# Can we decompose a correlated random variable into correlated plus uncorrelated?

Assume a random sequence $Y_k = X_k \cdot H_k + W_k$ where $X_k$ is a deterministic quantity, $H_k$ is a correlated random sequence, $W_k$ is Additive White Gaussian random sequence.

Can we decompose/rewrite the above sequence into a form like below? $Y_k = X_k \cdot \hat H_k + X_k \cdot \tilde H_k + W_k$

where $\hat H_k$ is deterministic/white and $\tilde H_k$ is correlated random - something like that?

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.