# Why can the :$\mathbf W^*_{MMSE}$ be written as $E[\mathbf s \mathbf y^*]E[\mathbf y \mathbf y^*]^{-1}$

I read this paper: https://arxiv.org/pdf/1305.2460.pdf

And there is a formula about MMSE:$$\mathbf W^*_{MMSE}= E[\mathbf s \mathbf y^*]E[\mathbf y \mathbf y^*]^{-1}$$

$$\mathbf W _{BB}$$:Baseband combiner

$$\mathbf W _{RF}$$:RF combiner

$$\mathbf W$$:The combination of $$\mathbf W_{RF}$$ and $$\mathbf W _{BB}$$

$$\mathbf W^*$$ : the hermitian of $$\mathbf W$$

$$\mathbf s$$:the signal on the left

$$\mathbf y$$:the signal when $$\mathbf s$$ goes through the channel

Why can the :$$\mathbf W^*_{MMSE}$$ be written as $$E[\mathbf s \mathbf y^*]E[\mathbf y \mathbf y^*]^{-1}$$.why should we multiply the these two signal ,$$\mathbf s$$ and $$\mathbf y$$,and average them ?Does anyone have information about this formula ?

• There is a reference in the paper, adressing this formula: T. Kailath, A. H. Sayed, and B. Hassibi, Linear estimation. Prentice Hall New Jersey, 2000, vol. 1. Mar 28, 2019 at 14:26

$$J(\mathbf{W})=\mathbb{E}\left[\|\mathbf{s}-\mathbf{W}\mathbf{y}\|^2\right]$$
which is the equalizer that minimizes the Euclidean distance between the transmitted signal $$\mathbf{s}$$ and the equalized received signal $$\mathbf{Wy}$$. To find the value of $$\mathbf{W}$$ the minimizes $$J(\mathbf{W})$$, take the derivative with respect to $$\mathbf{W}$$ and put it equal to $$\mathbf{0}$$, which will result in
$$-2\mathbb{E}\left[\left(\mathbf{s}-\mathbf{Wy}\right)\mathbf{y}^H\right]=\mathbf{0}$$ Solve the above equation with respect to $$\mathbf{W}$$ you get the MMSE equalizer.
Note: $$(.)^H$$ is the conjugate transpose, which is equivalent to $$(.)^*$$ in the paper.