A Hybrid Beamforming Algorithm Based on Riemannian Manifold Optimization with Non-Monotonic Line Search

YAN Junrong; SHI Weitao; LI Pei

doi:10.11999/JEIT250396

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YAN Junrong, SHI Weitao, LI Pei. A Hybrid Beamforming Algorithm Based on Riemannian Manifold Optimization with Non-Monotonic Line Search[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250396

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YAN Junrong, SHI Weitao, LI Pei. A Hybrid Beamforming Algorithm Based on Riemannian Manifold Optimization with Non-Monotonic Line Search[J]. Journal of Electronics & Information Technology. doi: 10.11999/JEIT250396

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A Hybrid Beamforming Algorithm Based on Riemannian Manifold Optimization with Non-Monotonic Line Search

doi: 10.11999/JEIT250396 cstr: 32379.14.JEIT250396

School of Communication Engineering, Hangzhou Dianzi University, Hangzhou 310000, China

Funds: The National Natural Science Foundation of China (U21A20450, 62301204)

Received Date: 2025-05-09
Rev Recd Date: 2025-08-29

Available Online: 2025-09-08

Abstract

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 in Fig. 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 in Fig. 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 in Fig. 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 in Fig. 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.
- Hybrid precoding,
- Riemannian manifold,
- Massive MIMO,
- Overlapping subarray structure

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References(19)

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