BIT Numerical Mathematics

, Volume 38, Issue 1, pp 34–43 | Cite as

Rank-deficient prewhitening with quotientSVD andULV decompositions

  • Per Christian Hansen


This paper deals with certain theoretical and numerical aspects of prewhitening, which is a technique frequently used in signal processing when dealing with signals degraded by colored noise. In particular, we demonstrate how to prewhiten a signal contaminated by an interfering noisy signal whose covariance matrix is rank deficient. The formulation of our technique is based on the quotient (or generalized) singular value decomposition, and we also show that a quotient-version of theULV decomposition can be used to provide an efficient updatable implementation.

AMS subject classifications

65F20 65F30 

Key words

Prewhitening QSVD quotientULV decomposition signal processing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. L. R. De Moor,The singular value decomposition and long and short spaces of noisy matrices, IEEE Trans. Signal Proc., 41 (1993), pp. 2826–2838.MATHCrossRefGoogle Scholar
  2. 2.
    L. Eldén,A weighted pseudoinverse, generalized singular values, and constrained least squares problems, BIT, 22 (1982), 487–501.MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    P. C. Hansen,Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, 1997.MATHGoogle Scholar
  4. 4.
    S. H. Jensen, P. C. Hansen, S. D. Hansen, and J. Aa. Sørensen,Reduction of broad-band noise in speech by trucated QSVD, IEEE Trans. Audio Speech Proc., 3 (1995), pp. 439–448.CrossRefGoogle Scholar
  5. 5.
    F. T. Luk and S. Qiao,A new matrix decomposition for signal processing, Automatica, 30 (1994), 39–43.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    F. T. Luk and S. Qiao,An adaptive algorithm for interference cancelling in array processing; in Advanced Signal Processing Algorithms, Architectures, and Implementations VI, F. T. Luk ed. SPIE Proceedings, Vol. 2846 (1996), pp. 151–161.Google Scholar
  7. 7.
    L. L. Scharf and D. W. Tufts,Rank reduction for modeling stationary signals, IEEE Trans. Acoust., Speech, Signal Proc., 35 (1987), pp. 350–355.CrossRefGoogle Scholar
  8. 8.
    G. W. Stewart,UTV decompositions; in Numerical Analysis 1993, D. F. Griffith and G. A. Watson, eds., Pitman Notes in Mathematical Sciences, New York, 1994, pp. 225–236.Google Scholar

Copyright information

© Swets & Zeitlinger 1998

Authors and Affiliations

  • Per Christian Hansen
    • 1
  1. 1.Department of Mathematical ModellingTechnical University of DenmarkLyngbyDenmark

Personalised recommendations