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Definitions

Persistence Excitation on page 121 here or shortly here and here. A signal is PE if this limit exists $r_u(\tau)=\lim_{N\rightarrow\infty}\frac 1 N \sum_{t=1}^{N} u(t+\tau)u^T(t)$

And it is of order $n$ if some condition for unknown matrix $R_u(\tau)$ is satisfied. I cannot understand this part of the definition, more in comments.

How can the limit exist with the PE? If it does, why is it not always zero with most signals? Could someone open the examples a bit to show the non-zero limits? What is the difference between capital $R$ and small $r$?

Observations

  • This article here states that ARMA processes are PE of any finite order.

  • My university defines PE in terms of the spectrum: if the condition $\Phi(t)>0$ for the spectrum $\Phi(\omega)$ of the signal $u(t)$ almost evewhere in the range $(-\pi, \pi)$, then $u(t)$ is continuously PE -- source here.

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    @hhh: can you visit the Tagging room and answer some questions about (system-identification)?2012-12-18

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