# Spectral decomposition of the DC part of the signal

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: $$I=\sum _{\omega _i} I(\omega _i)$$?