A Hybrid Beamforming Algorithm Based on Riemannian Manifold Optimization with Non-Monotonic Line Search
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摘要: 针对毫米波Massive MIMO系统中混合波束成形算法存在着的复杂度高、运行时间长、能量效率低的问题，该文提出一种基于黎曼流形优化与非单调线性搜索的混合波束成形算法(MO-NMLS)。该算法首先采用数字预编码器的最小二乘解重构目标函数，以降低求解维度；其次构建黎曼流形计算新目标函数的黎曼梯度，从而将模拟域的恒模约束转化为无约束优化；然后基于当前迭代点与历史迭代点梯度信息的非单调线性搜索算法计算动态步长因子，以提升数字预编码器的收敛速度；最后采用数字预编码器优化模拟预编码器，以实现混合预编码器的迭代更新。仿真结果表明，在全连接结构下，所提算法在保持等效频谱效率时，相比CG算法减少了75.3%的运行时间；尤其是在重叠子阵结构子阵偏移量为8时，所提算法的能量效率较全连接结构提升了10.9%。
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关键词:
- 混合预编码 /
- 黎曼流形 /
- Massive MIMO /
- 重叠子阵结构
Abstract:Objective Fully digital beamforming architectures provide high spectral efficiency but demand one Radio-Frequency (RF) chain per antenna element, resulting in substantial cost, power consumption, and hardware complexity. These limitations hinder their practical deployment in large-scale antenna systems. Hybrid beamforming offers a feasible alternative by reducing hardware requirements while retaining much of the performance. In such systems, analog beamforming modules follow a reduced number of RF chains to control massive antenna arrays. Analog phase shifters are energy-efficient and cost-effective but restricted to constant modulus constraints, which are essential for hardware implementation. In contrast, digital phase shifters offer flexible control over amplitude and phase. The central challenge is to approximate the spectral efficiency of fully digital systems while adhering to analog-domain constraints and minimizing energy and hardware demands. To overcome this challenge, this study proposes a novel hybrid beamforming algorithm that integrates Riemannian manifold optimization with a non-monotonic line search strategy (MO-NMLS). This approach achieves improved trade-offs among spectral efficiency, energy consumption, and hardware complexity. Methods The proposed methodology proceeds as follows. First, the joint matrix optimization problem for maximizing spectral efficiency in hybrid beamforming is decomposed into separate transmitter and receiver subproblems by formulating an appropriate objective function. This objective is then reformulated using a least squares approach, reducing the dimensionality of the search space from two to one. To accommodate the constant modulus constraints of analog beamforming, the problem is transformed into an unconstrained optimization on Riemannian manifolds. Both the Euclidean and Riemannian gradients of the modified objective function are derived analytically. Step sizes are adaptively determined using a MO-NMLS, which incorporates historical gradient information to compute dynamic step factors. This mechanism guides the search direction while avoiding convergence to suboptimal local minima due to fixed step sizes. Distinct update rules for the step factor are applied depending on whether the iteration count is odd or even. In each iteration, the current objective function value is compared with those from the preceding L iterations to decide whether to accept the new step and iteration point. After updating the step size, tangent vectors are retracted onto the manifold to generate new iterates until convergence criteria are satisfied. Once the analog precoder is fixed based on the optimized search direction, the corresponding digital precoder is derived in closed form. The dynamic step factor is computed using gradient data from the current and preceding L iterations, allowing the objective function to exhibit non-strict monotonicity within bounded ranges. This adaptive strategy results in faster convergence compared with conventional fixed-step methods. Results and Discussions The relationship between internal iteration count and Signal-to-Noise Ratio (SNR) for different beamforming algorithms is shown in Fig. 4 . The MO-NMLS algorithm requires significantly fewer iterations than the conventional Conjugate Gradient (CG) method under both fully connected and overlapping subarray architectures. This improved efficiency arises from the use of Riemannian manifold optimization, which inherently satisfies the constant modulus constraints without necessitating computationally intensive Hessian matrix evaluations. Runtime performance is benchmarked inFig. 5 . The MO-NMLS algorithm reduces runtime by 75.3% relative to CG in the fully connected structure and by 79.2% in the overlapping subarray structure. Additionally, MO-NMLS achieves a further 21.1% reduction in runtime under the overlapping subarray architecture compared with the fully connected one, owing to simplified hardware requirements. Spectral efficiency as a function of SNR is presented inFig. 6 . In fully connected systems, MO-NMLS achieves a 0.64% improvement in spectral efficiency over CG while maintaining comparable stability in overlapping subarray architectures. This performance gain stems from the algorithm’s ability to avoid local optima, a key limitation of Orthogonal Matching Pursuit (OMP), which selects paths based solely on residual correlation. The scalability of MO-NMLS with respect to the number of antennas and data streams is demonstrated inFig. 7 . In fully connected systems, MO-NMLS outperforms CG by 1.94%, 2.16%, and 2.74% in spectral efficiency at antenna and data stream configurations of (32, 2), (64, 4), and (128, 8), respectively. While spectral efficiency increases across all algorithms as system scale grows, MO-NMLS exhibits the most substantial gains at higher scales. Energy efficiency improvements under the overlapping subarray architecture are shown inFig. 8 . Compared with the fully connected configuration, MO-NMLS yields energy efficiency gains of 1.2%, 10.9%, and 25.9% at subarray offsets of 1, 8, and 16, respectively. These improvements are attributed to the reduced number of required phase shifters and power amplifiers, which decreases total system power consumption as the subarray offset increases.Conclusions The proposed MO-NMLS algorithm achieves an effective balance among spectral efficiency, hardware complexity, and energy consumption in hybrid beamforming systems, while substantially reducing computational runtime. Moreover, the overlapping subarray architecture attains spectral efficiency comparable to that of fully connected systems, with significantly lower execution times. These results highlight the practical advantages of the proposed approach for large-scale antenna systems operating under resource constraints. -
Key words:
- Hybrid precoding /
- Riemannian manifold /
- Massive MIMO /
- Overlapping subarray structure
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1 基于MO-NMLS的模拟预编码器设计(全连接结构/重叠子阵结构)
输入：最优数字预编码矩阵$ {{\boldsymbol{V}}_{{\text{opt}}}} $，初始黎曼梯度$ {\text{grad}}{f_{{{\boldsymbol{x}}_0}}} $，初始点$ {{\boldsymbol{x}}_0} \in \mathcal{M} $ 1.全连接结构根据式(13)计算欧式梯度，重叠子阵结构根据式(14)计算欧式梯度 2.根据式(16)计算初始梯度$ {{\boldsymbol{g}}_0} = {\text{grad}}{f_{{{\boldsymbol{x}}_0}}} $ 3.设置迭代计数器$ k = 0 $ 4.while未满足终止条件(梯度范数$ {\left\| {{{\boldsymbol{g}}_k}} \right\|_{\text{F}}} < \mu $，步长$ {\alpha _k} < {\alpha _{\min }} $或达到最大迭代次数) 5.分别根据式(20)和式(21)计算${{\boldsymbol{s}}_k}$和${{\boldsymbol{y}}_k}$ 6.根据式(19)计算交替步长$ \alpha _k^{{\text{BB}}} $ 7.步长约束$ \alpha _{k + 1}^{{\text{BB}}} = \left\{ \begin{gathered} \min \{ {\alpha _{\max }},\max \{ {\alpha _{\min }},\alpha _k^{{\text{BB}}}\} \} {\text{ if}}\left\langle {{{\boldsymbol{s}}_k},{{\boldsymbol{y}}_k}} \right\rangle > 0 \\ {\alpha _{\max }}{\text{ otherwise}} \\ \end{gathered} \right. $ 8.更新黎曼梯度$ {{\boldsymbol{g}}_{k + 1}} = {\text{grad}}{f_{{{\boldsymbol{x}}_{k + 1}}}} $，搜索方向$ {\eta _k} = - {{\alpha }_k}{{\boldsymbol{g}}_k} $，回缩迭代点$ {{\boldsymbol{x}}_{k + 1}} = {\text{Retr}}( - {\alpha _k}{{\boldsymbol{g}}_k}) $ 9.$ k = k + 1 $ 10.end 
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2 基于MO-NMLS的混合波束成形算法
输入：最优数字预编码矩阵$ {{\boldsymbol{V}}_{{\text{opt}}}} $，模拟预编码器$ {\boldsymbol{V}}_{{\text{RF}}}^k $具有随机
相位1.设置迭代计数器$ k = 0 $，计算黎曼梯度$ {\text{grad}}{J_{{\boldsymbol{V}}_{{\text{RF}}}^k}} $ 2.while未满足终止条件(同算法1) 3.固定$ {\boldsymbol{V}}_{{\text{RF}}}^k $，$ {\boldsymbol{V}}_{\text{B}}^k = {({\boldsymbol{V}}_{{\text{RF}}}^k)^\dagger }{{\boldsymbol{V}}_{{\text{opt}}}} $ 4.使用算法1优化$ {\boldsymbol{V}}_{{\text{RF}}}^{k + 1} $ 5.$k = k + 1$ 6.end 7.计算数字预编码器$ {{\boldsymbol{V}}_{\text{B}}} = {({\boldsymbol{V}}_{{\text{RF}}}^{\text{H}}{{\boldsymbol{V}}_{{\text{RF}}}})^{ - 1}}{\boldsymbol{V}}_{{\text{RF}}}^{\text{H}}{{\boldsymbol{V}}_{{\text{opt}}}} $
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