Performance Analysis of RIS-Based nT-FSO Link Over G-G Turbulence With Pointing Errors
Alain R. Ndjiongue, Telex M. N. Ngatched, Octavia A. Dobre, Ana G. Armada, Harald Haas
11 Performance Analysis of RIS-Based nT-FSO LinkOver G-G Turbulence With Pointing Errors
Alain R. Ndjiongue,
Senior Member, IEEE , Telex M. N. Ngatched,
Senior Member, IEEE , Octavia A. Dobre,
Fellow, IEEE , Ana G. Armada,
Senior Member, IEEE , and Harald Haas,
Fellow, IEEE
Abstract โOne of the main problems faced by communicationsystems is the presence of skip-zones in the targeted areas. Withthe deployment of the ๏ฌfth-generation mobile network, solutionsare proposed to solve the signal loss due to obstruction bybuildings, mountains, and atmospheric or weather conditions.Among these solutions, re-con๏ฌgurable intelligent surfaces (RIS),which are newly proposed modules, may be exploited to re๏ฌect theincident signal in the direction of dead zones, increase communi-cation coverage, and make the channel smarter and controllable.In this paper, we tackle the skip-zone problem in near-terrestrialfree-space optical (nT-FSO) systems using RIS. We carry outa performance analysis of RIS-aided nT-FSO links affected byturbulence and pointing errors, for both heterodyne detection(HD) and intensity modulation-direct detection (IM/DD) tech-niques. Turbulence is modeled using the Gamma-Gamma (G-G)distribution. We analyze the model and provide exact closed-form expressions of the probability density function, cumulativedistribution function, and moment generating function of the end-to-end signal-to-noise ratio, ๐พ . Capitalizing on these statistics, weevaluate the system performance through the outage probability, ๐ ๐๐ข๐ก , ergodic channel capacity, ๐ถ , and average bit-error-rate, ๐ ๐ , for selected binary modulation schemes. Numerical resultsobtained for different turbulence levels and pointing errorscon๏ฌrm that the HD technique outperforms IM/DD even inRIS-aided nT-FSO systems. These results also show that usinga blue color offers better channel capacity and communicationperformance compared to red and green colors. Index Terms โFree-space optical communications, re-con๏ฌgurable intelligent surfaces, uni๏ฌed GammaโGammaturbulence channels, average bit-error-rate, ergodic channelcapacity, outage probability, red, green, and blue transmissions.
I. I
NTRODUCTION
The recent extensive investigation of optical wireless com-munications in the outdoor environment, also called free-spaceoptical (FSO), is motivated by its advantages compared to itsradio frequency (RF) counterpart, especially in point-to-pointnetworks. These advantages include larger bandwidth, higherchannel capacity, and cost-effectiveness due to an unlicensedenvironment [1], which can be leveraged to solve the band-width limitation in the RF technology. Its most prominentapplications are satellite-to-ground, satellite-to-satellite, andnear-terrestrial FSO (nT-FSO) systems such as building-to-building (B2B) communications. Besides turbulence, pointing
Alain. R. Ndjiongue, T. M. N. Ngatched, and O. A. Dobre are withthe Faculty of Engineering and Applied Science, Memorial University ofNewfoundland, Canada.Ana G. Armada is with the Signal Theory and Communications Department,Universidad Carlos III de Madrid, Spain.Harald Haas is with the LiFi Research and Development Center, Departmentof Electronic and Electrical Engineering, the University of Strathclyde,Glasgow, United Kingdom. errors, and attenuation that affect optical signals over theFSO channel, signal obstruction due to buildings or trees canprevent the transmitted message to reach the destination. Weattempt to solve this obstructionโs problem in nT-FSO systems,affected by moderate-to-strong turbulence levels and pointingerrors, using re-con๏ฌgurable intelligent surfaces (RIS). RISare electromagnetic devices with electronically controllablecharacteristics. They can re๏ฌect, refract, extinct, or scatterthe incoming signal with an impact on its amplitude, phase,and polarization. The design of RIS modules depends on theapplication.The RIS module is a planar array of multiple mirrors usedto guide the incoming signal toward a targeted area andre-con๏ฌgure the transmission channel [2]. It offers wirelessnetworks several advantages over competing technologies suchas relay systems. In addition to their low power consump-tion, the RIS module is made of electronically controllableelements. These advantages have recently triggered intensiveinvestigations of the technology. It has lately been proposedto solve the dead zone problems in RF networks and createsmart communication channels and environments [3]โ[5], orserve as a wave-guard in visible light communications [6].To create a smart RF channel, authors in [3] investigatedprinciple, challenges, and opportunities related to the use ofRIS modules in an indoor environment. Most works in RIS-aided communication networks focus on simplex systems. Theauthors in [4] analyzed a two-way communication systemassisted by RIS modules. The RIS in๏ฌuence on spatial modula-tions is investigated in [7], where the authors proposed a low-complexity and fast antenna selection algorithm, while in [5],the authors introduced RIS-space shift keying and RIS-spatialmodulation schemes. As part of the channel, the RIS elementsmay decisively impact wireless communication systemsโ per-formance, leading to the need for new pre-coding designs [8].The RIS concept can be extended to re-con๏ฌgurable opticalcomponents. For example, in [9], the authors proposed re-con๏ฌgurable photo-detectors and explored the use of blindinterference alignment to achieve a multiplexing gain withoutcooperation among light-emitting diodes.Due to the presence and locations of obstacles, the use ofRIS in FSO are suitable for nT-FSO communication systemssuch as B2B . Investigation on using RIS in nT-FSO systemsis still in its infancy; however, it is predicted that it willattract signi๏ฌcant research interests. Early work on using a RIS A B2B environment is an nT-FSO data transmission environment wherethe information is transferred between buildings. module in nT-FSO systems is proposed in [10], [11]. In [10],the authors discuss the implementation of a RIS-based FSOsystem considering controllable multi-branches. They neglectturbulence over the channels because of the length shortnessof branches, which was less than 500 m, allowing the analysisto consider only pointing errors. In [11], based on the centrallimit theorem, the authors exploit the Gaussian distributionto approximate a Gamma-Gamma (G-G) channel for a largenumber of transmitting signals. In contrast with [10] and[11], this paper considers a RIS-based nT-FSO communicationsystem exploiting a single light-ray to transmit data over a G-Gchannel with pointing errors.To the best of our knowledge, using RIS in an nT-FSOsystem characterized by G-G turbulence with pointing errors,has not yet been proposed in the open literature and repre-sents the motivation of this paper. The main goals of thepaper are to show the numerical analysis and performanceof the considered nT-FSO system. To this end, we makethe following contributions: ( i ) we derive closed-form uni-๏ฌed statistical expressions of the probability density function(PDF), cumulative distribution function (CDF), and momentgenerating function (MGF) of the end-to-end signal-to-noiseratio (SNR), ๐พ ; ( ii ) based on these results, we derive the outageprobability (OP), ๐ ๐๐ข๐ก , the average ergodic channel capacity, ๐ถ , and the average bit-error-rate (BER), ๐ ๐ , for selected binarymodulation schemes including coherent binary frequency-shiftkeying (CBFSK), non-coherent binary frequency-shift keying(NBFSK), coherent binary phase-shift keying (CBPSK), anddifferential binary phase-shift keying (DBPSK); ( iii ) next,we derive closed-form expressions of the diversity order andcoding gain for the proposed RIS-based nT-FSO; ( iv ) ๏ฌnally,we present numerical results for different turbulence andpointing error levels, and compare the performance betweentransmissions using the red, green, and blue light colors.II. S YSTEM AND C HANNEL M ODELS
A. System Model
The environment under study is a cascaded system of asingle light ray traveling from source (S) to destination (D)after re๏ฌection on a RIS element, as shown in Fig. 1. There isno direct link between S and D owing to obstructions. The RISmodule, located at the top of a building, serves as a re๏ฌectorto the incoming signal and ensures that the transmitted lightpoints to the receiver. We assume that both channel portions,which will be denoted as the system sub-channels, exhibitmoderate-to-strong turbulence levels, and the light intensityover them undergoes the same attenuation level. It is alsoassumed that, at each receiving end, the detectors face similarpointing errors.
B. The FSO Channel
The FSO link is subject to three main signal impairmentfactors: pointing errors, atmospheric turbulence, and atten-uation. These impairment sources, each in its way, affectthe transmitted optical signal, ๐ผ , which can be expressedas ๐ผ = ๐ผ ๐ ๐ผ ๐ ๐ผ ๐ , where ๐ผ ๐ , ๐ผ ๐ , and ๐ผ ๐ represent the receivedintensity affected by pointing errors, atmospheric turbulence, (cid:53)(cid:72)(cid:70)(cid:82)(cid:81)(cid:73)(cid:76)(cid:74)(cid:88)(cid:85)(cid:68)(cid:69)(cid:79)(cid:72)(cid:3)(cid:76)(cid:81)(cid:87)(cid:72)(cid:79)(cid:79)(cid:76)(cid:74)(cid:72)(cid:81)(cid:87)(cid:3)(cid:86)(cid:88)(cid:85)(cid:73)(cid:68)(cid:70)(cid:72)(cid:86)(cid:3)(cid:11)(cid:53)(cid:44)(cid:54)(cid:12) (cid:75) (cid:74) (cid:39)(cid:54) (cid:53)(cid:44)(cid:54)(cid:16)(cid:71)(cid:72)(cid:86)(cid:87)(cid:76)(cid:81)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3) (cid:47) (cid:3)(cid:32)(cid:3)(cid:20)(cid:19)(cid:19)(cid:19)(cid:3)(cid:80) (cid:54)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72)(cid:16)(cid:53)(cid:44)(cid:54)(cid:15)(cid:3) (cid:47) (cid:3)(cid:32)(cid:3)(cid:20)(cid:19)(cid:19)(cid:19)(cid:3)(cid:80) Fig. 1: A model of RIS-aided nT-FSO system.and attenuation, respectively. The PDF describing pointingerrors, ๐ ๐ ( ๐ผ ๐ ) , is given by [1, Eq. (3)], [12] ๐ ๐ ( ๐ผ ๐ ) = ๐ ๐ด ๐ ๐ ๐ผ ๐ โ ๐ , โค ๐ผ ๐ โค ๐ด , (1)where ๐ is the ratio of equivalent beam radius at the receiverand pointing displacement standard deviation, ๐ด ๐ = [ erf ( ๐ฃ )] ,erf (ยท) being the error function and ๐ฃ = ๐ โ ๐ /โ ๐ค ๐ง . Here, ๐ and ๐ค ๐ง are the radius of the receiver aperture and the beamwaist, respectively [1], [12]. In general, the G-G distribution isused to model channels characterized by moderated-to-strongturbulence levels. Its PDF is given by [11] ๐ ๐ผ ๐ ( ๐ผ ๐ ) = ( ๐ผ๐ฝ ) ๐ผ + ๐ฝ ฮ ( ๐ผ ) ฮ ( ๐ฝ ) ๐ผ ๐ผ + ๐ฝ โ ๐ ๐พ ๐ผ โ ๐ฝ (cid:16) p ๐ผ๐ฝ๐ผ ๐ (cid:17) , (2)where ๐พ ๐ (ยท) is the ๐ ๐กโ -order modi๏ฌed Bessel function ofsecond kind. The values of ๐ผ and ๐ฝ can be calculated usingthe Rytov variance, ๐ = . ๐ถ ๐ ๐ ๐ฟ , which dependson the altitude-dependent index, ๐ถ ๐ , characterized by thetransmission environment, the angular wavenumber, ๐ = ๐๐ ,the aperture diameter, ๐ท , and the transmission distance, ๐ฟ .They are respectively given by [13] ๐ผ = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ exp ยฉ ยซ . ๐ (cid:16) + . ๐ + . ๐ / (cid:17) / ยชยฎยฎยฌ โ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป โ , (3)and ๐ฝ = ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ exp ยฉ ยซ . ๐ (cid:16) + . ๐ / (cid:17) โ / (cid:16) + . ๐ + . ๐ / (cid:17) / ยชยฎยฎยฌ โ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป โ , (4)where ๐ = p ๐๐ท / ๐ฟ . Finally, the path loss, which is con-sidered as constant for a given weather condition and linkdistance [1], is given by the Beer-Lambert law as ๐ผ ๐ = ๐ โ ๐ฟ๐ฟ ,where ๐ฟ is the attenuation factor.III. C LOSED - FORM S TATISTICAL A NALYSIS
In this section, we derive uni๏ฌed closed-form expressionsfor the PDF, CDF, and MGF of the end-to-end SNR.
A. End-to-End Signal-to-Noise Ratio (SNR)
We assume that the RIS module plays only a re๏ฌective func-tion and does not allow light through. We also assume perfectknowledge of the channel phases at RIS and destination. The detected signal can be expressed as ๐ฆ = โ ๐ธ ๐ ( โ๐๐ ๐ ๐ ๐ ) ๐ฅ + ๐ ,where ๐ธ ๐ is the symbol energy, โ and ๐ are respectively theS-RIS and RIS-D complex channel vectors, ๐๐ ๐ ๐ character-izes the RIS element with ๐ being its amplitude re๏ฌectioncoef๏ฌcient and ๐ its induced phase [5], [14]. ๐ฅ and ๐ฆ arethe transmitted and received symbols, respectively, and ๐ isthe additive white Gaussian noise at the destination. At RIS,the main goal is to re๏ฌect the original signal in such away to optimize signal reception at D, which can be donethrough end-to-end SNR maximization. This SNR is de๏ฌnedby ๐พ = ๐พ | โ๐๐ ๐ ๐ ๐ | , where ๐พ = ๐ธ ๐ / ๐ represents the averageSNR in both S-RIS and RIS-D sub-channels, and ๐ is thenoise power spectral density at D. B. PDF of the End-to-End SNR
The overall systemโs gain is given by โ๐๐ ๐ ๐ ๐ , where thequantity ๐๐ ๐ ๐ is deterministic in contrast to โ and ๐ , whichare random variables. Thus, the SNRโs PDF, ๐ ๐พ ( ๐พ ) , can becalculated from the SNRs, ๐พ โ and ๐พ ๐ , and can be evaluatedas [15, Eq. (5)] ๐ ๐พ ( ๐พ ) = โซ โ ๐ ๐พ โ ( ๐ก ) ๐ ๐พ ๐ (cid:16) ๐พ๐ก (cid:17) ๐ก ๐๐ก, (5)where ๐ ๐พ โ (ยท) and ๐ ๐พ ๐ (ยท) are respectively the PDFs of the S-RISand RIS-D sub-channelโs SNRs, ๐พ โ and ๐พ ๐ . With the assump-tion of a constant weather condition over the environment, bothparts of the channel can be modeled by a combined distributionincluding pointing errors and turbulence levels. The resultinguni๏ฌed PDF, ๐ ๐พ ๐ ( ๐พ ๐ ) is expressed as [1, Eq. (7)] ๐ ๐พ ๐ ( ๐พ ๐ ) = ๐๐พ ๐ G , , " ๐ (cid:18) ๐พ ๐ ๐พ ๐ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + ๐ , ๐ผ, ๐ฝ , (6)where ๐ โ { โ , ๐ }, ๐ = ๐ ๐ ฮ ( ๐ผ ) ฮ ( ๐ฝ ) , ๐ = ๐ ๐ผ๐ฝ ( + ๐ ) , ๐ โ {1, 2}indicates whether the transmission exploits the HD ( ๐ = 1) orIM/DD ( ๐ = 2) techniques [1], and G ๐,๐๐,๐ h ๐ง (cid:12)(cid:12)(cid:12) ๐ ๐ ๐ ๐ i is the Meijer-G function. We sequentially substitute ๐พ ๐ by ๐ก and ๐พ๐ก in Eq.(6), and obtain ๐ ๐พ โ ( ๐ก ) and ๐ ๐พ ๐ ! ๐พ๐ก (cid:1) respectively as ๐ ๐พ โ ( ๐ก ) = ๐๐ก G , , " ๐ (cid:18) ๐ก๐พ โ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + ๐ , ๐ผ, ๐ฝ , (7)and ๐ ๐พ ๐ (cid:16) ๐พ๐ก (cid:17) = ๐๐ก๐พ G , , " ๐ (cid:18) ๐พ๐พ ๐ ๐ก (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + ๐ , ๐ผ, ๐ฝ , (8)where ๐พ โ and ๐พ ๐ are average values of the SNRs ๐พ โ and ๐พ ๐ , respectively. In Eq. (8), the variable ๐ก appears at thedenominator. To obtain a Meijer-G function with a numerator-based variable ๐ก , we apply the re๏ฌection property of the Meijer-G function, given by [16] G ๐,๐๐,๐ (cid:20) ๐ง (cid:12)(cid:12)(cid:12)(cid:12) ๐ด ๐ ๐ต ๐ (cid:21) = G ๐,๐๐, ๐ (cid:20) ๐ง โ (cid:12)(cid:12)(cid:12)(cid:12) โ ๐ต ๐ โ ๐ด ๐ (cid:21) , (9)to Eq. (8) and obtain ๐ ๐พ ๐ (cid:16) ๐พ๐ก (cid:17) = ๐๐ก๐พ G , , ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ (cid:18) ๐พ ๐ ๐พ (cid:19) ๐ ๐ก ๐ (cid:12)(cid:12)(cid:12)(cid:12) โ ๐ , โ ๐ผ, โ ๐ฝ โ ๐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป . (10) To get the end-to-end SNRโs PDF, ๐ ๐พ ( ๐พ ) , we substitute Eqs.(7) and (10) into Eq. (5), which leads to ๐ ๐พ ( ๐พ ) = ๐ ๐พ โซ โ ๐ก G , , " ๐ (cid:18) ๐ก๐พ โ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + ๐ , ๐ผ, ๐ฝ ร G , , ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ (cid:18) ๐พ ๐ ๐พ (cid:19) ๐ ๐ก ๐ (cid:12)(cid:12)(cid:12)(cid:12) โ ๐ , โ ๐ผ, โ ๐ฝ โ ๐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ๐๐พ. (11)Applying the change of variable ๐ = ๐ก ๐ โ ๐ก = ๐ ๐ , and ๐๐ก = ๐๐ ๐ โ ๐๐ , we obtain ๐ ๐พ ( ๐พ ) = ๐๐ ๐พ โซ โ ๐ G , , ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ ๐๐พ ๐ โ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + ๐ , ๐ผ, ๐ฝ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ร G , , ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ (cid:18) ๐พ ๐ ๐พ (cid:19) ๐ ๐ (cid:12)(cid:12)(cid:12)(cid:12) โ ๐ , โ ๐ผ, โ ๐ฝ โ ๐ ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ๐๐. (12)With the help of [17, Eq. (07.34.21.0011.01)], we solve theintegral in Eq. (12), apply the identity in Eq. (9), and obtainthe exact uni๏ฌed PDF of end-to-end SNR, ๐ ๐พ ( ๐พ ) , as ๐ ๐พ ( ๐พ ) = ๐๐ ๐พ G , , " ๐ (cid:18) ๐พ๐พ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + , ๐ + ๐ , ๐ผ, ๐ฝ, ๐ , ๐ผ, ๐ฝ , (13)where ๐พ = ๐พ ๐ ๐พ โ . C. CDF of the End-to-End SNR
The CDF of the end-to-end SNR, ๐น ๐พ ( ๐พ ) , can be calculatedas ๐น ๐พ ( ๐พ ) = โซ โ ๐ ๐พ ( ๐พ ) ๐๐พ . Substituting the expression of ๐ ๐พ ( ๐พ ) (Eq. (13)), and permuting the variables ๐พ , ๐ฅ , and โ , we obtain ๐น ๐พ ( ๐พ ) = ๐๐ โซ ๐พ ๐ฅ G , , " ๐ (cid:18) ๐ฅ๐พ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + , ๐ + ๐ , ๐ผ, ๐ฝ, ๐ , ๐ผ, ๐ฝ ๐๐ฅ. (14)With the help of [17, Eq. (07.34.21.0084.01)], we solve theintegral in Eq. (14), to get the closed-form expression of ๐น ๐พ ( ๐พ ) as ๐น ๐พ ( ๐พ ) = ๐ G ๐, ๐ + , ๐ + (cid:20) ๐ (cid:18) ๐พ๐พ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) , ฮ ฮ , (cid:21) , (15)where ๐ = ๐ ๐ ( ๐ผ + ๐ฝ โ ) ( ๐ ) ( ๐ โ ) , ๐ = ๐ ๐ ๐ ๐ , ฮ = ๐ + ๐ , . . . , ๐ + ๐๐ , ๐ + ๐ , . . . , ๐ + ๐๐ with ๐ terms, and ฮ = ๐ ๐ , . . . , ๐ + ๐ โ ๐ , ๐ผ๐ , . . . , ๐ผ + ๐ โ ๐ , ๐ฝ๐ , . . . , ๐ฝ + ๐ โ ๐ , ๐ ๐ , . . . , ๐ + ๐ โ ๐ , ๐ผ๐ , . . . , ๐ผ + ๐ โ ๐ , ๐ฝ๐ , . . . , ๐ฝ + ๐ โ ๐ with ๐ terms. D. Moment Generating Function (MGF)
The MGF, ฮฉ ๐พ ( ๐ ) , is readily calculated from the CDF as [1,Eq. (15)] ฮฉ ๐พ ( ๐ ) = ๐ โซ โ exp (โ ๐พ๐ ) ๐น ๐พ ( ๐พ ) ๐๐พ. (16)Substituting Eq. (15) into Eq. (16), we obtain ฮฉ ๐พ ( ๐ ) = ๐ ๐ โซ โ ๐ โ ๐พ๐ G ๐, ๐ + , ๐ + (cid:20) ๐ (cid:18) ๐พ๐พ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) , ฮ ฮ , (cid:21) ๐๐พ. (17)Using [18, Eq. (7.813.1)], we solve the integral in Eq. (17) andobtain the closed-form and uni๏ฌed expression of the MGF as ฮฉ ๐พ ( ๐ ) = ๐ G ๐, ๐ + , ๐ + (cid:20) ๐ ๐พ๐ (cid:12)(cid:12)(cid:12)(cid:12) , , ฮ ฮ , (cid:21) . (18) IV. A
PPLICATIONS
In this section, we analyze the performance of the proposedRIS-aided nT-FSO system based on the OP, ๐ ๐๐ข๐ก , ergodicchannel capacity, ๐ถ , and average BER, ๐ ๐ , for selected binaryschemes. A. Outage Probability
Outage occurs when the end-to-end SNR, ๐พ , falls below athreshold value, ๐พ ๐กโ = ๐ ๐ โ , prede๏ฌned for a speci๏ฌc quality-of-service, ๐ being the transmission rate. This implies thatunder such conditions, the system does not reach the speci๏ฌcrate ๐ . The OP, ๐ ๐๐ข๐ก , which de๏ฌnes this failure, can be readilycalculated from Eq. (15) by ๏ฌnding ๐น ๐พ ( ๐พ ๐กโ ) . B. Ergodic Channel Capacity
In the proposed system, the channel state information isnot available at the transmitter and data is transmitted withoutinstantaneous feedback, which reduce system capacity [1]. Thetransmitted symbol is long enough so that data is encodedover all the possible channel fading states, and the atmosphericturbulence, which is slow-fading in nT-FSO, remains constantover the symbol transmission, combined with the effects ofthe pointing errors that make the signal ๏ฌuctuate at a veryhigh rate [1]. Thus, the overall channel statistical propertiescan be evaluated during the transmission of a single symbol.Therefore, the ergodic channel analysis can be performed [1],[19]. The ergodic channel capacity, ๐ถ , is given by ๐ถ = ( ) โซ โ ln ( + ๐๐พ ) ๐ ๐พ ( ๐พ ) ๐๐พ, (19)where ๐ = for HD and ๐ = ๐ ๐ for IM/DD [1]. Exploitingthe Meijerโs G-function representation of ln ( + ๐ฅ ) [17, Eq.(07.34.03.0456.01)] and substituting Eq. (13) in Eq. (19), ๐ถ becomes ๐ถ = ๐๐ ln ( ) โซ โ ๐พ G , , (cid:20) ๐๐พ (cid:12)(cid:12)(cid:12)(cid:12) , , (cid:21) ร G , , " ๐ (cid:18) ๐พ๐พ (cid:19) ๐ (cid:12)(cid:12)(cid:12)(cid:12) ๐ + , ๐ + ๐ , ๐ผ, ๐ฝ, ๐ , ๐ผ, ๐ฝ ๐๐พ. (20)With the help of [17, Eq. (07.34.21.0013.01)], we evaluate theintegral in Eq. (20) and obtain a closed-form uni๏ฌed expressionof ๐ถ as ๐ถ = ๐ ln ( ) G ๐ + , ๐ + , ๐ + (cid:20) ๐ ๐๐พ (cid:12)(cid:12)(cid:12)(cid:12) , , ฮ ฮ , , (cid:21) . (21) C. Average Bit-Error-Rate (BER) for Selected Binary Schemes
In data transmission, the BER is a classical metric usedto evaluate the system performance. Considering that in theproposed system, binary schemes are used to modulate the databefore transmission, the average BER, ๐ ๐ , can be evaluatedusing [20, Eq. (13)] ๐ ๐ = ๐ ๐ ฮ ( ๐ ) โซ โ ๐ โ ๐๐พ ๐พ ๐ โ ๐น ๐พ ( ๐พ ) ๐๐พ, (22)where the pair ( ๐ , ๐ ) de๏ฌnes the binary modulation schemes[1]. The values of ๐ and ๐ for selected modulation schemes, TABLE I: Values of ๐ and ๐ for Selected Binary ModulationSchemes. Scheme ๐ ๐
CBFSK 0.5 0.5NBFSK 1 0.5CBPSK 0.5 1DBPSK 1 1 namely, CBFSK, NBFSK, CBPSK, and DBPSK, are providedin Table I. Substituting Eq. (15) into Eq. (22) leads to ๐ ๐ = ๐ ๐ ๐ ฮ ( ๐ ) โซ โ ๐ โ ๐๐พ ๐พ โ ๐ G ๐, ๐ + , ๐ + ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ (cid:18) ๐พ๐พ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) , ฮ ฮ , ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป ๐๐พ. (23)Using [18, Eq. (7.813.1)], the integral in Eq. (23) can be solvedto obtain a closed-form uni๏ฌed expression of the average BER, ๐ ๐ , as ๐ ๐ = ๐ ฮ ( ๐ ) G ๐, ๐ + , ๐ + ๏ฃฎ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฏ๏ฃฐ ๐ ๐๐พ (cid:12)(cid:12)(cid:12)(cid:12) โ ๐, , ฮ ฮ , ๏ฃน๏ฃบ๏ฃบ๏ฃบ๏ฃบ๏ฃป . (24) D. Diversity Order and Coding Gain
The diversity order de๏ฌnes the increase in SNR due to somediversity schemes. Practically, it determines the slope of the ๐ ๐ = ๐ ( ๐พ ) curve. On the other hand, the coding gain is theshift between SNR levels for coded and un-coded systems,required to reach the same ๐ ๐ . At high SNR, the average BER, ๐ ๐ , can be approximated as ๐ ๐ = ( ๐บ ๐ ๐พ ) โ ๐บ๐ [21]. We usethe Meijer-G function expansion [17, Eq. (07.34.06.0017.01)],associated with the unity of lim ๐ฅ โโ ๐ ๐น ๐ [ ๐ ; ๐ ; ๐ฅ ] [1], [22], to๏ฌnd the uni๏ฌed asymptotic expression of ๐ ๐ , as ๐ ๐ โ ๐ ฮ ( ๐ ) ๐ ร ๐ = ๐ ( ๐, ๐, ๐ ) (cid:20) ๐๐พ๐ (cid:21) โ( ฮ ,๐ ) , (25)where ๐ ( ๐, ๐, ๐ ) is expressed as ๐ ( ๐, ๐, ๐ ) = ฮ ! ฮ ,๐ + ๐ (cid:1) ร ๐๐ = ๐ โ ๐ ฮ ! ฮ , ๐ โ ฮ ,๐ (cid:1) ฮ ,๐ ร ๐ + ๐ = ฮ ! ฮ ,๐ โ ฮ ,๐ (cid:1) , (26)where ฮ ,๐ = ฮ , , ฮ , , . . . , ฮ , ๐ + with ๐ + terms, ฮ , ๐ = ฮ , , ฮ , , . . . , ฮ , ๐ with ๐ terms, and ฮ ,๐ = ฮ , , ฮ , , . . . , ฮ , ๐ with ๐ terms. By comparing Eq.(25) to ๐ ๐ = ( ๐บ ๐ ๐พ ) โ ๐บ๐ , we obtain the uni๏ฌed expres-sions of the diversity order and coding gain as ๐บ ๐ = min (cid:16) ๐ ๐ , . . . , ๐ + ๐ โ ๐ , ๐ผ๐ , . . . , ๐ผ + ๐ โ ๐ , ๐ฝ๐ , . . . , ๐ฝ + ๐ โ ๐ (cid:17) and ๐บ ๐ = ๐๐ " ๐ ร ๐ = ๐ ( ๐, ๐, ๐ ) ๐ ฮ ( ๐ ) โ ฮ ,๐ , respectively. Note that ๐บ ๐ will be found between ๐ , ๐ผ , and ๐ฝ for the HD technique, and between ๐ , ๐ + , ๐ผ๐ , ๐ผ + , ๐ฝ , ๐ฝ + for the IM/DD technique. TABLE II: Values of ๐ผ and ๐ฝ used in the analysis. ๐ฟ = 1 kmand ๐ท = 1 mm. Color Red Blue Green ๐
700 470 530 ๐ถ ๐ = ร โ ๐ผ ๐ฝ ๐ถ ๐ = ร โ ๐ผ ๐ฝ ๐ถ ๐ = ร โ ๐ผ ๐ฝ V. N
UMERICAL R ESULTS
We consider a nT-FSO transmission environment in whichS and D are situated at the same distance, ๐ฟ = 1 km, fromthe RIS (see Fig. 1). Three transmitters, each equipped witha different colored light source, are available to be used.The corresponding wavelengths are ๐ ๐ = 700 nm, ๐ ๐ = 530nm, and ๐ ๐ = 470 nm, for the red, green, and blue light,respectively. The two systemโs sub-channels are characterizedby the same refractive structure and index, ๐ถ ๐ , which remainsconstant during the transmission of one symbol. The value of ๐ถ ๐ de๏ฌnes how moderate or strong the atmospheric turbulenceis. For such scenarios, its values ranges from โ to โ ,where โ represents the strongest turbulence levels [23]. Weconsider ๐ถ ๐ = 1.2 ร โ m โ / , 2.2 ร โ m โ / , and 3 ร โ m โ / . Using Eqs. (3), (4), and the expression of theRytov variance provided in Section II-A, we obtain the valuesof ๐ผ and ๐ฝ , which are given in Table II. These values arecalculated considering that the RIS structure is made in sucha way that its elements and the receiver aperture have the samediameter, ๐ท = 1 mm. The ยซsymengineยป function in Matlab isused to generate the Meijer-G function.First, we analyze the OP, ๐ ๐๐ข๐ก , the average channel capacity, ๐ถ , and the average BER, ๐ ๐ , for a single color (the blue), thenwe compare the system performance for red, green, and bluecolors. Figures 2, 3, and 4 depict the OP against normalizedaverage threshold SNR, ๐พ ๐กโ , and end-to-end electrical SNR, ๐พ ,respectively, while Figs. 5 and 6 present the ergodic channelcapacity. Figures 7 and 8 give BER results for CBFSK,NBFSK, CBPSK, and DBPSK, and ๏ฌnally, Fig. 9 comparesBER and average channel capacity results for the red, green,and blue colors for selected values of ๐ผ and ๐ฝ . The analysis iscarried out for ๐ = 1.1 and ๐ = 6.1, representing the pointingerror level, where 1.1 is the worst case.Figures 2, 3, and 4 respectively depict the OP acrossthe normalized threshold SNR, ๐พ ๐กโ , the normalized averageelectrical SNRs, ๐พ ๐ and ๐พ โ , for both detection techniques.Figure 2 con๏ฌrms that increasing ๐พ ๐กโ worsens the systemโsprobability of failure for all considered turbulence levels andpointing errors. As an example, for ๐ = 6.1, ๐ผ = 10.9537, and ๐ฝ = 2.9833, adopting the HD technique, a ๐ ๐๐ข๐ก target of 10 โ . can be obtained for ๐พ ๐กโ = 0 dB, and a ๐ ๐๐ข๐ก target of 10 โ ฮณ th , dB -4 -3 -2 -1 O u t ag e P r o b a b ili t y , P o u t ฮณ g ฮณ h = 9 dBHD, a = 1IM/DD, a = 2 ฮถ = 1.1 ฮถ = 1.1 ฮถ = 6.1 ฮถ = 6.1 ฮถ = 1.1 ฮถ = 1.1 ฮถ = 6.1 ฮถ = 6.1 ฮฑ = 10.9537, 2.9428 ฮฒ = 2.9833, 2.5605 Fig. 2: OP results showing the performance of both HD andIM/DD techniques under different turbulence conditions interms of normalized average threshold SNR, ๐พ ๐กโ , for harsh( ๐ = 1.1) and moderate ( ๐ = 6.1) pointing error levels. ๐พ ๐ = ๐พ โ = 9 dB. ฮณ , dB -5 -4 -3 -2 -1 O u t ag e P r o b a b ili t y , P o u t ฮถ = 6.1Heterody detection, a = 1 ฮณ th = 9 dB ฮถ = 1.1 ฮฑ = 10.9537, 2.9428, 4.9477 ฮฒ = 2.9833, 2.5605, 1.2310 Fig. 3: OP results showing the performance of the HD tech-nique under different turbulence conditions in terms of theaverage electrical SNR, ๐พ , for harsh ( ๐ = 1.1) and moderate( ๐ = 6.1) pointing error levels. ๐พ ๐กโ = 9 dB. ฮณ , dB -5 -4 -3 -2 -1 O u t ag e P r o b a b ili t y , P o u t ฮถ = 6.1 ฮถ = 1.1 ฮณ th = 9 dB IM/DD, a = 2 ฮฑ = 10.9537, 2.9428, 4.9477 ฮฒ = 2.9833, 2.5605, 1.2310 Fig. 4: OP results showing the performance of the IM/DDtechnique under different turbulence conditions in terms of theaverage electrical SNR, ๐พ , for harsh ( ๐ = 1.1) and moderate( ๐ = 6.1) pointing error levels. ๐พ ๐กโ = 9 dB. ฮณ , dB E r go d i c C a p a c i t y , C , b i t s / s ec / H ez ฮถ = 1.1 ฮถ = 6.1 ฮฑ = 10.9537, 2.9428, 4.9477 ฮฒ = 2.9833, 2.5605, 1.2310Heterodyne detection, a = 1 Fig. 5: Ergodic channel capacity results for the HD techniqueunder different turbulence conditions in terms of the averageelectrical SNR, ๐พ , for harsh ( ๐ = 1.1) and moderate ( ๐ = 6.1)pointing error levels. ฮณ , dB E r go d i c C a p a c i t y , C , b i t s / s ec / H ez ฮถ = 1.1 ฮถ = 6.1IM/DD, a = 2 ฮฑ = 10.9537, 2.9428, 4.9477 ฮฒ = 2.9833, 2.5605, 1.2310 Fig. 6: Ergodic channel capacity results for the IM/DD tech-nique under different turbulence conditions in terms of theaverage electrical SNR, ๐พ , for harsh ( ๐ = 1.1) and moderate( ๐ = 6.1) pointing error levels.requires that ๐พ ๐กโ is set to 5.5 dB. The results in Fig. 3 showthat using the HD technique for a ๏ฌxed ๐พ ๐กโ = 9 dB, a target ๐ ๐๐ข๐ก = โ is obtained at 26 dB, 33 dB, and 44 dB of ๐พ ,respectively for atmospheric conditions characterized by ( ๐ผ , ๐ฝ )= (10.9537, 2.9833), (4.9477, 1.2310), and (2.9428, 2.5605),and a pointing error of ๐ = 6.1; for the same ๐ ๐๐ข๐ก = โ target, the system at ๐ = 6.1 outperforms its counterpart at ๐ = 1.1 of 16 dB, 13 dB, and 9 dB, respectively. This is also seenin Fig. 4, which depicts the performance of the system underthe same conditions as previously, for the IM/DD technique,for the values of pointing errors ๐ = 1.1 and 6.1. In this case,the curves lead to the conclusion that systems employing theHD technique outperform those using the IM/DD technique ofat least 12 dB in all turbulence levels. For example, for ๐ ๐๐ข๐ก = โ , ๐พ = 35 dB (23 dB for the HD technique) is requiredif ๐ = 6.1, ๐ผ = 10.9537, and ๐ฝ = 2.9833.Figures 5 and 6 depict the systemโs ergodic channel capacityfor multiple atmospheric conditions considering and pointingerrors, ๐ = 1.1 and 6.1. The curves in Fig. 5 represent results ฮณ g ฮณ h , dB -6 -5 -4 -3 -2 -1 A v e r ag e B i t E rr o r R a t e ( B E R ) , P b CBFSKNBFSKCBPSKDBPSKCBFSKNBFSKCBPSKDBPSKAsymptotes
Heterodyne detection, a = 1 ฮฑ = 10.9537, ฮฒ = 2.9833 ฮถ = 6.1 ฮถ = 1.1 Fig. 7: Average BER of CBFSK, NBFSK, CBPSK, andDBPSK binary modulation schemes showing the HD tech-niqueโs performance under different turbulence conditions forextreme ( ๐ = 1.1) and moderate ( ๐ = 6.1) pointing error levels. ฮณ g ฮณ h , dB -6 -5 -4 -3 -2 -1 A v e r ag e B i t E rr o r R a t e ( B E R ) , P b CBFSKNBFSKCBPSKDBPSKCBFSKNBFSKCBPSKDBPSKAsymptotes ฮฑ = 10.9537, ฮฒ = 2.9833 ฮถ = 1.1IM/DD, a = 2 ฮถ = 6.1 Fig. 8: Average BER of CBFSK, NBFSK, CBPSK, andDBPSK binary modulation schemes showing the IM/DD tech-niqueโs performance under different turbulence conditions forextreme ( ๐ = 1.1) and moderate ( ๐ = 6.1) pointing error levels.for the HD technique while the performance of the IM/DDtechnique is given in Fig. 6. As in the case of ๐ ๐๐ข๐ก , Figs. 5 and6 con๏ฌrm that an RIS-based nT-FSO system utilizing the HDtechnique outperforms the IM/DD based counterpart. They arealso in agreement with the results of OP shown in Figs. 2, 3,and 4. The system offers the best ergodic channel capacity forturbulence parameters at which the OP is lower. For example,with a target ๐ ๐๐ข๐ก of 10 โ , the best result is obtained at ๐พ = 35dB for a pointing error characterized by ๐ = 6.1 and turbulencebehavior with parameters ๐ผ = 10.9537, and ๐ฝ = 2.9833. Forthe same turbulence level and pointing errors at ๐พ = 35 dB, thebest ergodic channel capacity for the HD detection, which isabout 10.2 bits/sec/Hez, is obtained for ๐ = 6.1, ๐ผ = 10.9537,and ๐ฝ = 2.9833.Figures 7 and 8 depict the BER performance of the FSOsystem analyzed in this paper. They highlight the impact ofpointing errors on the RIS-aided nT-FSO data transmissionsystems. In both ๏ฌgures, there is clearly a difference betweenthe BER performance for ๐ = 6.1 and ๐ = 1.1. This is valid for all studied schemes. A CBPSK under ๐ = 6.1 outperformsthe CBPSK under ๐ = 1.1. It is also clear that DBPSK slightlyoutperforms CBFSK, and NBFSK is the least attractive from aperformance point of view. Figures 7 and 8 also illustrate theasymptotes of BER curves for selected schemes to highlightthe diversity gains given by their negative slopes.Finally, Fig. 9 compares the BER (Fig. 9a) and ergodicchannel capacity (Fig. 9b) of the system assuming DBPSKfor red, green, and blue colors. The values of ๐ผ and ๐ฝ ,obtained from the wavelengths, are ( ๐ผ , ๐ฝ ) = (10.9537, 2.9833),(12.5331, 4.6787), and (13.2818, 5.7795), respectively. As the๏ฌgures con๏ฌrm, the blue light offers better performance asits corresponding BER curve is below those of green and redlights. For example, for a BER of 10 โ , ๐พ = 35 dB is requiredwhile transmitting using a blue light. This value increases to38 dB and 39 dB for green and red colors, respectively.VI. C ONCLUSION
This paper has presented uni๏ฌed and closed-form expres-sions for the PDF, CDF, and MGF of a RIS-based nT-FSOlink operating over G-G turbulence and pointing errors. Byexploiting these expressions, the system performance has beenevaluated through metrics such as the OP, ergodic channelcapacity, and average BER, for selected binary schemes interms of the Meijer-G function. The uni๏ฌed diversity orderand coding gain for the proposed RIS-based nT-FSO sys-tem have been also derived. It has been shown, throughnumerical results, that RIS-assisted nT-FSO systems using theHD technique outperform those using the IM/DD technique.The in๏ฌuence of turbulence level and pointing errors on thetransmitted signal has been highlighted. Finally, it has beenshown that the transmission using the blue color outperformsthose with red and green colors over a RIS-aided nT-FSOchannel. In our future work, we will investigate the scenariowhere the line-of-sight path is non negligible and also the caseof multiple RIS links. R
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