Research on 2-D Direction of Arrival (DOA) Estimation for an L-Shaped Array

Two-dimensional (2-D) direction of arrival (DOA) estimation has a wide range of applications in the fields of communication, sonar, hydroacoustic localization, etc. [1,2,3,4,5,6,7,8,9]. Circular arrays, L-shaped arrays, rectangular arrays, etc., have been proposed in recent decades [10,11,12,13,14,15,16,17,18]. The L-shaped array configuration boasts superior angular resolution capabilities and is straightforward to construct [19,20,21], making 2-D direction determination for this type of array the most widely pursued topic in research.

In order to realize 2-D DOA estimation more accurately, many scholars have proposed various 2-D DOA estimation algorithms. In [22] introduces an innovative technique for automatic pairing, which employs the collaborative singular value decomposition technique to handle a pair of cross-correlation matrices, aiming to enhance the accuracy of 2-D DOA estimation. This approach surpasses the traditional CCM-ESPRIT algorithm, particularly in scenarios with a low SNR and minimal angular disparity. In [23], a new sparse L-shaped electromagnetic vector sensor MIMO radar framework for 2-D DOA estimation is proposed. Existing electromagnetic vector sensor—multiple inputs and multiple outputs—radar arrays struggle to strike a good balance between estimation accuracy and computational burden. The sparse L-shaped array topology in this paper has a large sensor spacing, which can effectively reduce the mutual coupling effect and improve the angular resolution. A fast algorithm is proposed in this article to achieve more accurate DOA estimation by combining ambiguous and high-resolution directional cosine estimation with unambiguous but low-resolution estimation. Ultimately, the theoretical derivation is validated by simulation results, showing that the method outperforms existing techniques with better flexibility and accuracy. In [24], a 2-D DOA-assisted framework based on electromagnetic vector sensors and monostatic multiple-input–multiple-output radar for anonymous unmanned aerial vehicle localization is proposed. The framework considers multipath effects and employs the rearrangement multiple signal classification algorithm, which significantly improves the localization accuracy and robustness for antenna arrays with different geometric configurations. In [14], 2-D DOA estimation using polarized uniform rectangular arrays is proposed to address the challenges of coherent signals under multipath propagation. The rank defect of the source matrix is solved by modeling and rearranging the parallel factors, and the vector cross-product-assisted rotation invariant technique is used to achieve high-accuracy estimation. In [25] presents an improved algorithm for 2-D DOA estimation using electromagnetic vector sensor rectangular arrays under multipath propagation conditions. The method solves the rank deficiency problem by matrix rearrangement and is applicable to single snapshot scenarios, demonstrating superior accuracy and computational efficiency. In [26], a novel parallel synthetic mutual mass array for 2-D DOA and polarization estimation is proposed, aiming to improve the multiparameter estimation performance. By reducing the mutual coupling between antennas and the hardware cost, combined with the compressed sensing method, it demonstrates superior performance compared to existing techniques for target detection and identification in complex electromagnetic environments. In [27] presents a low-computational-complexity 2-D DOA estimation method designed for L-shaped arrays. The technique leverages the symmetrical properties of the array manifold matrix’s conjugate to amplify the functional array opening, merging the propagator approach with the ESPRIT technique for spontaneous pairing, thereby eliminating the need for spectral scanning and repetitive cycles. Tests have confirmed that this approach surpasses the performance of traditional algorithms. In [28], a new 2-D DOA estimation method is proposed for non-coherent distributed sources considering gain-phase perturbations in massive multiple-input and multiple-output systems. By establishing a subspace framework and constrained optimization, the authors jointly estimate the nominal DOA, angular expansion, and gain-phase perturbations, providing a solution that is efficient and suitable for real-time applications. In [29] proposed a joint 2-D DOA estimation method for L-shaped arrays based on a compressed perception framework. By introducing a separable observation model, the authors decompose the joint spatial–temporal parameters into three small matrices, which are estimated using L1-paradigm minimization and Bayesian compressed perception algorithms, and numerical simulations verify their effectiveness and superiority. Using sparse L-shaped arrays, in [30] presents a novel technique for 2-D DOA estimation. The technique lowers the computational cost and increases the accuracy of the estimation by utilizing the cross-covariance matrix to create a signal subspace from a combination of a sparse and a uniform linear array. In [31], an innovative scheme for 2-D DOA and polarization detection is introduced for millimeter-wave polarized MIMO arrays, utilizing a compact sampling methodology. The method provides an efficient estimation scheme with low computational complexity and robustness to sensor position errors for a wide range of sensor geometries by reducing the data dimensions, computing the signal subspace, and exploiting the rotational invariance property. In [32] proposed a new method for 2-D DOA estimation based on active hypersurfaces that utilizes non-uniform periodic time modulation to reduce hardware complexity. With amplifier integration, the system achieves high-accuracy (maximum error of 0.31°) multi-source direction estimation for applications such as IoT, wireless communication and radar. In [33] presents a novel polarized smart reflective surface architecture designed for the 2D-DOA estimation of non-line-of-sight signals. The authors develop a normalized vector cross-product estimator that avoids complex data recovery and grid searches, thus improving efficiency. By deploying electromagnetic vector sensor arrays at both the base station and the IRS, the article demonstrates that the method is capable of efficient 2-D-DOA estimation without requiring any a priori knowledge of the base station-IRS channel. In [34] proposed a deep-learning-based 2-D DOA estimation method that uses an L-shaped array to simultaneously estimate the elevation and azimuth angles of a signal. With two classification neural networks, the method performs superiorly in low SNR and highly correlated signal environments, significantly outperforming existing conventional and deep learning methods. In [35] presents a resource-efficient tensorized deep neural network for the 2-D DOA estimation of uniform rectangular arrays. The trainable parameters are compressed by propagating the covariance tensor to the hidden state tensor via an inverse tucker decomposition. The network significantly reduces the number of parameters, improves training speed, and reduces GPU memory usage while maintaining good estimation accuracy under non-ideal conditions. Simulation results show that the method outperforms traditional matrix-based neural networks and model-based methods in terms of training efficiency and computational performance. In [36] introduced an innovative design of a triple-concurrent compact nested antenna array to enhance the precision in 2-D DOA estimation. This configuration addresses the issue of angular resolution ambiguity through the expansion of the array’s effective area and the minimization of element interaction. By integrating the principles of BOMP with the CMT-BCS algorithm, the experimental outcomes indicate a superior performance compared to conventional methodologies.

In order to obtain a more accurate 2-D angle estimation, in this paper, a 2-D DOA algorithm based on interpolation complementation for L-shaped arrays is proposed, which first utilizes larger apertures and higher degrees of freedom. The complementary zeroes are utilized to complement the pairs of matrices. Finally, the DOA estimation of 2-D corners is realized by using the effective array aperture expansion technique. The computational modeling outcomes indicate that the proposed method excels in both the precision of estimation and the quality of angular discrimination. The primary contributions presented in this article can be outlined as follows:

(1) This paper proposes a covariance matrix reconstruction algorithm based on zero completion. The algorithm makes full use of the independent virtual array elements of the L-type array structure based on coprime arrays for DOA estimation.

(2) We utilize the zero-complement reconstructed covariance matrix and then combine the propagator method and ESPRIT method for DOA estimation. The method realizes 2-D angle auto-matching.

(3) Computational modeling results show that the proposed 2-D DOA estimation algorithm performs well in improving angular accuracy. In addition, the algorithm outperforms conventional algorithms in terms of direction-of-arrival estimation performance.

The remainder of this paper is organized as follows: The second part presents the signal model. The third part presents the proposed methodology. The fourth part presents the simulation. The paper concludes with a summary.

A uniform linear orthogonal array with a d spacing between two sensors forms an $x-z$ plane L-shaped array configuration, as shown in Figure 1. The e L-shaped coprime array located in the $x-z$ plane consists of two coprime arrays. Each conjugate array $S_i$ has $|S_i| =2M_i+N_i-1$ physical sensors and consists of two sparse ULAs. Note that $M_i$ and $N_i$ are mutually prime integers, and the two sparse ULAs contain $2M_i$ and $2N_i$ physical sensors with corresponding spacings of $N_{i}d$ and $M_{i}d$, where $d=\lambda /2$ and $\lambda$ denotes the carrier wavelength.

Suppose there are Q unrelated far-field narrowband source signals incident onto an L-shaped array from the $\{({\theta }_q,{\phi }_q),q\in \langle 1,Q\rangle \}$ directions, where ${\theta }_q\in [0,\pi ]$ and phi ${\phi }_q\in [0,\pi ]$ represent the azimuth and elevation angles of the qth source signal, respectively. Denote ${\mathbf{S}}_x\left(t\right)$ and ${\mathbf{S}}_z\left(t\right)$ as the information vectors captured by the coprime array ${\mathbf{S}}_x$ and the coprime array ${\mathbf{S}}_z$ at the instant t, respectively, and these can be formulated as follows: where ${\mathbf{n}}_{\mathbf{x}}\left(t\right)$ and ${\mathbf{n}}_{\mathbf{z}}\left(t\right)$ are additive Gaussian white noise obeying $\mathcal{CN}(\mathbf{0},{\sigma }_n^2\mathbf{I})$,

Next, $x\left(t\right)$ and $z\left(t\right)$ can be represented as where ${\mathbf{R}}_{\mathbf{ss}}=E\left[\mathbf{s}\left(t\right){\mathbf{s}}^H\left(t\right)\right]=diag\left(\mathbf{p}\right),$ and $\mathbf{p}={[{\sigma }_1^2,{\sigma }_2^2,\cdots ,{\sigma }_Q^2]}^T$ is the source signal power vector.

Considering that there is no precise covariance matrix $R_x$, in practice, its sample version can be used to approximate it where T is the number of snapshots.

In this section, we first use the mutual prime array arrival method of virtual array interpolation to achieve completion based on the virtual common array signal y. Then, the effective array aperture expansion technique is used for 2-D angle estimation.

Initially, the efficacy of the 2-D direction detection is assessed utilizing the root mean square error (RMSE), and its formulation is specified as where $K=500$ represents the number of Monte Carlo trials and ${\widehat{\theta }}_q^k$ and ${\widehat{\phi }}_q^k$ represent the qth estimated azimuth and elevation angles for the kth trial, respectively. The operating system used is Windows, running on an Intel(R) Core(TM) i7-8700 CPU at 3.20 GHz. The programming language was developed under Matlab 2022 (a).

We evaluated the various algorithms’ resolutions in the first simulation. Ten sensors were used, and T = 500 snapshots were taken. The signal-to-noise ratio (SNR) is fixed to 0 dB, and the 10 sources are uniformly distributed in the range [30°–150°], i.e., $({30}^{\circ },{30}^{\circ })$, $(43.{33}^{\circ },43.{33}^{\circ })$, $(56.{67}^{\circ },56.{67}^{\circ })$, $({70}^{\circ },{70}^{\circ })$, $(83.{33}^{\circ },83.{33}^{\circ })$, $(96.{67}^{\circ },96.{67}^{\circ })$, $({110}^{\circ },{110}^{\circ })$, $(123.{33}^{\circ },123.{33}^{\circ })$, $(136.{67}^{\circ },136.{67}^{\circ })$, $({150}^{\circ },{150}^{\circ })$.

The outcomes of DOA estimation utilizing the discrete inverse Fourier transform (DFT) [4] technique are depicted in Figure 2a, where two out of ten sources are not recognized and three sources are recognized with large errors. The recognition results of the joint singular value decomposition (JSVD) [22] algorithm are shown in Figure 2b. For the JSVD recognition results, there are five signal sources with large errors. the recognition results of the low-computation-complexity method (LCCM) [27] algorithm are shown in Figure 2c. It can be seen that there are three signal sources with large recognition errors. The recognition angles of the proposed method are shown in Figure 2d, and only one estimated angle has a slight error. It is evident that the suggested approach surpasses current techniques in terms of a more precise estimation of the DOA.

In the experiments in the spatial spectrum, we consider three uncorrelated source signals [25°,15°], [45°,35°] and [70°,55°]. The SNR and snapshots are configured as 5 dB and 1000, respectively. Figure 3 shows that the DFT and JSVD algorithms have large sidelobes, and the sidelobes of the proposed algorithm are minimized. In summary, the proposed algorithm achieves better DOA estimation performance.

The DOA performance is compared with the known algorithms, including DFT, JSVD, and LCCM, to validate the DOA accuracy of the proposed algorithm. Assuming 10 physical arrays are used, it is estimated that 10 independent sources are $({30}^{\circ },{30}^{\circ })$, $(43.{33}^{\circ },43.{33}^{\circ })$, $(56.{67}^{\circ },56.{67}^{\circ })$, $({70}^{\circ },{70}^{\circ })$, $(83.{33}^{\circ },83.{33}^{\circ })$, $(96.{67}^{\circ },96.{67}^{\circ })$, $({110}^{\circ },{110}^{\circ })$, $(123.{33}^{\circ },123.{33}^{\circ })$, $(136.{67}^{\circ },136.{67}^{\circ })$, $({150}^{\circ },{150}^{\circ })$. A consistent total of 500 snapshots is maintained, while the SNR is adjusted across a range from −10 dB up to 30 dB to assess the system’s performance under diverse SNR conditions. In parallel, a variation of sample sizes from 100 to 1000 is tested, with the SNR held steady at 5 dB, to evaluate the system’s effectiveness with varying snapshot counts. Here, we also introduce CRB as a reference curve [40].

The illustrations in Figure 4a,b depict that the RMSE for DOA estimation diminishes as the SNR or the count of snapshots rises, confirming the efficacy of the suggested DOA estimation technique. Additionally, the RMSE metrics for the suggested approach are consistently lower compared to those of the DFT, JSVD, and LCCM algorithms in both scenarios, suggesting that the proposed technique outperforms the other three in terms of DOA estimation accuracy across varying SNR levels and snapshot counts.

In the simulation of the effect of the number of arrays on the algorithm, 8, 10, 12, and 14 physical sensors were used. The parameters were the same as in the previous simulation. As can be seen from Figure 5a,b, in both the SNR and the number of snapshots, a smaller RMSE is obtained with the increase in the number of physical arrays.

In the RMSE simulation with a CA structure and an ACA structure, for 10 physical sensors, CA takes M = 5, N = 6, while ACA takes M = 3, N = 5. Other simulation parameters are consistent with the above simulation. As can be seen from Figure 6a,b, the RMSE performance of the proposed zero-complement structure is better than the traditional array structure, indicating that the complement structure can increase the number of available virtual arrays and thus obtain better performance.

In the RMSE performance experiments at 1/Sqrt(T) and 1/Sqrt(SNR), we consider three uncorrelated source signals, [25°,15°], [45°,35°], and [70°,55°]. The snap count is set to 100 to 800 intervals of 100, and SNR is set to 1 to 20 intervals of 4. Figure 7 and Figure 8 show that the DFT algorithm has the largest RMSE followed by the JSVD algorithm, and the proposed algorithm has the smallest RMSE. The RMSE increases and tends to 1 as 1/Sqrt(T) and 1/Sqrt(SNR) increase, which is in line with the theoretical principles.

Finally, we compared the computation times of various algorithms. In this example, we considered five uncorrelated source signals impacting the array structure. The SNR and snapshots were configured to 5 dB and 500, respectively. Table 1 shows the computation times of various algorithms after executing 1000 Monte Carlo trials. It can be observed that the DFT algorithm has the highest computational complexity, followed by the JSVD algorithm. The proposed algorithm is close to the LCCM. In summary, the proposed algorithm demonstrates the best overall performance.

This study presents an innovative 2-D DOA estimation technique based on the zero-complement method. The method spontaneously pairs 2-D angles by reconstructing the covariance matrix. The proposed algorithm is compared with DFT, JSVD, and LCCM algorithms. The results of computational simulations show that the proposed technique has excellent angular resolution capability. Moreover, in terms of DOA estimation accuracy, the proposed technique surpasses the performance of conventional methods.