to achieve accurate DOA estimation. A source localization technique called coarray MUSIC-Group delay was developed in [17], utilizing the high-resolution properties of the group delay in the phase spectrum of the coarray MUSIC algorithm.A computationally efficient adaptive method was developed in [18], utilizing the least mean square (LMS) principle to perform coarray-based DOA estimation. Contemporary NULA off-grid estimators demonstrate superior source resolvability, with techniques like search-based methods and Enhanced Principal-Singular-Vector Utilization for Modal Analysis (EPUMA) successfully identifying more sources than the number of available array elements [19,20]. However, when multiple target within the field of view change rapidly, the number of data snapshot that can be collected becomes limited, intensifying the problem of limited snapshots. Under these conditions, the above-mentioned methods perform poorly due to the significant corruption of the signal and noise subspaces.

Coarray-based methodology suffers from significant computational burdens, representing a major practical constraint [21,22,23,24,25,26,27,28,29]. Recently, numerous off-grid estimation techniques have been proposed, including sparse recovery, sparse Bayesian learning (SBL), and maximum-likelihood-estimation (MLE)-based methods [21,22,23]. Our prior research introduced an advanced coarray processing approach that incorporates Bayesian inference for enhanced off-grid DOA estimation [24]. These methods can overcome the off-grid problem. However, the use of dense sampling grids for DOA search in these approaches imposes a significant computational burden. A Nuclear-Norm-Minimization-based MUSIC (NNM-MUSIC) algorithm was developed in [25] for DOA estimation using a coprime electromagnetic vector sensor (EMVS) array, leveraging coarray interpolation techniques. An advanced two-dimensional (2D) parameter estimation framework that combines virtual array interpolation techniques with coprime EMVS array processing to simultaneously estimate both DOA and polarization states was presented [26]. It is worth noting that an advantage of these approaches is their gridless nature, which avoids the requirement for dense sampling grids. However, they usually require computationally expensive steps, such as $l_1$-norm minimization, NNM minimization, and the convex semi-definite programming (SDP) solving problem.

A computationally efficient coarray-based approach for off-grid DOA estimation that achieves accurate results with minimal snapshot requirements is proposed. The main contributions of this paper are threefold: (i) A noise elimination technique is proposed, in which a selection matrix is left-multiplied to reduce noise effects in DOA estimation. (ii) An iterative correction method for angle and power estimation is developed. In each iteration, the target signal is isolated by removing other signals from the noise cancellation data, and the parameters are refined using an alternating minimization framework to improve the estimation performance.(iii) The CRB is theoretically derived for a deterministically orthogonal signal model, providing a rigorous performance metric for the proposed estimator. Comprehensive simulations are conducted to validate the enhanced estimation accuracy and robustness of the proposed approach compared to state-of-the-art techniques.

This paper proceeds with the following structure. Section 2 formulates the coarray-based signal model for off-grid DOA estimation, providing mathematical foundations for the subsequent algorithm. Section 3 introduces the proposed iterative correction method. The theoretical analysis is given in Section 4, including the derivation of the CRB for the deterministically orthogonal signal model. Section 5 presents comprehensive simulation results validating the proposed method, while Section 6 provides concluding remarks and outlines promising directions for future research.

2. Signal Model

The linear array element positions are indexed by,

$$\mathbf{S}=\{x_i,i=1,2,\dots ,N\},$$

(1)

where $x_i=c_{i}d$ denote the sensor positions. $d=\lambda /2$ represents the unit inter-element spacing and $c_i$ are the integer indices. The corresponding difference coarray comprises the set of all possible inter-element spacing for arbitrary sensor pairs [30].

Assume there are K far-field, narrow-band sources with incident angles $\mathit{\theta }=[{\theta }_1,\dots ,{\theta }_K]$. The array output at time instant t is given by

$$\mathbf{y}\left(t\right)=\mathbf{A}\mathbf{s}\left(t\right)+\mathbf{n}\left(t\right),t=1,\dots ,T,$$

(2)

where $\mathbf{s}\left(t\right)={[s_1\left(t\right),\cdots ,s_K\left(t\right)]}^T$ is the source signal, assumed to be deterministic and orthogonal, and $\mathbf{n}\left(t\right)={[n_1\left(t\right),\dots ,n_N\left(t\right)]}^T$ is zero-mean complex white Gaussian noise. The array manifold matrix $\mathbf{A}=[\mathbf{a}\left({\theta }_1\right),\dots ,\mathbf{a}\left({\theta }_K\right)]$ comprises steering vectors $\mathbf{a}\left({\theta }_k\right)=[e^{j{c}_1\pi sin{\theta }_k},$$\dots ,e^{j{c}_N\pi sin{\theta }_k}{]}^T,k=1,\dots ,K$. Thus, the covariance matrix $\mathbf{R}$ is written as

$$\mathbf{R}=E\left[\mathbf{y}\left(t\right){\mathbf{y}}^H\left(t\right)\right]=\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H+{\sigma }_n^2\mathbf{I}.$$

(3)

Here, the diagonal matrix ${\mathbf{R}}_s=diag\{{\sigma }_1^2,\cdots ,{\sigma }_K^2\}$ is the source covariance matrix, where the diagonal entries correspond to the power of each source signal. ${\sigma }_n^2$ represents the noise power. ${\mathbf{A}}^H$ denotes the conjugate transpose of the matrix $\mathbf{A}$.

Then, the covariance matrix can be vectorized and calculated as,

$$\tilde{\mathbf{y}}=\mathrm{vec}\left(\mathbf{R}\right)=\tilde{\mathbf{A}}\mathbf{p}+{\sigma }_n^2\tilde{\mathbf{i}},$$

(4)

where $\mathbf{p}={[{\sigma }_1^2,{\sigma }_2^2,\cdots ,{\sigma }_K^2]}^T,\tilde{\mathbf{i}}=\mathrm{vec}\left(\mathbf{I}\right)$ and $\tilde{\mathbf{A}}=[\tilde{\mathbf{a}}\left({\theta }_1\right),\cdots ,$$\tilde{\mathbf{a}}\left({\theta }_K\right)]$. Each column vector $\tilde{\mathbf{a}}\left({\theta }_k\right)$ is expressed as,

$$\tilde{\mathbf{a}}\left({\theta }_k\right)={\mathbf{a}}^{*}\left({\theta }_k\right)\otimes \mathbf{a}\left({\theta }_k\right).$$

(5)

Here, ${\mathbf{a}}^{*}\left({\theta }_k\right)$ refers to the conjugate of $\mathbf{a}\left({\theta }_k\right)$ and ⊗ stands for the Kronecker product.

By removing the repeated elements, we can obtain,

$$\widehat{\mathbf{y}}=\mathbf{J}\tilde{\mathbf{y}}=\overline{\mathbf{A}}\mathbf{p}+{\sigma }_n^2\overline{\mathbf{i}},$$

(6)

where the array manifold matrix is $\overline{\mathbf{A}}=[\overline{\mathbf{a}}\left({\theta }_1\right),\cdots ,\overline{\mathbf{a}}\left({\theta }_k\right)]$. The length of the coarray is $N_c$ and the element index is denoted as ${\overline{c}}_i$. The central entry of the vector $\overline{\mathbf{i}}$ is equal to one and the other elements are equal to zero. The matrix $\mathbf{J}$ is defined as

$$\mathbf{J}={\mathbf{\Psi }\mathbf{G}}^{-1}{(\mathbf{\Phi }\otimes \mathbf{\Phi })}^{-1},$$

(7)

where $\mathbf{\Psi }$ is a $N\times (c_N+1)$ matrix, and the $({\overline{c}}_i+c_N+1)$-th entry of the i-th row is equal to one. $\mathbf{\Phi }$ is a $N_c\times (2c_N+1)$ matrix with the $(c_i+1)$-th element of the i-th row being equal to one. The matrix $\mathbf{G}$ contains an identity matrix positioned along its anti-diagonal.

Under limited snapshot conditions, the received data are modeled as,

$$\overline{\mathbf{y}}=\mathbf{J}\mathrm{vec}\left(\widehat{\mathbf{R}}\right)=\widehat{\mathbf{y}}+\overline{\mathit{ϵ}},$$

(8)

where the sample covariance matrix $\widehat{\mathbf{R}}=(1/T){\sum }_{t=1}^T\mathbf{y}\left(t\right){\mathbf{y}}^H\left(t\right)$. $\overline{\mathit{ϵ}}$ characterizes the estimation error resulting from the limited number of snapshots.

To accurately estimate the off-grid DOAs using the coarray data $\overline{\mathbf{y}}$, we propose a two-stage algorithm including initial DOA estimation and fine DOA estimation in the following section.

3. The Proposed Method

3.1. Initial DOA Estimation

Preliminary DOA estimates can be computed using the coarray-based SS-MUSIC algorithm. This approach addresses the inherent signal coherence artifacts induced by the covariance augmentation process in coarray domain transformation. That is,

$${\tilde{\mathbf{R}}}_y={\displaystyle \frac{1}{U}}\sum _{i=1}^U{\mathbf{Z}}_i^T\overline{\mathbf{y}}{\overline{\mathbf{y}}}^H{\mathbf{Z}}_i,$$

(9)

where $U=(L+1)/2$ and ${\mathbf{Z}}_i=[{\mathbf{e}}_i,{\mathbf{e}}_{i+1},\cdots ,{\mathbf{e}}_{i+U-1}]$. Here, ${\mathbf{e}}_i$ denotes the ith column of the $L\times L$ identity matrix. ${\mathbf{Z}}_i^T$ denotes the transpose of the matrix ${\mathbf{Z}}_i$.

Initial DOA estimation is carried out via the MUSIC algorithm, where the covariance matrix ${\tilde{\mathbf{R}}}_y$ undergoes eigendecomposition to obtain

$${\tilde{\mathbf{R}}}_y={\mathbf{U}}_s{\mathbf{\Lambda }}_s{\mathbf{U}}_s^H+{\mathbf{U}}_n{\mathbf{\Lambda }}_n{\mathbf{U}}_n^H,$$

(10)

where ${\mathbf{\Lambda }}_s=\mathrm{diag}[{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_K]$ is a diagonal matrix containing the signal eigenvalues, with the corresponding eigenvalues forming the columns of ${\mathbf{U}}_s$. Similarly, ${\mathbf{\Lambda }}_n=\mathrm{diag}[{\lambda }_{K+1},$${\lambda }_{K+2},\cdots ,{\lambda }_U]$ is the diagonal noise eigenvalue matrix, and ${\mathbf{U}}_n$ is the noise eigenvector matrix.Detection of the spectral peaks enables estimation of the K on-grid DOAs $\widehat{\mathit{\theta }}=[{\widehat{\theta }}_1,\cdots ,{\widehat{\theta }}_K]$, that is,

$$\widehat{\mathit{\theta }}=\underset{{\theta }_n\in \tilde{\mathit{\theta }}}{argmax}{\displaystyle \frac{1}{\mathbf{a}{\left({\theta }_n\right)}^H{\mathbf{U}}_n{\mathbf{U}}_n^H\mathbf{a}\left({\theta }_n\right)}},$$

(11)

where the angular set $\tilde{\mathit{\theta }}=[{\tilde{\theta }}_1,{\tilde{\theta }}_2,\dots ,{\tilde{\theta }}_Q]$ is constructed through uniform discretization of the potential DOA space $[-{90}^{\circ },{90}^{\circ }]$ with a grid interval. Here, Q denotes the grid number and $Q>>K$.

3.2. Fine Estimation

The initial DOA estimation suffers from quantization-induced estimation accuracy degradation due to the grid mismatch effect. To address this limitation, a refined DOA estimation method is developed in the following section.

Proceeding from (8), the noise effect can be eliminated by left-multiplying the expression with a selection matrix $\mathbf{F}$, that is,

$$\mathbf{r}=\mathbf{F}\overline{\mathbf{y}}=\mathbf{F}\overline{\mathbf{A}}\mathbf{p}+\mathbf{F}\overline{\mathit{ϵ}},$$

(12)

where

$$\mathbf{F}=\left[\begin{array}{cc}{\mathbf{I}}_{{\displaystyle \frac{L-1}{2}}}& {\mathbf{0}}_{{\displaystyle \frac{L-1}{2}}\times ({\displaystyle \frac{L-1}{2}}+1)}\\ {\mathbf{0}}_{{\displaystyle \frac{L-1}{2}}\times ({\displaystyle \frac{L-1}{2}}+1)}& {\mathbf{I}}_{{\displaystyle \frac{L-1}{2}}}\end{array}\right].$$

(13)

Here, ${\mathbf{I}}_{{\displaystyle \frac{L-1}{2}}}$ denotes the ${\displaystyle \frac{L-1}{2}}\times {\displaystyle \frac{L-1}{2}}$ identity matrix.

Assume there are $\overline{K}$ signals. The k-th signal can be obtained by isolating it from the contributions of multiple sources, that is,

$${\mathbf{r}}_k=\mathbf{r}-\sum _{\begin{array}{c}{\displaystyle \genfrac{}{}{0pt}{}{l=1,}{l\ne k}}\end{array}}^{\overline{K}}\mathbf{F}\overline{\mathbf{a}}\left({\widehat{\theta }}_l\right){\widehat{p}}_l,$$

(14)

where ${\{{\widehat{p}}_l,{\widehat{\theta }}_l\}}_{l=1,i\ne k}^{\overline{K}}$ are inherited from the estimates obtained in the preceding iterations. The current estimate ${\widehat{p}}_k$ is then derived through a least squares solution, given by

$${\widehat{p}}_k={\left({\left(\mathbf{F}\overline{\mathbf{a}}\left({\widehat{\theta }}_k\right)\right)}^H\left(\mathbf{F}\overline{\mathbf{a}}\left({\widehat{\theta }}_k\right)\right)\right)}^{-1}{\left(\mathbf{F}\overline{\mathbf{a}}\left({\widehat{\theta }}_k\right)\right)}^H{\mathbf{r}}_k.$$

(15)

The orthogonal projection onto $\mathbf{F}\overline{\mathbf{a}}$ is given by

$$P_{\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)}^{\perp }=\mathbf{I}-\left(\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)\right){\left({\left(\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)\right)}^H\left(\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)\right)\right)}^{-1}{\left(\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)\right)}^H.$$

(16)

Therefore, the noise subspace can be represented as ${\mathbf{r}}_k^H{P}_{\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)}^{\perp }{\mathbf{r}}_k$.

The DOA estimate of the kth source, denoted as ${\widehat{\theta }}_k$, can be obtained by minimizing the noise subspace projection, that is ${\widehat{\theta }}_k={argmin}_{\theta }{\mathbf{r}}_k^H{P}_{\mathbf{F}\overline{\mathbf{a}}\left(\theta \right)}^{\perp }{\mathbf{r}}_k$. The formulated minimization problem can be rigorously solved using various traditional optimization methods.

To summarize, the proposed methodology uses a dual-layer recursive estimation scheme. In the first loop, the current source data are separated from multiple sources. Then, the DOA and power estimates are corrected by alternating optimization. In the second loop, the DOA of a new source is obtained by utilizing the estimated angle and power information. The termination criterion is satisfied when the iterative updates to the DOA estimates fall below a predefined convergence threshold, indicating stability in the solution.

4. Performance Analysis

4.1. The CRB Derivation

Next, we derive the CRB of the deterministically orthogonal signal model, as stated in Lemma 1.

Lemma 1. 

Consider K far-field sources with DOAs $\mathit{\theta }=[{\theta }_1,\dots ,{\theta }_K]$ arriving at an N-element linear array. For the case of the deterministically orthogonal signal, the CRB lower bound is given by the following expression,

$$\begin{array}{ccc}\hfill CRB\left(\mathit{\theta }\right)& =& {[{\mathbf{J}}_{\mathit{\theta }\mathit{\theta }}-{\mathbf{J}}_{\mathit{\theta }\mathbf{p}}{\mathbf{J}}_{\mathbf{p}\mathbf{p}}^{-1}{\mathbf{J}}_{\mathbf{p}\mathit{\theta }}]}^{-1}\hfill \\ & =& {\displaystyle \frac{1}{T}}\Re \left[{\displaystyle \frac{{\tilde{\mathbf{A}}}^H{\mathbf{B}}^{-1}\tilde{\mathbf{A}}}{{\mathbf{R}}_s\mathbf{H}{\mathbf{R}}_s}}\right],\hfill \end{array}$$

(17)

where $\mathbf{H}={\tilde{\mathbf{A}}}^H{\mathbf{B}}^{-1}\tilde{\mathbf{A}}{\mathbf{D}}^H{\mathbf{B}}^{-1}\mathbf{D}-{\mathbf{D}}^H{\mathbf{B}}^{-1}\tilde{\mathbf{A}}{\tilde{\mathbf{A}}}^H{\mathbf{B}}^{-1}\mathbf{D}$. Here, $\mathbf{B}={\left(\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H\right)}^T\otimes \left({\sigma }_n^2\mathbf{I}\right)+\left({\sigma }_n^2\mathbf{I}\right)\otimes \left(\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H\right)+\left({\sigma }_n^2\mathbf{I}\right)\otimes \left({\sigma }_n^2\mathbf{I}\right)$, $\mathbf{D}=[\dot{\tilde{\mathbf{a}}}\left({\theta }_1\right),\dots ,\dot{\tilde{\mathbf{a}}}\left({\theta }_K\right)]$ and $\dot{\tilde{\mathbf{a}}}\left({\theta }_i\right)={\displaystyle \frac{\partial \tilde{\mathbf{a}}\left({\theta }_i\right)}{\partial {\theta }_i}}$. ℜ is the real part of the complex variable.

Proof. 

The expected value of the vector $\mathbf{r}$ is given by,

$$\mathbf{m}=\mathbf{F}\mathbf{J}\mathrm{vec}\left(\mathbf{R}\right)=\mathbf{F}\mathbf{J}\tilde{\mathbf{A}}\left(\mathit{\theta }\right)\mathbf{p}.$$

(18)

The covariance matrix of the vector $\mathbf{r}$ is,

$$\begin{array}{ccc}\hfill \mathbf{\Sigma }& =& {\displaystyle \frac{1}{T}}\left\{\mathbf{F}\mathbf{J}\right\{{\mathbf{R}}^T\otimes \mathbf{R}-[{\left(\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H\right)}^T\hfill \\ & \otimes & \left(\mathbf{A}{\mathbf{R}}_s{\mathbf{A}}^H\right)\left]\right\}{\mathbf{J}}^H{\mathbf{F}}^H\}\hfill \\ & =& {\displaystyle \frac{1}{T}}\left\{\mathbf{F}\mathbf{J}\mathbf{B}{\mathbf{J}}^H{\mathbf{F}}^H\right\}.\hfill \end{array}$$

(19)

We define the parameter vector $\mathit{\xi }={[\mathit{\theta },{\mathbf{p}}^T]}^T$, which concatenates the DOAs and the signal powers of the K sources.

Considering the concatenated parameter vector $\mathit{\xi }={[\mathit{\theta },{\mathbf{p}}^T]}^T$, the Fisher information matrix (FIM) admits the representation,

$$\mathbf{J}\left(\mathit{\xi }\right)=\left[\begin{array}{cc}{\mathbf{J}}_{\mathit{\theta }\mathit{\theta }}& {\mathbf{J}}_{\mathit{\theta }\mathbf{p}}\\ {\mathbf{J}}_{\mathit{\theta }\mathbf{p}}^T& {\mathbf{J}}_{\mathbf{p}\mathbf{p}}\end{array}\right],$$

(20)

where ${\mathbf{J}}_{\mathit{\theta }\mathbf{p}}^T={\mathbf{J}}_{\mathbf{p}\mathit{\theta }}$.

Each component in FIM is determined by the following formula,

$${\left[\mathbf{J}\right]}_{ij}={\displaystyle \frac{1}{2}}\left\{\mathrm{tr}\left[{\mathbf{\Sigma }}^{-1}{\displaystyle \frac{\partial \mathbf{\Sigma }}{\partial {\xi }_i}}{\mathbf{\Sigma }}^{-1}{\displaystyle \frac{\partial \mathbf{\Sigma }}{\partial {\xi }_j}}\right]\right\}+\Re \left[{\displaystyle \frac{\partial {\mathbf{m}}^H}{\partial {\xi }_i}}{\Sigma }^{-1}{\displaystyle \frac{\partial \mathbf{m}}{\partial {\xi }_j}}\right],$$

(21)

where ${\left[\mathbf{J}\right]}_{ij}$ refers to the entry located at the i-th row and j-th column of the matrix.

As the first term of (21) is negligible compared to the remaining terms, the equation can be approximated as,

$${\left[\mathbf{J}\right]}_{ij}=\Re \left[{\displaystyle \frac{\partial {\mathbf{m}}^H}{\partial {\xi }_i}}{\mathbf{\Sigma }}^{-1}{\displaystyle \frac{\partial \mathbf{m}}{\partial {\xi }_j}}\right].$$

(22)

The derivative of $\mathbf{m}$ with respect to ${\theta }_i$ is,

$${\displaystyle \frac{\partial \mathbf{m}}{\partial {\theta }_i}}=\mathbf{F}\mathbf{J}\dot{\tilde{\mathbf{a}}}\left({\theta }_i\right)p_i.$$

(23)

The derivative of $\mathbf{m}$ with respect to $p_i$ is,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial \mathbf{m}}{\partial p_i}}& =& \mathbf{F}\mathbf{J}\tilde{\mathbf{A}}\left(\mathit{\theta }\right){\displaystyle \frac{\partial \mathbf{p}}{\partial p_i}}\hfill \\ & =& \mathbf{F}\mathbf{J}\tilde{\mathbf{a}}\left({\theta }_i\right).\hfill \end{array}$$

(24)

Substituting (23) and (24) into (22), we obtain

$$\begin{array}{ccc}\hfill {\left[{\mathbf{J}}_{\mathit{\theta }\mathit{\theta }}\right]}_{ij}& =& T\{\Re \left[p_i^{*}{\dot{\tilde{\mathbf{a}}}}^H\left({\theta }_i\right){\mathbf{J}}^H{\mathbf{F}}^H{\mathbf{\Sigma }}^{-1}\mathbf{F}\mathbf{J}\dot{\tilde{\mathbf{a}}}\left({\theta }_j\right)p_j\right]\}\hfill \\ & =& T\{\Re \left[p_i^{*}{\dot{\tilde{\mathbf{a}}}}^H\left({\theta }_i\right){\mathbf{B}}^{-1}\dot{\tilde{\mathbf{a}}}\left({\theta }_j\right)p_j\right]\}.\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\left[{\mathbf{J}}_{\mathit{\theta }\mathbf{p}}\right]}_{ij}& =& T\{\Re \left[p_i^{*}{\dot{\tilde{\mathbf{a}}}}^H\left({\theta }_i\right){\mathbf{J}}^H{\mathbf{F}}^H{\mathbf{\Sigma }}^{-1}\mathbf{F}\mathbf{J}\tilde{\mathbf{a}}\left({\theta }_j\right)\right]\}\hfill \\ & =& T\{\Re \left[p_i^{*}{\dot{\tilde{\mathbf{a}}}}^H\left({\theta }_i\right){\mathbf{B}}^{-1}\tilde{\mathbf{a}}\left({\theta }_j\right)\right]\}.\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\left[{\mathbf{J}}_{\mathbf{p}\mathbf{p}}\right]}_{ij}& =& T\{\Re \left[{\tilde{\mathbf{a}}}^H\left({\theta }_i\right){\mathbf{J}}^H{\mathbf{F}}^H{\mathbf{\Sigma }}^{-1}\mathbf{F}\mathbf{J}\tilde{\mathbf{a}}\left({\theta }_j\right)\right]\}\hfill \\ & =& T\{\Re \left[{\tilde{\mathbf{a}}}^H\left({\theta }_i\right){\mathbf{B}}^{-1}\tilde{\mathbf{a}}\left({\theta }_j\right)\right]\}.\hfill \end{array}$$

(27)

According to (25)–(27), CRB can be derived as,

$$\begin{array}{ccc}\hfill CRB\left(\mathit{\theta }\right)& =& {[{\mathbf{J}}_{\mathit{\theta }\mathit{\theta }}-{\mathbf{J}}_{\mathit{\theta }\mathbf{p}}{\mathbf{J}}_{\mathbf{p}\mathbf{p}}^{-1}{\mathbf{J}}_{\mathbf{p}\mathit{\theta }}]}^{-1}\hfill \\ & =& {\displaystyle \frac{1}{T}}\Re \left[{\displaystyle \frac{{\tilde{\mathbf{A}}}^H{\mathbf{B}}^{-1}\tilde{\mathbf{A}}}{{\mathbf{R}}_s\mathbf{H}{\mathbf{R}}_s}}\right].\hfill \end{array}$$

(28)

□

4.2. Computational Complexity Analysis

A comprehensive computational complexity analysis, including the search-based, EPUMA, SS-MUSIC, and the proposed methods, is summarized in

Table 1. Computational complexity of the four methods.

Algorithm	Computational Complexity
Search-based [19]	$O(N_c{log}_2{N}_c+N_c+DK{N}_c)$
EPUMA [20]	$O(N_c^{2}T+G{N}_c^3)$
SS-MUSIC [15]	$O(T{N}_c+N_c^3)$
Proposed method	$O(T{N}_c+N_c^3+HK{N}_c^3)$

. A dense angular sampling number D is utilized for the search-based method to improve the estimation performance. EPUMA and SS-MUSIC demand substantial snapshots T for reliable covariance matrix computation. Our method requires only limited snapshots and few iterations H, resulting in substantially reduced complexity compared to existing techniques.

5. Simulation Results

Through rigorous simulation studies, this section provides an in-depth evaluation of the proposed algorithm’s performance and effectiveness. Specifically, the performance of the proposed method for ULA and NLA configurations is thoroughly analyzed in Section 5.1, Section 5.2, Section 5.3 and Section 5.4, particularly focusing on multi-source scenarios and varying snapshot conditions. The number of snapshots is set to 10 for the low-snapshot case and 100 for the high-snapshot case. Further, the resolution and the convergence performance of the proposed method are provided in Section 5.5 and Section 5.6, respectively. The simulation platform consists of an Intel i7-6700 processor with 16 GB RAM running a 64-bit operating system. The RMSE metrics are computed through 200 independent Monte Carlo trials, with the corresponding CRBs provided as theoretical performance benchmarks.

5.1. Fewer Sources than Sensors Case for ULA

The considered array configuration consists of $N=10$ ULA elements spaced at $d=\lambda /2$. The estimation accuracy is measured using the RMSE performance metric, defined as,

$$e=\sqrt{{\displaystyle \frac{1}{K}}\sum _{k=1}^K{({\theta }_k-{\widehat{\theta }}_k)}^2},$$

(29)

where ${\widehat{\theta }}_k$ indicates the estimated angle parameter of the k-th source.

In the following, the performance of the proposed method is evaluated against three state-of-the-art algorithms, including the search-based [19], EPUMA [20], and SS-MUSIC [15] methods. SS-MUSIC is a classical algorithm for DOA estimation. Nevertheless, its applicability is limited to on-grid DOA estimation problems. Although the search-based and EPUMA method can guarantee the accuracy of the off-grid DOAs, the algorithm’s exhaustive search grids and numerous snapshot computations result in a high computational burden. The RMSE of four approaches, including the search-based, EPUMA, SS-MUSIC, and the proposed methods, are evaluated against SNR for ULA configurations with fewer sources than sensors, as shown in Figure 1 and Figure 2 for different snapshot quantities. The simulation considers two distinct snapshot scenarios: 10 snapshots and 100 snapshots, respectively. Three source signals impinge on the array from distinct directions specified by ${\theta }_1=0.5^{\circ },{\theta }_2=30.5^{\circ }, \mathrm{and} {\theta }_3=60.5^{\circ }$. In the search-based method, we varied the search angle from 0 to $\pi$ in discrete steps of $\pi /50N$. Compared with the other three methods, the proposed method performed the best, regardless of the snapshot number. Moreover, the RMSE of the proposed method attains the CRB when SNR exceeds $-5$ dB. The reason the search-based and SS-MUSIC methods cannot approach the CRB lies in the finite number of angle-searching grids.

5.2. Fewer Sources than Sensors Case for NLA

The antenna array configuration consists of 8 elements arranged in a nested structure, as depicted in Figure 3. Filled circles denote the selected elements, while crosses denote the discarded elements. The corresponding coarray is a ULA with 39 elements.

To assess estimation accuracy, Figure 4 and Figure 5 present the RMSE of the search-based, EPUMA, SS-MUSIC, and the proposed methods, plotted against the SNR for an NLA configuration where the number of sources is less than the number of sensors, utilizing different numbers of snapshots. We employed the same snapshot quantity as specified in the preceding simulation scenario. The source angles were set as ${\theta }_1=0.5^{\circ },{\theta }_2=30.5^{\circ },{\theta }_3=60.5^{\circ }$. For the search-based method, the same search angle was utilized as in Figure 1. It was found that the proposed approach exhibited the best estimation performance, whereas the RMSE of the search-based and SS-MUSIC methods remained constant at a high SNR. The proposed method showed consistent performance advantages over EPUMA, particularly under low-SNR conditions. In summary, it can be seen from example A and example B that the proposed method is applicable to both ULA and NLA.

5.3. More Sources than Sensors Case for NLA

The subsequent analysis utilized the aforementioned nested array configuration. The RMSEs of the four methods mentioned above versus the SNR in the case of more sources than sensors using different numbers of snapshots are shown in Figure 6 and Figure 7. We considered two distinct snapshot scenarios: 10 snapshots and 100 snapshots, respectively. Nine sources impinged on the array from $[-45.5^{\circ },-35.5^{\circ },-25.5^{\circ },-15.5^{\circ },5.5^{\circ },15.5^{\circ },25.5^{\circ },35.5^{\circ },45.5^{\circ }]$. The same angular search grid was employed for the comparative analysis.The proposed method exhibited two distinct advantages. One was a consistently better estimation performance than the benchmarks, and the other was attaining CRB when SNR exceeded a certain threshold, demonstrating optimal estimation capability. Overall, the effectiveness of the proposed method for off-grid DOA estimation with NLA in the case of fewer sources than sensors and more sources than sensors was validated, as shown in both example B and example C.

5.4. Estimation Performance Versus Snapshot Number for NLA

The following analysis examined the RMSE characteristics of the proposed method across varying snapshot conditions ranging from 10 to 10,000. Figure 8 displays the results for the three-source scenario, while Figure 9 presents the corresponding results for the more challenging nine-sourse case.The same three sources as in Section 5.1 were considered and the same nine sources as in Section 5.3 were considered. Remarkably, the proposed technique reached the theoretical CRB limit in all SNR regimes, maintaining statistical efficiency even under snapshot-constrained scenarios.

5.5. Resolution Performance for NLA

To further examine the resolution of the proposed method, the RMSE of the search-based, EPUMA, SS-MUSIC, and the proposed methods versus source separation are compared in Figure 10 and Figure 11 at both 0 dB SNR and 10 dB SNR, respectively. The proposed algorithm demonstrated a superior spatial resolution capability, outperforming all benchmark methods in our comparative evaluation. In addition, the RMSE of the search-based, EPUMA, and SS-MUSIC methods did not approach CRB when the sources were closely spaced, whereas the proposed technique maintained a CRB-attaining RMSE performance with just $2^{\circ }$ source separation.

5.6. Convergence Performance for NLA

Finally, four off-grid sources coming from ${\theta }_1=0.5^{\circ }$, ${\theta }_2=10.5^{\circ }$, ${\theta }_3=20.5^{\circ }$, and ${\theta }_4=30.5^{\circ }$ and 20 snapshots were utilized in the convergence evaluation. The RMSE of the proposed method converged within three iterations across all of the tested SNRs, as shown in Figure 12, with further iterations yielding no improvement.

5.7. Performance Comparison

Table 2. RMSE of the four methods in the case of different conditions. Unit: degree.

Algorithm	Search-Based	EPUMA	SS-MUSIC	Proposed
Conditions
ULA (fewer sources) (10 snapshots)	0.475	0.242	0.255	0.214
ULA (fewer sources) (100 snapshots)	0.393	0.070	0.124	0.068
NLA (fewer sources) (10 snapshots)	0.294	0.118	0.181	0.110
NLA (fewer sources) (100 snapshots)	0.264	0.037	0.105	0.036
NLA (more sources) (10 snapshots)	0.214	0.160	0.205	0.140
NLA (more sources) (100 snapshots)	0.154	0.046	0.122	0.043

presents a comprehensive performance comparison between the proposed method and the four existing approaches, with the RMSE results evaluated under the varying conditions specified in Section 5.1, Section 5.2 and Section 5.3. The proposed method consistently surpassed the search-based, EPUMA, and SS-MUSIC methods in performance, particularly in underdetermined and snapshot-limited scenarios, while maintaining compatibility with both uniform and non-uniform array geometries.

6. Conclusions

This paper presented an effective coarray-based method for off-grid DOA estimation, demonstrating a superior performance in challenging scenarios. A noise elimination technique was developed to reduce the noise effect by left-multiplying a matrix to the vectorized sample covariance matrix. An angle and power iterative correction method was proposed to estimate the source angle in each iteration. The proposed method successfully addressed the conventional trade-off between the snapshot requirements and estimation accuracy, delivering precise DOA estimation with significantly reduced snapshot demands. Furthermore, we derived the CRB for the deterministic orthogonal signal model, providing a theoretical performance benchmark. Although the search-based, EPUMA, and SS-MUSIC methods can be utilized to perform off-grid DOA estimation, the proposed method provided an optimal performance, delivering a high-precision DOA estimation and excellent resolution with minimal snapshot requirements. Moreover, the RMSE of the proposed method can coincide with the CRB, and three iterations were sufficient for convergence, which manifested the effectiveness and efficiency of the proposed method for coarray-based off-grid DOA estimation for both cases of fewer sources than sensors and more sources than sensors with few snapshots. The limitation of the proposed method is that it does not consider the effects of different noise characteristics and array configurations, which provides practical guidelines for subsequent investigations in array signal processing.