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Why can we know the best precoding direction after we do the SVD to the channel?for example

$\mathbf y=\mathbf H\mathbf s+\mathbf n$,now we do the SVD to the $\mathbf H$

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If the $\sigma _1$ is the biggest singular value,after we normalize $V_1$,why can we say the $V_1$ is the best precoding direction?Is there any theorem can prove this?

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  • $\begingroup$ You don't really need any theorem – $\sigma_1$ is exactly the factor with which the power is scaled. $\endgroup$ – Marcus Müller Nov 3 at 17:19
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First you have to know that the vector in $U$ and the vector in $V$ are orthonomal,that is ,$\vec u \vec u^H=I$,and $\vec v \vec v^H=I$.

So now $H=U\Sigma V^H$,and $y=Hs+n=(U\Sigma V^H)s+n$

and $U^H y=\Sigma V^Hs+n$,see the $\Sigma V^Hs$,isn't it like " channel gain $\times$ beamforming $\times$ signal $s$ "?

Now you can also find that according to the rule of multiplication of matrix,the first element of $\Sigma$,which is the $\sigma_1$,will multiple every element of $v_1^H$,that is why we say the $v^H_1$ is the best precoding ,because its elements will multiply the biggest singular value ,$\sigma_1$, every time.

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