Wiener theory, formulated by Norbert Wiener, forms the foundation of data-dependent linear least squared error filters. Wiener filters play a central role in a wide range of applications such as linear prediction, signal coding, echo cancellation, signal restoration, channel equalisation, system identification etc. The coefficients of a Wiener filter are calculated to minimise the average squared distance between the filter output and a desired signal. In its basic form, the Wiener theory assumes that the signals are stationary processes. However, if the filter coefficients are periodically recalculated, for every block of N samples, then the filter adapts to the average characteristics of the signals within the blocks and becomes block-adaptive. A block-adaptive filter can be used for signals that can be considered stationary over the duration of the block. In this chapter we study the theory of Wiener filters, and consider alternative methods of formulation of the Wiener filter problem. We consider the application of Wiener filters in channel equalisation, time-delay estimation, and additive noise suppression. A case study of the frequency response of a Wiener filter, for additive noise reduction, provides useful insight into the operation of the filter. We also deal with some implementation issues of Wiener filters.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akaike H. (1974), A New Look at Statistical Model Identification, IEEE Trans, on Automatic Control, AC-19 Pages 716–23, Dec.Google Scholar
  2. Alexander S.T. (1986), Adaptive Signal Processing Theory and Applications. Springer-Verlag, New York.zbMATHGoogle Scholar
  3. Anderson B.D., Moor J.B. (1979) Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N. J.Google Scholar
  4. Bjorck A. (1967), Solving Linear Least Squares Problems by Gram-Schmidt Orthogonalisation, BIT, Vol. 7, Pages 1–21.MathSciNetCrossRefGoogle Scholar
  5. Dorny C.N. (1975), A Vector Space Approach to Models and Optimisation, Wiley, New York.Google Scholar
  6. Durbin J. (1959), Efficient Estimation of Parameters in Moving Average Models, Biometrica Vol. 46, Pages 306–16.MathSciNetzbMATHGoogle Scholar
  7. Giordano A.A., Hsu F.M. (1985), Least Square Estimation with Applications to Digital Signal Processing, Wiley, New York.Google Scholar
  8. Givens W. (1958), Computation of Plane Unitary Rotations Transforming a General Matrix to Triangular Form, SIAM J. Appl. Math. Vol. 6, Pages 26–50.MathSciNetzbMATHGoogle Scholar
  9. Golub G.H., Reinsch (1970), Singular Value Decomposition and Least Squares Solutions, Numerical Mathematics, Vol. 14, Pages 403–20.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Golub G.H., Van Loan C.F. (1983). Matrix Computations, Johns Hopkins University Press, Baltimore, MD.zbMATHGoogle Scholar
  11. Golub G.H., Van Loan C.F. (1980). An Analysis of the Total Least Squares Problem, SIAM Journal of Numerical Analysis, Vol. 17, Pages 883–93.zbMATHCrossRefGoogle Scholar
  12. Halmos P. R. (1974), Finite-Dimensional Vector Spaces. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  13. Haykin S. (1991), Adaptive Filter Theory, 2nd Edition, Prentice-Hall, Englewood Cliffs, N. J.zbMATHGoogle Scholar
  14. Householder A.S. (1964), The Theory of Matrices in Numerical Analysis, Blaisdell, Waltham, Mass.zbMATHGoogle Scholar
  15. Kailath T. (1974), A View of Three Decades of Linear Filtering Theory, IEEE Trans. Information Theory, Vol. IT-20, Pages 146–81.MathSciNetCrossRefGoogle Scholar
  16. Kailath T. (1977), Linear Least Squares Estimation, Benchmark Papers in Electrical Engineering and Computer science, Dowden, Hutchinson &Ross.Google Scholar
  17. Kailath T. (1980), Linear Systems, Prentice-Hall, Englewood Cliffs, N. J.zbMATHGoogle Scholar
  18. Klema V.C., Laub A. J. (1980), The Singular Value Decomposition: Its Computation and Some Applications, IEEE Trans. Automatic Control, Vol. AC-25, Pages 164–76.MathSciNetCrossRefGoogle Scholar
  19. Kolmogrov A.N. (1939), Sur 1’ Interpolation et Extrapolation des Suites Stationaires, Comptes Rendus de l’Academie des Sciences, Vol. 208, Pages 2043–45.Google Scholar
  20. Lawson C.L., Hanson R.J. (1974), Solving Least Squares Problems, Prentice-Hall, Englewood Cliffs, N. J.zbMATHGoogle Scholar
  21. Orfanidis S. J. (1988), Optimum Signal Procesing: An introduction, 2nd Edition, Macmillan, New York.Google Scholar
  22. Scharf L.L. (1991), Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, Addison Wesley, Reading, Mass.zbMATHGoogle Scholar
  23. Strang G.(1976), Linear Algebra and Its Applications, 3rd ed., Harcourt Brace Jovanovich, San Diego, California.zbMATHGoogle Scholar
  24. Wiener N. (1949), Extrapolation, Interpolation and Smoothing of Stationary Time Series, MIT Press Cambridge, Mass.zbMATHGoogle Scholar
  25. Wilkinson J. H. (1965), The Algebraic Eigenvalue Problem, Oxford University Press, Oxford.Google Scholar
  26. Whittle P.W. (1983), Prediction and Regulation by Linear Least-Squares Methods, University of Minnesota Press, Minneapolis, Minnesota.Google Scholar
  27. Wold H., (1954), The Analysis of Stationary Time Series, 2nd ed. Almquist and Wicksell, Uppsala, Sweden.zbMATHGoogle Scholar

Copyright information

© John Wiley & Sons Ltd. and B.G. Teubner 1996

Authors and Affiliations

  • Saeed V. Vaseghi
    • 1
  1. 1.Queen’s UniversityBelfastUK

Personalised recommendations