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A New Analytical Approximation of the Fluid Antenna System Channel

Malek Khammassi, Abla Kammoun, Mohamed-Slim Alouini

arXiv:2203.09318v3cs.ITeess.SPeess.SY

TL;DR

FAS performance analysis requires a channel model that captures highly correlated ports while remaining analytically tractable. This paper proposes a two-stage approximation, reducing multi-fold outage integrals and producing a single-integral representation; numerical results validate the approximations but show limited performance gain under realistic assumptions.

  • Problem

    Highly correlated FAS ports following Jake’s model make outage analysis difficult, while prior simplified parameterizations may not accurately capture that correlation.

  • Method

    The paper introduces a two-stage FAS channel approximation with more model parameters, first reducing multi-fold outage integrals and then approximating the result using a single integral.

  • Results

    Numerical results validate the proposed approximations and show limited performance gain under the less-idealized correlation model.

  • Takeaways & Limitations

    FAS outage performance can saturate as the number of ports increases, constraining performance through space limitations.

  • Takeaways & Limitations

    The analysis uses a less-idealized correlation model in which highly correlated ports produce a detrimental effect on diversity gain.

Abstract

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Fluid antenna systems (FAS) are an emerging technology that promises a significant diversity gain even in the smallest spaces. Motivated by the groundbreaking potentials of liquid antennas, researchers in the wireless communication community are investigating a novel antenna system where a single antenna can freely switch positions along a small linear space to pick the strongest received signal. However, the FAS positions do not necessarily follow the ever-existing rule separating them by at least half the radiation wavelength. Previous work in the literature parameterized the channels of the FAS ports simply enough to provide a single-integral expression of the probability of outage and various insights on the achievable performance. Nevertheless, this channel model may not accurately capture the correlation between the ports, given by Jake's model. This work builds on the state-of-the-art and accurately approximates the FAS channel while maintaining analytical tractability. The approximation is performed in two stages. The first stage approximation considerably reduces the number of multi-fold integrals in the probability of outage expression, while the second stage approximation provides a single integral representation of the FAS probability of outage. Further, the performance of such innovative technology is investigated under a less-idealized correlation model. Numerical results validate our approximations of the FAS channel model and demonstrate a limited performance gain under realistic assumptions. Further, our work opens the door for future research to investigate scenarios in which the FAS provides a performance gain compared to the current multiple antennas solutions.

I. INTRODUCTION

Fluid antenna systems let one antenna switch among closely spaced ports to select the strongest signal, potentially providing diversity without conventional spatial separation. This paper develops a more accurate, analytically tractable FAS channel approximation because prior models may not capture Jake’s port correlation and can overstate performance.

  • FAS concept: FAS lets a single antenna switch among positions in a small linear space and select the port with the strongest signal.The possible antenna positions are treated as ports, following selection diversity.
  • Motivation: FAS can provide multiple-antenna diversity at the receiver without conventional space limitations.The system exploits multipath and closely spaced antenna positions rather than requiring ports separated by at least half a radiation wavelength.
  • Practical challenge: Selecting the strongest port may require SNR observations from every port, causing potentially unbearable switching delays in practice.Prior work addressed this issue using machine learning and analytical approximation from a subset of observed ports; observing 10% of the ports provided more than an order of magnitude outage-probability reduction.
  • Channel-model gap: Prior FAS analysis used a simplified channel parameterization that may not accurately capture dependence between ports under Jake’s model.That imposed covariance structure can produce an optimistic performance analysis.
  • Proposed approach: The paper adds channel-model parameters to approximate Jake’s correlation more flexibly while retaining a tractable outage-probability expression.Its two stages first reduce the number of multi-fold integrals, then approximate the result with a power of a single integral.
  • Findings: Numerical results validate the approximations but show limited performance gain compared with the prior model under the more complex correlation assumptions.The authors attribute the limited gain to the detrimental effect of highly correlated ports on diversity gain, motivating careful modeling to identify when FAS can outperform traditional multiple-antenna solutions.

II. FAS CHANNEL

The FAS models a single antenna that switches among N positions along a linear space to select the strongest channel magnitude. Port spacing induces channel correlation following Jake’s model, while practical switching can introduce delays.

  • FAS uses one antenna that can move among N equally distributed ports along a linear space of length Wλ.The distance from the first port to port k is specified by the port-position parameterization.
  • The received signal at port k is modeled as r_k = g_kq + n_k, with q as the transmitted symbol and n_k as complex AWGN.
  • The channel coefficients are correlated complex Gaussian variables with covariance matrix Σ_g under isotropic scattering.Spatial separation creates phase differences among arriving paths, inducing correlation through Jake’s model.
  • The proposed model represents correlated Rayleigh fading channels using additional parameters, increasing flexibility at the cost of complexity.The earlier channel model is identified as a particular case of the more general representation.
  • Instant switching between ports can be difficult in practice because moving physical materials causes delay.Smaller antennas at higher frequencies and digitally controlled mini pixels are discussed as possible routes toward faster switching.

B. Exact Model

The exact model represents the correlated FAS channel vector through the eigenstructure of its covariance matrix. Although exact, its FAS-channel CDF requires N-fold integrals for N > 3.

  • The exact representation chooses model parameters from the eigenvalues and eigenvectors of Σ_g so that g and h have the same joint distribution.The representation uses the eigenvalues s_l and associated eigenvector components u_k,l.
  • Theorem 1 provides an exact representation of the correlated channel vector g using the general model.
  • For N > 3, the exact model’s CDF of the FAS selection combiner can only be written as N-fold integrals.This motivates approximating the joint distribution while retaining analytical tractability.

1) First Stage Approximation:

The first-stage approximation truncates the covariance representation using an ϵ-rank below N, preserving tractability while controlling approximation accuracy. It yields explicit CDF and outage expressions with fewer multi-fold integrals.

  • The approximation ˆg is constructed to remain close to g while making the CDF of g_FAS analytically tractable.The approximation is controlled by the threshold ϵ and its associated ϵ-rank.
  • Theorem 3 establishes convergence of max{|ˆg_1|, …, |ˆg_N|} to g_FAS as ϵ approaches zero.
  • Including independent Gaussian variables makes channel magnitudes conditionally independent, so their conditional joint CDF becomes a product of conditional CDFs.This structure enables the first step toward deriving the FAS-channel CDF.
  • For ϵ-rank < N, the approximated CDF and outage probability contain 2 × ϵ-rank multi-fold integrals instead of N-fold integrals.Increasing ϵ reduces ϵ-rank and improves tractability, whereas decreasing ϵ improves approximation accuracy.
  • The tractability–accuracy trade-off requires choosing ϵ small enough to reduce multi-fold integrals while maintaining high approximation accuracy.
  • The asymptotic estimate ϵ-rank ≈ 2W N/(N−1) becomes accurate only for very large N beyond the tested FAS range, motivating a fitted constant a.

2) Second Stage Approximation:

The second-stage approximation replaces the first-stage matrix structure with one that supports a product of single integrals. An optimized parameter R controls the approximation of the maximum channel magnitude distribution.

  • ˆG has dependent rows and independent columns, whereas ˜G has independent rows and dependent columns, with matched column distributions.
  • The parameter R is selected by minimizing a covariance-based distance between the distributions of ˆG and ˜G.The relaxed optimization restricts candidate values to divisors of N, and a further approximation simplifies its objective.
  • Theorem 8 identifies the optimal relaxed value R* as the greatest divisor of N satisfying the stated condition.
  • Simulation results show that the second-stage approximation remains accurate even when R* is not a divisor of N.
  • The resulting CDF and outage approximations use the selected R and the first-stage ϵ-rank to approximate the FAS channel distribution.
  • The second-stage approximation designs ˜G so that the CDF of the maximum magnitude is a power of single integrals rather than N multi-fold integrals.

IV. NUMERICAL RESULTS

Numerical results show that Jake’s correlation model has a sharply decaying eigenvalue profile, enabling accurate low-rank approximation and substantial reduction in outage-probability computation. Accuracy improves with larger ϵ-rank, while dominant-eigenvalue counts vary with port number and antenna-array length.

  • Motivation: Jake’s correlation model has only 3% of eigenvalues exceeding 10^-4 for W = 0.2, compared with 97% for Ref..
  • Eigenvalue profile: For N = 200 and W = 0.2, more than 95% of Σg eigenvalues are below 3 × 10^-15, supporting ϵ-rank = 9.
  • Parameter effects: At fixed W = 2, eigenvalues exceeding 7.5 × 10^-15 decreased from 30% to 8% as N increased from 50 to 200.
  • Parameter effects: At fixed N = 200, eigenvalues exceeding the threshold increased from 6.5% to 8% as W increased from 1 to 2.
  • First-stage approximation: Using only dominant eigenvalues yields ϵ-rank ≪ N, reducing multi-fold outage integrals while maintaining a satisfactory approximation.The eigenvalue count increases with W and decreases with N because larger W decorrelates ports, whereas larger N makes them closer and more correlated.
  • Approximation accuracy: The first-stage approximation improves as ϵ decreases and ϵ-rank increases, because more dominant eigenvalues are retained.
  • First-stage approximation: For W = 1, N = 100, and σ = 10, high accuracy was obtained with ϵ-rank = 5, reducing the integral count from 100 to 10.The reduction is reported as 0.9 and is described as a considerable computational gain.

2. On the other hand,

The paper develops two approximations of the FAS channel: a first stage that reduces multi-fold integrals and a second stage that yields a single-integral CDF. Simulations show improved fidelity over [2] under tested port-density settings, while outage performance often saturates with increasing port count.

  • First Stage Approximation: The first-stage approximation uses ϵ-rank ≈⌈3.1935W N N−1⌉, with the coefficient obtained by minimizing mean squared error and validated by simulation.For N = 100 and W = 1, ϵ-rank = 4 provides a satisfactory CDF approximation.
  • First Stage Approximation: For N ∈{10, . . . , 300} and W ∈[0.1, 5], ϵ-rank is much smaller than N, considerably reducing the number of multi-fold integrals.The approximation varies rapidly with W and slowly with N.

V. CONCLUSION

The paper develops a two-stage FAS channel approximation that more closely follows Jake’s correlation model while retaining analytical tractability. Under less-idealized correlation, outage probability saturates with increasing N, limiting performance gains but leaving scope for future designs.

  • Conclusion: The proposed two-stage approximation incorporates more parameters to model FAS-port correlation closely according to Jake’s model.It addresses the analytical challenge created by highly correlated ports and simplified correlation models in prior work.
  • Conclusion: The first approximation reduces the number of multi-fold integrals in the outage-probability expression.
  • Conclusion: The second approximation represents FAS outage probability as a power of a single integral.
  • Conclusion: Numerical results assess approximation accuracy and compare the proposed model with previous related work.
  • Conclusion: Outage probability saturates rather than decreasing without a floor as N increases, constraining FAS performance under space-limited, less-idealized correlation.
  • Conclusion: The work leaves open implementation limitations, including delay and frequency deviations affecting CSI estimation at the ports.
  • Conclusion: Future research can investigate designs where diversity gain is guaranteed and where FAS gains can be realized despite implementation limitations.

APPENDIX A

Appendix A derives the outage expression by transforming the covariance structure and conditioning on auxiliary random variables. Conditional independence yields Rician channel magnitudes and enables a product-form maximum-distribution expression.

  • Appendix A: Diagonalizing the covariance matrix represents Σg through its eigenvectors and eigenvalues, with a truncated diagonal matrix used for approximation.
  • Appendix A: Because g and h have zero means and the same covariance matrix, they have the same joint distribution.
  • Appendix A: Conditioned on auxiliary vectors a and b, the approximated channel components are independent and each magnitude follows a Rician distribution.
  • Appendix A: Independence permits writing the joint distribution as a product involving the conditional component distributions and the density of a and b.
  • Appendix A: The maximum-channel CDF is obtained by evaluating the joint CDF at equal arguments and substituting the auxiliary density and conditional CDF.
  • Appendix A: The appendix also introduces a matrix T_N and a Fourier-transform argument showing that only the zero integer index satisfies the stated condition for 0 < c < 1.
  • Appendix A: The covariance derivation establishes zero means for the approximated variables and derives the covariance structure used in Proposition 2.

APPENDIX F

Appendix F derives the distribution of transformed channel variables by conditioning on shared random vectors. It then relates the resulting covariance matrices through norm and block-structure arguments.

  • Appendix F: The appendix defines the maximum CDF of the approximated channel variables and auxiliary vectors used in its expectation representation.
  • Appendix F: Conditioned on a(k) and b(k), the variables ˜gk,r are independent across r, so their magnitudes have conditional Rician distributions.
  • Appendix F: The auxiliary vectors are identically distributed with the earlier variables, allowing their density to be reused in the distribution calculation.
  • Appendix F: The vectors γk and the scalar variables Zk are introduced to determine the distribution of the transformed channel quantity.
  • Appendix F: The appendix uses definitions of Σ˜G(R) and I(R), matrix norms, and covariance differences to bound or compare the approximating covariance structure.
  • Appendix F: The covariance comparison distinguishes matrices with matching off-diagonal elements but different diagonal behavior, motivating the subsequent block analysis.

APPENDIX H

Appendix H analyzes the block structure of covariance matrices and selects a divisor R of N for the approximation. The choice is tied to minimizing a matrix norm after subtracting σ2 from diagonal blocks.

  • Appendix H: The relevant covariance matrix is organized into block forms, including zero and R × R identity blocks in the displayed structure.
  • Appendix H: ΣG(R) is block diagonal with R equal N × N blocks, whereas I(R) has N equal R × R blocks.
  • Appendix H: For R dividing N, ΣG(R) − σ2I(R) is block diagonal with R equal N × N blocks.
  • Appendix H: R determines the diagonal-block size from which σ2 is subtracted in the covariance approximation.
  • Appendix H: The norm calculation reduces to examining one diagonal block because all diagonal blocks are equal.
  • Appendix H: Increasing R does not necessarily decrease the maximum absolute column sum of the resulting matrix.
  • Appendix H: Subtracting σ2 decreases diagonal-block entries’ absolute values exactly when those entries exceed σ2.
  • Appendix H: The selected R is the greatest divisor of N satisfying the stated norm condition.
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