what is the covariance matrix for deterministic signal+normal noise

Question

Say that we have a signal that is written as follow $y=y_0+r$ where $y$ and $y_0$ are n-dimensional vectors and $r$ is n-dimensional noise vector. I would like to have $r\sim \mathcal{N}(0,\Sigma)$ where $\Sigma=\sigma^2I_{n,n}\in \mathcal{M}_{n,n}$ is the covariance matrix of the vector $r$. So my question is how can I generate $r$ such that $r\sim \mathcal{N}(0,\sigma^2I_{n,n})$.

Answer 1

If $I_{n,n}$ is the ientity matrix, start from $n$ independent standard normal random variables $x_k$ and define the random vector $r$ by $r=(\sigma x_k)_{1\leqslant k\leqslant n}$.