Error Performance Analysis of FSO Links with Equal Gain Diversity Receivers over Double Generalized Gamma Fading Channels
aa r X i v : . [ c s . I T ] M a y Error Performance Analysis of FSO Links withEqual Gain Diversity Receivers over DoubleGeneralized Gamma Fading Channels
Mohammadreza Aminikashani and Mohsen KavehradDepartment of Electrical EngineeringThe Pennsylvania State University, University Park, PA 16802Email: { mza159, mkavehrad } @psu.edu Abstract —Free space optical (FSO) communication has beenreceiving increasing attention in recent years with its ability toachieve ultra-high data rates over unlicensed optical spectrum. Amajor performance limiting factor in FSO systems is atmosphericturbulence which severely degrades the system performance. Toaddress this issue, multiple transmit and/or receive aperturescan be employed, and the performance can be improved viadiversity gain. In this paper, we investigate the bit error rate(BER) performance of FSO systems with transmit diversity orreceive diversity with equal gain combining (EGC) over atmo-spheric turbulence channels described by the Double GeneralizedGamma (Double GG) distribution. The Double GG distribution,recently proposed, generalizes many existing turbulence modelsin a closed-form expression and covers all turbulence conditions.Since the distribution function of a sum of Double GG randomvariables (RVs) appears in BER expression, we first derive aclosed-form upper bound for the distribution of the sum ofDouble GG distributed RVs. A novel union upper bound forthe average BER as well as corresponding asymptotic expressionis then derived and evaluated in terms of Meijers G-functions.
Index Terms —Free-space optical systems, atmospheric tur-bulence, Double GG distribution, bit error rate, equal gaincombining, spatial diversity.
I. I
NTRODUCTION F REE-SPACE optical (FSO) communication, which hasbeen receiving growing attention, refers to terrestrial line-of-sight optical transmission through the atmosphere usinglasers or light emitting diodes (LEDs). FSO systems offer cost-effective, license-free and high capacity communication whichis a promising solution for the ”last mile” problem. The uniqueproperties of these systems also make them appealing for anumber of other applications including fiber backup, wirelessmetropolitan area network extensions, broadband access toremote or underserved areas, and disaster recovery. [1]–[3].A major performance impairing factor in FSO systems isthe effect of atmospheric turbulence-induced fading resultingin random fluctuations in the received signal [4]. Atmosphericturbulence occurs as a result of the variations in the refrac-tive index due to inhomogenities in the temperature and thepressure of the atmosphere. In the literature, many statisticalmodels have been proposed to model this random phenomenon for different degrees of turbulence severity. Under weak tur-bulence conditions, log-normal distribution has been widelyused to model the random irradiance fluctuations [5]–[10]. Asthe strength of turbulence increases, log-normal distributionshows large deviations from the experimental data. Therefore,other statistical models such as K [11], I-K [12], log-normalRician [13], Gamma-Gamma [14], M [15] and Double Weibull[16] distributions have been proposed to cover a wide rangeof turbulence conditions. Recently, a unifying statistical distri-bution named Double Generalized Gamma (Double GG) wasproposed in [17] which is valid under all range of turbulenceconditions and contains most of the existing models in theliterature as special cases.Spatial diversity techniques provide a promising approachto mitigate turbulence-induced fading. Over the years, theperformance of FSO systems with spatial diversity havebeen extensively studied over the most commonly utilizedturbulence-induced fading models including log-normal, K,negative exponential and Gamma-Gamma [18]–[22]. In [23],a unified closed-form expression for the BER of single-inputmultiple-output (SIMO) FSO systems with optimal combining(OC) receiver over Double GG channel was proposed. How-ever, the BER performance of equal gain combining (EGC)receivers, which is equivalent to that of multiple-input single-output (MISO) FSO systems, was presented in integral form.In fact, the main difficulty in studying EGC receivers is thedistribution of the sum of Double GG R.Vs required to bederived in order to obtain a closed-form solution.An analytical solution for the distribution of the sum ofDouble GG RVs is a very cumbersome task if not impossible.Thus, in this paper, we derive a novel upper bound for thedistribution of the sum of Double GG distributed RVs. Partic-ularly, a useful expression for the distribution of the productof Double GG distributed RVs is first obtained. Then, basedon a well-known inequality between arithmetic and geometricmeans, a closed-form union upper bound for the distribution ofthe sum of Double GG distributed RVs is derived. This deriveddistribution is used to study the error rate performance of FSOsystems with MISO/SIMO with EGC receivers employing c (cid:13) I n ( I n ) = γ ,n p n p nm ,n − / q m ,n − / I − (2 π ) ( p n + q n ) / − Γ ( m ,n ) Γ ( m ,n ) G ,p n + q n p n + q n , "(cid:18) Ω ,n I γ ,n (cid:19) p n p p n n q q n n Ω q n ,n m q n ,n m p n ,n | ∆ ( q n : 1 − m ,n ) , ∆ ( p : 1 − m ,n ) − (1)intensity modulation/direct detection (IM/DD) with on-offkeying (OOK) over independent and not necessarily identicallydistributed (i.n.i.d.) Double GG turbulence channels.The rest of the paper is organized as follows: In Section II,the statistical characteristics of the Double GG distribution areprovided, and an upper bound on the pdf of the sum of DoubleGG variates is presented. In Section III, we introduce theSIMO FSO system model. In Section IV, the BER expressionsfor SIMO FSO links are provided. In Section V, we presentnumerical results to confirm the accuracy of the derivedexpressions and demonstrate the advantages of employingspatial diversity over SISO links. Finally, Section VI concludesthe paper.II. D EFINITION AND S TATISTICAL C HARACTERISTICS
A. Double GG distribution
Let { I n } Nn =1 be N statistically independent but not neces-sarily identically distributed Double GG RVs whose probabil-ity density function (pdf) follows (1) at the top of the page[17, Eq. (5)].In (1), G [ . ] is the Meijers G-function [24, Eq.(9.301)], ∆ ( j ; x ) is defined as ∆ ( j ; x ) , x/j , ..., ( x + j − /j , p n and q n are positive integer numbers that satisfy p n /q n = γ ,n /γ ,n , and m i,n ≥ . , i = 1 , , is a distribution shapingparameter. The distribution parameters γ i,n and Ω i,n of theDouble GG model can be identified using the followingequations Ω i,n = (cid:18) Γ ( m i,n )Γ ( m i,n + 1 /γ i,n ) (cid:19) γ i,n m i,n , i = 1 , (2) σ x,n = Γ ( m ,n + 2 /γ ,n ) Γ ( m ,n )Γ ( m ,n + 1 /γ ,n ) − (3a) σ y,n = Γ ( m ,n + 2 /γ ,n ) Γ ( m ,n )Γ ( m ,n + 1 /γ ,n ) − (3b)where σ x,n and σ y,n are normalized variances of small andlarge scale irradiance fluctuations, respectively. The DoubleGG distribution accurately describes irradiance fluctuationsover atmospheric channels under a wide range of turbulenceconditions (weak to strong) [17]. Furthermore, it is verygeneric and includes most commonly used fading modelsproposed in the literature as special cases such as Gamma-Gamma ( γ i,n = 1 , Ω i,n = 1 ), Double-Weibull ( m i,n = 1 ),and K channel ( γ i,n = 1 , Ω i,n = 1 , m ,n = 1 ). In addition,for the limiting case of γ i,n → and m i,n → ∞ , Double GGpdf coincides with the log-normal pdf. Meijers G-function is a standard built-in function in mathematicalsoftware packages such as MATLAB, MAPLE and MATHEMATICA. Ifrequired, this function can be also expressed in terms of the generalizedhypergeometric functions using [24, Eqs.(9.303-304)].
The cumulative distribution function (cdf) of Double GGdistribution can be derived from (1) as in (4) at the top of thenext page [17].
B. An Upper-Bound for the Distribution of the Sum of DoubleGG Variates
Let us define a new R.V R , as the product of N DoubleGG R.Vs I n , i.e., R , N Y n =1 I n . (5)The Double GG distribution considers irradiance fluctua-tions as the product of small-scale and large-scale fluctua-tions which are both governed by Generalized Gamma (GG)distributions, i.e. I n = U x,n U y,n , where U x,n and U y,n arestatistically independent arising respectively from large-scaleand small scale turbulent eddies. Thus, R can be expressed asthe product of N GG R.Vs U n , i.e., R = N Y n =1 U n . (6)The pdf of R can be obtained using the statistical modelproposed in [25] as f R ( r ) = αξr G β, ,β (cid:20) r α ω (cid:12)(cid:12)(cid:12)(cid:12) − J α ( γ N , m N ) (cid:21) (7)where ξ , ω and J α ( γ N , m N ) are defined as ξ = (cid:16) √ π (cid:17) N − β N Y l =1 ( α/γ l ) m l − / Γ ( m l ) (8) ω = N Y l =1 (cid:18) α Ω l m l γ l (cid:19) α/γ l (9) J α ( γ N , m N ) , (10) ∆ ( α / γ ; m ) , ∆ ( α / γ ; m ) , . . . , ∆ ( α / γ N ; m N ) In (7), α and β are two positive integers defined as α , N Y l =1 k l (11) β , α N X l =1 γ l (12)under the constraint that l l = 1 γ l l Y i =1 k i (13)is a positive integer with k i being also a positive integer. I n ( I n ) = p m − / n q m − / n (2 π ) − ( p n + q n ) / Γ ( m ,n ) Γ ( m ,n ) G p n + q n , ,p n + q n +1 "(cid:18) I γ ,n n Ω ,n (cid:19) p n m q n ,n m p n ,n p p n n q q n n Ω q n ,n |
1∆ ( q n : m ,n ) , ∆ ( p n : m ,n ) , (4)The cdf of R can be derived from (7) as F R ( r ) = ξG β, ,β +1 (cid:20) r α ω (cid:12)(cid:12)(cid:12)(cid:12) J α ( γ N , m N ) , (cid:21) . (14)Using the well-known inequality between arithmetic andgeometric means, i.e. A N ≥ G N , with A N = 1 N n X n =1 I n (15)and G N = N Y n =1 I /Nn (16)a lower-bound for R.V Z dened as the sum of Double GGR.Vs, i.e., Z , N X n =1 I n (17)can be obtained as Z ≥ N R /N . (18)Considering Eqs. (14) and (18), the cdf of Z is upper boundedas F Z ( r ) ≤ ξG β, ,β +1 " ( r/N ) αN ω (cid:12)(cid:12)(cid:12)(cid:12) J α ( γ N , m N ) , . (19)By taking the first derivative of (19) with respect to r , an upperbound for the pdf of Z can be obtained in closed-form as f Z ( r ) ≤ f ∗ Z ( r ) (20)where f ∗ Z ( r ) is defined as f ∗ Z ( r ) = N αξr G β, ,β " ( r/N ) αN ω (cid:12)(cid:12)(cid:12)(cid:12) − J α ( γ N , m N ) . (21)III. S YSTEM M ODEL
We consider an FSO system employing IM/DD with OOKwhere the information signal is transmitted via one apertureand received by N apertures (i.e., SIMO) over the Double GGchannel. We assume EGC receivers where the receiver addsthe receiver branches. The received signal is then given by r = ηx N X n =1 I n + υ n , n = 1 , . . . , N (22)where x represents the information bits and can either be 0 or1, υ n is the Additive White Gaussian noise (AWGN) at the n th receive aperture with zero mean and variance σ υ = N / , and η is the optical-to-electrical conversion coefficient. Here, I n is the normalized irradiance from the transmitter to the n th receive aperture whose pdf follows (1). We should emphasizethat the performance of SIMO under the assumption of equalgain combining is equivalent to that of MISO FSO links. IV. BER P ERFORMANCE
A. Upper bound expression
The optimum decision metric for OOK is given by [20] P ( r | on,I n ) on ≶ off P ( r | off,I n ) (23)where r is the received signal vector. Following the sameapproach as [19], [20], the conditional bit error probabilitiesare given by (see [19] for details of derivation) P e, MISO ( off | I n ) = P e, MISO ( on | I n ) = 12 erfc √ ¯ γ N N X n =1 I n ! (24)where ¯ γ is the average electrical SNR obtained as ¯ γ = η /N .Therefore, the average error rate can be expressed as P SIMO,ECG = 12 Z I f I ( I ) erfc √ ¯ γ N N X n =1 I n ! d I (25)where f I ( I ) is the joint pdf of vector I = ( I , I , . . . , I N ) .The factor N is used to ensure that the sum of the N receiveaperture areas is the same as the area of the receive apertureof the SISO link for a fair comparison. The integral expressedin (25) does not yield a closed-form solution even for simplerturbulence distributions. However, an upper bound on (25) canbe obtained by considering (20) and (21) as P MISO ≤ ∞ Z f ∗ Z ( z ) erfc (cid:18) √ ¯ γz N (cid:19) dz. (26)The above integral can be evaluated in closed form by firstexpressing the erfc ( . ) in terms of the Meijer G-functionpresented in [26, eq. (11)] as erfc (cid:0) √ x (cid:1) = 1 √ π G , , (cid:20) x (cid:12)(cid:12)(cid:12)(cid:12) , / (cid:21) . (27)Then, a closed-form expression for (26) is obtained using [26,Eq. (21)] as P SIMO,ECG ≤ N αξq µ √ s (cid:0) √ π (cid:1) s +( q − β (28) × G qβ, s s,qβ + s " (cid:0) γ − sN (cid:1) s ( ωq β N αN ) q (cid:12)(cid:12)(cid:12)(cid:12) ∆ ( s, , ∆ ( s, / K q ( α/γ N , m N ) , ∆ ( s, where s and q are positive integer numbers that satisfy s/q = αN / , and µ and K q ( α/γ N , m N ) are definedas K q ( α/γ N , m N ) = (29) ( J α ( γ N , m N ) q = 1 n J α ( γ N ,m N )2 , J α ( γ N ,m N ) − o q = 2 = N X l =1 m l − N + 1 . (30)The derived upper-bounded BER expression in (28) for SIMOFSO systems with ECG can be seen as a generalization of BERresults over other atmospheric turbulence models. Specially, ifwe insert γ i = 1 and Ω i = 1 in (28), we obtain an upper boundon BER expression over Gamma-Gamma channel. Setting m i = 1 in (28), we obtain an upper bound for BER for DoubleWeibull channel. Similarly, for γ i = 1 , Ω i = 1 and m i = 1 ,an upper-bounded BER expression for K-channel is obtained. B. Diversity Gain and Asymptotic analysis
Although Meijers G-function can be expressed in termsof more tractable generalized hypergeometric functions, (28)appears to be complex and the impact of the basic system andchannel parameters on performance is not very clear. However,for large SNR values, the asymptotic behavior of the systemperformance is dominated by the behavior of the pdf near theorigin, i.e. f ∗ Z ( z ) at z → [27]. Thus, employing a seriesexpansion corresponding to the Meijers G-function [28, Eq.(07.34.06.0006.01)], f ∗ Z ( z ) given in (21) can be approximatedby a single polynomial term as f ∗ Z ( z ) ≈ N αξ ( ωN αN ) min { m γ α , ··· , m N γ Nα } β Y j =1 j = k Γ ( c j − c k ) z N min { m γ , ··· ,m N γ N }− (31)where c k and c j are defined as c k = min n m γ α , · · · , m N γ N α o (32) c j ∈ { ∆ ( α / γ ; m ) , . . . , ∆ ( α / γ N ; m N ) } \ min n m γ α , · · · , m N γ N α o . (33)It must be noted that (31) is only valid for independent and notidentically distributed Double GG turbulence channels. Basedon Eqs. (26) and (31) and at high SNRs, the average BER canbe well approximated as P SIMO,ECG ≈ Γ ((1 +
N αc k ) / ξ √ πc k ( ωN αN ) c k β Y j =1 j = k Γ ( c j − c k ) (cid:18) N √ ¯ γ (cid:19) Nαc k (34)Therefore, the diversity order of FSO links with N receiveapertures employing equal gain combining is obtained as . N min { m γ · · · , m N γ N } .V. N UMERICAL R ESULTS
In this section, we present analytical and Monte-Carlosimulation results using the previous mathematical analysis forthe performance of SIMO FSO links employing EGC receiversover Double GG channels. The performance improvementsover SISO systems are further quantified. Similar as in [23],
Fig. 1. Average BER of EGC and SISO for plane wave assuming i.i.d.turbulent channel defined as channel b .Fig. 2. Average BER of EGC and SISO for spherical wave assuming i.i.d.turbulent channel defined as channel c . we consider the following four scenarios of atmosphericturbulence conditions reported in [17] • Channel a : Plane wave and moderate irradiance fluc-tuations with γ = 2 . , γ = 0 . , m = 0 . , m = 2 . , Ω = 1 . , Ω = 0 . , p = 28 and q = 11 • Channel b : Plane wave and strong irradiance fluctu-ations with γ = 1 . , γ = 0 . , m = 0 . , m = 1 . , Ω = 1 . , Ω = 0 . , p = 17 and q = 7 . • Channel c : Spherical wave and moderate irradiancefluctuations with γ = 0 . , γ = 1 . , m = 2 . , m = 0 . , Ω = 0 . and Ω = 1 . , p = 7 and ig. 3. Average BER of EGC over two i.n.i.d. atmospheric turbulencechannels defined as channel a and channel b . q = 11 . • Channel d : Spherical wave and strong irradiance fluc-tuations with γ = 0 . , γ = 0 . , m = 3 . , m = 2 . , Ω = 0 . and Ω = 0 . , p = 7 and q = 11 .Figs. 1-2 demonstrate the average BER over i.i.d. channelsdefined as channel b and channel c , respectively. In order toverify the tightness of the bound, we present upper boundanalytical results obtained through (28) along with the Monte-Carlo simulation of (25). As a benchmark, the average BERof SISO FSO link obtained through [17, Eq. 24] is alsoincluded in these figures. As clearly seen from Figs. 1-2, thenumerical results for the bounds are close to the equivalentsimulated ones which represent the exact BER. For instance,at a target bit error rate of − , gaps between the exact andthe upper bound curves are 5.2 dB and 6.6 dB respectivelyfor N = 2 and receive apertures employing EGC forchannel b . Similarly, for channel c , at a BER of − , thegaps are respectively 3.8 dB and 4.3 dB for N = 2 and receive apertures employing EGC. This observation clearlydemonstrates the accuracy of the proposed bound. It is alsoillustrated that the upper bound becomes tighter as the numberof receive apertures decreases. This is expected as the upperbound and the exact BER curves coincide for N = 1 . Inaddition, we observe that multiple receive apertures deploy-ment employing EGC signicantly improves the performance.Specially, at a target bit error rate of − , we observeperformance improvements of 46.8 dB and 66.8 dB for SIMOFSO links with N = 2 and receive apertures with respect tothe SISO transmission over channel b . Similarly, for channel c , at a BER of − , impressive performance improvements of51.1 dB and 63.9 dB are achieved for SIMO links with N = 2 and 3 employing EGC compared to the SISO deployment. Fig. 4. Average BER of EGC over two i.n.i.d. atmospheric turbulencechannels defined as channel c and channel d . Figs. 3-4 illustrate the BER performance of SIMO FSOlinks employing EGC receivers over non-identically dis-tributed (i.n.i.d.) Double GG channels. Similar to i.i.d. results,our upper bound closed-form expression yields a close matchto simulation results. For example, at a BER of − in SIMOlinks with N = 2 over i.n.i.d. channels a and b , the differencebetween the exact and the upper bound curves is 7 dB. Fori.n.i.d. channels c and d and at a BER of − , this gap is 5.8dB with N = 2 . We also compare the performance of i.n.i.d.case with respect to i.i.d. case presented in Figs. 1-2. As anexample, to achieve a BER of − in SIMO links over i.n.i.d.channels a and b , we need 8.5 dB less in comparison to i.i.d.case as channel a is less severe than channel b . Note that in Fig.1, we assume that both of the two channels between transmitterand receivers are described by channel b . On the other hand,to achieve a BER of − for SIMO links with N = 2 overi.n.i.d. channels c and d , we need 6.8 dB more in comparisonto i.i.d channels as the channel d exhibits harsher conditionsthan channel c . Note that in Fig. 2, both of the channelsbetween the transmitter and receivers are described by channel c . It can be further observed that asymptotic bounds on theBER become tighter at high enough SNR values confirmingthe accuracy and usefulness of the asymptotic expression givenin (34). VI. C ONCLUSIONS
In this paper, we have derived a closed-form union upperbound for the pdf of the sum of Double GG distributed RVs.Using this bound, we have investigated the BER performanceof FSO links with receive diversity employing equal gaincombining over Double GG turbulence channels. An efficientand unified upper bound for the average BER of SIMO FSOsystems with EGC receiver has been obtained which general-izes BER results over other atmospheric turbulence models aspecial cases. Based on the asymptotical performance analysis,we have further derived diversity gains for SIMO FSO systemsunder consideration. We have presented BER performancebased on analytical and numerical simulation results. Furthercomparisons between numerical and analytical results haveconfirmed the accuracy and usefulness of the derived results.R
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