Application Of Matrix/Tensor Completion Theory In Signal Processing

The low-rank completion theory developes rapidly after the compressed sensing theory.It is mainly dedicated to achieving the filling of missing data from incomplete data and is widely applied in image inpainting,signal processing,machine learning,computer vision and other fields.It mainly utilizes the low-rank property of matrice or tensor to transform the completion of missing data into the solution of optimization problems by constructing corresponding kernel norm minimization optimization model.The paper combines the application background of array signal processing and spectrum data processing.It firstly solves the problem of two-dimensional DOA estimation effectively when partial sensors are damaged in uniform rectangular array,and then realizes effective DOA estimation under nonuniform noise.A robust long-term spectrum prediction algorithm is proposed for high-dimensional spectrum data with missing values and sparse anomalies,which has strong theoretical value and practical application significance.The main content of the paper is as follows:(1)At the situation where partial sensors are not completely damaged in a uniform rectangular array,the traditional algorithms' performance for two-dimensional DOA estimation is severely degraded or even fails.To solve the problem,a two-dimensional DOA estimation algorithm based on matrix completion is proposed.The algorithm uses inexact augmented Largarian method(IALM),which is a solution to the problem of matrix completion,to implement the filling of missing data in the received signal matrix.Based on this,it utilizes propagator method(PM)to estimate direction of arrival.The simulation confirms that the proposed algorithm can effectively achieve the DOA estimation in this scenario,and it is superior to the propagator method based on singular value threshold(SVT-PM).(2)Aiming at the situation where partial sensors are completely damaged in a uniform rectangular array,a two-dimensional DOA estimation algorithm based on matrix completion by tensor reconstruction is firstly proposed.Based on the tensor model of the received signal,the proposed algorithm first adapts the tensor reconstruction strategy and IALM of matrix completion to realize the filling of missing data in the received signal matrix,and then it utilizes ESPRIT to estimate twodimensional direction of arrival effectively.Then a two-dimensional DOA estimation algorithm based on low-rank tensor completion is proposed.It exploits ESPRIT to realize the DOA estimation on the basis of Ha LRTC,which is an algorithm designed for tensor completion.When partial sensors are completely damaged,the corresponding row data in the received signal matrix is entirely missing.The low-rank matrix completion cannot effectively achieve the completion of missing data.Meanwhile,the two proposed algorithms can realize effective two-dimensional DOA estimation in this scenario.(3)For the one-dimensional DOA estimation problem with nonuniform noise in a uniform linear array,the nonuniform noise power matrix has a serious impact on the singular value distribution of the signal covariance matrix,so that the signal subspace acquired from the signal covariance matrix and direction vector space are no longer isomorphic.To solve the problem,the estimating signal parameter via rotational invariance techniques based on unitaray inexact augmented Lagrangian method is proposed.The algorithm not only achieves effective DOA estimation with nonuniform noise,but also reduces the complexity.It first introduces the unitary transformation,which transforms the covariance matrix in the complex domain into a real matrix,and then the low-rank matrix completion in the real domain is applied to separate the nonuniform noise power matrix,and adapts the unitary ESPRIT algorithm to implement an effective the direction estimation.(4)A robust long-term spectrum prediction algorithm(RLSP)is proposed for high-dimensional spectrum data with missing values and sparse anomalies.The algorithm adapts the iterative architecture of pre-fill process and robust tensor recovery,which greatly reduces the excessive dependence of longterm spectrum prediction performance on pre-fill accuracy.Compared with the existing long-term spectrum prediction scheme based on tensor completion(LSP-TC),this algorithm not only achieves effective long-term spectrum prediction with sparse anomalies,but also has better performance than the LSP-TC algorithm even in the absence of anomalies.