A Hybrid Beamforming Algorithm Based on Limited-Broyden-Fletcher-Goldfarb-Shanno
-
摘要: 针对现有混合波束成形算法运行时间长、频谱效率低、误码率高的问题，该文提出一种基于有限内存拟牛顿法的混合波束成形算法(LBFGS)。该算法首先通过数字预编码器的最小二乘解构建单变量目标函数；然后采用目标函数的梯度近似黑塞矩阵的逆得到搜索方向并沿搜索方向更新模拟预编码器，直到满足停止条件；最后固定模拟预编码器得到数字预编码器。MATLAB仿真结果表明，LBFGS算法较现有MO算法减少了28%的运行时间，频谱效率提高了1.05%，误码率降低了1.06%。Abstract: To solve the problems of long runtime, low spectral rate and high bit error rate, which exist in conventional hybrid beamforming schemes, a hybrid beamforming algorithm based on Limited-Broyden-Fletcher-Goldfarb-Shanno (LBFGS) is proposed. Firstly, a single variable objective function is constructed through the least squares solution of the digital precoder. Then, the gradient of the objective function is used to approximate the inverse of the Hessian matrix for obtaining the search direction and the analog precoder is updated along the search direction until the stop condition is satisfied. Finally, the analog precoder is fixed to obtain the digital precoder. The MATLAB simulation analysis indicate that LBFGS algorithm reduces the running time by 28%, increases spectral rate by 1.05%, and reduces bit error rate by 1.06%, compared to MO algorithm.
-
Key words:
- Millimeter wave /
- Massive MIMO /
- Hybrid beamforming /
- Overlapped antenna arrays
-
算法1 计算搜索方向 输入：${\gamma _0}{\text{ = }}1$，${\mathrm{grad}}{{\boldsymbol{\mathcal{J}}}_{{{\boldsymbol{v}}_k}}}$，已存储的梯度向量
$ \{ {\boldsymbol{s}}_i^{(k)},{\boldsymbol{y}}_i^{(k)}\} _{i = k - m}^{k - 1} $输出： 搜索方向${{\boldsymbol{\eta}} _k}$ ${\boldsymbol{\mathcal{H}}}_k^0 = {\gamma _k}$； $ {\boldsymbol{d}} \leftarrow {\mathrm{grad}}{{\boldsymbol{\mathcal{J}}}_{{v_k}}} $； If $m \ne 0$ for $i = k - 1,k - 2, \cdots ,k - m$do ${\boldsymbol{\alpha}} \leftarrow {\rho _i} < {\boldsymbol{s}}_i^{(k)},{\boldsymbol{d}} > $ ${\boldsymbol{d}} \leftarrow {\boldsymbol{d}} - {\alpha _i}{\boldsymbol{y}}_i^{(k)}$ end for ${\boldsymbol{e}} \leftarrow {\boldsymbol{\mathcal{H}}}_k^0{\boldsymbol{d}}$ for $i = k - m,k - m + 1, \cdots ,k - 1$ $\beta \leftarrow {\rho _i} < {\boldsymbol{y}}_i^{(k)},{\boldsymbol{e}} > $ $ {\boldsymbol{e}} \leftarrow {\boldsymbol{e}} + {\boldsymbol{s}}_i^{(k)} < {\varepsilon _i} - \beta > $ end for ${{\boldsymbol{\eta}} _k} = - {\boldsymbol{e}}$ else ${{\boldsymbol{\eta}} _k} = - {\boldsymbol{\mathcal{H}}}_k^0{\boldsymbol{d}}$ end if 
下载: 导出CSV
算法2 基于有限内存拟牛顿法的模拟预编码器算法 输入：最优全数字矩阵${{\boldsymbol{V}}_{\rm{opt}}}$，初始模拟波束成形矩阵${\boldsymbol{V}}_{\rm{RF}}^0$，内存
容量$\forall M \in \mathbb{Z}$且$M > 0$。输出：${{\boldsymbol{V}}_{\rm{RF}}}$ 初始化：内部循环次数$k = 0$，内存占用量$m = 0$ 根据式(10)计算黎曼梯度${\mathrm{grad}}{{\boldsymbol{\mathcal{J}}}_{{{\boldsymbol{v}}_0}}}$ while梯度的范数达到阈值${\mu _2}$ 根据算法1计算搜索方向${{\boldsymbol{\eta}} _k}$ 线搜索并回缩得到${{\boldsymbol{v}}_{k + 1}}{\text{ = }}{{{R}}_{{{\boldsymbol{v}}_k}}}({\alpha _k}{{\boldsymbol{\eta}} _k})$ 计算黎曼梯度${\mathrm{grad}}{{\boldsymbol{\mathcal{J}}}_{{{\boldsymbol{v}}_{k + 1}}}}$ 计算$ {\boldsymbol{s}}_k^{(k + 1)} $，$ {\boldsymbol{y}}_k^{(k + 1)} $，${\rho _{k + 1}} = 1/ < {\boldsymbol{s}}_k^{(k + 1)},{\boldsymbol{y}}_k^{(k + 1)} > $ if 满足存储条件 计算${\gamma _{k + 1}} = < {\boldsymbol{s}}_k^{(k + 1)},{\boldsymbol{y}}_k^{(k + 1)} > /||{\boldsymbol{y}}_k^{(k + 1)}|{|^2}$ if溢出 丢弃$ {\boldsymbol{s}}_{k - M}^{(k)} $，$ {\boldsymbol{y}}_{k - M}^{(k)} $； end if 历史梯度向量传输 存储$ {\text{\{ }}{\boldsymbol{s}}_k^{(k + 1)},{\boldsymbol{y}}_k^{(k + 1)}{\text{\} }} $ 若$m < M$，那么$m = m + 1$，否则$m = M$ else ${\gamma _{k + 1}} = {\gamma _k}$； end if $k \leftarrow k + 1$ end
下载: 导出CSV
-
[1] SAAD W, BENNIS M, and CHEN Mingzhe. A vision of 6G wireless systems: Applications, trends, technologies, and open research problems[J]. IEEE Network, 2020, 34(3): 134–142. doi: 10.1109/MNET.001.1900287. [2] ZHANG Jun, YU Xianghao, and LETAIEF K B. Hybrid beamforming for 5G and beyond millimeter-wave systems: A holistic view[J]. IEEE Open Journal of the Communications Society, 2020, 1: 77–91. doi: 10.1109/OJCOMS.2019.2959595. [3] ABDULLAH Q, SALH A, MOHDSHAH N S, et al. A brief survey and investigation of hybrid beamforming for millimeter waves in 5G massive MIMO systems[J]. arXiv: 2105.00180, 2021. doi: 10.48550/arXiv.2105.00180. [4] RIHAN M, SOLIMAN T A, XU Chen, et al. Taxonomy and performance evaluation of hybrid beamforming for 5G and beyond systems[J]. IEEE Access, 2020, 8: 74605–74626. doi: 10.1109/ACCESS.2020.2984548. [5] UWAECHIA A N and MAHYUDDIN N M. A comprehensive survey on millimeter wave communications for fifth-generation wireless networks: feasibility and challenges[J]. IEEE Access, 2020, 8: 62367–62414. doi: 10.1109/ACCESS.2020.2984204. [6] ULLAH M S, SARKER S C, ASHRAF Z B, et al. Spectral efficiency of multiuser massive MIMO-OFDM THz wireless systems with hybrid beamforming under inter-carrier interference[C]. Proceedings of the 12th International Conference on Electrical and Computer Engineering (ICECE), Dhaka, Bangladesh, 2022: 228–231. doi: 10.1109/ICECE57408.2022.10089136. [7] LIU Ying and WANG Jintao. Low-complexity OFDM-based hybrid precoding for multiuser massive MIMO systems[J]. IEEE Wireless Communications Letters, 2020, 9(3): 263–266. doi: 10.1109/lwc.2019.2929518. [8] ZHAO Xingyu, LIN Tian, ZHU Yu, et al. Partially-connected hybrid beamforming for spectral efficiency maximization via a weighted MMSE equivalence[J]. IEEE Transactions on Wireless Communications, 2021, 20(12): 8218–8232. doi: 10.1109/TWC.2021.3091524. [9] 赵峰, 何晓华. 混合动态连接结构的多用户多流混合预编码[J]. 电子与信息学报, 2021, 43(9): 2647–2653. doi: 10.11999/JEIT 200441.ZHAO Feng and HE Xiaohua. Multi-user multi-stream hybrid precoding with hybrid dynamic connection structure[J]. Journal of Electronics & Information Technology, 2021, 43(9): 2647–2653. doi: 10.11999/JEIT 200441. [10] AYACH O E, RAJAGOPAL S, ABU-SURRA S, et al. Spatially sparse precoding in millimeter wave MIMO systems[J]. IEEE Transactions on Wireless Communications, 2014, 13(3): 1499–1513. doi: 10.1109/TWC.2014.011714.130846. [11] YU Xianghao, SHEN J C, ZHANG Jun, et al. Alternating minimization algorithms for hybrid precoding in millimeter wave MIMO systems[J]. IEEE Journal of Selected Topics in Signal Processing, 2016, 10(3): 485–500. doi: 10.1109/JSTSP.2016.2523903. [12] JIN Juening, ZHENG Y R, CHEN Wen, et al. Hybrid precoding for millimeter wave MIMO systems: A matrix factorization approach[J]. IEEE Transactions on Wireless Communications, 2018, 17(5): 3327–3339. doi: 10.48550/arXiv.2003.11972. [13] LI Jinmeng, CHENG Zhiqun, and LI Hang. Hybrid precoding scheme in millimeter wave massive MIMO based on stochastic gradient descent[C]. Proceedings of 2021 IEEE Asia Conference on Information Engineering (ACIE), Sanya, China, 2021: 22–26. doi: 10.1109/ACIE51979.2021.9381072. [14] 刘文龙, 黄雯静, 王本巍, 等. 毫米波单用户多天线系统的混合预编码设计[J]. 电子与信息学报, 2022, 44(2): 620–626. doi: 10.11999/JEIT202019.LIU Wenlong, HUANG Wenjing, WANG Benwei, et al. Hybrid precoding design in millimeter wave single-user large-scale multiple-input multiple-output system[J]. Journal of Electronics & Information Technology, 2022, 44(2): 620–626. doi: 10.11999/JEIT202019. [15] HUANG Wen. Optimization algorithms on Riemannian manifolds with applications[D]. [Ph. D. dissertation], Florida State University, 2013. [16] ZANG Guangda, HU Lingna, YANG Feng, et al. Partially-connected hybrid beamforming for multi-user massive MIMO systems[J]. IEEE Access, 2020, 8: 215287–215298. doi: 10.1109/ACCESS.2020.3040508. [17] SONG Nuan, YANG Tao, and SUN Huan. Overlapped subarray based hybrid beamforming for millimeter wave multiuser massive MIMO[J]. IEEE Signal Processing Letters, 2017, 24(5): 550–554. doi: 10.1109/LSP.2017.268 1689. [18] BOUMAL N. An Introduction to Optimization on Smooth Manifolds[M]. Cambridge: Cambridge University Press, 2023: 5–151. doi: 10.1017/9781009166164. [19] KASAI H. Fast optimization algorithm on complex oblique manifold for hybrid precoding in millimeter wave MIMO systems[C]. Proceedings of 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Anaheim, USA, 2018: 1266–1270. doi: 10.1109/GlobalSIP.2018.8646553. [20] LIN Tian, CONG Jiaqi, ZHU Yu, et al. Hybrid beamforming for millimeter wave systems using the MMSE criterion[J]. IEEE Transactions on Communications, 2019, 67(5): 3693–3708. doi: 10.1109/TCOMM.2019.2893632. -
图(5) / 表(2)
计量
- 文章访问数: 188
- HTML全文浏览量: 183
- PDF下载量: 27
- 被引次数: 0
下载:
