HAPS based FSO Links Performance Analysis and Improvement with Adaptive Optics Correction

(1)

Item Type Article

Authors Ata, Yalcin;Alouini, Mohamed-Slim

Citation Ata, Y., & Alouini, M.-S. (2022). HAPS based FSO Links

Performance Analysis and Improvement with Adaptive Optics Correction. IEEE Transactions on Wireless Communications, 1–1.

https://doi.org/10.1109/twc.2022.3230737 Eprint version Post-print

DOI 10.1109/twc.2022.3230737

Publisher Institute of Electrical and Electronics Engineers (IEEE) Journal IEEE Transactions on Wireless Communications

Rights (c) 2022 IEEE. Personal use of this material is permitted.

Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.

Download date 2024-01-16 18:38:50

Link to Item http://hdl.handle.net/10754/686672

(2)

HAPS based FSO Links Performance Analysis and Improvement with Adaptive Optics Correction

Yalc¸ın Ata, and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—This paper investigates the performance of high altitude platforms high altitude platforms (HAPS) based free- space optical (FSO) communication links including HAPS-to- ground station (downlink), ground-to-HAPS (uplink) and HAPS- to-HAPS (horizontal link) communications. The effects of attenuation loss, atmospheric turbulence, pointing error and angle- of-arrival (AOA) are taken into account. Also, the application of adaptive optics correction, one of the most effective turbulence mitigation techniques, is analyzed using the Zernike polynomials representation. Analytical expressions are obtained for probability density function (PDF), cumulative distribution function (CDF), Rytov variance, adaptive optics filter function and outage probability mainly in terms of Meijer’s G functions when both no adaptive optics correction is used and adaptive optics correction is applied. Some selected results are presented depending on the various parameters such as the HAPS altitude, the ratio of vertical and horizontal deviations, beam waist, Zenith angle, height of ground station, receiver aperture diameter, channel state threshold and wind speed. The performance improvement with adaptive optics correction is investigated by removing different Zernike modes. We show that (i) the downlink outperforms the uplink, (ii) the performance of the horizontal link sharply increases above a certain altitude, and, (iii) the communication links benefit from the adaptive optics correction up to a certain level in terms of performance improvement.

Index Terms—Free-space optical communication, outage probability, uplink, downlink, horizontal link, high altitude platforms, adaptive optics.

I. INTRODUCTION

H

IGH altitude platforms (HAPS) ¹ assisted free-space optical (FSO) communications have become a promising solution in terms of improving the data rate performance of communication links. Although conventional wireless communication applications using radio frequency (RF) spectrum found common usage; increasing demands for wide bandwidth, licence free communication, noise immunity and large capacity resulting from technological developments make FSO communication attractive among the researchers. However, FSO communication is more prone to be affected from en- vironmental conditions. Indeed, it is well known that optical wireless communication and imaging systems particularly suffer from atmospheric turbulence.

Using an aerial platform such as airship, aircraft, unmanned

Manuscript created June, 2022. (Corresponding author: Yalc¸ın Ata) Yalc¸ın Ata is with Electrical and Electronics Engineering Department, OSTIM Technical University, OSTIM, 06374 Yenimahalle, Ankara, Turkey (e-mail: [email protected]).

Mohamed-Slim Alouini is with Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia (e-mail: [email protected]).

1Known also as high altitude pseudo-satellites

aerial vehicle (UAV) or balloon in the stratosphere up to 28 km altitude can provide a reliable and performance improving alternative for wireless communication service [1], [2]. In [3], the uplink performance between ground to HAPS was analyzed for Log-normal and Gamma-Gamma distributed channel models and it was shown that the uplink performance is significantly affected from receiver field of view (FOV) and beam waist. A FSO-based backhaul and fronthaul vertical network frame was investigated in terms of link budget and achievable data rate [4] and it was shown that a remarkable performance improvement can be obtained by reducing the divergence angle. An overview of the UAV-to-UAV, UAV- to-ground and ground-to-UAV communication links including networking architecture and design considerations was presented in [5]. The throughput coverage area improvement using unmanned aerial base stations discussed in [6] and it was shown with simulations that the coverage area and network capacity can be maximized by optimal locating of the aerial stations. To overcome the pointing error and angle of arrival (AOA) fluctuations and, improve the performance of HAPS FSO communication systems, a beam optimization and adaptation method was studied to find the optimum beam size at the transmitter and receiver [7]. The bit-error-rate (BER) performance and capacity of ground-to-ground HAPS FSO network were analyzed as function of the attenuation loss, atmospheric turbulence, and pointing error [8].

All the studies mentioned above show that many phenomena are involved in HAPS based FSO communication and they need to be carefully analyzed and modeled to provide a reliable communication. Atmospheric attenuation is a phe- nomenon showing the combined effect of absorption and scattering effects caused by constituents, gases, particulates and molecules in the atmosphere. Both absorption and scattering are wavelength dependent and the optimum wavelength selection remains an important issue to provide an effective wireless communication service. Another very important phe- nomenon is the atmospheric turbulence that shows its effect in various ways such as scintillation, beam wander, AOA fluctuations, etc [9]. These effects vary with the strength of the refractive index fluctuations and the performance of FSO communication systems degrade in various levels [9].In this study, the Gamma-Gamma distribution is used to characterize the turbulent channel. The Gamma-Gamma distribution, which assumes that the small- and large-scale irradiance fluctuations are governed by Gamma distributions [9], yields very accurate results in wide range of turbulent regimes. Due to its efficient results especially in moderate to strong turbulent regimes, the Gamma-Gamma channel model has found

(3)

a wide usage in terms of performance analysis of optical wireless communication (OWC) systems operating in turbulent mediums [10]–[13]. Another factor is the aligment of optical wave in FSO communication system between transmitter and receiver. A hovering HAPS yields fluctuations and causes misalignments [14]. Misalignments are generally modeled by Rayleigh distribution [3], [15] assuming that the vertical and horizontal deviations are identical. However, the average BER performance of a terrestrial FSO link was examined modeling the misalignments by Hoyt distribution instead of Rayleigh distribution providing the capability of analysing different deviations in vertical and horizontal axes [16].

Mitigation techniques are also reported to reduce the atmospheric turbulence induced impairments including adaptive optics correction, spatial diversity, aperture averaging, special coding and decoding schemes [17]–[20]. Adaptive optics correction is one of the effective turbulence mitigation techniques. It was found that Zernike polynomials can be used to compensate the wavefront corruptions and a spatial filter depending on the Zernike polynomials was constructed on the turbulence power spectrum [21], [22]. The benefit of adaptive optics correction on average BER for uplink, downlink and horizontal link FSO communication systems using plane and spherical waves was presented in [17]

Growing number of applications are increasing the demand for slant or horizontal links operating in optical spectrum.

However, since optical communication systems are particularly affected by atmospheric conditions, correct analysis and optimum use of the communication links become critical issues.

Our motivation in this study is to characterize the behaviour of FSO systems operating in atmospheric medium when exposed to various effects such as atmospheric turbulence, beam attenuation, pointing error, and AOA fluctuations. This motivation is propelled by the possible usage of the HAPS assisted communication links in integrated ground/air/space networks to improve the performance of FSO systems and provide higher bandwidth for various applications. Besides, using the adaptive optics (AO) correction, one of turbulence mitigation techniques, we aim to reveal the effect of AO correction on the performance of FSO communication systems.

We can summarize the novelty and contribution of our paper as follows:

- Modeling beam displacement with Hoyt distribution allowing not only symmetrical deviations but asymmetrical displacement in x- and y- directions,

- Combining the effects of atmospheric turbulence, pointing error, AOA fluctuations and attenuation loss for slant path and horizontal link,

- Obtaining analytical expressions in terms of Meijer’s G function,

- Expressing AO correction filters in analytical form by using Meijer’s G function,

- Studying the aperture averaging effect on FSO communication link under a Gamma-Gamma turbulent channel.

The sctructure of this paper is arranged as follows. The system and the channel models including attenuation loss, atmospheric turbulence, pointing error and AOA fluctuations are presented in Section II and III, respectively.

TABLE I: List of Symbols

Symbol Definition

α Effective numbers of large-scale eddies β Effective numbers of small-scale eddies ζ Zenith angle

ηs Normalized beam waist with respect toσs

θ Angle of the wavenumber in polar coordinates θ_d Orientation deviation

θF OV Field of view

Θ Beam curvature parameter at the receiver Θ0 Beam curvature parameter at the transmitter

Θ Complementary parameter

κ Magnitude of the spatial frequency λ Wavelength

Λ Fresnel ratio of Gaussian beam at receiver Λ0 Fresnel ratio of Gaussian beam at transmitter

ξ Normalized distance parameter

σs Displacement variance in polar s coordinates σz Displacement variance in polar z coordinates σ₀ Variance of orientation deviation

σ²_lnX Large-scale log variance σ_lnY² Small-scale log variance

Φn Turbulence power spectrum ϖ_n Propagation parameter

A Nominal value of ground turbulence C(λ) Attenuation coefficient

DG Receiver aperture size

F₀ Phase front radius of curvature h Channel state

h₀ Height of ground station haf Angle of arrival

hal Attenuation loss hat Atmospheric turbulence hpl Pointing error

hth Threshold for channel state H Altitude of HAPS

k Wavenumber

l Height parameter L Link length

m Azimuthal frequency n Radial frequency

N Number of Zernike mode removed q Visibility parameter

qH Ratio of vertical and horizontal deviations r_a Receiver aperture radius

V Visibility

W0 Gaussian beam radius ω Wind speed

ωb Beam waist

The mitigation of atmospheric turbulence effect with adaptive optics correction is analyzed in Section IV. The outage propability of HAPS FSO communication links is introduced in Section V. Obtained results are presented and discussed in Section VI while concluding remarks are given in Section VII. Finally, the details of analytical derivations are given in

(4)

Appendix A, B and C.

The list of symbols used in this paper is given in Table I II. SYSTEMMODEL

In this study, we investigate the performance of the downlink, the uplink, and the horizontal link. The system model under consideration is shown in Fig.1

Fig. 1: System model

HAPSs are hovering at the height of H that varies between 18-28 Km in the stratospheric layer. The ground station stands at the height of h0 above the ground level. The link length between transmitter and receiver in downlink, uplink and horizontal cases is denoted by L. The zenith angle between the direction of the propagation and vertical axis is ζ. When the HAPS hovers, AOA fluctuations occurs at the received optical beam resulting from transceiver vibrations and turbulence effect. Another angular impairment is the pointing error caused by the beam misalignment. The Hoyt distribution is used for modeling the deviations. Here orientation deviations are included in the pointing error effect.

The attenuation due to the combined effect of absorption and scattering is modeled depending on the visibility range.

The atmospheric turbulence effect is analyzed assuming Gamma-Gamma distribution model. Finally, the adaptive optics correction effect on the outage probability of uplink, downlink and horizontal link is examined.

III. CHANNELMODEL

Combining the effects of attenuation loss, atmospheric turbulence, pointing error and AOA fluctuations; the channel state for “ground to HAPS”, “HAPS to ground” or “HAPS to HAPS” communication can be modeled by

h=halhathplhaf, (1) where hal, hat, hpl, and haf denote the effects of the attenuation loss, atmospheric turbulence, pointing error, and AOA fluctuations, respectively.

A. Attenuation Loss

According to the Beer-Lambert law, the attenuation due to the absorption and scattering effects in atmosphere can be given as

hal = exp[−C(λ)L], (2) where C(λ) is the attenuation coefficient that is dependent on the wavelengthλ. The attenuation coefficient in Eq. (2) is expressed as a function of the visibility [23]

C(λ) = 3.912 V

λ 550

−q

, (3)

whereV is the visibility (in Km), the wavelength is here in nm, and the parameterq is

q=







1.6, V >50Km

1.3, 6Km< V <50Km 0.585V^1/3, V <6Km

. (4)

B. Atmospheric Turbulence

According to the conventionally used Hufnagel-Valley (HV) model, the turbulence structure constant is empirically found to be ( [9], pp. 481)

C_n²(l) = 0.00594(ω/27)²(10⁻⁵l)¹⁰exp(−l/1000)

+ 2.7×10⁻¹⁶exp(−l/1500) +Aexp(−l/1000), (5) wherel is the height in m, ω is the root mean square (rms) value of the wind speed,Ais the nominal value ofC_n²(0)at the ground in m^−2/3 and it is given by A= 1.7×10⁻¹⁴m^−2/3 and A = 1.7×10⁻¹³m^−2/3 depending on the wind speed ω = 10,21,and30m/s in the HV5/7 model in which the obtained turbulence structure constantC_n²resulting from input parameters yields a coherence length of 5 cm and an iso- planatic angle of 7µm from earth to space link [24]–[26].

The probability density function (PDF) of the atmospheric turbulence channel is modeled by the Gamma-Gamma channel distribution and it is given by ( [9], pp. 462)

fh_at= 2(αβ)^(α+β)/2

Γ(α)Γ(β) h^{(α+β)/2−1}_at K_α−β 2p

αβhat

, (6) where hat > 0, α = _exp(σ2¹

lnX)−1, σ²_lnX is the large-scale log variance, β = _exp(σ2¹

lnY)−1, σ_lnY² is the small-scale log variance, and Ka(.) is the modified Bessel function of the second kind with the order a. The large- and small-scale log variances are defined as [9]

σ²_{lnX i j} = 0.49σ_{B i j}²

1 + 0.56(1 + Θ)σ_{B i j}^12/5 7/6, (7) σ²_{lnY i j} = 0.51σ²_{B i j}

1 + 0.69σ_{B i j}^12/5 ^5/6, (8) where σ²_B is the Rytov variance for Gaussian beam, the index i∈d, h, u denoting the downlink, horizontal or uplink communications, and j ∈ AO showing the application of adaptive optics correction, and the index j will not be used

(5)

for the no adaptive optics correction cases. We also note that, the denominator of Eq.(7) changes to

1 + 1.11σ_{B i j}^12/5 ^7/6 for downlink communication.

As the Rytov variance represents the scintillation index in weak turbulence conditions, the longitudinal component of the scintillation index that is called on-axis scintillation will be used as Rytov variance of Gaussian beam and is given by [9]

σ_B² =8π²k²sec(ζ)

H

Z

h0

∞

Z

0

κΦn(l, κ) exp −ΛLκ²ξ²/k

×Re

1−exp

iκ²L

k ξ 1−Θξ

dκ dl,

(9)

wherek= 2π/λ is the wave number, where Λ = Λ0/(Θ²₀+ Λ²₀)is the Fresnel ratio of Gaussian beam at receiver, Λ0 = 2L/kW₀², W₀ is the beam radius, Θ₀ = 1−L/F₀ is the beam curvature parameter at the transmitter, F₀ is the phase front radius of curvature, Θ = 1−Θis the complementary parameter,Θ = Θ₀/(Θ²₀+Λ²₀)is the beam curvature parameter at receiver, ξ is the normalized distance parameter, κ is the magnitude of the spatial frequency and Re represents the real part of the argument. The term Φn(l, κ) in Eq. (9) is the turbulence power spectrum and using the turbulence structure constant in Eq. (5), the Kolmogorov power spectrum becomes Φn(l, κ) = 0.033C_n²(l)κ^−11/3, (10) and is used in Eq. (9)

1) Downlink Channel: The normalized distance parameter is ξ= _(H−h^(l−h⁰⁾

0) for the downlink propagation case. Using Eq.

(9), the Rytov variance for a Gaussian beam propagating in downlink propagation is approximated as [9], [27]

σ_{B d}² =2.25k^7/6(H−h₀)^5/6sec^11/6(ζ)

×

H

Z

h₀

C_n²(l)

l−h0

H−h0

^5/6

dl. (11) 2) Uplink Channel: The normalized distance parameter is ξ = 1− _(H−h^(l−h⁰⁾

0) for the uplink propagation case. Using Eq.

(9), the Rytov variance for a Gaussian propagation is found to be [9]

σ²_{B u}=8.70k^7/6(H−h0)^5/6sec^11/6(ζ)Re





H

Z

h0

C_n²(l)

×n

Λξ²+iξ 1−Θξ^5/6

−Λ^5/6ξ^5/3o dl

. (12)

3) Horizontal Channel: From [9], the Rytov variance of a Gaussian beam is given in this case by

σ²_{B h}=8π²k²L

1

Z

0

dξ

∞

Z

0

κΦn(l, κ) exp −ΛLκ²ξ²/k

×Re

1−exp

iκ²L

k ξ 1−Θξ

dκ dl.

(13)

Here, there is no height dependent variation. Then,C_n²(l)will be obtained from Eq. (5) for a fixed height l and it will be

constant over the propagation distance. Using Eq. (13), the Rytov variance of Gaussian beam for horizontal path can be found as

σ_{B h}² =4.75C_n²(l)k⁷⁶L¹¹⁶

0.40h

(1 + 2Θ)²+ 4Λ²i₁₂⁵

×cos 5

6tan⁻¹

1 + 2Θ 2Λ

−11 16Λ^5/6

.

(14)

C. Pointing Error

Assuming that the jitter variance in vertical and horizontal direction are different and using the Hoyt distribution to determine the displacement at the receiver, the PDF of misalignment in polar coordinates is given by [16]

fr,φ(r, φ) = r 2πq_Hσ²_sexp

−r²ξ(φ) 2q²_Hσ_s²

, (15) where

ξ(φ) =1−(1−q_H²) cos²φ

q_H² . (16)

and q_H = σ_z/σ_s ∈ (0,1], σ_s and σ_z are the variances of beam jitters in s and z orthogonal directions, s = rcos(φ) andz=rsin(φ). Then the distribution of pointing error given in Eq. (15) is obtained in integral form as [16]

f_pl(h_pl) = η_s² 2πqH

π

Z

−π

h^η_pl²^s^ξ(φ)−1 A^η₀²^s^ξ(φ)

dφ, (17)

where η_s = ω_e/(2σ_s), ω_e = ω_bp√

πerf(υ)/(2υe^−υ²),ω_b is beamwaist, erf(.) is error function, υ = p

π/2ra/ωb, ra is receiver aperture radius, D_G = 2r_a is the receiver aperture diameter andA₀=erf²(υ).

Defining the deviation angle from the normal vector of the receiver aperture as θ_d then, the conditional probability for the PDF ofh_pl can be found

f_pl|θ_d(h_pl) = η²_s 2πqH

π

Z

−π

h^η_pl²^s^ξ(φ)−1 A^η₀²^s^ξ(φ)

cos(θ_d)dφ. (18) For the special caseq_H = 1, Eq. (18) becomes f_pl|θ_d(h_pl) = η_s²^h

η2 s−1 pl

A^η₀²^s

cos(θd).

D. AOA Fluctuations

Assuming that the deviation angle is smaller than the field of view (θd≤θF OV), the fading resulting from AOA fluctuations are given by [3]

haf = 1−[J0(πra/λ)]²−[J1(πra/λ)]², (19) where Jn(.) is the Bessel function of the first kind having order of n. Assuming that the random variableθd is Rayleigh distributed, the PDF ofθ_d is given by [3]

fθ_d(θd) = θd

σ₀²exp

− θ²_d 2σ₀²

, θd≥0, (20) where σ²₀ is the variance of Gaussian distributed random variable θ_d. As it is done in [3], if the channel states for

(6)

attenuation lossh_al, atmospheric turbulenceh_atand pointing error h_pl are conditioned on h_pl by using the relationship as h_ag =h_alh_ath_pl we can write

f_h_ag_|θ_d(hag) =

∞

Z

hag A0hal

f_h_pl_|θ_d hag

halhat

fh_at(hat) halhat

dhat (21)

The PDF of the combined channel state can be found after derivations given in APPENDIX-A as

fh(h) =

(αβ)^(α+β)/2η²_shafexp(−σ₀²/2)1F1

−¹₂,¹₂;^σ₂⁰²

2πqHA^(α+β)/2₀ Γ(α)Γ(β)h^(α+β)/2_al

×h^α+β−2²

π

Z

−π

G^3,0_1,3

"

αβh A₀h_al

2−α−β+2η2 s ξ(φ) 2

−α−β+2η2 s ξ(φ)

2 ,^α−β₂ ,^β−α₂

# dφ.

(22) To validate our results and verify the accuracy of our derivations, we compared the initial ve final equations in MATLAB simulation environment. For this purpose, we first validated the conditional probability fh_ag|θd(hag) given in Eqs. (A.1) and its derivation given in Eq. (A.4) in APPENDIX-A. To validate the obtaind PDFf_h(h)given in Eq. (22), we compared Eqs.

(A.6) and (22). In both comparison cases, compared equations matched perfectly indicating the validity of obtained results.

For the special case of Hoyt distribution q_H = 1 where the Rayleigh distribution is obtained, the integration in Eq.

(22) disappears and Eq. (22) simplifies to a simpler analytical expression in terms of the Meijer’s G function as

fh(h) =

(αβ)^(α+β)/2η_s²hafexp(−σ₀²/2)1F1

−¹₂,¹₂;^σ₂²⁰

A^(α+β)/2₀ Γ(α)Γ(β)h^(α+β)/2_al

×h^{(α+β)/2−1}G^3,0_1,3

"

αβ A₀h_alh

2−α−β+2η2 s 2

−α−β+2η2 s

2 ,^α−β₂ ,^−α+β₂

# . (23) IV. ADAPTIVEOPTICSCORRECTION TOMITIGATE THE

TURBULENCEEFFECT

Adaptive optics correction behaves as a spatial filter on turbulence power spectrum. When adaptive optics correction is applied then, the power spectrum given in Eq. (10) becomes [17], [28]

Φn AO(l, κ) = Φn(l, κ)

"

1−

N

X

l=1

Fl(κ, ϖDG, θ)

# , (24) whereF_l(κ, ϖD_G, θ)is the filter function in isotropic turbulence, θ is polar angle, N is the number of Zernike-modes removed, ϖ is the propagation parameter varying with the beam type and is equal to ϖ = 1−(θ+iΛ)ξ for Gaussian beam and the filter function is

Fm,n(κ, ϖDG, θ) = (n+ 1)

2Jn+1(0.5κϖDG) 0.5κϖD_G

² θterm,

(25) where m is the azimuthal frequency, n is the radial degree.

θ_termin Eq. (25) isθ_term= 1form= 0,θ_term= 2 cos²(mθ)

when m takes even values and θ_term= 2 sin²(mθ)when m takes odd values.

The representation of the wavefront aberrations with Zernike polynomials shows the aberration strength for optical system using circular aperture. Each mode can be defined depending on the radial polynomials and angular functions and some of them are given in Table II. Zernike polynomials are orthogonal and they define wavefront aberrations over a unit circle [29]. Wavefront corrugations are corrected by applying adaptive optics correction of each mode. We note that the piston mode is not applied and excluded in this study.

Then, selecting the number of Zernike modes removed that is denoted by N in the result section as N = 1, N = 2, N = 3,N = 4,N = 5corresponds to the application of tip, tip+tilt, tip+tilt+defocus, tip+tilt+defocus+astigmatism and tip+tilt+defocus+astigmatism+astigmatism, respectively. This means that eachm, npairs corresponding the related Zernike modes are substituted into Eq. (24) and added up to N number.

TABLE II: Zernike Polynomials

Mode Radial Degreen Azimuthal Frequencym

Piston 0 0

Tip 1 -1

Tilt 1 1

Defocus 2 0

Astigmatism 2 -2

Astigmatism 2 2

Coma 3 -1

Coma 3 1

Trefoil 3 -3

Trefoil 3 3

The lower order tip and tilt corrections are applied to compensate the wavefront tilts by moving the tip-tilt mirror. The higher order aberrations such as defocus, astigmatism, coma are corrected by using deformable mirrors. Required number of adaptive optics corrections can be applied sequentially to reduce the residual wavefront aberrations to the minimum level. Zernike polynomials given in the Table II defines the tip, tilt, defocus, astigmatism, coma, and trefoil wavefront aberrations.

A. Down Link and Up Link

Inserting Eq. (24) into Eq. (9), Rytov variance for a Gaus- sian beam over a slant path is given by

σ_{B AO}² =4πk²sec(ζ)

2π

Z

0 H

Z

h0

∞

Z

0

"

1−

N

X

l=1

Fl(κ, ϖDG, θ)

# κ

×Φn(l, κ)e⁻^ΛLκ

2ξ2

k Re

1−eⁱ

κ2Lξ(^1−Θξ)

k

dκdldθ.

(26) We can divide Eq. (26) into two parts for downlink and uplink communications as

σ_{B i AO}² =σ_B1² _i−σ²_B2 _i, (27)

(7)

whereσ_B1² _iis equal toσ²_{B d}for downlink andσ²_{B u}for uplink propagations as given in Eqs. (11) and (12), respectively.σ_B2² _i is

σ_B2² _i=4πk²sec(ζ)

2π

Z

0

dθ

H

Z

h₀

dl

∞

Z

0

κ

N

X

l=1

F_l(κ, ϖD_G, θ)

×Φ_n(l, κ)e⁻^ΛLκ

2ξ2

k Re

1−eⁱ

κ2Lξ(1−Θξ)

k

dκ.

(28) We now solve σ_B2² _i that shows the adaptive optics effect in Eq. (28). Inserting the power spectrumΦ_n(κ, l)given in Eq.

(10) and the filter function given in Eq. (25) into Eq. (28) yields

σ_B2² _i=1.056πk^1/6L^11/6sec(ζ)

√πD_G² Re





N

X

l=1

(n+ 1)

2π

Z

0

θ_termdθ

×





 Λ¹¹⁶

H

Z

h₀

C_n²(l)ξ¹¹³

ϖ² G^1,2_2,3 ϖ²D_G²k 4ΛLξ²

17/6,1/2 n+1,0,−n−1

! dl

−

H

Z

h₀

C_n²(l)ξ¹¹⁶ Λξ−i(1−Θξ)¹¹₆ ϖ²

× G^1,2_2,3 ϖ²D²_Gk 4L Λξ²−iξ(1−Θξ)

17/6,1/2 n+1,0,−n−1

! dl

)!

, (29) where G^m,n_p,q (.)is the Meijer’s G function. The detail for the derivation of Eq. (29) is given in APPENDIX-B. The only difference between downlink and uplink propagation in Eq.

(29) is the representation of the normalized distance parameter that is ξ= _(H−h^(l−h⁰⁾

0) for the downlink and is ξ= 1−_(H−h^(l−h⁰⁾

0)

for the uplink.

B. Horizontal Link

When adaptive optics correction is utilized, the Rytov variance for a Gaussian beam in Eq. (13) with the power spectrum in Eq. (10) becomes

σ_{B h AO}² =4πk²L

2π

Z

0 1

Z

0

∞

Z

0

"

1−

N

X

l=1

F_l(κ, ϖD_G, θ)

#

×Re

1−exp

iκ²L

k ξ 1−Θξ

×κΦ_n(l, κ) exp −ΛLκ²ξ²/k dκdξdθ

(30)

We can divide Eq. (30) into two parts as

σ²_{B h AO}=σ_B1² _h−σ²_B2_h. (31) In Eq. (30), the first partσ_B1² _his obtained from the horizontal path equation in Eq. (14). Inserting the spectral filter function

given in Eq. (25) in Eq. (30), σ_B2² _h will be σ²_{B h AO}= 2.112πk²L

D²_G

N

X

l=1

(n+ 1)Re





2π

Z

0

θ_termdθ

×

1

Z

0

dξ ϖ²

∞

Z

0

κ⁻¹⁴³C_n²(l) [Jn+1(0.5κϖDG)]²

×e⁻^ΛLκ

2ξ2 k

1−exp

iκ²L

k ξ 1−Θξ

. (32) After some mathematical manipulations and integrations (see APPENDIX-C), Eq. (32) becomes as

σ²_{B h AO}=2.112πk^1/6L^17/6C_n²(l) 2√

πD_G²

N

X

l=1

(n+ 1)

2π

Z

0

θ_termdθ

×Re







1

Z

0

Λξ²^11/6

ϖ² G^1,2_2,3 ϖ²D²_Gk 4ΛLξ²

17/6,1/2 n+1,0,−n−1

!

−

1

Z

0

Λξ²−iξ(1−Θξ)^11/6 ϖ²

×G^1,2_2,3 ϖ²D²_Gk 4L Λξ²−iξ(1−Θξ)

17/6,1/2 n+1,0,−n−1

! dξ

) . (33)

V. OUTAGEPERFORMANCEANALYSIS

In this section, the outage probability for “ground-to- HAPS”, “HAPS to ground” and “HAPS to HAPS” is investigated depending on the channel state and signal-to-noise ratio (SNR). The cumulative distribution function (CDF) of the channel stateh can be found by using the PDF given in Eq. (22) as

F_h(h) =

h

Z

0

f_h(x)dx. (34) Taking the integration over x in Eq. (34) and using the Eq.

(26) of [30], the CDF ofhcan be derived by Fh(h) =

(αβ)^α+β² η²_sh_afe⁻^σ

20 2 1F₁

−¹₂,¹₂;^σ₂²⁰ 2πqHA^(α+β)/2₀ Γ(α)Γ(β)h^(α+β)/2_al

π

Z

−π

h^α+β²

×G^3,1_2,4 αβh A₀h_al

2−α−β

2 ,^{2−α−β+2η}

s ξ2 (φ) 2

2 ,^α−β₂ ,^β−α₂ ,^−α−β₂

! dφ.

(35) The outage probability, which is defined the probability of instantaneous channel valuehfalls below the defined threshold channel levelhth, can be found by

Pout =Pr(h≤hth) =Fh(hth). (36)

(8)

Inserting Eq. (35) into Eq. (36), the outage probability is found to be

Pout=

(αβ)^α+β² η_s²h_afe⁻^σ

2 0 2 1F₁

−¹₂,¹₂;^σ₂²⁰ 2πqHA^(α+β)/2₀ Γ(α)Γ(β)h^(α+β)/2_al

h

α+β 2

th

×

π

Z

−π

G^3,1_2,4 αβhth

A₀h_al

2−α−β

2 ,^{2−α−β+2η}

2s ξ(φ) 2

2 ,^α−β₂ ,^β−α₂ ,^−α−β₂

! dφ.

(37) For the special case qH = 1, Eq. (37) simplifies to the analytical expression

Pout=

(αβ)^α+β² η_s²hafe⁻^σ

02 2 1F1

−¹₂,¹₂;^σ₂⁰²

A^(α+β)/2₀ Γ(α)Γ(β)h^(α+β)/2_al h

α+β 2

th

×G^3,1_2,4 αβ A0hal

hth

2−α−β

2 ,^{2−α−β+2η}₂ ²^s

−α−β+2η2 s

2 ,^α−β₂ ,^β−α₂ ,^−α−β₂

! .

(38)

VI. RESULTS ANDDISCUSSION

In this section, the numerical results are presented for horizontal link, downlink and uplink HAPS based FSO communications depending on various parameters. Results are obtained by using MATLAB software package. The Meijer’s G function in our analytical expressions is evaluated by using the ”meijerG” function in the MATLAB’s library. Fixed parameters are given in Table III. The parameters different from those in Table III are given either in the figure captions or on the plots.

TABLE III: Fixed Parameters

Symbol Value

λ 1550nm

ξ 40^◦

H 20km

h0 10m

V 3km

DG 5cm

W₀ 2cm

F0 ∞

ω 21m/s

A 1.7×10⁻¹³m^−2/3

N 10

qH 1

r_a D_G/2 σs 4×ra

σ_z q_H×σ_s σ0 15mrad ω_b 5×r_a ηs ωe/(2σs)

A. Horizontal Link

Fig. 2 shows the variation of the PDF of the horizontal link with the channel state for different values of qH. Since qH denotes the ratio of the horizontal and vertical deviations, it can be seen from Fig. 2 that the higher the assymmetry

of deviations the lower the PDF of horizontal link. Rayleigh distributed deviations (q_H = 1) yield the highest PDF while the PDF takes the lowest value for Hoyt distributed deviations (qH = 0.1). Keeping the channel state ash= 1×10⁻⁴, the channel PDF takes the value of fh(h) ≈ 668 for Rayleigh distributed deviations however, the channel PDF falls to the level of fh(h) ≈ 329 for Hoyt distributed deviations. It is also observed from Fig. 2 that the values of the channel PDF starts to approach each other with the increase of channel state. Another conclusion that can be drawn from Fig. 2 is that the trend changes for higher values of the channel state.

For example, the case ofqH = 0.8 shows a larger value for its PDF than for q_H = 1. This shows that PDF maintains its higher level for symmetric deviations with the increase of the channel state. However, after certain level of the channel state, the PDF saturates and starts to be smaller for symmetric deviations than that of asymmetric deviations.

In Fig. 3, the outage probability between two low altitue

10^-4 h 10^-3

10² 10³

f h(h)

qH=1 qH=0.8 qH=0.6 qH=0.4 qH=0.3 qH=0.1 q_H=1.0 (Rayleigh)

qH=0.1 (Hoyt)

Fig. 2: PDF of horizontal link for various values ofqH (H = 10000m,L= 20km, andσ_s= 10×r_a)

platforms (LAPS) is depicted as a function of threshold value of channel state for different ratios of beam waist and receiver aperture radius (ωb/ra). Assuming that two LAPs hoverH = 500m above and the link distance isL = 5km, the outage probability of the FSO link between two LAPs increases with the rise of the threshold of the channel state.

It is also inferred from Fig. 3, the outage probability stands smaller for the higher values of the ratios of beam waist and receiver aperture radius. When the threshold of channel state is h_th = 4×10⁻⁴, the outage probability drops from

∼ 1.2×10⁻¹ to ∼ 2.2×10⁻⁵ with the increase of ω_b/r_a from 5 to15. Although a decrease is observed in the outage probability with low value of ω_b/r_a for the parameters used, the outage probability variation with the beam waist may not be monotonic and an optimization may be required for the larger values of beam waist [3]. The proper selection of beam width at the receiver and controlling theωb/raratio can decrease the outage probability to optimum levels. Fig. 4 gives the variation of the outage probability of a FSO link between two HAPSs with threshold of the channel state. Both HAPSs are locatedH= 10km above and the distance between them isL= 20km. As it is seen in Fig. 3, an increase in the outage probability with the increase ofhthand the decrease ofωb/ra

is also observed in Fig. 4. However, the outage probability

(9)

maintains in lower levels at the higher altitudes. For example, the outage probability takes the value of ∼ 2.2×10⁻⁵ for link distance L = 5km and height H = 500m in Fig. 3.

The approximate value of outage probability is obtained for link distance L = 20km and height H = 10km in Fig. 4.

This shows that the atmospheric turbulence effect becomes less effective when altitude increases. Changing altitude from H = 500m toH= 10km extends the distance producing the same outage probability approximately four times for same conditions. Our simulation results show that (not shown here), increasing the altitude of HAPSs to H = 20km and above causes drastic fall in the outage probability and it can be said that atmospheric turbulence effect can be neglected for horizontal link between HAPSs above H = 20km up to several ten kilometers. The observed benefit of higher beam waist in Figs. 3 and 4 can be physically explained as the result of increasing the probability of optical beam being captured by the receiver aperture with the increase of beam waist. The beam waist up to a certain level causes more of the optical beam landing on the photo detector and increase as such the intensity level to be processed.

10^-4 h 10^-3

th 10^-6

10^-4 10^-2 10⁰

Outage Probability ^b^/r^a⁼¹⁵

b/r a=10 b/r

a=8 b/r

a=5

b/r a=12

Fig. 3: LAP-LAP horizontal link outage probability (H = 500m,L= 5km, andσs= 4×ra)

10^-4 h 10^-3

th 10^-6

10^-4 10^-2 10⁰

Outage Probability

b/r a=5

b/r a=8

b/r a=10

b/r a=12

b/r a=15

Fig. 4: HAPS-HAPS horizontal link outage probability for various values of ωb/ra (H = 10000m, L = 20km, and σs= 4×ra)

B. Downlink

The performance of the HAPS to ground FSO link as function of different parameters is presented in Figs. 5-11.

From Fig. 5, it can be seen that outage probability of the downlink increases with the increase of the threshold value of channel state (h_th) and decrease of the ratio of the beam waist and receiver aperture radius (ωb/ra). Increase in the beam waist causes reduction in the outage probability. When hth = 1×10⁻⁵, the outage probability is ∼ 3.3 ×10⁻² for ωb/ra = 5 while outage probability is ∼ 7.83×10⁻⁶ for ωb/ra = 12. The effect of adaptive optics correction on the outage probability variation is also illustrated in Fig. 5.

It can be observed that applying adaptive optics correction yields a considerable improvement in the performance of downlink especially when beam waist is higher. Removing five Zernike modes (N = 5), changes the outage probability from ∼7.83×10⁻⁶ to ∼2.8×10⁻⁶ for ω_b/r_a = 12when h_th= 1×10⁻⁵.

10^-6 10^-5 10^-4 10^-3 10^-2

h_th 10^-6

10^-4 10^-2 10⁰

b/r a=5 b/r

a=8 b/r

a=10 b/r

a=12

without AO with AO

Fig. 5: Downlink outage probability versus channel state threshold for various values ofω_b/r_a

To show the impact of the SNR on the outage probability, the variation of the dependence of the outage probability of a downlink as function of the SNR threshold is illustrated in Fig. 6. The instantaneous electrical SNR can be defined as γ=^η²^P_σ^t2²^h²

n whereη is the responsivity of the detector, Pt is the transmitted power, and σ²_n is the noise variance. In Fig.

6, the parameters are selected as η = 0.9,Pt= 10dBm and σ²_n = 1×10⁻¹⁶. It can be seen from Fig. 6 that the outage probability increases when the threshold level of the average SNR takes higher values. When ωb/ra = 12, the outage probability increases from∼8.8×10⁻⁷ to∼1×10⁻⁴ with the rise of SNR threshold from γth= 10dB toγth= 30dB.

It can also be seen from Fig. 6 that the AO correction brings an improvement in the performance of communication system up to a certain level.

In Fig. 7, the outage probability variation versus the number of Zernike modes removes is given for different values of Zenith angle. One can see from Fig. 7 that outage probability raises with the increase of the Zenith angle. The performance of the downlink communication degrades severely when the Zenith angle approaches to 90^◦, for example, the outage probability takes the value of∼6.5×10⁻⁴ for Zenith angle ζ = 85^◦ when no adaptive optics correction is applied. The outage probability falls around ∼ 10⁻⁶ levels when Zenith angle is smaller than ζ= 60^◦ that is atmospheric turbulence is in weak regime when Zenit angle between tranmitter and

(10)

10 15 20 25 30 th (dB)

10^-8 10^-6 10^-4 10^-2

b/r a=8 b/r

a=10 b/r

a=12 b/r

a=15 without AO

with AO

Fig. 6: Downlink outage probability versus SNR threshold for various values of ωb/ra

receiver is below the mentioned value. It is seen from Fig. 7 that adaptive optics correction can improve the performance of downlink. Adaptive optics application reduces outage probability up to approximately N = 5 Zernike modes removed then, continuing the removal of Zernike modes causes slight reduction in outage probability. The highest improvement in outage probability is obtained for highes value of Zenith angle.

The downlink performance improvement with smaller Zenith angle values and adaptive optics correction is also shown in Fig. 8.

The altitude dependency of the outage probability is plotted in Figs.8-10. As the altitude of HAPS increases, the downlink performance degrades. Keeping Zenith angle ζ = 70^◦ in Fig.8, the outage probability varies from ∼ 8.3 ×10⁻⁶ to

∼ 1.2×10⁻⁵ with the change of the HAPS altitude from H = 5km toH = 28km. The similar upward trend of outage probability with the increase of HAPS altitude can also be seen from Figs. 9-10. To show the effect of the height of the ground station on downlink communication, the outage probability variation depending on the height of the ground station h0 is plotted in Fig. 9. It is seen that keeping the ground station at higher height benefits the performance of FSO downlink. Leaving the hovering HAPS at the altitude H = 25km, the outage probability of the downlink takes the values of ∼ 6.1×10⁻⁵, ∼ 4.4×10⁻⁵, ∼ 3.6×10⁻⁵ and

∼ 2.9×10⁻⁵ for the height of ground station as h0 = 0 (ground level), h₀ = 50m, h₀ = 100m and h₀ = 200m, respectively. At the same altitude of HAPS and h₀ = 0, outage probability drops from∼6.1×10⁻⁵ to∼4.1×10⁻⁵ when adaptive optics correction is applied andN = 5Zernike modes removed.Since atmospheric turbulence becomes more effective at lower altitudes, installing the ground station at higher heights will result in less exposure of optical beam to the turbulence so the performance of FSO communication systems will improve.

Fig. 10 demonstrates the effect of both the aperture averaging and adaptive optics correction on the performance of downlink FSO communication. The monotonic decrease in the outage probability is seen together with the larger aperture diameter size. Using a receiver with DG = 10cm aperture diameter decreases the outage probability from ∼ 4.8 × 10⁻⁵ to

∼ 2.4×10⁻⁵ for the downlink between ground station and

HAPS at the altitude ofH = 20km. This shows the benefit of the aperture averaging on the performance of FSO downlink communication. Results in Fig.10 also indicate that aperture averaging, that is using larger aperture at the receiver side and collecting the optical beam on the larger area, remains an important tool in terms of compensating the intensity fluctuations (scintillation) and mitigating the turbulence effect.

0 10 20

Zernike modes removed (N) 10^-5

10^-4 10^-3

=85^°

=70^°

=60^°

=50^°

0 10 20

Zernike modes removed (N) 10^-5

10^-4 10^-3

=85^°

=70^°

=60^°

=50^°

without AO with AO

(A) (B)

Fig. 7: Downlink outage probability versus Zernike modes removed for various values of the Zenith angleζA) Comparison of AO and no AO cases B) only AO results

5 10 15 20 25

Altitude (H) 10^-5

10^-4 10^-3

=85^°

=70^°

=60^°

=50^°

without AO with AO

Fig. 8: Downlink outage probability versus altitude for various Zenith angle values

5 10 15 20 25

Altitude (H) 10^-5

10^-4

h0=0 h0=50 m

h0=100 m h0=200 m

without AO with AO

Fig. 9: Downlink outage probability versus HAPS altitude for various heights of ground station

In Fig. 11, the variation of downlink outage probability is indicated against both Zenith angle and the average wind