Kalman Filters

  • Paulo S. R. DinizEmail author


This chapter describes the Kalman filters that provide an optimal estimate of hidden signals through a linear combination of previous estimates of these signals and with the newest available measurement signals. The Kalman filters can be considered an extension of the Wiener filtering concept [1, 2], in the sense that it allows for an estimate of non-directly measurable state variables of dynamic systems. The Kalman filter has as objective the minimization of the estimation square errors of nonstationary signals buried in noise. The estimated signals themselves are modeled utilizing the so-called state–space formulation [3] describing their dynamical behavior. While the Wiener filter provides the minimum MSE solution for the hidden parameters, leading to the optimal solution for an environment with wide-sense stationary signals, the Kalman filter offers a minimum MSE solution for time-varying environments involving linear dynamic systems whose noise processes involved are additive Gaussian noises. In the latter case of Kalman filters, the parameters of the dynamic systems can be time-varying.


  1. 1.
    R.E. Kalman, A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82, 34–45 (1960)MathSciNetGoogle Scholar
  2. 2.
    R.E. Kalman, Mathematical description of linear dynamical systems. SIAM J. Control 1, 152–192 (1963)MathSciNetzbMATHGoogle Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    M.H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, New York, 1996)Google Scholar
  5. 5.
    T. Kailath, A.H. Sayed, B. Hassibi, Linear Estimation (Prentice Hall, Englewood Cliffs, 2000)zbMATHGoogle Scholar
  6. 6.
    S.M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice Hall, Englewood Cliffs, 1993)zbMATHGoogle Scholar
  7. 7.
    D.G. Manolakis, V.K. Ingle, S.M. Kogon, Statistical and Adaptive Signal Processing (McGraw Hill, New York, 2000)Google Scholar
  8. 8.
    Y. Bar-Shalom, X.R. Li, T. Kirubarajan, Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software (Wiley, New York, 2001)CrossRefGoogle Scholar
  9. 9.
    X.-H. Wen, W.H. Chen, Real-time reservoir model updating using ensemble Kalman filter. SPE J. 11, 431–442 (2006)CrossRefGoogle Scholar
  10. 10.
    G. Evensen, The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)CrossRefGoogle Scholar
  11. 11.
    G. Evensen, The ensemble Kalman filter: theoretical formulation and practical implementation. IEEE Control Syst Mag. 29, 83–104 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    G. Burgers, P.J. van Leeuwen, G. Evensen, Analysis scheme in the ensemble Kalman filter. Mon. Weather Rev. 126, 1719–1724 (1998)CrossRefGoogle Scholar
  13. 13.
    M. Roth, G. Hendeby, C. Fritsche, F. Gustafsson, The ensemble Kalman filter: a signal processing perspective. EURASIP J. Adv. Signal Process. 2017(56), 1–17 (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Universidade Federal do Rio de JaneiroNiteróiBrazil

Personalised recommendations