# Why is the $G_{rx}=\sqrt{\frac{N_{rx}}{\hat{N}_{rx}}}S^T_{rx}\tilde{U}^H$ ? How does author calculate it?

I search for some information about Null Space Projection,and saw some information from this pdf: http://taneli.riihonen.fi/presentations/C027slides.pdf ,in the page 16,the author begin explaining something about Null Space Projection,however,there is something important formula that i can't understand.

Here is the figure about Null Space Projection

Why is the $$G_{rx}=\sqrt{\frac{N_{rx}}{\hat{N}_{rx}}}S^T_{rx}\tilde{U}^H$$ ? How does author calculate it ?

I saw something clues:https://ieeexplore.ieee.org/document/5757642 about the the explanation of formula ,but still i don't understand how does the author calculate it.

$$\mathcal E\{||\Delta \tilde{H}_I||^2_F\}=\epsilon_H^2||\tilde{H}_I||^2_F=N_{rx}N_{tx}\epsilon_H^2$$

$$R_t=\mathcal E\{\Delta \tilde{t} \Delta \tilde{t}^H\}=\epsilon ^2_t \frac{tr\{R_{\tilde t}\}}{N_{tx}}I=\epsilon ^2_t \frac{P_{tx}}{N_{TX}}I$$

$$H_I= \tilde H_I+\Delta \tilde H_I ;t= \tilde t+\Delta \tilde t$$ ,and they are both matrices

$$\tilde H_I, \tilde t$$ are the side information available ,and $$\Delta \tilde H_I, \Delta \tilde t$$ are the respective unknown errors