# What is the maximum likelihood for this distribution(received signal)? [duplicate]

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

• 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. Commented Feb 1, 2019 at 0:27
• e.g. 1. stats.stackexchange.com/questions/124576/… (details, though not necessarily in the most straightforward way) 2. stats.stackexchange.com/questions/144495/… (outline of approach) 3. stats.stackexchange.com/questions/173621/… (simple case) ... I'll try to find a more canonical version though Commented Feb 1, 2019 at 0:36