CN105577249A - A Precoding Method for MIMO Relay System with Channel Estimation Error and Antenna Correlation - Google Patents

Abstract

The invention discloses a pre-coding method of a MIMO relay system having a channel estimation error and antenna correlation. Under a condition that both the power of a transmitting end and the power of a relay end are limited, closed-form solutions of a pre-coding matrix of the transmitting end, a pre-coding matrix of the relay end and a processing matrix of a receiving end are obtained by derivation in a manner of taking a minimum mean square error as a criterion; therefore, the system performance can be improved well; a joint iterative algorithm for calculating the pre-coding matrix of the transmitting end, the pre-coding matrix of the relay end and the processing matrix of the receiving end is given; the joint iterative algorithm has good convergence and practical value and is easy to implement; the MIMO relay system is combined with problems of the channel estimation error and the antenna correlation; the condition that incomplete channel state information exists in a practical condition is considered; the pre-coding method of the MIMO relay system under the condition of the channel estimation error and the antenna correlation is implemented to improve the performance of the MIMO relay system based on the incomplete channel state information condition; and thus, the pre-coding method has wide application prospect.

Description

Translated fromChinese

一种存在信道估计误差及天线相关的MIMO中继系统的预编码方法A Precoding Method for MIMO Relay System with Channel Estimation Error and Antenna Correlation

技术领域：Technical field:

本发明属于无线通信领域，涉及MIMO中继系统的预编码方法，更具体的涉及存在信道估计误差及天线相关的MIMO中继系统的预编码方法。The invention belongs to the field of wireless communication, relates to a precoding method of a MIMO relay system, and more specifically relates to a precoding method of a MIMO relay system with channel estimation error and antenna correlation.

背景技术：Background technique:

随着用户对各种实时多媒体业务需求的增加，特别是21世纪以来互联网技术的迅猛发展，传统的单天线传输技术已经无法满足无线业务的要求。多输入多输出(Multiple-InputMultiple-Output，MIMO)技术极大地提高了通信系统的频率效率并改善了通信链路的可靠性，已成为无线通信领域一种关键的核心技术。中继技术能够有效扩大通信网络覆盖范围、提高通信系统容量，将中继引入MIMO通信系统，可带来容量增益和覆盖面扩展等优势。With the increase of users' demand for various real-time multimedia services, especially the rapid development of Internet technology since the 21st century, the traditional single-antenna transmission technology has been unable to meet the requirements of wireless services. Multiple-Input Multiple-Output (MIMO) technology greatly improves the frequency efficiency of communication systems and improves the reliability of communication links, and has become a key core technology in the field of wireless communication. The relay technology can effectively expand the coverage of the communication network and improve the capacity of the communication system. The introduction of the relay into the MIMO communication system can bring advantages such as capacity gain and coverage expansion.

实际MIMO中继系统中，不可避免地会存在信道估计误差和天线相关性。因此，在信道中考虑信道估计误差和天线相关性的情况，对于改善通信系统的性能会有很大的帮助。近年来，关于MIMO中继的研究层出不穷，但大多都是基于完全信道的中继结构，而对于信道中存在的信道估计误差和天线相关性的相关MIMO中继系统的研究还处于甚少阶段。In practical MIMO relay systems, channel estimation errors and antenna correlations inevitably exist. Therefore, considering the channel estimation error and antenna correlation in the channel will be of great help to improve the performance of the communication system. In recent years, researches on MIMO relays emerge in an endless stream, but most of them are based on the complete channel relay structure, while the research on MIMO relay systems related to the channel estimation error and antenna correlation existing in the channel is still in a very small stage.

发明内容：Invention content:

本发明为了解决现有技术存在的不足，提供一种存在信道估计误差及天线相关的MIMO中继系统的预编码方法，与传统预编码方法相比，本发明的方法能进一步改善MIMO中继系统的误码性能。In order to solve the deficiencies in the prior art, the present invention provides a precoding method for a MIMO relay system with channel estimation error and antenna correlation. Compared with the traditional precoding method, the method of the present invention can further improve the MIMO relay system error performance.

本发明采用如下技术方案：一种存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其包括如下步骤：The present invention adopts the following technical scheme: a precoding method for a MIMO relay system with channel estimation error and antenna correlation, which includes the following steps:

第一步：针对MIMO中继系统，构建存在信道估计误差及天线相关的信道模型，假设发射-中继端及中继-接收端信道均存在信道估计误差及天线相关，用分别表示发射-中继端和中继-接收端信道矩阵，n_s，n_r，n_d分别是发射端，中继端，接收端的天线数并且满足n_s≤min(n_r,n_d)条件；Step 1: For the MIMO relay system, construct a channel model with channel estimation error and antenna correlation, assuming that there are channel estimation errors and antenna correlation in both the transmitter-relay and relay-receiver channels, using Represent the transmitter-relay end and relay-receiver channel matrix respectively, n_s , n_r , n_d are the number of antennas of the transmitter, relay end, and receiver end respectively and satisfy n_s ≤min(n_r ,n_d ) condition;

第二步：符号子流经发射端预编码矩阵处理后转发给中继，其中发射端发送信号须满足发射端功率约束，发送信号经过发射-中继端信道可得中继接收信号Step 2: Symbolic Subflow Transmitter precoding matrix After processing, it is forwarded to the relay, where the signal sent by the transmitter must meet the power constraints of the transmitter, and the signal sent by the transmitter can be received by the relay through the channel of the transmitter-relay

第三步：接收信号y_r经中继端预编码矩阵处理得到并转发给接收端，其中转发信号Y满足中继功率约束，转发信号经过中继-接收端信道，得到接收端接收信号接收端通过接收端处理矩阵得到还原信号Step 3: The received signal y_r is precoded by the relay terminal dealt with And forward it to the receiving end, where the forwarding signal Y satisfies the relay power constraint, and the forwarding signal passes through the relay-receiving end channel to obtain the receiving end receiving signal The receiving end processes the matrix through the receiving end get back signal

第四步：以最小均方误差为设计准则，比较发送信号x与接收端还原信号构建MSE代价函数以此实时迭代更新发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵，最终得到三者的最优解以改善系统的误比特率。Step 4: Using the minimum mean square error as the design criterion, compare the transmitted signal x with the restored signal at the receiving end Build the MSE cost function In this way, the precoding matrix at the transmitting end, the precoding matrix at the relay end, and the processing matrix at the receiving end are iteratively updated in real time, and finally the optimal solution of the three is obtained to improve the bit error rate of the system.

进一步地，所述第一步构建信道估计误差及天线相关的信道模型包括：Further, the first step of constructing a channel estimation error and an antenna-related channel model includes:

H和G的天线相关矩阵用Kronecker模型表示，定义和和Ψ_G是发射天线相关矩阵，Σ_H和Σ_G是H和G的接收天线相关矩阵，天线相关矩阵是半正定及完全已知的，和的元素服从均值为0方差为1的复高斯分布，定义和和是和的估计矩阵，ΔH和ΔG是信道估计误差矩阵，其元素服从均值为0方差分别为和的复高斯分布，因此，信道模型可以表示为：The antenna correlation matrices of H and G are represented by the Kronecker model, defining and and Ψ_G are the transmitting antenna correlation matrices, Σ_H and Σ_G are the receiving antenna correlation matrices of H and G, the antenna correlation matrices are positive semi-definite and fully known, and The elements of obey the complex Gaussian distribution with a mean of 0 and a variance of 1, defined and and yes and The estimation matrix of , ΔH and ΔG are channel estimation error matrices, and its elements obey the mean value of 0 and the variance is respectively and The complex Gaussian distribution of , therefore, the channel model can be expressed as:

$\mathrm{<span><span>H</span>h</span>}<span><span>=</span>=</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>\mathrm{<span><span>\&Delta;</span>\&Delta;</span>}\mathrm{<span><span>H</span>h</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;H\&Psi;</span>\&Delta;H\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}$

$\mathrm{<span><span>G</span>G</span>}<span><span>=</span>=</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>\mathrm{<span><span>\&Delta;</span>\&Delta;</span>}\mathrm{<span><span>G</span>G</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}$

进一步地，所述第二步信号经过发射端预编码并发送至中继端采用如下公式得到：Further, the second step signal is precoded by the transmitting end and sent to the relay end using the following formula to obtain:

发射端节点发射信息x_i(i＝1,2,…,n_s)至中继端，令ε[·]表示期望；中继端接收信号可以表示为：The transmitting end node transmits information x_i (i=1,2,…,n_s ) to the relay end, so that ε[·] means expectation; the relay terminal receives the signal It can be expressed as:

y_r＝HBx+wy_r =HBx+w

其中为信号x的预编码矩阵，满足tr(Bxx^HB^H)＝tr(BB^H)≤P_s，tr[·]表示矩阵的迹，P_s是发射端最大功率，是发射端至中继端的MIMO信道矩阵，w是中继端的加性高斯白噪声，均值为零，方差矩阵为为其噪声功率，中继端通过处理矩阵对接收信号进行处理，然后向接收端转发信号，转发信号为：in is the precoding matrix of signal x, satisfying tr(Bxx^H B^H )=tr(BB^H )≤P_s , tr[ ] represents the trace of the matrix, P_s is the maximum power of the transmitting end, is the MIMO channel matrix from the transmitting end to the relay end, w is the additive white Gaussian noise of the relay end, the mean value is zero, and the variance matrix is For its noise power, the relay end processes the matrix Process the received signal, then forward the signal to the receiving end, and forward the signal for:

Y＝Fy_rY=Fy_r

其中功率约束满足tr(YY^H)≤P_r，P_r是中继端最大功率。The power constraint satisfies tr(YY^H )≤P_r , where P_r is the maximum power of the relay.

进一步地，所述第三步中继转发和接收端还原发射信号是根据以下公式得到：Further, the third step of relay forwarding and receiving end restoration of the transmitted signal is obtained according to the following formula:

接收端接收信号可以表示为：The receiving end receives the signal It can be expressed as:

r＝GY+n＝GFHS+GFw+nr=GY+n=GFHS+GFw+n

其中是均值为零，方差矩阵为的加性高斯白噪声，是n的方差；in is the mean is zero, and the variance matrix is The additive white Gaussian noise of is the variance of n;

接收端的处理矩阵用来恢复出发射端的发射信号，令接收处理矩阵为则估计信号矢量为：The processing matrix at the receiving end is used to restore the transmitted signal at the transmitting end, so that the receiving processing matrix is Then estimate the signal vector for:

$\begin{array}{c}\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>r</span>r</span>}<span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>S</span>S</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}\\ <span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>a</span>a</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}\end{array}<span><span>.</span>.</span>$

进一步地，所述第四步求取发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵最优解的处理方法是根据以下公式得到：Further, in the fourth step, the processing method for obtaining the optimal solution of the transmitting end precoding matrix, the relay end precoding matrix, and the receiving end processing matrix is obtained according to the following formula:

(1).以最小均方误差为设计准则，求取MSE代价函数：(1). Taking the minimum mean square error as the design criterion, find the MSE cost function:

$\begin{array}{c}\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>F</span>f</span>}<span><span>,</span>,</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>x</span>x</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>F</span>f</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;H\&Psi;</span>\&Delta;H\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>x</span>x</span>}\\ <span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>+</span>+</span>{\mathrm{<span><span>QR</span>QR</span>}}_{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\\ <span><span>+</span>+</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\\ <span><span>-</span>-</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>B</span>B</span>}<span><span>+</span>+</span>{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>\end{array}$

其中$Z=\&lsqb;\stackrel{\&OverBar;}{H}{\mathrm{BB}}^H{\stackrel{\&OverBar;}{H}}^H+{\mathrm{\&sigma;}}_H^{2}tr\left({\mathrm{BB}}^H{\mathrm{\&Psi;}}_H\right){\mathrm{\&Sigma;}}_H+R_w\&rsqb;.$in$Z=\&lsqb;\stackrel{\&OverBar;}{h}{\mathrm{BB}}^h{\stackrel{\&OverBar;}{h}}^h+{\mathrm{\&sigma;}}_h^{2}tr\left({\mathrm{BB}}^h{\mathrm{\&Psi;}}_h\right){\mathrm{\&Sigma;}}_h+R_w\&rsqb;.$

功率约束为：The power constraints are:

tr(BB^H)≤P_str(BB^H )≤P_s

tr(YY^H)＝tr[F(HBB^HH^H+R_w)F^H]tr(YY^H )=tr[F(HBB^H H^H +R_w )F^H ]

＝tr[FZF^H]≤P_r=tr[FZF^H ]≤P_r

(2).由于发射端与中继端需满足功率约束，则对于最小化代价函数MSE的最优化问题可以表示为：(2). Since the transmitting end and the relay end need to satisfy power constraints, the optimization problem for minimizing the cost function MSE can be expressed as:

$\begin{array}{c}\mathrm{<span><span>arg</span>arg</span>}\underset{\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>F</span>f</span>}<span><span>,</span>,</span>\mathrm{<span><span>Q</span>Q</span>}}{\mathrm{<span><span>min</span>min</span>}}<span><span>\{</span>\{</span>\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>\}</span>\}</span>\\ \begin{array}{cc}\mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}& \mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>BB</span>BB</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}\end{array}\\ \begin{array}{cc}& \mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>r</span>r</span>}}\end{array}\end{array}$

本发明具有如下有益效果：The present invention has following beneficial effect:

(1)本发明提出了一种适用于MIMO中继系统的预编码方法，在发射端和中继端功率都受限条件下，以最小均方误差MMSE为准则，推导得到了发射端预编码矩阵、中继端预编码矩阵和接收端处理矩阵的闭式解，使其能极好地提高系统性能；(1) The present invention proposes a precoding method suitable for MIMO relay systems. Under the condition that the power of both the transmitting end and the relay end is limited, and the minimum mean square error MMSE is used as the criterion, the precoding of the transmitting end is derived. The closed-form solution of the matrix, the relay-end precoding matrix and the receiving-end processing matrix can greatly improve the system performance;

(2)给出了计算发射端预编码矩阵、中继端预编码矩阵和接收端处理矩阵的联合迭代算法，此迭代算法具有良好的收敛性，易于实现，具有很好的实用价值；(2) A joint iterative algorithm for calculating the precoding matrix at the transmitting end, the precoding matrix at the relay end and the processing matrix at the receiving end is given. This iterative algorithm has good convergence, is easy to implement, and has good practical value;

(3)本发明将MIMO中继系统与信道存在估计误差及天线相关性的问题相结合，考虑了实际情况中存在的不完全信道状态信息的情况，具有良好的实用性，因此，基于信道存在估计误差及天线相关性条件下的MIMO中继系统预编码方法的实施在基于不完全信道状态信息条件下提升MIMO中继系统的性能有着广泛的应用前景。(3) The present invention combines the MIMO relay system with the problems of channel estimation error and antenna correlation, and considers the situation of incomplete channel state information existing in the actual situation, which has good practicability. Therefore, based on channel existence The implementation of MIMO relay system precoding method under the condition of estimation error and antenna correlation has a broad application prospect in improving the performance of MIMO relay system based on incomplete channel state information.

附图说明：Description of drawings:

图1为本发明中的MIMO中继系统的原理图。FIG. 1 is a schematic diagram of a MIMO relay system in the present invention.

图2是在图1所示的MIMO中继系统中采用本发明的方法进行信号发送的示意图。FIG. 2 is a schematic diagram of signal transmission using the method of the present invention in the MIMO relay system shown in FIG. 1 .

图3给出了SNR_sr＝SNR_rd时基于不同信道估计误差的MIMO中继系统采用联合迭代设计法与其他设计方法的误比特率比较图。Fig. 3 shows the bit error rate comparison diagram of the joint iterative design method and other design methods for the MIMO relay system based on different channel estimation errors when SNR_sr =SNR_rd .

图4给出了SNR_sr＝SNR_rd时基于不同天线相关性的MIMO中继系统采用联合迭代设计法与其他设计方法的误比特率比较图。Fig. 4 shows the bit error rate comparison diagram of the joint iterative design method and other design methods for the MIMO relay system based on different antenna correlations when SNR_sr =SNR_rd .

具体实施方式：detailed description:

本发明是针对存在信道估计误差及天线相关的MIMO中继系统，研究发射端与中继端的预编码设计方法。目的是通过考虑实际情况中信道存在估计误差及天线相关的问题来得到更为优化的系统误比特率性能。The invention aims at the MIMO relay system with channel estimation error and antenna correlation, and studies the precoding design method of the transmitting end and the relay end. The purpose is to obtain a more optimized system bit error rate performance by considering channel estimation errors and antenna-related problems in actual situations.

为了使本发明的原理更加清楚，首先对本发明采用的MIMO中继系统的工作原理进行简单介绍。MIMO中继系统模型如图1所示，它由发射端、中继端和接收端三个部分组成，其中发射端、中继端和接收端分别有n_s，n_r，n_d根天线，并且满足n_s≤min(n_r,n_d)条件。结合图2信号发送的原理图，发射端、中继端及接收端分别配置4根天线，发射信息为随机生成的QPSK调制符号，为降低中继工作的复杂度，中继传输采用半双工方式，一次传输由2个时隙组成，假设信道为平坦瑞利衰落，信道状态信息在一次传输的2个时隙内保持不变。In order to make the principle of the present invention clearer, firstly, the working principle of the MIMO relay system adopted in the present invention is briefly introduced. The MIMO relay system model is shown in Figure 1. It consists of three parts: the transmitter, the relay and the receiver. The transmitter, the relay and the receiver have n_s , n_r , and n_d antennas respectively. And satisfy the condition of n_s ≤ min(n_r , n_d ). Combined with the schematic diagram of signal transmission in Figure 2, the transmitting end, the relay end and the receiving end are respectively equipped with 4 antennas, and the transmission information is randomly generated QPSK modulation symbols. In order to reduce the complexity of the relay work, the relay transmission adopts half-duplex In this way, a transmission consists of 2 time slots, assuming that the channel is flat Rayleigh fading, and the channel state information remains unchanged within 2 time slots of a transmission.

本发明采用如下技术方案，一种存在信道估计误差及天线相关的MIMO中继系统的预编码方法，具体步骤为：The present invention adopts the following technical scheme, a precoding method for a MIMO relay system with channel estimation error and antenna correlation, and the specific steps are:

第一步：针对MIMO中继系统，构建一种存在信道估计误差及天线相关的信道模型。本发明假设发射-中继端及中继-接收端信道均存在天线相关及信道估计误差。用分别表示发射-中继端和中继-接收端信道矩阵。Step 1: Aiming at the MIMO relay system, construct a channel model with channel estimation error and antenna correlation. The present invention assumes that there are antenna correlation and channel estimation errors in both the transmitting-relay-end and relay-receiving-end channels. use Denote the transmitter-relay and relay-receiver channel matrices, respectively.

第二步：符号子流经发射端预编码矩阵处理后转发给中继，其中发射端发送信号须满足发射端功率约束。发送信号经过发射-中继端信道可得中继接收信号Step 2: Symbolic Subflow Transmitter precoding matrix After processing, it is forwarded to the relay, and the signal sent by the transmitter must meet the power constraint of the transmitter. Send the signal through the transmitter-relay channel to get the relay receiving signal

第三步：接收信号y_r经中继端预编码矩阵处理得到并转发给接收端。其中转发信号Y满足中继功率约束。转发信号经过中继-接收端信道，得到接收端接收信号接收端通过接收端处理矩阵得到还原信号Step 3: The received signal y_r is precoded by the relay terminal dealt with and forwarded to the receiver. The forwarded signal Y satisfies the relay power constraint. The forwarded signal passes through the relay-receiver channel to obtain the signal received by the receiver The receiving end processes the matrix through the receiving end get back signal

第四步：以最小均方误差(MinimumMeanSquaredError,MMSE)为设计准则，比较发送信号x与接收端还原信号构建MSE代价函数以此迭代更新发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵，最终得到三者的最优解，以此有效地改善系统的BER。Step 4: Using Minimum Mean Squared Error (MMSE) as the design criterion, compare the transmitted signal x with the restored signal at the receiving end Build the MSE cost function In this way, the precoding matrix at the transmitting end, the precoding matrix at the relay end, and the processing matrix at the receiving end are updated iteratively, and finally the optimal solution of the three is obtained, thereby effectively improving the BER of the system.

其中第一步构建信道估计误差及天线相关的信道模型包括：用分别表示发射-中继端和中继-接收端信道矩阵，n_s，n_r，n_d分别是发射端，中继端，接收端的天线数并且满足n_s≤min(n_r,n_d)条件。H和G的天线相关矩阵常用Kronecker模型表示，定义和Ψ_H和Ψ_G是发射天线相关矩阵，Σ_H和Σ_G是H和G的接收天线相关矩阵，天线相关矩阵采用指数相关模型，天线相关矩阵是半正定及完全已知的。Σ_H的元素为Σ_H(m,n)＝ρ^|m-n|，1≤m,n≤n_r，ρ表示相关系数，ρ＝0.4。同理，Ψ_H，Σ_G和Ψ_G采用相同的相关系数。和的元素服从均值为0方差为1的复高斯分布。然而,在实际通信系统中，很难获得完全CSI，故定义和和是和的估计矩阵，ΔH和ΔG是估计误差矩阵，其元素服从均值为0方差分别为和的复高斯分布，因此，信道模型可以表示为:The first step is to construct the channel estimation error and antenna-related channel model including: Represent the transmitter-relay terminal and relay-receiver channel matrix respectively, n_s , n_r , n_d are the antenna numbers of the transmitter, relay terminal, and receiver respectively and satisfy n_s ≤min(n_r ,n_d ) condition. The antenna correlation matrices of H and G are usually represented by the Kronecker model, defined as and Ψ_H and Ψ_G are the transmitting antenna correlation matrices, Σ_H and Σ_G are the receiving antenna correlation matrices of H and G, the antenna correlation matrix adopts an exponential correlation model, and the antenna correlation matrix is positive semi-definite and fully known. The elements of Σ_H are Σ_H (m,n)=ρ^|mn| , 1≤m,n≤n_r , ρ represents the correlation coefficient, and ρ=0.4. Similarly, Ψ_H , Σ_G and Ψ_G use the same correlation coefficient. and The elements of obey a complex Gaussian distribution with mean 0 and variance 1. However, in actual communication systems, it is difficult to obtain complete CSI, so the definition and and yes and The estimated matrix of , ΔH and ΔG are estimated error matrices, whose elements obey the mean value of 0 and the variance are respectively and The complex Gaussian distribution of , Therefore, the channel model can be expressed as:

$\mathrm{<span><span>H</span>h</span>}<span><span>=</span>=</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>\mathrm{<span><span>\&Delta;</span>\&Delta;</span>}\mathrm{<span><span>H</span>h</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;H\&Psi;</span>\&Delta;H\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span>$

$\mathrm{<span><span>G</span>G</span>}<span><span>=</span>=</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>(</span>(</span>\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>\mathrm{<span><span>\&Delta;</span>\&Delta;</span>}\mathrm{<span><span>G</span>G</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>2</span>2</span>}<span><span>)</span>)</span><span><span>.</span>.</span>$

其中第二步信号经过发射端预编码并发送至中继端采用如下公式得到：In the second step, the signal is precoded by the transmitting end and sent to the relay end using the following formula:

发射端节点发射信息x_i(i＝1,2,…,n_s)至中继端，令ε[·]表示期望；中继端接收信号可以表示为：The transmitting end node transmits information x_i (i=1,2,…,n_s ) to the relay end, so that ε[·] means expectation; the relay terminal receives the signal It can be expressed as:

y_r＝HBx+w(3)y_r =HBx+w(3)

其中为信号x的预编码矩阵，满足tr(Bxx^HB^H)＝tr(BB^H)≤P_s，tr[·]表示矩阵的迹，P_s是发射端最大功率。是发射端至中继端的MIMO信道矩阵，w是中继端的加性高斯白噪声，均值为零，方差矩阵为为其噪声功率。in is the precoding matrix of signal x, satisfying tr(Bxx^H B^H )=tr(BB^H )≤P_s , tr[·] represents the trace of the matrix, and P_s is the maximum power of the transmitting end. is the MIMO channel matrix from the transmitting end to the relay end, w is the additive white Gaussian noise of the relay end, the mean value is zero, and the variance matrix is for its noise power.

中继端通过预编码矩阵对接收信号进行处理，然后向接收端转发信号，转发信号为：The relay end passes the precoding matrix Process the received signal, then forward the signal to the receiving end, and forward the signal for:

Y＝Fy_r(4)Y=Fy_r (4)

其中功率约束满足tr(YY^H)≤P_r，P_r是中继端最大功率。The power constraint satisfies tr(YY^H )≤P_r , where P_r is the maximum power of the relay.

其中第三步中继转发和接收端还原发射信号是根据以下公式得到：In the third step, the relay forwarding and the receiving end restoring the transmitted signal are obtained according to the following formula:

令为中继端至接收端的MIMO信道矩阵，接收端接收信号可以表示为：make is the MIMO channel matrix from the relay end to the receiving end, and the receiving end receives the signal It can be expressed as:

r＝GY+n＝GFHS+GFw+n(5)r=GY+n=GFHS+GFw+n (5)

其中是均值为零，方差矩阵为的加性高斯白噪声(AWGN)，是n的方差。in is the mean is zero, and the variance matrix is Additive White Gaussian Noise (AWGN) of is the variance of n.

接收端的处理矩阵用来恢复出发射端的发射信号，令接收处理矩阵为则估计信号矢量为：The processing matrix at the receiving end is used to restore the transmitted signal at the transmitting end, so that the receiving processing matrix is Then estimate the signal vector for:

$\begin{array}{c}\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>r</span>r</span>}<span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>S</span>S</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>W</span>W</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}\\ <span><span>=</span>=</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>a</span>a</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>6</span>6</span>}<span><span>)</span>)</span>$

所述第四步求取发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵最优解的处理方法是根据以下公式得到：The processing method for obtaining the optimal solution of the transmitting end precoding matrix, the relay end precoding matrix, and the receiving end processing matrix in the fourth step is obtained according to the following formula:

以MMSE为设计准则，求取MSE代价函数：Taking MMSE as the design criterion, the MSE cost function is obtained:

$\begin{array}{c}\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>F</span>f</span>}<span><span>,</span>,</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\stackrel{<span><span>~</span>~</span>}{\mathrm{<span><span>x</span>x</span>}}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>H</span>h</span>}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>x</span>x</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>G</span>G</span>}\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}<span><span>(</span>(</span><span><span>|</span>|</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>F</span>f</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;H\&Psi;</span>\&Delta;H\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>x</span>x</span>}\\ <span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>(</span>(</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>\&Delta;G\&Psi;</span>\&Delta;G\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\mathrm{<span><span>F</span>f</span>}\mathrm{<span><span>w</span>w</span>}<span><span>+</span>+</span>\mathrm{<span><span>Q</span>Q</span>}\mathrm{<span><span>n</span>no</span>}<span><span>-</span>-</span>\mathrm{<span><span>x</span>x</span>}{<span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}<span><span>)</span>)</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>+</span>+</span>{\mathrm{<span><span>QR</span>QR</span>}}_{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\\ <span><span>+</span>+</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\\ <span><span>-</span>-</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>B</span>B</span>}<span><span>+</span>+</span>{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>7</span>7</span>}<span><span>)</span>)</span>$

其中$Z=\&lsqb;\stackrel{\&OverBar;}{H}{\mathrm{BB}}^H{\stackrel{\&OverBar;}{H}}^H+{\mathrm{\&sigma;}}_H^{2}tr\left({\mathrm{BB}}^H{\mathrm{\&Psi;}}_H\right){\mathrm{\&Sigma;}}_H+R_w\&rsqb;.$in$Z=\&lsqb;\stackrel{\&OverBar;}{h}{\mathrm{BB}}^h{\stackrel{\&OverBar;}{h}}^h+{\mathrm{\&sigma;}}_h^{2}tr\left({\mathrm{BB}}^h{\mathrm{\&Psi;}}_h\right){\mathrm{\&Sigma;}}_h+R_w\&rsqb;.$

功率约束为The power constraint is

tr(BB^H)≤P_str(BB^H )≤P_s

tr(YY^H)＝tr[F(HBB^HH^H+R_w)F^H](8)tr(YY^H )=tr[F(HBB^H H^H +R_w )F^H ](8)

＝tr[FZF^H]≤P_r=tr[FZF^H ]≤P_r

由于发射端与中继端需满足功率约束，则对于最小化代价函数MSE的最优化问题可以表示为Since the transmitting end and the relay end need to satisfy the power constraints, the optimization problem for minimizing the cost function MSE can be expressed as

$\begin{array}{c}\mathrm{<span><span>arg</span>arg</span>}\underset{\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>F</span>f</span>}<span><span>,</span>,</span>\mathrm{<span><span>Q</span>Q</span>}}{\mathrm{<span><span>min</span>min</span>}}<span><span>\{</span>\{</span>\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>\}</span>\}</span>\\ \begin{array}{cc}\mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}& \mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>BB</span>BB</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}\end{array}\\ \begin{array}{cc}& \mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>r</span>r</span>}}\end{array}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>9</span>9</span>}<span><span>)</span>)</span>$

其中第四步求取基站预编码矩阵、中继线性处理矩阵、终端解码矩阵最优解的步骤为：The fourth step is to obtain the optimal solution of the base station precoding matrix, the relay linear processing matrix, and the terminal decoding matrix:

(1).发射端预编码矩阵：(1). Transmitter precoding matrix:

固定F和Q最优化B，将公式转化为：Fixing F and Q optimizes B, transforming the formula into:

$\begin{array}{c}\underset{\mathrm{<span><span>B</span>B</span>}}{\mathrm{<span><span>min</span>min</span>}}{\mathrm{<span><span>b</span>b</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}<span><span>\&times;</span>\&times;</span>\mathrm{<span><span>R</span>R</span>}<span><span>(</span>(</span>{{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>D</span>D.</span>}}_{\mathrm{<span><span>1</span>1</span>}}\\ \begin{array}{cc}\mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}& {\mathrm{<span><span>b</span>b</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}<span><span>\&times;</span>\&times;</span>\mathrm{<span><span>R</span>R</span>}<span><span>(</span>(</span>{{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>2</span>2</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>D</span>D.</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>\&le;</span>\&le;</span>\mathrm{<span><span>0</span>0</span>}\end{array}\\ \begin{array}{cc}& {\mathrm{<span><span>b</span>b</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>3</span>3</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}<span><span>\&times;</span>\&times;</span>\mathrm{<span><span>R</span>R</span>}<span><span>(</span>(</span>{{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>3</span>3</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>D</span>D.</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>\&le;</span>\&le;</span>\mathrm{<span><span>0</span>0</span>}\end{array}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>10</span>10</span>}<span><span>)</span>)</span>$

参数定义为：The parameters are defined as:

$\mathrm{<span><span>b</span>b</span>}<span><span>=</span>=</span>\mathrm{<span><span>v</span>v</span>}\mathrm{<span><span>e</span>e</span>}\mathrm{<span><span>c</span>c</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>11</span>11</span>}<span><span>)</span>)</span>$

$\begin{array}{c}{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>\\ <span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>\\ <span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>12</span>12</span>}<span><span>)</span>)</span>$

${\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>=</span>=</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>\mathrm{<span><span>I</span>I</span>}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>13</span>13</span>}<span><span>)</span>)</span>$

${\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>3</span>3</span>}}<span><span>=</span>=</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>H</span>h</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>F</span>f</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>F\&Sigma;</span>F\&Sigma;</span>}}_{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>(</span>(</span>\mathrm{<span><span>I</span>I</span>}<span><span>\&CircleTimes;</span>\&CircleTimes;</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>14</span>14</span>}<span><span>)</span>)</span>$

${\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>=</span>=</span>\mathrm{<span><span>v</span>v</span>}\mathrm{<span><span>e</span>e</span>}\mathrm{<span><span>c</span>c</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{<span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>F</span>f</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>15</span>15</span>}<span><span>)</span>)</span>$

C₂＝0(16)C₂ =0(16)

C₃＝0(17)C₃ =0(17)

D₁＝0(18)D₁ =0(18)

D₂＝-P_s(19)D₂ =-P_s (19)

D₃＝-P_r+tr(FR_wF^H)(20)D₃ ＝-P_r +tr(FR_w F^H )(20)

引入变量t,该问题可以转化为半正定规划问题：Introducing the variable t, the problem can be transformed into a semi-positive definite programming problem:

$\begin{array}{c}\begin{array}{cc}\underset{\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>t</span>t</span>}}{\mathrm{<span><span>min</span>min</span>}}& \mathrm{<span><span>t</span>t</span>}\end{array}\\ \begin{array}{cc}\mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}<span><span>.</span>.</span>& \left[\begin{array}{cc}\mathrm{<span><span>I</span>I</span>}& {{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>b</span>b</span>}\\ {<span><span>(</span>(</span>{{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span>}^{\mathrm{<span><span>H</span>h</span>}}& \mathrm{<span><span>t</span>t</span>}<span><span>+</span>+</span>\mathrm{<span><span>2</span>2</span>}<span><span>\&times;</span>\&times;</span>\mathrm{<span><span>R</span>R</span>}<span><span>(</span>(</span>{{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>D</span>D.</span>}}_{\mathrm{<span><span>1</span>1</span>}}\end{array}\right]<span><span>\&GreaterEqual;</span>\&Greater\; Equal;</span>\mathrm{<span><span>0</span>0</span>}\end{array}\\ \begin{array}{cc}& \left[\begin{array}{cc}\mathrm{<span><span>I</span>I</span>}& {{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>i</span>i</span>}}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>b</span>b</span>}\\ {<span><span>(</span>(</span>{{\mathrm{<span><span>A</span>A</span>}}_{\mathrm{<span><span>1</span>1</span>}}}^{\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span>\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span>}^{\mathrm{<span><span>H</span>h</span>}}& \mathrm{<span><span>2</span>2</span>}<span><span>\&times;</span>\&times;</span>\mathrm{<span><span>R</span>R</span>}<span><span>(</span>(</span>{{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>i</span>i</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>b</span>b</span>}<span><span>)</span>)</span><span><span>-</span>-</span>{\mathrm{<span><span>D</span>D.</span>}}_{\mathrm{<span><span>i</span>i</span>}}\end{array}\right]<span><span>\&GreaterEqual;</span>\&Greater\; Equal;</span>\mathrm{<span><span>0</span>0</span>}<span><span>,</span>,</span>\mathrm{<span><span>i</span>i</span>}<span><span>=</span>=</span>\mathrm{<span><span>2</span>2</span>}<span><span>,</span>,</span>\mathrm{<span><span>3</span>3</span>}\end{array}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>21</span>twenty\; one</span>}<span><span>)</span>)</span>$

该问题可以通过内点法或者MATLABCVX工具箱解决。This problem can be solved by interior point method or MATLABCVX toolbox.

(2).中继预编码矩阵F(2). Relay precoding matrix F

将最优化问题(9)转化为固定B和Q求解F的子问题：Transform the optimization problem (9) into a subproblem of solving F with fixed B and Q:

$\begin{array}{c}\mathrm{<span><span>arg</span>arg</span>}\underset{\mathrm{<span><span>F</span>f</span>}}{\mathrm{<span><span>min</span>min</span>}}<span><span>\{</span>\{</span>\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>\}</span>\}</span>\\ \begin{array}{cc}\mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}& \mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>FZF</span>FZF</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span><span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>r</span>r</span>}}\end{array}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>22</span>twenty\; two</span>}<span><span>)</span>)</span>$

很容易证明该问题是凸优化问题。因此，F可以用KKT算法求得，公式(23)的拉格朗日函数为：It is easy to show that the problem is a convex optimization problem. Therefore, F can be obtained by the KKT algorithm, and the Lagrangian function of formula (23) is:

L(F,λ)＝MSE(B,F,Q)+λ(tr[FZF^H]-P_r)(23)L(F,λ)=MSE(B,F,Q)+λ(tr[FZF^H ]-P_r )(23)

其中，λ是拉格朗日乘子。通过对L(F,λ)求导，并要满足如下功率约束：where λ is the Lagrangian multiplier. By taking the derivative of L(F,λ), and satisfying the following power constraints:

$\frac{<span><span>\&part;</span>\&part;</span>\mathrm{<span><span>L</span>L</span>}<span><span>(</span>(</span>\mathrm{<span><span>F</span>f</span>}<span><span>,</span>,</span>\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}<span><span>)</span>)</span>}{<span><span>\&part;</span>\&part;</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}}{<span><span>|</span>|</span>}_{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>F</span>f</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>F</span>f</span>}}^{<span><span>*</span>*</span>}}<span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>24</span>twenty\; four</span>}<span><span>)</span>)</span>$

tr[FZF^H]-P_r＜0(25)tr[FZF^H ]-P_r <0(25)

λ＞0(26)λ>0(26)

λ(tr[FZF^H]-P_r)＝0(27)λ(tr[FZF^H ]-P_r )=0(27)

因此，可得中继预编码矩阵F：Therefore, the relay precoding matrix F can be obtained:

$\begin{array}{c}\mathrm{<span><span>F</span>f</span>}<span><span>=</span>=</span>{<span><span>(</span>(</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&lambda;I</span>\&lambda;\; I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>)</span>)</span>}^{<span><span>+</span>+</span>}\\ {\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>28</span>28</span>}<span><span>)</span>)</span>$

最优化λ必须满足和，λ的约束条件为式中最优的λ可以采用二分法求解。Optimizing λ must satisfy and, and the constraint condition of λ is The optimal λ in the formula can be solved by the dichotomy method.

为了简化最优化问题，在接收端处理矩阵中引入线性系数η＞0，并将Q替换为η^-1Q。经过一系列计算，可得：In order to simplify the optimization problem, a linear coefficient η>0 is introduced in the receiving end processing matrix, and Q is replaced by η^-1 Q. After a series of calculations, we can get:

$\begin{array}{c}\mathrm{<span><span>F</span>f</span>}<span><span>=</span>=</span>\mathrm{<span><span>\&eta;</span>\&eta;</span>}{<span><span>(</span>(</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>)</span>)</span>}^{<span><span>+</span>+</span>}\\ {\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>29</span>29</span>}<span><span>)</span>)</span>$

令make

$\mathrm{<span><span>\&theta;</span>\&theta;</span>}<span><span>=</span>=</span>\frac{\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>QR</span>QR</span>}}_{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>}{{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>30</span>30</span>}<span><span>)</span>)</span>$

$\begin{array}{c}{\mathrm{<span><span>MSE</span>MSE</span>}}_{\mathrm{<span><span>min</span>min</span>}}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>,</span>,</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>)</span>)</span><span><span>-</span>-</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\\ {<span><span>(</span>(</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>)</span>)</span>}^{<span><span>+</span>+</span>}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>31</span>31</span>}<span><span>)</span>)</span>$

(3).接收端处理矩阵Q(3). Receiver processing matrix Q

将最优化问题(9)转化为固定B和F求解Q的子问题：Transform the optimization problem (9) into a subproblem for solving Q with fixed B and F:

$\begin{array}{c}\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>)</span>)</span><span><span>-</span>-</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>B</span>B</span>}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}\\ {<span><span>(</span>(</span>{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&psi;</span>\&psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;\; I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>)</span>)</span>}^{<span><span>+</span>+</span>}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\\ <span><span>=</span>=</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span><span><span>+</span>+</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span><span><span>-</span>-</span>\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>\&lsqb;</span>\&lsqb;</span>{\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>\&rsqb;</span>\&rsqb;</span>\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>32</span>32</span>}<span><span>)</span>)</span>$

定义：definition:

${\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>H</span>h</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>-</span>-</span>{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>B</span>B</span>}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>33</span>33</span>}<span><span>)</span>)</span>$

$\begin{array}{c}{\mathrm{<span><span>\&beta;</span>\&beta;</span>}}_{\mathrm{<span><span>n</span>no</span>}}<span><span>=</span>=</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}\\ <span><span>=</span>=</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}\\ <span><span>\&le;</span>\&le;</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}\mathrm{<span><span>t</span>t</span>}\mathrm{<span><span>r</span>r</span>}<span><span>(</span>(</span>{\mathrm{<span><span>Q</span>Q</span>}}^{\mathrm{<span><span>H</span>h</span>}}\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}_{\mathrm{<span><span>max</span>max</span>}}<span><span>(</span>(</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>\&theta;I</span>\&theta;I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}\\ <span><span>=</span>=</span><span><span>(</span>(</span>{\mathrm{<span><span>\&sigma;</span>\&sigma;</span>}}_{\mathrm{<span><span>G</span>G</span>}}^{\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>P</span>P</span>}}_{\mathrm{<span><span>r</span>r</span>}}{\mathrm{<span><span>\&lambda;</span>\&lambda;</span>}}_{\mathrm{<span><span>max</span>max</span>}}<span><span>(</span>(</span>{<span><span>\&Sigma;</span>\&Sigma;</span>}_{\mathrm{<span><span>G</span>G</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>R</span>R</span>}}_{\mathrm{<span><span>n</span>no</span>}}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>\&Psi;</span>\&Psi;</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>r</span>r</span>}}}<span><span>)</span>)</span>{\mathrm{<span><span>\&theta;</span>\&theta;</span>}}_{\mathrm{<span><span>n</span>no</span>}}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>34</span>34</span>}<span><span>)</span>)</span>$

${\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>G</span>G</span>}}<span><span>=</span>=</span>{<span><span>(</span>(</span>{\mathrm{<span><span>I</span>I</span>}}_{{\mathrm{<span><span>n</span>no</span>}}_{\mathrm{<span><span>s</span>the\; s</span>}}}<span><span>+</span>+</span>{\mathrm{<span><span>Q</span>Q</span>}}_{\mathrm{<span><span>n</span>no</span>}}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}{{\mathrm{<span><span>\&beta;</span>\&beta;</span>}}_{\mathrm{<span><span>n</span>no</span>}}}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{{\mathrm{<span><span>Q</span>Q</span>}}_{\mathrm{<span><span>n</span>no</span>}}}^{\mathrm{<span><span>H</span>h</span>}}<span><span>)</span>)</span>}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>35</span>35</span>}<span><span>)</span>)</span>$

该无约束最优化问题可采用梯度线性搜索法解决This unconstrained optimization problem can be solved using the gradient linear search method

$\begin{array}{c}{\mathrm{<span><span>\&Delta;Q</span>\&Delta;Q</span>}}_{\mathrm{<span><span>n</span>no</span>}}<span><span>=</span>=</span><span><span>-</span>-</span>{<span><span>\&dtri;</span>\&dtri;</span>}_{{\mathrm{<span><span>Q</span>Q</span>}}^{<span><span>*</span>*</span>}}\mathrm{<span><span>M</span>m</span>}\mathrm{<span><span>S</span>S</span>}\mathrm{<span><span>E</span>E.</span>}<span><span>(</span>(</span>\mathrm{<span><span>Q</span>Q</span>}<span><span>)</span>)</span>\\ <span><span>=</span>=</span><span><span>-</span>-</span>{{\mathrm{<span><span>\&eta;</span>\&eta;</span>}}_{\mathrm{<span><span>n</span>no</span>}}}^{<span><span>-</span>-</span>\mathrm{<span><span>2</span>2</span>}}{\mathrm{<span><span>QR</span>QR</span>}}_{\mathrm{<span><span>n</span>no</span>}}<span><span>+</span>+</span>{\mathrm{<span><span>E</span>E.</span>}}_{\mathrm{<span><span>G</span>G</span>}}{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>H</span>h</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}}^{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>Z</span>Z</span>}}^{<span><span>+</span>+</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>H</span>h</span>}}{\mathrm{<span><span>BE</span>BE</span>}}_{\mathrm{<span><span>G</span>G</span>}}\mathrm{<span><span>Q</span>Q</span>}\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}{{\mathrm{<span><span>\&beta;</span>\&beta;</span>}}_{\mathrm{<span><span>n</span>no</span>}}}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>G</span>G</span>}}}^{\mathrm{<span><span>H</span>h</span>}}\end{array}<span><span>-</span>-</span><span><span>-</span>-</span><span><span>-</span>-</span><span><span>(</span>(</span>\mathrm{<span><span>36</span>36</span>}<span><span>)</span>)</span>$

对应求解过程如算法1所示：The corresponding solution process is shown in Algorithm 1:

(4).联合迭代设计(4). Joint iterative design

该算法如下：The algorithm is as follows:

下表为本实施例采用的仿真条件。The following table shows the simulation conditions used in this embodiment.

为了验证本文提出的联合迭代算法的优越性，将该方法与其他方法进行对比，仿真中对比的方法为：In order to verify the superiority of the joint iterative algorithm proposed in this paper, this method is compared with other methods. The methods compared in the simulation are:

迭代B和F预编码方案:Q∝I. Iterative B and F precoding scheme: Q∝I.

迭代F和Q预编码方案: Iterative F and Q precoding schemes:

联合迭代预编码方案 Joint Iterative Precoding Scheme

图3和图4分别给出了SNR_sr＝SNR_rd时基于不同估计误差及不同天线相关性下MIMO中继系统的误比特率比较图，从仿真结果看出，在低信噪比时，迭代F和Q方法的性能要优于迭代B和F方法，而随着信噪比的增大，联合迭代法显著好于其他两种方法，并保持2-4dB的信噪比增益，由此可知本文方案确实可以获得更低的BER，验证了所提算法的有效性和优越性。Figure 3 and Figure 4 respectively show the bit error rate comparison diagrams of the MIMO relay system based on different estimation errors and different antenna correlations when SNR_sr = SNR_rd . It can be seen from the simulation results that when the SNR is low, the iterative The performance of the F and Q methods is better than that of the iterative B and F methods, and as the SNR increases, the joint iterative method is significantly better than the other two methods, and maintains a 2-4dB SNR gain. It can be seen that The scheme in this paper can indeed obtain a lower BER, which verifies the effectiveness and superiority of the proposed algorithm.

以上所述仅是本发明的优选实施方式，应当指出，对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下还可以作出若干改进，这些改进也应视为本发明的保护范围。The above is only a preferred embodiment of the present invention, it should be pointed out that for those of ordinary skill in the art, some improvements can also be made without departing from the principle of the present invention, and these improvements should also be regarded as the invention. protected range.

Claims (5)

Translated fromChinese

1.一种存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其特征在于：包括如下步骤1. There is a precoding method of channel estimation error and antenna related MIMO relay system, it is characterized in that: comprise the following steps第一步：针对MIMO中继系统，构建存在信道估计误差及天线相关的信道模型，假设发射-中继端及中继-接收端信道均存在信道估计误差及天线相关，用分别表示发射-中继端和中继-接收端信道矩阵，n_s，n_r，n_d分别是发射端，中继端，接收端的天线数并且满足n_s≤min(n_r,n_d)条件；Step 1: For the MIMO relay system, construct a channel model with channel estimation error and antenna correlation, assuming that there are channel estimation errors and antenna correlation in both the transmitter-relay and relay-receiver channels, using Represent the transmitter-relay end and relay-receiver channel matrix respectively, n_s , n_r , n_d are the number of antennas of the transmitter, relay end, and receiver end respectively and satisfy n_s ≤min(n_r ,n_d ) condition;第二步：符号子流经发射端预编码矩阵处理后转发给中继，其中发射端发送信号须满足发射端功率约束，发送信号经过发射-中继端信道可得中继接收信号Step 2: Symbolic Subflow Transmitter precoding matrix After processing, it is forwarded to the relay, where the signal sent by the transmitter must meet the power constraints of the transmitter, and the signal sent by the transmitter can be received by the relay through the transmitter-relay channel第三步：接收信号y_r经中继端预编码矩阵处理得到并转发给接收端，其中转发信号Y满足中继功率约束，转发信号经过中继-接收端信道，得到接收端接收信号接收端通过接收端处理矩阵得到还原信号Step 3: The received signal y_r is precoded by the relay terminal dealt with And forward it to the receiving end, where the forwarding signal Y satisfies the relay power constraint, and the forwarding signal passes through the relay-receiving end channel to obtain the receiving end receiving signal The receiving end processes the matrix through the receiving end get back signal第四步：以最小均方误差为设计准则，比较发送信号x与接收端还原信号构建MSE代价函数以此实时迭代更新发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵，最终得到三者的最优解以改善系统的误比特率。Step 4: Using the minimum mean square error as the design criterion, compare the transmitted signal x with the restored signal at the receiving end Build the MSE cost function In this way, the precoding matrix at the transmitting end, the precoding matrix at the relay end, and the processing matrix at the receiving end are iteratively updated in real time, and finally the optimal solution of the three is obtained to improve the bit error rate of the system.2.如权利要求1所述的存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其特征在于：所述第一步构建信道估计误差及天线相关的信道模型包括：2. the precoding method of the MIMO relay system that exists channel estimation error and antenna correlation as claimed in claim 1, it is characterized in that: described first step constructs channel estimation error and antenna correlation channel model and comprises:H和G的天线相关矩阵用Kronecker模型表示，定义和Ψ_H和Ψ_G是发射天线相关矩阵，Σ_H和Σ_G是H和G的接收天线相关矩阵，天线相关矩阵是半正定及完全已知的，和的元素服从均值为0方差为1的复高斯分布，定义和和是和的估计矩阵，ΔH和ΔG是信道估计误差矩阵，其元素服从均值为0方差分别为和的复高斯分布，因此，信道模型可以表示为：The antenna correlation matrices of H and G are represented by the Kronecker model, defining and Ψ_H and Ψ_G are the transmitting antenna correlation matrices, Σ_H and Σ_G are the receiving antenna correlation matrices of H and G, the antenna correlation matrices are positive semi-definite and fully known, and The elements of obey the complex Gaussian distribution with a mean of 0 and a variance of 1, defined and and yes and The estimation matrix of , ΔH and ΔG are channel estimation error matrices, and its elements obey the mean value of 0 and the variance is respectively and The complex Gaussian distribution of , therefore, the channel model can be expressed as:3.如权利要求1所述的存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其特征在于：所述第二步信号经过发射端预编码并发送至中继端采用如下公式得到：3. The precoding method of channel estimation error and antenna-related MIMO relay system as claimed in claim 1, wherein the second step signal is precoded by the transmitting end and sent to the relay end using the following formula get:发射端节点发射信息x_i(i＝1,2,…,n_s)至中继端，令ε[·]表示期望；中继端接收信号可以表示为：The transmitting end node transmits information x_i (i=1,2,…,n_s ) to the relay end, so that ε[·] means expectation; the relay terminal receives the signal It can be expressed as:y_r＝HBx+wy_r =HBx+w其中为信号x的预编码矩阵，满足tr(Bxx^HB^H)＝tr(BB^H)≤_sP，tr[·]表示矩阵的迹，P_s是发射端最大功率，是发射端至中继端的MIMO信道矩阵，w是中继端的加性高斯白噪声，均值为零，方差矩阵为为其噪声功率，中继端通过处理矩阵对接收信号进行处理，然后向接收端转发信号，转发信号为：in is the precoding matrix of signal x, satisfying tr(Bxx^H B^H )=tr(BB^H )≤_s P, tr[ ] represents the trace of the matrix, P_s is the maximum power of the transmitter, is the MIMO channel matrix from the transmitting end to the relay end, w is the additive white Gaussian noise of the relay end, the mean value is zero, and the variance matrix is For its noise power, the relay end processes the matrix Process the received signal, then forward the signal to the receiving end, and forward the signal for:Y＝Fy_rY=Fy_r其中功率约束满足tr(YY^H)≤P_r，P_r是中继端最大功率。The power constraint satisfies tr(YY^H )≤P_r , where P_r is the maximum power of the relay.4.如权利要求1所述的存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其特征在于：所述第三步中继转发和接收端还原发射信号是根据以下公式得到：4. the precoding method of the MIMO relay system that exists channel estimation error and antenna correlation as claimed in claim 1, it is characterized in that: described 3rd step relay forwards and receiving end restores transmission signal and obtains according to the following formula:接收端接收信号可以表示为：The receiving end receives the signal It can be expressed as:r＝GY+n＝GFHS+GFw+nr=GY+n=GFHS+GFw+n其中是均值为零，方差矩阵为的加性高斯白噪声，是n的方差；in is the mean is zero, and the variance matrix is The additive white Gaussian noise of is the variance of n;接收端的处理矩阵用来恢复出发射端的发射信号，令接收处理矩阵为则估计信号矢量为：The processing matrix at the receiving end is used to restore the transmitted signal at the transmitting end, so that the receiving processing matrix is Then estimate the signal vector for:。 .5.如权利要求1所述的存在信道估计误差及天线相关的MIMO中继系统的预编码方法，其特征在于：所述第四步求取发射端预编码矩阵、中继端预编码矩阵、接收端处理矩阵最优解的处理方法是根据以下公式得到：5. the precoding method that there is channel estimation error and antenna correlation MIMO relay system as claimed in claim 1, it is characterized in that: described 4th step asks for transmitting terminal precoding matrix, relay terminal precoding matrix, The processing method of the receiving end to process the optimal solution of the matrix is obtained according to the following formula:(1).以最小均方误差为设计准则，求取MSE代价函数：(1). Taking the minimum mean square error as the design criterion, find the MSE cost function:其中in功率约束为：The power constraints are:tr(BB^H)≤P_str(BB^H )≤P_str(YY^H)＝tr[F(HBB^HH^H+R_w)F^H]tr(YY^H )=tr[F(HBB^H H^H +R_w )F^H ]＝tr[FZF^H]≤P_r=tr[FZF^H ]≤P_r(2).由于发射端与中继端需满足功率约束，则对于最小化代价函数MSE的最优化问题可以表示为：(2). Since the transmitting end and the relay end need to satisfy power constraints, the optimization problem for minimizing the cost function MSE can be expressed as:s.ttr(BB^H)≤P_s。s.ttr(BB^H )≤P_s .tr[FZF^H]≤P_r。tr[FZF^H ]≤P_r .