Tensor Approach for Eigenvector-Based Multi-Dimensional Harmonic Retrieval

In this paper, we propose an eigenvector-based frequency estimator for R-dimensional (R-D) sinusoids with R ≥ 2 in additive white Gaussian noise. Our underlying idea is to utilize the tensorial structure of the received data and then apply higher-order singular value decomposition (HOSVD) and structure least squares (SLS) to perform estimation. After obtaining the tensor-based signal subspace from HOSVD, we decompose it into a set of single-tone tensors from which single-tone vectors can be constructed by another HOSVD. In doing so, the R-D multiple sinusoids are converted to a set of single-tone sequences whose frequencies are individually estimated according to SLS. The mean and variance of the frequency estimator are also derived. Computer simulations are also included to compare the proposed approach with conventional R -D harmonic retrieval schemes in terms of mean square error performance and computational complexity particularly in the presence of identical frequencies.

[1]  Christoph F. Mecklenbräuker,et al.  Multidimensional Rank Reduction Estimator for Parametric MIMO Channel Models , 2004, EURASIP J. Adv. Signal Process..

[2]  Rémy Boyer.  Deterministic asymptotic Cramér-Rao bound for the multidimensional harmonic model , 2008, Signal Process..

[3]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[4]  Joos Vandewalle,et al.  A Multilinear Singular Value Decomposition , 2000, SIAM J. Matrix Anal. Appl..

[5]  Josef A. Nossek,et al.  Simultaneous Schur decomposition of several nonsymmetric matrices to achieve automatic pairing in multidimensional harmonic retrieval problems , 1998, IEEE Trans. Signal Process..

[6]  A. Bax.  A Simple Description of Two-Dimensional NMR Spectroscopy , 1986 .

[7]  Florian Roemer,et al.  Tensor-Based Spatial Smoothing (TB-SS) Using Multiple Snapshots , 2010, IEEE Transactions on Signal Processing.

[8]  Martin Haardt,et al.  Structured least squares to improve the performance of ESPRIT-type algorithms , 1997, IEEE Trans. Signal Process..

[9]  Frankie K. W. Chan,et al.  An Efficient Approach for Two-Dimensional Parameter Estimation of a Single-Tone , 2010, IEEE Transactions on Signal Processing.

[10]  Tao Jiang,et al.  Multidimensional Harmonic Retrieval with Applications in MIMO Wireless Channel Sounding , 2005 .

[11]  Florian Roemer,et al.  Higher-Order SVD-Based Subspace Estimation to Improve the Parameter Estimation Accuracy in Multidimensional Harmonic Retrieval Problems , 2008, IEEE Transactions on Signal Processing.

[12]  F. Li,et al.  Performance analysis for DOA estimation algorithms: unification, simplification, and observations , 1993 .

[13]  M.D. Zoltowski,et al.  Uni-vector-sensor ESPRIT for multi-source azimuth-elevation angle-estimation , 1996, IEEE Antennas and Propagation Society International Symposium. 1996 Digest.

[14]  L. Lathauwer,et al.  On the Best Rank-1 and Rank-( , 2004 .

[15]  Florian Roemer,et al.  Analytical performance evaluation for HOSVD-based parameter estimation schemes , 2009, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[16]  Nikos D. Sidiropoulos,et al.  Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO Radar , 2010, IEEE Transactions on Signal Processing.

[17]  Joos Vandewalle,et al.  On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors , 2000, SIAM J. Matrix Anal. Appl..

[18]  K. J. Liu,et al.  A high-resolution technique for multidimensional NMR spectroscopy , 1998, IEEE Transactions on Biomedical Engineering.

[19]  Weize Sun,et al.  Accurate and Computationally Efficient Tensor-Based Subspace Approach for Multidimensional Harmonic Retrieval , 2012, IEEE Transactions on Signal Processing.

[20]  Josef A. Nossek,et al.  Unitary ESPRIT: how to obtain increased estimation accuracy with a reduced computational burden , 1995, IEEE Trans. Signal Process..

[21]  Xiaoli Ma,et al.  First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition , 2007, 2007 IEEE/SP 14th Workshop on Statistical Signal Processing.

[22]  Xiaoli Ma,et al.  Ieee Transactions on Signal Processing Multidimensional Frequency Estimation with Finite Snapshots in the Presence of Identical Frequencies , 2022 .

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

[24]  Rémy Boyer,et al.  Asymptotic Cramér-Rao Bound for Multi-Dimensional Harmonic Models , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[25]  Florian Roemer,et al.  Analytical performance assessment of 1-D Structured Least Squares , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[26]  Nikos D. Sidiropoulos,et al.  A PARAFAC-based technique for detection and localization of multiple targets in a MIMO radar system , 2009, 2009 IEEE International Conference on Acoustics, Speech and Signal Processing.

[27]  K. T. Wong,et al.  Uni-vector-sensor ESPRIT for multisource azimuth, elevation, and polarization estimation , 1997 .

[28]  Jun Liu,et al.  An Eigenvector-Based Approach for Multidimensional Frequency Estimation With Improved Identifiability , 2006, IEEE Transactions on Signal Processing.

[29]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.