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Statistical Modeling of the FSO Fronthaul Channel for UAV-based Communications

Marzieh Najafi, Hedieh Ajam, Vahid Jamali, Panagiotis D. Diamantoulakis, George K. Karagiannidis, Robert Schober

arXiv:1905.12424v2eess.SPcs.ITmath.IT

TL;DR

UAV fronthaul links combine non-orthogonal beams with fluctuating position and orientation, requiring channel models that reflect these characteristics. The paper models GML under different fluctuation scenarios and finds that beam width trades outage risk against collected power.

  • Problem

    For hovering UAVs, position and orientation fluctuate, and the laser beam may not be orthogonal to the receiver lens plane.

    Source: 2) UAV · p. 3, A. Models for the Random Position and… · p. 7

  • Method

    The paper derives GML models for independent Gaussian, correlated Gaussian, and correlated uniform UAV position and orientation fluctuations.

    Source: §VII · p. 13

  • Results

    For large fluctuation variance, a wider beam is preferable to avoid outages, while reducing average and maximum received power; for small variance, a narrower beam increases collected power.

    Source: §I · p. 2

  • Takeaways & Limitations

    Beam width should be adjusted to the fluctuation variance, balancing outage probability against the amount of power collected by the receiver.

    Source: §I · p. 2

Abstract

from arXiv · show

In this paper, we investigate the statistics of the free space optics (FSO) communication channel between a hovering unmanned aerial vehicle (UAV) and a central unit. Two unique characteristics make UAV-based FSO systems significantly different from conventional FSO systems with static transceivers. First, for UAV-based FSO systems, the incident laser beam is not always orthogonal to the receiver lens plane. Second, both position and orientation of the UAV fluctuate over time due to dynamic wind load, inherent random air fluctuations in the atmosphere around the UAV, and internal vibrations of the UAV. On the contrary, for conventional FSO systems, the laser beam is always perpendicular to the receiver lens plane and the relative movement of the transceivers is limited. In this paper, we develop a novel channel model for UAV-based FSO systems by quantifying the corresponding geometric and misalignment losses (GML), while taking into account the non-orthogonality of the laser beam and the random fluctuations of the position and orientation of the UAV. In particular, for diverse weather conditions, we propose different fluctuation models for the position and orientation of the UAV and derive corresponding statistical models for the GML. We further analyze the performance of a UAV-based FSO link in terms of outage probability and ergodic rate and simplify the resulting analytical expressions for the high signal-to-noise ratio (SNR) regime. Finally, simulations validate the accuracy of the presented analysis and provide important insights for system design. For instance, we show that for a given variance of the fluctuations, the beam width should be properly adjusted to minimize the outage probability.

I. INTRODUCTION

UAV-based FSO fronthaul is affected by fluctuations in UAV position and orientation and by beams that may not be orthogonal to the receiver lens. The paper develops a channel framework for these conditions, analyzes link performance, and describes a receiver architecture and coordinate representation for the system.

I. INTRODUCTION

  • The proposed statistical GML models cover independent Gaussian, correlated Gaussian, and correlated uniform fluctuations associated with calm, weakly windy, and strongly windy conditions, respectively.The models account for non-orthogonal beams and random UAV position and orientation; correlated uniform fluctuations are proposed for large-variance conditions.Source: §I · p. 2
  • The paper analyzes outage probability and ergodic rate, including high-SNR asymptotic expressions, assuming GML dominates atmospheric turbulence for links several hundred meters long.The paper also simplifies statistical GML expressions for special cases, including an orthogonal beam and receiver lens plane.Source: §I · p. 2
  • For large fluctuation variance, a wider beam is preferable to avoid outages, while for small variance a narrower beam increases power collected by the receiver lens.The beam-width choice trades outage probability against average and maximum received power.Source: §I · p. 2

A. System Model

  • A lens in front of the CU’s photodetector array separates signals by spatial angle, supporting reception from UAVs at different positions and from multiple UAVs.The system uses RF multiple-access links from UAVs to mobile users and FSO fronthaul links from UAVs to the CU.Source: Figure 1 · p. 3, A. System Model · p. 3

2) UAV:

  • The UAV’s position is represented by the laser-source location and its orientation by the beam direction, with position and orientation fluctuations modeled as random variables.The CU is treated as fixed; the paper drops rotation around the beam because it does not affect the signal under rotational beam symmetry.Source: 2) UAV · pp. 3–4

B. FSO Channel Model

The channel model treats the received optical signal as affected by atmospheric loss, turbulence, and geometric and misalignment loss. It then characterizes the geometric and misalignment loss for a given UAV state, including the beam footprint on a receiver lens that need not be orthogonal to the beam.

B. FSO Channel Model

  • The received optical signal is modeled with a channel coefficient affected by atmospheric loss, atmospheric turbulence fading, and geometric and misalignment loss.The model assumes intensity modulation/direct detection, background-illumination noise independent of the signal, and an average transmitted-power constraint.Source: B. FSO Channel Model · p. 4
  • For links several hundred meters long, atmospheric turbulence is treated as negligible compared with GML, and the model approximates turbulence by its mean for typical parameters.The text gives an example turbulence variance of 3 × 10^-2 for L = 500 m, h_d = 120 m, and λ = 1550 nm.Source: B. FSO Channel Model · pp. 4–5

III. THE CONDITIONAL GML MODEL

  • In this section, the paper derives the channel parameter h_g for a given UAV state, specified by position r and orientation ω.Source: §III · p. 5

A. Center of the Beam Footprint

  • The beam footprint center on the receiver lens is defined by the intersection of the beam line and the lens plane x = 0.The lens-plane power-density model accounts for cases where the receiver lens plane is not orthogonal to the beam direction.Source: A. Center of the Beam Footprint · p. 5

B. Power Density on the Lens Plane

The paper models power density and collected power when the beam is not necessarily perpendicular to the receiver lens, then approximates the resulting geometric and misalignment loss (GML). It bounds the lens-area integral and validates a closed-form approximation for practical parameters.

B. Power Density on the Lens Plane

  • The Gaussian beam’s power density on a non-orthogonal lens plane is given under conditions typical of links spanning several hundred meters.The density depends on the angle between the beam and lens plane and on the elliptical footprint geometry.Source: B. Power Density on the Lens Plane · p. 5

C. GML

  • The collected-power fraction is an integral over the lens area, and the paper bounds it because the integral has no closed-form evaluation.The bounds are based on rotated elliptical footprints; in the orthogonal case, they coincide with the exact GML.Source: C. GML · pp. 5–6, Figure 3 · p. 6
  • A0 decreases as beam width increases, creating a trade-off when selecting beam width for the GML approximation.A0 is the maximum fraction of optical power captured when the beam footprint is centered on the receiver lens.Source: C. GML · p. 6
  • The approximation separates maximum captured power A0 from an exponential misalignment attenuation, with wider beams collecting less power.The arithmetic-mean and geometric-mean choices for t give accurate approximations over the practical parameter range shown in Fig. 4.Source: C. GML · p. 6

IV. STATISTICAL MODELS FOR THE GML

The paper treats the hovering UAV’s position and orientation as random because tracking errors and environmental disturbances prevent perfect alignment. It maps distinct fluctuation distributions to calm, weak-wind, and strong-wind conditions.

A. Models for the Random Position and Orientation Fluctuations of the UAV

  • Position and orientation fluctuations make statistical models necessary because hovering UAVs are not perfectly stable and tracking errors persist.The text attributes errors to control-system delay or inaccuracy and imperfect wind estimation, and expects them to be more severe than building-sway errors in conventional FSO links.Source: A. Models for the Random Position and… · pp. 6–7
  • The UAV’s position and orientation are represented as mean vectors plus zero-mean random fluctuations.The representation is r = µr + ϵr and ω = µω + ϵω.Source: A. Models for the Random Position and… · p. 7

IV. STATISTICAL MODELS FOR THE GML

1) First Moments of RVs:

The section relates UAV stability to the variances of its position and orientation fluctuations and develops an independent-Gaussian statistical GML model. It characterizes how small-channel-coefficient behavior depends on fluctuation variance and beam geometry.

1) First Moments of RVs:

  • The tracking system selects the mean orientation so the beam line intersects the receiver-lens center, maximizing collected power.This condition gives E{b} = (0, 0, 0).Source: 1) First Moments of RVs · p. 7

2) Second Moments of RVs:

  • For the second-moment measure, smaller position and orientation variances indicate a more stable UAV.The section evaluates FSO fronthaul performance in terms of this stability measure.Source: 2) Second Moments of RVs · p. 7

B. Statistical GML Model for Independent Gaussian Fluctuations

  • Independent Gaussian fluctuations are motivated by many contributing sources, including atmospheric air fluctuations and internal UAV vibrations.The model assumes the resulting position and orientation fluctuations are independent and zero-mean.Source: B. Statistical GML Model for… · p. 7
  • For an orthogonal beam, the GML density near zero vanishes when ϖ > 1, is uniform on [0, A0] when ϖ = 1, and diverges when ϖ < 1.For the general non-orthogonal case, the threshold involves qϖ; wider beams and smaller fluctuation variances can meet qϖ ≥ 1, but a wider beam reduces A0.Source: B. Statistical GML Model for… · p. 8
  • With an orthogonal beam, orientation variance affects GML more than position variance because it is scaled by the average UAV–CU distance, while x-axis position variance does not affect GML.The orthogonal-case distribution has a form similar to conventional FSO geometric-spread and pointing-error models with position fluctuations only.Source: B. Statistical GML Model for… · p. 8

C. Statistical GML Model for Correlated Gaussian Fluctuations

The correlated-Gaussian model represents wind-driven fluctuations along particular position and orientation directions, alongside independent variations. The resulting small-fluctuation GML model combines the independent and correlated covariance contributions.

C. Statistical GML Model for Correlated Gaussian Fluctuations

  • The correlated-Gaussian model captures weak-wind fluctuations that are larger along a direction v for position and a direction τ for orientation, in addition to independent variations.The direction of rotational fluctuation depends on UAV geometry and wind direction.Source: C. Statistical GML Model for… · pp. 8–9
  • For small fluctuations, the GML density has the independent-Gaussian form with covariance matrix ΣIG replaced by ΣT = ΣIG + ΣCG.The model therefore includes both independent and correlated Gaussian contributions.Source: C. Statistical GML Model for… · p. 9
  • When wind’s impact dominates, the misalignment variable follows a one-sided Gaussian distribution and the GML density diverges near A0.For ϖ ≥ 1, the density also diverges as h approaches zero; the near-A0 behavior differs from the independent and general correlated Gaussian cases, where it is bounded for q ≠ 0.Source: C. Statistical GML Model for… · p. 9

D. Statistical GML Model for Correlated Uniform Fluctuations

For strong wind, the model treats UAV position and orientation fluctuations as bounded and uniformly distributed, with wind dominating and fluctuations stronger in a particular direction. The section then defines outage probability through the GML CDF.

D. Statistical GML Model for Correlated Uniform Fluctuations

  • For strong wind, bounded position and orientation fluctuations are modeled as uniform rather than Gaussian, with wind dominant and fluctuations stronger in a particular direction.The model also describes orientation fluctuation caused by wind in a specific direction.Source: D. Statistical GML Model for… · p. 9
  • Theorem 4 gives the GML probability density function for correlated uniform fluctuations under the assumption ξ → 0.Source: D. Statistical GML Model for… · p. 9

A. Outage Probability

  • Outage probability is the probability that SNR γ falls below threshold γ_thr, and is expressed using the GML cumulative distribution function.The SNR is defined for the channel model in (1).Source: A. Outage Probability · p. 9

1) Independent Gaussian Fluctuations:

The section gives outage-probability expressions for independent Gaussian fluctuations, including a high-SNR simplification, and discusses correlated Gaussian and uniform cases. For correlated uniform fluctuations, outage probability is zero above a critical transmit SNR.

1) Independent Gaussian Fluctuations:

  • For independent Gaussian fluctuations, the outage probability is given by (31), with a special expression for an orthogonal beam and a high-SNR simplification.The section also uses the special-case expression (32) to identify the link's diversity gain.Source: 1) Independent Gaussian Fluctuations · pp. 9–10

2) Correlated Gaussian Fluctuations:

  • With independent and correlated Gaussian fluctuations, the outage probability has the same expression as for the independent Gaussian case; when wind dominates, the section gives an alternative expression and derives diversity gain.Source: 2) Correlated Gaussian Fluctuations · p. 10

3) Correlated Uniform Fluctuations:

B. Ergodic Rate

The analysis relates high-SNR ergodic-rate loss to UAV misalignment and fluctuation statistics. Simulations validate the channel and performance analyses and show a beam-width trade-off between outage probability and collected power.

B. Ergodic Rate

  • The high-SNR ergodic-rate loss ΔR̄g depends on fluctuation statistics, while R̄max without misalignment is independent of them.Across the considered fluctuation models, ΔR̄g is evaluated using the expected squared misalignment; the loss decreases and the rate increases as the UAV becomes more stable.Source: B. Ergodic Rate · p. 10

VI. SIMULATIONS AND DISCUSSIONS

  • Within the practical regime |αd| ≤ π/4, the conditional-GML approximation agrees with simulations, and non-orthogonality adds attenuation to the misaligned-beam case.At |αd| = π/2, the channel coefficient approaches zero; the stated non-orthogonality loss is small in the practical regime.Source: §VI · p. 11
  • The statistical GML model agrees with the simulated histogram; larger fluctuation variance increases the probability of smaller channel coefficients, while orthogonal beams yield larger coefficients than non-orthogonal beams.For the orthogonal special case, increasing position and orientation fluctuations makes the PDF more heavy tailed.Source: §VI · p. 11
  • For correlated Gaussian fluctuations, the GML PDF has higher probabilities at both small and large channel coefficients than for independent Gaussian fluctuations.Combining independent and correlated Gaussian fluctuations makes the PDF more heavy tailed, with larger values near zero for the stated parameters.Source: §VI · p. 11
  • For ξ = 4r0 and hg ≤ 0.03, widening the beam from 3r0 to 4r0 changes A0 from 0.16 to 0.1 and Pout from 0.1 to 0.02.The wider beam reduces outage probability while collecting a smaller maximum fraction of power.Source: §VI · pp. 12–13
  • For link lengths of several hundred meters, the reported gap in outage and ergodic-rate curves with and without GG turbulence is negligible for Gaussian fluctuations.The outage simulations also agree with the analytical results, and the high-SNR asymptotic outage expression is accurate for Gaussian fluctuations.Source: §VI · p. 13
  • At high SNR, the simulated ergodic rates approach their asymptotic values, with rate losses of 0.83, 0.9, and 0.9 bits/symbol for independent Gaussian, correlated Gaussian, and correlated uniform fluctuations, respectively.The correlated Gaussian and uniform rates are equal for the stated parameter set because ζ = ξ.Source: §VI · p. 13

VII. CONCLUSIONS

The paper develops statistical GML models for UAV-based FSO links that account for non-orthogonal beams and random UAV position and orientation. Its conclusions compare fluctuation models and identify a beam-width trade-off: wider beams can reduce outages under large fluctuations, while narrower beams collect more power when fluctuations are small.

VII. CONCLUSIONS

  • The channel models account for non-orthogonal laser beams and random UAV position and orientation, with independent Gaussian, correlated Gaussian, and correlated uniform fluctuation cases.The paper associates these fluctuation models, respectively, with calm, weakly windy, and strongly windy conditions, and derives high-SNR asymptotic expressions for outage probability and ergodic rate.Source: §VII · p. 13
  • Correlated fluctuations produce a higher outage probability than independent fluctuations, while beam width trades outage avoidance against collected power.For relatively large fluctuation variance, a wider beam is preferable to avoid outages, although average and maximum collected power decrease; with small variance, a narrower beam increases collected power.Source: §VII · p. 14

APPENDIX A

The appendices support the geometric-loss model by deriving the non-orthogonal-beam power relation, bounding the conditional GML, and approximating the resulting integrals.

APPENDIX A

  • The power collected on the tilted lens plane is related to the orthogonal-beam intensity using the angle between the beam and lens plane.The derivation uses the beam-line distance from a point on the lens and approximates the source-to-perpendicular-plane distance by the UAV-to-CU distance when that link distance is much larger than the relevant offsets.Source: Appendix A · p. 14

APPENDIX B

  • The power-density contours are ellipses, and replacing them with rotated ellipses of matching axis lengths yields lower and upper bounds for conditional GML.The coordinate system is rotated so the ellipse center lies on the rotated y axis; the circular lens retains the same description under this rotation.Source: Appendix B · pp. 14–15

APPENDIX C

  • The lower-bound integral is approximated by replacing the circular photodetector with an equal-area square and using a Taylor-series approximation.A related rectangular-area approximation is used for the upper-bound integral.Source: Appendix C · p. 15

APPENDIX D

The appendices derive GML distributions by expressing a displacement magnitude in terms of position and orientation fluctuations. They use Gaussian structure to obtain a Hoyt distribution for that magnitude in the independent and correlated Gaussian cases.

APPENDIX D

  • For independent Gaussian fluctuations, linearizing the displacement components yields a bivariate Gaussian vector whose transformed magnitude u follows a Hoyt distribution.The GML PDF is then obtained from the distribution of u using the relation in (19).Source: Appendix D · p. 15

APPENDIX E

  • For correlated Gaussian fluctuations, the displacement components remain Gaussian and u again follows a Hoyt distribution, giving the same formal GML PDF expression as for independent Gaussian fluctuations.The Hoyt parameters use the eigenvalues of the covariance matrix specified for the correlated case.Source: Appendix E · pp. 15–16

APPENDIX F

These appendices derive the GML distribution for a degenerate correlated-Gaussian case and for correlated uniform fluctuations, then simplify the high-SNR outage expression.

APPENDIX F

  • When the correlated-Gaussian covariance has one non-zero eigenvalue, u follows a single-sided Gaussian distribution, from which the GML PDF is obtained.The case has Hoyt parameter q = 0.Source: Appendix F · p. 16

APPENDIX G

  • For correlated uniform fluctuations, substituting the derived uniform distribution into (19) gives the GML PDF in (29).Source: Appendix G · p. 16

APPENDIX H

  • The high-SNR outage-probability expression is simplified using an approximation to the symmetric difference of the first-order Marcum Q-function.Source: Appendix H · p. 16
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