# Ergodic Capacity Analysis of Free-Space Optical Links With Nonzero Boresight Pointing Errors

A unified capacity analysis of a free-space optical (FSO) link that accounts for nonzero boresight pointing errors and both types of detection techniques (i.e. intensity modulation/direct detection as well as heterodyne detection) is addressed in this work. More specifically, an exact closed-form expression for the moments of the end-to-end signal-to-noise ratio (SNR) of a single link FSO transmission system is presented in terms of well-known elementary functions. Capitalizing on these new moments expressions, we present approximate and simple closed-form results for the ergodic capacity at high and low SNR regimes. All the presented results are verified via computer-based Monte-Carlo simulations.

## Authors

• 4 publications
• 167 publications
• 32 publications
• ### Performance Analysis of Free-Space Optical Links Over Málaga (M) Turbulence Channels with Pointing Errors

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• ### Capacity Analysis of Index Modulations over Spatial, Polarization and Frequency Dimensions

Determining the capacity of a modulation scheme is a fundamental topic o...
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• ### Analysis of the reliability of LoRa

This letter studies the performance of a single gateway LoRa system in t...

• ### Lower Bound on the Capacity of the Continuous-Space SSFM Model of Optical Fiber

The capacity of a discrete-time model of optical fiber described by the ...
11/23/2020 ∙ by Milad Sefidgaran, et al. ∙ 0

• ### Photon-Flooded Single-Photon 3D Cameras

Single photon avalanche diodes (SPADs) are starting to play a pivotal ro...
03/20/2019 ∙ by Anant Gupta, et al. ∙ 4

• ### On the Effect of Correlation on the Capacity of Backscatter Communication Systems

We analyse the effect of correlation between the forward and backward li...
03/11/2020 ∙ by J. L. Matez-Bandera, et al. ∙ 0

• ### Simple Closed-Form Approximations for Achievable Information Rates of Coded Modulation Systems

The intuitive sphere-packing argument is used to obtain analytically-tra...
12/02/2020 ∙ by Maria Urlea, et al. ∙ 0

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## I Introduction

### I-a Background

In recent times, radio frequency (RF) spectrum scarcity has become one of the biggest and prime concern in the arena of wireless communications. Due to this RF spectrum scarcity, additional RF bandwidth allocation, as utilized in the recent past, is not anymore a viable solution to fulfill the demand for higher data rates [haykin]. Of the many other popular solutions, free-space optical (FSO) systems have gained an increasing interest due to their advantages including higher bandwidth and higher capacity compared to the traditional RF communication systems.

FSO links are license-free and hence are cost-effective relative to the traditional RF links. FSO is indeed a promising technology as it offers full-duplex Gigabit Ethernet throughput in certain applications and environment offering a huge license-free spectrum, immunity to interference, and high security [popoola]. These features of FSO communication systems potentially enable solving the issues that the RF communication systems face due to the expensive and scarce spectrum (see [peppas1, peppas4, ansari6] and references therein). Additionally, FSO communications does offer bandwidth as the world record stands at 1.2 Tbps or 1200 Gbps [hogan]. With the correct setup, much higher speeds may be possible as the approach utilizes multiple wavelengths acting like separate channels. Hence, in this concept, the signals are sent down a fiber and launched through the air (known as fiber over the air) and then they travel through a lens before ending up back in fiber [hogan]. Besides these nice characteristic features of FSO communication systems, they span over long distances of 1Km or longer. However, the atmospheric turbulence may lead to a significant degradation in the performance of the FSO communication systems [andrews].

Thermal expansion, dynamic wind loads, and weak earthquakes result in the building sway phenomenon that causes vibration of the transmitter beam leading to a misalignment between transmitter and receiver known as pointing error. These pointing errors may lead to significant performance degradation and are a serious issue in urban areas, where the FSO equipments are placed on high-rise buildings [sandalidis2]. It is worthy to learn that intensity modulation/direct detection (IM/DD) is the main mode of detection in FSO systems but coherent communications have also been proposed as an alternative detection mode. Among these, heterodyne detection is a more complicated detection method but has the ability to better overcome the turbulence effects (see [tsiftsis] and references cited therein).

### I-B Motivation

Over the last couple of decades, a good amount of work has been done on studying the performance of a single FSO link operating over weak turbulence channels modeled by lognormal (LN) distribution (see [fried, niu2, zhu1, cheng, liu3, ansari13] and references cited therein), operating over composite turbulence channels (such as Rician-lognormal (RLN) (see [yang2, churnside, song1, yang3, ansari13] and references cited therein)), and operating over generalized turbulence channels modeled by Málaga () distribution (see [navas, balsells, wang2, ansari12] and references therein) and Gamma-Gamma (GG) distribution (as a special case to distribution) (see [andrews, peppas1, popoola, park, safari, navidpour, kedar, ansari6, liu, sandalidis3, ansari11] and references therein) under heterodyne detection as well as IM/DD techniques. However, as per authors’ best knowledge, there are no unified exact expressions nor asymptotic expressions that capture the ergodic capacity performance of both these detection techniques with nonzero boresight pointing errors under such turbulence channels.

### I-C Contributions

The key contributions of this work are stated as follows.

• The integrals are setup for the ergodic capacity of the LN, the RLN, and the (also GG as a special case of ) turbulence models in composition with nonzero boresight pointing errors. On analyzing these integrals, it is realized that most of these integrals are very complex to solve and to the authors’ best knowledge, an exact closed-form solution to most of these integrals is not achievable. Hence, it is required to look into alternative solutions to analyze the ergodic capacity for such turbulence models.

• A unified approach for the calculation of the moments of a single FSO link is presented in exact closed-form in terms of simple elementary functions for the LN, the RLN, and the (also GG as a special case of ) turbulence models. These unified moments are then utilized, as an alternative solution, to perform the ergodic capacity analysis for such turbulence models.

• A general methodology is presented for simplifying the ergodic capacity analysis of composite FSO turbulence models by independently integrating the various constituents of the composite turbulence model thereby trying to reduce the number of integrals. If succeeded to reduce to a single integral (that is not solvable further) then various techniques such as Gauss-Hermite formula can be utilized to obtain the required results.

• Asymptotic closed-form expressions for the ergodic capacity of the LN, the RLN, and the (also GG as a special case of ) FSO turbulence models, applicable to high as well as low signal-to-noise ratio (SNR) regimes, are derived in terms of simple elementary functions via utilizing the derived unified moments.

### I-D Structure

The remainder of the paper is organized as follows. Sections II presents the channel and system model inclusive of the nonzero boresight pointing error model and the various turbulence models applicable to both the types of detection techniques (i.e. heterodyne detection and IM/DD) utilized in this work. Section III presents the derivation of the exact closed-form channel statistic in terms of the moments in simple elementary functions for the various turbulence models introduced in Section II under the effects of nonzero boresight effects. Ergodic capacity analysis in terms of approximate though closed-form expressions is presented along with some simulation results to validate these analytical results in Section IV for these turbulence channels in terms of simple elementary functions. Finally Section V makes some concluding remarks.

## Ii Channel and System Model

We consider a FSO system with either of the two types of detection techniques i.e. heterodyne detection (denoted in our formulas by ) or IM/DD (denoted in our formulas by ). The transmitted data propagates through an atmospheric turbulence channel in the presence of pointing errors. The received optical power is converted into an electrical signal through either of the two types of detection technique (i.e. heterodyne detection or IM/DD) at the photodetector. Assuming additive white Gaussian noise (AWGN) for the thermal/shot noise, the received signal can be expressed as

 y=Ix+N, (1)

where is the transmit intensity and is the channel gain. Following [farid, yang3], we assume that the off-axis scintillation varies slowly near the spot of boresight displacement and uses a constant value of scintillation index to characterize the atmospheric turbulence. Hence, the atmospheric turbulence and the pointing error are independent. Subsequently, the channel gain can be expressed as , where is the path loss that is a constant in a given weather condition and link distance,

is a random variable that signifies the atmospheric turbulence loss factor, and

is another random variable that represents the pointing error loss factor.

### Ii-a Pointing Error Models

#### Ii-A1 Nonzero Boresight Pointing Error Model

Pointing error impairments are assumed and employed to be present for which the probability density function (PDF) of the irradiance

with nonzero boresight effects is given by 111For detailed information on this model of the pointing error and its subsequent derivation, one may refer to [yang3]. [yang3, Eq. (5)]

 fp(Ip) =ξ2/Aξ20exp{−s2/(2σ2s)}Iξ2−1p (2) ×I0(s/σs√−2ξ2ln{Ip/A0}),0≤Ip≤A0,

where

is the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter)

at the receiver, is a constant term that defines the pointing loss, is the boresight displacement, and represents the -order modified Bessel functions of an imaginary argument of the first kind [gradshteyn, Sec. (8.431)].

#### Ii-A2 Zero Boresight Pointing Error Model

The PDF of the irradiance with zero boresight effects (i.e. in (2)) is given by 222For detailed information on this model of the pointing error and its subsequent derivation, one may refer to [farid]. [farid, Eq. (11)]

 fp(Ip)=ξ2Iξ2−1p/Aξ20,0≤Ip≤A0. (3)

### Ii-B Atmospheric Turbulence Models

#### Ii-B1 Lognormal (LN) Turbulence Scenario

The optical turbulence can be modeled as LN distribution when the optical channel is considered as a clear-sky atmospheric turbulence channel [niu2]. Hence, for weak turbulence conditions, reference [andrews] suggested a LN PDF to model the irradiance that is the power density of the optical beam. Employing weak turbulence conditions, with a log-scale parameter , the LN PDF of the irradiance is given by (please refer to [andrews, niu2] and references therein)

 fL(IaL)=1IaL√2πσexp⎧⎨⎩−[ln{IaL}−λ]22σ2⎫⎬⎭,IaL>0, (4)

where is defined as the scintillation index [niu2, Eq. (1)]

or the Rytov variance

and is related to the log-amplitude variance by , and is the log-scale parameter [niu2].

Now, the joint distribution of

can be derived by utilizing

 f(ILN) =∫∞ILN/A0fILN|IaL(ILN|IaL)fL(IaL)dIaL (5) =∫∞ILN/A01IlIaLfp(ILNIlIaL)fL(IaL)dIaL.

On substituting (4) and (2) appropriately into the integral in (5), following PDF under the influence of nonzero boresight effects is obtained as [yang3, Eq. (10)]

 f(ILN) =ξ2/[2(IlA0)ξ2]Iξ2−1LNexp{ξ2[ξ2σ2/2−λ]+s2/σ2s} (6)

where is the complementary error function [abramowitz, Eq. (7.1.2)]. As a special case, for , the integral in (5) results into the PDF that is in absence of the boresight effects as

 f(ILN) =ξ2/[2(IlA0)ξ2]Iξ2−1LNexp{ξ2[ξ2σ2/2−λ]} (7) ×erfc{[ξ2σ2−λ+ln{ILN/(IlA0)}]/[√2σ]}.

#### Ii-B2 Rician-Lognormal (RLN) Turbulence Scenario

In FSO communication environments, the received signals can also be modeled as the product of two independent random processes i.e. a Rician small-scale turbulence process and a lognormal large-scale turbulence process [churnside, yang2]. The Rician PDF (amplitude PDF) of the irradiance is given by [slim_book, Eq. (2.16)]

 fR(IaR) =(k2+1)/Ωexp{−k2−[(k2+1)/Ω]IaR} (8) ×I0(2k√(k2+1)/ΩIaR),IaR>0,

where is the mean-square value or the average power of the irradiance being considered and is the turbulence parameter. This parameter is related to the Rician factor by that corresponds to the ratio of the power of the line-of-sight (LOS) (specular) component to the average power of the scattered component. The LN PDF is as given in (4).

Now, with the presence of the nonzero boresight pointing errors whose PDF is given in (2), the combined PDF of is given as

 f(IRLN)=(k2+1)ξ2/[2(IlA0)ξ2]exp{−k2} (9) ×exp{ξ2[ξ2σ22−λ]+s2σ2s}∫∞01zξ2exp{−k2+1zIRLN} ×I0(2k√k2+1zIRLN)erfc⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ξ2σ2−λ+3s2ξ2σ2s+ln{zIlA0}√2(s2σ2sξ4+σ2)⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎬⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎭dz.

Similarly, the combined PDF of , in presence of zero boresight pointing errors whose PDF is given in (3), is given as

 f(IRLN)=(k2+1)ξ2/[2(IlA0)ξ2]exp{−k2} (10) ×exp{ξ2[ξ2σ22−λ]}∫∞01zξ2exp{−k2+1zIRLN} ×I0(2k√k2+1zIRLN)erfc⎧⎪ ⎪⎨⎪ ⎪⎩ξ2σ2−λ−ln{zIlA0}√2σ⎫⎪ ⎪⎬⎪ ⎪⎭dz.

The integrals in (9) and (10), to the best of our knowledge, are not easy to solve and hence the analysis will be resorted based on moments as will be seen in the upcoming sections.

#### Ii-B3 Málaga (M) Turbulence Scenario

The optical turbulence can be modeled as distribution when the irradiance fluctuating of an unbounded optical wavefront (plane or spherical waves) propagates through a turbulent medium under all irradiance conditions in homogeneous, isotropic turbulence [navas]. As a special case, the optical turbulence can be modeled as GG distribution when the optical channel is considered as a cloudy/foggy-sky atmospheric turbulence channel [sandalidis2, tsiftsis, sandalidis, gappmair, wang1]. Hence, employing generalized turbulence conditions, the PDF of the irradiance is given by [navas]

 fM(IaM)=Aβ∑m=1amIaMKα−m(2√αβIaMgβ+Ω′),IaM>0, (11)

where

 A ≜2αα/2g1+α/2Γ(α)(gβgβ+Ω′)β+α/2, (12) am ≜(β−1m−1)(gβ+Ω′)1−m/2(m−1)!(Ω′g)m−1(αβ)m/2,

is a positive parameter related to the effective number of large-scale cells of the scattering process, is the amount of fading parameter and is a natural number 333A generalized expression of (14) is given in [navas, Eq. (22)] for

being a real number though it is less interesting due to the high degree of freedom of the proposed distribution (Sec. III of

[navas])., denotes the average power of the scattering component received by off-axis eddies, is the average power of the total scatter components, the parameter represents the amount of scattering power coupled to the LOS component, represents the average power from the coherent contributions, is the average power of the LOS component, and are the deterministic phases of the LOS and the coupled-to-LOS scatter terms, respectively, is the Gamma function as defined in [gradshteyn, Eq. (8.310)], and is the -order modified Bessel function of the second kind [gradshteyn, Sec. (8.432)]. It is interesting to know here that denotes the average power of the coupled-to-LOS scattering component and .444Detailed information on the distribution, its formation, and its random generation can be extracted from [navas, Eqs. (13-21)].

Now, with the presence of the nonzero boresight pointing errors whose PDF is given in (2), the combined PDF of is given as

 f(IM)=ξ2AIξ2−1MIξ2lAξ20exp{−s22σ2s}β∑m=1∫∞I/A0I1−ξ2aM (13) ×I0⎛⎜⎝sσs ⎷−2ξ2ln{IMIlIaMA0}⎞⎟⎠Kα−m(2√αβIaMgβ+Ω′)dIaM.

The integral in (13), to the best of our knowledge, is not easy to solve in closed-form and hence the analysis will be resorted based on moments as will be seen in the upcoming sections. Similarly, the combined PDF of , in presence of zero boresight pointing errors (i.e. in (13)) whose PDF is given in (3), is known to be given by [navas]

 f(IM)=ξ2A2IMβ∑m=1bm (14)