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Compressed Sensing Based Channel Estimation for Movable Antenna Communications

Wenyan Ma, Lipeng Zhu, Rui Zhang

arXiv:2306.04333v1cs.ITeess.SP

TL;DR

MA communications require channel responses throughout a transmitter or receiver region, although measurements are available only at finite locations. The paper proposes successive transmitter-receiver compressed sensing using sparse multipath field-response information, and reports lower pilot overhead with higher channel reconstruction accuracy than benchmark schemes.

  • Problem

    MA positioning requires reconstructing complete channel state information across a region from finite measurements, a task conventional fixed-position-antenna methods cannot directly solve.

  • Method

    The STRCS method exploits sparse multipath representations to sequentially estimate angular-domain FRI, including AoDs, AoAs, and complex multipath coefficients.

  • Results

    The proposed channel reconstruction method outperforms benchmark schemes in both pilot overhead and channel reconstruction accuracy.

  • Takeaways & Limitations

    Sparse multipath FRI enables high-accuracy channel reconstruction with only moderate pilot overhead for wireless applications with slow-varying channels.

Abstract

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In this letter, we study the channel estimation for wireless communications with movable antenna (MA), which requires to reconstruct the channel response at any location in a given region where the transmitter/receiver is located based on the channel measurements taken at finite locations therein, so as to find the MA's location for optimizing the communication performance. To reduce the pilot overhead and computational complexity for channel estimation, we propose a new successive transmitter-receiver compressed sensing (STRCS) method by exploiting the efficient representation of the channel responses in the given transmitter/receiver region (field) in terms of multi-path components. Specifically, the field-response information (FRI) in the angular domain, including the angles of departure (AoDs)/angles of arrival (AoAs) and complex coefficients of all significant multi-path components are sequentially estimated based on a finite number of channel measurements taken at random/selected locations by the MA at the transmitter and/or receiver. Simulation results demonstrate that the proposed channel reconstruction method outperforms the benchmark schemes in terms of both pilot overhead and channel reconstruction accuracy.

I. INTRODUCTION

Movable antennas create a channel-estimation problem because CSI is needed throughout their continuous operating regions, whereas finite measurements must support position optimization. The proposed STRCS method exploits sparse multipath structure to reconstruct channels with reduced pilot overhead and high accuracy.

  • Motivation: MAs can move within continuous transmitter or receiver regions to improve wireless channel conditions through spatial-domain degrees of freedom.This is especially relevant to slow-varying, narrow-band scenarios such as machine-type communications, where time- or frequency-domain diversity may be unavailable and antenna or RF-chain counts may be small.
  • Problem: MA positioning requires CSI across the entire region, creating a channel-reconstruction problem from finite measurements at random or selected locations.Conventional FPA channel estimation acquires CSI only at fixed antenna positions and therefore cannot be directly applied to this setting.
  • Problem: Measuring every possible MA location would incur prohibitively high pilot overhead and large data-transmission delay.This motivates channel reconstruction methods that infer unmeasured responses rather than spatially sampling the region at a very high rate.
  • Method: The proposed method represents field responses sparsely using multipath components and recovers angular-domain FRI, including AoDs, AoAs, and complex coefficients.The estimated FRI is then used to reconstruct the channel response between arbitrary transmitter and receiver locations.
  • Method: STRCS estimates transmitter-side AoDs, receiver-side AoAs, and multipath coefficients successively using compressed sensing and least squares.The transmitter MA moves while the receiver MA is fixed, then the receiver MA moves while the transmitter MA is fixed, followed by joint movement for coefficient estimation.
  • Results: The reconstructed channels outperform benchmark schemes in pilot overhead and channel reconstruction accuracy.The method is designed to reduce both pilot overhead and computational complexity while retaining high reconstruction accuracy.

A. System Model

The system uses movable antennas whose transmitter and receiver positions can be adjusted within given two-dimensional regions. The received signal depends on both antenna positions and includes transmit power, a unit-power signal, and AWGN.

  • A. System Model: The transmitter and receiver each employ a movable antenna whose position can be flexibly adjusted within a square A × A region.Their coordinates are represented by t and r in regions C_t and C_r, respectively.
  • A. System Model: The channel h(t, r) depends on the transmitter and receiver antenna positions.The position-dependent channel connects the transmitter to the receiver.
  • A. System Model: For notation simplicity, the transmit signal is normalized to unit power and the pilot is set to s = 1 for FRI estimation.The noise variance σ^2 denotes the average noise power.
  • A. System Model: The received signal is modeled as Ph(t, r)s + z.Here, P is transmit power, s is the transmitted signal, and z is additive white Gaussian noise.

B. Field-Response Based Channel Model

The channel is modeled through transmitter- and receiver-side field responses and a path response matrix, with angular-domain multipath information enabling reconstruction across the antenna regions. Direct estimation at every transmitter-receiver position is infeasible, motivating efficient FRI estimation.

  • B. Field-Response Based Channel Model: The far-field model treats each multipath component’s AoD, AoA, and coefficient amplitude as approximately constant across the antenna regions, while its phase varies with position.The far-field assumption relies on the antenna-region size being much smaller than the transmitter-receiver distance.
  • B. Field-Response Based Channel Model: The channel response is represented using transmitter and receiver field-response vectors together with a path response matrix Σ characterizing multipath responses.The numbers of transmitter- and receiver-side multipath components are L_t and L_r.
  • B. Field-Response Based Channel Model: Directly estimating h(t, r) over all position pairs is infeasible, while joint estimation of all FRI components can require prohibitively high computational complexity.Efficient estimation is therefore designed to reduce pilot or training overhead and computational complexity.
  • B. Field-Response Based Channel Model: The channel h(t, r) can be reconstructed for any transmitter and receiver positions from the angular-domain field-response information.The proposed approach first estimates FRI components and then reconstructs h(t, r) over the regions using the channel representation.

III. PROPOSED METHOD FOR FRI ESTIMATION

The proposed STRCS method estimates angular-domain FRI in three successive stages using movable-antenna measurements. It separately estimates transmitter-side and receiver-side angular information before estimating multipath coefficients jointly.

  • III. PROPOSED METHOD FOR FRI ESTIMATION: STRCS estimates angular-domain FRI with affordable complexity through three successive steps.The method is illustrated in Fig. 2.
  • III. PROPOSED METHOD FOR FRI ESTIMATION: The transmitter movable antenna moves while the receiver antenna remains fixed to estimate transmitter multipath components’ AoDs through compressed sensing.The AoD estimation is formulated as a sparse signal recovery problem.
  • III. PROPOSED METHOD FOR FRI ESTIMATION: The receiver movable antenna then moves while the transmitter antenna remains fixed to estimate receiver multipath components’ AoAs similarly.This forms the second stage of the successive transmitter-receiver procedure.
  • III. PROPOSED METHOD FOR FRI ESTIMATION: Finally, both movable antennas are jointly moved to estimate the complex coefficients of all multipath components using a least-squares-based algorithm.The joint movement uses the previously obtained AoD and AoA information.

A. AoD Estimation

AoD estimation moves the transmitter antenna across measurement positions while the receiver remains fixed, transforming the angular search into a sparse recovery problem. Compressed sensing then estimates the transmitter-side AoD pairs from pilot measurements.

  • A. AoD Estimation: The transmitter antenna is moved over M positions, with one pilot sent at each position while the receiver antenna remains fixed.The resulting measurements form the received signal vector used for transmitter-side AoD estimation.
  • A. AoD Estimation: Each sensing-matrix column corresponds to a transmitter-side steering vector associated with a fixed antenna position and varying AoDs.The steering vector is parameterized by an AoD pair and the propagation-distance differences induced by antenna movement.
  • A. AoD Estimation: Uniformly discretizing the angular variables into G grids produces an over-complete sensing matrix and a sparse vector with L_t nonzero elements.The approximation uses G ≫ L_t to represent the transmitter-side angular response.
  • A. AoD Estimation: When G ≫ L_t, AoD estimation becomes sparse signal recovery by finding a sparse vector that minimizes the received-signal reconstruction error.Orthogonal matching pursuit can be used to estimate the AoD pairs corresponding to nonzero coefficients.

B. AoA Estimation

The receiver estimates AoAs from measurements collected while the R-MA moves across selected positions, using sparse recovery over an angular-domain steering dictionary. Position count and spatial arrangement affect angular resolution and estimation performance.

  • The receiver-side estimation procedure parallels the transmitter-side AoD estimation procedure, with the T-MA position fixed during this step.
  • The R-MA moves over N−1 positions while the T-MA remains fixed, collecting one channel measurement per position for AoA estimation.
  • The received measurements are modeled with a receiver-side steering matrix whose columns correspond to candidate AoA pairs.
  • OMP estimates the sparse receiver-side path components by selecting dictionary columns associated with nonzero coefficients.
  • The MA position count increases pilot number and received signal power, while spatial distribution determines angular-domain resolution for AoD/AoA estimation.

C. PRM Estimation

After estimating angular information, the method estimates the path-response matrix using additional transmitter–receiver measurements and optimized MA positions. The optimization improves conditioning, while the full algorithm has polynomial complexity.

  • The first two steps generally cannot exactly estimate the path-response matrix because the channel measurement matrix is usually rank-deficient.
  • The method uses estimated AoDs and AoAs to choose the minimum number of additional MA positions for estimating the path-response matrix.
  • K additional transmitter–receiver position pairs provide measurements that are stacked with the earlier transmitter- and receiver-side observations.
  • The receiver optimizes additional MA positions to reduce the measurement-matrix condition number before estimating the vectorized path-response matrix by least squares.
  • Problem (20) is highly non-convex, so the implementation finds a local optimum using MATLAB’s fmincon function.
  • The total algorithmic complexity is O(M L̂_t G^2 + N L̂_r G^2 + K^3.5 log(1/ε) + (L̂_r L̂_t)^2J), polynomial in the listed parameters.

IV. NUMERICAL RESULTS

Simulations evaluate reconstruction across MA position setups and third-step selection methods using NMSE. The proposed method supports multiple setups, with UPA positioning and optimized additional positions yielding the strongest reported performance.

  • The simulations average results over 10^4 independent channel realizations and evaluate reconstruction on spatial grids separated by Δ=λ/5.
  • For the first two steps, the proposed method is tested with square-, cross-, UPA-, and circle-shaped MA position setups.
  • The proposed reconstruction method applies to different MA position setups, while the UPA-shaped setup achieves lower NMSE than the alternatives.
  • At SNR=25 dB, UPA positioning improves performance by 65.8%, 94.7%, and 54.3% over square-, cross-, and circle-shaped setups, respectively.
  • For SNR<12 dB, UPA positioning has lower NMSE than exhaustive measurement while using 527 measurements instead of 1.6×10^5.
  • With K=10, optimized third-step positions improve performance by 88.9%, 52.2%, and 88.6% over RP, RPS, and PEAM, respectively.
  • Optimizing transmitter- and receiver-side positions produces a better-conditioned measurement matrix and higher path-response-matrix estimation accuracy than RP and RPS.

V. CONCLUSIONS

The paper proposes STRCS channel reconstruction for MA systems by exploiting sparse multipath representations. It sequentially estimates angular field-response information and achieves accurate reconstruction with moderate pilot overhead.

  • The proposed channel reconstruction method exploits sparse channel-response representations in the transmitter and receiver regions through multipath components.
  • To reduce pilot overhead and computational complexity, the method sequentially estimates angular-domain field-response information from finite MA measurements.
  • The angular-domain information includes AoDs, AoAs, and complex coefficients of significant multipath components.
  • Numerical results show high channel reconstruction accuracy with only moderate pilot overhead, supporting use in slow-varying wireless channels.
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