Stochastic analysis and H2—norm of linear periodic operators

  • Efim N. Rossenwasser
  • Bernhard P. Lampe
Part of the Communications and Control Engineering book series (CCE)

Abstract

In this chapter we investigate the response of a linear periodic operator to an input signal x(t), that is a centered, and in a loose sense, stationary stochastic process. The last property means that, Åström (1970)
$$E[x(t)] = 0$$
(8.1)
$$E[x({{t}_{1}})x({{t}_{2}})] = {{K}_{x}}({{t}_{2}} - {{t}_{1}})$$
(8.2)
where E denotes the operator of mathematical expectation and K x (t) is the autocorrelation function of the signal x(t). If we take t1 = t and t2 = t + τ, then
$$E[x(t)x(t + \tau )] = {{K}_{x}}(\tau )$$
(8.3)
and, as is well known
$${{K}_{x}}(t) = {{K}_{x}}( - t).$$
(8.4)

Keywords

Autocorrelation 

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Copyright information

© Springer-Verlag London Limited 2000

Authors and Affiliations

  • Efim N. Rossenwasser
    • 1
  • Bernhard P. Lampe
    • 2
  1. 1.Department of Automatic ControlSt Petersburg State Marine Technical UniversitySt PetersburgRussia
  2. 2.Institute of AutomationUniversity of RostockRostockGermany

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