Hadamard Product Perspective on Source Resolvability of Spatial-smoothing-based Subspace Methods

Spatial smoothing is a common preprocessing scheme for subspace methods that resolves their sensitivity to coherent sources. The source resolvability problem of spatial-smoothing-based subspace methods has been extensively investigated using different analysis techniques. In this paper, a unified Hadamard product technique is provided to recover these results. This is done by answering a long-standing question in linear algebra as to under what conditions the Hadamard product of two singular positive-semidefinite matrices is positive definite.

[1]  P. P. Vaidyanathan,et al.  Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom , 2010, IEEE Transactions on Signal Processing.

[2]  S. Unnikrishna Pillai,et al.  Forward/backward spatial smoothing techniques for coherent signal identification , 1989, IEEE Trans. Acoust. Speech Signal Process..

[3]  J. Schur.  Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. , 1911 .

[4]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[5]  Kah-Chye Tan,et al.  Estimating directions-of-arrival of coherent signals in unknown correlated noise via spatial smoothing , 1997, IEEE Trans. Signal Process..

[6]  Shing-Chow Chan,et al.  DOA estimation of coherent signals for uniform linear arrays with mutual coupling , 2011, 2011 IEEE International Symposium of Circuits and Systems (ISCAS).

[7]  Yoram Bresler,et al.  On the number of signals resolvable by a uniform linear array , 1986, IEEE Trans. Acoust. Speech Signal Process..

[8]  C. Ballantine.  On the Hadamard product , 1968 .

[9]  V. Pisarenko.  The Retrieval of Harmonics from a Covariance Function , 1973 .

[10]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[11]  L. Imhof.  Matrix Algebra and Its Applications to Statistics and Econometrics , 1998 .

[12]  Thomas Kailath,et al.  On spatial smoothing for direction-of-arrival estimation of coherent signals , 1985, IEEE Trans. Acoust. Speech Signal Process..

[13]  Jinkuan Wang,et al.  Spatial Differencing Method for DOA Estimation Under the Coexistence of Both Uncorrelated and Coherent Signals , 2012, IEEE Transactions on Antennas and Propagation.

[14]  G. Styan.  Hadamard products and multivariate statistical analysis , 1973 .

[15]  Jian Li,et al.  Sparse Methods for Direction-of-Arrival Estimation , 2016, ArXiv.

[16]  P. P. Vaidyanathan,et al.  Remarks on the Spatial Smoothing Step in Coarray MUSIC , 2015, IEEE Signal Processing Letters.

[17]  Shihua Zhu,et al.  A CANDECOMP/PARAFAC Perspective on Uniqueness of DOA Estimation Using a Vector Sensor Array , 2011, IEEE Transactions on Signal Processing.

[18]  James E. Evans,et al.  Application of Advanced Signal Processing Techniques to Angle of Arrival Estimation in ATC Navigation and Surveillance Systems , 1982 .

[19]  P. Stoica,et al.  Direction-of-arrival estimation in applications with multipath and few snapshots , 1991 .

[20]  T. Kailath,et al.  A subspace rotation approach to signal parameter estimation , 1986, Proceedings of the IEEE.

[21]  Ralph Otto Schmidt,et al.  A signal subspace approach to multiple emitter location and spectral estimation , 1981 .

[22]  Surendra Prasad,et al.  An improved spatial smoothing technique for bearing estimation in a multipath environment , 1988, IEEE Trans. Acoust. Speech Signal Process..

[23]  Ed F. Deprettere,et al.  Analysis of joint angle-frequency estimation using ESPRIT , 2003, IEEE Trans. Signal Process..