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Modeling and Performance Analysis for Movable Antenna Enabled Wireless Communications

Lipeng Zhu, Wenyan Ma, Rui Zhang

arXiv:2210.05325v2cs.ITeess.SP

TL;DR

The paper addresses how fixed antenna locations limit exploitation of spatial channel variation. It proposes a far-field field-response model for movable antennas and analyzes their maximum channel gain and outage behavior, finding performance gains over fixed-position and antenna-selection systems, especially with more paths.

  • Problem

    Fixed-position antennas cannot fully exploit spatial channel variation, while antenna selection needs increasingly more elements to achieve higher diversity orders.

  • Method

    The paper develops a far-field field-response channel model and analyzes a single receive MA's maximum channel gain against an FPA in deterministic and stochastic channels.

  • Results

    The analysis and simulations show that MAs can achieve considerable gains over FPA and antenna-selection systems, with stronger gains as the number of channel paths increases.

  • Takeaways & Limitations

    Movable antennas are especially applicable to slowly varying, low-mobility systems where spatial diversity can complement unavailable time and frequency diversity.

Abstract

from arXiv · show

In this paper, we propose a novel antenna architecture called movable antenna (MA) to improve the performance of wireless communication systems. Different from conventional fixed-position antennas (FPAs) that undergo random wireless channel variation, the MAs with the capability of flexible movement can be deployed at positions with more favorable channel conditions to achieve higher spatial diversity gains. To characterize the general multi-path channel in a given region or field where the MAs are deployed, a field-response model is developed by leveraging the amplitude, phase, and angle of arrival/angle of departure (AoA/AoD) information on each of the multiple channel paths under the far-field condition. Based on this model, we then analyze the maximum channel gain achieved by a single receive MA as compared to its FPA counterpart in both deterministic and stochastic channels. First, in the deterministic channel case, we show the periodic behavior of the multi-path channel gain in a given spatial field, which can be exploited for analyzing the maximum channel gain of the MA. Next, in the case of stochastic channels, the expected value of an upper bound on the maximum channel gain of the MA in an infinitely large receive region is derived for different numbers of channel paths. The approximate cumulative distribution function (CDF) for the maximum channel gain is also obtained in closed form, which is useful to evaluate the outage probability of the MA system. Moreover, our results reveal that higher performance gains by the MA over the FPA can be acquired when the number of channel paths increases due to more pronounced small-scale fading effects in the spatial domain. Numerical examples are presented which validate our analytical results and demonstrate that the MA system can reap considerable performance gains over the conventional FPA systems with/without antenna selection (AS).

I. INTRODUCTION

The paper introduces movable antennas (MAs), whose positions can be adjusted within a spatial region to exploit channel variation beyond fixed-position antennas. It develops a far-field field-response model and studies MA channel gains, implementation, and practical operating conditions.

  • Motivation: Movable antennas adjust their positions within a spatial region to improve channel conditions and exploit continuous spatial-domain degrees of freedom.Unlike fixed-position antennas, MAs can move locally and continuously; antenna selection requires increasingly more fixed elements for higher diversity orders.
  • Motivation: MA systems can obtain full spatial diversity with fewer or even one moving antenna, whereas antenna selection uses increasingly more antenna elements as diversity orders grow.The paper contrasts this flexibility with the cost associated with larger antenna-selection systems.
  • Practical scope: MA performance gains depend on mechanically moving antennas fast enough relative to channel variation, making slowly varying or low-mobility scenarios especially suitable.The paper identifies IoT, smart-city, industrial, smart-home, NB-IoT, LoRa, and M2M settings as relevant examples.
  • Channel modeling: The proposed field-response model represents multi-path channels using path amplitudes, phases, and AoA/AoD information under far-field propagation.The model describes channel variation as antennas move through transmit and receive regions.
  • System model: The channel coefficient depends on the transmit and receive antenna positions, and the receive signal model includes transmit power, normalized signal, and Gaussian noise.The transmit and receive regions are represented with Cartesian coordinates, while flexible cables and drive components enable mechanical movement.

B. Relationship with Conventional Channel Models

The proposed channel model encompasses several conventional channel models under specific path-structure and scattering assumptions. These include LoS, geometric, Rayleigh, and Rician fading channels.

  • LoS channel: The model reduces to an LoS channel when there is exactly one transmit path and one receive path, Lt = Lr = 1.The path-response matrix and field-response vectors become scalars under this condition.
  • Geometric channel: The model represents a geometric channel when transmit and receive paths have one-to-one correspondence and the path-response matrix is diagonal.This requires Lt = Lr = L, with each diagonal coefficient corresponding to one path component.

3) Rayleigh Fading Channel:

For Rayleigh and Rician fading, the model uses many paths with specified scattering and path-response assumptions, while the deterministic analysis examines channel-gain behavior across the receive region. The paper also links the model to maximum-gain and SNR analysis for a movable receive antenna.

  • Rayleigh fading channel: Rayleigh fading requires infinitely many statistically independent transmit and receive paths, with isotropically distributed angles and i.i.d. CSCG path-response coefficients.Under these assumptions, superimposed path responses become CSCG through the central limit theory.
  • Rician fading channel: Rician fading adds an LoS path with constant amplitude to the many-path, uniformly scattered non-LoS components used for Rayleigh fading.The Rician factor is determined from the distinguished LoS entry of the path-response matrix.
  • Performance analysis: For a fixed transmit antenna and movable receive antenna, the effective path-response vector determines how receive channel gain and SNR vary with receive position.The analysis compares the maximum gain of one receive MA with an FPA at the reference position.
  • Deterministic channel: The deterministic multi-path channel gain is analyzed separately for one-path, two-path, three-path, and multiple-path cases, with periodic behavior illustrated for the receive region.The multiple-path analysis identifies a minimum real period for the channel-gain pattern.

1) One-Path Case:

The receive channel gain is constant for one path, but becomes spatially periodic with multiple paths, enabling position-based maximization. The achievable upper bound depends on path amplitudes and sufficient receive-region size.

  • 1) One-Path Case:: One receive path yields a constant channel gain, so changing the antenna position cannot provide an SNR gain over an FPA.The position changes only the channel phase in this case.
  • 2) Two-Path Case:: For two paths, the channel gain is periodic because the superimposed path power contains a cosine of the position-dependent phase difference.Along x_r, the period is λ/(ϕ_r,1 − ϕ_r,2); along y_r, it is λ/(ϑ_r,1 − ϑ_r,2).
  • 2) Two-Path Case:: For two paths, maximum-gain locations satisfy an integer phase-alignment condition and form parallel lines in the x_r-y_r plane.Adjacent maximum lines have a spacing determined by the path-response and virtual-AoA differences.
  • 2) Two-Path Case:: The tight upper bound for two paths is not achievable in a very small region, but a circular area with diameter larger than d_2 is sufficient.A larger difference between the two paths’ AoAs requires a smaller area to achieve the bound.

3) Three-Path Case:

With three receive paths, the channel gain combines pairwise cosine terms and remains periodic in the receive region. Its tight upper bound occurs when the path-phase arguments are simultaneously maximized, subject to sufficient region size.

  • 3) Three-Path Case:: For three receive paths, the channel gain is the sum of individual path powers and three pairwise cosine-interference terms.The pairwise terms are weighted by the products of the corresponding path amplitudes.
  • 3) Three-Path Case:: The three-path channel gain is periodic when a period vector preserves every pairwise cosine argument across the receive region.The period vector r_p satisfies |h_3(r)|^2 ≡ |h_3(r + r_p)|^2.
  • 3) Three-Path Case:: The tight upper bound is obtained when all pairwise cosine arguments reach their maximum value of one.The maximizing positions are intersections of the lines imposed by the phase-alignment constraints.
  • 3) Three-Path Case:: The tight three-path upper bound is not achievable in a very small region; a sufficient receive-region diameter condition is required.The cited text introduces a geometric diameter threshold but does not state its complete value.
  • 3) Three-Path Case:: The Fig. 5 example uses four receive paths with equal path-response magnitudes and specified receive-path phases, illustrating the channel-gain pattern.For more than three paths, the exact period generally cannot be obtained explicitly because the virtual AoAs are randomly distributed.

4) Multiple-Path Case:

For more than three receive paths, explicit channel-gain periods generally are unavailable, but quantized AoAs yield approximate periods useful for locating maximum gains. Small spatial movements can still improve channel conditions because multiple paths create many local maxima.

  • Multiple-Path Case:: The angular-domain representation resembles a 2D DTFT, with virtual AoAs corresponding to time and antenna positions corresponding to frequency.This analogy provides an interpretation of the spatial channel-gain periodicity.
  • Multiple-Path Case:: For Lr > 3, random virtual AoAs generally prevent obtaining an explicit channel-gain period.An approximate period can be derived by quantizing the virtual AoAs.
  • Multiple-Path Case:: The minimum period X is determined by requiring Xλ(ϕr,m −ϕr,n) to be an integer for every path pair.This condition follows from treating angular differences through the quantized-AoA representation.
  • Multiple-Path Case:: A receive region extending at least one period along xr can always contain a maximum channel gain for fixed yr.This conclusion applies under the period construction used for the quantized virtual AoAs.
  • Multiple-Path Case:: Higher angle-quantization resolution makes the estimated period more accurate and increases the likelihood that the maximum gain occurs within period X.Small T causes larger AoA approximation errors, while large T reduces them.
  • Multiple-Path Case:: The channel gain has a periodic character for Lr = 2, 3, whereas Lr ≥4 produces many local maxima without an explicit period.For Lr ≥4, prominent spatial small-scale fading enables performance improvement through sub-wavelength movement.

B. Stochastic Channel

The stochastic model assumes identically distributed complex Gaussian path responses and independent physical AoAs. Under far-field conditions, the expected gain is position-invariant, and movable antennas can exceed fixed-position antennas on average.

  • B. Stochastic Channel: The stochastic environment models path coefficients as i.i.d. CSCG variables with variance σ2/Lr and physical AoAs as i.i.d. random variables.The identical average path power assumption supports the i.i.d. coefficient model.
  • B. Stochastic Channel: Under the far-field assumption, the expected channel gain is the same at every point in the receive region.The reference point r0 = (0, 0) can therefore represent the expected gain throughout the region.
  • B. Stochastic Channel: The FPA has expected channel gain σ2, while the MA can find a position with larger gain and thereby increase average channel gain.The comparison is between a fixed antenna position and flexible placement within the receive region.

1) Single-Path Case:

With one receive path, the channel gain is spatially constant, so an MA provides no SNR gain over an FPA. With two paths, sufficiently large regions can attain the coherent upper bound, enabling average-SNR and outage analysis.

  • 1) Single-Path Case:: For one receive path, the channel gain is constant throughout the receive region.Consequently, moving the antenna cannot change the maximum gain.
  • 1) Single-Path Case:: Gmax,1 = G0 = σ2, so the MA acquires no SNR gain over the FPA in the single-path case.The maximum MA gain equals the expected gain at the reference point.
  • 1) Single-Path Case:: The single-path channel-gain CDF and corresponding outage probability are derived for arbitrary transmit power, noise power, and SNR threshold.The outage expression is obtained from the single-path gain distribution.
  • 2) Two-Path Case:: For two paths, the upper bound (|b1| + |b2|)2 is attainable when the receive-region diameter exceeds d2.This condition holds for any specified pair of receive virtual AoAs.
  • 2) Two-Path Case:: As the receive-region size approaches infinity, the two-path upper bound is attainable for arbitrary AoAs.The expected maximum gain and average-SNR gain can therefore be analyzed using the Rayleigh path-amplitude model.
  • 2) Two-Path Case:: The two-path maximum-gain CDF and its outage probability are derived from the upper-bound distribution.The outage expression uses transmit power, noise power, and the receive SNR threshold.

3) Three-Path Case:

For three receive paths, sufficiently large regions attain the coherent upper bound, whose average-SNR gain and outage behavior are characterized probabilistically. The CDF is approximated because the relevant square-sum has no closed form, and more than three paths require upper-bound analysis.

  • 3) Three-Path Case:: As the receive-region size approaches infinity, the three-path upper bound is attainable for arbitrary AoAs.The expected maximum gain and resulting average-SNR gain are then characterized using Rayleigh path amplitudes.
  • 3) Three-Path Case:: The three-path maximum gain is the square-sum of three i.i.d. Rayleigh variables, whose CDF has no closed-form expression.The paper therefore uses an approximation for the CDF.
  • 3) Three-Path Case:: The approximate three-path CDF yields an approximate outage probability for any transmit power, noise power, and receive SNR threshold.The outage expression is tied to the approximate distribution of the maximum channel gain.
  • 4) Multiple-Path Case:: For Lr > 3, the explicit maximum-gain expression and the CDF of the square-sum of independent Rayleigh variables are difficult or unavailable.The analysis instead derives an upper bound and an approximate CDF for that bound.
  • 4) Multiple-Path Case:: For multiple paths, the MA’s average-SNR gain over the FPA is upper-bounded using the maximum-gain upper bound.This extends the stochastic analysis beyond the cases with directly characterized gains.

5) Infinite-Path Case:

The infinite-path analysis bounds the maximum channel gain by modeling spatial samples as independent exponential variables. The resulting expected gain grows approximately logarithmically with receive-region size, and closed-form CDF bounds support outage analysis.

  • Spatial independence: Under isotropic scattering, channel coefficients at sufficiently separated positions are treated as independent CSCG variables.Positions spaced by at least λ/2 are statistically independent under the sinc-based correlation model.
  • Expected maximum gain: The expected maximum channel gain increases monotonically with receive-region size and approaches infinity for an infinite region.The lower-bound expectation is expressed through largest-order statistics of independent exponential random variables.
  • Infinite-path upper bound: The receive region is discretized into grids, whose centers provide an upper-bound construction for the maximum channel gain.The grid count satisfies NUB = ⌈PA + 1⌉^2 for a square region of size A × A.
  • Outage analysis: Closed-form CDF bounds for the maximum channel gain yield corresponding outage-probability bounds for any transmit power, noise power, and SNR threshold.The upper-bound CDF produces a lower outage-probability bound, while the lower-bound CDF produces an upper bound.
  • Expected maximum gain: The lower and upper expected-gain bounds increase approximately logarithmically with receive-region size.This behavior follows from approximating the relevant harmonic-series terms.

C. Implementation Issues of MA Systems

The implementation discussion identifies region size and antenna movement as practical design issues. It distinguishes fundamental performance relations from engineering constraints that affect deployment.

  • Implementation Issues of MA Systems: The analysis establishes relations between channel-gain improvement, the number of channel paths, and the region size available for antenna movement.The discussion then turns to operation frequency, antenna movement, and position optimization.
  • Implementation Issues of MA Systems: Larger movement regions can improve performance, but practical deployment must also account for the space required to install and move the antennas.The paper treats these considerations as implementation issues rather than changes to the analytical model.

1) Frequency Band:

The paper evaluates MA systems across frequency-related channel settings and numerical scenarios. MA gains increase with region size and path richness, while movement and optimization introduce practical costs and constraints.

  • Frequency Band: Frequency-band channel characteristics differ through their distributions of AoD, AoA, and path-response measures.The paper contrasts richer scattering at sub-6 GHz with different characteristics at higher frequencies.
  • Antenna Position Optimization: Position optimization can use exhaustive grid search for small systems, whereas large regions and multiple MAs favor channel-map-based optimization.SCA, gradient methods, and alternating optimization are identified as candidate techniques.
  • Numerical Results: MA position optimization increases the relative SNR gain as the receive region grows, with larger regions needed when more channel paths are present.For Lr = 2 and Lr = 3, upper bounds are reached at approximately 4λ and 5λ, while gains continue increasing beyond 8λ for Lr = 5.
  • Numerical Results: MA performance is comparable to DBF and better than AS despite AS and DBF using M times more antennas than MA.DBF has higher hardware cost because its receive antennas require multiple RF chains.
  • Numerical Results: The analytical CDF matches simulations exactly for Lr = 2 and deviates slightly for Lr = 3 and Lr = 5 because approximations are used.The reported deviation is small enough for the approximations to support outage analysis.
  • Numerical Results: MA achieves lower outage probability than FPA and AS at fixed SNR thresholds, and can outperform DBF in the low-SNR region.The performance gap over benchmarks increases with the number of receive paths.
  • Numerical Results: Relative SNR gain and maximum SNR gain increase with the number of paths because stronger small-scale fading creates larger spatial channel fluctuations.For sufficiently many paths, gain approaches the analytical bound as the receive region expands.
  • Numerical Results: Multiple-MA relative SNR gain increases with region size and approaches an analytical upper bound when antennas can occupy distinct favorable positions.The multiple-MA evaluation uses four MAs separated by at least half a wavelength.

C. Performance Evaluation Based on 3GPP Channel Model

The 3GPP wideband evaluation studies achievable rates for MA, FPA, AS, and DBF in indoor-factory LoS and NLoS settings. MA achievable rates increase with the movement-region size.

  • C. Performance Evaluation Based on 3GPP Channel Model: The evaluation uses a 3GPP indoor-factory dense-clutter, high-BS scenario measuring 20 × 20 × 10 m^3, with BS and UT heights of 10 m and 1.5 m.The horizontal BS–UT distance is fixed at 20 m.
  • C. Performance Evaluation Based on 3GPP Channel Model: The smart-factory motivation targets fixed or low-mobility machines, vehicles, and robots whose propagation environments vary slowly.These settings are identified as suitable for installing MAs.
  • C. Performance Evaluation Based on 3GPP Channel Model: Achievable rates are compared among MA, FPA, AS, and DBF under both 3GPP LoS and NLoS wideband channel models.The figures report effective maximum achievable rate as a function of the movement-region size.
  • C. Performance Evaluation Based on 3GPP Channel Model: The effective achievable rate of the MA system increases with the size of its movement region under the evaluated wideband channel models.This extends the region-size performance trend to the 3GPP channel evaluation.
  • Conclusion: The paper concludes that flexible MA movement can place antennas in more favorable spatial channel conditions and improve spatial diversity gains.The field-response model characterizes multipath channels using path amplitudes, phases, and AoA/AoD information under far-field conditions.
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