In my research work, I came across the following statement:
$\textbf{v}_{c,l}\in \mathbb{C}^{M \times 1}$ is noise vector (random vector) at time $l$ at receiver having $M$ antennas.
The $\textbf{v}_{c,l}$ is assumed as Circularly Symmetric Complex Gaussian (CSCG) random vector with $\textbf{v}_{c,l} \sim \mathcal{C}\mathcal{N} (\mathbb{0},\sigma^2 \mathbf{I}_M)$, where $\mathbb{0}$ indicates zero vector, $\sigma^2$ is noise variance at each antenna of the receiver and $\mathbf{I}_M$ identity Matrix.
I understand that mean vector (in this case it is zero vector) consists of means at each antenna of the receiver.
My query is that I am not getting why Identity matrix is included in the above expression.
The identity matrix means that there is no cross-correlation between the antennas, that is, the observations in antenna $i$ are completely independent of the observations in antenna $j$. For $M$ antennas, you have a correlation matrix $\mathbf{C}\in \mathbb{C}^{M \times M}$, where element $(i,j)$ is the correlation coefficient between antenna $i$ and antenna $j$. If $\mathbf{C} = \sigma^2\mathbf{I}$, then only the diagonal elements are non-zero and, thus, $\mathbf{C}_{ij} = 0$ if $i \neq j$.