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  应用科技  2018, Vol. 45 Issue (1): 73-76  DOI: 10.11991/yykj.201612010
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引用本文  

张卓, 刘彤. 基于空间相关性的大规模多用户MIMO预编码[J]. 应用科技, 2018, 45(1): 73-76. DOI: 10.11991/yykj.201612010.
ZHANG Zhuo, LIU Tong. Research on massive multi-user MIMO precoding based on spatial correlation[J]. Applied Science and Technology, 2018, 45(1): 73-76. DOI: 10.11991/yykj.201612010.

通信作者

张卓, E-mail:zhangzhuo214@outlook.com

作者简介

张卓(1991-), 男, 硕士研究生;
刘彤(1977-), 男, 教授, 博士

文章历史

收稿日期:2016-12-08
网络出版日期:2018-04-28
基于空间相关性的大规模多用户MIMO预编码
张卓, 刘彤    
哈尔滨工程大学 信息与通信工程学院, 黑龙江 哈尔滨 150001
摘要:由于FDD(frequency-division duplex)系统的信道不具有互易性,使得大规模MIMO(multiple-input multiple-output)系统发射端对多天线信道信息的获取变得十分困难。为了解决基站获取信道状态信息的问题,可以采用双重结构预编码方案,将对长期CSIT处理的波束成形和随后对短期CSIT处理的线性预编码进行级联,可以有效地降低反馈开销。预编码矩阵由基于空间相关性的预波束成形矩阵和基于短期CSIT的传统预编码矩阵决定。通过将具有空间相关性的用户进行分组,并将分组后的用户的信道矩阵与基于空间相关性的相同的预处理矩阵相乘,使预编码信号空间的维度降低。
关键词大规模MIMO    预编码    空间相关性    级联    双重结构    FDD    信道状态信息    和速率    
Research on massive multi-user MIMO precoding based on spatial correlation
ZHANG Zhuo, LIU Tong    
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: Because the channel of FDD system lacks reciprocity, it becomes very difficult to obtain the multi-antenna channel information by the transmitting terminal of the massive MIMO system. In order to solve the problem that the base station obtains the state information of channel, a dual-structure precoding scheme can be used. The long-term CSIT beamforming and the linear precoding of the short-term CSIT processing are cascaded to effectively reduce the feedback overhead. The precoding matrix is determined by the pre-beamforming matrix based on spatial correlation and the traditional precoding matrix based on short-term CSIT. By grouping the users with spatial correlation, multiplying the channel matrix of the grouped users with the same pre-processing matrix based on spatial correlation, the dimensionality of the precoding signal space can be reduced.
Key words: massive MIMO    precoding    spatial correlation    concatenate    dual structure    FDD    state information of channel    sum speed    

大规模MIMO技术是MIMO技术的演进,通过在基站配置大量的天线,获得更高的频谱效率。由于大规模MIMO系统可以有效地提高频谱利用率和系统容量,近几年来已经成为了学者们的研究热点[1-5]。另外,预编码技术可以在已知信道状态信息(channel state information,CSI)情况下,通过在发送端对发送的信号做一个预先的处理,以方便接收机进行信号检测,从而进一步提升系统性能。但是在FDD传输模式中,大规模多用户MIMO系统中大量的用户也会导致获取CSI困难的问题[6-8]。因此,如何设计一种预编码方案,使得在FDD传输模式下的大规模MIMO系统中获得准确的CSI,从而减小反馈开销,成为了需要去解决的问题[9-13]

1 系统模型

假定系统为FDD传输模式下单小区下行链路系统,其中有一个配置有M个垂直/水平极化分集天线单元的基站和N个配置有一个垂直或水平极化天线单元的活跃用户,且MN均为偶数。由于人们的通常活动在建筑物这种小集群区域,因此将用户的位置视为空间集群,并划为G个群组。在平坦衰落信道下,第g个群组的接收信号如式(1):

$ {\mathit{\boldsymbol{y}}_g} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{y}}_g^v}\\ {\mathit{\boldsymbol{y}}_g^h} \end{array}} \right] = \mathit{\boldsymbol{H}}_g^{\rm{H}}\mathit{\boldsymbol{x}} + {\mathit{\boldsymbol{n}}_g} $ (1)

式中:ygvygh分别是用户垂直极化和水平极化的接收信号;ng=[ngv ngh]T是均值为0、方差为INg的复高斯噪声向量,即服从分布ng~CN(0, INg);第g个群组中的用户数量为Ng,为简单起见,假设N1=N2=…=NG=N,且N为偶数。ygvygh均为N/2×1的向量;Hg为第g个群组的信道矩阵,其中第k个用户的信道向量为hgk=[Hg]kM×1的向量x为经过线性预编码后的发射信号,其表达式为

$ \mathit{\boldsymbol{x}} = \sum\limits_{g = 1}^G {{\mathit{\boldsymbol{P}}_g}{\mathit{\boldsymbol{d}}_g}} $

式中:$ {\mathit{\boldsymbol{P}}_g} \in {C^{M \times \bar N}} $为线性预编码矩阵,dg=[dgv dgh]T为第g个群组中用户的信息符号向量。经过预编码后的信号x应满足功率限制E[‖x‖]2P

2 信道模型

利用Karhunen-Loeve变换和文献[1-2]中提出的部署有无穷小的天线单元的极化MIMO信道模型,第g个群组的下行链路信道Hg可表示为

$ {\mathit{\boldsymbol{H}}_g} = \left( {\left[ {\begin{array}{*{20}{c}} 1&{{r_{xp}}}\\ {{r_{xp}}}&1 \end{array}} \right] \otimes \left( {{\mathit{\boldsymbol{U}}_g}\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_g^{\frac{1}{2}}} \right)} \right)\left( {{\mathit{\boldsymbol{G}}_g} \odot \left( {\mathit{\boldsymbol{X}} \otimes {1_{r \times \frac{{\bar N}}{2}}}} \right)} \right) $ (2)

式中:rxp为垂直和水平极化天线单元之间的相关系数;Λg是由第g个群组中的空间协方差矩阵Rgs的非零特征值组成的rg×rg的对角矩阵,通常rgM$ {\mathit{\boldsymbol{U}}_g} \in {C^{\frac{M}{2} \times {r_g}}} $的列由Rgs的特征值组成。矩阵Gg定义为

$ {\mathit{\boldsymbol{G}}_g} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{G}}_g^{vv}}&{\mathit{\boldsymbol{G}}_g^{hv}}\\ {\mathit{\boldsymbol{G}}_g^{vh}}&{\mathit{\boldsymbol{G}}_g^{hh}} \end{array}} \right] $

式中$ G_{_g}^{^{pq}} \in {C^{{r_g} \times \frac{{\bar N}}{2}}} $p, q∈{h, v},Ggpq中的元素独立同分布且服从具有零均值和单位方差的复高斯分布。矩阵X描述了正交极化的功率失衡,定义为

$ \mathit{\boldsymbol{X}} = \left[ {\begin{array}{*{20}{c}} 1&{\sqrt \chi }\\ {\sqrt \chi }&1 \end{array}} \right] $ (3)

式中:参数χ为XPD的倒数,且1≤XPD≤∞,即0≤χ≤1。根据文献[9-10],rxp≈0,因此式(2)可化简为

$ {\mathit{\boldsymbol{H}}_g} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{H}}_g^{vv}}&{\mathit{\boldsymbol{H}}_g^{hv}}\\ {\mathit{\boldsymbol{H}}_g^{vh}}&{\mathit{\boldsymbol{H}}_g^{hh}} \end{array}} \right] $

Hg的协方差矩阵为

$ {\mathit{\boldsymbol{R}}_g} = {\mathit{\boldsymbol{R}}_{gv}} + {\mathit{\boldsymbol{R}}_{gh}} $

式中Rgv和分Rgh为垂直和水平共极化用户子群组的协方差矩阵。长期参数Rgsχ变化缓慢,因此本文假定它们均可在低反馈开销时准确地获得,而短期CSI参数Gg在短期相干时间内独立变化。基站获得反馈的CSI是非完美的,并且产生了巨大的开销。

3 预编码方案

BD双重结构预编码由2部分组成,分别为利用空间相关性的块对角化(block diagonalization,BD)预处理和对每个解耦群组进行的正则化迫零(regularized zero force,RZF)预编码。

基于长期/短期CSIT的双重预编码方案可以有效降低运算复杂度和反馈开销,其预编码矩阵为

$ {\mathit{\boldsymbol{V}}_g} = {\mathit{\boldsymbol{B}}_g} + {\mathit{\boldsymbol{P}}_g} $

式中$ {\mathit{\boldsymbol{B}}_g} \in {C^{M \times \bar B}}(\bar N \le \bar B \le 2{r_g} \le M) $为基于长期信道统计量的预处理矩阵。$ {\mathit{\boldsymbol{P}}_g} \in {C^{\bar B \times \bar N}} $为有效信道HgHBg的预编码矩阵。B为设计参数,用于决定利用长期CSIT的转换信道的维数。式(1)可写成

$ {\mathit{\boldsymbol{y}}_g} = \mathit{\boldsymbol{H}}_g^{\rm{H}}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{P}}_g}{\mathit{\boldsymbol{d}}_g} + \sum\limits_{l = 1,l \ne g}^G {\mathit{\boldsymbol{H}}_g^{\rm{H}}{\mathit{\boldsymbol{B}}_l}{\mathit{\boldsymbol{P}}_l}{\mathit{\boldsymbol{d}}_l}} + {\mathit{\boldsymbol{n}}_g} $ (4)

为了消除对于其他群组的干扰,基于空间相关性的BD预处理矩阵Bg应满足条件:

$ \mathit{\boldsymbol{H}}_l^{\rm{H}}{\mathit{\boldsymbol{B}}_g} \approx 0,l \ne g $ (5)

利用块对角化方法可以获得满足式(5)的预处理矩阵Bg。由式(3)的块对角化结构,首先定义

$ {\mathit{\boldsymbol{U}}_{ - g}} = \left[ {U_1^a,U_2^a, \cdots ,U_{g - 1}^a,U_{g + 1}^a, \cdots ,U_G^a} \right] \in {C^{\frac{M}{2} \times \sum\nolimits_{l \ne g} {r_l^a} }} $ (6)

式中$ \mathit{\boldsymbol{U}}_{_g}^{^a} = {\left[{{\mathit{\boldsymbol{U}}_g}} \right]_{1:r_{_g}^{^a}}} $rga(≤rg)为反映Rgs主导特征值的设计参数。rga越接近rg ,BD预处理越能使子空间与跨越其他群组信道的信号子空间正交,当rga=rg时完美正交,同时相应的正交子空间的维数随$ M/2-{\sum _{l \ne g}}r_{_l}^{^a} $而减小。

式(6)中的矩阵Ug进行奇异值分解后的形式为

$ {\mathit{\boldsymbol{U}}_{ - g}} = \left[ {\mathit{\boldsymbol{E}}_{ - g}^{\left( 1 \right)},\mathit{\boldsymbol{E}}_{ - g}^{\left( 0 \right)}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{ - g}^{\left( 1 \right)}}&{}\\ {}&{\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}_{ - g}^{\left( 0 \right)}} \end{array}} \right]\mathit{\boldsymbol{V}}_{ - g}^{\rm{H}} $
$ \mathit{\boldsymbol{E}}_{ - g}^{\left( 0 \right)} \in {C^{\frac{M}{2} \times \frac{M}{2} - \sum\nolimits_{l \ne g} {r_l^a} }} $

式中:$ \mathit{\boldsymbol{E}}_{_{-g}}^{^{(1)}} $是对应$ {\sum _{l \ne g}}r_{_l}^{^a} $的主导奇异值Λg(1)的左奇异值向量,Eg(0)是对应$ M/2-{\sum _{l \ne g}}r_{_l}^{^a} $的非主导奇异值Λg(0)的右奇异值向量。由于$ {\left( {\mathit{\boldsymbol{E}}_{_{-g}}^{^{(0)}}} \right)^{\rm{H}}}{\mathit{\boldsymbol{U}}_{-g}} = 0 $,通过定义$ {\mathit{\boldsymbol{\tilde H}}_g} = {({I_2} \otimes \mathit{\boldsymbol{E}}_{_{-g}}^{^{(0)}})^{\rm{H}}}{\mathit{\boldsymbol{H}}_g} $$ {\mathit{\boldsymbol{\tilde H}}_g}$可与被其他群组信道跨越的主导特征空间正交。$ {\mathit{\boldsymbol{\tilde H}}_g} $的协方差矩阵为

$ {{\mathit{\boldsymbol{\tilde R}}}_g} = {\left( {{\mathit{\boldsymbol{I}}_2} \otimes \mathit{\boldsymbol{E}}_{ - g}^{\left( 0 \right)}} \right)^{\rm{H}}}{\mathit{\boldsymbol{R}}_g}\left( {{\mathit{\boldsymbol{I}}_2} \otimes \mathit{\boldsymbol{E}}_{ - g}^{\left( 0 \right)}} \right) $

定义$ \mathit{\boldsymbol{\tilde R}}_{_g}^{^s}= {\left( {\mathit{\boldsymbol{E}}_{_{-g}}^{^{(0)}}} \right)^{\rm{H}}}\mathit{\boldsymbol{R}}_{_g}^{^s}\mathit{\boldsymbol{E}}_{_{-g}}^{^{(0)}} $,其特征值分为

$ \mathit{\boldsymbol{\tilde R}}_g^s = {\mathit{\boldsymbol{F}}_g}{{\mathit{\boldsymbol{ \boldsymbol{\tilde \varLambda} }}}_g}\mathit{\boldsymbol{F}}_g^{\rm{H}} $

式中Fg$ \mathit{\boldsymbol{\tilde R}}_{_g}^{^s} $的特征值向量。令$\mathit{\boldsymbol{F}}_{_g}^{^{(1)}} = {\left[{{\mathit{\boldsymbol{F}}_g}} \right]_{1:\frac{B}{2}}} $,则可得预处理矩阵为

$ {\mathit{\boldsymbol{B}}_g} = {\mathit{\boldsymbol{I}}_2} \otimes \mathit{\boldsymbol{B}}_g^s,\mathit{\boldsymbol{B}}_g^s = \mathit{\boldsymbol{E}}_{ - g}^{\left( 0 \right)}\mathit{\boldsymbol{F}}_g^{\left( 1 \right)} $

通过预处理矩阵Bg可以将第g个群组的发送信号转换为维数为B的主导特征空间,从而与被其他群组信道跨越的子空间正交。Brga的选取应满足条件$ \bar N \le \bar B \le 2\left( {M/2-{\sum _{l \ne g}}r_{_l}^{^a}} \right) $以及B≤2rg。不失一般性,假设$ r_{_1}^{^a} = r_{_2}^{^a} = \cdots = r_{_G}^{^a} = r $,且r为定值,以满足上述两个约束条件。接下来,式(4)中的Pg可以利用群组间干扰已经被消除的解耦系统$ {\mathit{\boldsymbol{y}}_g} \approx \mathit{\boldsymbol{H}}_{_g}^{\rm{H}}{\mathit{\boldsymbol{B}}_g}{\mathit{\boldsymbol{P}}_g}{\mathit{\boldsymbol{d}}_g} + {\mathit{\boldsymbol{n}}_g} $来计算获得。

由式(4),第g个群组的有效信道为$ {\mathit{\boldsymbol{\bar H}}_g} = \mathit{\boldsymbol{H}}_{_g}^{\rm{H}}{\mathit{\boldsymbol{B}}_g} $,对应的协方差矩阵为

$ \begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\bar R}}}_g} = \mathit{\boldsymbol{B}}_g^{\rm{H}}{\mathit{\boldsymbol{R}}_g}{\mathit{\boldsymbol{B}}_g} = \mathit{\boldsymbol{B}}_g^{\rm{H}}\left( {{\mathit{\boldsymbol{R}}_{gv}} + {\mathit{\boldsymbol{R}}_{gh}}} \right){\mathit{\boldsymbol{B}}_g} = }\\ {{{\mathit{\boldsymbol{\bar R}}}_{gv}} + {{\mathit{\boldsymbol{\bar R}}}_{gh}} = \left( {1 + \chi } \right)\left[ {\begin{array}{*{20}{c}} {{{\left( {\mathit{\boldsymbol{B}}_g^s} \right)}^{\rm{H}}}\mathit{\boldsymbol{R}}_g^s\mathit{\boldsymbol{B}}_g^s}&0\\ 0&{{{\left( {\mathit{\boldsymbol{B}}_g^s} \right)}^{\rm{H}}}\mathit{\boldsymbol{R}}_g^s\mathit{\boldsymbol{B}}_g^s} \end{array}} \right]} \end{array} $

通过前面的预处理,式(4)中来自其他群组的干扰几乎被完全消除。因此,接下来将对基于第g个群组短期CSIT的预编码矩阵Pg进行设计,以消除群组内的干扰。假设功率均等分配,具有非完美CSIT的RZF预编码矩阵公式为

$ {\mathit{\boldsymbol{P}}_g} = {\xi _g}{{\mathit{\boldsymbol{\hat {\bar K}}}}_g}{{\mathit{\boldsymbol{\hat {\bar H}}}}_g} $

式中:

$ {{\mathit{\boldsymbol{\hat {\bar K}}}}_g} = {\left( {{{\mathit{\boldsymbol{\hat {\bar H}}}}_g}\mathit{\boldsymbol{\hat {\bar H}}}_g^{\rm{H}} + \bar B\alpha I_B^ - } \right)^{ - 1}},{{\mathit{\boldsymbol{\hat {\bar H}}}}_g} = B_g^{\rm{H}}{{\mathit{\boldsymbol{\hat H}}}_g} $

式中:$ {{\mathit{\boldsymbol{\hat{\bar{H}}}}}_{g}} $为在基站端可用的有效信道估计,α为正则化参数。本文设定$ \alpha = \frac{N}{{\bar BP}} $,等效于MMSE线性滤波器。归一化因子ξg公式为

$ \begin{array}{*{20}{c}} {\xi _g^2 = \frac{{\bar N\frac{P}{N}}}{{\frac{P}{N}{\rm{tr}}\left( {\mathit{\boldsymbol{\hat {\bar H}}}_g^{\rm{H}}\mathit{\boldsymbol{\hat {\bar K}}}_g^{\rm{H}}\mathit{\boldsymbol{B}}_g^{\rm{H}}{\mathit{\boldsymbol{B}}_g}{{\mathit{\boldsymbol{\hat {\bar K}}}}_g}{{\mathit{\boldsymbol{\hat {\bar H}}}}_g}} \right)}} = }\\ {\frac{{\bar N}}{{{\rm{tr}}\left( {\mathit{\boldsymbol{\hat {\bar H}}}_g^{\rm{H}}\mathit{\boldsymbol{\hat {\bar K}}}_g^{\rm{H}}{{\mathit{\boldsymbol{\hat {\bar K}}}}_g}{{\mathit{\boldsymbol{\hat {\bar H}}}}_g}} \right)}}} \end{array} $

$ {{\hat{\bar{h}}}_{gk}}={{\left[{{{\mathit{\boldsymbol{\hat{\bar{H}}}}}}_{g}} \right]}_{k}} $表示第g个群组内第k个用户的有效信道估计,则该用户p极化的SINR表达式为

$ \begin{array}{*{20}{c}} {\gamma _{gpk}^{BD} = \frac{{\frac{P}{N}\xi _g^2{{\left| {\mathit{\boldsymbol{h}}_{gk}^{\rm{H}}{\mathit{\boldsymbol{B}}_g}{{\mathit{\boldsymbol{\hat {\bar K}}}}_g}{{\mathit{\boldsymbol{\hat {\bar h}}}}_{gk}}} \right|}^2}}}{{\frac{P}{N}\Delta + \frac{P}{N}\sum\nolimits_{l \ne g} {\sum\nolimits_j {\xi _l^2{{\left| {\mathit{\boldsymbol{h}}_{gk}^{\rm{H}}{\mathit{\boldsymbol{B}}_l}{{\mathit{\boldsymbol{\hat {\bar K}}}}_l}{{\mathit{\boldsymbol{\hat {\bar h}}}}_{lj}}} \right|}^2}} } + 1}} = }\\ {\frac{{\frac{P}{N}\xi _g^2{{\left| {\mathit{\boldsymbol{h}}_{gk}^{\rm{H}}{\mathit{\boldsymbol{B}}_g}{{\mathit{\boldsymbol{\hat {\bar K}}}}_g}\mathit{\boldsymbol{B}}_g^{\rm{H}}{{\mathit{\boldsymbol{\hat {\bar h}}}}_{gk}}} \right|}^2}}}{{\frac{P}{N}\Delta + \frac{P}{N}\sum\nolimits_{l \ne g} {\sum\nolimits_j {\xi _l^2{{\left| {\mathit{\boldsymbol{h}}_{gk}^{\rm{H}}{\mathit{\boldsymbol{B}}_l}{{\mathit{\boldsymbol{\hat {\bar K}}}}_l}\mathit{\boldsymbol{B}}_l^{\rm{H}}{{\mathit{\boldsymbol{\hat {\bar h}}}}_{lj}}} \right|}^2}} } + 1}}} \end{array} $

式中$ \Delta =\sum\nolimits_{j\ne k}{\xi _{g}^{2}}{{\left| \mathit{\boldsymbol{h}}_{gk}^{\rm{H}}{{\mathit{\boldsymbol{B}}}_{g}}{{{\mathit{\boldsymbol{\hat{\bar{K}}}}}}_{g}}{{{\mathit{\boldsymbol{\hat{\bar{h}}}}}}_{gk}} \right|}^{2}} $

和速率表达式为

$ {R_{BD}} = \sum\limits_{g = 1}^G {\sum\limits_{p \in \left\{ {v,h} \right\}} {\sum\limits_{k = 1}^{\underline 2 } {{{\log }_2}\left( {1 + \gamma _{gpk}^{BD}} \right)} } } $
4 仿真

为了评估BD双重结构预编码方案的性能,采用蒙特卡洛(monte-carlo,MC)方法进行仿真。同样将部署了双重极化线性阵列的基站天线个数设置为M=120,且N=32,G=4,N=8。对于预处理,设置B=min(2N, 2r),其中r$ \mathit{\boldsymbol{R}}_{g}^{s}\left( g=1, 2, \cdots, G \right) $的阶数中的最小值。假设当基站部署单极化线性阵列时,所有用户的天线单元与基站的天线单元均为共极化,这种情况为单极化线性阵列时的最佳场景。对于双极化线性阵列的情况,每个人群组中均有N/2个垂直天线单元和N/2水平天线单元。

图 1给出了单极化和双极化线性阵列的双重预编码的和速率曲线。从图中可以看出,在ds=λ0/2的相同条件下,双极化阵列的性能优于单极化阵列。如果减小单极化阵列中天线的距离,即ds=λ0/4的情况下,单极化阵列的性能反而更差。因此多极化天线可以成为针对大规模MIMO系统空间限制问题的解决方案。

Download:
图 1 单极化与多极化预编码方案性能比较

图 2给出了具有非完美CSIT,τ2=0.1的情况下,并且天线单元间不存在交叉极化现象,即χ=0的情况下,BD双重预编码与经典的ZF预编码、BD预编码随信噪比变化的和速率曲线。可以看出,BD是双重结构预编码方案可以获得更好的和速率性能,并且随着信噪比的增加,其展现的优势也越来越大。

Download:
图 2 BD双重结构预编码与经典预编码性能比较
5 结论

将基于空间相关的BD预处理与RZF预编码进行级联,设计了BD双重结构预编码方案,并对该方案的和速率进行推导。该预编码方案首先进行BD预处理,即利用波束成形,消除来自其他群组的干扰,然后利用RZF预编码技术,进一步消除群组内用户间的干扰。仿真结果表明,在FDD传输模式下,相比于经典的ZF预编码和MMSE预编码,BD双重结构预编码方案可以有效地减小反馈开销,获得更佳的和速率性能。

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