MUSIC for joint frequency estimation: Stability with compressive measurements

This paper studies the application of MUtiple Signal Classification (MUSIC) algorithm on Multiple Measurement Vector (MMV) problem for the purpose of frequency parameter estimation while s true frequencies are located in the continuum of a bounded domain and sensors are randomly selected from a Uniform Linear Array (ULA). The MUSIC algorithm amounts to identifying a noise subspace from measurements, forming a noise-space correlation function and searching the s smallest local minima of the noise-space correlation function. Under the assumption that the true frequencies are separated by at least one Rayleigh Length (RL), we show that with high probability the noise-space correlation function is stably perturbed by noise if the number of sensors n ~ O(s) up to a logarithmic factor by means of a compressive version of discrete Ingham inequalities. As the theory implies, our numerical experiments demonstrate that the reconstruction error of MUSIC with n random sensors makes little difference once n is above a point of transition.

[1]  Vilmos Komornik,et al.  Fourier Series in Control Theory , 2005 .

[2]  Petre Stoica,et al.  MUSIC, maximum likelihood, and Cramer-Rao bound , 1989, IEEE Transactions on Acoustics, Speech, and Signal Processing.

[3]  M. Viberg,et al.  Two decades of array signal processing research: the parametric approach , 1996, IEEE Signal Process. Mag..

[4]  Yuxin Chen,et al.  Robust Spectral Compressed Sensing via Structured Matrix Completion , 2013, IEEE Transactions on Information Theory.

[5]  Wenjing Liao,et al.  Super-resolution by compressive sensing algorithms , 2012, 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR).

[6]  S. Frick,et al.  Compressed Sensing , 2014, Computer Vision, A Reference Guide.

[7]  Sergiy A. Vorobyov,et al.  Maximum likelihood direction-of-arrival estimation in unknown noise fields using sparse sensor arrays , 2005, IEEE Transactions on Signal Processing.

[8]  Emmanuel J. Cand.  Towards a Mathematical Theory of Super-Resolution , 2012 .

[9]  Ali Pezeshki,et al.  Threshold Effects in Parameter Estimation From Compressed Data , 2013, IEEE Transactions on Signal Processing.

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

[11]  D. Donoho.  Superresolution via sparsity constraints , 1992 .

[12]  Albert Fannjiang,et al.  The MUSIC algorithm for sparse objects: a compressed sensing analysis , 2010, ArXiv.

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

[14]  Lena Jaeger,et al.  Fourier Series In Control Theory , 2016 .

[15]  Xavier Mestre,et al.  MUSIC, G-MUSIC, and Maximum-Likelihood Performance Breakdown , 2008, IEEE Transactions on Signal Processing.

[16]  A. Ingham.  Some trigonometrical inequalities with applications to the theory of series , 1936 .

[17]  A. Robert Calderbank,et al.  Sensitivity to Basis Mismatch in Compressed Sensing , 2011, IEEE Trans. Signal Process..

[18]  J. Devaney,et al.  1 Super-resolution Processing of Multi-static Data Using Time Reversal and MUSIC A , 2000 .

[19]  Joel A. Tropp,et al.  User-Friendly Tail Bounds for Sums of Random Matrices , 2010, Found. Comput. Math..

[20]  Michael D. Zoltowski,et al.  Closed-form eigenstructure-based direction finding using arbitrary but identical subarrays on a sparse uniform Cartesian array grid , 2000, IEEE Trans. Signal Process..

[21]  A. Lee Swindlehurst,et al.  A Performance Analysis ofSubspace-Based Methods in thePresence of Model Errors { Part I : The MUSIC AlgorithmA , 1992 .

[22]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[23]  John K. Thomas,et al.  The probability of a subspace swap in the SVD , 1995, IEEE Trans. Signal Process..

[24]  T. Tao.  An uncertainty principle for cyclic groups of prime order , 2003, math/0308286.

[25]  Wenjing Liao,et al.  Coherence Pattern-Guided Compressive Sensing with Unresolved Grids , 2011, SIAM J. Imaging Sci..

[26]  Wenjing Liao,et al.  Mismatch and resolution in compressive imaging , 2011, Optical Engineering + Applications.

[27]  Adriaan van den Bos,et al.  Resolution: a survey , 1997 .

[28]  Wenjing Liao,et al.  MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution , 2014, ArXiv.

[29]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[30]  Yuxin Chen,et al.  Spectral Compressed Sensing via Structured Matrix Completion , 2013, ICML.

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

[32]  Emmanuel J. Candès,et al.  Towards a Mathematical Theory of Super‐resolution , 2012, ArXiv.

[33]  Jean-Jacques Fuchs.  Extension of the Pisarenko method to sparse linear arrays , 1997, IEEE Trans. Signal Process..

[34]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[35]  H. Weyl.  Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung) , 1912 .

[36]  Alexei Gorokhov,et al.  Comparison of DOA estimation performance for various types of sparse antenna array geometries , 1996, 1996 8th European Signal Processing Conference (EUSIPCO 1996).

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

[38]  Emmanuel J. Candès,et al.  Super-Resolution from Noisy Data , 2012, Journal of Fourier Analysis and Applications.

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