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One of the primitive aims of the two-dimensional BTW model had been toexplain the $1/f^{\alpha}$ noise which is widely seen in the natural systems.In this paper we study some time signals, namely the activity inside anavalanche ($x(t)$), the avalanches sizes ($s(T)$) and the rare events waitingtime (REWT) ($\tau(n)$ as a new type of noise). The latter is expected to beimportant in predicting the period and also the vastness of the upcoming largescale events in a sequence of self-organized natural events. Especially wereport some exponential anti-correlation behaviors for $s(T)$ and $\tau(n)$which are finite-size effects. Two characteristic time scales $\delta T_{s}$and $\delta T_{\tau}$ emerge in our analysis. It is proposed that the powerspectrum of $s(T)$ and $\tau(n)$ behave like $\left(b_{s,\tau}(L)^2+\omega^2\right)^{-\frac{1}{2}}$, in which $b_{s}$ and$b_{\tau}$ are some $L$-dependent parameters and $\omega$ is the angularfrequency. The $1/f$ noise is therefore obtained in the limit $\omega\ggb_{s,\tau}$. $b_{s}$ and $b_{\tau}$ decrease also in a power-law fashion withthe system size $L$, which signals the fact that in the thermodynamic limit thepower spectrum tends to the Dirac delta function.

Power spectrum of rare events in two-dimensional BTW model, violation of $1/f$ noise hypothesis