Can we decompose a correlated random variable into correlated plus uncorrelated?
Using Karhunen-loève expansion (principal component analysis) on a centered stochastic process $X_k$ you will get :
$$
X_k=\sum_{i=1}^{N}Z_ie_i(k) \quad k\in[1,N]
$$
Where $N$ is the process's length and $e_i$ are the $N$ eigenvectors of length $N$ from ACP. The $Z_i$ will be uncorrelated.

I'm not sure it is answering your question, I advise you to ask this on cross validated.