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Capacity Analysis and Joint Gaussian Beam Pattern Optimization for Positioning-Assisted Communications

Yuanbo Liu, Rui Wang, Han Zhang, Shuojin Huang, Zaichen Zhang, Bingcheng Zhu

arXiv:2609.28196v1cs.ITmath.IT

TL;DR

Positioning error makes capacity analysis for positioning-assisted beamforming difficult, while prior approaches leave gaps in theoretical beam-design guidance across 2D and 3D. The paper derives closed-form capacity approximations and optimal beam patterns, with numerical results verifying the expressions and pattern.

  • Problem

    Channel capacity analysis under positioning error is difficult, and existing CSI-free approaches lack sufficient theoretical guidance for beam-pattern design across 2D and 3D.

    Source: §I · p. 1

  • Method

    The paper models positioning-assisted beamforming in 2D and 3D and derives closed-form approximations for instantaneous and ergodic capacity and closed-form optimal beam patterns.

    Source: §I · p. 2

  • Results

    Numerical results verify the derived capacity expressions and optimal beam patterns, which maximize asymptotic ergodic capacity in both 2D and 3D.

    Source: §I · p. 2, §VII · p. 10

  • Takeaways & Limitations

    The derived expressions provide a basis for designing beam patterns to maximize channel capacity in positioning-assisted systems.

    Source: §VII · p. 10

Abstract

from arXiv · show

Positioning-assisted beamforming is a novel enabling technology for massive multiple-input multiple-output in modern wireless communications, as it provides reliable beam steering without requiring explicit channel state information. However, the channel capacity analysis under positioning error remains mathematically intractable, which limits both performance characterization and beamforming design. In this work, we derive instantaneous and ergodic channel capacity approximations in closed forms for both two-dimensional and three-dimensional positioning-assisted beamforming systems. Based on the new expressions, we provide closed-form optimal joint Gaussian beam pattern that maximizes the asymptotic ergodic capacity. Numerical results verify the theoretical capacity expressions and the optimal beam pattern. The derived expressions enable efficient beam design to maximize channel capacity and provide a theoretical basis for positioning-assisted beamforming design.

I. INTRODUCTION

The paper addresses the lack of theoretical guidance for capacity and beam design under positioning error in 2D and 3D positioning-assisted systems. It models those systems and derives capacity expressions and capacity-maximizing beam patterns.

I. INTRODUCTION

  • Prior CSI-free beamforming lacked sufficient theoretical guidance for beam design, while existing approaches had limits in positioning-error models, dimensional scope, or computational demands.The introduction identifies numerical optimization for 3D designs and an open problem of simple capacity-maximizing beam design in 2D and 3D.Source: §I · p. 1
  • Closed-form instantaneous and ergodic capacity expressions and capacity-maximizing beam patterns are derived for positioning-assisted beamforming in both 2D and 3D.The expressions consider link distance, transmit power, positioning error, and beam pattern.Source: §I · p. 2

II. SYSTEM MODEL

  • The system model expresses received power using transmitter peak power density, receiver effective area, antenna gain, and free-space propagation loss.The propagation loss depends on link distance.Source: §II · p. 2

A. 2D Positioning-Assisted Beamforming Model

  • The 2D model represents positioning error with a bivariate Gaussian distribution and covariance whose eigenvalues give variances along the PDF contour's major and minor axes.The beam points toward the estimated receiver position; its gain depends on angular error and 3-dB beamwidth.Source: A. 2D Positioning-Assisted Beamforming… · p. 2

B. 3D Positioning-Assisted Beamforming Model

  • The 3D model represents positioning error with a trivariate Gaussian distribution and a rotated covariance whose eigenvalues describe variance along three axes.Its antenna gain depends jointly on horizontal and vertical angular deviation, with beamwidths and rotation describing the beam pattern; the beam points toward the estimated position.Source: B. 3D Positioning-Assisted Beamforming… · pp. 2–3

III. 2D ASYMPTOTIC CAPACITY EXPRESSIONS

The 2D analysis expresses ergodic capacity as an integral over positioning error and develops closed-form approximations because that integral is difficult to evaluate analytically. Two limiting beamwidth regimes motivate a polynomial expansion that preserves both asymptotic cases.

A. 2D Ergodic Capacity in an Integral Form

1) Case 1:

  • For a sufficiently narrow beam, the instantaneous capacity concentrates near the origin, allowing the positioning-error PDF in the ergodic-capacity integral to be approximated by its value at the origin.This is the regime θ3dBd/σx → 0.Source: 1) Case 1 · p. 3

2) Case 2:

  • For a sufficiently wide beam, the beam nearly covers the possible positioning-error range and instantaneous capacity is nearly flat around the origin.This is the regime θ3dBd/σx → +∞.Source: 2) Case 2 · p. 3

C. Approximations to C2D

  • The paper proposes an approximation to the 2D instantaneous capacity that satisfies both asymptotic cases and yields a closed-form ergodic-capacity expression.A non-negative second-order Taylor approximation alone has no remaining degrees of freedom to satisfy both cases, motivating an added fourth-order term.Source: C. Approximations to C2D · p. 4

D. Calculation of κ and Asymptotic Approximation to C2D

The proposed 2D approximation uses a coefficient κ to match both asymptotic cases and produce a closed-form ergodic-capacity approximation. It improves the instantaneous-capacity fit, although it has a discontinuity at x0.

D. Calculation of κ and Asymptotic Approximation to C2D

  • The coefficient κ is chosen using the two asymptotic cases, enabling a closed-form approximation to the 2D ergodic capacity.The expression uses truncated second and fourth moments of the positioning-error variable.Source: D. Calculation of κ and Asymptotic… · pp. 4–5
  • In the narrow-beam case, instantaneous capacity is concentrated near the origin while the positioning-error PDF is nearly constant there, so the PDF can be approximated by its value at the origin.The paper notes that the positioning error is highly concentrated around the origin and the beam can cover most of that region.Source: D. Calculation of κ and Asymptotic… · p. 5
  • The approximation in (38) is two-sided asymptotic and remains accurate under both narrow-beam and wide-beam conditions, covering both beamwidth regimes.Source: D. Calculation of κ and Asymptotic… · p. 5
  • The approximation has a discontinuity at x0, but still satisfies both asymptotic cases.The text attributes the discontinuity to using the root of the quadratic rather than the fourth-order approximation.Source: D. Calculation of κ and Asymptotic… · p. 5
  • In the wide-beam case, instantaneous capacity is nearly constant while the positioning-error PDF behaves like an impulse around the origin, allowing capacity to be approximated by its value there.The second-order approximation deviates away from the origin, but the deviation has negligible impact when the beamwidth is sufficiently large.Source: D. Calculation of κ and Asymptotic… · p. 5

IV. 3D ASYMPTOTIC CAPACITY EXPRESSION

The 3D analysis reduces ergodic capacity to an integral over the joint positioning-error distribution and constructs a polynomial approximation to obtain a closed-form expression. The resulting capacity depends on link distance, beamwidth, and positioning error, which the paper says should be jointly optimized.

A. 3D Ergodic Capacity in an Integral Form

  • The 3D ergodic capacity is computed by marginalizing the positioning-error distribution and integrating instantaneous capacity over the joint PDF of the two transverse error coordinates.The 3D approximation begins from instantaneous capacity under the assumption that positioning error is small compared with link distance.Source: A. 3D Ergodic Capacity in an Integral… · pp. 5–6

B. Approximations to C3D

  • A polynomial expansion approximates 3D instantaneous capacity while matching two asymptotic cases, and the coefficient η is selected to enforce that match.The closed-form ergodic-capacity approximation uses truncated zero, second, and fourth moments of a Gaussian distribution.Source: B. Approximations to C3D · p. 6
  • The 3D ergodic capacity is governed by link distance, beamwidth, and positioning error, so these parameters should be jointly optimized to maximize capacity.The paper describes this behavior as consistent with its 2D case.Source: B. Approximations to C3D · p. 7

V. BEAM PATTERN OPTIMIZATION

The paper derives asymptotic capacity-based Gaussian beam optima for 2D and 3D systems. The 2D optimum depends on link distance and x-axis error variance, while the 3D optimum matches the beam and positioning-error distribution’s spatial characteristics.

A. 2D Optimal Beam Pattern

  • For 2D optimization, the authors approximate capacity asymptotically to avoid direct optimization involving Q and Li functions, then set (62) to zero to obtain the optimal beamwidth.Source: A. 2D Optimal Beam Pattern · p. 7
  • The 2D optimal beam is determined by link distance and x-axis positioning-error variance, and remains invariant to transmit-power changes.Source: A. 2D Optimal Beam Pattern · p. 7

B. 3D Optimal Beam Pattern

  • In 3D, the asymptotic case expands the integration region to R2 and takes η toward zero; gradient-zero optimization then yields A*.Source: B. 3D Optimal Beam Pattern · p. 7
  • In 3D, capacity is optimized when the beam pattern and positioning-error distribution share the same spatial characteristics, enabling more effective coverage of the user distribution.Source: B. 3D Optimal Beam Pattern · p. 7
  • The 3D quantity m depends on beam rotation angle ψ, and the optimal ψ is evaluated by combining the cited expressions.Source: B. 3D Optimal Beam Pattern · p. 7
  • Longer link distances favor narrower 3D beams, whereas larger positioning errors require wider beams to cover the potential user distribution.Source: B. 3D Optimal Beam Pattern · pp. 7–8

VI. NUMERICAL RESULTS

Numerical results compare the derived capacity expressions with simulation and characterize how distance, positioning error, transmit power, and beam design affect capacity in 2D and 3D. The conclusion states that appropriately designed closed-form beam patterns can maximize capacity.

VI. NUMERICAL RESULTS

  • Capacity varies nonmonotonically with 2D link distance, peaking at a moderate distance as beam coverage and path loss trade off.Smaller positioning errors and higher transmit power increase capacity; beamwidth affects coverage and interference.Source: §VI · p. 8
  • 2D capacity peaks at an intermediate beamwidth, while lower positioning error or higher transmit power raises capacity across beamwidths.The effect of link distance varies with beamwidth.Source: §VI · p. 8
  • In 2D, capacity decreases as positioning error grows; wider beam coverage slows the degradation compared with a narrow beam.For small divergence angles, capacity degrades rapidly at large error; the optimal beamwidth depends jointly on distance and positioning error.Source: §VI · p. 8
  • The simulated and analytical capacity curves agree, and the approximation remains sufficiently accurate beyond the two asymptotic conditions used to establish equivalence.The figures compute capacity using (28); “Exact” denotes (38) for 2D and (58) for 3D.Source: §VI · p. 8
  • In 3D, lower positioning error or higher transmit power increases capacity across link distances, while beamwidth effects vary with distance.The distance trends are similar to those observed in 2D.Source: §VI · pp. 8–9
  • For 3D beamwidth, lower positioning error shifts the optimal beamwidth smaller; higher transmit power improves capacity without changing the optimal beam pattern.The optimal point also shifts smaller as link distance increases.Source: §VI · p. 9, Figure 8 · p. 9
  • In 3D, wider divergence angles make capacity more robust to positioning error, whereas narrow beams show rapid capacity loss as error grows.The reported narrow-beam case uses θ3dB = 0.01 rad and ϕ3dB = 0.02 rad; the wide-beam case uses θ3dB = 0.45 rad and ϕ3dB = 0.55 rad.Source: §VI · p. 9, Figure 9 · p. 9
  • In 3D, the beam rotation angle that maximizes capacity should align with the positioning-error ellipse’s directional angle.When the error direction is zero, the maximum occurs at zero rotation.Source: §VI · p. 9, Figure 10 · p. 9

APPENDIX A PROOF OF THEOREM 1

The proof uses upper and lower bounds to establish the stated asymptotic relation. It applies the squeeze theorem under the condition θ3dB d/σx → 0.

APPENDIX A PROOF OF THEOREM 1

  • The proof bounds the relevant integral using the even, monotonically decreasing positioning-error density, then applies the squeeze theorem.The density is bounded above by its value at zero and below by its value at δ within |x| ≤ δ.Source: Appendix A · p. 10
  • Under θ3dB d/σx → 0, substituting δ = √σx · θ3dB d yields T1 → 1, and the transformed integral yields T2 → 1.Together, the bounds establish the asymptotic relation referenced as (73).Source: Appendix A · p. 10

APPENDIX B PROOF OF THEOREM 1

This proof uses the squeeze theorem to establish an asymptotic relation in the regime θ3dB d/σx → +∞. It bounds the integral using the even, monotonically decreasing instantaneous-capacity function.

APPENDIX B PROOF OF THEOREM 1

  • The proof bounds the integral using the even, monotonically decreasing instantaneous-capacity function and applies the squeeze theorem.The function is bounded above by its value at zero and below by its value at δ within |x| ≤ δ.Source: Appendix B · p. 10
  • For θ3dB d/σx → +∞, the proof obtains T3 → 1 and T4 → 1, establishing the asymptotic relation referenced as (82).The argument combines these limits with the bounds in (86).Source: Appendix B · pp. 10–11

APPENDIX C DERIVATION OF CLOSED-FORM EXPRESSION OF M 3D

The derivation reduces a 3D integral using eigenvalue decompositions and expresses the resulting distribution through the Hoyt distribution. It treats equal and unequal eigenvalues separately.

APPENDIX C DERIVATION OF CLOSED-FORM EXPRESSION OF M 3D

  • An eigenvalue decomposition transforms the integral involving the joint positioning-error density into a form organized by the eigenvalues of a matrix.The derivation defines the transformed vector q using the eigenvectors.Source: Appendix C · p. 11
  • The derivation separates the cases λ1 < λ2 and λ1 = λ2 before evaluating the transformed expression.For unequal eigenvalues, it defines q = λ1/λ2 ∈ (0, 1) and ω as specified in the text.Source: Appendix C · p. 11
  • The resulting expression uses the Hoyt distribution’s probability density and cumulative distribution function.The text identifies I0 as the first-order modified Bessel function and the integral as the first-order Marcum Q-function.Source: Appendix C · p. 11

APPENDIX D DERIVATION OF CLOSED-FORM EXPRESSION OF M 3D

The appendix derives a closed-form expression for M 3D² by establishing moment relationships and evaluating the remaining integral terms. It handles the case λ1 = λ2 separately.

  • For λ1 = λ2, the derivation defines λ = λ1 = λ2 and calculates the expression separately.Source: Appendix D · pp. 11–12
  • The derivation differentiates the density fs(s) and integrates the resulting relation to establish a relationship between M 3D² terms.It uses integration by parts and the Newton-Leibniz theorem in evaluating the integrals.Source: Appendix D · pp. 11–12
  • Equation (104) relates M 3D² to an expression that still contains integrals Φ(R0, q, ω).Source: Appendix D · p. 12
  • The remaining Φ(R0, q, ω) terms are solved for and substituted into the relationship to obtain M 3D² in closed form.Source: Appendix D · p. 12

APPENDIX E DERIVATION OF CLOSED-FORM EXPRESSION OF M 3D

The appendix derives a closed-form expression for M 3D⁴ by evaluating its density-moment integral and eliminating the remaining integral terms. It treats λ1 = λ2 as a separate case.

  • For λ1 = λ2, the derivation defines λ = λ1 = λ2 and calculates the expression separately.Source: Appendix E · pp. 12–13
  • To evaluate the integral for M 3D⁴, the derivation multiplies a density relation by s² and applies integration by parts.Source: Appendix E · p. 12
  • Equation (115) establishes a relationship for M 3D⁴ that still contains integrals Ψ(R0, q, ω).Source: Appendix E · p. 12
  • The derivation solves for Ψ(R0, q, ω) and substitutes it into the relationship to obtain M 3D⁴ in closed form.Source: Appendix E · pp. 12–13
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