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I saw orthogonal beamforming (OBFM) in this paper : https://ieeexplore.ieee.org/document/7959644

Let me just explain what does the paper said about OBFM

enter image description here

First ,in the picture above,the channel is

$\mathbf H_m=\mathbf B_m^{LOS} \mathbf A_m^{LOS}+\sum\limits ^L_{l=1}\mathbf B_l^{NLOS}\mathbf A_l^{NLOS}= a (\mathbf \theta_m) \mathbf A_m^{LOS}+\sum\limits ^L_{l=1} a(\mathbf \theta_{m,l})\mathbf A_l^{NLOS}$

enter image description here

And the received signals at the BS after performing beam formation is

$\mathbf y_{out,m}=\sqrt{P} \mathbf w \mathbf H \mathbf x + \mathbf n_s=\sqrt{P} \mathbf w_m \mathbf H_m \mathbf x_m + \sum\limits_{j \neq m}^M \sqrt{P} \mathbf w_m \mathbf H_j \mathbf x_j+\mathbf n_{s,m}$

Honestly,if $\mathbf w = \mathbf H^{-1},$ it would be perfect,however,in the real world ,we all know that it is impossible to know $\mathbf H$ all the time.And it is better to know $\mathbf B^{LOS}$,so we can assume $\mathbf w = (\mathbf B^{LOS})^{-1},$ so the $\mathbf B^{LOS}$ must be nonsingular matrix

Unfortunately,However, if the system employs a higher number of antennas compared to the number of users, the matrix $\mathbf B^{LOS}$ cannot be directly inversed,so we will use the Moore-Penrose pseudoinverse to let

$\mathbf w = (\mathbf B^{LOS})^+=(\mathbf B^{LOS})^*( \mathbf B^{LOS}(\mathbf B^{LOS})^* )^{-1}$

If we design $\mathbf w$,we can let all beamforming are orthogonal to one another, and so users do not encounter interference.

My question are

  1. Why if the system employs a higher number of antennas compared to the number of users, the matrix $\mathbf B^{LOS}$ cannot be directly inversed ?

  2. It seems that all $w_m$ will be the same,at first,all beamforming are not orthogonal to one another,why do they multiply the same thing and become orthogonal to one another?Where do i misunderstand?

  3. What is the formula of $a (\mathbf \theta_m)$ ?

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