How to find covariance matrix of a AWGN noise vector $n$ with variance =$\sigma^2$

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Suppose $n=\sigma*randn(1000,1)+ u$ is a noise vector. How can i simulate the covariance matrix $E[n.n^T]$ in MATLAB such that we get $E[n.n^T]= \sigma^2I$ in the simulation with rank=1000.

If we multiply vector $n$ with its transpose, rank will be $1$ only,but the rank of $E[n.n^T]$ should be 1000,right?

In general how do you take expectation E of a $[n.n^T]$ matrix. I have seen some answers but didn't understand correctly. Thanks.