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.
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- Akaike H. (1974), A New Look at Statistical Model Identification, IEEE Trans, on Automatic Control, AC-19 Pages 716–23, Dec.Google Scholar
- Anderson B.D., Moor J.B. (1979) Linear Optimal Control, Prentice-Hall, Englewood Cliffs, N. J.Google Scholar
- Dorny C.N. (1975), A Vector Space Approach to Models and Optimisation, Wiley, New York.Google Scholar
- Giordano A.A., Hsu F.M. (1985), Least Square Estimation with Applications to Digital Signal Processing, Wiley, New York.Google Scholar
- Kailath T. (1977), Linear Least Squares Estimation, Benchmark Papers in Electrical Engineering and Computer science, Dowden, Hutchinson &Ross.Google Scholar
- 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
- Orfanidis S. J. (1988), Optimum Signal Procesing: An introduction, 2nd Edition, Macmillan, New York.Google Scholar
- Wilkinson J. H. (1965), The Algebraic Eigenvalue Problem, Oxford University Press, Oxford.Google Scholar
- Whittle P.W. (1983), Prediction and Regulation by Linear Least-Squares Methods, University of Minnesota Press, Minneapolis, Minnesota.Google Scholar