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Prony Method for Reconstruction of Structured Functions

  • Gerlind Plonka
  • Daniel Potts
  • Gabriele Steidl
  • Manfred Tasche
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

The recovery of a structured function from sampled data is a fundamental problem in applied mathematics and signal processing. In Sect. 10.1, we consider the parameter estimation problem, where the classical Prony method and its relatives are described. In Sect. 10.2, we study frequently used methods for solving the parameter estimation problem, namely MUSIC (MUltiple Signal Classification), APM (Approximate Prony Method), and ESPRIT (Estimation of Signal Parameters by Rotational Invariance).

References

  1. 5.
    F. Andersson, M. Carlsson, ESPRIT for multidimensional general grids (2017). arXiv e-printsGoogle Scholar
  2. 11.
    C. Aubel, H. Bölcskei, Vandermonde matrices with nodes in the unit disk and the large sieve. Appl. Comput. Harmon. Anal. (to appear, 2019)Google Scholar
  3. 20.
    H.H. Bauschke, P.L. Combettes, D.R. Luke, Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Am. A 19(7), 1334–1345 (2002)MathSciNetCrossRefGoogle Scholar
  4. 21.
    F.S.V. Bazán, Conditioning of rectangular Vandermonde matrices with nodes in the unit disk. SIAM J. Matrix Anal. Appl. 21, 679–693 (2000)MathSciNetCrossRefGoogle Scholar
  5. 22.
    F.S.V. Bazán, P.L. Toint, Error analysis of signal zeros from a related companion matrix eigenvalue problem. Appl. Math. Lett. 14(7), 859–866 (2001)MathSciNetCrossRefGoogle Scholar
  6. 25.
    R. Beinert, G. Plonka, Ambiguities in one-dimensional discrete phase retrieval from Fourier magnitudes. J. Fourier Anal. Appl. 21(6), 1169–1198 (2015)MathSciNetCrossRefGoogle Scholar
  7. 26.
    R. Beinert, G. Plonka, Sparse phase retrieval of one-dimensional signals by Prony’s method. Front. Appl. Math. Stat. 3, 5 (2017)CrossRefGoogle Scholar
  8. 29.
    J. Berent, P.L. Dragotti, T. Blu, Sampling piecewise sinusoidal signals with finite rate of innovation methods. IEEE Trans. Signal Process. 58(2), 613–625 (2010)MathSciNetCrossRefGoogle Scholar
  9. 45.
    Y. Bresler, A. Macovski, Exact maximum likelihood parameter estimation of superimposed exponential signals in noise. IEEE Trans. Acoust. Speech Signal Process. 34(5), 1081–1089 (1986)CrossRefGoogle Scholar
  10. 46.
    W.L. Briggs, V.E. Henson, The DFT. An Owner’s Manual for the Discrete Fourier Transform (SIAM, Philadelphia, 1995)Google Scholar
  11. 66.
    O. Christensen, An Introduction to Frames and Riesz Bases, 2nd edn. (Birkhäuser/Springer, Cham, 2016)Google Scholar
  12. 76.
    A. Cuyt, W.-S. Lee, Multivariate exponential analysis from the minimal number of samples. Adv. Comput. Math. 44(4), 987–1002 (2018)MathSciNetCrossRefGoogle Scholar
  13. 79.
    C. de Boor, A Practical Guide to Splines, revised edn. (Springer, New York, 2001)Google Scholar
  14. 82.
    G.R. de Prony, Essai éxperimental et analytique: sur les lois de la dilatabilité des fluides élastiques et sur celles de la force expansive de la vapeur de l’eau et de la vapeur de l’alkool, à différentes températures. J. Ecole Polytech. 1, 24–76 (1795)Google Scholar
  15. 87.
    P.L. Dragotti, M. Vetterli, T. Blu, Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang–Fix. IEEE Trans. Signal Process. 55, 1741–1757 (2007)MathSciNetCrossRefGoogle Scholar
  16. 99.
    M. Ehler, S. Kunis, T. Peter, C. Richter, A randomized multivariate matrix pencil method for superresolution microscopy (2018). ArXiv e-printsGoogle Scholar
  17. 104.
    A.C. Fannjiang, The MUSIC algorithm for sparse objects: a compressed sensing analysis. Inverse Prob. 27(3), 035013 (2011)MathSciNetCrossRefGoogle Scholar
  18. 112.
    F. Filbir, H.N. Mhaskar, J. Prestin, On the problem of parameter estimation in exponential sums. Constr. Approx. 35(2), 323–343 (2012)MathSciNetCrossRefGoogle Scholar
  19. 134.
    G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996)zbMATHGoogle Scholar
  20. 135.
    G.H. Golub, P. Milanfar, J. Varah, A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21(4), 1222–1243 (1999–2000)MathSciNetCrossRefGoogle Scholar
  21. 136.
    N. Golyandina, A. Zhigljavsky, Singular Spectrum Analysis for Time Series (Springer, Heidelberg, 2013)CrossRefGoogle Scholar
  22. 137.
    N. Golyandina, V. Nekrutkin, A. Zhigljavsky, Analysis of Time Series Structure. SSA and Related Techniques (Chapman & Hall/CRC, Boca Raton, 2001)CrossRefGoogle Scholar
  23. 165.
    G. Heinig, K. Rost, Algebraic Methods for Toeplitz-Like Matrices and Operators (Akademie-Verlag, Berlin, 1984)CrossRefGoogle Scholar
  24. 169.
    R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd edn. (Cambridge University Press, Cambridge, 2013)zbMATHGoogle Scholar
  25. 170.
    Y. Hua, T.K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process. 38(5), 814–824 (1990)MathSciNetCrossRefGoogle Scholar
  26. 172.
    A.E. Ingham, Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367–379 (1936)MathSciNetCrossRefGoogle Scholar
  27. 202.
    A. Kirsch, The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media. Inverse Prob. 18(4), 1025–1040 (2002)MathSciNetCrossRefGoogle Scholar
  28. 204.
    V. Komornik, P. Loreti, Fourier Series in Control Theory (Springer, New York, 2005)zbMATHGoogle Scholar
  29. 215.
    S. Kunis, T. Peter, T. Römer, U. von der Ohe, A multivariate generalization of Prony’s method. Linear Algebra Appl. 490, 31–47 (2016)MathSciNetCrossRefGoogle Scholar
  30. 216.
    S. Kunis, H.M. Möller, T. Peter, U. von der Ohe, Prony’s method under an almost sharp multivariate Ingham inequality. J. Fourier Anal. Appl. 24(5), 1306–1318 (2018)MathSciNetCrossRefGoogle Scholar
  31. 225.
    P. Lemke, S.S. Skiena, W.D. Smith, Reconstructing sets from interpoint distances, in Discrete and Computational Geometry (Springer, Berlin, 2003), pp. 597–631CrossRefGoogle Scholar
  32. 228.
    W. Liao, A. Fannjiang, MUSIC for single-snapshot spectral estimation: stability and super-resolution. Appl. Comput. Harmon. Anal. 40(1), 33–67 (2016)MathSciNetCrossRefGoogle Scholar
  33. 231.
    R.D. Luke, Relaxed averaged alternating reflections for diffraction imaging. Inverse Prob. 21(1), 37–50 (2005)MathSciNetCrossRefGoogle Scholar
  34. 235.
    D.G. Manolakis, V.K. Ingle, S.M. Kogon, Statistical and Adaptive Signal Processing (McGraw-Hill, Boston, 2005)Google Scholar
  35. 236.
    I. Markovsky, Structured low-rank approximation and its applications. Autom. J. IFAC 44(4), 891–909 (2008)MathSciNetCrossRefGoogle Scholar
  36. 237.
    I. Markovsky, Low-Rank Approximation: Algorithms, Implementation, Applications, 2nd edn. (Springer, London, 2018)zbMATHGoogle Scholar
  37. 243.
    A. Moitra, The threshold for super-resolution via extremal functions. Massachusetts Institute of Technology, Cambridge (2014, preprint)Google Scholar
  38. 244.
    A. Moitra, Super-resolution, extremal functions and the condition number of vandermonde matrices, in Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing (2015), pp. 821–830Google Scholar
  39. 245.
    H.L. Montgomery, R.C. Vaughan, Hilbert’s inequality. J. Lond. Math. Soc. 8, 73–82 (1974)MathSciNetCrossRefGoogle Scholar
  40. 250.
    F. Natterer, F. Wübbeling, Mathematical Methods in Image Reconstruction (SIAM, Philadelphia, 2001)CrossRefGoogle Scholar
  41. 259.
    M. Osborne, G. Smyth, A modified Prony algorithm for exponential function fitting. SIAM J. Sci. Comput. 16(1), 119–138 (1995)MathSciNetCrossRefGoogle Scholar
  42. 261.
    H. Pan, T. Blu, M. Vetterli, Towards generalized FRI sampling with an application to source resolution in radioastronomy. IEEE Trans. Signal Process. 65(4), 821–835 (2017)MathSciNetCrossRefGoogle Scholar
  43. 264.
    V. Pereyra, G. Scherer, Exponential Data Fitting and Its Applications (Bentham Science Publishers, Sharjah, 2010)Google Scholar
  44. 265.
    T. Peter, G. Plonka, A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators. Inverse Prob. 29, 025001 (2013)MathSciNetCrossRefGoogle Scholar
  45. 266.
    T. Peter, D. Potts, M. Tasche, Nonlinear approximation by sums of exponentials and translates. SIAM J. Sci. Comput. 33, 1920–1947 (2011)MathSciNetCrossRefGoogle Scholar
  46. 267.
    T. Peter, G. Plonka, R. Schaback, Prony’s method for multivariate signals. Proc. Appl. Math. Mech. 15(1), 665–666 (2015)CrossRefGoogle Scholar
  47. 276.
    G. Plonka, M. Wischerhoff, How many Fourier samples are needed for real function reconstruction? J. Appl. Math. Comput. 42(1–2), 117–137 (2013)MathSciNetCrossRefGoogle Scholar
  48. 277.
    G. Plonka, K. Stampfer, I. Keller, Reconstruction of stationary and non-stationary signals by the generalized Prony method. Anal. Appl. (to appear, 2019)Google Scholar
  49. 278.
    G. Plonka, K. Wannenwetsch, A. Cuyt, W.-S. Lee, Deterministic sparse FFT for m-sparse vectors. Numer. Algorithms 78(1), 133–159 (2018)MathSciNetCrossRefGoogle Scholar
  50. 282.
    D. Potts, M. Tasche, Parameter estimation for exponential sums by approximate Prony method. Signal Process. 90, 1631–1642 (2010)CrossRefGoogle Scholar
  51. 283.
    D. Potts, M. Tasche, Parameter estimation for multivariate exponential sums. Electron. Trans. Numer. Anal. 40, 204–224 (2013)MathSciNetzbMATHGoogle Scholar
  52. 284.
    D. Potts, M. Tasche, Parameter estimation for nonincreasing exponential sums by Prony-like methods. Linear Algebra Appl. 439(4), 1024–1039 (2013)MathSciNetCrossRefGoogle Scholar
  53. 285.
    D. Potts, M. Tasche, Sparse polynomial interpolation in Chebyshev bases. Linear Algebra Appl. 441, 61–87 (2014)MathSciNetCrossRefGoogle Scholar
  54. 286.
    D. Potts, M. Tasche, Fast ESPRIT algorithms based on partial singular value decompositions. Appl. Numer. Math. 88, 31–45 (2015)MathSciNetCrossRefGoogle Scholar
  55. 287.
    D. Potts, M. Tasche, Error estimates for the ESPRIT algorithm, in Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics (Birkhäuser/Springer, Cham, 2017), pp. 621–648CrossRefGoogle Scholar
  56. 298.
    D. Potts, M. Tasche, T. Volkmer, Efficient spectral estimation by MUSIC and ESPRIT with application to sparse FFT. Front. Appl. Math. Stat. 2, Article 1 (2016)Google Scholar
  57. 305.
    J. Ranieri, A. Chebira, Y.M. Lu, M. Vetterli, Phase retrieval for sparse signals: uniqueness conditions (2013). arXiv:1308.3058v2Google Scholar
  58. 312.
    R. Roy, T. Kailath, ESPRIT - estimation of signal parameters via rotational invariance techniques, in Signal Processing, Part II, IMA Volumes in Mathematics and its Applications, vol. 23 (Springer, New York, 1990), pp. 369–411Google Scholar
  59. 316.
    S. Sahnoun, K. Usevich, P. Comon, Multidimensional ESPRIT for damped and undamped signals: algorithm, computations, and perturbation analysis. IEEE Trans. Signal Process. 65(22), 5897–5910 (2017)MathSciNetCrossRefGoogle Scholar
  60. 318.
    T.K. Sarkar, O. Pereira, Using the matrix pencil method to estimate the parameters of a sum of complex exponentials. IEEE Antennas Propag. 37, 48–55 (1995)CrossRefGoogle Scholar
  61. 319.
    T. Sauer, Prony’s method in several variables: symbolic solutions by universal interpolation. J. Symbolic Comput. 84, 95–112 (2018)MathSciNetCrossRefGoogle Scholar
  62. 323.
    R. Schmidt, Multiple emitter location and signal parameter estimation. IEEE Trans. Antennas Propag. 34, 276–280 (1986)CrossRefGoogle Scholar
  63. 326.
    B. Seifert, H. Stolz, M. Tasche, Nontrivial ambiguities for blind frequency-resolved optical gating and the problem of uniqueness. J. Opt. Soc. Am. B 21(5), 1089–1097 (2004)CrossRefGoogle Scholar
  64. 327.
    B. Seifert, H. Stolz, M. Donatelli, D. Langemann, M. Tasche, Multilevel Gauss-Newton methods for phase retrieval problems. J. Phys. A 39(16), 4191–4206 (2006)MathSciNetCrossRefGoogle Scholar
  65. 330.
    P. Shukla, P.L. Dragotti, Sampling schemes for multidimensional signals with finite rate of innovation. IEEE Trans. Signal Process. 55(7, Pt 2), 3670–3686 (2007)MathSciNetCrossRefGoogle Scholar
  66. 331.
    M.R. Skrzipek, Signal recovery by discrete approximation and a Prony-like method. J. Comput. Appl. Math. 326, 193–203 (2017)MathSciNetCrossRefGoogle Scholar
  67. 338.
    G. Steidl, A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3–4), 337–353 (1998)MathSciNetCrossRefGoogle Scholar
  68. 364.
    M. Vetterli, P. Marziliano, T. Blu, Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50(6), 1417–1428 (2002)MathSciNetCrossRefGoogle Scholar
  69. 370.
    L. Weiss, R.N. McDonough, Prony’s method, Z-transforms, and Padé approximation. SIAM Rev. 5, 145–149 (1963)MathSciNetCrossRefGoogle Scholar
  70. 388.
    R.M. Young, An Introduction to Nonharmonic Fourier Series, revised 1st edn. (Academic, San Diego, 2001)zbMATHGoogle Scholar
  71. 391.
    R. Zhang, G. Plonka, Optimal approximation with exponential sums by maximum likelihood modification of Prony’s method (2018, preprint)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gerlind Plonka
    • 1
  • Daniel Potts
    • 2
  • Gabriele Steidl
    • 3
  • Manfred Tasche
    • 4
  1. 1.University of GöttingenGöttingenGermany
  2. 2.Chemnitz University of TechnologyChemnitzGermany
  3. 3.TU KaiserslauternKaiserslauternGermany
  4. 4.University of RostockRostockGermany

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