Multidimensional folding for sinusoidal order selection

The order estimation algorithm relies on multidimensional folding combined with ESTimation ERror criterion.Multidimensional folding and unfolding construct two matrix slices that satisfy the shift invariance equality.The ESTER criterion estimates number of harmonics based on shift invariance equality.The algorithm is able to identify significantly more harmonics.The algorithm is robust against colored noise. Estimation of the number of harmonics in multidimensional sinusoids is studied in this paper. The ESTimation ERror (ESTER) is a subspace based detection approach that is robust against colored noise. However, the number of signals it can detect is very limited. To improve the identifiability, we propose to combine the multidimensional folding (MDF) techniques with ESTER for multidimensional sinusoidal order selection. Our algorithm development is inspired by the shift invariance properties of the two matrix slices resulting from multidimensional folding and unfolding, which have been exploited to extract the spatial frequencies in the literature. The maximum identifiable number of signals of the MDF-ESTER is of the order of magnitude of product of the lengths of all spatial dimensions with uniform spacing, which is significantly larger than that of the conventional multidimensional ESTER methods. Meanwhile, it inherits the robustness of the ESTER against colored noise, and performs comparably to state-of-the-art schemes when the number of signals is small.

[1]  Josef A. Nossek,et al.  Efficient high-resolution 3-D channel sounding , 1998, VTC '98. 48th IEEE Vehicular Technology Conference. Pathway to Global Wireless Revolution (Cat. No.98CH36151).

[2]  Harry L. Van Trees,et al.  Optimum Array Processing , 2002 .

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

[4]  Nikos D. Sidiropoulos,et al.  Almost-sure identifiability of multidimensional harmonic retrieval , 2001, IEEE Trans. Signal Process..

[5]  Thomas Kailath,et al.  ESPRIT-estimation of signal parameters via rotational invariance techniques , 1989, IEEE Trans. Acoust. Speech Signal Process..

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

[7]  Roland Badeau,et al.  A new perturbation analysis for signal enumeration in rotational invariance techniques , 2006, IEEE Transactions on Signal Processing.

[8]  Angzhao Di,et al.  Multiple source location-A matrix decomposition approach , 1985, IEEE Trans. Acoust. Speech Signal Process..

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

[10]  Martin Haardt,et al.  Multidimensional prewhitening for enhanced signal reconstruction and parameter estimation in colored noise with Kronecker correlation structure , 2013, Signal Process..

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

[12]  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.

[13]  Charles Van Loan,et al.  Block Tensor Unfoldings , 2011, SIAM J. Matrix Anal. Appl..

[14]  Weize Sun,et al.  Subspace approach for two-dimensional parameter estimation of multiple damped sinusoids , 2012, Signal Process..

[15]  Hsien-Tsai Wu,et al.  Source number estimators using transformed Gerschgorin radii , 1995, IEEE Trans. Signal Process..

[16]  Xianda Zhang,et al.  Harmonic retrieval in mixed Gaussian and non-Gaussian ARMA noises , 1994, IEEE Trans. Signal Process..

[17]  Yan Zhang,et al.  Harmonic retrieval in colored non-Gaussian noise using cumulants , 2000, IEEE Trans. Signal Process..

[18]  Harry L. Van Trees,et al.  Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory , 2002 .

[19]  Boaz Nadler,et al.  Non-Parametric Detection of the Number of Signals: Hypothesis Testing and Random Matrix Theory , 2009, IEEE Transactions on Signal Processing.

[20]  R. Carlson,et al.  Two-Dimensional Nmr Spectroscopy , 1998 .

[21]  Nikos D. Sidiropoulos,et al.  On 3D harmonic retrieval for wireless channel sounding , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[22]  Lei Huang,et al.  Subspace based Multidimensional Model Order Selection in Colored Noise , 2013 .

[23]  Lei Huang,et al.  Subspace techniques for multidimensional model order selection in colored noise , 2013, Signal Process..

[24]  Roland Badeau,et al.  Selecting the modeling order for the ESPRIT high resolution method: an alternative approach , 2004, 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[25]  Nikos D. Sidiropoulos,et al.  Almost sure identifiability of constant modulus multidimensional harmonic retrieval , 2002, IEEE Trans. Signal Process..

[26]  H. Akaike.  A new look at the statistical model identification , 1974 .

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

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

[29]  Lei Huang,et al.  A multi-dimensional model order selection criterion with improved identifiability , 2012, 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[30]  Eric M. Dowling,et al.  Efficient direction-finding methods employing forward/backward averaging , 1994, IEEE Trans. Signal Process..

[31]  B. Nadler,et al.  Determining the number of components in a factor model from limited noisy data , 2008 .

[32]  Martin Haardt,et al.  Single Snapshot R-D Unitary Tensor-ESPRIT Using an Augmentation of the Tensor Order , 2009, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[33]  Y. Selen,et al.  Model-order selection: a review of information criterion rules , 2004, IEEE Signal Processing Magazine.

[34]  Lei Huang,et al.  Source Enumeration for High-Resolution Array Processing Using Improved Gerschgorin Radii Without Eigendecomposition , 2008, IEEE Transactions on Signal Processing.

[35]  Alan Edelman,et al.  Sample Eigenvalue Based Detection of High-Dimensional Signals in White Noise Using Relatively Few Samples , 2007, IEEE Transactions on Signal Processing.

[36]  A. Bax,et al.  Two-dimensional nuclear magnetic resonance spectroscopy. , 1986, Science.

[37]  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..

[38]  Thomas Kailath,et al.  Detection of signals by information theoretic criteria , 1985, IEEE Trans. Acoust. Speech Signal Process..

[39]  Sabine Van Huffel,et al.  A Shift Invariance-Based Order-Selection Technique for Exponential Data Modelling , 2007, IEEE Signal Processing Letters.

[40]  Nikolaus M. Szeverenyi,et al.  TWO‐DIMENSIONAL NMR SPECTROSCOPY , 1984 .

[41]  Brian M. Sadler,et al.  Noise subspace techniques in non-gaussian noise using cumulants , 1995 .

[42]  Ying-Chang Liang,et al.  A hybrid approach to harmonic retrieval in non-Gaussian ARMA noise , 1994, IEEE Trans. Inf. Theory.

[43]  Jun Fang,et al.  Source enumeration for large array using shrinkage-based detectors with small samples , 2015, IEEE Transactions on Aerospace and Electronic Systems.

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