# When i have already found beamforming,do i still need to use MRC?i have some problems for them

I have some problem when i learning MIMO,first thing is that i know designing beamforming and MRC can ensure better receive signals,but they are not the same,so i have some question about them.

1.I know the MRC is the one of method to ensure better received signal,so if I have already found the beamforming direction now,do i still do MRC when calculating the SNR?,like the example below

2.If the transmitter use $$N_T$$ antenna to transmit one signal to the receiver which is with $$N_R$$ antenna,now i do SVD to the MIMO channel $$\mathbf H$$ ($$N_R \times N_T$$ matrix ),and find the best beamfoming direction $$f_A,(N_T \times 1$$ column vector), from it.Now i have beamforming now ,so i don't need to do MRC?

3.Now the received signal is $$y_t=\sqrt{P}\mathbf H \vec f_Ax+\vec n$$,so now the $$SNR=\frac{P||H\vec f_A||^2}{\sigma^2 _n}$$?

Because i heard my classmate said that $$SNR$$ is not $$\frac{P||H\vec f_A||^2}{\sigma ^2 _n}$$,but $$\frac{P||\vec u^HH\vec f_A||^2}{\sigma ^2_n}$$,and this $$\vec u^H$$ is calculate the SVD too.The first column of $$U$$,SVD=$$U\Sigma V^H$$.Because $$H\vec f_Ax$$ is still a vetor,and we can't not calculate SNR from a vector,it must be calculate from a value,and $$\vec u^HH\vec f_A$$ is a value

Is my SNR formula right or my classmate's is right?Does anyone know the answer about my question?It confused me for lots of months

You are doing transmit beamforming and choosing your vector $$\mathbf{f}_A$$ so that you beamforming in the "best" direction. So you transmit the signal: $$\mathbf{z}=\sqrt{P} \mathbf{f}_A x$$, where $$\mathbf{z}$$ is a length $$N_T$$ vector. Now you receive the signal: $$\mathbf{y} = \sqrt{P}\mathbf{H}\mathbf{f}_A x + \mathbf{n}$$, which is a length $$N_R$$ vector.
The post-MRC SNR should still be a vector. You will have a SNR for each transmit stream, so $$N_T$$ SNR values. To compute the SNR of the $$k^{\text{th}}$$ transmit stream after MRC you take the channel gains from transmit antenna $$k$$ to each of the $$N_R$$ receive antennas, $$\mathbf{h}_k$$, and compute its output power: $$\gamma_k = P ||\mathbf{h}_k^H \mathbf{H} \mathbf{f}_A||^2$$. Or, you can also equivalently do it all at once by: $$||\mathbf{H}^H \mathbf{H} \mathbf{f}_A||^2$$.