A Hybrid Beamforming Algorithm Based on Limited-Broyden-Fletcher-Goldfarb-Shanno
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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.
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Key words:
- Millimeter wave /
- Massive MIMO /
- Hybrid beamforming /
- Overlapped antenna arrays
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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 
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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
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