Tensor Theory And Its Application To Array Signal Processing

With the mathematical tool, tensor, which represents a multilinear relationship between multivariables and also multidimensional data, this thesis exploits the multidimensional structure of polarization sensitive array measurement data for spatial electromagnetic source’s DOA and polarization estimation.The origins, definitions, basic operations and two important decompositions of tensor, i.e., Tucker and PARAFAC decompositions, are introduced. The mode-R product and its matrix expansion are proposed, from which Tucker decomposition can be generalized. In addition, a suboptimal step of alternating least squares iterative algorithm based on the Gauss-Newton approximation with adaptive extrapolation direction is proposed. The goal is to have a fast PARAFAC decomposition algorithm by a better trade-off between iteration number and the computation cost of each iteration in order to achieve less overall time consumption.In order to model the multidimensional structures within the polarization sensitive electromagnetic vector sensor array data explicitly, we propose a corresponding tensorial model from the perspective of Tucker decomposition. The mode-R signal subspace is defined based on this model. With the projection to the mode-R signal subspaces, a novel algorithm, which incorporates the refined signal subspace estimation and the traditional MUSIC algorithm, is then presented. The proposed technique exploits the inherent multidimensional structure information for increased parameter estimation accuracy. We further extend mode-R projection to muiltidimensional harmonic retrieval (MHR) problems. The measurement tensor is proposed to be projected to the mode-R signal subspaces in a bottom-up way, and the long-vector signal subspace required by many signal subspace based parameter estimation algorithms can be refined. As an example, a mode-R projection based Tensor-ESPRIT algorithm is presented. It is shown that mode-R subspace projections can reduce the perturbation when calculating signal subspace. The analyses also generate two criteria on how the mode-R subspace based projection technique should be carried out.We also discuss parameter estimation method based on PARAFAC decomposition Specifically, a unified tensorial framework of smoothing is first proposed, of which the traditional spatial, polarization, and weighted smoothing techniques are special cases. According to the tensorial smoothing, further identification analysis in snapshot and second-order statistics domain are carried out. As applications, a single snapshot PARAFAC fitting for partially polarized wave parameter estimation in the snapshot domain and second-order covariance PARAFAC model fitting for greatly improved identifiability of a nested electromagnetic vector sensor array are presented.To reduce the mutual coupling effects among the electromagnetic vector sensor components, we also propose two types of spatially spread electromagnetic vector sensor array(SSEMVS), whose special geometry enables the application of cross product algorithm. The cross product vector can be fitted directly for higher estimation accuracy, while the amplitude of the cross product vector can be used to initialize the corresponding optimization problem. The PARAFAC decomposition is introduced for array manifold estimation, and can exceed the limitation of traditional ESPRIT algorithms that can only estimate up to 5 sources. To further analyze the array structure and the cross product of the array manifold, we define the g parameters, based on which the DOA ambiguities are analyzed. It is also found that the array geometry of SSEMVS should be designed carefully to avoid ambiguity.A two-stage quaternion joint eigenvalue decomposition (QJEVD) algorithm is devised through the joint quaternion Schur decomposition and tensorial representation of the quaternion product, baesd on which we propose a novel 2-D quaternion ESPRIT algorithm. The analyses show that our 2-D Q-ESPRIT estimator can achieve enhanced robustness and is computationally more efficient than the MUSIC like counterparts.To sum up, this thesis makes use of tensor, and also proposes new inventive techniques in tensor processing. The results are applied to the polarization sensitive array for parameter estimation, where improved performance as well as new insights are obtained.