# Explain the existence of limit in Persistence Excitation — mostly zero and non-existent?

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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There's no order in your definition. – joriki Dec 17 '12 at 12:05
@joriki I haven't yet understood the notation particularly the (ii) part with $R_u(\tau)$ in comparison to $r_u(\tau)$. This is how I understand it -- a signal is PE if the limit exist. And it is of some order $n$ if some other condition $n$ is satisfied but cannot understand the notation differente with capital $R$ and small $r$ -- perhaps misunderstanding the definition. – hhh Dec 17 '12 at 12:18
@hhh: can you visit the Tagging room and answer some questions about (system-identification)? – Willie Wong Dec 18 '12 at 16:20