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Channel Estimation for Movable Antenna Communication Systems: A Framework Based on Compressed Sensing
Zhenyu Xiao, Songqi Cao, Lipeng Zhu, Yanming Liu, Xiang-Gen Xia, Rui Zhang
TL;DR
Movable-antenna systems need complete CSI over continuous transmitter and receiver regions, which cannot be obtained by measuring every position. This paper jointly estimates multipath parameters with compressed sensing, analyzes measurement-position coherence, and reports high-accuracy CSI reconstruction, with optimized-position receive SNR nearly matching perfect-CSI performance.
Problem
Accurate CSI over continuous transmitter and receiver regions is required, but exhaustive measurements are infeasible because they impose excessive pilot overhead and time consumption.
Method
The paper jointly estimates AoDs, AoAs, and complex MPC coefficients through compressed sensing using channel measurements at movable-antenna positions, while analyzing measurement-matrix coherence.
Results
The framework reconstructs complete CSI between the transmitter and receiver regions with high accuracy, and optimized-position receive SNR is almost the same as with perfect CSI.
Takeaways & Limitations
Compressed sensing enables complete movable-antenna CSI reconstruction from a limited number of channel measurements within the proposed framework.
Abstract
from arXiv · showhide
Movable antenna (MA) is a new technology with great potential to improve communication performance by enabling local movement of antennas for pursuing better channel conditions. In particular, the acquisition of complete channel state information (CSI) between the transmitter (Tx) and receiver (Rx) regions is an essential problem for MA systems to reap performance gains. In this paper, we propose a general channel estimation framework for MA systems by exploiting the multi-path field response channel structure. Specifically, the angles of departure (AoDs), angles of arrival (AoAs), and complex coefficients of the multi-path components (MPCs) are jointly estimated by employing the compressed sensing method, based on multiple channel measurements at designated positions of the Tx-MA and Rx-MA. Under this framework, the Tx-MA and Rx-MA measurement positions fundamentally determine the measurement matrix for compressed sensing, of which the mutual coherence is analyzed from the perspective of Fourier transform. Moreover, two criteria for MA measurement positions are provided to guarantee the successful recovery of MPCs. Then, we propose several MA measurement position setups and compare their performance. Finally, comprehensive simulation results show that the proposed framework is able to estimate the complete CSI between the Tx and Rx regions with a high accuracy.
I. INTRODUCTION
Movable antennas exploit spatial channel variation through local movement, but realizing their gains requires complete CSI across continuous transmitter and receiver regions. The paper introduces a compressed-sensing framework that jointly estimates multipath information and analyzes measurement-position design.
- Fixed-position MIMO antennas do not fully exploit continuous spatial channel variation, limiting available spatial degrees of freedom.
- Movable antennas adjust positions to exploit spatial diversity, improve channel gain, mitigate interference, and provide additional beamforming and multiplexing degrees of freedom.
- Complete CSI across continuous transmitter and receiver regions is required, but measuring every candidate position would incur excessive pilot overhead and time consumption.
- The proposed framework jointly estimates AoDs, AoAs, and complex MPC coefficients by discretizing angles and solving a compressed-signal recovery problem with OMP.
- Measurement positions determine the compressed-sensing matrix; Fourier-based mutual-coherence analysis yields two position criteria and five deterministic or random setups for evaluation.
- Simulations report high-accuracy complete CSI reconstruction, with optimized-position receive SNR almost matching perfect-CSI performance.
III. PROPOSED FRAMEWORK FOR CHANNEL ESTIMATION
The framework converts the multipath channel into a discrete angular representation suitable for compressed sensing. Quantized virtual angles and path responses form a sparse discrete model, while quantization introduces approximation error.
- The framework quantizes AoDs and AoAs to derive a discrete-form channel representation and then estimates angular information and the path-response matrix from finite measurements.
- Virtual AoDs and AoAs are mapped to four coordinates in the range −1 to 1, providing a universal representation for arbitrary propagation angles.
- For L = L_t × L_r multipath components, the path-response function is represented as a superposition of L discrete impulse functions.
- The discrete model uses angular resolution 2/N, so larger N permits more accurate virtual-angle estimates.
- Quantization creates a mismatch between the discrete and actual channel responses, represented by an explicit quantization-error term.
- The discrete path-response matrix contains quantized angles and complex coefficients, while the associated discrete field-response vectors encode transmitter and receiver positions.
B. Channel Estimation
Channel measurements collected at multiple movable-antenna positions form a sparse recovery problem. OMP jointly recovers multipath parameters, enabling channel-response reconstruction from comparatively few measurements when the angular representation is sparse.
- The transmitter and receiver movable antennas change positions across measurement slots, and their received pilot signals are stacked into a measurement vector.
- The MPC information is recovered by solving a sparse signal-recovery problem that accounts for quantization error and measurement noise.
- Because L_t × L_r ≪ N^4, compressed-sensing theory allows the high-dimensional path-response vector to be recovered from a small number of measurements.
- OMP jointly estimates virtual AoDs, virtual AoAs, and channel coefficients, then reconstructs the channel response from the recovered path information.
- OMP iteratively selects path indices, estimates their coefficients by least squares, updates the residual, and stops when normalized residual power falls below a threshold.
IV. DESIGN OF MEASUREMENT MATRIX
The measurement matrix is central to compressed-sensing channel estimation, but MA systems can modify it only through measurement-position design. The paper analyzes its mutual coherence and proposes deterministic and random position setups.
- MA measurement positions determine the measurement matrix and significantly affect channel-estimation performance.
- The paper analyzes the measurement matrix and its mutual coherence to guide MA measurement-position design.
- Five MA measurement setups are proposed using deterministic or random antenna positions.
- Deterministic and random compressed-sensing matrices use distinct strategies to satisfy the restricted isometry property.
A. Mutual Coherence
The analysis interprets MA channel measurements as a transform between angular and locational domains, then derives sampling and region criteria for recovering multipath responses.
- The measurement matrix columns encode virtual AoD–AoA pairs, and their cross-correlations determine mutual coherence.
- The effective measurement matrix transforms sparse angular-domain path responses into locational-domain channel responses similarly to a Fourier transform.
- Continuous measurements over an infinite region can recover path responses with arbitrarily high angular resolution.
- Uniform sampling requires a spacing below λ/2 for full recovery with arbitrarily high angular resolution under infinite-region measurements.
- With no prior information, the measurement region should cover the entire available limited region, r = R, to minimize angular path-response spread.
- Five deterministic or randomized MA position setups are constructed from the two measurement criteria.
B. Deterministic-Position Setups
Deterministic MA measurement positions implement uniform sampling throughout the Tx and Rx regions, with the Tx-MA and Rx-MA traversing their respective measurement positions.
- Deterministic-position setups correspond to uniform sampling in both the Tx and Rx regions.
- The Tx-MA traverses all Tx measurement positions, while the Rx-MA traverses all Rx positions for each Tx-MA position.
1) UPA-Shape:
The UPA-shape setup samples the entire Tx and Rx regions with MA positions arranged similarly to uniform planar arrays.
- The UPA-shape setup places MA measurement positions similarly to uniform planar arrays.
- The setup is regarded as sampling throughout the entire Tx and Rx regions.
- The setup defines Tx-MA and Rx-MA positions for each of the M channel measurements.
2) Edge of Region:
The edge-of-region setup uses uniform sampling at the edges of the Tx and Rx regions. Its measurement positions are specified for each channel measurement.
- Edge of Region: The setup samples uniformly at the edges of the Tx and Rx regions.
- Edge of Region: The MA measurement positions are defined as a set for the edge-of-region setup.
- Edge of Region: For measurement m, the Rx-MA and Tx-MA positions are specified explicitly.
3) Cross-shape:
The cross-shape setup samples uniformly along the coordinate axes of the Tx and Rx regions. Its measurement positions are specified for each channel measurement.
- Cross-shape: The setup samples uniformly along the coordinate axes of the Tx and Rx regions.
- Cross-shape: The MA measurement positions are defined as a set for the cross-shape setup.
- Cross-shape: For measurement m, the Rx-MA and Tx-MA positions are specified explicitly.
C. Random-Position Setups
The random-position setups introduce randomness into Tx-MA and Rx-MA measurement locations to reduce mutual coherence. Random distribution samples uniformly, while random walk constrains movement between consecutive measurements.
- Random-Position Setups: Random-position setups use randomness in Tx-MA and Rx-MA locations to achieve low mutual coherence.This follows the compressed-sensing idea of random matrices.
- Random Distribution: The random distribution setup generates Tx-MA and Rx-MA positions from a two-dimensional uniform distribution.
- Random Walk: The random walk setup fixes the moving distance between adjacent measurements while varying the moving direction.It addresses the potentially long movements required by random distribution.
- Random Walk: Random walk uses boundary bounce-back actions to keep both MAs within their feasible regions.
- Comparison of Setups: With 0.4λ deterministic spacing, the UPA, edge, and cross setups use 1296, 400, and 144 measurements, respectively.The random distribution and random walk setups also use 144 measurements.
- Mutual Coherence: All five setups reach mutual coherence 1 only when n = n′ = 1, while main and side lobes arise from angular path-response spread.The observations indicate Criterion 1 is satisfied when adjacent-position spacing is less than half a wavelength.
- Mutual Coherence: Random-position setups have mutual coherence closer to ideal continuous sampling than deterministic-position setups.The comparison indicates more accurate angular channel information and better channel-estimation performance for random positions.
V. SIMULATION RESULTS
The simulations evaluate compressed-sensing channel estimation and measurement-position setups under a geometry channel model. Performance is assessed across complete regional channel responses, path angles, and path coefficients.
- Simulation Setup: The simulation study evaluates the proposed compressed-sensing channel estimation method and compares MA measurement-position setups.
- Simulation Setup: The geometry channel model assumes one corresponding receive path for each transmit path, with L paths and a diagonal path-response matrix.
- Evaluation Metrics: NMSE measures estimation reliability across channel responses sampled throughout the Tx and Rx regions.H contains responses between all sampled Tx and Rx points, while Ĥ denotes the estimate.
- Evaluation Metrics: Angle error and coefficient error additionally measure recovery accuracy for virtual AoDs, virtual AoAs, and individual path coefficients.Coefficient error is the NMSE of the complex coefficients.
- Simulation Parameters: The default simulation uses average SNR 20 dB, N = 24 quantization angles, L = 3 paths, and 10^3 Monte Carlo results.The random walk moves λ/2 between adjacent measurements, with λ = 0.01 m.
B. Numerical Results
The numerical results show that random MA measurement positions generally improve channel-estimation accuracy with fewer measurements, while deterministic setups depend more strongly on mutual coherence, angular resolution, and quantization error. These estimation differences affect angle and MPC-coefficient recovery and the SNR achieved by subsequent MA position optimization.
- NMSE versus channel measurements: Random-position setups achieve better NMSE than deterministic-position setups, with estimation improving as the number M of channel measurements increases.Random positions enable traversal of the full Tx and Rx regions with a small mutual coherence.
- NMSE versus channel measurements: Deterministic setups reach NMSE lower bounds after Criterion 1 is satisfied; UPA-shape has the lowest bound, whereas cross-shape has high NMSE from high mutual coherence.UPA-shape samples the entire regions, while cross-shape samples only coordinate axes.
- Angle errors: Random-position and UPA-shape setups achieve similar angle-recovery performance, while cross-shape and edge-of-region setups have higher angle errors from larger mutual coherence.UPA-shape requires a large number of measurements to reach comparable performance.
- MPC coefficient errors: Coefficient errors decrease slowly with more measurements because noise and quantization errors remain in the measured channel responses.Edge-of-region positions produce large quantization errors, which increase recovered MPC coefficient errors.
- NMSE versus quantization angles: Increasing quantization angles significantly reduces NMSE for most setups, but UPA-shape is not implementable with M = 256 because Criterion 1 is not satisfied.At high angular resolution, cross-shape remains worse than edge-of-region, random-distribution, and random-walk setups.
- NMSE versus SNR: At M = 256, random-position setups outperform deterministic setups with few measurements, while high-SNR NMSE is mainly limited by quantization error.The STRCS method has lower NMSE at SNR = 0 dB but higher NMSE than random-position methods at high SNR.