Kalman Filters

  • Paulo S. R. Diniz


This section provides a brief description of Kalman filter that can be considered an extension of the Wiener filtering concept [4]. The Kalman filter has as objective the minimization of the estimation square error of a nonstationary signal buried in noise. The estimated signal itself is modeled utilizing the state–space formulation [1] describing its dynamical behavior. In summary, Kalman filtering deals with random processes described using state–space modeling which generate signals that can be measured and processed utilizing time recursive estimation formulas. The presentation here is brief and addresses the case of signals and noises represented in vector form; for more details in this subject the reader can consult many books available presenting Kalman filtering, including [3, 5]. There are many different ways to describe the Kalman filtering problem, and to derive its corresponding relations, here we follow the presentations of [2, 6].


Kalman Filter Space Formulation Observation Vector Error Covariance Matrix Kalman Gain 
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  1. 1.
    P.S.R. Diniz, E.A.B. da Silva, S.L. Netto, Digital Signal Processing: System Analysis and Design, 2nd edn. (Cambridge University Press, Cambridge, 2010)CrossRefMATHGoogle Scholar
  2. 2.
    M.H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, New York, 1996)Google Scholar
  3. 3.
    T. Kailath, A.H. Sayed, B. Hassibi, Linear Estimation (Prentice Hall, Englewood Cliffs, 2000)MATHGoogle Scholar
  4. 4.
    R.E. Kalman, A new approach to linear filtering and prediction problem. Trans. ASME J. Basic Eng. 82, 34–45 (1960)Google Scholar
  5. 5.
    S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, 1993)MATHGoogle Scholar
  6. 6.
    D.G. Manolakis, V.K. Ingle, S.M. Kogon, Statistical and Adaptive Signal Processing (McGraw Hill, New York, 2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil

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