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Flexible Precoding for Multi-User Movable Antenna Communications
Songjie Yang, Wanting Lyu, Boyu Ning, Zhongpei Zhang, Chau Yuen
TL;DR
Traditional precoding can fall short under multipath effects and inter-user coherence, motivating its adaptation to movable antennas. The paper formulates flexible RZF as sparse optimization and proposes OffG RLS-OMP, interpreting position selection through subspace projection. Flexible precoding achieves more than double the sum rate of fixed antenna positions when both the number of paths and movement-region size are large.
Problem
Traditional precoding can fall short of ideal performance because of multipath effects and inter-user coherence.
Source: §I · p. 1Method
The paper formulates flexible RZF antenna-position optimization as sparse optimization and proposes OffG RLS-OMP, with a subspace-projection interpretation.
Source: §I · p. 2Results
Flexible precoding delivers more than double the sum rate of fixed antenna positions when both the number of paths and movement-region size are large.
Source: §V · p. 4Takeaways & Limitations
The authors frame antenna-position optimization as finding the most effective antenna subspace and report gains over fixed antennas and classical antenna selection.
Source: §VI · p. 4
Abstract
from arXiv · showhide
This letter rethinks traditional precoding in multi-user wireless communications with movable antennas (MAs). Utilizing MAs for optimal antenna positioning, we introduce a sparse optimization (SO)-based approach focusing on regularized zero-forcing (RZF). This framework targets the optimization of antenna positions and the precoding matrix to minimize inter-user interference and transmit power. We propose an off-grid regularized least squares-based orthogonal matching pursuit (RLS-OMP) method for this purpose. Moreover, we provide deeper insights into antenna position optimization using RLS-OMP, viewed from a subspace projection angle. Overall, our proposed flexible precoding scheme demonstrates a sum rate that exceeds more than twice that of fixed antenna positions.
I. INTRODUCTION
The introduction motivates movable antennas as a way to alter spatial channel paths, then proposes flexible RZF precoding that jointly optimizes antenna positions and precoding. It presents OffG RLS-OMP for the resulting sparse optimization and a subspace-projection interpretation of position selection.
I. INTRODUCTION
- Movable antennas can alter spatial channel paths through constructive or destructive interference and path orthogonalization.The introduction groups these effects as modifying channel conditions.Source: §I · p. 1
- Earlier work studied antenna selection and array synthesis, while MA/FAS studies examined channel gain, capacity, diversity, multiplexing, power, and user-rate objectives.The introduction notes that metaheuristics can have high computational complexity and limited theoretical insight.Source: §I · p. 1
- Flexible RZF adapts traditional precoding by optimizing antenna positions, with regularization tuning also accommodating MRT, ZF, or MMSE.The motivation is that traditional precoding can fall short because of multipath effects and inter-user coherence.Source: §I · pp. 1–2
- Off-grid RLS-OMP efficiently addresses the sparse optimization problem for flexible RZF under minimum inter-element spacing constraints.The formulation uses an on-grid dictionary for antenna positions and off-grid refinement.Source: §I · p. 2
- The subspace view casts optimizing N antenna positions as finding an optimal subspace spanned by the array-position manifolds at those locations.The authors use the relationship between column-subspace projection and RLS to explain OffG RLS-OMP's role.Source: §I · p. 2
II. SYSTEM MODEL
The system model considers a multi-user MISO downlink with movable base-station antennas and one fixed antenna per user. It describes received signals and SINR under a multipath channel model parameterized by the antenna positions and path angles.
II. SYSTEM MODEL
- The downlink has a base station with N movable antennas in the x-z plane, K single-antenna users, and one data stream per user.The model defines N as Nx × Nz.Source: §II · p. 2
- Each user's received signal is modeled from the channel, precoding matrix, user data streams, and additive white Gaussian noise.The model assumes unit-power uncorrelated data streams when defining SINR.Source: §II · p. 2
- User channels use an L-path spatial model, with path gains and virtual directions parameterizing the array-angle manifold.Antenna positions are represented by x- and z-axis coordinates; the model omits the array normalization factor when ignoring array gain.Source: §II · p. 2
III. SO-BASED FLEXIBLE RZF PRECODING
The section formulates flexible RZF as a joint antenna-position and precoding optimization, then casts position selection as sparse optimization over a virtual-channel dictionary. OffG RLS-OMP selects and refines antenna positions iteratively while updating precoding and the residual.
III. SO-BASED FLEXIBLE RZF PRECODING
- Flexible RZF optimizes antenna coordinates and the precoding matrix, with positions restricted to the feasible region and antennas separated by at least λ/2.RZF is the starting precoding scheme; in movable-antenna systems, the channel changes with the optimized positions.Source: §III · p. 2
- The position search is formulated as off-grid sparse optimization: a discretized virtual-channel dictionary represents feasible positions, with refinement toward continuous locations.The grid spacing is d = λ/2, and the dictionary is a discretized counterpart of the continuous feasible region.Source: §III · pp. 2–3
- OffG RLS-OMP runs for N iterations, selecting and refining an antenna position, calculating coefficients with RLS, and updating the residual after each selection.The method adapts an OMP-like greedy process to the position-optimization problem.Source: §III · p. 3
- Before the next antenna is selected, the support is checked against the minimum-spacing constraint and any antenna that fails is excluded.The algorithm initializes the residual as I_K and returns the optimized position set and precoding matrix.Source: §III · p. 3
IV. INSIGHTS INTO FLEXIBLE RZF PRECODING
The analysis interprets RLS coefficient updates as regularized projections and residual updates as regularized complements. It uses this connection to explain the greedy antenna-position search as successive subspace construction and residual updating.
IV. INSIGHTS INTO FLEXIBLE RZF PRECODING
- For α ≠ 0, the RZF expression is equivalent to the RLS coefficient calculation; for ZF, the limit α → 0 is required rather than α = 0.The coefficient calculation uses the present optimized channel at each iteration.Source: §IV · p. 3
- The regularized channel expression acts as a directionally adjusted projection, while its complement corresponds to the algorithm’s residual update.The regularized expression is not strictly a projection matrix because it does not satisfy P² = P.Source: §IV · p. 3
- Each greedy iteration matches and refines an antenna using a column of I_K, then uses the resulting subspace projection to update the residual for the next iteration.The process seeks an antenna subspace suited to precoding.Source: §IV · p. 4, Figure 1 · p. 4
V. SIMULATION RESULTS
The simulations compare flexible precoding with fixed antenna positions and fast antenna selection across precoding schemes, channel-path counts, and movable-region sizes. Flexible precoding exceeds twice the fixed-position sum rate when both the number of paths and region size are large.
V. SIMULATION RESULTS
- The evaluation uses a 3 GHz system with four antennas serving four users, unit total transmit and noise power, and uniformly distributed user and scatterer locations.The compared methods are fixed-position RZF, fast AS with RZF, and the proposed flexible precoding.Source: §V · p. 4
- Flexible precoding for ZF and MMSE outperforms the other methods, whereas for MRT it is similar to fixed positions and trails fast AS.The comparison uses fixed-position RZF and fast AS with RZF as baselines.Source: §V · p. 4
- Roughly twice the sum rate of fixed antenna positions is achieved by flexible precoding at G = 9, and the gap grows as G increases.Both fast AS and flexible precoding show increasing sum rate with larger G, more markedly for L = 15 than for L = 3.Source: §V · p. 4
- More than double the sum rate of fixed antenna positions is achieved by flexible precoding when both channel-path count and movable-region size are large.In this setting, flexible precoding also outperforms fast AS.Source: §V · p. 4
VI. CONCLUSIONS
The proposed flexible RZF framework optimizes movable-antenna positions through sparse optimization and the OffG RLS-OMP algorithm. It uses a subspace-projection view of position optimization and is reported to outperform fixed antennas and classical antenna selection.
VI. CONCLUSIONS
- The proposed strategy significantly outperforms traditional fixed antennas and the classical antenna selection method.The conclusion does not provide a numerical comparison here.Source: §VI · p. 4
- The framework optimizes movable-antenna positions for flexible RZF precoding through sparse optimization, using the OffG RLS-OMP algorithm under minimum inter-element spacing constraints.Source: §VI · p. 4
- The authors interpret antenna-position optimization as finding the most effective antenna subspace through subspace projection.Source: §VI · p. 4
- The proposed approach may be explored for movable-antenna applications beyond the RZF precoding considered in this letter.Source: §VI · p. 4
APPENDIX A DERIVATION OF OFFG POSITION REFINEMENT
OffG position refinement adjusts each antenna's coordinates through a regularized projection-based perturbation step. After each position update, the channel and precoding are updated before the next iteration.
APPENDIX A DERIVATION OF OFFG POSITION REFINEMENT
- First-order Taylor expansion turns coordinate perturbations into a fine-tuning optimization for the current antenna position.Source: Appendix A · p. 5
- The position update uses regularized projections of the residual onto coordinate-specific vectors, with the regularization factor chosen to satisfy inter-element spacing constraints.The resulting coordinate increments are applied to the antenna's x_n and z_n positions, retaining their real parts.Source: Appendix A · p. 5
- After each antenna-position update, the method updates the channel and precoding before proceeding to the next iteration.Source: Appendix A · p. 5