工程科学与技术   2018, Vol. 50 Issue (2): 112-117

An Improved Block Diagonalization Precoding Algorithm
GAO Ming, SUN Chengyue, LIN Shaoxing, WANG Yong
State Key Lab. of Integrated Services Networks,Xidian Univ.,Xi’an 710071,China
Abstract: In order to improve the performance of block diagonalization precoding algorithm,an improved algorithm was proposed for multiuser multiple-input multiple-output (MIMO) downlink systems,which employs QR decomposition based on Givens transformation.In the block diagonalization precoding algorithm,the first half of the precoding matrix is required to solve the problem of multi-user interference,and the latter one to reduce the interference between the antennas of each user.The core algorithm of the first half matrix is orthogonalization algorithm,which will directly affect the bit error rate (BER) performance of the block diagonalization precoding algorithm.Therefore,the original orthogonalization algorithm was replaced by QR decomposition algorithm based on Givens transformation.Using this algorithm to solve the first half of the precoding matrix,the orthogonal basis of the interference matrix’s zero space can be obtained.By using the Givens transformation,the matrix with better orthogonality can be obtained,so as to further reduce the BER of the system.The simulation results showed that compared with the traditional block diagonalization algorithms,the complexity of the system was decreased and the BER performance can be improvedgreatly.Compared with the block diagonalization algorithms based on Gram-Schmidt orthogonalization,the BER performance can be improved by 3~5 dB,with a slight increase in the algorithm complexity.
Key words: multiple-input multiple-output (MIMO)    precoding    block diagonalization    computational complexity    bit error rate (BER)

1 系统模型

 ${{{y}}_k} = {{{H}}_k}{{{W}}_k}{{{x}}_k} + \sum\limits_{i = 1,i \ne k}^K {{{{H}}_i}{{{W}}_i}{{{x}}_i}} + {{{n}}_k}$ (1)

 图1 多用户MIMO预编码系统模型 Fig. 1 Multiuser MIMO precoding system model

2 传统块对角化算法

2.1 求取传统块对角化算法的预编码矩阵 ${{{W}}^{{o}}}$

 ${{\widetilde{ H}}_k} = {[{{H}}_{{1}}^{{T}},{{H}}_{{2}}^{{T}}, \cdots \!,\;{{H}}_{k - 1}^{{T}},{{H}}_{k + 1}^{{T}}, \cdots\! ,\;{{H}}_K^{{T}}]^{{T}}}$ (2)

 ${{\widetilde{ H}}_k} = {{\widetilde{ U}}_k}\left[ {\begin{array}{*{20}{c}} {{{{\widetilde{ \varSigma }}}_k}} & 0 \\ 0 & 0 \end{array}} \right]{\left[ {{\widetilde{ V}}_k^{(1)},{\widetilde{ V}}_k^{(0)}} \right]^{{H}}}$ (3)

 ${{W}}_k^{{o}} = {\widetilde{ V}}_k^{(0)}$ (4)

 ${{W}}_k^{{o}} = [{{W}}_1^{{o}},{{W}}_2^{{o}}, \cdots ,{{W}}_K^{{o}}]$ (5)
2.2 求取传统块对角化算法的预编码矩阵 ${{{W}}^{{g}}}$

 ${{H}}_k^{{{eff}}} = {{{H}}_k}{{W}}_k^{{o}} = {{{U}}_k}{{{\varSigma }}_k}{[{{V}}_k^{(1)},{{V}}_k^{(0)}]^{{H}}}$ (6)

 ${{W}}_k^{{g}} = {{V}}_k^{(1)}$ (7)

 ${{{W}}^{{g}}} = {{diag}}\{ {{W}}_1^{{g}},{{W}}_2^{{g}}, \cdots ,{{W}}_K^{{g}}\}$ (8)

 ${{W}} = {{{W}}^{{o}}}{{{W}}^{{g}}}$ (9)

 ${{{B}}_k} = {{U}}_k^{{H}}$ (10)
3 改进块对角化算法

3.1 求取改进块对角化算法的预编码矩阵 ${{{W}}^{{o}}}$

 ${{{H}}^{{H}}} = {{QR}}$ (11)

 \begin{aligned}[b] {{{H}}^{\text{†}} } = & {{{H}}^{{H}}}{({{H}}{{{H}}^{{H}}})^{ - 1}} = {{QR}}{({{{R}}^{{H}}}{{{Q}}^{{H}}}{{QR}})^{ - 1}} = \\& {{QR}}{{{R}}^{ - 1}}{({{{R}}^{{H}}})^{ - 1}} = {{Q}}{({{{R}}^{{H}}})^{ - 1}}\end{aligned} (12)

${({{{R}}^{{H}}})^{ - 1}} = {{L}} = [{{{L}}_1},{{{L}}_2}, \cdots ,{{{L}}_K}]$ ，其中， ${{{L}}_k} \in {\mathbb{C}^{{N_{ R}} \times {N_k}}}$ ${{L}} \in {\mathbb{C}^{{N_{ R}} \times {N_{ R}}}}$ 中用户 $k$ 所对应的子矩阵。由于 ${{H}}{{{H}}^{\text{†}} } = {{I}}$ ，可以看出对于任意用户 $k$ ，都有 ${{\widetilde{ H}}_k}{{Q}}{{{L}}_k} = 0$ ，这表明矩阵 ${{Q}}{{{L}}_k}$ 位于信道干扰矩阵 ${{\widetilde{ H}}_k}$ 的零空间内。此时，还需保证 ${{Q}}{{{L}}_k}$ 为酉矩阵，以不改变总的发射功率。

 ${{Q}}{{{L}}_k} = {{\overline{ Q}}_k}{{\overline{ R}}_k}$ (13)

 ${{\widetilde{ H}}_k}{{Q}}{{{L}}_k} = {{\widetilde{ H}}_k}{{\overline{ Q}}_k}{{\overline{ R}}_k} = 0$ (14)

 ${{{W}}^{{o}}} = {{\overline{ Q}}_k}$ (15)

 ${{W}}_k^{{o}} = [{{W}}_1^{{o}},{{W}}_{{2}}^{{o}}, \cdots ,{{W}}_{{K}}^{{o}}]$ (16)

 ${{H}}{{{W}}^{{o}}} = {{diag}}\{ {{{H}}_1}{{W}}_{{1}}^{{o}},{{{H}}_1}{{W}}_2^{{o}}, \cdots ,{{{H}}_K}{{W}}_K^{{o}}\}$ (17)
3.2 求取改进块对角化算法的预编码矩阵 ${{{W}}^{{g}}}$

 ${{H}}_k^{{eff}} = {{{H}}_k}{{W}}_k^{{o}} = {{{U}}_k}{{{\varSigma }}_k}{[{{V}}_k^{(1)},{{V}}_k^{(0)}]^{{H}}}$ (18)

 ${{W}}_k^{{g}} = {{V}}_k^{(1)}$ (19)

 ${{{W}}^{{g}}} = { diag}\{ {{W}}_1^{{g}},{{W}}_2^{{g}}, \cdots ,{{W}}_K^{{g}}\}$ (20)

 ${{W}} = {{{W}}^{{o}}}{{{W}}^{{g}}}$ (21)

 ${{{B}}_k} = {{U}}_k^{{H}}$ (22)
4 性能分析与仿真 4.1 复杂度分析

1）与 $m \times p$ 维复矩阵相乘所需浮点运算数为 $8nmp - 2np$

2）矩阵奇异值分解所需浮点运算数为 $24n{m^2} +$ $48{n^2}m + 54{n^3}$

3）基于修正Gram-Schmidt正交化的QR分解所需浮点运算数为 $8{n^2}m$

4）基于Givens变换的QR分解所需浮点运算数为 $24{n^2}m - 8{n^3}$

5） $n \times n$ 维实矩阵使用高斯消元法求逆，所需浮点运算数为 $4{n^3}/3$

 \begin{aligned}[b]{\varPsi _{{{BD}}}} = & 24{K^2}{N_k}N_{{T}}^2 + (56{K^2} - 40K + 48)N_k^2{N_{{T}}} - \\& 2K{N_k}{N_{{T}}} + 54({K^3} - 3{K^2} + 4K - 1)N_k^3 - 2KN_k^2{{ = }}\\& O({K^2}{N_k}N_{{T}}^2)\end{aligned} (23)

 \begin{aligned} {\varPsi _{{{QR}} - {{GSO}} - {{BD}}}} = & 24K{N_k}N_{{T}}^2 + (24{K^2} + 56K)N_k^2{N_{{T}}} - 4K{N_k}{N_{{T}}} + \\& (4{K^3}/3 \!+\! 8{K^2} \!+\! 54K)N_k^3 - 2KN_k^2 \!=\! O(K{N_k}N_{{T}}^2)\end{aligned} (24)

 \begin{aligned}[b]{\varPsi _{{{QR}} \!-\! {{Givens}} \!-\! {{BD}}}} \!= & 24K{N_k}N_{{T}}^2 \!+\! (24{K^2} + 56K)N_k^2{N_{{T}}} - 4K{N_k}{N_{{T}}} + \\& (4{K^3}/3 \!+\! 24{K^2} \!+\! 48K)N_k^3 \!-\! 2KN_k^2 \!=\! O(K{N_k}N_{{T}}^2)\end{aligned} (25)

 图2 BD与QR-Givens-BD复杂度比较 Fig. 2 Comparison of complexity between BD and QR-Givens-BD

 图3 QR-GSO-BD与QR-Givens-BD复杂度比较 Fig. 3 Comparison of complexity between QR-GSO-BD and QR-Givens-BD

4.2 误码性能仿真

 图4 误码率仿真结果（ ${{{N}}_{{T}}}= 6,{{{N}}_{k}} = 2, {{K}}= 3$ ） Fig. 4 Simulation of bit error rate ( ${{{N}}_{{T}}}= 6,{{{N}}_{k}} = 2, {{K}}= 3$ )

 图5 误码率仿真结果（ ${{{N}}_{{T}}}= 18,{{{N}}_{k}} = 3,{{K}}= 6$ ） Fig. 5 Simulation of bit error rate ( ${{{N}}_{{T}}}= 18,{{{N}}_{k}} = 3,{{K}}= 6$ )

5 结　论

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