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.

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


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