A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

Wang, Wei; Zhang, Ming; Yao, Bobin; Zheng, Tongxing; Fan, Xiaojiao

doi:https://doi.org/10.1155/2023/8622428

International Journal of Antennas and Propagation

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 8622428 | https://doi.org/10.1155/2023/8622428

A Low-Complexity Direction-of-Arrival Estimation Algorithm for Noncircular Signals via Subspace Rotation Technique

Wei Wang,¹Ming Zhang,²Bobin Yao,³Tongxing Zheng,⁴and Xiaojiao Fan¹

Academic Editor: Atsushi Mase

Received07 Nov 2022

Revised07 Mar 2023

Accepted08 Mar 2023

Published21 Mar 2023

Abstract

In this paper, we investigate the problem of the heavy computational burden of the direction-of-arrival (DoA) estimation for the noncircular (NC) signals. A novel low-complexity direction-of-arrival estimation algorithm for NC signals via subspace rotation technique (SRT) is proposed. The proposed algorithm divides the noise subspace matrix along its row direction into two submatrices, and the SRT is performed to get a new reduced-dimension noise subspace. Then, utilizing the separation of variables and the orthogonality between the reduced-dimension noise subspace and the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is derived to estimate DoAs. As the size of the block matrices of the noise subspace matrix has a great impact on the computational complexity of the spectral search, the optimal number of rows of the block matrices is determined. The proposed algorithm not only avoids the two-dimensional spectral search but also efficiently removes the redundancy computations in the one-dimensional spectral search. Theoretical analysis and simulation results show that the proposed algorithm can significantly improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation for the NC signals, especially in scenarios where large numbers of sensors are applied.

1. Introduction

Direction-of-arrival (DoA) estimation has been widely used in various communication applications such as radar, sonar, target location, and wireless communication, which is an extremely important topic in the field of array signal processing [1–9]. High resolution methods for DoA estimation include multiple signal classification (MUSIC) [10], root-multiple signal classification (Root-MUSIC) [11], estimating signal parameters via rotational invariance techniques (ESPRIT) [12], and so forth. However, these traditional algorithms based on the premise of the narrow-band or wide-band sources have not exploited the properties of the incoming sources [13–16]. As the noncircular (NC) signals such as binary phase shift keying (BPSK) signals, amplitude modulation (AM) signals, and multiple amplitude shift keying (MASK) signals have been more and more widely utilized in practical communication, the NC property of incoming sources has been considered in DOA estimation [17–20]. The knowledge of the elliptic covariance matrix, which is not equal to zero, can be used to improve the estimation performance by increasing the dimension of the received signal matrix and extending the array aperture. A two-dimensional noncircular MUSIC (2D-NC-MUSIC) algorithm exploiting the NC property of the incoming sources was proposed in [21], which needs a two-dimensional spectral search, bringing huge computational complexity. In order to decrease the computational complexity of the 2D-NC-MUSIC algorithm, the noncircular MUSIC (NC-MUSIC) algorithm was also presented in [21] to convert the two-dimensional spectral search into the one-dimensional spectral search. A polynomial rooting NC-MUSIC algorithm proposed in [22] can replace the two-dimensional spectral search with the polynomial rooting so as to reduce the computational complexity of the 2D-NC-MUSIC algorithm to some extent. In [23–25], convex optimization was utilized to avoid the two-dimensional spectral search and greatly improves the computational effectively. However, the required one-dimensional spectral search still has the high computational cost. A SRTRD-NC-MUSIC algorithm was presented in [26], which enhances the computational efficiency by means of the conversion of the two-dimensional peak searching into the one-dimensional peak searching and the removal of some redundancy computations of the one-dimensional peak searching, but still remains redundancy computations in peak searching. Based on the sparsity of target azimuth information in the spatial domain, the methods proposed in [27, 28] use the spatial dictionary and the sparse signal reconstruction to finish the DoA estimation, which can reduce the computational complexity only on the premise of precisely compensating for the NC phase caused by transmission delay.

In order to reduce the computational complexity of the 2D-NC-MUSIC algorithm, we improve the SRTRD-NC-MUSIC algorithm and propose a novel low-complexity DoA estimation algorithm called the SRTRD-p-NC-MUSIC algorithm, which can further avoid redundancy computations in spectral search. In this paper, we divide the noise subspace matrix along its row direction into two submatrices and the subspace rotation technique (SRT) [29] is used to construct a new reduced-dimension noise subspace. As the new reduced-dimension noise subspace is also orthogonal to the space spanned by the columns of the extended manifold matrix, a new one-dimensional spectral search function is finally derived to estimate DOAs by means of the separation of variables. Considering that the size of the block matrices of the noise subspace matrix has a great impact on the dimensions of the new reduced-dimension noise subspace and the computational complexity of the spectral search, we have a discussion about the numbers of rows of the block matrices, and the optimal number of rows of the block matrices is derived. The SRTRD-p-NC-MUSIC algorithm can reduce the computational complexity to less than 3% as compared to the 2D-NC-MUSIC algorithm on the premise of ensuring the accuracy of DOA estimation. Especially, the efficiency advantage of the new algorithm is more obvious for the case that large numbers of sensors are required.

1.1. Notation

Throughout the paper, we utilize upper-case bold letters to denote matrices (e.g., A), lower-case bold letters for vectors (e.g., a) and upper-case double-line characters for sets (e.g., ℂ). (·), (·)^T, (·)^H, and (·)^† represent complex conjugate, transpose, conjugate transpose, and pseudoinverse, respectively. E {·} is used to represent the statistical expectation. ‖.‖ and det (·), respectively, indicate the norm of a vector and the determinant of a matrix. diag {.} stands for a diagonal matrix composed of the embraced elements. Span (·) is the space spanned by the columns of a matrix, dim (·) is the dimension of a linear space and rank (·) is the rank of a matrix. I_M represents a M × M identity matrix and 0_M×N denotes a M × N matrix with all zero elements.

2. Data Model and 2D-NC-MUSIC Algorithm

2.1. Data Model

The noncircular signals have been extensively used in various modern communication systems. As the noncircular signals have the property that the pseudocovariance is nonzero, the effective array aperture can be extended by means of increasing the number of virtual sensors [17–20]. Considering the impact of the actual transmission channel, the received signals can be represented aswhere S_R (t) is the real signal, and the phase φ is the sum of the initial phase of the signal and the additional phase of the transmission channel. Consider an M-element uniform linear array (ULA) with the interelement spacing d and select the leftmost sensor of the ULA as the reference one. K independent narrow-band far-field noncircular signals S_k (t) (k = 1, 2, … , K) are impinging on the ULA from the different directions θ_k (k = 1, 2, … , K), as shown in Figure 1. The received signals of the ULA can be expressed aswhere is the noncircular incident signal vector, is the manifold matrix with the steering vector (k = 1, 2, … , K) and λ is the wavelength of the carrier wave. represents the white Gaussian noise vector with zero mean and variance σ². and with () being the noncircular phase of the kth incident signal.

Considering that a noncircular signal has the characteristics of E {S (t) S^T (t)} ≠ 0 and E { (t) S^H (t)} ≠ 0, we concatenate the array output and its conjugation to increase the number of available sensors [13–20]. So a new extended vector is obtained, as shown in the following:where and .

2.2. 2D-NC-MUSIC Algorithm

The extended covariance matrix is expressed as

According to the eigenvalue decomposition (EVD) of the matrix , the label (4) can be further rewritten aswhere , , , and (i = 1, 2, …, 2M) is the eigenvector corresponding to the eigenvalue λ_i (i = 1, 2, …, 2M). and are the signal subspace and the noise subspace, respectively. In practice, we usually use the L snapshots of the received signal X (t) to estimate the extended covariance matrix R_Y as follows:

Based on the orthogonality property between the signal subspace and the noise subspace, the two-dimensional NC-MUSIC function can be constructed to estimate the DOAs of the noncircular signals as follows [21]:where

The K maxima referring to source DoAs can be obtained by the two-dimensional spectral search. It is observed that although the 2D-NC-MUSIC algorithm can extend the effective array aperture and greatly improve the resolution of the DoA estimation, the expensive computation cost of two-dimension spectral search limits the practical application of the 2D-NC-MUSIC algorithm in the high-real-time processing scenarios [30–32].

3. SRTRD-p-NC-MUSIC Algorithm

In this section, we propose a new algorithm to reduce the computational complexity of the 2D-NC-MUSIC algorithm. We first use the block matrix method and SRT to construct a reduced-dimensional noise space. Then, based on the orthogonality between the reduced-dimensional noise space and the space spanned by the columns of the extended manifold matrix, the separation of variables is utilized to convert the two-dimensional spectral search into the one-dimensional spectral search. Finally, because the number of rows in the block matrices has a great impact on the computational complexity of the proposed algorithm, we discuss about the selection of the number of rows in the block matrices and give the optimum value.

3.1. SRT and Reconstruction of the Noise Subspace

The noise subspace can be divided along its row direction into two block matrices, as shown in the following:where and . Obviously, , and if 1 ≤ p ≤ M − K, the matrix is row full rank. According to the matrix theory [33], we know that the matrix has a right inverse matrix satisfying . In order to construct the new noise space span , we use the matrix to weight the column vectors of the noise space with the following form:

The abovementioned label (10) means we rotate the space span by an angle in the 2M-dimensional space. Consider , then the new constructed space span is a subset of the space span (V_N):

As we know, and , so the space span (B) is orthogonal to the reconstructed space span :where i = 1,2, …, K.

3.2. Construction of the Reduced-Dimension Spectrum Function

Based on the orthogonality property between the reconstructed noise subspace and the column space spanned by the matrix B, the two-dimensional spectrum function is defined according to label (13) [27]:

The estimation of two-dimensional DOAs of noncircular signals can be acquired by searching the minimum of . Based on the separation of variables, is represented aswhere

The label (13) can be further rewritten as

Since is a Hermitian matrix and is a nonnegative-definite quadratic form, the minimum of can be determined by the minimum eigenvalue of the matrix . When θ is one of the real DOAs of the received signals, the minimum eigenvalue of the matrix is zero. Then the minimum of the determinant of the matrix is also zero [31]:

We construct a new reduced-dimension spectrum function as follows:

By only one-dimensional peak searching, the DOAs of the noncircular signals can be obtained by searching K minimum values of (18).

Compared with the 2D-NC-MUSIC algorithm, the proposed algorithm can not only reduce the dimension of the noise subspace, but also need only one-dimensional peak searching to obtain the estimation of the DOAs of the received signals. The following analysis will show that the proposed SRTRD-p-NC-MUSIC algorithm can significantly reduce the computational complexity of the 2D-NC-MUSIC algorithm and dramatically improve the computational efficiency on the premise of ensuring the accuracy of DoA estimation.

3.3. Simplification of the SRTRD-p-NC-MUSIC Algorithm

Substituting label (9) into label (10), label (10) can be further represented aswith ∈ℂ^{(M−P)×(M+P)}. Then, we can divide the matrix into block matrices as follows:where

Based on label (19) and label (20), we can obtain

Furthermore, we substitute label (22) into label (18) to getwhere the tag θ representing the angle is omitted for notational convenience. Then, the proposed SRTRD-NC-MUSIC algorithm can be further simplified, as shown in the following:where . From label (24), we can see that the computational complexity of the proposed SRTRD-NC-MUSIC algorithm is greatly affected by the selection of because of the relation between the value of and the dimensions of , , , and . The following analysis shows that when the value of is optimal, the proposed SRTRD-NC-MUSIC algorithm can dramatically improve the computational efficiency as compared to the 2D-NC-MUSIC algorithm.

3.4. Optimal Selection of

In this section, we further discuss about the selection of the value of to dramatically improve the computational efficiency. Note that the complexities are given in terms of complex-valued flops [26–29]. For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the EVD of the covariance matrix R_Y can use the fast subspace decomposition (FSD) technique [34], and O (4M²K) flops are required to use the FSD technique of the covariance matrix R_Y. Compared with the 2D-NC-MUSIC, we need to additionally construct the new noise subspace and calculate the matrix for the SRTRD-p-NC-MUSIC. As we know, the dimensions of the matrix and the matrix are 2M × (2M − K) and (M + p) × (2M − K), respectively. Then, the construction of the matrix needs Ο [(M − p) (M + p) (2M − K)] + O [2M (2M − K)] flops [29].

For the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC, the spectral search step is needed to estimate the DOAs of the received signals. Assuming there are J points within the search range [−90° and 90°] of the incident angle θ and the search range [0, 2π] of the noncircular phase φ, respectively, the 2D-NC-MUSIC requires O [J² (2M + 1) (2M − K)] flops to finish the spectral search. For the SRTRD-p-NC-MUSIC, it has to compute , , and for each search point. Then, the cost of computing , , and are O [(M − + 1) (M + p)] flops, O ((M − + 1) ) flops, and O ((M − + 1) M + 1) flops, respectively. Consequently, the spectral search of the SRTRD-p-NC-MUSIC costs O [J (−2² + 2 + 2 M² + 2M + 1)] flops.

Above all, we can obtain the number of total flops of the 2D-NC-MUSIC and the SRTRD-p-NC-MUSIC with the following form:

As it usually satisfies J >> M > K in practice, the label (25) can be further approximated as follows:

Obviously, the selection of the value of p depends on the following optimization problem:

As shown in Figure 2, the label (26) is a quadratic function of p which is monotone decreasing if . Considering that p satisfies 1 ≤ p ≤ M − K, we can get since the number of sensors and the number of sources usually satisfy M > K. Therefore, we can further get the optimal choice of p, which is represented as

3.5. Analysis of Computational Complexity

When = _opt, we substitute label (28) into label (26) and the computational complexity of the proposed SRTRD-p-NC-MUSIC can be further written, as shown in the following:

Figure 3 illustrates the computational complexity comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC [21], RD-NC-MUSIC [23], and SRTRD-NC-MUSIC [26] with different numbers of sensors and different numbers of sources, where J = 1000 is considered. The RD-NC-MUSIC and the SRTRD-NC-MUSIC require flops [23] and flops [26], respectively.

From Figure 3, we can see that the computational complexities of different algorithms increase with the number of sensors. Compared with the 2D-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can reduce the computational cost dramatically. Especially, the efficiency advantage of the proposed SRTRD-p-NC-MUSIC is more obvious in scenarios where the large numbers of sensors are applied. Moreover, the proposed SRTRD-p-NC-MUSIC can reduce the computational complexity to about 20% as compared to the RD-NC-MUSIC. Compared with the SRTRD-NC-MUSIC, the proposed SRTRD-p-NC-MUSIC can enhance the computational efficiency significantly when the large numbers of sensors are applied. In addition, unlike the 2D-NC-MUSIC and the RD-NC-MUSIC, the computational complexity of the proposed SRTRD-NC-MUSIC increases as the number of sources increases.

3.6. Summary of the Proposed Algorithm

The main steps of the proposed SRTRD-p-NC-MUSIC are shown as follows: Step 1. Estimate the extended covariance matrix R_Y according to label (6). Step 2. Perform the eigenvalue decomposition of the matrix R_Y according to label (5) to acquire the noise subspace . Step 3. Obtain the optimal choice of according to label (28). Divide the matrix according to label (9) to obtain . Step 4. Construct the new noise subspace according to label (19). Divide the matrix according to label (20). Step 5. Construct the one-dimensional spectral search function according to label (24). Estimate the DoAs of the noncircular signals by searching K minimum values of label (24).

4. Simulation Results

In this section, we will demonstrate the effectiveness of the proposed SRTRD-p-NC-MUSIC, compared with the 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC. For all simulations, a ULA with spacing d = 0.5λ is used. In all experiments except for Figure 4, we set the number of sensors M = 8. We assume that the sources are narrow-band BPSK signals with equal power in the presence of the additive white Gaussian noise. All the simulation results are averaged via 3000 Monte Carlo runs. The Root Mean Square Error (RMSE) is defined aswhere N_m is the total number of the Monte Carlo simulations, and , are the real value and estimated result of the kth source in the ith simulation, respectively. For the signals incoming from two similar directions, it is considered as the successful estimation when satisfyingwhere (·) is the spectrum function, and θ₁, θ₂ are the incoming directions of the signals, respectively [29].

Figure 5 shows the DoA estimation results of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC, where K = 3 with θ = {−45°, −15°, 5°} and the snapshots L = 1000. The signal-to-noise ratio (SNR) is set to 5 dB. From Figure 5, the proposed SRTRD-p-NC-MUSIC can resolve all sources sufficiently well. Compared with the 2D-NC-MUSIC, the spatial spectrum of the SRTRD-p-NC-MUSIC has a sharper peak. In addition, the SRTRD-p-NC-MUSIC performs close to the 2D-NC-MUSIC and improves the computational efficiency significantly.

Figures 4, 6, and 7 indicate the RMSE performance comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC, where K = 2 sources with θ = {−15°, 20°}. In Figure 6, the number of snapshots is fixed at 500, and the SNR is varied from −10 dB to 20 dB. It is clearly shown that the RMSEs of all algorithms in Figure 6 decrease as the SNR increases. Furthermore, the SRTRD-p-NC-MUSIC shows close RMSEs to the 2D-NC-MUSIC, and it has the better RMSE performance than the other three algorithms. In Figure 7, the SNR is fixed at 10 dB, and the number of snapshots is varied. From Figure 7, when the number of snapshots increases, the extended covariance matrix can be estimated more accurately, and the RMSEs of all algorithms in Figure 7 decrease. At the same time, the SRTRD-p-NC-MUSIC can achieve a better RMSE performance, close to that of the 2D-NC-MUSIC, than the other three algorithms. In Figure 4, the number of sensors is varied from 5 to 15, where SNR = 10 dB and L = 500. It is indicated that the RMSE performance of all algorithms in Figure 4 becomes better with increasing the number of sensors because the increasing number of sensors expands the aperture of the ULA. In addition, the SRTRD-p-NC-MUSIC still has a closer RMSE performance to the 2D-NC-MUSIC in comparison with the other three algorithms. Obviously, it can be seen clearly that the computational efficiency of the SRTRD-p-NC-MUSIC is improved without sacrificing too much estimation accuracy.

In Sections 3.4 and 3.5, we have analysed how the choice of the value of p affects the computational complexity of the proposed SRTRD-p-NC-MUSIC. In order to investigate the estimation accuracy of the proposed SRTRD-p-NC-MUSIC, in Figure 8, we further compare the RMSE performance of the proposed algorithm with different choices of the value of , where K = 2 sources with θ = {−15°, 20°}. In Figure 8(a), we set the number of snapshots to be 500 and the SNRs from −10 dB to 20 dB. It can be seen that as the SNR increases, the RMSEs of all choices of the value of decrease. In Figure 8(b), we set the SNR to be 10 dB, and the number of snapshots is varied. It is observed that the RMSEs of all choices of the value of p decrease with increasing the number of snapshots. In addition, from Figure 8(a) and Figure 8(b), we can see that when the choice of the value of p is closer to the optimum p_opt, the RMSE performance of the proposed algorithm is much better. This is because as the value of p is closer to the optimum _opt, the orthogonality between the reconstructed noise subspace and the space spanned by the columns of the extended manifold matrix is more improved [29, 35].

(a)

(b)

In Figures 9 and 10, there are two signals impinging upon the ULA from the directions θ₁ = 10° and θ₂ = 11°. Here, the number of the snapshots is 500. Figure 9 illustrates the DoA resolution performance versus different SNRs, where two sources are located at similar directions. It is can be observed that these four algorithms in Figure 9 have a 100% resolution probability to distinguish the two sources when the SNR is higher than 5 dB. However, the SRTRD-p-NC-MUSIC can obtain much better spatial resolution than the other three algorithms when the SNR is lower than 0 dB. This is because the reconstruction of the noise subspace improves the orthogonality between the signal subspace and the noise subspace and the dimension of the reconstructed noise subspace is reduced significantly, which leads to a sharper peak of the spatial spectrum [29, 35]. In Figure 10, the DoA estimation results of the proposed SRTRD-p-NC-MUSIC and the 2D-NC-MUSIC are depicted for 10 simulations, where SNR = 5 dB. We can see that the spectral peak of the SRTRD-p-NC-MUSIC is sharper than that of the 2D-NC-MUSIC, and the SRTRD-p-NC-MUSIC achieves a better performance than the 2D-NC-MUSIC when distinguishing two signals from similar directions. The simulation results in Figure 10 also verify the experimental results in Figure 9.

Figure 11 depicts the resolution probability comparison versus different intersource spacing ∆θ among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC, where SNR = 0 dB and the snapshots L = 500. It is considered that two sources are located at 20° and (20+Δθ)°, where Δθ represents the intersource spacing. We can see that when the intersource spacing Δθ is not less than 1.8°, these four algorithms have a 100% resolution probability to identify the two sources. Especially, the proposed SRTRD-p-NC-MUSIC can distinguish the two sources accurately when the intersource spacing Δθ is greater than 1°. Compared with the other three algorithms, the SRTRD-p-NC-MUSIC can achieve much better spatial resolution when the intersource spacing is less than 1° due to the better orthogonality between the signal subspace and the noise subspace and the sharper peak of the spatial spectrum [29, 35], which is also verified by the simulation results in Figure 10.

Figure 12 indicates the running time comparison among the proposed SRTRD-p-NC-MUSIC, 2D-NC-MUSIC, RD-NC-MUSIC, and SRTRD-NC-MUSIC versus different number of sensors. The simulation parameters are set as SNR = 10 dB, L = 500, K = 2 and θ = {−10°, 25°}. The searching interval is fixed at 0.05°. The running times in Figure 12 are given by running the MATLAB codes on a PC with an Intel (R) Core (TM) i5-5300U processor, a 2.3 GHz CPU, and 2 GB RAM in the same environment. It can be observed that the SRTRD-p-NC-MUSIC costs much lower running time than the other three algorithms. Specifically, as the number of sensors increases, the running times of all algorithms in Figure 12 increase. Simultaneously, the efficiency advantage of the SRTRD-p-NC-MUSIC is more obvious with increasing the number of sensors, which are consistent with the aforementioned computational complexity analysis. In Figure 12(a), compared with the 2D-NC-MUSIC, the SRTRD-p-NC-MUSIC only needs a one-dimensional peak searching, and furthermore, the reconstruction of the noise space efficiently avoids the redundancy computations in the one-dimensional peak searching due to the rank deficiency of the noise subspace. Therefore, the SRTRD-p-NC-MUSIC can reduce the computational complexity to less than 3% as compared to the 2D-NC-MUSIC. In Figure 12(b), the dimension of the reconstructed noise space of the SRTRD-p-NC-MUSIC is descended and the redundancy computations of the one-dimensional peak searching of the SRTRD-p-NC-MUSIC are removed, and so, the computational complexity of the SRTRD-p-NC-MUSIC is reduced to less than 20% as compared to the RD-NC-MUSIC. In Figure 12(c), compared with the SRTRD-NC-MUSIC, the SRTRD-p-NC-MUSIC can more completely remove the redundancy computations in the one-dimensional peak searching according to a novel block matrix method. Consequently, from Figure 12(c), we can see that the computational complexity of the SRTRD-p-NC-MUSIC is lower than that of the SRTRD-NC-MUSIC.

(a)

(b)

(c)

5. Conclusions

In this paper, we have proposed the SRTRD-p-NC-MUSIC algorithm for the DoA estimation of NC signals in order to reduce the computational complexity of the conventional 2D-NC-MUSIC algorithm. The proposed SRTRD-p-NC-MUSIC algorithm not only converts the two-dimensional peak searching into the one-dimensional peak searching, but also utilizes the block matrix method and SRT to efficiently avoid the redundancy computations in the one-dimensional peak searching due to the rank deficiency of the noise subspace. Compared with the 2D-NC-MUSIC algorithm, the proposed SRTRD-p-NC-MUSIC algorithm can dramatically improve the computational efficiency on the premise of ensuring the accuracy of DOA estimation, especially in scenarios where large numbers of sensors are applied.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the Natural Science Special Foundation of Shaanxi Provincial Department of Education (Grant no.20JK0966), the Natural Science Foundation of Shaanxi Province (Grant no. 2023-JC-QN-0756), and the National Natural Science Foundation of China (Grant nos. 62172338 and 62072378).

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Copyright © 2023 Wei Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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