/ 科研向  

【预编码论文阅读(一)】传统方法

预编码论文阅读(一)——传统方法

由于没有非常系统地看完MIMO的相关内容,整理中必定有很多的问题,欢迎在评论区批评指正。

整理很乱。。。

由于网页公式渲染器KaTeX不支持公式交叉引用,我的前端水平就不足以把我这个模板加入mathjax。故将所有公式交叉引用均删除了,有的是在显示不出来的建议贴到markdown里面去吧

线性预编码

大规模MIMO下行链路预编码(1)_月半 月半的博客-CSDN博客

大规模MIMO下行链路预编码(2)_月半 月半的博客-CSDN博客

MRT预编码

由于可以平衡系统性能和计算复杂度,最大比传输(MRT)预编码【又称作匹配滤波器(MF)预编码】是最简单易实现的预编码算法,通过最大化接收信噪比(SNR)实现。在大规模MIMO系统中,当基站天线数M MM足够大时,最简单的MRT线性预编码方案便可以得到最优的系统性能。

在发射端已知完美信道状态信息的前提下,MRT线性预编码矩阵为:

V=βMRTHH\mathbf V=\beta_{MRT}\mathbf H^H

(注:以上编码矩阵后可以加上功率分配矩阵组成整个预编码矩阵)

式中, βMRT=1tr(HHH)\beta_{MRT}=\sqrt{\frac{1}{tr(HH^H )}} 为约束基站发送功率的约束因子。大规模MIMO系统中基站天线数的不断增加使得信道矩阵列向量之间逐渐呈现正交性,即不同终端间的干扰逐渐降低甚至被完全消除,因此最简单的 MRT 预编码下便可以获得最优的频谱效率和最好的信号传输质量,且复杂度最低。

传统MIMO系统中,匹配滤波预编码方案的侧重点在于接收端用户的信号增益最大化,但在多用户系统的场景下,随着传输信道相关性的提升,此方案由于没有考虑如何对用户间的干扰进行处理,将会导致整个系统性能快速下降

Zero-forcing precoding

转化为互相独立的并行信道,不考虑其它信道的干扰。hiHw~j=0,ij{\bf h}_i^H\tilde{\bf w}_j=0,i\neq j(5)——achieve virtually optimal

WZF=HH(HHH)1\mathbf{W}_{\mathrm{ZF}}=\mathbf{H}^{H}\left(\mathbf{H} \mathbf{H}^{H}\right)^{-1}

Equal power scaling allocation(ZF-EPS)

平均分配能量——normalized=power allocate

w~i=ηwiη=Ptr{WZFWZFH}\begin{gathered} \tilde{\bf w}_i=\eta{\bf w}_i\\ \eta=\sqrt{\frac{P}{tr\{\mathbf{W}_{\mathrm{ZF}}\mathbf{W}_{\mathrm{ZF}}^H\}}} \end{gathered}

Optimal power allocation - water-filling solution(ZF-WF)

信噪比高的多分配能量

maximizep1,,pKi=1Klog(1+ρipi) subject to i=1KγipiP,pi0\begin{aligned} &\operatorname{maximize}_{p_{1}, \ldots, p_{K}} &\sum_{i=1}^{K} \log \left(1+\rho_{i} p_{i}\right) \\ &\text { subject to } & \sum_{i=1}^{K} \gamma_{i} p_{i} \leq P , p_{i} \geq 0 \end{aligned}

其中,γi=[(HHH)1]\gamma_i=[({\bf HH}^H)^{-1}]。注水法功率控制(6)——ηi=pi\eta_i=\sqrt{p_i}

pi=[μγi1ρi]+,ip_i=\left[\frac{\mu}{\gamma_i}-\frac{1}{\rho_i}\right]^+,\forall i

其中, [x]+=max(x,0)[x]^+=max(x,0) , 总功率限制

i=1K=[μγiρi1]+=P\sum_{i=1}^K=\left[\mu-\gamma_i\rho_i^{-1}\right]^+=P

Regularized zero-forcing precoding(RZF)

(HHH)1(\mathbf{H} \mathbf{H}^{H})^{-1}最大奇异值的不良性质,即使增加天线,也无法增加最大速率。加入正则项,(7)

WRZF=HH(HHH+αI)1\mathbf{W}_{\mathrm{RZF}}=\mathbf{H}^{H}\left(\mathbf{H} \mathbf{H}^{H}+\alpha \mathbf{I}\right)^{-1}

W~=ηWRZFη=Ptr{WRZFWRZFH}\begin{gathered} \tilde{\bf W}=\eta{\bf W}_\text{RZF}\\ \eta=\sqrt{\frac{P}{tr\{\mathbf{W}_{\mathrm{RZF}}\mathbf{W}_{\mathrm{RZF}}^H\}}} \end{gathered}

  1. homogeneous SNR conditions(ρi=1\rho_i=1

α=KP \alpha^{\star}=\frac KP

  1. non-homogeneous SNR conditions——non-weighted sum-MSE minimization(9)

    WRZF=HH(HHH+i=1K(1/ρi)PI)1\mathbf{W}_{\mathrm{RZF}}=\mathbf{H}^{H}\left(\mathbf{HH}^{H}+\frac{\sum_{i=1}^{K}\left(1 / \rho_{i}\right)}{P} \mathbf{I}\right)^{-1}

Iterative weighted minimization ofmean squared error algorithm(IWMMSE)

解决非凸问题的数值算法。weighted MSE problem(10)

minimizeΛ,Ω,W~E{Ω12(uΛy)2}logdetΩ subject to Tr{W~W~H}P\begin{aligned} &\underset{\Lambda, \Omega, \tilde{\mathbf{W}}}{\operatorname{minimize}} \mathbb{E}\left\{\left\|\Omega^{\frac{1}{2}}(\mathbf{u}-\Lambda \mathbf{y})\right\|^{2}\right\}-\log \operatorname{det} \Omega \\ &\text { subject to } \operatorname{Tr}\left\{\tilde{\mathbf{W}} \tilde{\mathbf{W}}^{H}\right\} \leq P \end{aligned}

其中,Ω,Λ\mathbf{\Omega},\mathbf{\Lambda}均为对角阵,Ω\mathbf{\Omega}表示K个UE的权重(weights),Λ\mathbf{\Lambda}表示接收因子(receive coefficients)。W~,Ω,Λ\tilde{\mathbf{W}},\mathbf{\Omega},\mathbf{\Lambda}是优化变量。

S1:随机初始化变量

w~i2=PK||\tilde{\mathbf{w}}_i||^2=\frac PK

S2:迭代直到convergence

  • 确定W~,Ω\tilde{\mathbf{W}},\mathbf{\Omega},优化Λ\mathbf{\Lambda}(11)

    λi=argminλiE{uiλiyi2}=(j=1KρihiHw~j2+1)1ρiw~iHhi.\begin{aligned} \lambda_{i} &=\arg \min _{\lambda_{i}} \mathbb{E}\left\{\left|u_{i}-\lambda_{i} y_{i}\right|^{2}\right\} \\ &=\left(\sum_{j=1}^{K} \rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{j}\right|^{2}+1\right)^{-1} \sqrt{\rho_{i}} \tilde{\mathbf{w}}_{i}^{H} \mathbf{h}_{i} . \end{aligned}

  • 确定W~,Λ\tilde{\mathbf{W}},\mathbf{\Lambda},优化Ω\mathbf{\Omega}(12)

    ωi=argminωiωieilogωi=ei1=1+SINRi=j=1KρihiHw~j2+1jiKρihiHw~j2+1\begin{aligned} \omega_{i} &=\arg \min _{\omega_{i}} \omega_{i} e_{i}-\log \omega_{i} \\ &=e_{i}^{-1}=1+\operatorname{SINR}_{i}=\frac{\sum_{j=1}^{K} \rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{j}\right|^{2}+1}{\sum_{j \neq i}^{K} \rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{j}\right|^{2}+1} \end{aligned}

  • 确定Ω,Λ\mathbf{\Omega},\mathbf{\Lambda},优化W~\tilde{\mathbf{W}}(14)

    W~=(HHΛHΩΣΛH+μI)1HHΛHΩΣ12=HH[HHH+μ(ΛHΩΣΛ)1]1Λ1Σ12\begin{aligned} \tilde{\mathbf{W}} &=\left(\mathbf{H}^{H} \Lambda^{H} \Omega \Sigma \Lambda \mathbf{H}+\mu \mathbf{I}\right)^{-1} \mathbf{H}^{H} \Lambda^{H} \Omega \Sigma^{\frac{1}{2}} \\ &=\mathbf{H}^{H}\left[\mathbf{H} \mathbf{H}^{H}+\mu\left(\Lambda^{H} \Omega \Sigma \Lambda\right)^{-1}\right]^{-1} \Lambda^{-1} \Sigma^{-\frac{1}{2}} \end{aligned}

WMMSE【数字预编码】——2014

MMSE precoding for multiuser MISO downlink transmission with non-homogeneous user SNR conditions

MU-MISO模型(M天线、K用户)

u\bf u是向量,每次给UEiUE_i发送的是单个字符uiu_i——多用户

y=Σ12HW~u+nx=i=1Kw~iui=uW~\begin{gathered} \mathbf{y}=\mathbf{\Sigma}^{\frac 12}\mathbf{H\tilde Wu}+\mathbf{n}\\ \mathbf{x}=\sum_{i=1}^K\tilde{\mathbf{w}}_iu_i=\mathbf{u\tilde W} \end{gathered}

其中,Σ=diag(ρ1,,ρK)\mathbf{\Sigma}=\mathrm{diag}(\rho_1,\cdots,\rho_K)是对角阵,表示非同质信噪比条件的差异;H=[h1,,hK]H\mathbf{H}=[\mathbf{h}_1,\cdots,\mathbf{h}_K]^HW~=[w~1,,w~K]\tilde{\mathbf{W}}=[\tilde{\mathbf{w}}_1,\cdots,\tilde{\mathbf{w}}_K]u=[u1,,uK]Tu=[u_1,\cdots,u_K]^T

SINR(信干噪比)

SINRi=ρihiHw~i2jiKρihiHw~j2+1\mathrm{SINR}_i=\frac{\rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{i}\right|^{2}}{\sum_{j \neq i}^{K} \rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{j}\right|^{2}+1}

sum-rate的优化问题——非凸问题

maximizew~1,,w~Ki=1Klog(1+ρihiHw~i2jiKρihiHw~j2+1) subject to i=1Kw~i2P\begin{aligned} &\underset{\tilde{\mathbf{w}}_{1}, \ldots, \tilde{\mathbf{w}}_{K}}{\operatorname{maximize}} \sum_{i=1}^{K} \log \left(1+\frac{\rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{i}\right|^{2}}{\sum_{j \neq i}^{K} \rho_{i}\left|\mathbf{h}_{i}^{H} \tilde{\mathbf{w}}_{j}\right|^{2}+1}\right) \\ &\text { subject to } \sum_{i=1}^{K}\left\|\tilde{\mathbf{w}}_{i}\right\|^{2} \leq P \end{aligned}

perfect CSI and average SNR knowledge

方法

take advantage of the non-homogeneous SNR conditions at the UE to predetermine the weights and receive coefficients and thus remove the iterative procedure of the IWMMSE algorithm.(利用UE的信噪比非同质这一特点,事先确定IWMMSE中的两个迭代参数Ω,Λ\mathbf{\Omega},\mathbf{\Lambda}

思路

用effective small-scale channel gain G(gi=hiσni)\mathbf{G}(g_i=\frac{||\mathbf{h}_i||}{\sigma_{n_i}})表示两个迭代参数(16)

MSE=E{GΣ12Ω12(uη1G1Σ12Λy)2}\mathrm{MSE}=\mathbb{E}\left\{\left\|\underbrace{\mathbf{G} \boldsymbol{\Sigma}^{\frac{1}{2}}}_{\mathbf{\Omega}^{\frac12}}\left(\mathbf{u}-\underbrace{\eta^{-1} \mathbf{G}^{-1} \boldsymbol{\Sigma}^{-\frac{1}{2}}}_{\Lambda} \mathbf{y}\right)\right\|^{2}\right\}

使用lagrange乘数法求lagrange因子μ\mu^{\star}和波束成形矩阵W\mathbf{W}^{\star}

结果

μ=KPW=HH(HHH+KPΣ1)1G\begin{gathered}\mu^\star=\frac KP\\ \mathbf{W}^\star=\mathbf{H}^H\left(\mathbf{HH}^H+\frac KP\mathbf{\Sigma}^{-1}\right)^{-1}\mathbf{G} \end{gathered}

结果与IWMMSE形式类似,与RZF相比采用non-identity regularizer matrix(非同一正则矩阵)

quantized CDI and CQI feedbacks

对平均SNR无knowledge。

  • CDI(channel direction indicator):量化方向矢量h^i\mathbf{\hat h}_i的编号。用量化矢量h^i\mathbf{\hat h}_i代替SINR表达式中的hi\mathbf{h}_i
  • CQI(channel quality indicator):用有效信道增益表示瞬时信噪比g^iρ^i=MPSNRi^\hat g_i \hat \rho_i=\frac MP\widehat{SNR_i}代替SINR中平均信噪比ρi\rho_i
  • CDI和CQI综合即可改写式\eqref{eq:16} ,但量化误差会带来non-robust。

方法——Robust

分解归一化信道h~i=hihi\mathbf{\tilde h}_i=\frac{\mathbf{h}_i}{||\mathbf{h}_i||},

h~i=1zih^i+ziszi=1h~iHh^i2\begin{gathered} \mathbf{\tilde h}_i=\sqrt{1-z_i}\,\mathbf{\hat h}_i+\sqrt{z_i}\,\mathbf{s}\\ z_i=1-|\mathbf{\tilde h}_i^H\mathbf{\hat h}_i|^2 \end{gathered}

其中,ziz_i是量化误差,h~i\mathbf{\tilde h}_i是实际方向,h^i\mathbf{\hat h}_i是量化方向的投影,s\mathbf{s}是量化矢量h^i\mathbf{\hat h}_i的核空间上的各向同性的单位矢量。【1-[16]】

思路

和perfect CSI and average SNR knowledge的MSE一致,但是其中H~\mathbf{\tilde H}有所变化:

H=GH~=G(IZ)12H^+GZ12S\mathbf{H}=\mathbf{G} \tilde{\mathbf{H}}=\mathbf{G}(\mathbf{I}-\mathbf{Z})^{\frac{1}{2}} \hat{\mathbf{H}}+\mathbf{G} \mathbf{Z}^{\frac{1}{2}} \mathbf{S}

化开,同样使用lagrange乘数法求lagrange因子μ\mu^{\star}和波束成形矩阵W\mathbf{W}^{\star}

结果

μ\mu^\star不变

W=ζ1δH^H(H^H^H+δPTr{ΣG2}+KMPM(1δ)Σ1G2)1\mathbf{W}=\frac{\zeta}{1-\delta} \hat{\mathbf{H}}^{H}\left(\hat{\mathbf{H}} \hat{\mathbf{H}}^{H}+\frac{\delta P \operatorname{Tr}\left\{\mathbf{\Sigma G}^{2}\right\}+K M}{P M(1-\delta)} \mathbf{\Sigma}^{-1} \mathbf{G}^{-2}\right)^{-1}

系数不影响归一化的波束成形矩阵W~\mathbf{\tilde W},信噪比SNR=PMG2Σ\mathbf{SNR}=\frac PM\mathbf{G}^2\mathbf{\Sigma}(37)

W=H^H(H^H^H+δTr{SNR}+KM(1δ)SNR1)1\mathbf{W}^\star=\hat{\mathbf{H}}^{H}\left(\hat{\mathbf{H}} \hat{\mathbf{H}}^{H}+\frac{\delta \operatorname{Tr}\left\{\mathbf{SNR}\right\}+K }{M(1-\delta)} \mathbf{SNR}^{-1}\right)^{-1}

  • 由quantized CDI and CQI feedbacks可以推perfect CSI and average SNR knowledge
  • CQI量化可能无法很好估计SNR
  • 可以通过WMMSE推出non-weighted MMSE【1-[11]】

Contribution

用计算的effective small-scale channel gain却代替迭代的过程。


Iteratively Weighted MMSE【数字预编码,迭代】——2011-TSP

An Iteratively Weighted MMSE Approach to Distributed Sum-Utility Maximization for a MIMO Interfering Broadcast Channel

MIMO

每次给UEiki_k发送的是s\bf s,而非单个字符——讨论单用户

image-20211104143313382

s^ik=UikHyik\mathbf{\hat s}_{i_k}=\mathbf{U}_{i_k}^H\mathbf{y}_{i_k}

优化问题

Sum-rate problem

在点对点单用户信道中,如果我们已知发送信号的协方差矩阵Q=E{xkxkH}\mathbf{Q}=\mathbb{E}\{\mathbf{x}_k\mathbf{x}_k^H\} ,假设干扰加噪声协方差矩阵是单位矩阵,那么单用户的信道容量就是

logI+HQHH.\log |\mathbf{I}+\mathbf{H}\mathbf{Q}\mathbf{H}^H|.

扩展到多用户IC信道,此时干扰加噪声协方差矩阵(interference-plus-noise covariance matrix)就不会再成为单位矩阵,它是Ri=ijHjiQjHjiH+I\mathbf{R}_i = \sum_{i \neq j} \mathbf{H}_{ji}\mathbf{Q}_j \mathbf{H}_{ji}^H+\mathbf{I},多用户信道容量就变成了:

i=1KlogI+Ri1HiiQiHiiH.\sum_{i=1}^K \log |\mathbf{I}+\mathbf{R}^{-1}_i\mathbf{H}_{ii}\mathbf{Q}_i\mathbf{H}^H_{ii}|.

考虑它更广义的形式ii个用户,加入效用因子λi\lambda_i(文章1的ρi\rho_i)。当效用因子都是1时,就和上式等价,问题就变成了:

i=1KλilogI+Ri1HiiQiHiiH\sum_{i=1}^K \lambda_i\log |\mathbf{I}+\mathbf{R}^{-1}_i\mathbf{H}_{ii}\mathbf{Q}_i\mathbf{H}^H_{ii}|

【优质信源】计划02–多用户通信中总速率优化问题的一些凸优化模式 - 知乎 (zhihu.com)

  • sum-rate problem(1)

    maxVk=1Kik=1IkαikRik s.t. i=1IkTr(VikVikH)Pk,k=1,2,,K\begin{aligned} \max_{\mathbf{V}} & \sum_{k=1}^{K} \sum_{i_{k}=1}^{I_{k}} \alpha_{i_{k}} R_{i_{k}} \\ \text { s.t. } & \sum_{i=1}^{I_{k}} \operatorname{Tr}\left(\mathbf{V}_{i_{k}} \mathbf{V}_{i_{k}}^{H}\right) \leq P_{k}, \forall k=1,2, \ldots, K \end{aligned}

    其中,αik\alpha_{i_k}是优先级

  • sum-MSE minimization(4)

    minU,Vk=1Ki=1IkTr(Eik)=k=1Ki=1Iks^iksik s.t. i=1IkTr(VikVikH)Pk,k=1,2,,K.\begin{aligned} \min _{\mathbf{U}, \mathbf{V}} & \sum_{k=1}^{K} \sum_{i=1}^{I_{k}} \operatorname{Tr}\left(\mathbf{E}_{i_{k}}\right) =\sum_{k=1}^{K} \sum_{i=1}^{I_{k}} ||\mathbf{\hat s }_{i_k}-\mathbf{s}_{i_k}||\\ \text { s.t. } & \sum_{i=1}^{I_{k}} \operatorname{Tr}\left(\mathbf{V}_{i_{k}} \mathbf{V}_{i_{k}}^{H}\right) \leq P_{k}, \quad k=1,2, \ldots, K . \end{aligned}

    Uikmmse=Jik1HikkVikEikmmse=IVikHHikkHJik1HikkVik\begin{aligned} \mathbf{U}_{i_{k}}^{\mathrm{mmse}}=&\mathbf{J}_{i_{k}}^{-1} \mathbf{H}_{i_{k} k} \mathbf{V}_{i_{k}}\\ \mathbf{E}_{i_{k}}^{\mathrm{mmse}}=&\mathbf{I}-\mathbf{V}_{i_{k}}^{H} \mathbf{H}_{i_{k} k}^{H} \mathbf{J}_{i_{k}}^{-1} \mathbf{H}_{i_{k} k} \mathbf{V}_{i_{k}} \end{aligned}

    其中,Jikj=1K=1IjHikjVjVjHHikH+σik2I\mathbf{J}_{i_{k}} \triangleq \sum_{j=1}^{K} \sum_{\ell=1}^{I_{j}} \mathbf{H}_{i_{k} j} \mathbf{V}_{\ell_{j}} \mathbf{V}_{\ell_{j}}^{H} \mathbf{H}_{i_{k}}^{H}+\sigma_{i_{k}}^{2} \mathbf{I}

  • 两个问题的统一性(7)——【1-[5]、2-[13]】梯度、KKT条件引出

    minW,U,Vk=1Ki=1Ikαik(Tr(WikEik)logdet(Wik)) s.t. i=1IkTr(VikVikH)Pk,k=1,2,,K\begin{aligned} \min _{\mathbf{W}, \mathbf{U}, \mathbf{V}} & \sum_{k=1}^{K} \sum_{i=1}^{I_{k}} \alpha_{i_{k}}\left(\operatorname{Tr}\left(\mathbf{W}_{i_{k}} \mathbf{E}_{i_{k}}\right)-\log \operatorname{det}\left(\mathbf{W}_{i_{k}}\right)\right) \\ \text { s.t. } & \sum_{i=1}^{I_{k}} \operatorname{Tr}\left(\mathbf{V}_{i_{k}} \mathbf{V}_{i_{k}}^{H}\right) \leq P_{k}, k=1,2, \ldots, K \end{aligned}

    (7) is in the space of (u,v,w)(u,v,w) and is easier to handle since optimizing each variable while holding others fixed is convex and easy (e.g., closed form).

Rik=logdet((Eikmmse)1)R_{i_k}=\log\det\left(\left(\mathbf{E}_{i_k}^{mmse}\right)^{-1}\right)

迭代优化的方法

要解决sum-rate问题,即解决式\eqref{eq:2-7}的优化问题,需要优化U,V,W\mathbf{U,V,W}

1 Initialize Vik\mathbf{V}_{i_{k}} 's such that Tr(VikVikH)=pkIk\operatorname{Tr}\left(\mathbf{V}_{i_{k}} \mathbf{V}_{i_{k}}^{H}\right)=\frac{p_{k}}{I_{k}}
2 repeat
3WikWik,ikI3 \quad \mathbf{W}_{i_{k}}^{\prime} \leftarrow \mathbf{W}_{i_{k}}, \quad \forall i_{k} \in \mathcal{I}
4Uik((j,)HikjVjVjHHikjH+σik2I)1HikkVk,ikI4 \quad \mathbf{U}_{i_{k}} \leftarrow\left(\sum_{(j, \ell)} \mathbf{H}_{i_{k} j} \mathbf{V}_{\ell_{j}} \mathbf{V}_{\ell_{j}}^{H} \mathbf{H}_{i_{k} j}^{H}+\sigma_{i_{k}}^{2} \mathbf{I}\right)^{-1} \mathbf{H}_{i_{k} k} \mathbf{V}_{k}, \forall i_{k} \in \mathcal{I}
5Wik(IUikHHikkVik)1,ikI5\quad\mathbf{W}_{i_{k}} \leftarrow\left(\mathbf{I}-\mathbf{U}_{i_{k}}^{H} \mathbf{H}_{i_{k} k} \mathbf{V}_{i_{k}}\right)^{-1}, \forall i_{k} \in \mathcal{I}
6Vikαik((j,)αjHjkHUjWjUjHHjk+μkI)1HikkHUikWik,ik6 \quad \mathbf{V}_{i_{k}} \leftarrow \alpha_{i_{k}}\left(\sum_{(j, \ell)} \alpha_{\ell_{j}} \mathbf{H}_{\ell_{j} k}^{H} \mathbf{U}_{\ell_{j}} \mathbf{W}_{\ell_{j}} \mathbf{U}_{\ell_{j}}^{H} \mathbf{H}_{\ell_{j} k}+\mu_{k}^{*} \mathbf{I}\right)^{-1} \mathbf{H}_{i_{k} k}^{H} \mathbf{U}_{i_{k}} \mathbf{W}_{i_{k}}, \forall i_{k}
7 until (j,)logdet(Wj)(j,)logdet(Wj)ϵ\left|\sum_{(j, \ell)} \log \operatorname{det}\left(\mathbf{W}_{\ell_{j}}\right)-\sum_{(j, \ell)} \log \operatorname{det}\left(\mathbf{W}_{\ell_{j}}^{\prime}\right)\right| \leq \epsilon

  • W\mathbf{W}的优化来自对\eqref{eq:2-7}的求解——对W\mathbf{W}求一阶导数得到W=(EikMMSE)1\mathbf{W}^\star=(\mathbf{E}_{i_k}^{MMSE})^{-1}EikMMSE\mathbf{E}_{i_k}^{MMSE}来自MMSE问题的求解,可由U\mathbf{U}表示
  • U\mathbf{U}的优化来自MMSE的求解——对MSE表达式对U\mathbf{U}求一阶导,求极值
  • V\mathbf{V}的优化来自对\eqref{eq:2-7}的求解——将MSE表达式代入\eqref{eq:2-7},对V\mathbf{V}利用Lagrange乘数法,解出V\mathbf{V}μk\mu_k有关,根据约束条件求μk\mu_k^\star,回代

这种方法也适用于general utility maximization

Contributions

(7) is in the space of (u,v,w)(u,v,w) and is easier to handle since optimizing each variable while holding others fixed is convex and easy (e.g., closed form).

the sum-rate maximization problem is first equivalently transformed into an MMSE problem and then a block coordinate descent (BCD) method is proposed to solve the resultant MMSE problem.


Low-Complexity Hybrid Precoding in Massive Multiuser MIMO Systems【混合预编码】——2014-LWC

混合预编码

混合预编码——传统数字预编码的RF chain数量需要和NtN_t相等,通过WF\mathbf{WF}的统一考虑,减少RF chain。

yk=hkHFNt×KWK×KsK×1+nky_k=\mathbf{h}_k^H\mathbf{F}_{N_t\times K}\mathbf{W}_{K\times K}\mathbf{s}_{K\times 1}+n_k

方法

apply phase-only control to couple the KK RF chain outputs with NtN_t transmit antennas, using cost-effective RF phase shifters.(F\mathbf{F}只相移,将K条射频链和NtN_t个发射天线耦合)

思路

  • F\bf F只调相

    Fi,j=1Ntejφi,j\mathbf{F}_{i,j}=\frac{1}{\sqrt{N_t}}e^{j\varphi_{i,j}}

    This is to align the phases of channel elements and can thus harvest the large array gain provided by the massive multiuser MIMO systems.

  • W\bf W调幅、调相:将HF\bf HF看作等效的H\bf H,利用非注水的ZF(块对角化)

    W=HeqH(HeqHeqH)1Λ\mathbf{W}=\mathbf{H}_{eq}^H(\mathbf{H}_{eq}\mathbf{H}_{eq}^H)^{-1}\mathbf{\Lambda}

    Λ\mathbf{\Lambda}为列功率归一化。类似【1】的ηi\eta_i

  • Quantized RF Phase Control:由于F\mathbf{F}控制相位,实际中移相器位数有限,需要量化,则再计算W\mathbf{W}时利用量化后的F^\mathbf{\hat F}

结果

频谱利用率(Spectral Efficiency)分析

——当NtN_t足够大,用户间干扰可以忽略

Phased-ZF的上界RKRR\leq K \mathcal{R},且

limNtRlog2(1+π4PNtK)=1R=E[1+PKhkHfk2]\begin{gathered} \lim_{N_t\to \infty}\frac{\mathcal{R}}{\log_2\left(1+\frac{\pi}{4}\frac{PN_t}{K}\right)}=1\\ \mathcal{R}=\mathbb{E}\left[1+\frac PK |\mathbf{h}_k^H\mathbf{f}_k|^2\right] \end{gathered}

仿真1:Rayleigh信道

1
2
3
4
5
6
F = 1/sqrt(Nt)*exp(j.*angle(H))';   % RF precoder 128x4
Fb = CalBDPrecoder(H*F); % 4x(128)x4, baseband, same as inverse with column normalization W = Fb
% (BD precoder is a method of ZF)
wt = F*Fb; % aggregate precoder wt=FW
WPR = wt*inv(sqrt(diag(diag(wt'*wt)))); % normalized columns
rateHyb(isnr) = rateHyb(isnr) + CalRate((P/K)*eye(K), H, WPR);% ZF-PRP

仿真2:mmWave信道

毫米波信道的特点:limited multipath components.—— poor scattering nature

hkH=NtNpl=1NpαlkaH(ϕlk,θlk)\mathbf{h}_k^H=\sqrt{\frac{N_t}{N_p}}\sum_{l=1}^{N_p}\alpha_l^k\mathbf{a}^H(\phi_l^k,\theta_l^k)

Contribution

在性能减弱不大的情况下,减少射频链(RF chains)的数量


Hybrid Digital and Analog Beamforming Design for Large-Scale Antenna Arrays【混合预编码】——2016-JSTSP

image-20211109103444712

x=VRFN×NtRFVDNtRF×Nss==1KVRFVDs\mathbf{x}=\underbrace{\mathbf{V}_{RF}}_{N\times N_t^{RF}}\underbrace{\mathbf{V}_D}_{N_t^{RF}\times N_s}\mathbf{s}=\sum_{\ell=1}^K\mathbf{V}_{RF}\mathbf{V}_{D_\ell}\mathbf{s}_\ell

其中,Ns=KdN_s=KdKK个用户,每个用户dd个符号

接收信号

s~k=y~k=WtkHHkVtkskdesired signals +WtkHHkkVtseffective interference +WtkHzkeffective noise \mathbf{\tilde s}_k=\tilde{\mathbf{y}}_{k}=\underbrace{\mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{H}_{k} \mathbf{V}_{\mathrm{t}_{k}} \mathbf{s}_{k}}_{\text {desired signals }}+\underbrace{\mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{H}_{k} \sum_{\ell \neq k} \mathbf{V}_{\mathrm{t}_{\ell}} \mathbf{s}_{\ell}}_{\text {effective interference }}+\underbrace{\mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{z}_{k}}_{\text {effective noise }}

其中,基站的预编码Vtk=VRFVDk\mathbf{V}_{\mathrm{t}_k}=\mathbf{V}_{RF}\mathbf{V}_{D_k},同理,用户侧的预编码Wtk=VRFkVDk\mathbf{W}_{\mathrm{t}_k}=\mathbf{V}_{RF_k}\mathbf{V}_{D_k}

优化问题sum-rate problem(4)

Rk=log2IM+WtkCk1WtkHHkVtkVtkHHkH where Ck=WtkHHk(kVtVtH)HkHWtk+σ2WtkHWtk\begin{aligned} &R_{k}=\log _{2}\left|\mathbf{I}_{M}+\mathbf{W}_{\mathrm{t}_{k}} \mathbf{C}_{k}^{-1} \mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{H}_{k} \mathbf{V}_{\mathrm{t}_{k}} \mathbf{V}_{\mathrm{t}_{k}}^{H} \mathbf{H}_{k}^{H}\right|\\ &\text { where } \quad \mathbf{C}_{k}=\mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{H}_{k}\left(\sum_{\ell \neq k} \mathbf{V}_{\mathrm{t}_{\ell}} \mathbf{V}_{\mathrm{t}_{\ell}}^{H}\right) \mathbf{H}_{k}^{H} \mathbf{W}_{\mathrm{t}_{k}}+\sigma^{2} \mathbf{W}_{\mathrm{t}_{k}}^{H} \mathbf{W}_{\mathrm{t}_{k}} \end{aligned}

  • Point-to-Point MIMO——两侧都是大规模天线阵列
  • 下行链路MU-MIMO——基站侧多天线,用户侧单天线

Fully Digital Beamformer(全数字波束赋形)

VFDCN×Ns\mathbf{V}_{FD}\in \mathbb{C}^{N\times N_s}

  • 必要条件:NRFNsN^{RF}\geq N_s
  • 充分条件:NRF2NsN^{RF}\geq 2N_s
  • Ns2NRFN_s\geq 2N^{RF}的构造方法:it is in fact possible to realize any fully digital beamformer using the hybrid structure with NsN_s RF chains and 2NsN2N_sN phase shifters.
  • NRFNN^{RF}\approx N,可以达到近似最优解,此时移相器数量为NsNN_sN
  • 在低信噪比环境中,若VFD\mathbf{V}_{FD}非满秩矩阵,则先进行满秩分解(VFD=AN×rBr×Ns\mathbf{V}_{FD}=\mathbf{A}_{N\times r}\mathbf{B}_{r\times N_s}),将A=VRFVD\mathbf{A}=\mathbf{V}_{RF}\mathbf{V}'_D作为预编码矩阵,射频链数量为2r2r,此时模拟预编码VRF\mathbf{V}_{RF},数字预编码VDB\mathbf{V}'_D\mathbf{B}

SU-Point-to-Point MIMO

假设NtRF=NrRF=NRFN_t^{RF}=N_r^{RF}=N^{RF}

优化目标:由\eqref{eq:4-4}化简:

R=log2IM+1σ2Wt(WtHWt)1WtHHVtVtHHHR=\log _{2}\left|\mathbf{I}_{M}+\frac{1}{\sigma^2}\mathbf{W}_{\mathrm{t}} (\mathbf{W}_t^H\mathbf{W}_t)^{-1} \mathbf{W}_{\mathrm{t}}^{H} \mathbf{H} \mathbf{V}_{\mathrm{t}} \mathbf{V}_{\mathrm{t}}^{H} \mathbf{H}^{H}\right|

  • 先考虑必要条件NRF=NsN^{RF}=N_s的启发式算法

    • 数字预编码:由注水法VD=(VRFHVRF)12UeΓe\mathbf{V}_D=(\mathbf{V}_{RF}^H\mathbf{V}_{RF})^{-\frac12}\mathbf{U}_e\mathbf{\Gamma}_e(11),得VDVDH=γ2I,γ=PNNRF\mathbf{V}_D\mathbf{V}_D^H=\gamma^2\mathbf{I},\gamma=\frac{P}{NN^{RF}}

    • RF预编码:优化函数(12)

      maxVRFlog2I+γ2σ2VRFHF1VRF s.t. VRF(i,j)2=1,i,j\begin{aligned} \max _{\mathbf{V}_{\mathrm{RF}}}\quad & \log _{2}\left|\mathbf{I}+\frac{\gamma^{2}}{\sigma^{2}} \mathbf{V}_{\mathrm{RF}}^{H} \mathbf{F}_{1} \mathbf{V}_{\mathrm{RF}}\right| \\ \text { s.t. }\quad &\left|\mathbf{V}_{\mathrm{RF}}(i, j)\right|^{2}=1, \forall i, j \end{aligned}

      目标函数也可写为(13)

      log2Cj+log2(2Re{VRF(i,j)ηij}+ζij+1)\log _{2}\left|\mathbf{C}_{j}\right|+\log _{2}\left(2 \operatorname{Re}\left\{\mathbf{V}_{\mathrm{RF}}^{*}(i, j) \eta_{i j}\right\}+\zeta_{i j}+1\right)

      \begin{aligned} &\hline \text { Algorithm 1. Design of } \mathbf{V}_{\mathrm{RF}} \text { by solving (12) } \\ &\hline \text { Given: } \mathbf{F}_{1}, \gamma^{2}, \sigma^{2} \\ &\text { 1: Initialize } \mathbf{V}_{\mathrm{RF}}=1_{N \times N^{\mathrm{RF}}} \\ &\text { 2: \textbf{for} } j=1 \rightarrow N^{\mathrm{RF}} \textbf { do } \\ &\text { 3: Calculate } \mathbf{C}_{j}=\mathbf{I}+\frac{\gamma^{2}}{\sigma^{2}}\left(\overline{\mathbf{V}}_{\mathrm{RF}}^{j}\right)^{H} \mathbf{F}_{1} \overline{\mathbf{V}}_{\mathrm{RF}}^{j} \\ &\text { 4: Calculate } \mathbf{G}_{j}=\frac{\gamma^{2}}{\sigma^{2}} \mathbf{F}_{1}-\frac{\gamma^{4}}{\sigma^{4}} \mathbf{F}_{1} \overline{\mathbf{V}}_{\mathrm{RF}}^{j} \mathbf{C}_{j}^{-1}\left(\overline{\mathbf{V}}_{\mathrm{RF}}^{j}\right)^{H} \mathbf{F}_{1} . \\ &\text { 5: } \quad \textbf { for } i=1 \rightarrow N \\ &\text { 6: } \quad \text { Find } \eta_{i j}=\sum_{\ell \neq i} \mathbf{G}_{j}(i, \ell) \mathbf{V}_{\mathrm{RF}}(\ell, j) . \\ &\text { 7: } \quad \mathbf{V}_{\mathrm{RF}}(i, j)=\left\{\begin{array}{cc} 1, & \text { if } \eta_{i j}=0, \\ \frac{\eta_{i j}}{\left|\eta_{i j}\right|}, \text { otherwise. } \end{array}\right. \\ &\text { 8: } \quad \textbf { end for } \\ &\text { 9: \textbf{end for} } \\ &\text { 10: Check convergence. If yes, stop; if not go to Step } 2 . \\ \hline \end{aligned}

      算法运行中,目标函数不减

    • RF合并:将优化函数中的WRFHWRFMI\mathbf{W}_{RF}^H\mathbf{W}_{RF}\approx M\mathbf{I},后续类似上述算法1(16)

    • 数字合并:MMSE方法,类似【2】-\eqref{eq:2-5}

  • 后考虑将NRF=NsN^{RF}=N_s扩展到Ns<NRF<2NsN_s<N^{RF}<2N_s:多余的射频链可用作相移器精度的折中

    \begin{aligned} &\hline \begin{array}{l} \text { Algorithm 2. Design of Hybrid Beamformers for Point-to- } \\ \text { Point MIMO systems } \end{array} \\ &\hline \text { Given: } \sigma^{2}, P \\ &\text { 1: Assuming } \mathbf{V}_{\mathrm{D}} \mathbf{V}_{\mathrm{D}}^{H}=\gamma^{2} \mathbf{I} \text { where } \gamma=\sqrt{P /\left(N N^{\mathrm{RF}}\right)}, \text { find } \\ &\mathbf{V}_{\mathrm{RF}} \text { by solving the problem in (12) using Algorithm 1. } \\ &\text { 2: Calculate } \mathbf{V}_{\mathrm{D}}=\left(\mathbf{V}_{\mathrm{RF}}^{H} \mathbf{V}_{\mathrm{RF}}\right)^{-1 / 2} \mathbf{U}_{e} \boldsymbol{\Gamma}_{e} \text { where } \mathbf{U}_{e} \text { and } \boldsymbol{\Gamma}_{e} \\ &\text { are defined as following }(11) \text {. } \\ &\text { 3: Find } \mathbf{W}_{\mathrm{RF}} \text { by solving the problem in (16) using } \\ &\text { Algorithm 1. } \\ &\text { 4: Calculate } \mathbf{W}_{\mathrm{D}}=\mathbf{J}^{-1} \mathbf{W}_{\mathrm{RF}}^{H} \mathbf{H V}_{\mathrm{RF}} \mathbf{V}_{\mathrm{D}} \text { where } \mathbf{J}=\mathbf{W}_{\mathrm{RF}}^{H} \mathbf{H} \\ &\mathbf{V}_{\mathrm{RF}} \mathbf{V}_{\mathrm{D}} \mathbf{V}_{\mathrm{D}}^{H} \mathbf{V}_{\mathrm{RF}}^{H} \mathbf{H}^{H} \mathbf{W}_{\mathrm{RF}}+\sigma^{2} \mathbf{W}_{\mathrm{RF}}^{H} \mathbf{W}_{\mathrm{RF}} \text {. } \\ \hline \end{aligned}

MU-MISO

  • 考虑用户间干扰的因素
  • 考虑streams的优先级
  • proposes a design for the scenarios where NRF>KN^{RF} > K with practical NN and show numerically that adding a few more RF chains can increase the overall performance of the system and reduce the gap to capacity.

方法:VRF\mathbf{V}_{RF}P\mathbf{P}的设计之间迭代

  • 先优化VRF=ejθi,j\mathbf{V}_{RF}=e^{-j\theta_{i,j}}

  • VRF\mathbf{V}_{RF}收敛,注水法功控

    P=diag(p1,,pk)pk=1q~kk(βkλq~kkσ2)+\begin{gathered} \mathbf{P}=\mathrm{diag}(p_1,\cdots,p_k)\\ p_k=\frac{1}{\tilde q_{kk}}\left(\frac{\beta_k}{\lambda}-\tilde q_{kk}\sigma^2\right)^+ \end{gathered}

    其中,q~kk\tilde q_{kk} 是 $\mathbf{\tilde Q}=\mathbf{\tilde V}DH\mathbf{V}_{RF}H\mathbf{V}{RF}\mathbf{\tilde V}_D $ 的主对角线元素。同时 λ\lambda 使 k=1K(βkλq~kkσ2)+=P\sum_{k=1}^K\left(\frac{\beta_k}{\lambda}-\tilde q_{kk}\sigma^2\right)^+=PV~D\mathbf{\tilde V}_DHeq=HVRF\mathbf{H}_{eq}=\mathbf{HV}_{RF}的ZF数字预编码。

  • 整个算法收敛,应用注水功控的ZF数字预编码

    VDZF=VRFHHH(HVRFVRFHHH)1P12=V~DP12\mathbf{V}_{\mathrm{D}}^{\mathrm{ZF}}=\mathbf{V}_{\mathrm{RF}}^{H} \mathbf{H}^{H}\left(\mathbf{H V}_{\mathrm{RF}} \mathbf{V}_{\mathrm{RF}}^{H} \mathbf{H}^{H}\right)^{-1} \mathbf{P}^{\frac{1}{2}}=\tilde{\mathbf{V}}_{\mathrm{D}} \mathbf{P}^{\frac{1}{2}}

\begin{aligned} &\hline \begin{array}{l} \text { Algorithm 3. Design of Hybrid Precoders for MU-MISO } \\ \text { systems } \end{array} \\ &\hline \text { Given: } \beta_{k}, P, \sigma^{2} \\ &\text { 1: Start with a feasible } \mathbf{V}_{\mathrm{RF}} \text { and } \mathbf{P}=\mathbf{I}_{K} \text {. } \\ &\text { 2: for } j=1 \rightarrow N^{\mathrm{RF}} \\ &\text { 3:\quad Calculate } \mathbf{A}_{j}=\mathbf{P}^{-\frac{1}{2}} \mathbf{H} \overline{\mathbf{V}}_{\mathrm{RF}}^{j}\left(\overline{\mathbf{V}}_{\mathrm{RF}}^{j}\right)^{H} \mathbf{H}^{H} \mathbf{P}^{-\frac{1}{2}} \\ &\text { 4: } \quad \text { for } i=1 \rightarrow N \\ &\text { 5: } \quad\quad \text { Find } \zeta_{i j}^{B}, \zeta_{i j}^{D}, \eta_{i j}^{B}, \eta_{i j}^{D} \text { as defined in Appendix A. } \\ &\text { 6: } \quad\quad \text { Calculate } \theta_{i, j}^{(1)} \text { and } \theta_{i, j}^{(2)} \text { according to }(27) . \\ &\text { 7: } \quad\quad \text { Find } \theta_{i, j}^{\text {opt }}=\arg \min \left(\hat{f}\left(\theta_{i, j}^{(1)}\right), \hat{f}\left(\theta_{i, j}^{(2)}\right)\right) . \\ &\text { 8:\quad Set } \mathbf{V}_{\mathrm{RF}}(i, j)=e^{-j \theta_{i, j}^{\text {opt }}} \text {. } \\ &\text { 9: \quad end for } \\ &\text { 10: end for } \\ &\text { 11: Check convergence of RF precoder. If yes, continue; if not } \\ &\text { go to Step } 2 . \\ &\text { 12: Find } \mathbf{P}=\operatorname{diag}\left[p_{1}, \ldots, p_{k}\right] \text { using water-filling as in (23). } \\ &\text { 13: Check convergence of the overall algorithm. If yes, stop; } \\ &\text { if not go to Step } 2 . \\ &\text { 14: Set } \mathbf{V}_{\mathrm{D}}=\mathbf{V}_{\mathrm{RF}}^{H} \mathbf{H}^{H}\left(\mathbf{H V}_{\mathrm{RF}} \mathbf{V}_{\mathrm{RF}}^{H} \mathbf{H}^{H}\right)^{-1} \mathbf{P}^{\frac{1}{2}} \text {. }\\\hline \end{aligned}

相移器精度有限

与第三篇先计算波束赋形矢量后再进行量化不同,在迭代优化过程中就开始量化,尤其是max Re{VRF(i,j)ηij}\max\ \mathrm{Re}\{\mathbf{V}_{RF}^*(i,j)\eta_{ij}\}即最小化复平面上VRF\mathbf{V}_{RF}ηij\eta_{ij}的夹角。


Weighted Sum-Rate Maximization for Reconfigurable Intelligent Surface Aided Wireless Networks【智能反射面】——2020-TWC

MU-MISO

image-20211111150451134

基站MM根天线,RIS有NN个反射单元,KK个单天线用户

yk=hd,kHxDirect link +hr,kHΘGxRIS-aided link +uk=(hd,kH+hr,kHΘG)k=1Kwksk+uk=(hd,kH+θHHr,k)k=1Kwksk+uk\begin{aligned} y_{k} &=\underbrace{\mathbf{h}_{\mathrm{d}, k}^{\mathrm{H}} \mathbf{x}}_{\text {Direct link }}+\underbrace{\mathbf{h}_{\mathrm{r}, k}^{\mathrm{H}} \Theta \mathbf{G} \mathbf{x}}_{\text {RIS-aided link }}+u_{k} \\ &=\left(\mathbf{h}_{\mathrm{d}, k}^{\mathrm{H}}+\mathbf{h}_{\mathrm{r}, k}^{\mathrm{H}} \Theta \mathbf{G}\right) \sum_{k=1}^{K} \mathbf{w}_{k} s_{k}+u_{k}\\ &=\left(\mathbf{h}_{d,k}^H+\boldsymbol{\theta}^H\mathbf{H}_{r,k}\right)\sum_{k=1}^K\mathbf{w}_ks_k+u_k \end{aligned}

其中,Θ=diag(θ1,,θN),θ=[θ1,,θN]H\boldsymbol{\Theta}=diag(\theta_1,\cdots,\theta_N),\boldsymbol{\theta}=[\theta_1,\cdots,\theta_N]^HHr,k=diag(hr,kH)GCN×M\mathbf{H}_{r,k}=diag(\mathbf{h}_{r,k}^H)\mathbf{G}\in\mathbb{C}^{N\times M}

  • perfect CSI:weighted sum-rate maximization

    maxW,θfA(W,θ)=k=1Kωklog(1+γk) s.t. θn=1,n=1,,Nk=1Kwk2PT\begin{aligned} \max _{\mathbf{W}, \boldsymbol{\theta}} \quad &f_{\mathrm{A}}(\mathbf{W}, \boldsymbol{\theta})=\sum_{k=1}^{K} \omega_{k} \log \left(1+\gamma_{k}\right) \\ \text { s.t. }\quad&\left|\theta_{n}\right|=1, \quad \forall n=1, \cdots, N \\ &\sum_{k=1}^{K}\left\|\mathbf{w}_{k}\right\|^{2} \leq P_{\mathrm{T}} \end{aligned}

    γk\gamma_k即SINR。

  • imperfect CSI:$ \mathbf{h}_{d,k}$ , G\mathbf{G}hr,k\mathbf{h}_{r,k}可以看成由非完美CSI (h^d,k({\hat{\mathbf{h}}}_{d,k}G^\hat{\mathbf{G}}h^r,k{\hat{\mathbf{h}}}_{r,k})与信道估计误差(zd, k\mathbf{z}_{d,\ k}ZG\mathbf{Z}_Gzr,k\mathbf{z}_{r,k})分布的样本空间的一个随机样本。

    maxθfB(θ)=Eξ[maxWξfA(W(ξ),θ;ξ)] s.t. θn=1,n=1,,Nk=1Kwk(ξ)2PT,ξ.\begin{aligned} \max _{ \boldsymbol{\theta}} \quad &f_{\mathrm{B}}(\boldsymbol{\theta})=\mathbb{E}_\xi\left[\max_{\mathbf{W}_\xi}f_A\left(\mathbf{W}(\xi) ,\boldsymbol{\theta};\xi \right)\right] \\ \text { s.t. }\quad&\left|\theta_{n}\right|=1, \quad \forall n=1, \cdots, N \\ &\sum_{k=1}^{K}\left\|\mathbf{w}_{k}(\xi)\right\|^{2} \leq P_{\mathrm{T}},\quad\forall\xi. \end{aligned}


本文标题:【预编码论文阅读(一)】传统方法

文章作者:Levitate_

发布时间:2021年12月07日 - 19:57:07

原始链接:https://levitate-qian.github.io/2021/12/08/precoding-1/

许可协议: 署名-非商业性使用-禁止演绎 4.0 国际 转载请保留原文链接及作者。