基于稀疏线阵协方差矩阵重构的DOA估计方法研究

DOA Estimation Method Based on Sparse Linear Covariance Matrix Reconstruction

相比均匀线阵(Uniform Linear Array,ULA),相同阵元数目下稀疏线阵(Sparse Linear Array,SLA)的抗耦合效应更好,阵列孔径更大,到达方向(Direction of Arrival,DOA)估计的自由度(Degrees Of Freedom,DOF)更高,因而近年来得到了广泛的研究。为了可以进行高DOF的DOA估计,学者们开始研究SLA的差分虚拟阵元,差分虚拟阵元对应的协方差矩阵相比原阵元对应的协方差矩阵维度更大,因而估计的DOF更高。当SLA的差分虚拟阵元连续取值时,可以利用已有阵元的接收信息,得到SLA的协方差矩阵,在该矩阵的基础之上构建差分虚拟阵元的协方差矩阵进而进行DOA估计。然而,当SLA的差分虚拟阵元存在孔洞时,即差分虚拟阵元不能连续取值时,不能直接利用重构的协方差矩阵进行DOA估计,需要恢复完全增广协方差矩阵的信息再进行DOA估计。对于该问题,本文基于矢量化后原协方差矩阵和虚拟差分阵协方差矩阵的误差分布情况,并结合完全增广协方差矩阵的低秩特性和半正定特性来构建优化问题。通过求解该问题来恢复维度更高的完全增广协方差矩阵。最后对该矩阵进行奇异值分解,利用多重信号分类(Multiple Signal Classification,MUSIC)算法就可以获得多源的空间谱。本文最后通过数值仿真试验验证了所提算法可以实现高DOF的DOA估计,并且相比于现有算法,本文所提算法对欠定DOA估计的效果更好,多源DOA估计的精度更高,产生的误差更小。

The Sparse Linear Array(SLA)exhibits superior anti-coupling effects and a larger array aperture than the Uniform Linear Array(ULA)with the same number of elements.Direction of Arrival(DOA)estimation,which offers higher Degrees Of Freedom(DOFs),has been extensively researched in recent years.Among various DOA estimation algorithms,the Multiple Signal Classification(MUSIC)algorithm provides high-resolution estimations but requires the covariance matrix of the original array.To achieve accurate DOA estimation with high DOFs,scholars have begun investigating the covariance matrix corresponding to the differential virtual elements of the SLA.This augmented covariance matrix has larger dimensions than the covariance matrix of the primitive element,resulting in increased estimated DOFs.When the differential virtual elements of the SLA are continuously valued,we can obtain its covariance matrix by utilizing information from existing elements and construct a covariance matrix of differential virtual elements based on this for DOA estimation.However,if the SLA differential virtual element values have holes where continuity is disrupted,direct usage of the reconstructed covariance matrix becomes unfeasible for DOA estimation.In such cases,information should be recovered from a fully augmented covariance matrix before performing DOA estimation.To address this problem,considering that errors between vectorized SLA and covariances of virtual difference matrices follow an asymptotic normal distribution,we use normalization and further transformation to formulate an inequality constraint problem that can be easily solved.Consequently,we establish a relationship between the fully augmented covariance matrix,SLA covariance matrix,and the covariances of virtual difference matrices.Finally,leveraging the low rank and semi-definite properties of the fully augmented covariance matrix,we construct a convex optimization problem that can be solved using the MATLAB CVX toolkit.Through this approach,we can recover a higher-dimensional fully augmented covariance matrix.Subsequently,we decompose this completely augmented covariance matrix using singular value decomposition(SVD)and employ the MUSIC algorithm to estimate the spatial spectrum of our target signal source.Numerical simulation results demonstrate that our proposed algorithm enables underdetermined DOA estimation(i.e.,when more sources than elements exist)while achieving high DOFs with respect to DOA estimation through the reconstruction of the fully augmented covariance matrix.Compared with existing algorithms,our method exhibits superior resolution towards targets and performs closer to Cramér-Rao bound(CRB)under varying SNR levels and snapshots scenarios.For multi-source DOA estimates,spectral peaks become sharper while bias is reduced.In conclusion,our paper presents an effective approach for achieving highly precise DOA estimation with ample DOFs for sparse linear arrays with holes.