Let's assume we are sending signal $\mathbf{x}$ (which is a vector $N\times 1$) through channel $\mathbf{H}$ (a $M\times N$ matrix). Our model is $\mathbf{y}=\mathbf{Hx}+\mathbf{n}$. Note that the matrix $\mathbf{H}$ is given at the receiver and $\mathbf{n}$ is a normal distribution with $\mathbf{n}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbf{I}_M)$.

We wish to estimate the vector $\mathbf{x}$ by using the maximum likelihood method. How we can do it?

My solution was to calculate $P\left(Y|X\right)$ which is a normal distribution with mean $\mathbf{Hx}$ and variance $\sigma^2\mathbf{I}_M$ and then maximize it. I don't know where to go after this.




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