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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.

Thanks

Jacob
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    Write the likelihood for $\mathbf{n}$ in terms of $\mathbf{y}$ and $\mathbf{Hx}$. Minimize $-2\log L$ with respect to $\mathbf{x}$. You should find this reasonably straightforward. This (with slightly different notation) is done on site already. – Glen_b Feb 01 '19 at 00:27
  • e.g. 1. https://stats.stackexchange.com/questions/124576/deriving-the-maximum-likelihood-for-the-parameters-in-linear-regression (details, though not necessarily in the most straightforward way) 2. https://stats.stackexchange.com/questions/144495/maximum-likelihood-estimate-mle-equivalent-to-finding-hat-y-in-linear-regre?rq=1 (outline of approach) 3. https://stats.stackexchange.com/questions/173621/linear-regression-any-non-normal-distribution-giving-identity-of-ols-and-mle (simple case) ... I'll try to find a more canonical version though – Glen_b Feb 01 '19 at 00:36

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