2
$\begingroup$

Let R be an $(N+1)\times (N+1)$ Toeplitz matrix. I would be mostly interested in the case of $N\to \infty$. Let $x$ be a complex Gaussian random $N\times 1$ vector with mean zero and covariance matrix $I$, i.e., $x\sim \mathcal{CN}(0,I)$. Let $X$ be a diagonal matrix with $x$ along its main diagonal. Let $\lambda$ be an arbitrary but fixed positive number. Let $\tilde{R}$ be the $N\times N$ sub matrix of $R$ constructed by removing the last row and the last column from $R$. Let $r$ be an $N\times 1$ vector that equals the first column of $R$, but with the first element removed.

Question: How to compute $$E_x\left[1-r^*X^*(X\tilde{R}X^*+\lambda I)^{-1}Xr\right].$$ This can be interpreted as the MMSE of an estimation of a correlated process observed in Gaussian noise through Gaussian training symbols. The diagonal of $R$ is assumed here to be $I$.

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.