Efficient Nyström-Based Unitary Single-Tone 2D DOA Estimation for URA Signals

Abstract

We propose an efficient Nyström-based unitary subspace method for low-complexity two-dimensional (2D) direction-of-arrival (DOA) estimation in uniform rectangular array (URA) signal processing systems. The conventional high-resolution DOA estimation methods often suffer from excessive computational complexity, particularly when dealing with large-scale antenna arrays. The proposed method addresses this challenge by combining the Nyström approximation with a unitary transformation to reduce the computational burden while maintaining estimation accuracy. The signal subspace is approximated using a partitioned covariance matrix, and a real-valued transformation is applied to further simplify the eigenvalue decomposition (EVD) process. Furthermore, the linear prediction coefficients are estimated via a weighted least squares (WLS) approach, enabling robust extraction of the angular parameters. The 2D DOA estimates are then derived from these coefficients through a closed-form solution, eliminating the need for exhaustive spectral searches. Numerical simulations demonstrate that the proposed method achieves comparable estimation performance to state-of-the-art techniques while significantly reducing computational complexity. For a fixed array size of $M=N=20$, the proposed method demonstrates significant computational efficiency, requiring less than $50\%$ of the running time compared to conventional ESPRIT, and only $6\%$ of the time required by ML methods, while maintaining similar performance. This makes it particularly suitable for real-time applications where computational efficiency is critical. The novelty lies in the integration of Nyström approximation and unitary subspace techniques, which jointly enable efficient and accurate 2D DOA estimation without sacrificing robustness against noise. The method is applicable to a wide range of array processing scenarios, including radar, sonar, and wireless communications.

1. Introduction

Direction of arrival (DOA) estimation techniques are commonly used to determine the angle or direction of a signal. This technique is applicable in various fields such as radar, sonar, and mobile communications [1,2,3,4]. The localization of signals in space using signal processing techniques involves employing a two-dimensional (2D) DOA estimation of a uniform rectangular array (URA). A URA is an antenna array that comprises multiple rows and columns of sensor elements. URA-based DOA estimation methods are useful when the signal arrives from a 2D plane, which may include signals from multiple angles or directions that require localization and identification. The use of a URA enables the separation and individual processing of signals from different directions, resulting in accurate localization information.

Over the past several decades, various approaches have been developed to address the DOA estimation problem. Among these techniques, subspace-based methods, including multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance technique (ESPRIT), along with their enhanced versions such as root-MUSIC and Unitary-ESPRIT, are the most prominent super-resolution methods [5,6]. The 2D MUSIC algorithm estimates 2D DOA by utilizing the noise subspace of a received dataset’s covariance matrix from a URA [7], but its computation is intensive due to the required 2D search for the spectrum. To address this issue, researchers have proposed efficient methods like search-free 2D ESPRIT [8] and the matrix pencil approach [9,10], which require extra pairing processes in some cases. Most of these methods focus on multi-snapshot cases for 2D DOA estimation. However, real-world scenarios may limit the available snapshots due to physical constraints like fast time-variant channel scenarios in mobile communication or sonar signal processing. Researchers have explored using compressed sensing to estimate parameters with small sample sizes [11,12]. Single-snapshot DOA estimation is crucial in applications like automotive radar systems, where only one snapshot is available to estimate multiple spatial sources. Therefore, several DOA estimation algorithms have been proposed for single snapshot cases [13,14,15]. It is important that any direction-finding algorithm used in real-world scenarios be computationally efficient and rely on only a limited number of array snapshots. Many computationally efficient methods have been proposed to solve DOA estimation, such as real-valued eigenvalue decomposition (EVD) or singular value decomposition (SVD)-based methods [16,17]. The problem of DOA estimation is addressed in [16] with real-valued computations by considering the real part of the array output covariance matrix and the imaginary part of the array output covariance matrix separately, which coincides with the intersection of the original noise subspace and its conjugate subspace. A real-valued root-MUSIC algorithm that reduces the computation is proposed in [17]; it exploits the EVD/SVD of a real-valued covariance matrix to extract a real-valued noise subspace, which reduces the complexity by a factor of about four compared to root-MUSIC. The problem of 2D DOA estimation for URA signals is linked to the compressive sensing trilinear model in [18], in which the proposed algorithm not only requires no spectral peak searching but also has better angle estimation performance than the estimation of the ESPRIT algorithm with low computational complexity. An efficient 2D DOA estimation algorithm based on OMP for URA is proposed in [19]. The main idea of it is to reduce the number of atoms in the complete dictionary of the OMP algorithm to reduce the amount of computation required. A low complexity 2D DOA estimation and self-calibration for URA with gain-phase error is developed in [20], the URA is modeled as the Kronecker product of two uniform linear arrays (ULAs) to decouple the 2D DOA estimation. Then, several well-calibrated auxiliary array elements are added in the two ULAs, based on which the rotation invariant factor of the URA destroyed by the gain-phase error is reconstructed by solving constrained optimization problems. The problem of 2D DOA estimation for coherent sources using an EMVS array with URA geometry is addressed in [21], where three novel PARAFAC approaches are proposed to eliminate the rank deficiency of the source matrix via data rearrangement. Unlike existing smoothing-based algorithms, this approach does not reduce the number of available measurements. Recently, deep neural network (DNN) studies on DOA estimations have attracted more and more attention, providing an alternative method for dealing with the DOA problem and successfully showing its potential application. However, the computational complexity may be higher compared with classical methods [22].

This paper introduces a novel approach for the fast 2D DOA estimation of URA signals in the single snapshot case. The approach is based on the Nyström approximation [23,24]. To simplify the denoising process, the Nyström method is initially applied to the received data, providing an approximation of the signal’s subspace. Subsequently, to further reduce the complexity, the 2D DOA is estimated using the obtained subspaces and a unitary subspace-based method, where the covariance matrix is computed from a real-valued computation. The main contributions of this manuscript are the following: (1) Utilize the Nyström method to obtain the signal subspace for estimating the 2D DOA, avoiding computing the EVD or SVD of the signal. (2) Utilize the unitary subspace-based algorithm, which makes use of the principal singular vectors and an iterative process to estimate the 2D DOA of the URA signals, avoiding the complex-valued decomposition of computation, which reduces the computational complexity.

The paper is structured as follows: Section 2 introduces the array signal model. Section 3 describes the application of the Nyström approximation method and the unitary subspace-based method for obtaining 2D DOA estimation. Section 4 presents the results of computer simulations conducted to evaluate the performance of the proposed method and compare it with existing algorithms. Finally, Section 5 concludes the paper.

Notation

In this paper, the boldface lowercase letters denote vectors and boldface uppercase letters denote matrices. Superscripts such as ${\left(\right)}^T$, ${\left(\right)}^{*}$, ${\left(\right)}^H$, ${\left(\right)}^{†}$, and ${\left(\right)}^{-1}$ represent transpose, complex conjugate, Hermitian transpose, pseudo inverse, and matrix inverse, respectively. $\mathbb{E}$ denotes the expectation operator and is the diagonal matrix operator. The $\widehat{a}$ denotes an estimate of a. ${\mathbf{I}}_m$ is the $m\times m$ identity matrix, and $\mathbb{C}$ denotes the set of complex numbers.

2. Signal Model

A URA comprising omnidirectional antennas arranged in an $M\times N$ configuration with interelement spacing d is under consideration [18,19]. The URA is being subjected to P far-field narrowband signal sources arriving from the directions $({\varphi }_p,{\theta }_p)$, with $p=1,2,\cdots ,P$, where p ranges from 1 to P. The first sensor is set as the reference sensor. The received signal of the sensor located at $(m,n)$ is defined as a single snapshot.

$$\begin{array}{cc}\hfill x_{m,n}=& s_{m,n}+q_{m,n}\hfill \\ \hfill =& \sum _{p=1}^P{s}_p{e}^{-j{\displaystyle \frac{2\pi }{\lambda }}(m-1)dsin{\varphi }_{p}cos{\theta }_p}{e}^{-j{\displaystyle \frac{2\pi }{\lambda }}(n-1)dsin{\varphi }_{p}sin{\theta }_p}\hfill \\ \hfill & +q_{m,n}.\hfill \end{array}$$

(1)

The received signal comprises the wavefront $s_p$ of the pth signal, with a wavelength of $\lambda$, and white Gaussian noise assumed to have a noise power of ${\sigma }^2$. The objective of the 2D DOA estimation problem is to estimate the DOA, represented by $({\varphi }_p,{\theta }_p)$, from the received signals. As this paper considers the single-tone case, $P=1$ is assumed known in the signal model; then, the signal becomes

$$\begin{array}{cc}\hfill x_{m,n}=& s_{m,n}+q_{m,n}\hfill \\ \hfill =& s{e}^{-j{\displaystyle \frac{2\pi }{\lambda }}(m-1)dsin\varphi cos\theta }{e}^{-j{\displaystyle \frac{2\pi }{\lambda }}(n-1)dsin\varphi sin\theta }\hfill \\ \hfill & +q_{m,n}.\hfill \end{array}$$

(2)

We assume $\mu ={\displaystyle \frac{2\pi }{\lambda }}dsin\varphi cos\theta$ and $\nu ={\displaystyle \frac{2\pi }{\lambda }}dsin\varphi sin\theta$ for simplicity.

3. Algorithm Development

3.1. The Nyström Approximation

To analyze the signal, we initially transform the signal represented in (1) to a matrix format.

$$\mathbf{X}=\mathbf{S}+\mathbf{Q},$$

(3)

the matrices are derived from the components of both the signal and the noise, and ${\left[\mathbf{X}\right]}_{m,n}=x_{m,n}$, ${\left[\mathbf{S}\right]}_{m,n}=s_{m,n}$, and ${\left[\mathbf{Q}\right]}_{m,n}=q_{m,n}$

In traditional approaches, the covariance matrix is usually calculated by utilizing numerous snapshots subsequent to receiving the signal $\mathbf{X}$. Nevertheless, in the scenario of single snapshot signals, there are insufficient snapshots to eliminate noise from the signal by computing the covariance matrix. As an alternative, we can decompose the signal matrix as

$$\mathbf{S}=s\mathbf{g}{\mathbf{h}}^T,$$

(4)

where s is the wavefront, and

$$\mathbf{g}={[\mu {\mu }^2\cdots {\mu }^M]}^T,$$

(5)

$$\mathbf{h}={[\nu {\nu }^2\cdots {\nu }^N]}^T,$$

(6)

where ${(·)}^T$ denotes the transpose operator, $\mu$ and $\nu$ are the same as that in (2). We see that the elements in ${\mathbf{g}}_P$ and ${\mathbf{h}}_P$ have linear factor (LP) relations. Here, LP is a scalar multiplier that scales a variable or signal linearly. In traditional ideas, we can perform the EVD or SVD on $\mathbf{X}$ to obtain the subspace with high complexity; therefore, we apply the Nyström approximation to obtain the signal subspace.

In the Nyström method, we first divide the signal matrix $\mathbf{X}={\left[{\mathbf{x}}_1,{\mathbf{x}}_2,\cdots ,{\mathbf{x}}_M\right]}^T$ into two parts [24]

$${\mathbf{X}}_{new}=\left[\begin{array}{c}{\mathbf{X}}_1\\ {\mathbf{X}}_2\end{array}\right],$$

(7)

where ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ are parts of $\mathbf{X}$

$$\begin{array}{ccc}\hfill {\mathbf{X}}_1& =& \mathbf{X}(1:K,:),\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\mathbf{X}}_2& =& \mathbf{X}(K+1:M,:).\hfill \end{array}$$

(9)

where K is a parameter defined by users, and $K\in (1,2,\cdots ,M)$. We have the covariance matrices of ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$

$${\mathbf{R}}_{11}=\mathbb{E}\left[{\mathbf{X}}_1{\mathbf{X}}_1^H\right],$$

(10)

$${\mathbf{R}}_{21}=\mathbb{E}\left[{\mathbf{X}}_2{\mathbf{X}}_1^H\right],$$

(11)

where ${\mathbf{R}}_{11}$ is the covariance matrix of ${\mathbf{X}}_1$, ${\mathbf{R}}_{21}$ is the covariance matrix of ${\mathbf{X}}_2$ and ${\mathbf{X}}_1$.

When noise is present, the covariance matrices cannot be computed with (10) and (11); hence, we use an approximation approach to estimate the covariance matrices [18,19].

$${\mathbf{R}}_{11}={\displaystyle \frac{1}{N}}\left[{\mathbf{X}}_1{\mathbf{X}}_1^H\right],$$

(12)

$${\mathbf{R}}_{21}={\displaystyle \frac{1}{N}}\left[{\mathbf{X}}_2{\mathbf{X}}_1^H\right].$$

(13)

Before proceeding with the covariance matrices, it is typically required for the matrix ${\mathbf{R}}_{11}$ to have a full rank. To accomplish this, we set the value of K to be greater than $P=1$ but less than the minimum of M and N, i.e., $1<K\le min\{M,N\}$. Our simulations have demonstrated that the value of K does not necessarily need to increase with an increase in the number of array sensors, M. Instead, a specific value of K is adequate for precise DOA estimation while simultaneously reducing computational complexity.

We can perform EVD or SVD on ${\mathbf{R}}_{11}$ and ${\mathbf{R}}_{21}$ to obtain the subspace [25]

$${\mathbf{R}}_{11}={\mathbf{U}}_{11}{\Lambda }_{11}{\mathbf{U}}_{11}^H,$$

(14)

the eigenvector matrix ${\mathbf{U}}_{11}\in {\mathbb{C}}^{K\times K}$ and the corresponding eigenvalue matrix ${\Lambda }_{11}\in {\mathbb{C}}^{K\times K}$ are obtained, where ${\Lambda }_{11}$ contains the eigenvalues in descending order. Additionally, we define ${\mathbf{U}}_{21}={\mathbf{R}}_{21}{\mathbf{U}}_{11}{\Lambda }_{11}^{-1}$. By utilizing (14), we can establish the relationship among ${\mathbf{R}}_{21}$, ${\mathbf{U}}_{11}$, and ${\Lambda }_{11}$.

$${\mathbf{R}}_{21}{\mathbf{U}}_{11}={\mathbf{U}}_{21}{\Lambda }_{11}.$$

(15)

We use ${\mathbf{U}}_{11}$ and ${\mathbf{U}}_{21}$ to obtain another matrix

$$\mathbf{U}=\left[\begin{array}{c}{\mathbf{U}}_{11}\\ {\mathbf{U}}_{21}\end{array}\right],$$

(16)

with the use of (15), (16) is written as

$$\mathbf{U}=\left[\begin{array}{c}{\mathbf{U}}_{11}\\ {\mathbf{R}}_{21}{\mathbf{U}}_{11}{\Lambda }_{11}^{-1}\end{array}\right].$$

(17)

To orthogonalize $\mathbf{U}$, we use ${\mathbf{U}}_{11}$ and ${\Lambda }_{11}$ to construct $\mathbf{D}$

$$\mathbf{D}=\mathbf{U}{\Lambda }_{11}^{{\displaystyle \frac{1}{2}}},$$

(18)

and $\mathbf{Z}$

$$\mathbf{Z}={\mathbf{D}}^H\mathbf{D}.$$

(19)

we perform EVD on $\mathbf{Z}$, then

$$\mathbf{Z}={\mathbf{U}}_D{\Lambda }_D{\mathbf{U}}_D^H.$$

(20)

The eigenvalue matrix ${\Lambda }_D=\mathrm{diag}\left[{\lambda }_{D1},{\lambda }_{D2},\cdots ,{\lambda }_{DK}\right]$ is obtained with ${\lambda }_{D1}\le {\lambda }_{D2}\le \cdots \le {\lambda }_{DK}$, and ${\mathbf{U}}_D$ is derived from the corresponding eigenvector matrix ${\mathbf{U}}_D=[{\approxeq }_{D1},{\approxeq }_{D2},\cdots ,{\approxeq }_{DK}]$, where ${\approxeq }_{Di}(i=1,2,\cdots ,K)$ represents the ith eigenvector. Since the signal number $P=1$ is known, we use the first column vectors of matrix $\mathbf{Y}$ to construct the signal subspace, i.e., $\mathbf{Y}=\mathbf{D}{\mathbf{U}}_D$ [24]. Furthermore, as ${\tilde{\mathbf{U}}}_s=\mathbf{Y}(:,1)$ satisfies the Equation (21), where

$$span\left\{{\tilde{\mathbf{U}}}_s\right\}=span\left\{\mathbf{A}\right\}.$$

(21)

Then, we obtain the subspace of the signal. We summarize the process to obtain the signal subspace in

Table 1. Nyström approximation.

Step 1: Decompose X into two sub-matrices ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ to obtain ${\mathbf{X}}_{new}$ as (7).

Step 2: Compute the covariance matrices ${\mathbf{R}}_{11}$ and ${\mathbf{R}}_{21}$ with (12) and (13).

Step 3: Perform EVD on ${\mathbf{R}}_{11}$ to obtain ${\mathbf{U}}_{11}$ and ${\Lambda }_{11}$ to construct $\mathbf{U}$ in (17).

Step 4: Use $\mathbf{U}$ in step 3 to construct $\mathbf{D}$ in (18) and obtain $\mathbf{Y}$; then, perform EVD on $\mathbf{Y}$ to obtain the subspace of the signal.

.

3.2. The Unitary Estimation of DOA

Upon acquiring the signal subspace through eigenvalue decomposition of the covariance matrix, various subspace-based algorithms can be implemented for 2D DOA estimation. While conventional methods like MUSIC theoretically achieve superior resolution by exploiting orthogonality between signal and noise subspaces, their computational complexity grows exponentially with the search grid density—which is particularly challenging for 2D angle estimation requiring joint azimuth–elevation searches. This issue becomes more pronounced in multi-tone scenarios where additional parameter pairing procedures are mandatory. However, in our single-tone signal case, this parameter matching complexity is inherently avoided, allowing direct subspace utilization. To address the computational burden while preserving the accuracy of the estimation, we adopt a subspace method based on unit transformation [26]. This approach transforms the complex-valued array manifold into a real-valued domain through unitary matrix operations. First, we introduce the unitary transform process.

It is a well-known fact that the covariance matrix ${\mathbf{R}}_{11}$ is a matrix with complex values, as well as ${\mathbf{R}}_{21}$. To simplify computational complexity, we can transform it into a real-valued covariance $\mathbf{C}$ using a unitary transform [27]. For simplicity, we use $\mathbf{R}$ other than ${\mathbf{R}}_{11}$ or ${\mathbf{R}}_{21}$ in the following process. First, we define

$$\mathbf{C}={\displaystyle \frac{1}{2}}{\mathbf{Q}}_M^H(\mathbf{R}+{\mathbf{J}}_M{\mathbf{R}}^{*}{\mathbf{J}}_M){\mathbf{Q}}_M=\mathcal{R}\left\{{\mathbf{Q}}_M^H\mathbf{R}{\mathbf{Q}}_M\right\},$$

(22)

where ^* is the conjugate operator, $\mathcal{R}$ means the real part of the matrix, ${\mathbf{J}}_M$ is an $M\times M$ exchange matrix with ones in its antidiagonal and 0 otherwise. ${\mathbf{Q}}_M$ is defined as [27]:

$${\mathbf{Q}}_M=\left\{\begin{array}{c}{\displaystyle \frac{\sqrt{2}}{2}}\left[\begin{array}{cc}{\mathbf{I}}_l& j{\mathbf{I}}_l\\ {\mathbf{J}}_l& j/{\mathbf{J}}_l\end{array}\right],\mathrm{for}M=2l\\ {\displaystyle \frac{\sqrt{2}}{2}}\left[\begin{array}{ccc}{\mathbf{I}}_l& {\mathbf{0}}_l& j{\mathbf{I}}_l\\ {\mathbf{0}}_l^T& \sqrt{(}2)& {\mathbf{0}}_l^T\\ {\mathbf{J}}_l& {\mathbf{0}}_l& -j/{\mathbf{J}}_l\end{array}\right],\mathrm{for}M=2l+1,\end{array}\right.$$

(23)

where ${\mathbf{0}}_l$ is an $l\times 1$ zero vector. The SVD of the real-valued covariance matrix $\mathbf{C}$ is,

$$\mathbf{C}={\mathbf{U}}_{cs}{\Lambda }_{cs}{\mathbf{V}}_{cs}^H+{\mathbf{U}}_{cn}{\Lambda }_{cn}{\mathbf{V}}_{cn}^H,$$

(24)

where ${\mathbf{U}}_{cs}$ and ${\mathbf{U}}_{cn}$ are the signal and noise subspaces of the real value, respectively. To obtain the estimate of DOA, we can use subspace methods such as root-MUSIC [28] or Unitary ESPRIT [27]. Since $\mathbf{Y}$ in the Nyström approximation, Section 3.1 is now obtained from real-valued ${\mathbf{R}}_{11}$ and ${\mathbf{R}}_{21}$, and we use ${\mathbf{Y}}_c$ to denote the real-valued $\mathbf{Y}$. Then, the EVD of the real-valued ${\mathbf{Y}}_c$ is

$${\mathbf{Y}}_c={\mathbf{U}}_{ycs}{\Lambda }_{ycs}{\mathbf{U}}_{ycs}^H+{\mathbf{U}}_{ycn}{\Lambda }_{ycn}{\mathbf{V}}_{ycn}^H,$$

(25)

Since we are considering a single-tone signal, the first column of the signal subspace ${\mathbf{U}}_{ycs}$, denoted as ${\approxeq }_{\mu ,1}$, contains the information of the signal. In traditional methods, the signal subspace can be directly used to estimate the DOA, such as the MUSIC method, which needs a great time consumption due to the grid search of the spectrum peak. To reduce the computation complexity, we use a real-valued method to estimate the DOA. In order to estimate the DOA from the real-valued eigenvector matrix, we first define two selection matrices.

$${\mathbf{J}}_{\mu ,1}=\left[{\mathbf{I}}_{M-1}{\mathbf{0}}_{M-1}\right],$$

(26)

$${\mathbf{J}}_{\mu ,2}=\left[{\mathbf{0}}_{M-1}{\mathbf{I}}_{M-1}\right].$$

(27)

By utilizing the property of shift invariance and assuming that the noise disturbance is significantly smaller compared to the signal power, we can deduce

$${\mathbf{J}}_{\mu ,2}{\approxeq }_{\mu ,1}\approx e^{j\mu }{\mathbf{J}}_{\mu ,1}{\approxeq }_{\mu ,1}.$$

(28)

Then

$${\mathbf{Q}}_{M-1}^H{\mathbf{J}}_{\mu ,2}{\mathbf{Q}}_M={\left({\mathbf{Q}}_{M-1}^H{\mathbf{J}}_{\mu ,1}{\mathbf{Q}}_M\right)}^{*}.$$

(29)

We use the real and image part of ${\mathbf{Q}}_{M-1}^H{\mathbf{J}}_{\mu ,2}{\mathbf{Q}}_M$, denoted as ${\mathbf{K}}_{\mu ,1}$ and ${\mathbf{K}}_{\mu ,2}$. Then, we have

$$({\mathbf{K}}_{\mu ,1}+j{\mathbf{K}}_{\mu ,2}){\mathbf{e}}_{\mu ,1}=e^{j\mu }({\mathbf{K}}_{\mu ,1}-j{\mathbf{K}}_{\mu ,2}){\mathbf{e}}_{\mu ,1}.$$

(30)

We follow the LP property to have

$${\mathbf{x}}_{\mu ,2}\approx a_{\mu }{\mathbf{x}}_{\mu ,1},$$

(31)

and $a_{\mu }=tan(\mu /2)$, ${\mathbf{x}}_{\mu ,1}={\mathbf{K}}_{\mu ,1}{\mathbf{e}}_{\mu ,1}$ and ${\mathbf{x}}_{\mu ,2}={\mathbf{K}}_{\mu ,2}{\mathbf{e}}_{\mu ,1}$. To estimate the LP factor, we use the WLS approach, which is

$$\underset{a_{\mu }}{min}={({\mathbf{x}}_{\mu ,2}-a_{\mu }{\mathbf{x}}_{\mu ,1})}^T{\mathbf{W}}^{-1}({\mathbf{x}}_{\mu ,2}-a_{\mu }{\mathbf{x}}_{\mu ,1})$$

(32)

and $\mathbf{W}$ is the weighting matrix defined as [29]

$$\begin{array}{cc}\hfill \mathbf{W}\left(\mu \right)& =\mathbb{E}\left\{({\mathbf{x}}_{\mu ,2}-a_{\mu }{\mathbf{x}}_{\mu ,1}){({\mathbf{x}}_{\mu ,2}-a_{\mu }{\mathbf{x}}_{\mu ,1})}^T\right\}\hfill \\ \hfill & =\mathbf{A}\left(\mu \right)\mathbb{E}{\mathbf{e}}_{\mu ,1}{\mathbf{e}}_{\mu ,1}^T{\mathbf{A}}^T\left(\mu \right),\hfill \end{array}$$

(33)

and $\mathbf{A}\left(\mu \right)={\mathbf{K}}_{\mu ,2}-a_{\mu }{\mathbf{K}}_{\mu ,1}$. Then, the LP factor is estimated as

$${\widehat{a}}_{\mu }={\displaystyle \frac{{\mathbf{x}}_{\mu ,1}^T{\mathbf{W}}^{-1}{\mathbf{x}}_{\mu ,2}}{{\mathbf{x}}_{\mu ,1}^T{\mathbf{W}}^{-1}{\mathbf{x}}_{\mu ,1}}}.$$

(34)

And we substitute ${\mathbf{e}}_{\mu ,1}={\tilde{\mathbf{e}}}_{\mu ,1}+\Delta {\mathbf{e}}_{\mu ,1}$ into (33), where $\Delta {\mathbf{e}}_{\mu ,1}$ is the error of the real one and the estimate of ${\mathbf{e}}_{\mu ,1}$

$$\mathbf{W}\left(\mu \right)=\mathbf{A}\left(\mu \right)\mathbb{E}\Delta {\mathbf{e}}_{\mu ,1}\Delta {\mathbf{e}}_{\mu ,1}^T{\mathbf{A}}^T\left(\mu \right).$$

(35)

At the beginning of the iteration, we use

$$\mathbf{W}\left(\mu \right)=\mathbf{A}\left(\mu \right){\mathbf{A}}^T\left(\mu \right).$$

(36)

For the LP factor $a_{\mu }$, we define it as

$${\widehat{a}}_{\mu ,int}={\mathbf{x}}_{\mu ,1}^{†}{\mathbf{x}}_{\mu ,2}.$$

(37)

Then,

$$\widehat{\mathbf{A}}\left(\mu \right)={\mathbf{K}}_{\mu ,2}-{\widehat{a}}_{\mu ,int}{\mathbf{K}}_{\mu ,1}.$$

(38)

Since $a_{\mu }=tan(\mu /2)$, we get

$$\widehat{\mu }=2tan{\displaystyle \frac{{\mathbf{x}}_{\mu ,1}^T{\widehat{\mathbf{W}}}^{-1}\left(\mu \right){\mathbf{x}}_{\mu ,2}}{{\mathbf{x}}_{\mu ,1}^T{\widehat{\mathbf{W}}}^{-1}\left(\mu \right){\mathbf{x}}_{\mu ,1}}}.$$

(39)

After we get the estimate of $\mu$, we can also perform the same process to estimate $\nu$ by performing the real-valued SVD on ${\mathbf{Y}}^T$. Then,

$$\widehat{\nu }=2tan{\displaystyle \frac{{\mathbf{x}}_{\nu ,1}^T{\widehat{\mathbf{W}}}^{-1}\left(\nu \right){\mathbf{x}}_{\nu ,2}}{{\mathbf{x}}_{\nu ,1}^T{\widehat{\mathbf{W}}}^{-1}\left(\nu \right){\mathbf{x}}_{\nu ,1}}},$$

(40)

where ${\mathbf{x}}_{\nu ,1}$, ${\mathbf{x}}_{\nu ,2}$, and ${\widehat{\mathbf{W}}}^{-1}\left(\nu \right)$ are calculated from the same process as $\mu$. Since we are considering single-tone signals, no parameter pairing method is needed here.

Then, the estimations of the 2D DOA are computed from $\widehat{\mu }$ and $\widehat{\nu }$ as

$$\widehat{\varphi }={\displaystyle \frac{\lambda }{2\pi d}}arcsin\left(\sqrt{{\widehat{\mu }}^2+{\widehat{\nu }}^2}\right),$$

(41)

and

$$\widehat{\theta }=arctan\left({\displaystyle \frac{\widehat{\nu }}{\widehat{\mu }}}\right).$$

(42)

Since all of the parameters are estimated from a real-valued process, the complexity is significantly reduced compared with a complex-valued process. The estimation process is summarized in Figure 1.

3.3. Computational Complexity Analysis

The computational complexity in our analysis is primarily evaluated in terms of matrix multiplication operations, which serve as the standard metric in computational mathematics. Our detailed analysis reveals that the principal computational burden of the proposed method stems from two key operations: (1) the Nyström approximation for obtaining the equivalent signal subspace, and (2) the unitary estimation process of DOA. To compute ${\mathbf{R}}_{11}$ and ${\mathbf{R}}_{12}$, we require $\mathcal{O}\left(N{K}^2\right)$ and $\mathcal{O}(MNK-N{K}^2)$ flops, respectively. Meanwhile, we use the Nyström approximation that just needs $\mathcal{O}\left(M{K}^2\right)$ flops to construct the signal subspace. Then, the total flops of the Nyström approximation is $\mathcal{O}(MNK+M{K}^2)$. The process of unitary estimation of DOA needs $\mathcal{O}(M{N}^2+N{M}^2+M^3+N^3)$ if we do not use the Nyström approximation [26]; it becomes $\mathcal{O}(M{K}^2+N{K}^2+M^3+N^3)$ when we use the subspaces obtained from the Nyström approximation. As established in [30], real-valued matrix operations require only approximately one-fourth (1/4) of the computational resources compared to their complex-valued counterparts. While the ESPRIT method needs $\mathcal{O}\left(M^2{N}^2\right)$ [31] and ML needs $\mathcal{O}\left(M^3{N}^3\right)$ multiplications [32], respectively. The simulation results are shown in the last simulation of Section 4.

4. Simulation Results

In this section, we validate the effectiveness of the proposed approach through computer simulations. We compare its performance with the ESPRIT method [31] and the ML method [32]. To assess performance, we utilized the RMSE, which is calculated by comparing the estimated DOA with the actual DOA. The SNR is defined as SNR = $10log({\sigma }_s^2/{\sigma }_n^2)$ (dB), where ${\sigma }_s^2$ represents the signal power and ${\sigma }_n^2$ represents the noise power.

We set the 2D DOA of the URA as $\theta =50°$ and $\varphi =30°$. The RMSE is computed as

$$\begin{array}{c}\hfill \mathrm{MSE}=\mathbb{E}\left\{{\displaystyle \frac{1}{P_1}}\sum _{p_1=1}^{P_1}[{({\widehat{\theta }}_{p_1}-\theta )}^2+{({\widehat{\varphi }}_{p_1}-\varphi )}^2]\right\}.\end{array}$$

We perform computer simulations using $P_1=200$ Monte Carlo trials, as we have found that increasing the number of trials does not contribute significantly to improving the results.

We start by comparing the MSE versus SNR of the Nyström approximation method for different values of K. SNR ranges from −10 dB to 30 dB, and the array size is $M=N=20$. In this method, we need to define the parameter K, so we use $K_1=8$, $K_2=12$, and $K_3=16$ in the simulation to evaluate the performance of our approach under different K values. The results are illustrated in Figure 2 and Figure 3. From these figures, we observe that the MSE of the proposed algorithm remains consistent for different K values when SNR is larger than $-5$ dB. This suggests that the performance does not significantly change with increasing K in higher SNR scenarios. Therefore, in the following simulations, we choose to use $K=8$. Unless otherwise stated, the remaining parameters remain unchanged in the remaining experiments.

To further evaluate the performance of our proposed method, we compare it with the ESPRIT and ML techniques, using $K=8$ in the Nyström approximation. The remaining data settings remain the same as in the first simulation. The results of this comparison are presented in Figure 4 and Figure 5. From these figures, we see that all three evaluated methods—our proposed algorithm, ML, and ESPRIT—demonstrate statistically indistinguishable estimation accuracy when SNR exceeds the −5 dB threshold. These findings indicate that the proposed algorithm has equivalent accuracy compared to the benchmark ML, where precision matters most.

In the following experiment, we conducted a comprehensive MSE analysis to systematically assess estimation accuracy under varying array configurations, spanning array numbers $M=N$ from 12 to 30. The results are shown in Figure 6 and Figure 7. These figures demonstrate that the proposed method achieves superior estimation accuracy compared to ESPRIT while maintaining statistically equivalent performance to the ML benchmark. We find that the array numbers increase, and the performance of all methods improves.

To rigorously evaluate the angular discrimination capability of our proposed algorithm, we conducted a targeted resolution test under controlled conditions. The fourth experiment is designed to assess the method’s precision in distinguishing closely spaced sources while maintaining azimuthal accuracy with fixed elevation angle and azimuth angles $\theta$ varying from ${10}^{°}$ to ${90}^{°}$, while the other data settings remain the same as in the previous simulation. As demonstrated in Figure 8, the proposed method achieves angular resolution performance statistically equivalent to the ML benchmark while outperforming ESPRIT across the tested azimuth range, which again verifies the accuracy of the proposed method.

The fundamental computational advantage of our proposed method lies in its exclusive use of real-valued operations for signal subspace derivation. Traditional subspace-based algorithms rely extensively on complex-number arithmetic throughout their processing chains—from covariance matrix construction to eigenvalue decomposition and spectral search. Each complex multiplication requires four real multiplications and two real additions, while complex matrix decompositions demand substantially higher floating-point operations than their real-valued equivalents. By systematically transforming the entire computational workflow into the real domain through unitary matrix operations, our approach bypasses this inherent inefficiency. To quantitatively validate these theoretical advantages, we designed a comprehensive runtime benchmark comparing our method against two techniques: ESPRIT, ML. Experiments were conducted on a standardized computing platform with fixed hardware specifications to ensure fair comparison. The average CPU runtime is measured across 500 independent trials under identical signal conditions. This metric serves as a direct proxy for computational complexity, reflecting the aggregate costs of matrix construction, decomposition, and parameter estimation. Figure 9 shows that the proposed algorithm exhibits exceptional stability in computational load when the array number increases from 12 to 30. As seen from the figure, the running time of the ML method increases quickly, with the array number becoming larger, while the ESPRIT method grows slowly. Our proposed method maintains near-constant execution time. To give a further detailed view, we also give the running time under different array sizes in

Table 2. Comparison of Running Time (Second).

Method	M = 18	M = 24	M = 28
Proposed	0.007	0.009	0.009
ESPRIT	0.011	0.035	0.056
ML	0.051	0.255	0.354

, which again, unequivocally confirms the lowest computational complexity of our approach among the methods of comparison.

5. Conclusions

In conclusion, a novel and efficient technique has been proposed for estimating the 2D DOA of signals from a URA. The proposed method combines the Nyström method for obtaining the signal subspace and a unitary subspace-based algorithm for estimating the 2D DOA. By utilizing the Nyström method, we are able to acquire the signal subspace efficiently. The obtained subspace is then utilized in the unitary subspace-based algorithm, which makes use of the principal singular vectors and an iterative process to estimate the 2D DOA of the URA signals. Since the covariance matrix is computed with real-valued decomposition, the complexity of DOA estimation is significantly reduced. Simulation results have been presented to demonstrate the effectiveness and performance of the proposed method. These results validate the capability of the proposed technique in accurately estimating the 2D DOA of signals from a URA while achieving improved computational efficiency. The integration of the Nyström approximation with unitary subspace techniques represents a novel approach to balancing accuracy and complexity in 2D DOA estimation. With a fixed array size of M = N = 20, the proposed method demonstrates significant computational efficiency, requiring less than $50\%$ of the running time compared to conventional ESPRIT and only $6\%$ of the time required by ML methods, while maintaining similar performance. The method’s efficiency stems from both algorithmic innovations and careful exploitation of array geometry properties. Future work could explore adaptive parameter selection and robustness enhancements while maintaining the core computational benefits demonstrated in this study and multi-tone signal cases. The proposed technique offers a computationally efficient high-resolution angle estimation solution, particularly suitable for resource-constrained systems, such as modern radar and 5G beamforming applications, where both accuracy and real-time processing are paramount.