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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

Spectral Density Green Function Output Variance Mathematical Expectation Stochastic Analysis 
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 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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