Spatial-Temporal-Based Underdetermined Near-Field 3-D Localization Employing a Nonuniform Cross Array

In this paper, an underdetermined three-dimensional (3-D) near-field source localization method is proposed, based on a two-dimensional (2-D) symmetric nonuniform cross array. Firstly, by utilizing the symmetric coprime array along the x-axis, a fourth-order cumulant (FOC) based matrix is constructed, followed by vectorization operation to form a single virtual snapshot, which is equivalent to the received data of a virtual array observing from virtual far-field sources, generating an increased number of degrees of freedom (DOFs) compared to the original physical array. Meanwhile, multiple delay lags, named as pseudo snapshots, are introduced to address the single snapshot issue. Then, the received data of the uniform linear array along the y-axis is similarly processed to form another virtual array, followed by a cross-correlation operation on the virtual array observations constructed from the coprime array. Finally, the 2-D angles of the near-field sources are jointly estimated by employing the recently proposed sparse and parametric approach (SPA) and the Vandermonde decomposition technique, eliminating the need for parameter discretization. To estimate the range term, the conjugate symmetry property of the signal's autocorrelation function is used to construct the second-order statistics based received data with the whole array elements, and subsequently, the one-dimensional (1-D) MUSIC algorithm is applied. Moreover, some properties of the proposed array are analyzed. Compared with existing algorithms, the proposed one has better estimation performance given the same number of sensor elements, which can work in an underdetermined and mixed sources situation, as shown by simulation results with 3-D parameters automatically paired.