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Channel Estimation for Extremely Large-Scale Massive MIMO Systems

Yu Han, Shi Jin, Chao-Kai Wen, Xiaoli Ma

arXiv:1910.05432v1cs.ITeess.SP

TL;DR

Extremely large-scale massive MIMO creates near-field and spatially non-stationary channels, challenging channel estimation and transceiver design. This letter models mappings between subarrays and scatterers and proposes subarray-wise and scatterer-wise estimators. The subarray-wise method provides accurate MSE performance with low complexity, while the scatterer-wise method positions scatterers and identifies almost all mappings.

  • Problem

    Near-field and spatially non-stationary channels from extremely large aperture arrays create channel-estimation requirements that traditional LS and linear minimum mean square error methods cannot satisfy.

  • Method

    The letter models last-hop scatterers under spherical wavefronts and proposes subarray-wise and scatterer-wise channel estimation methods based on refined OMP.

  • Results

    The subarray-wise method achieves accurate MSE performance with low complexity, whereas the scatterer-wise method accurately positions scatterers and identifies almost all mappings between subarrays and scatterers.

  • Takeaways & Limitations

    The two methods provide complementary ways to estimate near-field non-stationary channels: subarray-wise estimation favors low complexity, while scatterer-wise estimation recovers scatterer locations and mappings.

Abstract

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Extremely large-scale massive multiple-input multiple-output (MIMO) has shown considerable potential in future mobile communications. However, the use of extremely large aperture arrays has led to near-field and spatial non-stationary channel conditions, which result in changes to transceiver design and channel state information that should be acquired. This letter focuses on the channel estimation problem and describes the non-stationary channel through mapping between subarrays and scatterers. We propose subarray-wise and scatterer-wise channel estimation methods to estimate the near-field non-stationary channel from the view of subarray and scatterer, respectively. Numerical results demonstrate that subarray-wise method can derive accurate channel estimation results with low complexity, whereas the scatterer-wise method can accurately position the scatterers and identify almost all the mappings between subarrays and scatterers.

I. INTRODUCTION

Extremely large-scale massive MIMO introduces near-field and spatial non-stationary channels that require revised transceiver design and channel-state information acquisition. The letter proposes subarray-wise and scatterer-wise estimation methods with complementary performance advantages.

  • Motivation: Extremely large aperture arrays and short array–user distances produce near-field and spatially non-stationary channel conditions.Near-field propagation involves spherical rather than planar wavefronts, while different array regions can receive different power levels or miss paths.
  • Problem: Traditional LS and linear minimum mean square error channel estimators cannot satisfy the resulting estimation requirements.The requirements arise from changed transceiver design and the need to acquire channel state information under these channel conditions.
  • Problem: Few studies simultaneously position scatterers and identify the visible regions linking them to subarrays.Prior work addressed scatterer positioning in near-field stationary channels and visible-region estimation separately.
  • Proposed methods: The letter models the multipath channel with last-hop scatterers under spherical wavefronts and divides the large aperture array into subarrays.The subarray-wise method estimates visible scatterers per subarray using refined OMP, whereas the scatterer-wise method uses multiple subarrays jointly.
  • Results: The subarray-wise method achieves excellent MSE performance with low complexity, while the scatterer-wise method accurately positions scatterers and identifies almost all subarray–scatterer mappings.These methods estimate the near-field non-stationary channel from subarray and scatterer perspectives, respectively.

II. SYSTEM MODEL

The system model considers a single-user uplink to an extremely large ULA, whose aperture is divided into subarrays to represent spatially non-stationary propagation. Scatterers, including the user antenna for the LoS path, are assigned coordinates within the ULA’s visible region.

  • Array geometry: The base station uses an M-element ULA, where M can be 10^3 or larger, with aperture (M − 1)d.The ULA center is the origin, and the array lies along the y-axis.
  • Array geometry: The ULA is uniformly divided into N subarrays, each containing M/N antennas.This division supports the model of spatial non-stationarity across the large aperture.
  • Propagation: A single-antenna user lies in the positive x-axis region and communicates through a line-of-sight path or reflections from multiple last-hop scatterers.The user antenna is treated as a scatterer for the LoS path.
  • Scatterer representation: Scatterer coordinates (x_s, y_s) lie within the ULA’s visible-region bounds, with S scatterers counted including the user antenna.Distances and coordinates are normalized by the carrier wavelength, and the single-user formulation relies on identical independent user-channel estimations.
  • Scope: The ULA model can be extended to UPA cases by using a three-dimensional coordinate system.The letter reduces the more widely used UPA to a ULA for analysis.

1) Near-field property:

The near-field non-stationary model uses spherical-wave array responses and explicit mappings between subarrays and visible scatterers. Pilots enable the base station to estimate the channel, scatterer coordinates, and visibility mappings for later design.

  • 1) Near-field property:: A scatterer at (x, y) stimulates an M-dimensional array response under spherical-wave propagation.The response depends on distances from the scatterer to the ULA and to each array element.
  • 2) Non-stationarity:: Non-stationarity permits a subarray to miss some scatterers while each scatterer is visible to at least one but not all subarrays.Subarray-wise and scatterer-wise visibility sets encode the two directions of the mapping.
  • 2) Non-stationarity:: The channel model combines each scatterer’s complex attenuation, spherical-wave response, and a selection vector identifying the subarrays that see it.The selection vector enters through a Hadamard product in the multipath channel expression.
  • 2) Non-stationarity:: All-1 pilots are transmitted during estimation, and the received signal is modeled as √P h + w.Given the received pilot, the base station estimates h and the scatterer coordinates and visibility mappings.
  • 2) Non-stationarity:: With more subarrays observing a scatterer, the ULA array pattern becomes more directional.This directionality motivates estimating the channel from the scatterer view.

III. CHANNEL ESTIMATION METHODS

The methods estimate near-field non-stationary channels using sparse signal recovery and array-pattern radiation power over spatial coordinates. They identify scatterer positions and mappings between subarrays and scatterers.

  • The paper develops two methods to estimate the near-field non-stationary channel and determine subarray–scatterer mappings.
  • The array pattern evaluates radiation power from a signal toward spatial directions in the ULA’s visible region.
  • The spatial search uses a coordinate grid Ξ defined by sampled x- and y-axis positions, subject to nonempty visible-subarray mappings.
  • The strongest radiation power appears at the target position, demonstrating array-pattern directionality and spatial channel sparsity.
  • Near-field channel sparsity in the spatial domain permits OMP-based extraction of paths from the noisy mixture.

B. Subarray-wise channel estimation

The subarray-wise method exploits stationarity within each subarray and estimates each subchannel separately using a refined OMP procedure. Multilayer grids address the accuracy–complexity trade-off in spatial localization.

  • A subarray is the smallest unit assumed to experience stationarity, so each subchannel can be treated individually.
  • The method applies OMP separately to each subarray’s received signal, extracting one path per iteration from the residual.
  • The spatial-domain representation projects the antenna-domain channel onto uniformly sampled xy-directions.
  • Dense spatial grids improve coordinate accuracy but make exhaustive search time-consuming.
  • The refined OMP algorithm uses multilayer grids to obtain coarse and refined coordinate estimates before estimating each path’s attenuation factor.

1) Coarse estimation:

Coarse estimation selects a high-layer grid point with maximum radiation power, then refines the coordinate locally on a denser lower-layer grid. The stopping procedure reconstructs subarray channels and records visible scatterers.

  • Coarse estimation:: The higher-layer grid ΞH uses substantially larger x- and y-axis steps than the lower-layer grid.
  • Coarse estimation:: The coarse coordinate estimate is the ΞH point with the largest radiation power from the residual-derived signal.
  • Coarse estimation:: The lower-layer grid densely samples a local region around the coarse estimate, and the refined coordinate maximizes the corresponding radiation-power measure.
  • The refined OMP iterations terminate when only noise remains in the residue, after which the estimated subarray channel is reconstructed.
  • The procedure identifies the scatterers visible to each subarray and estimates their associated channel contributions.
  • When subarray size M/N is small, spatial-resolution loss greatly affects coordinate accuracy, preventing the base station from recognizing identical scatterers across subarrays.

C. Scatterer-wise channel estimation

The scatterer-wise method jointly processes all subarrays that observe a scatterer, using their combined array gain and directionality to estimate scatterers and their mappings.

  • Joint manipulation of multiple subarrays can maximize array gain and improve system efficiency.
  • The method estimates the channel directly from the scatterer perspective rather than treating subarrays independently.
  • Using more subarrays that see a scatterer makes the array-pattern directionality more distinct.
  • The method jointly uses all observing subarrays to position each scatterer and determine its visible-subarray mapping.
  • The scatterer-wise procedure is based on refined OMP and estimates a scatterer’s coordinate, attenuation factor, and observing-subarray set from the residual.

1) Coarse estimation:

The scatterer-wise method first coarsely estimates each scatterer’s position, visible subarrays, and gain, then refines these estimates.

  • Coarse estimation:: The method derives each scatterer’s visible-subarray set from its coarse position and radiation power across candidate subarrays.Subarrays with the largest normalized radiation power are selected when their cumulative power exceeds threshold δ, with 0 < δ < 1.
  • Coarse estimation:: It avoids estimating the scatterer gain directly from coarse position and visibility because the resulting gain accuracy would be low.The gain is instead estimated after the position has been refined.
  • Coarse estimation:: The algorithm refines coarse estimates of scatterer coordinates, visible subarrays, and gains in subsequent steps.The refined quantities are denoted (˜x_s, ˜y_s), ˜Φ_s, and ˜g_s.

2) Refined estimation:

The refined estimation procedure iteratively updates scatterer coordinates, visible-subarray mappings, and gains before reconstructing the channel, with higher complexity than the subarray-wise method.

  • Refined estimation:: Scatterer coordinates are refined after the coarse estimate, while visible-subarray sets are refined using the array-pattern directionality.The refined visible set includes subarrays whose calculated radiation power satisfies the prescribed criterion.
  • Refined estimation:: Algorithm 2 iteratively detects active subarrays, estimates scatterer coordinates and visible regions, refines them, and reconstructs the channel.Its steps alternate coarse and refined estimates of (x_s, y_s), Φ_s, and g_s until the stopping condition is met.
  • Refined estimation:: The final reconstructed channel is obtained by inserting the refined scatterer parameters into the channel model.The refined coordinates, gains, and visible-subarray mappings are used together for reconstruction.
  • Refined estimation:: The subarray-wise method has lower complexity because it avoids estimating several scatterer-wise parameters.It estimates each subarray channel from its own observation, whereas the scatterer-wise method uses observations containing interference from other subarrays.
  • Refined estimation:: The scatterer-wise method uses multiple-subarray array gain to achieve more accurate scatterer positioning and mapping.This supports comprehensive globalized transceiver design, while the subarray-wise method suits low-complexity subarray-based design.

IV. NUMERICAL RESULTS

Numerical results show that both proposed methods outperform LS in MSE, while their positioning and mapping strengths depend on SNR and the number of subarrays.

  • IV. NUMERICAL RESULTS: The proposed methods achieve considerably lower MSEs than LS, whose MSE exceeds 10^-2 at SNR = 20 dB.The LS drawback is attributed to increasing noise, while Fig. 3 compares the methods for N ∈ {4, 16}.
  • IV. NUMERICAL RESULTS: In the high-SNR region, the lower-complexity subarray-wise method has better MSE performance than the scatterer-wise method.This agrees with the paper’s complexity and estimation analysis.
  • IV. NUMERICAL RESULTS: When SNR is less than 10 dB, neither method works well for scatterer positioning and non-stationary mapping.Successful detection is evaluated using a detected scatterer within distance 10, normalized by wavelength, and visible to the subarray.
  • IV. NUMERICAL RESULTS: When N = 4, the subarray-wise method can accurately position scatterers, whereas the scatterer-wise method outperforms it as N increases.The gap widens with N because smaller subarrays degrade refined OMP accuracy, while integrated array gain preserves scatterer-wise positioning and mapping accuracy.
  • IV. NUMERICAL RESULTS: When N = 16 and SNR ≥ 15 dB, the subarray-wise method’s successful detection ratio exceeds 0.9.The passage describes this result as demonstrating the effectiveness of the scatterer-wise method.

V. CONCLUSION

The paper models near-field non-stationary massive MIMO channels through scatterer–subarray mappings and proposes two estimation perspectives with different strengths.

  • V. CONCLUSION: The introduced channel model describes near-field non-stationary properties in extremely large-scale massive MIMO systems.The model represents non-stationarity through mappings between scatterers and subarrays.
  • V. CONCLUSION: The subarray-wise method uses stationarity within each subarray to position the scatterers visible to that subarray.It is designed for subarray-based transceivers.
  • V. CONCLUSION: The scatterer-wise method uses array gain to position each scatterer and determine its mapping with subarrays simultaneously.It is designed for joint subarray transceivers.
  • V. CONCLUSION: The low-complexity subarray-wise method provides better MSE performance, whereas the scatterer-wise method accurately positions scatterers and finds almost all mappings.The conclusion separates the methods’ estimation-accuracy and mapping strengths.
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