0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.