A simultaneous sparse approximation method for multidimensional harmonic retrieval

In this paper, a new method for the estimation of the parameters of multidimensional (R-D) harmonic and damped complex signals in noise is presented. The problem is formulated as R simultaneous sparse approximations of multiple 1-D signals. To get a method able to handle large size signals while maintaining a sufficient resolution, a multigrid dictionary refinement technique is associated to the simultaneous sparse approximation. The refinement procedure is proved to converge in the single R-D mode case. Then, for the general multiple modes case, the signal tensor model is decomposed in order to handle each mode separately in an iterative scheme. The proposed method does not require an association step since the estimated modes are automatically "paired". We also derive the Cramer-Rao lower bounds of the parameters of modal R-D signals. The expressions are given in compact form in the single tone case. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed method. HighlightsThe simultaneous sparse approximation concept is well-suited for R-D modal retrieval.A new multigrid scheme of two-step refinement of 1-D grids is proposed.Derivation of a new algorithm for estimating parameters of R-D damped signals.Theoretical analysis.Derivation of the Cramer-Rao bounds for the parameters of R-D signal model.

[1]  Petre Stoica,et al.  Sparse Estimation of Spectral Lines: Grid Selection Problems and Their Solutions , 2012, IEEE Transactions on Signal Processing.

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

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

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

[5]  Yonina C. Eldar,et al.  Block-Sparse Signals: Uncertainty Relations and Efficient Recovery , 2009, IEEE Transactions on Signal Processing.

[6]  Joel A. Tropp,et al.  ALGORITHMS FOR SIMULTANEOUS SPARSE APPROXIMATION , 2006 .

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

[8]  Yonina C. Eldar,et al.  Robust Recovery of Signals From a Structured Union of Subspaces , 2008, IEEE Transactions on Information Theory.

[9]  A. TroppJ..  Greed is good , 2006 .

[10]  Tapan K. Sarkar,et al.  Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise , 1990, IEEE Trans. Acoust. Speech Signal Process..

[11]  Zhao Tan,et al.  Discrete and Continuous Sparse Recovery Methods and Their Applications , 2015 .

[12]  Louis L. Scharf,et al.  Two-dimensional modal analysis based on maximum likelihood , 1994, IEEE Trans. Signal Process..

[13]  Souleymen Sahnoun,et al.  Sparse modal estimation of 2-D NMR signals , 2013, 2013 IEEE International Conference on Acoustics, Speech and Signal Processing.

[14]  Petre Stoica,et al.  Introduction to spectral analysis , 1997 .

[15]  J. Tropp.  Algorithms for simultaneous sparse approximation. Part II: Convex relaxation , 2006, Signal Process..

[16]  Dmitry M. Malioutov,et al.  A sparse signal reconstruction perspective for source localization with sensor arrays , 2005, IEEE Transactions on Signal Processing.

[17]  Andreas Jakobsson,et al.  High resolution sparse estimation of exponentially decaying two-dimensional signals , 2014, 2014 22nd European Signal Processing Conference (EUSIPCO).

[18]  El-Hadi Djermoune,et al.  Perturbation Analysis of Subspace-Based Methods in Estimating a Damped Complex Exponential , 2009, IEEE Transactions on Signal Processing.

[19]  Parikshit Shah,et al.  Compressed Sensing Off the Grid , 2012, IEEE Transactions on Information Theory.

[20]  Yoram Bresler,et al.  Subspace Methods for Joint Sparse Recovery , 2010, IEEE Transactions on Information Theory.

[21]  Daniel Pérez Palomar,et al.  Space-Time Processing for MIMO Communications , 2004 .

[22]  Pierre Comon,et al.  Tensors : A brief introduction , 2014, IEEE Signal Processing Magazine.

[23]  Bhaskar D. Rao,et al.  An Empirical Bayesian Strategy for Solving the Simultaneous Sparse Approximation Problem , 2007, IEEE Transactions on Signal Processing.

[24]  Yingbo Hua,et al.  Estimating two-dimensional frequencies by matrix enhancement and matrix pencil , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[25]  Steven Kay,et al.  Fundamentals Of Statistical Signal Processing , 2001 .

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

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

[28]  Joel A. Tropp,et al.  Algorithms for simultaneous sparse approximation. Part I: Greedy pursuit , 2006, Signal Process..

[29]  Marco F. Duarte,et al.  Spectral compressive sensing , 2013 .

[30]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[31]  Wen-Hsien Fang,et al.  Efficient Multidimensional Harmonic Retrieval: A Hierarchical Signal Separation Framework , 2013, IEEE Signal Processing Letters.

[32]  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.

[33]  Charles Soussen,et al.  Sparse multidimensional modal analysis using a multigrid dictionary refinement , 2012, EURASIP J. Adv. Signal Process..

[34]  Yonina C. Eldar,et al.  Reduce and Boost: Recovering Arbitrary Sets of Jointly Sparse Vectors , 2008, IEEE Transactions on Signal Processing.

[35]  Bhaskar D. Rao,et al.  Sparse solutions to linear inverse problems with multiple measurement vectors , 2005, IEEE Transactions on Signal Processing.

[36]  Andreas Jakobsson,et al.  High resolution sparse estimation of exponentially decaying signals , 2014, 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[37]  Randolph L. Moses,et al.  Two-dimensional Prony modeling and parameter estimation , 1993, IEEE Trans. Signal Process..

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

[39]  Nikos D. Sidiropoulos,et al.  Almost sure identifiability of multidimensional harmonic retrieval , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[40]  Jian Li,et al.  SPICE: A Sparse Covariance-Based Estimation Method for Array Processing , 2011, IEEE Transactions on Signal Processing.

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

[42]  Martin Vetterli,et al.  Atomic signal models based on recursive filter banks , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

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

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

[45]  Jian Li,et al.  New Method of Sparse Parameter Estimation in Separable Models and Its Use for Spectral Analysis of Irregularly Sampled Data , 2011, IEEE Transactions on Signal Processing.

[46]  Yingbo Hua.  Estimating two-dimensional frequencies by matrix enhancement and matrix pencil , 1992, IEEE Trans. Signal Process..

[47]  S. Kay.  Fundamentals of statistical signal processing: estimation theory , 1993 .

[48]  Stéphanie Rouquette-Léveil,et al.  Estimation of frequencies and damping factors by two-dimensional ESPRIT type methods , 2001, IEEE Trans. Signal Process..

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

[50]  Stephen J. Wright,et al.  Computational Methods for Sparse Solution of Linear Inverse Problems , 2010, Proceedings of the IEEE.

[51]  Sudhakar M. Pandit,et al.  Cramer-Rao lower bounds for a damped sinusoidal process , 1995, IEEE Trans. Signal Process..

[52]  Lei Huang,et al.  Multidimensional Sinusoidal Frequency Estimation Using Subspace and Projection Separation Approaches , 2012, IEEE Transactions on Signal Processing.

[53]  Alain Rakotomamonjy,et al.  Surveying and comparing simultaneous sparse approximation (or group-lasso) algorithms , 2011, Signal Process..

[54]  Souleymen Sahnoun,et al.  Sparse multigrid modal estimation: Initial grid selection , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).