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 = β M R T H H \mathbf V=\beta_{MRT}\mathbf H^H
V = β M R T 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, β M R T = 1 t r ( H H H ) \beta_{MRT}=\sqrt{\frac{1}{tr(HH^H )}} β M R T = t r ( H H H ) 1 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。h i H w ~ j = 0 , i ≠ j {\bf h}_i^H\tilde{\bf w}_j=0,i\neq j h i H w ~ j = 0 , i = j (5)——achieve virtually optimal
W Z F = H H ( H H H ) − 1 \mathbf{W}_{\mathrm{ZF}}=\mathbf{H}^{H}\left(\mathbf{H} \mathbf{H}^{H}\right)^{-1}
W Z F = H H ( H H H ) − 1
Equal power scaling allocation(ZF-EPS)
平均分配能量——normalized=power allocate
w ~ i = η w i η = P t r { W Z F W Z F H } \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}
w ~ i = η w i η = t r { W Z F W Z F H } P
Optimal power allocation - water-filling solution(ZF-WF)
信噪比高的多分配能量
maximize p 1 , … , p K ∑ i = 1 K log ( 1 + ρ i p i ) subject to ∑ i = 1 K γ i p i ≤ P , p i ≥ 0 \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}
m a x i m i z e p 1 , … , p K subject to i = 1 ∑ K log ( 1 + ρ i p i ) i = 1 ∑ K γ i p i ≤ P , p i ≥ 0
其中,γ i = [ ( H H H ) − 1 ] \gamma_i=[({\bf HH}^H)^{-1}] γ i = [ ( H H H ) − 1 ] 。注水法功率控制(6)——η i = p i \eta_i=\sqrt{p_i} η i = p i
p i = [ μ γ i − 1 ρ i ] + , ∀ i p_i=\left[\frac{\mu}{\gamma_i}-\frac{1}{\rho_i}\right]^+,\forall i
p i = [ γ i μ − ρ i 1 ] + , ∀ i
其中, [ x ] + = m a x ( x , 0 ) [x]^+=max(x,0) [ x ] + = m a x ( x , 0 ) , 总功率限制
∑ i = 1 K = [ μ − γ i ρ i − 1 ] + = P \sum_{i=1}^K=\left[\mu-\gamma_i\rho_i^{-1}\right]^+=P
i = 1 ∑ K = [ μ − γ i ρ i − 1 ] + = P
Regularized zero-forcing precoding(RZF)
( H H H ) − 1 (\mathbf{H} \mathbf{H}^{H})^{-1} ( H 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)
W R Z F = H H ( H H H + α I ) − 1 \mathbf{W}_{\mathrm{RZF}}=\mathbf{H}^{H}\left(\mathbf{H} \mathbf{H}^{H}+\alpha \mathbf{I}\right)^{-1}
W R Z F = H H ( H H H + α I ) − 1
W ~ = η W RZF η = P t r { W R Z F W R Z F H } \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}
W ~ = η W RZF η = t r { W R Z F W R Z F H } P
homogeneous SNR conditions(ρ i = 1 \rho_i=1 ρ i = 1 )
α ⋆ = K P \alpha^{\star}=\frac KP
α ⋆ = P K
non-homogeneous SNR conditions——non-weighted sum-MSE minimization(9)W R Z F = H H ( H H H + ∑ i = 1 K ( 1 / ρ i ) P I ) − 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}
W R Z F = H H ( H H H + P ∑ i = 1 K ( 1 / ρ i ) I ) − 1
Iterative weighted minimization ofmean squared error algorithm(IWMMSE)
解决非凸问题的数值算法。weighted MSE problem(10)
minimize Λ , Ω , W ~ E { ∥ Ω 1 2 ( u − Λ y ) ∥ 2 } − log det Ω 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}
Λ , Ω , W ~ m i n i m i z e E { ∥ ∥ ∥ Ω 2 1 ( u − Λ y ) ∥ ∥ ∥ 2 } − log d e t Ω subject to T r { W ~ W ~ H } ≤ P
其中,Ω , Λ \mathbf{\Omega},\mathbf{\Lambda} Ω , Λ 均为对角阵,Ω \mathbf{\Omega} Ω 表示K个UE的权重(weights),Λ \mathbf{\Lambda} Λ 表示接收因子(receive coefficients)。W ~ , Ω , Λ \tilde{\mathbf{W}},\mathbf{\Omega},\mathbf{\Lambda} W ~ , Ω , Λ 是优化变量。
S1:随机初始化变量
∣ ∣ w ~ i ∣ ∣ 2 = P K ||\tilde{\mathbf{w}}_i||^2=\frac PK
∣ ∣ w ~ i ∣ ∣ 2 = K P
S2:迭代直到convergence
确定W ~ , Ω \tilde{\mathbf{W}},\mathbf{\Omega} W ~ , Ω ,优化Λ \mathbf{\Lambda} Λ (11)
λ i = arg min λ i E { ∣ u i − λ i y i ∣ 2 } = ( ∑ j = 1 K ρ i ∣ h i H w ~ j ∣ 2 + 1 ) − 1 ρ i w ~ i H h i . \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}
λ i = arg λ i min E { ∣ u i − λ i y i ∣ 2 } = ( j = 1 ∑ K ρ i ∣ ∣ h i H w ~ j ∣ ∣ 2 + 1 ) − 1 ρ i w ~ i H h i .
确定W ~ , Λ \tilde{\mathbf{W}},\mathbf{\Lambda} W ~ , Λ ,优化Ω \mathbf{\Omega} Ω (12)
ω i = arg min ω i ω i e i − log ω i = e i − 1 = 1 + SINR i = ∑ j = 1 K ρ i ∣ h i H w ~ j ∣ 2 + 1 ∑ j ≠ i K ρ i ∣ h i H w ~ j ∣ 2 + 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}
ω i = arg ω i min ω i e i − log ω i = e i − 1 = 1 + S I N R i = ∑ j = i K ρ i ∣ ∣ h i H w ~ j ∣ ∣ 2 + 1 ∑ j = 1 K ρ i ∣ ∣ h i H w ~ j ∣ ∣ 2 + 1
确定Ω , Λ \mathbf{\Omega},\mathbf{\Lambda} Ω , Λ ,优化W ~ \tilde{\mathbf{W}} W ~ (14)
W ~ = ( H H Λ H Ω Σ Λ H + μ I ) − 1 H H Λ H Ω Σ 1 2 = H H [ H H H + μ ( Λ H Ω Σ Λ ) − 1 ] − 1 Λ − 1 Σ − 1 2 \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}
W ~ = ( H H Λ H Ω Σ Λ H + μ I ) − 1 H H Λ H Ω Σ 2 1 = H H [ H H H + μ ( Λ H Ω Σ Λ ) − 1 ] − 1 Λ − 1 Σ − 2 1
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 u 是向量,每次给U E i UE_i U E i 发送的是单个字符u i u_i u i ——多用户
y = Σ 1 2 H W ~ u + n x = ∑ i = 1 K w ~ i u i = u W ~ \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}
y = Σ 2 1 H W ~ u + n x = i = 1 ∑ K w ~ i u i = u W ~
其中,Σ = d i a g ( ρ 1 , ⋯ , ρ K ) \mathbf{\Sigma}=\mathrm{diag}(\rho_1,\cdots,\rho_K) Σ = d i a g ( ρ 1 , ⋯ , ρ K ) 是对角阵,表示非同质信噪比条件的差异;H = [ h 1 , ⋯ , h K ] H \mathbf{H}=[\mathbf{h}_1,\cdots,\mathbf{h}_K]^H H = [ h 1 , ⋯ , h K ] H ;W ~ = [ w ~ 1 , ⋯ , w ~ K ] \tilde{\mathbf{W}}=[\tilde{\mathbf{w}}_1,\cdots,\tilde{\mathbf{w}}_K] W ~ = [ w ~ 1 , ⋯ , w ~ K ] , u = [ u 1 , ⋯ , u K ] T u=[u_1,\cdots,u_K]^T u = [ u 1 , ⋯ , u K ] T
SINR(信干噪比)
S I N R i = ρ i ∣ h i H w ~ i ∣ 2 ∑ j ≠ i K ρ i ∣ h i H w ~ j ∣ 2 + 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}
S I N R i = ∑ j = i K ρ i ∣ ∣ h i H w ~ j ∣ ∣ 2 + 1 ρ i ∣ ∣ h i H w ~ i ∣ ∣ 2
sum-rate的优化问题——非凸问题
maximize w ~ 1 , … , w ~ K ∑ i = 1 K log ( 1 + ρ i ∣ h i H w ~ i ∣ 2 ∑ j ≠ i K ρ i ∣ h i H w ~ j ∣ 2 + 1 ) subject to ∑ i = 1 K ∥ w ~ i ∥ 2 ≤ P \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}
w ~ 1 , … , w ~ K m a x i m i z e i = 1 ∑ K log ( 1 + ∑ j = i K ρ i ∣ ∣ h i H w ~ j ∣ ∣ 2 + 1 ρ i ∣ ∣ h i H w ~ i ∣ ∣ 2 ) subject to i = 1 ∑ K ∥ w ~ i ∥ 2 ≤ P
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} Ω , Λ )
思路
用effective small-scale channel gain G ( g i = ∣ ∣ h i ∣ ∣ σ n i ) \mathbf{G}(g_i=\frac{||\mathbf{h}_i||}{\sigma_{n_i}}) G ( g i = σ n i ∣ ∣ h i ∣ ∣ ) 表示两个迭代参数(16)
M S E = E { ∥ G Σ 1 2 ⏟ Ω 1 2 ( u − η − 1 G − 1 Σ − 1 2 ⏟ Λ y ) ∥ 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\}
M S E = E ⎩ ⎪ ⎨ ⎪ ⎧ ∥ ∥ ∥ ∥ ∥ ∥ Ω 2 1 G Σ 2 1 ⎝ ⎛ u − Λ η − 1 G − 1 Σ − 2 1 y ⎠ ⎞ ∥ ∥ ∥ ∥ ∥ ∥ 2 ⎭ ⎪ ⎬ ⎪ ⎫
使用lagrange乘数法求lagrange因子μ ⋆ \mu^{\star} μ ⋆ 和波束成形矩阵W ⋆ \mathbf{W}^{\star} W ⋆
结果
μ ⋆ = K P W ⋆ = H H ( H H H + K P Σ − 1 ) − 1 G \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}
μ ⋆ = P K W ⋆ = H H ( H H H + P K Σ − 1 ) − 1 G
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 h ^ i 的编号。用量化矢量h ^ i \mathbf{\hat h}_i h ^ i 代替SINR表达式中的h i \mathbf{h}_i h i
CQI(channel quality indicator):用有效信道增益表示瞬时信噪比g ^ i ρ ^ i = M P S N R i ^ \hat g_i \hat \rho_i=\frac MP\widehat{SNR_i} g ^ i ρ ^ i = P M S N R i 代替SINR中平均信噪比ρ i \rho_i ρ 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 = h i ∣ ∣ h i ∣ ∣ \mathbf{\tilde h}_i=\frac{\mathbf{h}_i}{||\mathbf{h}_i||} h ~ i = ∣ ∣ h i ∣ ∣ h i ,
h ~ i = 1 − z i h ^ i + z i s z i = 1 − ∣ h ~ i H h ^ i ∣ 2 \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}
h ~ i = 1 − z i h ^ i + z i s z i = 1 − ∣ h ~ i H h ^ i ∣ 2
其中,z i z_i z i 是量化误差,h ~ i \mathbf{\tilde h}_i h ~ i 是实际方向,h ^ i \mathbf{\hat h}_i h ^ i 是量化方向的投影,s \mathbf{s} s 是量化矢量h ^ i \mathbf{\hat h}_i 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} H ~ 有所变化:
H = G H ~ = G ( I − Z ) 1 2 H ^ + G Z 1 2 S \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}
H = G H ~ = G ( I − Z ) 2 1 H ^ + G Z 2 1 S
化开,同样使用lagrange乘数法求lagrange因子μ ⋆ \mu^{\star} μ ⋆ 和波束成形矩阵W ⋆ \mathbf{W}^{\star} W ⋆
结果
μ ⋆ \mu^\star μ ⋆ 不变
W = ζ 1 − δ H ^ H ( H ^ H ^ H + δ P Tr { Σ G 2 } + K M P M ( 1 − δ ) Σ − 1 G − 2 ) − 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}
W = 1 − δ ζ H ^ H ( H ^ H ^ H + P M ( 1 − δ ) δ P T r { Σ G 2 } + K M Σ − 1 G − 2 ) − 1
系数不影响归一化的波束成形矩阵W ~ \mathbf{\tilde W} W ~ ,信噪比S N R = P M G 2 Σ \mathbf{SNR}=\frac PM\mathbf{G}^2\mathbf{\Sigma} S N R = M P G 2 Σ (37)
W ⋆ = H ^ H ( H ^ H ^ H + δ Tr { S N R } + K M ( 1 − δ ) S N R − 1 ) − 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}
W ⋆ = H ^ H ( H ^ H ^ H + M ( 1 − δ ) δ T r { S N R } + K S N R − 1 ) − 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UEi k i_k i k 发送的是s \bf s s ,而非单个字符——讨论单用户
s ^ i k = U i k H y i k \mathbf{\hat s}_{i_k}=\mathbf{U}_{i_k}^H\mathbf{y}_{i_k}
s ^ i k = U i k H 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 { x k x k H } \mathbf{Q}=\mathbb{E}\{\mathbf{x}_k\mathbf{x}_k^H\} Q = E { x k 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
log ∣ I + H Q H H ∣ . \log |\mathbf{I}+\mathbf{H}\mathbf{Q}\mathbf{H}^H|.
log ∣ I + H Q 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R i = ∑ i ≠ j H j i Q j H j i H + I , \mathbf{R}_i = \sum_{i \neq j} \mathbf{H}_{ji}\mathbf{Q}_j \mathbf{H}_{ji}^H+\mathbf{I}, R i = ∑ i = j H j i Q j H j i H + I , 多用户信道容量就变成了:
∑ i = 1 K log ∣ I + R i − 1 H i i Q i H i i H ∣ . \sum_{i=1}^K \log |\mathbf{I}+\mathbf{R}^{-1}_i\mathbf{H}_{ii}\mathbf{Q}_i\mathbf{H}^H_{ii}|.
i = 1 ∑ K log ∣ I + R i − 1 H i i Q i H i i H ∣ .
考虑它更广义的形式i i i 个用户,加入效用因子λ i \lambda_i λ i (文章1的ρ i \rho_i ρ i )。当效用因子都是1时,就和上式等价,问题就变成了:
∑ i = 1 K λ i log ∣ I + R i − 1 H i i Q i H i i H ∣ \sum_{i=1}^K \lambda_i\log |\mathbf{I}+\mathbf{R}^{-1}_i\mathbf{H}_{ii}\mathbf{Q}_i\mathbf{H}^H_{ii}|
i = 1 ∑ K λ i log ∣ I + R i − 1 H i i Q i H i i H ∣
【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)
max V ∑ k = 1 K ∑ i k = 1 I k α i k R i k s.t. ∑ i = 1 I k Tr ( V i k V i k H ) ≤ P k , ∀ 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}
V max s.t. k = 1 ∑ K i k = 1 ∑ I k α i k R i k i = 1 ∑ I k T r ( V i k V i k H ) ≤ P k , ∀ k = 1 , 2 , … , K
其中,α i k \alpha_{i_k} α i k 是优先级
sum-MSE minimization(4)
min U , V ∑ k = 1 K ∑ i = 1 I k Tr ( E i k ) = ∑ k = 1 K ∑ i = 1 I k ∣ ∣ s ^ i k − s i k ∣ ∣ s.t. ∑ i = 1 I k Tr ( V i k V i k H ) ≤ P k , 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}
U , V min s.t. k = 1 ∑ K i = 1 ∑ I k T r ( E i k ) = k = 1 ∑ K i = 1 ∑ I k ∣ ∣ s ^ i k − s i k ∣ ∣ i = 1 ∑ I k T r ( V i k V i k H ) ≤ P k , k = 1 , 2 , … , K .
U i k m m s e = J i k − 1 H i k k V i k E i k m m s e = I − V i k H H i k k H J i k − 1 H i k k V i k \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}
U i k m m s e = E i k m m s e = J i k − 1 H i k k V i k I − V i k H H i k k H J i k − 1 H i k k V i k
其中,J i k ≜ ∑ j = 1 K ∑ ℓ = 1 I j H i k j V ℓ j V ℓ j H H i k H + σ i k 2 I \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} J i k ≜ ∑ j = 1 K ∑ ℓ = 1 I j H i k j V ℓ j V ℓ j H H i k H + σ i k 2 I 。
两个问题的统一性(7)——【1-[5]、2-[13]】梯度、KKT条件引出
min W , U , V ∑ k = 1 K ∑ i = 1 I k α i k ( Tr ( W i k E i k ) − log det ( W i k ) ) s.t. ∑ i = 1 I k Tr ( V i k V i k H ) ≤ P k , 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}
W , U , V min s.t. k = 1 ∑ K i = 1 ∑ I k α i k ( T r ( W i k E i k ) − log d e t ( W i k ) ) i = 1 ∑ I k T r ( V i k V i k H ) ≤ P k , k = 1 , 2 , … , K
(7) is in the space of ( u , v , w ) (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).
R i k = log det ( ( E i k m m s e ) − 1 ) R_{i_k}=\log\det\left(\left(\mathbf{E}_{i_k}^{mmse}\right)^{-1}\right)
R i k = log det ( ( E i k m m s e ) − 1 )
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} U , V , W 。
1 Initialize V i k \mathbf{V}_{i_{k}} V i k 's such that Tr ( V i k V i k H ) = p k I k \operatorname{Tr}\left(\mathbf{V}_{i_{k}} \mathbf{V}_{i_{k}}^{H}\right)=\frac{p_{k}}{I_{k}} T r ( V i k V i k H ) = I k p k
2 repeat
3 W i k ′ ← W i k , ∀ i k ∈ I 3 \quad \mathbf{W}_{i_{k}}^{\prime} \leftarrow \mathbf{W}_{i_{k}}, \quad \forall i_{k} \in \mathcal{I} 3 W i k ′ ← W i k , ∀ i k ∈ I
4 U i k ← ( ∑ ( j , ℓ ) H i k j V ℓ j V ℓ j H H i k j H + σ i k 2 I ) − 1 H i k k V k , ∀ i k ∈ I 4 \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} 4 U i k ← ( ∑ ( j , ℓ ) H i k j V ℓ j V ℓ j H H i k j H + σ i k 2 I ) − 1 H i k k V k , ∀ i k ∈ I
5 W i k ← ( I − U i k H H i k k V i k ) − 1 , ∀ i k ∈ I 5\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} 5 W i k ← ( I − U i k H H i k k V i k ) − 1 , ∀ i k ∈ I
6 V i k ← α i k ( ∑ ( j , ℓ ) α ℓ j H ℓ j k H U ℓ j W ℓ j U ℓ j H H ℓ j k + μ k ∗ I ) − 1 H i k k H U i k W i k , ∀ i k 6 \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} 6 V i k ← α i k ( ∑ ( j , ℓ ) α ℓ j H ℓ j k H U ℓ j W ℓ j U ℓ j H H ℓ j k + μ k ∗ I ) − 1 H i k k H U i k W i k , ∀ i k
7 until ∣ ∑ ( j , ℓ ) log det ( W ℓ j ) − ∑ ( j , ℓ ) log det ( W ℓ j ′ ) ∣ ≤ 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 ∣ ∣ ∣ ∑ ( j , ℓ ) log d e t ( W ℓ j ) − ∑ ( j , ℓ ) log d e t ( W ℓ j ′ ) ∣ ∣ ∣ ≤ ϵ
W \mathbf{W} W 的优化来自对\eqref{eq:2-7}的求解——对W \mathbf{W} W 求一阶导数得到W ⋆ = ( E i k M M S E ) − 1 \mathbf{W}^\star=(\mathbf{E}_{i_k}^{MMSE})^{-1} W ⋆ = ( E i k M M S E ) − 1 , E i k M M S E \mathbf{E}_{i_k}^{MMSE} E i k M M S E 来自MMSE问题的求解,可由U \mathbf{U} U 表示
U \mathbf{U} U 的优化来自MMSE的求解——对MSE表达式对U \mathbf{U} U 求一阶导,求极值
V \mathbf{V} 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} V 利用Lagrange乘数法,解出V \mathbf{V} V 与μ k \mu_k μ k 有关,根据约束条件求μ k ⋆ \mu_k^\star μ k ⋆ ,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) ( 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——传统数字预编码的RF chain数量需要和N t N_t N t 相等,通过W F \mathbf{WF} W F 的统一考虑,减少RF chain。
y k = h k H F N t × K W K × K s K × 1 + n k y_k=\mathbf{h}_k^H\mathbf{F}_{N_t\times K}\mathbf{W}_{K\times K}\mathbf{s}_{K\times 1}+n_k
y k = h k H F N t × K W K × K s K × 1 + n k
方法
apply phase-only control to couple the K K K RF chain outputs with N t N_t N t transmit antennas, using cost-effective RF phase shifters.(F \mathbf{F} F 只相移,将K条射频链和N t N_t N t 个发射天线耦合)
思路
F \bf F F 只调相
F i , j = 1 N t e j φ i , j \mathbf{F}_{i,j}=\frac{1}{\sqrt{N_t}}e^{j\varphi_{i,j}}
F i , j = N t 1 e j φ 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 W 调幅、调相:将H F \bf HF H F 看作等效的H \bf H H ,利用非注水的ZF(块对角化)
W = H e q H ( H e q H e q H ) − 1 Λ \mathbf{W}=\mathbf{H}_{eq}^H(\mathbf{H}_{eq}\mathbf{H}_{eq}^H)^{-1}\mathbf{\Lambda}
W = H e q H ( H e q H e q H ) − 1 Λ
Λ \mathbf{\Lambda} Λ 为列功率归一化。类似【1】的η i \eta_i η i
Quantized RF Phase Control:由于F \mathbf{F} 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} W 时利用量化后的F ^ \mathbf{\hat F} F ^
结果
频谱利用率(Spectral Efficiency)分析
——当N t N_t N t 足够大,用户间干扰可以忽略
Phased-ZF的上界R ≤ K R R\leq K \mathcal{R} R ≤ K R ,且
lim N t → ∞ R log 2 ( 1 + π 4 P N t K ) = 1 R = E [ 1 + P K ∣ h k H f k ∣ 2 ] \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}
N t → ∞ lim log 2 ( 1 + 4 π K P N t ) R = 1 R = E [ 1 + K P ∣ h k H f k ∣ 2 ]
仿真1:Rayleigh信道
1 2 3 4 5 6 F = 1 /sqrt (Nt)*exp (j .*angle (H))'; Fb = CalBDPrecoder(H*F); wt = F*Fb; WPR = wt*inv(sqrt (diag (diag (wt'*wt)))); rateHyb(isnr) = rateHyb(isnr) + CalRate((P/K)*eye (K), H, WPR);
仿真2:mmWave信道
毫米波信道的特点:limited multipath components.—— poor scattering nature
h k H = N t N p ∑ l = 1 N p α l k a H ( ϕ l k , θ l k ) \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)
h k H = N p N t l = 1 ∑ N p α l k a H ( ϕ l k , θ l k )
Contribution
在性能减弱不大的情况下,减少射频链(RF chains)的数量
x = V R F ⏟ N × N t R F V D ⏟ N t R F × N s s = ∑ ℓ = 1 K V R F V D ℓ s ℓ \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
x = N × N t R F V R F N t R F × N s V D s = ℓ = 1 ∑ K V R F V D ℓ s ℓ
其中,N s = K d N_s=Kd N s = K d , K K K 个用户,每个用户d d d 个符号
接收信号
s ~ k = y ~ k = W t k H H k V t k s k ⏟ desired signals + W t k H H k ∑ ℓ ≠ k V t ℓ s ℓ ⏟ effective interference + W t k H z k ⏟ effective 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 }}
s ~ k = y ~ k = desired signals W t k H H k V t k s k + effective interference W t k H H k ℓ = k ∑ V t ℓ s ℓ + effective noise W t k H z k
其中,基站的预编码V t k = V R F V D k \mathbf{V}_{\mathrm{t}_k}=\mathbf{V}_{RF}\mathbf{V}_{D_k} V t k = V R F V D k ,同理,用户侧的预编码W t k = V R F k V D k \mathbf{W}_{\mathrm{t}_k}=\mathbf{V}_{RF_k}\mathbf{V}_{D_k} W t k = V R F k V D k
优化问题sum-rate problem(4)
R k = log 2 ∣ I M + W t k C k − 1 W t k H H k V t k V t k H H k H ∣ where C k = W t k H H k ( ∑ ℓ ≠ k V t ℓ V t ℓ H ) H k H W t k + σ 2 W t k H W t k \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}
R k = log 2 ∣ ∣ I M + W t k C k − 1 W t k H H k V t k V t k H H k H ∣ ∣ where C k = W t k H H k ⎝ ⎛ ℓ = k ∑ V t ℓ V t ℓ H ⎠ ⎞ H k H W t k + σ 2 W t k H W t k
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
V F D ∈ C N × N s \mathbf{V}_{FD}\in \mathbb{C}^{N\times N_s} V F D ∈ C N × N s
必要条件:N R F ≥ N s N^{RF}\geq N_s N R F ≥ N s
充分条件:N R F ≥ 2 N s N^{RF}\geq 2N_s N R F ≥ 2 N s
N s ≥ 2 N R F N_s\geq 2N^{RF} N s ≥ 2 N R F 的构造方法:it is in fact possible to realize any fully digital beamformer using the hybrid structure with N s N_s N s RF chains and 2 N s N 2N_sN 2 N s N phase shifters.
当N R F ≈ N N^{RF}\approx N N R F ≈ N ,可以达到近似最优解,此时移相器数量为N s N N_sN N s N 。
在低信噪比环境中,若V F D \mathbf{V}_{FD} V F D 非满秩矩阵,则先进行满秩分解(V F D = A N × r B r × N s \mathbf{V}_{FD}=\mathbf{A}_{N\times r}\mathbf{B}_{r\times N_s} V F D = A N × r B r × N s ),将A = V R F V D ′ \mathbf{A}=\mathbf{V}_{RF}\mathbf{V}'_D A = V R F V D ′ 作为预编码矩阵,射频链数量为2 r 2r 2 r ,此时模拟预编码V R F \mathbf{V}_{RF} V R F ,数字预编码V D ′ B \mathbf{V}'_D\mathbf{B} V D ′ B
SU-Point-to-Point MIMO
假设N t R F = N r R F = N R F N_t^{RF}=N_r^{RF}=N^{RF} N t R F = N r R F = N R F
优化目标:由\eqref{eq:4-4}化简:
R = log 2 ∣ I M + 1 σ 2 W t ( W t H W t ) − 1 W t H H V t V t H H H ∣ R=\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|
R = log 2 ∣ ∣ ∣ ∣ I M + σ 2 1 W t ( W t H W t ) − 1 W t H H V t V t H H H ∣ ∣ ∣ ∣
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 N R F > K N^{RF} > K N R F > K with practical N N N 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:在V R F \mathbf{V}_{RF} V R F 和P \mathbf{P} P 的设计之间迭代
先优化V R F = e − j θ i , j \mathbf{V}_{RF}=e^{-j\theta_{i,j}} V R F = e − j θ i , j
V R F \mathbf{V}_{RF} V R F 收敛,注水法功控
P = d i a g ( p 1 , ⋯ , p k ) p k = 1 q ~ k k ( β k λ − q ~ k k σ 2 ) + \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}
P = d i a g ( p 1 , ⋯ , p k ) p k = q ~ k k 1 ( λ β k − q ~ k k σ 2 ) +
其中,q ~ k k \tilde q_{kk} q ~ k k 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 λ 使 ∑ k = 1 K ( β k λ − q ~ k k σ 2 ) + = P \sum_{k=1}^K\left(\frac{\beta_k}{\lambda}-\tilde q_{kk}\sigma^2\right)^+=P ∑ k = 1 K ( λ β k − q ~ k k σ 2 ) + = P , V ~ D \mathbf{\tilde V}_D V ~ D 是H e q = H V R F \mathbf{H}_{eq}=\mathbf{HV}_{RF} H e q = H V R F 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
V D Z F = V R F H H H ( H V R F V R F H H H ) − 1 P 1 2 = V ~ D P 1 2 \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}}
V D Z F = V R F H H H ( H V R F V R F H H H ) − 1 P 2 1 = V ~ D P 2 1
\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 R e { V R F ∗ ( i , j ) η i j } \max\ \mathrm{Re}\{\mathbf{V}_{RF}^*(i,j)\eta_{ij}\} max R e { V R F ∗ ( i , j ) η i j } 即最小化复平面上V R F \mathbf{V}_{RF} V R F 和η i j \eta_{ij} η i j 的夹角。
Weighted Sum-Rate Maximization for Reconfigurable Intelligent Surface Aided Wireless Networks【智能反射面】——2020-TWC
MU-MISO
基站M M M 根天线,RIS有N N N 个反射单元,K K K 个单天线用户
y k = h d , k H x ⏟ Direct link + h r , k H Θ G x ⏟ RIS-aided link + u k = ( h d , k H + h r , k H Θ G ) ∑ k = 1 K w k s k + u k = ( h d , k H + θ H H r , k ) ∑ k = 1 K w k s k + u k \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}
y k = Direct link h d , k H x + RIS-aided link h r , k H Θ G x + u k = ( h d , k H + h r , k H Θ G ) k = 1 ∑ K w k s k + u k = ( h d , k H + θ H H r , k ) k = 1 ∑ K w k s k + u k
其中,Θ = d i a g ( θ 1 , ⋯ , θ N ) , θ = [ θ 1 , ⋯ , θ N ] H \boldsymbol{\Theta}=diag(\theta_1,\cdots,\theta_N),\boldsymbol{\theta}=[\theta_1,\cdots,\theta_N]^H Θ = d i a g ( θ 1 , ⋯ , θ N ) , θ = [ θ 1 , ⋯ , θ N ] H ,H r , k = d i a g ( h r , k H ) G ∈ C N × M \mathbf{H}_{r,k}=diag(\mathbf{h}_{r,k}^H)\mathbf{G}\in\mathbb{C}^{N\times M} H r , k = d i a g ( h r , k H ) G ∈ C N × M
perfect CSI:weighted sum-rate maximization
max W , θ f A ( W , θ ) = ∑ k = 1 K ω k log ( 1 + γ k ) s.t. ∣ θ n ∣ = 1 , ∀ n = 1 , ⋯ , N ∑ k = 1 K ∥ w k ∥ 2 ≤ P T \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}
W , θ max s.t. f A ( W , θ ) = k = 1 ∑ K ω k log ( 1 + γ k ) ∣ θ n ∣ = 1 , ∀ n = 1 , ⋯ , N k = 1 ∑ K ∥ w k ∥ 2 ≤ P T
γ k \gamma_k γ k 即SINR。
imperfect CSI:$ \mathbf{h}_{d,k}$ , G \mathbf{G} G , h r , k \mathbf{h}_{r,k} h r , k 可以看成由非完美CSI ( h ^ d , k ({\hat{\mathbf{h}}}_{d,k} ( h ^ d , k , G ^ \hat{\mathbf{G}} G ^ , h ^ r , k {\hat{\mathbf{h}}}_{r,k} h ^ r , k )与信道估计误差(z d , k \mathbf{z}_{d,\ k} z d , k , Z G \mathbf{Z}_G Z G ,z r , k \mathbf{z}_{r,k} z r , k )分布的样本空间的一个随机样本。
max θ f B ( θ ) = E ξ [ max W ξ f A ( W ( ξ ) , θ ; ξ ) ] s.t. ∣ θ n ∣ = 1 , ∀ n = 1 , ⋯ , N ∑ k = 1 K ∥ w k ( ξ ) ∥ 2 ≤ P T , ∀ 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}
θ max s.t. f B ( θ ) = E ξ [ W ξ max f A ( W ( ξ ) , θ ; ξ ) ] ∣ θ n ∣ = 1 , ∀ n = 1 , ⋯ , N k = 1 ∑ K ∥ w k ( ξ ) ∥ 2 ≤ P T , ∀ ξ .