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To improve the performance of two-dimensional direction-of-arrival (2D DOA) estimation in sparse array, this paper presents a Fixed Point Continuation Polynomial Roots (FPC-ROOT) algorithm. Firstly, a signal model for DOA estimation is established based on matrix completion and it can be proved that the proposed model meets Null Space Property (NSP). Secondly, left and right singular vectors of received signals matrix are achieved using the matrix completion algorithm. Finally, 2D DOA estimation can be acquired through solving the polynomial roots. The proposed algorithm can achieve high accuracy of 2D DOA estimation in sparse array, without solving autocorrelation matrix of received signals and scanning of two-dimensional spectral peak. Besides, it decreases the number of antennas and lowers computational complexity and meanwhile avoids the angle ambiguity problem. Computer simulations demonstrate that the proposed FPC-ROOT algorithm can obtain the 2D DOA estimation precisely in sparse array.

Two-dimensional direction-of-arrival (2D DOA) [

Matrix completion [

This study proposes a Fixed Point Continuation Polynomial Roots (FPC-ROOT) algorithm. The proposed algorithm establishes a signal model of matrix completion for 2D DOA estimation based on low-rank property of target in two-dimensional space domain, which turns out to meet the Null Space Property (NSP) and ensure the feasibility of 2D DOA estimation via matrix completion. By contrast to conventional algorithms, the proposed algorithm obtains left and right singular vectors of received signal matrix directly from the output of matrix completion algorithm instead of eigendecomposing autocorrelation matrix of received signal, for the benefit of lower dimensions. Besides, the proposed algorithm can avoid scanning of the 2D spectral peak by solving the polynomial roots, which reduces the computational complexity. In addition, it can estimate target angle accurately with less number of array units by adopting matrix completion.

Suppose that the uniform rectangular array (URA) [

The uniform rectangular array.

Assume that there are multiple targets in the space. The number of snapshots is

When the power of noise matrix

With regard to the sparsity of signals, CS can sample signals at far lower sampling frequency than Nyquist sampling frequency and meanwhile reconstruct original signals precisely. In CS, the target to be recovered is a vector; however, on some practical occasions, it normally refers to a matrix and is sensitive to data missing, data corruption, and so on.

Matrix completion is an extension of CS. CS exploits the sparsity of signals; nevertheless, matrix completion utilizes the sparsity of matrix singular values, namely, the low-rank property, and reconstructs full matrix by solving nuclear-norm optimization.

Suppose

We sample elements from URA at random. Let the number of sampled elements be

According to (

If a signal model meets NSP, the rank minimization of this model is equivalent to its nuclear-norm minimization [

According to NSP, we are unable to recover a matrix if it belongs to the null space of projection

Suppose

Firstly, to solve problem (

Based on singular value decomposition, it is easy to know that left and right singular value matrices

So the left and right singular vectors of received signal matrix can be obtained directly from the output of matrix completion algorithm instead of eigendecomposing autocorrelation matrix of received signal and the computational complexity of the proposed algorithm obtained a corresponding reduction.

Consider a polynomial as

By solving the polynomial roots, the proposed algorithm can avoid scanning of the 2D spectral peak. Taking use of the orthogonality of signal subspace, the above equation can be converted to

The dimensionality of

Thus,

Similarly, it can be obtained that

In summary, the proposed FPC-ROOT algorithm in this paper can be programmed as follows.

Solve (

Construct the polynomial roots (

Solve (

Determine the targets angles

End.

In normal 2D-DOA estimation, DOA estimation is achieved through autocorrelation matrix of a vector which is transformed from the signal matrix. Suppose the signal matrix is

In this section, several simulations for 2D DOA estimation are conducted to demonstrate the feasibility and effectiveness of the proposed algorithm in sparse array. In these experiments, we sample 1200 elements from full array at random to formulate a sparse array. The full array is an URA of

In the first examples, 2D DOA estimation of sparse array is shown. Let three targets be in the space domain, whose elevation and azimuth angles of 2D-DOA are

2D DOA estimation via FPC-ROOT algorithm.

In the second experiment, recovery errors by matrix completion in sparse array with different SNR are examined. We recover a full matrix from a sparse matrix using FPC algorithm. Suppose the full matrix is

Recovery errors by matrix completion with different elements and SNR.

In the third experiment, root mean square error (RMSE) of 2D DOA estimation based on matrix completion is discussed. When 500 Monte Carlo simulations are performed, Figure

RMSE of 2D DOA estimation based on matrix completion.

In the last experiment, RMSE by FPC-ROOT algorithm with different SNR and elements is demonstrated. Let the number of snapshots be 50 and let 500 Monte Carlo experiments be implemented. Figure

RMSE versus SNR with different elements by FPC-ROOT algorithm.

In this paper, a FPC-ROOT algorithm is proposed based on matrix completion, which can achieve high accuracy of 2D DOA estimation with reduced antenna units. The proposed algorithm obtains left and right singular vectors of received signal by the output of matrix completion algorithm directly instead of eigendecomposing the autocorrelation matrix of received signal, for the benefit of lower dimensions. Besides, by computing polynomial roots, the proposed algorithm can avoid the scanning of two-dimensional spectral peak, which cuts down the computational complexity.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (Grant no. 61401204), Postdoctoral Science Foundation of Jiangsu Province (Grant no. 1501104C), Technology Research and Development Program of Jiangsu Province (Grant no. BY2015004-03), and the Fundamental Research Funds for the Central Universities (Grant no. 30916011319).