最小RFchian数量以实现全数字预编码性能

Proposition 1$$rank(V_{RF}{V}_D)\le min(V_{RF},V_D)\le N^{RF},rank(V_{FD})=N_s$$
Therefore, hybrid beamforming structure requires at least $N^{RF}\ge N_s$RF chains to implement $V_{FD}$
Proposition 2 :To realize any fully digital beamforming matrix, it is sufficient that the number of RFchains in hybrid architecture is greater than or equal to twice the number of data streams, $N^{RF}>2N_s$
Remark 2：In the case that $V_{FD}$is a rank deficient matrix(low SNR regime), it can always be decomposed as $V_{FD}=A_{N\times r}{B}_{r\times N_s}$,where $r=rank(V_{FD})$.
Since A is a full-rank matrix ,it can be realized using the procedure in the proof of proposition 2 as $A=V_{RF}{V}_D^{\prime }$with hybrid structure using $2r$ RF chains.Therefore,$V_{FD}=V_{RF}(V_D^{\prime }B)$ can be realized by hybrid structure using $2r$ RF chains with $V_{RF}$as RF beamformer and $(V_D^{\prime }B)$ as the digital beamformer.
Proposition 3 : for large-scale MIMO systems,$Q=V_{RF}^H{V}_{RF}=NI$with high probability. Similar, RF combiner typically satisfies $W_{RF}^H{W}_{RF}=MI$
proof: the element of $Q=V_{RF}^H{V}_{RF}$ are exactly N while the off-diagonal elements can be approximated as a summation of N independent terms which is much less than N with high probability for large N.
Sherman Morrison formulation:
$$(A+B{)}^{-1}=A^{-1}-\frac{A^{-1}B{A}^{-1}}{1+tr(A^{-1}B)}$$

HYBRID BEAMFORMING DESIGN FOR MULTI-USER MASSIVE MISO SYSTEMS

MU-MISO 与点到点通信之间的不同：

• MU-MISO场景下，接收天线之间不在collocated（并置），需要考虑用户之间的干扰。因此无法在使用点到点的接收天线合作的速率公式
• MU-MISO场景下，不同stream之间可能不是平等的。

HYBRID BEAMFORMING WITH FINITE RESOLUTION PHASE SHIFTERS

For infinite resolution phase shifters, so there will be assumed that the elements of RF beamformer can have any arbitrary phase angles. However, components required for accurate phase control can be expensive.
Since the number of antennas, infinite resolution phase shifter assumption is not always practical for systems with large antenna array terminals.
$V_{RF}(i,j)\in F,W_{RF}(i,j)\in F$,where $F=1,w,w^2,...,W^{n_{PS}}-1$,and $w=e^{j2\pi /n_{PS}}$and $n_{PS}$ is the number of realizable phase angles which is typically $n_{PS}=2^b$,where $b$ is the number of bits in the resolution of phase shifters.

仿真实现

算法1： 求解$V_{RF}$

算法2：端到端整体混合预编码设计
接收端与发射端类似，只是能量参数稍微变化了下，答题思路仍是按照MMSE求解的。