Singular value decomposition:
Denoising an FID

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SVD with Java

Online tool

Java library

Toeplitz matrix

  • Wikipedia: Toeplitz matrix
    Toeplitz matrix

    matrix in which each descending diagonal from left to right is constant

  • Andrew E. Yagle
    A new algorithm for the nearest singular Toeplitz matrix to a given Toeplitz matrix, (PDF)
  • IMSL® Fortran Numerical Math Library: Solves a complex Toeplitz linear system. (LSLTC)

Hankel matrix

  • Wikipedia: Hankel matrix
    Hankel matrix

    matrix in which each descending diagonal from right to left is constant

    Hankel matrix element

  • Fast general Hankel/Toeplitz SVD package (MatLab)
  • Xuezhi Zhao and Bangyan Ye
    Similarity of signal processing effect between Hankel matrix-based SVD and wavelet transform and its mechanism analysis,
    Mechanical Systems and Signal Processing 23, 1062-1075 (2009).
  • Kevin Browne, Sanzheng Qiao, and Yimin Wei
    A Lanczos bidiagonalization algorithm for Hankel matrices,
    Linear Algebra Appl. 430, 1531-1543 (2009).

    For any square complex Hankel matrix A of order n, there exist a unitary Q and an order n nonnegative diagonal Σ such that A = QΣQT.

  • Wei Xua and Sanzheng Qiao
    A fast symmetric SVD algorithm for square Hankel matrices ,
    Linear Algebra Appl. 428, 550-563 (2008).
  • Vadim Olshevsky and Michael Stewart
    Stable factorization for Hankel and Hankel-like matrices,
    Numer. Linear Algebra Appl. 189, 401-434 (2001).
  • Ye Li, K. J. Ray Liu, and J. Razavilar
    A parameter estimation scheme for damped sinusoidal signals based on low-rank Hankel approximation,
    IEEE Trans. Signal Processing 45, 481-486 (1997).
  • Peter Strobach
    Square Hankel SVD subspace tracking algorithms,
    Signal Process. 57, 1-18 (1997).

    For a real square N x N Hankel data matrix, left and right orthonormal matrices are identical.

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  • Réemy Boyer and Roland Badeau
    Adaptive multilinear SVD for structured tensors, (PDF)
  • Timothy M. Toolan and Donald W. Tufts
    Detection and estimation in non-stationary environments, (PDF)
  • D. L. Boley, F. T. Luk, and D. Vandevoorde
    Vandermonde factorization of a Hankel matrix, (PDF)

Cadzow procedure

  • Stewart Trickett
    F-xy Cadzow noise suppression,
    2008 CSPG CSEG CWLS Convention
  • Simon Poulding, Adrian J. Charlton, James Donarski, and Julie C. Wilson
    Removal of t1 noise from metabolomic 2D 1H-13C HSQC NMR spectra by correlated trace denoising,
    J. Magn. Reson. 189, 190-199 (2007).
  • Sharif D. Kunikeev, Howard S. Taylor, Thorsten Schroer, Ralf Haiges, C. J. Bigler Jones, and Karl O. Christe
    New signal processing method for the faster observation of natural-abundance 15N NMR spectra and its application to N5+,
    Inorg. Chem. 45, 437-442 (2006).
  • Sharif D. Kunikeev, Howard S. Taylor, Jian-Jung Pan, Allan Kershaw, and Charles E. McKenna
    A new signal processing method to observe weak 31P and 17O NMR peaks,
    J. Organomet. Chem. 690, 2644-2650 (2005).
  • Alexey L. Kaledin, Sharif D. Kunikeev, and Howard S. Taylor
    An accurate theoretical prediction of the zero point vibrational energy of CH5+,
    J. Phys. Chem. A 108, 4995-4997 (2004).
  • Sharif D. Kunikeev and Howard S. Taylor
    Saving measurement time in 13C NMR spectroscopy,
    J. Phys. Chem. A 108, 743-753 (2004).
    Abstract (PDF file)
  • Sharif D. Kunikeev, Erdinç Atilgan, Howard S. Taylor, Alexey L. Kaledin, and Jörg Main
    An application of error reduction and harmonic inversion schemes to the semiclassical calculation of molecular vibrational energy levels,
    J. Chem. Phys. 120, 6478-6486 (2004).
  • Stewart Trickett
    F-x eigen noise suppression,
    CSEG Geophysics 2002 PDF file
  • Caroline Brissac, Thérèse E. Malliavin, and Marc A. Delsuc
    Use of the Cadzow procedure in 2D NMR for the reduction of t1 noise,
    J. Biomol. NMR 6, 361-365 (1995).
  • A. Diop, Y. Zaim-Wadghiri, A. Briguet, and D. Graveron-Demilly
    Improvements of quantitation by using the Cadzow enhancement procedure prior to any linear-prediction methods,
    J. Magn. Reson. B 105, 17-24 (1994).
  • Y. Y. Lin and L. P. Hwang
    NMR signal enhancement based on matrix property mappings,
    J. Magn. Reson. A 103, 109-114 (1993).
  • A. Diop, A. Briguet, and D. Graveron-Demilly
    Automatic in vivo nmr data processing based on an enhancement procedure and linear prediction method,
    Magn. Reson. Med. 27, 318-328 (1992).
  • J. A. Cadzow
    Signal enhancement–A composite property mapping algorithm,
    IEEE Trans. Acoustics, Speech, and Signal Processing 36, 49-62 (1988).
  • J. A. Cadzow
    Spectral estimation: An overdetermined rational model equation approach,
    Proceedings IEEE 70, 907-939 (1982).

Java applet performing SVD of a complex Hankel matrix: Denoising an FID

The size (600 complex numbers) of an FID (free-induction decay) signal is limited by the web browser. With a 3-GHz processor, the singular value decomposition of the associated complex Hankel matrix takes about 40 seconds.

It is simpler to input the real and imaginary values of the FID into two columns of a spreadsheet program such as MS EXCEL, then copy and paste each data column into the Java applet.

The fast Fourier transform of an FID requires a power of two for the number of complex numbers in an FID, therefore zero-filling the FID before FFT.

Java does not allow us to paste external data into an applet. We provide the corresponding fidsvd applet for local use, outside a web browser: download.

The change of color of View FID button from red to green after clicking Run SVD button means the number of iterations for singular value decomposition has reached its maximum. As a result, close then restart the applet. Provide a shorter FID.

Here is an MS Excel spread sheet containing simulated FIDs. The SVD of 1000 complex numbers takes about 180 seconds.

Spectrum from denoised 29-Si MAS FID by SVD

Comparison of spectra obtained from FID traited by line broadening with that obtained from the same FID denoised by SVD:

Silicium-29 MAS spectrum

Spectrum obtained from denoised FID by SVD presents better resolution.

Signal and noise

  • Stevens Institute for Innovation, University of Southern California, 2009
    “Denoising 1D solution NMR (DSNMR)” fewer transients - more sensitivity magnet resonance analysis by Singular Value Decomposition Harmonic Inversion,
    PDF file
  • François Chapeau-Blondeau and David Rousseau
    Raising the noise to improve performance in optimal processing,
    J. Stat. Mech: Theory Exp. P01003 (2009).
  • Christoph Kaiser, Jakob J. Lopez, Wolfgang Bermel, and Clemens Glaubitz
    Dual transformation of homonuclear solid-state NMR spectra—an option to decrease measuring time,
    Biochim. Biophys. Acta, Biomembr. 1768, 3107-3115 (2007).
  • Ulrich L. Günther
    Advanced NMR Processing,
    EuroLabCourse "Advanced Computing in NMR Spectroscopy", Florence, Sept. 2001
    Presentation PDF file, Lecture notes PDF file
  • Ulrich L. Günther, Christian Ludwig, and H. Rüterjans
    NMRLAB—Advanced NMR data processing in Matlab,
    J. Magn. Reson. 145, 201-208 (2000).
  • Martin Vetterli: Sparse sampling: Variations on a theme by Shannon
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