The title of the question might be misleading, but it shows how clueless I am about this problem.

I have a time series of some quantity $X(t)$. I can calculate autocorrelation of this quantity $Y(\tau)=\frac{1}{N}\sum_{t_i}X(t_i)X(t_i+\tau)$. I am looking for an integral of the autocorrelation function, which would be DC component of the signal $Y(\tau)$ (it is the zeroth frequency element of the Fourier transform of $Y(\tau)$): \begin{align*} I=\frac{1}{N}\sum _{\tau _i}Y(\tau _i)=\frac{1}{N}\sum _{\tau _i}Y(\tau _i)\exp(i\omega \tau _i)\rvert _{\omega=0}. \end{align*}

I know that my signal $Y(\tau)$ can be represented in a Fourier space, meaning it is a sum of some oscillatory functions. These oscillatory functions are decaying, hence the non-zero $I$. That means that somehow I should be able to decompose $I$ as a contribution of each of the oscillating functions.

Is there a standard way of obtaining $I(\omega _i)$ from my original signal $X(t)$ so it satisfies: $I=\sum _{\omega _i} I(\omega _i)$?


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