Two-dimensional DOA estimation of coherent sources using two parallel uniform linear arrays

2-D direction-of-arrival (DOA) estimation of incident coherent source signals has received increasing attention in radar, sonar, and seismic exploration [1–5]. Many high-resolution techniques, such as MUSIC [6] and ESPRIT [7], have achieved exciting estimation performance. However, the aforementioned methods assume the incident signals are independent, which would encounter performance degradation due to the rank deficiency when coherent signals exist. To decorrelate coherent signals, the spatial smoothing (SS) [8] or forward-backward spatial smoothing (FBSS) [9] are especially noteworthy. However, this technique generally reduces the effective array aperture, and the maximum number of resolvable signals cannot exceed the number of array sensors. In [10], an effective matrix decomposition method utilizing cross-correlation matrix is proposed to decorrelate coherent signals. Chen et al. [11] have proposed a 2-D ESPRIT-like method that realizes decorrelation by reconstructing a Toeplitz matrix. With the help of three correlation matrices, Wang et al. [12] have presented a 2-D DOA estimation method. Recently, Nie et al. [13] have introduced an efficient subspace algorithm for 2-D DOA estimation. In [14], a novel 2-D DOA estimation method using a sparse L-shaped array is proposed to obtain high performance and less complexity. Xia et al. [15] have proposed a polynomial root-finding-based method for 2-D DOA estimation by using two parallel uniform linear arrays (ULAs), which has less computational burden. Some decorrelation algorithms are proposed in [16–18] to achieve 2-D DOA estimation by utilizing two parallel ULAs. However, the limitation of the abovementioned algorithms is that the estimation performance cannot be satisfactory due to the fact that the structure of the array is not being fully exploited.

For the purpose of description, the following notations are used. Boldface italic lower/uppercase letters denote vectors/matrices. (·)*, (·)^T, (·)^†, and (·)^H stand for the conjugation, transpose, Moore-Penrose pseudo-inverse, and conjugate transpose of a vector/matrix, respectively. The notation E(x) and diag (·) separately denote the expectation operator and the diagonal matrix, respectively.

As illustrated in Fig. 1, the antenna array consists of two parallel ULAs (X _a and Y _a ) in the x − y plane. Each ULA has N omnidirectional sensors with spacing d _x , and the interelement spacing between the two ULAs is d _y . Suppose that M far-field narrowband coherent signals impinge on the two parallel ULAs from 2-D distinct directions (α _i , β _i )(1 ≤ i ≤ M), where α _i and β _i are measured relatively to the x axis and to the y axis, respectively.

Similarly as in (1), the superscript k(k = 1, 2, ⋯, N) of the \( {y}_m^k(t) \) stands for the number of the reference element in subarray Y _a , and the subscript m(m = 1, 2, ⋯, N) of the \( {y}_m^k(t) \) denotes the number of the element along the x positive axis in subarray Y _a . n _y,m (t) is the AGWN of the mth element in subarray Y _{a .}

From (18) and (19), it is easy to see that since α _i  ≠ α _j , (i ≠ j), A(α) is a full column rank matrix with rank (A(α)) = M. Similarly, since β _i  ≠ β _j , (i ≠ j), Ψ(β) is a full-rank diagonal matrix with rank (Ψ(β)) = M. According to (7) and (15), note that the incident signals s _i (t) ≠ 0, (i = 1, 2 ⋯ M), so g _i (t) ≠ 0. As a result, G is a full-rank diagonal matrix, namely, rank(G) = M. That is, if the narrowband far-field signals are statistically independent, the diagonal element g _i (t) of the matrix G represents the power of the ith incident signal. If the narrowband far-field signals are fully coherent, the diagonal element g _i (t) of the matrix G denotes the sum of the powers of the M incident signals. Notice that if the narrowband far-field signals are the coexistence of the uncorrelated and coherent signals, which means there are K coherent signals, the remaining are M − K statistically independent signals. Then, the diagonal element g _i (t) of the matrix G stands for the sum of the powers of the K coherent signals when the subscript of the diagonal element g _i (t) in matrix G corresponding to the source signal belongs to one of the K coherent signals. If the subscript of the diagonal element g _i (t) in matrix G corresponding to the source signal belongs to one of the remaining M − K mutually independent signals, the diagonal element g _i (t) of the matrix G denotes the power of the ith independent signal.

From the above theoretical analysis, the coherency of incident signals is decorrelated through matrices constructing no matter whether the signals are uncorrelated, coherent, or partially correlated.

Obviously, the columns of A(α) are the eigenvectors corresponding to the major diagonal elements of diagonal matrix Ψ(β). Therefore, by performing the EVD of R, the A(α) and Ψ(β) can be achieved. Then, the DOA estimation of the coherent signals can be achieved according to \( \upsilon \left({\beta}_i\right)={e}^{j\left(2\pi /\lambda \right){d}_y \cos {\beta}_i} \) and \( \mathtt{a}\left({\alpha}_i\right)={\left[1,{e}^{- j\left(2\pi /\lambda \right){d}_x \cos {\alpha}_i},\cdots, {e}^{- j\left(2\pi /\lambda \right)\left( N-1\right){d}_x \cos {\alpha}_i}\right]}^T \) without additional computations for parameter pair-matching and 2-D peak searching.

A novel decoupling algorithm for 2-D DOA estimation with two parallel ULAs has been presented. In the proposed algorithm, two equivalent covariance matrices are reconstructed to achieve the decorrelation of the coherent signals and the estimated angle parameters are pair-matched automatically. It has been shown that the proposed algorithm yields remarkably better estimation performance than the DMR-DOAM algorithm.