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Linear Filters

  • Robert Azencott
  • Didier Dacunha-Castelle
Part of the Applied Probability book series (APPLIEDPROB, volume 2)

Abstract

Let X be a centered second-order stationary process. Let ak, k ɛ ZZ be a sequence of complex numbers, equal to zero for |k| large enough. The process Y = AX defined by
$${Y_n} = \sum\limits_k {{a_k}{X_{n - k}}}$$
is then centered and second-order stationary, for if γ(m - n) = Г(X m ,X n ), the covariance of Y becomes
$$\Gamma \left( {{Y_m},{Y_n}} \right) = \sum\limits_{k,\ell } {{a_k}{{\bar a}_\ell }E\left( {{X_{m - k}}{{\bar X}_{n - \ell }}} \right) = \sum\limits_{k,\ell } {{a_k}{{\bar a}_\ell }\gamma \left( {m - n - k + \ell } \right)} }$$
which is clearly a function of (m - n).

Keywords

Response Function Spectral Density Spectral Measure Linear Filter Inverse Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Robert Azencott
    • 1
  • Didier Dacunha-Castelle
    • 1
  1. 1.Equipe de Recerche Associée au C.N.R.S. 532 Statistique Appliquée MathématiqueUniversité de Paris-SudOrsay CedexFrance

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