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

Covariance 

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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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