I Introduction
Ia 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, freespace 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 licensefree and hence are costeffective relative to the traditional RF links. FSO is indeed a promising technology as it offers fullduplex Gigabit Ethernet throughput in certain applications and environment offering a huge licensefree 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 highrise 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).
IB 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 Ricianlognormal (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 GammaGamma (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.
IC 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 closedform 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 closedform 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 GaussHermite formula can be utilized to obtain the required results.

Asymptotic closedform 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 signaltonoise ratio (SNR) regimes, are derived in terms of simple elementary functions via utilizing the derived unified moments.
ID 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 closedform 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 closedform 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
(1) 
where is the transmit intensity and is the channel gain. Following [farid, yang3], we assume that the offaxis 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.Iia Pointing Error Models
IiA1 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 ^{1}^{1}1For detailed information on this model of the pointing error and its subsequent derivation, one may refer to [yang3]. [yang3, Eq. (5)](2)  
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)].IiA2 Zero Boresight Pointing Error Model
The PDF of the irradiance with zero boresight effects (i.e. in (2)) is given by ^{2}^{2}2For detailed information on this model of the pointing error and its subsequent derivation, one may refer to [farid]. [farid, Eq. (11)]
(3) 
IiB Atmospheric Turbulence Models
IiB1 Lognormal (LN) Turbulence Scenario
The optical turbulence can be modeled as LN distribution when the optical channel is considered as a clearsky 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 logscale parameter , the LN PDF of the irradiance is given by (please refer to [andrews, niu2] and references therein)
(4) 
where is defined as the scintillation index [niu2, Eq. (1)]
or the Rytov variance
and is related to the logamplitude variance by , and is the logscale parameter [niu2].Now, the joint distribution of
can be derived by utilizing(5)  
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)]
(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
(7)  
IiB2 RicianLognormal (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 smallscale turbulence process and a lognormal largescale turbulence process [churnside, yang2]. The Rician PDF (amplitude PDF) of the irradiance is given by [slim_book, Eq. (2.16)]
(8)  
where is the meansquare 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 lineofsight (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
(9)  
Similarly, the combined PDF of , in presence of zero boresight pointing errors whose PDF is given in (3), is given as
(10)  
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.
IiB3 Málaga () 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/foggysky atmospheric turbulence channel [sandalidis2, tsiftsis, sandalidis, gappmair, wang1]. Hence, employing generalized turbulence conditions, the PDF of the irradiance is given by [navas]
(11) 
where
(12)  
is a positive parameter related to the effective number of largescale cells of the scattering process, is the amount of fading parameter and is a natural number ^{3}^{3}3A 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 offaxis 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 coupledtoLOS 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 coupledtoLOS scattering component and .^{4}^{4}4Detailed information on the distribution, its formation, and its random generation can be extracted from [navas, Eqs. (1321)].Now, with the presence of the nonzero boresight pointing errors whose PDF is given in (2), the combined PDF of is given as
(13)  
The integral in (13), to the best of our knowledge, is not easy to solve in closedform 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]
(14) 
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