Journal of ommuniations Vol., o. 7, July 7 Performane Analysis of FSO ommuniation Systems with Higher-Order Spatial Diversity Shemes Using BPSK- S over Log-ormal Atmospheri urbulene hannels Okikiade A. Layioye, homas J. O. Afullo, and Pius A. Owolawi University of KwaZulu-atal, Durban, 4, South Afria shwane University of ehnology, Pretoria, South Afria Email: okiki.layioye@yahoo.om; afullot@ukzn.a.za; p.owolawi@gmail.om Abstrat Free Spae Optial (FSO) ommuniation system is an optial wireless onnetivity operating at an unliensed optial spetrum, with lots of advantages over the onventional adio Frequeny (F) transmission. However, the performane of the FSO hannel is limited due to atmospheri turbulene and severe weather onditions. here are many authors who have worked on mitigating these effets by using Spatial Diversity (SD) with X-O systems and few have worked up to 4X4-O systems, blended with BPSK-Sub-arrier ntensity odulation (BPSK-S) under the log-normal atmospheri turbulene model. However, in this paper, we extended the SD tehnique to higher-order onfigurations suh as 8X8-O system in order to improve the performane obtained from lower-order FSO systems. his study was onsidered at logirradiane variane values of. and.9, representing the mildly weak and moderately weak atmospheri turbulene regimes respetively. his work has presented the performane analysis of various FSO-SD tehniques. As a result, at BE of -9 the X8, 4X4, 6X6 and 8X8 higher-order SD systems were 46%, 55%, 6% and 69% respetively better than the "nondiversity" FSO system during the moderately weak atmospheri ondition. Also, their hannel apaities were 74%, 77%, 85% and 89% respetively better than the "non-diversity" FSO system at S of 3 db. ndex erms Atmospheri turbulene, BPSK-Sub-arrier intensity modulation, free spae optial ommuniation systems, higher-order spatial diversity tehnique, log-normal atmospheri turbulene model. ODUO Free Spae Optial (FSO) ommuniation System is an Optial Wireless ommuniation (OW) system that an be desribed as a robust tehnology that provides users with a superior mobility, flexibility as well as high data rate. he FSO ommuniation system onveys information by transmitting laser beams through the atmosphere to a photodetetor. n the reent years, it has attrated appliations in both indoor and outdoor wireless ommuniations. t has proven to be a good and viable substitute to other traditional wireless ommuniation systems due to its large bandwidth, easy deployment and anusript reeived ay 5, 7; revised July 3, 7. orresponding author email: okiki.layioye@yahoo.om. doi:.7/jm..7.379-394 ommissioning, and ost effetiveness. he sophistiated outdoor wireless apabilities of the FSO system, with lots of other optimal servies mentioned above are part of its advantages over other types of wireless ommuniation systems. Based on its numerous and broad appliations, the FSO ommuniation system has attrated great attention and has beome a widely used OW system for the improvement of the Average hannel apaity (A) of ommuniation systems []-[5]. he motivation for alternative tehnology that meets today s wireless aess platform is as a result of the lastmile bottlenek. his is oupled with ontinually inreasing demand plaed on the need for higher bandwidth and transmission rate, along with ertain adio Frequeny (F) limitations due to frequeny spetrum ongestion [], [6], [7]. he FSO system has proved to be a ommuniation system with alloation for high speed, broad (wide) bandwidth, interferene-free, highly diretional, reasonable seurity, liense-free, robust ommuniation servies with less time of deployment and also importantly it has a low maintenane ost. Another advantage of the FSO is its ability to equal the speed of the Fiber optis, whih is as high as.5 Gbps and as a matter of fat, it has the ability to surpass this speed to reah Gbps. he reason being that it transmits data faster through air than in the glass fibers used in fiber optis [], [8], [9]. However, the effets of the atmospheri onditions greatly determine the reliability and effiieny of the FSO ommuniation system by ausing a flutuation in the irradiane of the optial signal whih is known as atmospheri sintillation []-[]. he main disadvantages of the FSO ommuniation system arise from the ourrenes assoiated with atmospheri onditions resulting into atmospheri turbulene or adverse weather onditions. he atmospheri turbulene effets are aused by the differenes deteted in the refrative index as a result of the in-homogeneities in temperature and also by flutuations obtained in air pressure along the transmission pathway of the laser beam [4], [], [3]. he effets of the index inhomogeneities obviously have a great deteriorating impat on the quality of the reeived signal and at the same time it an ause variations in both the reeived 7 Journal of ommuniations 379
Journal of ommuniations Vol., o. 7, July 7 signals intensity and phase. As a matter of fat, the long run effets of these variations an result into an inreasing link error probability, thereby ausing a limitation to the performane of the FSO ommuniation system. hese atmospheri turbulene onditions an be ategorized into weak, intermediate and strong atmospheri turbulene regimes depending on the level of the atmospheri turbulene strength [4], [5], [], [4]. he eeived Optial Signal Level (OSL) at the reeiver an be adversely affeted as a result of atmospheri disorders suh as the ourrene of Fog and haze whih leads to beam sattering, thereby reduing the OSL. Also, the higher the magnitude of the fog, the higher the attenuation that it produes. Attenuation exeeding 3 db/km is aused by heavy fog, whih in turn limits the length of the FSO link to a range that is less than m [6], [5], [6]. he effets of some atmospheri onditions suh as rain and snow are not quite harmful to FSO ommuniation systems, but it has been observed that they mainly ause serious attenuation to the radio and mirowave frequenies [6], [6], [7]. Overoming the orruption effets on optial signals aused by fading that has been indued by turbulene an be ahieved by several diversity tehniques. he employment of different methods under the diversity tehniques an be used to improve the hannel performane of the FSO ommuniation links. he onventional diversity tehniques used in FSO ommuniation system are spae diversity, time diversity, wavelength diversity, frequeny diversity and temporal diversity [8]-[]. As sited by several authors, spatial diversity tehnique an be onsidered as a promising mitigating tehnique for FSO ommuniation to produe a high data rate. his led to an investigation into the spatial orrelation amongst a pair of onfigured transmitters in a ultiple nput ultiple Output (O) struture as presented in []. However, the atmospheri effets are ontrollable and an be greatly minimized to a reasonable level through the use of the O tehnique. his inludes the appliation of multiple lasers at the transmitter end as well as multiple photodetetors at the reeiver end [8], [], [3]. Another work disussed the performane investigation of multi-beam Free Spae Optial system using diversity tehniques [4]. he hoie of the appropriate modulation sheme to ahieve an optimum FSO performane is a key fator in mitigating against the indued fading whih is as a result of atmospheri turbulene. he modulation tehniques that are the simplest as well as most extensively employed for FSO systems are the On-Off Keying (OOK) sheme and the Pulse Position odulation (PP) [6], [9], [3]. hough the OOK sheme has shown that it doesn t deliver any form of protetion or resistane to indued fading aused by the atmospheri turbulene [4], [6], [4], [5]. he level of the optial intensity obtained at the position of the reeiver sometimes experienes a random instability whih is aused by the non-preditive behavior of the atmospheri turbulene level. herefore, all these suggest that for the OOK tehnique to produe an optimum performane when it is applied, it will definitely need an adaptive thresholding sheme. However, due to the formation of this sheme, the implementation tends to be omplex and pratially not suitable [6], [7], [6]. eanwhile, the PP modulation tehnique on the other hand seems to have a poor bandwidth effiieny, despite its low ost and simpliity [3]. Applying the modulation shemes that onvey the required information in either the frequeny or the phase of the arrier signal has beome a reasonable and better approah. his is beause the level of the optial intensity in FSO ommuniation system is drastially affeted by fog and sintillation [6]. herefore, following the trend of urrent researhes, a proposition has been made that the limitations obtained from the employment of the OOK or PP an be overome by the employment of Sub-arrier (S) ntensity odulation (S) shemes. he various types of Sub-arrier ntensity odulation shemes that an be applied are: Sub-arrier Binary Phase Shift Keying (S- BPSK) and Sub-arrier Quadrature Amplitude odulation (S-QA). However, this paper employed the former. As a matter of fat, the Binary Phase Shift Keying (BPSK) based S does not need any adaptive thresholding sheme and it is moderately bandwidth effiient. his makes it a better option when ompared to the operations of the OOK and PP in the presene of fading hannels that have been indued by the atmospheri turbulene [4], [], [7], [8]. hus, a omprehensive investigation of the FSO ommuniation system performane using S-PSK has been shown in [7], [5], [8], [9], but without onsidering the spatial diversity tehnique. Wilson et al. in [3]-[3], onsidered the FSO O ommuniation system using the Pulse Position odulation (PP) and the Q-ary PP under two turbulent indued fading hannels whih are: the lognormal and ayleigh fading hannels, and then made expressions and analysis on the Bit Error Probability (BE) and the Symbol Error Probability (SE). Apart from the usual study of Average Bit Error ate (ABE) and Average Symbol Error ate (ASE) performane for O-FSO ommuniation systems, the average apaity performane of these systems are now reently studied in [3], [33]. he Average hannel apaity (A) is also a standard metri system for determining the maximum possible data rate (in b/s/hz) onsistently transmitted between the transmit lasers and the reeive photodetetors in the FSO ommuniation regime [34]. n addition to the BE and the A performane analysis studied in this paper for higherorder spatial diversity FSO systems, the Outage hannel apaities (O) of these FSO systems were also studied under the weak turbulene regime. hus, from the foregoing, the performane on the A of FSO-O ommuniation systems under different atmospheri turbulene indued fading hannels suh as the weak, intermediate and strong atmospheri turbulene fading hannels has not been fully investigated. his 7 Journal of ommuniations 38
Journal of ommuniations Vol., o. 7, July 7 work investigates the BPSK-S over O and other Spatial Diversity (SD) tehniques (suh as Single nput ultiple Output (SO) and ultiple nput Single Output (SO)); whereby the Subarrier ntensity odulation is employed to improve the apaity of the FSO system by making sure that the multiple soures are being modulated at different sub-arriers. hough, to obtain a preferred BE performane, the penalty that is paid is the higher S. herefore, in situations whereby we plae great priority on the inrease in apaity of the system more than the power requirement, then it will be proper to hoose the multiple S tehnique. he arrangement of this paper is given as follows: Setion explains expliitly the system and hannel models that were used in this work. he performane analysis showing the BE, A and O of the various higher-order spatial diversity shemes are presented in Setion 3. umerial results as well as the graphial analysis of the higher-order spatial diversity shemes under the mildly weak and moderately weak atmospheri turbulene regimes are shown in Setion 4. Finally, the last setion states the onlusion with remarks of this paper.. A. System odel SYSE AD HAEL ODELS ) FSO-SSO systems using BPSK he whole system model in this ase involves the Free Spae Optial SSO system using BPSK modulation tehnique with a struture that onsists of a single transmitter at the input and a single reeiver at the output of the optial link. his type of FSO system employs BPSK signaling to modulate the intensity of the optial signal produed by the single transmit laser by means of a arrier signal reated by the modulator. A simple blok diagram to desribe the Free Spae Optial SSO system whih uses the BPSK-S modulation tehnique is shown in Fig.. he soure input data (d i ) whih arries the information to be onveyed to a user at a distant destination, beomes modulated by the BPSK modulator onto the reated F subarrier. t is then relayed onto a laser whih is desribed as an optial arrier. he g( in the FSO SSO BPSK-S blok diagram denotes the retangular pulse shaping funtion of the BPSK modulator. n order to bias the optial arrier appropriately, the reated omplex F signal is being mixed with a D bias signal b o. his synhronization ensures that the optial arrier totally aepts the full swing of the time varying F subarrier signal. hus, after synhronization with the D bias signal, the intensity of the optial soure beomes modulated by the F subarrier. However, the modulated optial information produed by the laser is transmitted over the free spae whih is an atmospheri hannel. he atmospheri hannel onsidered in this paper is a weak turbulent regime. he log-normal distribution is used to model the weak turbulene effet whih results into fading of the transferred optial signal intensity. Using a oherent Detetion (D) tehnique at the reeiver, we an onveniently reover the omplex Sub-arrier ntensity odulated signal that has been superimposed on the overing of the inbound optial signal. n order to apture the sub-arriers, the Eletrial Bandpass filters are used. Following this is the standard F oherent detetor whih is also the BPSK demodulator that, thus reovers the transferred data series given as D i. Fig.. he blok diagram of BPSK sub-arrier intensity modulated SSO-FSO link under the log-normal atmospheri hannel. n this SSO ase, a single F sub-arrier that has BPSK sub-arrier amplitude, sub-arrier phase and subarrier frequeny as A, a, and f is being used [6]. t is onfirmed that as the photodetetor (PD) optially detets 7 Journal of ommuniations 38
Journal of ommuniations Vol., o. 7, July 7 and ollets optial signal at the reeiver end, this proess is done in the presene of the following: signal distortion, noise interferene and bakground radiation [8]. he expression for the transmitted optial signal intensity from the single laser driver is expressed as [4]: s( P a e b () where P a represents the average optial signal power per bit and κ denotes the index of modulation whih is given as. However, the Peak transmitted optial signal power (P p ) is related to the average optial signal power per bit as follows: P /. a P p he eletrial BPSK sub-arrier signal [ e b ( ] is given as [6], [5]: e ( A g( os( f t a ) () b where t is the time in seonds and all other parameters remain as earlier desribed. hus, () beomes: Pp s( [ A os(f t a )] (3) After modulation, the transmitted optial signal from the laser is transmitted over the atmospheri turbulene regime where it gets distorted by atmospheri effets, suh as sintillation, fog, rain, snow and so on. hus, in an FSO ommuniation system with the effets of sintillation and the likes, the reeived optial signal intensity r( at the input of the single photodetetor is given as [35]: r( a( P ( (4) n (4), parameter a denotes the atmospheri attenuation fator ontributed by other effets rather than sintillation, ( is the signal sintillation fator due to the effets of the atmospheri turbulene, whih an then be demonstrated as a stationary random proess. P o ( represents the reeived signal intensity without onsidering the effets of sintillation. However, it an be said that s( = P o (. herefore, for BPSK systems in an FSO ommuniation system, the reeived optial signal intensity at the input of the single photodetetor is given as [4]: o r( a( s( (5) Pp r( a( [ A os(f t a )] (6) his reeived optial signal intensity has already been distorted while being transmitted over the atmospheri turbulene hannel. onsequently, sine we are onsidering the weak atmospheri turbulene hannel in this paper, ( an be modelled with the Log-normal distribution as a stationary random proess. he mean amplitude ( a ( ) of the reeived eletrial signal whih is also alled the D term of the signal an be filtered out using a bandpass filter, whih is inorporated into the reeiving end of the FSO system. Hene, at the output of the photodetetor, after it has been filtered and onverted from the optial form, we an obtain the expression for the eletrial signal as follows [4]. Pp re ( a( eb ( ( Pp re ( a( A os(f t a ) ( P p (8) where is the Photodetetor s responsivity; this is a parameter that gives the responsivity of the photodetetor or how reative the photodetetor is. ( represents the total reeiver noise. his noise fator ontributed or aumulated at the reeiver an be modelled as an Additive White Gaussian oise (AWG) proess whih has a power spetral density. Finally, the reeived eletrial signal undergoes BPSK demodulation and then afterwards beomes sampled in order to reover the original information (data). Hene, for the BPSK demodulation of the reeived eletrial signal r e (, the output signal ( that ontains the original data an be finally obtained after the reeived eletrial signal r e ( is demodulated by the referene signal A os( f and given as [4]: ( r ( A os( f (9) e Pp A ( a( ( () herefore, after the BPSK-S oherent detetion demodulation, baseband eletrial signal that will be obtained at the end of the FSO SSO ommuniation link is given as [4], [35]: ( a( PpA ( () 4 ) FSO spatial diversity systems using BPSK he Spatial Diversity tehnique involves employing multiple antennas on either the transmitter or reeiver (SO or SO respetively), or having more than one antenna on both the transmitter and reeiver as in the ase of O. However, as shown in Fig., a universal FSO Diversity system using the BPSK-S modulation tehnique an be onsidered and dedued. Hene, this generi formulation an be used to handle the FSO-SO, FSO-SO and FSO-O systems. (7) 7 Journal of ommuniations 38
Journal of ommuniations Vol., o. 7, July 7 Fig.. he blok diagram of BPSK sub-arrier intensity modulated spatial diversity FSO links under the log-normal atmospheri hannel n this paper, the systems desribed above employed re (t ) [a][ transmitting lasers direted towards a set of aligned reeiving Photodetetors (PD). he transmitting lasers BPSK signaling with a set of m n whih desribes the atmospheri turbulene hannel between the mth transmit laser to the nth photodetetor, and (t ) denotes the Additive White Gaussian oise with zero mean and a noise power direted towards the photodetetors at the reeiving end. However, in order to justify the performane analysis, it is important to make ertain assumptions suh as assuming that eah transmit telesope s light beamwidth is suffiiently broad to make sure that the reeiver array is being ompletely illuminated. Also, it is essential ( n ). he transmitted eletrial signal an be estimated at the reeiving end by employing the means of a aximum atio ombining () Detetor. he was hosen in this paper due to the fat that in terms of optimum performane, it has a ombining gain advantage whih adds benefits to the FSO system. his is as a result of the fat that the required S expeted to reah a ertain BE performane is slightly lower for ompared to other ombining shemes suh as Equal Gain ombining (EG) and Seletion ombining (S). Unlike other ombiners, the performs its operation by estimating the gains of all the hannels of the diversity system. herefore, as a result of this, a finite sum of the individual sub-hannel Signal to oise atios an be obtained. his respetively gives an expression for the nstantaneous Eletrial S (E-S), and it is given as follows [3], [4]: to inlude the assumption that irrespetive of, the same total optial power is being produed by the transmitter s telesope array. his is done in order to impose a reasonable evaluation with the ase of a system having a single transmit telesope. he spatial orrelation has to be insignifiant when onsidering these spatial diversity tehniques. hus, for this to be negligible, it has to be assumed that the distane between the disrete transmit and reeive telesopes is suffiient. he model for the turbulene hannels of the various Diversity systems an be represented by an atmospheri turbulene hannel, having an matrix whih desribes the onfigurations of the Diversity turbulene hannel, and it is given as [3], [4]: ( mn ) (4) m n ( mn mn ) () m n However, at the input of the BPSK demodulator, the eletrial signal is given as [3], [4]: 7 Journal of ommuniations ][ ][ ][eb (t )] mn (t ) (t ) (3) m =,, and n =,,, where mn (t ) represents the stationary random proess individually provide an expliit perfet synhronized data transmission whih employs similar BPSK signals. hese data transmission, with seamless synhronization by eah of the transmitting lasers, provide a stream of data transmitted through an atmospheri turbulene hannel [mn (t )]m,n, Pp 383 (5)
Journal of ommuniations Vol., o. 7, July 7 he fator mn is obtained as andom Variables (Vs) of arbitrary sizes and it desribes the omponent of the nstantaneous Eletrial S. his omponent is aused by the signal that results from the sub-hannel. Eah of the sub-hannels that exist from the mth transmit laser to the nth photodetetor ontributes to the nstantaneous Eletrial S (E-S). hus, is expressed as follows [3], [4]: mn mn ( ap ) (6) mn a mn mn where is a representation of the total noise power of the system. However, from a ombination of the various instantaneous eletrial S omponents obtained from the various sub-hannels in the middle of the mth transmit laser and the nth photodetetor, the average eletrial S ( mn ) an be found and it is ontributed by the various sub-hannels. From (6), mn is obtained as [3], [4]: ( apa ) (7) Hene, for FSO SSO ommuniation system, the average eletrial signal to noise ratio is given as: ( ap ) a (8) (sine m = n =, for SSO systems). B. hannel odel ) Atmospheri turbulene model he irradiane flutuations that arise in the FSO system as a result of ertain atmospheri turbulene onditions within the link regime an be desribed using various statistial models. However, the irradiane flutuations an be desribed in various levels based on the impat of the atmospheri turbulene. his has led to the fat that the atmospheri turbulene ondition an be ategorized as weak, intermediate (moderate) or strong turbulent onditions. he atmospheri turbulene indued fading that desribes perfetly the weak atmospheri turbulene ondition used in this paper is expeted to be a random proess that has the same pattern like the log-normal distribution [3], [7]. On the other hand, the random proess for the atmospheri turbulene indued fading for the intermediate and the strong turbulene onditions follows the Gamma-Gamma distribution [3], []. ) he log-normal turbulene model he andom proess of the turbulene indued fading as a result of the influene of the weak atmospheri turbulene ondition, an be modelled using the lognormal distribution funtion. his Log-normal turbulene model offers an opportunity to desribe the weak atmospheri turbulene indued fading hannel during the propagation of signal from the transmitter to the photodetetor. he Probability Density Funtion (PDF) that mathematially expresses the log-normal fading hannel for an irradiane with log-normal andom Variable (V), when, is given as follows [5], [4], [35]: f mn (ln( mn ) ) ) exp( ) (9) ( ) mn ( mn mn where m =,, and n =,,. he variane parameter of the distribution used in this ase is alled the ytov Variane whih is denoted by the parameter, and it is expressed as follows [6]: (7 / 6) (/ 6).3 n k L () Under the weak sintillation theory, this ytov Variane expression in () is mostly used when we assume a plane wave propagation. However, it an be said that the ytov variane is proportional to the sintillation index. n the expression above showing the ytov variane, the atmospheri turbulene strength or the refrative index parameter is represented using. his parameter depends on the altitude of the optial link, and aording to the atmospheri turbulene ondition, it 7 has a range that varies from 3 to m / 3 [3], [4]. he expression for the normalized Log-irradiane variane also alled the sintillation index ( ) is given as [3], [4]: exp( ) s () where parameters and respetively are given as [3]: (.49 ) () (.8d.56 ) s / 5 7 / 6 (.5 ) (3) (.9d.6 ) / 5 5 / 6 From the above expressions, we have the following parameters: the optial parameter, d, whih is given as: ( kd ) / 4L, and also, the optial wave number, k, is given as: /. Where D represents the reeiver aperture diameter of the Photodetetor, the optial Link distane in meters is represented by L, and the optial wavelength in meters is denoted by. n 7 Journal of ommuniations 384
Journal of ommuniations Vol., o. 7, July 7. PEFOAE AALYSS A. Bit Error ate Analysis for Log-normal hannel odel Using BPSK-S onsidering the same transmitted data symbols, that is, when the system allows the transmission of a data symbol, the Bit Error ate (BE) an be derived as the probability of the output signal after demodulation to be less than zero ( ( ). herefore, it an be said that the theoretial unonditional BE per subarrier hannel is given as [4], [35]: P( ei) ) f ( ) Pe ( d (4) where P( e i )( ) represents the instantaneous probability of error that is the same as the probability of obtaining (, whereby ( is onsidered as a Gaussian random proess that has an instantaneous mean given as ( and a noise variane of. he 4 P p A parameter f () represents the PDF of the random proess whih is as a result of the atmospheri sintillation within the system. hus, the instantaneous error probability is given as [35]: PpA ( ( ) 4 ( ei) exp[ ] P PpA Q( ) 4 d (5) (6) Q (7) where Q (.) denotes the Gaussian Q-funtion and it is given as [35]: Q ( y) exp( t ) dt y (8) and the noise variane at the input of the BPSK modulator is given as [35]: qg FA( Pp) f (4kB ) F f (9) 4 Also,, suh that L represents the oise Power Spetral Density (PSD) and where q G, F,, f, F, k all represent the, A B and L harge of the eletron, average Avalanhe Photo-Detetor (APD) gain, additional noise fator, reeiver noise temperature, effetive noise bandwidth, noise figure of the amplifier, Boltzmann s onstant and load resistane respetively. During the weak atmospheri turbulene event in an FSO ommuniation system, it is already a fat that the log-normal distribution perfetly desribes the primary influene of turbulene ( ( ) whih is identified as a random proess [35]. he PDF ( f ( )) for an FSO system desribed by a log-normal distribution an be expressed using the expression given in (9). he variane parameter in this expression also represents the log-irradiane variane, whih determines the strength of the atmospheri turbulene ondition and it is a value that is dependent on the harateristis of the hannel. his log-normal PDF ontinues to be valid for the Sintillation ndex (S.) whih is given as [35]: S. e s. (3) hus, using Eq. 3, the sintillation indies hosen for this researh work were obtained from the seleted logirradiane variane values. hese sintillation indies values are within the range of values that the log-normal model an handle. herefore, we used these set of values to determine the minimum and maximum effets of weak atmospheri turbulene on the hannels of the FSO ommuniation links. Sine the limit of the log-irradiane variane values that the log-normal model an handle for the weak atmospheri turbulene regime is. [6], thus, in this paper we hose to limit ourselves to log-irradiane variane values from. to.9. he BE expression for the FSO system an be obtained by substituting (9) and (6) into (4), and this is shown below: PpA Pe Q( ) 4 (ln( ) / ) exp[ ] d (3) B. Average (Ergodi) hannel apaity Analysis ) Average Spetral Effiieny (ASE) he Average hannel apaity (A) is a very essential metri whih an be used to determine or evaluate the optimal performane of the optial link. t an as well be used to determine the data rate that ommuniates between the available -transmitter and -photodetetor. However, if the hannel frequeny response is known, and sine the A is a system for obtaining the maximum ahievable data rate, then we an also onveniently express the A of the systems in terms of the Average Spetral Effiieny (ASE) whih will then be given in bits/seonds/hertz [3], [4]. n the presene of weak atmospheri turbulenes with fading strength less than one, the A an be derived analytially as shown in this setion, for both the SSO- FSO link and the spatial diversity-fso links inluding the O-FSO link. 7 Journal of ommuniations 385
Journal of ommuniations Vol., o. 7, July 7 he following assumptions were taken into onsideration for this work in order to determine the ASE of the system: (a). he optial hannel of the FSO ommuniation system is presumed to be tratable, stationary, ergodi with independent and identially distributed (i.i.d.) turbulene statistis and memoryless. (b). A seamless (perfe hannel State nformation (S) is present at the -transmitting optial lasers as well as at the -reeiving apertures [4]. herefore, the ASE of the system an be expressed as follows [3], [4]: where where signal and. (36) f (x) is presented as the PDF of the transmitted optial signal vetor (x) in onsideration and the mutual where information for both the transmit and reeived signal is represented with ( x : y). herefore, the apaity of a hannel an be defined as the maximum mutual information obtained as a result of frequently hanging the PDF of the transmitted optial signal vetor. Using the priniple of information theory, the mutual information is given as [36]: optial hannels, f ( ) represents the joint Probability Density Funtion for the available array of optial hannels present in the atmospheri turbulent regime and then the parameter is a matrix representation of the O atmospheri turbulene indued optial hannels, and it is given as follows [5]: ( x : y) log det( Ex H )bps / Hz (37) H where the Hermitian symmetri matrix represents the hannel matrix obtained from the auto-orrelation { mn, m,, n,, } (33) matrix of the reeived signal y. and respetively represent the information about the reeive and transmit antennas. From (36) and (37), the hannel apaity of a O hannel in ases where a deterministi O hannel is assumed, is given in bps/hz as [36], [37]: he expression in (3) above whih an likewise be referred to as the A of the system is also defined as the ratio of the hannel s bit rate ( ) to the hannel s bandwidth (B). Sine we have assumed that the hannel is ergodi, therefore we an find the ergodi hannel apaity for SSO, SO, SO and O as presented in the next sub-setions. ) Ergodi hannel apaity of O-FSO hannels Aording to the arrangement of a O system with O max log det( Ex H ) (38) n ases where a random O hannel is assumed, thus, the hannel matrix is onsidered as a random matrix. herefore, this suggest that the apaity of the O hannel is also obtained as randomly time-varying. As a result of this, the apaity of the O hannel an best be desribed using the time average. Sine it is known that an ergodi proess pratially desribes a random hannel, then the ergodi hannel apaity of a random O hannel is given in terms of expeted value as [36], [37]: and reeive antennas, the hannel will be represented by a matrix whih is given as. However, the transmitted and multiple transmit reeived signals an both be represented as vetors given as [36], [37]: (34) O E O (x) for the transmitted signal Ex H ) E max log det( unrelated input symbol vetors: x, x,, x. he reeived signal ( y ) an be onsists of expressed in matrix system as [36], [37]: 7 Journal of ommuniations max f ( x ) ( x : y ) bits/hannel use is the hannel s bit rate, B is the Ex x n is the noise vetor and it is given as: However, the apaity of a hannel an be generally defined as follows [36], [37]: transmission bandwidth of the ergodi optial hannel, denotes the total Signal to oise atio (S) of the y n n ( n, n,, n ) (bits / s / Hz) log ( ) f ( )d (3) B he symbol vetor E x denotes the energy of the transmitted optial (39) hus, when S is available at the transmitter end, the ergodi hannel apaity expression for the O system is presented as shown in (39). However, when S is not available at the transmitter end, then the expression is simplified into the following [36], [37]: (35) 386
Journal of ommuniations Vol., o. 7, July 7 eanwhile, max Ex H E log ( ) O (4) F ( ) H r (4) i j he FSO hannel s total power gain is given as the squared Frobenius norm of the FSO-O hannel [36]. 3) Ergodi hannel apaity of SSO-FSO hannels For a SSO-FSO system that onsists of a single transmit antenna and a single reeive antenna, the reeived signal (y) an be written in linear form in terms of the transmitted signal x as: y E x x n (4) where and represents the single FSO hannel. hus, similarly to (38), the deterministi SSO hannel apaity an be expressed as: SSO E max x H log det( ) (43) n ases where maximum number of hannel =, and ij, therefore, the ergodi hannel apaity with hannel State nformation (S) at the transmitter beomes [36]: Ex E Elog ( ) F SSO SSO (44) Sine, and where the hannel s total power gain F, then the ergodi hannel apaity for a SSO hannel in (44) beomes [36]: Ex Elog ( ) bps Hz (45) SSO / herefore, regardless of S available either at the transmitter and/or reeiver sides, the hannel apaity for SSO remains the same. 4) Ergodi hannel apaity of SO-FSO and SO-FSO hannels he SO hannel to be desribed has a single transmit antenna along with multiple reeive antennas. Sine the hannel gain an be expressed as, then the time average of the hannel apaity for SO system (regardless of S availability at the transmitter) is given in terms of the expeted value as [36]: Ex E Elog ( ) F SO SO (46) where the hannel s total power gain F, then the ergodi SO hannel apaity expression beomes: Elog E ( ) bps (47) SO x / Hz However, for the ergodi SO hannel apaity, we have the following when S is not available at the transmitter, and when the hannel gain is given as where [36]: Ex E Elog ( ) F SO SO (48) F, then the expression beomes: Elog E ( ) bps (49) SO x / Hz Hene, it an be notied that the ergodi hannel apaity is the same for the SSO and SO hannels, if no S is available at the transmitter side. Finally, if the transmitter side of the SO hannel has the S, then the expression beomes [36]: Ex SO Elog ( ) bps / Hz (5). Outage hannel apaity Analysis he Outage hannel apaity (O),, is an alternative statistial onept of the hannel apaity whih is defined in terms of the outage probability of the FSO systems. herefore, to evaluate the O in terms of the rate of transmission ( b / s / Hz), we an use the following outage probability expression [36]: P out ( ) P( ( ) / Hz) bps (5) whereby for the FSO O hannels we an say that ( ). herefore, (5) beomes: P O out ( ) P( log ( Ex ) max H (5) n order words, the O an be defined as the maximum attainable data rate obtained when the outage probability as defined in (5) is less than the outage 7 Journal of ommuniations 387
Journal of ommuniations Vol., o. 7, July 7 of various FSO spatial diversity hannels from the less omplex ones (suh as and systems) up to very high-order O systems (suh as 4 4, 6 6 and 8 8 O systems) as a funtion of the S, for two various values of atmospheri turbulene strength (i.e.. and.9 ) representing the mildly weak and moderately weak atmospheri turbulene regimes respetively. he performane of the non-diversity FSO system (SSO) is also inluded in the analysis shown in Fig. 3 for the purpose of omparison and as a benhmark to measure the diversity gain of the BE or S. he performane analysis as learly observed from the figure, shows signifiant improvement as number of the transmit lasers and reeive photo-detetors inreases in a ertain order, but more signifiantly for FSO systems with higher-order antenna onfigurations suh as the 6 6 and 8 8 O systems. ore expliitly, a signifiant BE performane degradation results from an inrease in the strength of the atmospheri turbulene from. to.9. On the other hand, this performane degradation beomes absolutely insignifiant when the 8 8 O system is used, sine hannel apaity. t is required that the deoding error probability be made arbitrarily small in aordane with the rate of transmission. However, in ases where this is not so, this suggests that the FSO system may experiene an outage. hus, the ergodi hannel apaity of the spatial diversity FSO systems when there is no hannel State nformation (S) at the transmitter end, is given in terms of the umulative Distribution Funtion (DF). V. PEFOAE ESULS AD DSUSSO n this setion, as disussed earlier, the two various ases of weak atmospheri turbulene (i.e. mildly weak and moderately weak) analyzed in this paper are modelled using the Log-normal distribution model with BPSK signaling to provide a better performane. he analytial results obtained from the expressions derived in setion 3 are used to make omparative studies on the performane analysis of the SSO and spatial diversity FSO systems in Log-normal fading. he analysis shows the evaluation of the BE, A and O for various atmospheri turbulene onditions of turbulene strength:. and.9 for respetive ases of mildly weak and moderately weak atmospheri turbulene regimes. Here, the diameter of the Photo-detetor s reeiver aperture was taken to be m, the optial wavelength 85 nm, responsivity of the photodetetor and for the range of the atmospheri turbulene links we hose distanes of km, 4 km and 6 km. at BE of 9, the S required by the 8 8 O system is approximately the same for both the mildly weak and moderately weak atmospheri turbulene regimes. hus, as expeted, as the order of the spatial diversity inreases from the non-diversity ase ( ) to higher-order diversity ase ( 8), the BE dereases signifiantly. Also, in order to reah a BE of 9, it was notied that the required S dereased by approximately 8 db, 3 db and 35 db, as the system inreases from the nondiversity FSO system to 4 4, 6 6 and 8 8 O A. BE Performane esults able shows the input parameters used for obtaining the BE for the FSO systems onsidered. he predited and alulated BE performane as derived in (3) against the Signal to oise ratio in db for SSO ( nondiversity ) and spatial diversity shemes employing the BPSK-S through the weak atmospheri turbulene indued hannels is illustrated in Fig. 3. he logirradiane variane values of {. and.9} were onsidered for the log-normal fading hannels and also an ideal hannel (with no atmospheri turbulene) was used as a benhmark for the performane analysis. systems respetively at.9. herefore, this analysis has shown the various influene of the Logirradiane variane value of various antenna-order onfigurations of the FSO system. On the other hand, the AWG hannel (no turbulene hannel) an also be used as a benhmark to determine the required S loss. From the results in Fig. 3, we observed that in the non-atmospheri turbulene regime, the required S is approximately 6 db over the AWG hannel at a BE of 9, thus, the required S loss is approximately.4 db,.4 db, db and db for the 8, 4 4, 6 6 and 8 8 diversity systems to attain similar BE in the mildly weak atmospheri ABLE : PU PAAEES Parameter Value Operating Wavelength ytov Variane Values Photodetetor esponsivity efrative ndex rradiane odulation ndex Subarrier Signal Amplitude 85 nm at. and.9 9 6 m / 3. to 5 (5 samples) turbulene regime at.. But at that same BE, the S loss is.5 db, 7 db, 3.75 db and. db for the 8, 4 4, 6 6 and 8 8 diversity systems under a moderately weak atmospheri turbulene regime at o ahieve a better system performane, we need to ahieve Bit Error probability with values as low as the order of 9 or less. Fig. 3 shows the BE performane 7 Journal of ommuniations ( ) on the BE performane.9. However, the 8 8 O system possesses no signifiant S loss at both. and.9, sine 388
Journal of ommuniations Vol., o. 7, July 7 BE of 9. the required S is still approximately 6 db at the same Fig. 3. BE performane against the S for the FSO-SSO and FSO-Spatial diversity shemes of SO and O in log-normal hannel for. and.9. and 6 km). hese figures depit the influene of the various optial link distanes over the BE performane of various antenna order onfigurations of the spatial diversity FSO system. Fig. 4. BE performane against the S for FSO-SSO and FSOSpatial diversity shemes of SO, SO and O in log-normal hannel with refrative index n 9 6 m / 3 for distane L = m. Figs. 4, 5 and 6 show the BE performane analysis of various FSO spatial diversity hannels from the singleorder (SSO) FSO system up to very high-order O FSO systems (suh as 4 4, 6 6 and 8 8 O systems) as a funtion of the S, for atmospheri turbulene strength parameter of n 9 6 m / 3, representing the weak atmospheri turbulene regime, with three diverse optial link distanes (i.e. km, 4 km 7 Journal of ommuniations 389 Fig. 5. BE performane against the S for FSO-SSO and FSOSpatial diversity shemes of SO, SO and O in log-normal hannel with refrative index n 9 6 m / 3 for distane L = 4 m. From Fig. 4, the BE performane analysis for the SSO-FSO, SO-FSO, SO-FSO and the O-FSO systems over the Log-normal fading hannel with
Journal of ommuniations Vol., o. 7, July 7 refrative index n 9 6 m / 3 and distane are all ompared with the AWG hannel (no turbulene hannel) as the benhmark. he S penalties for these four (4) higher-order FSO systems are.5 db, 7 db, 3.75 db and. db respetively at a distane L 6m when ompared to the AWG hannel. However, in order to ahieve the same BE, an improvement was observed in terms of S gain of 5.5 db, 3 db, 33.5 db and 36.9 db for 8, 4 4, 6 6, and 8 8 FSO spatial diversity systems respetively when ompared to the non-diversity system. oreover, at an optial link distane of 6 km, no signifiant S loss was observed on the 8 8 O system when L m is illustrated. Fig. 4 shows that at this atmospheri turbulene strength parameter and the optial link distane, the antenna order of the spatial diversity system strongly determines the systems performane. 9 For instane, in order to ahieve a BE of, an improvement was observed in terms of S gain of.6 db, 3.6 db, 4 db and 5 db for 8, 4 4, 6 6, and 8 8 FSO spatial diversity systems respetively when ompared to the non-diversity system. However, at an optial link distane of km, no S loss was observed on the 8 8 O system when ompared to ompared to the AWG hannel at a BE of 9 the AWG hannel at a BE of. Fig. 5 shows that at atmospheri turbulene strength B. Average Ergodi hannel apaity esults Fig. 7 and Fig. 8 illustrate the Average Ergodi hannel apaity for various groupings of transmitters ( ) and reeivers ( ) in the presene of the weak atmospheri turbulene onditions. he various ombinations of and are onsidered under the n 9 6 m / 3 and optial link distane L 4 m, there exist a redution in the required S of 8.5 db, db,.5 db and 4 db for 8, 4 4, 6 6, and 8 8 FSO spatial diversity parameter systems respetively when ompared to the non 9 two log-irradiane variane values of diversity system in order to attain the BE of. Also, at this optial link distane, no S loss was observed on the 8 8 O system when ompared to the AWG hannel at a BE of 9. {. and.9}. he alulated Average Ergodi hannel apaity as derived from (3) to (5), against the Signal to oise atio in db for SSO ( non-diversity ) and spatial diversity shemes through the weak atmospheri turbulene indued hannels are illustrated in these figures. he ASE for SSO-FSO, SO-FSO, SOFSO and O-FSO hannels with respet to the S ( ), over the two different values of the log-irradiane variane were determined and analyzed. Fig. 7 shows the A of diverse FSO spatial diversity hannels with the same number of transmit and reeive 9. antennas (i.e. ) as a funtion of the S, for two different values of atmospheri turbulene strength (i.e. {. and.9} ) representing the mildly weak and moderately weak turbulene regimes respetively. t was observed from this illustration that the ASE whih suggests the bandwidth effiieny of the FSO systems depends strongly on the hanges in the strength of the atmospheri turbulene. Fig. 7 illustrates that irrespetive of the influene of atmospheri turbulene on the ASE, the higher-order O systems will still provide a signifiant bandwidth effiieny on the FSO system. However, it an be vividly notied that the influene of atmospheri turbulene beomes signifiant on the "nondiversity" systems. his results into the fat that as the turbulene inreases, there an arise a time whereby the bandwidth effiieny of the non-diversity system will approah a value lose to zero or atually beomes zero. On the other hand, it an be notied that 8 8 O systems gives the same A for both the mildly weak and moderately weak atmospheri turbulene regimes. t is worth mentioning that an 8 8 O system is Fig. 6. BE performane against the S for FSO-SSO and FSOSpatial diversity shemes of SO, SO and O in log-normal hannel with refrative index n 9 6 m / 3 for distane L = 6 m. Fig. 6 shows that at the atmospheri turbulene strength parameter n 9 6 m / 3 and the optial link distane L 6m, the 8, 4 4, 6 6 and 8 8 spatial diversity FSO systems, requires an additional S to reah the BE of 7 Journal of ommuniations 9, when they 39
Journal of ommuniations Vol., o. 7, July 7 unaffeted by the level of turbulene within the range of the weak atmospheri turbulene regime, whereby at S of 3 db it yields a bandwidth effiieny of up to 7 bps/hz. Using the "non-diversity" FSO system as the benhmark, at S of 3 db, we notied a bandwidth effiieny gain of 63 bps/hz, 45 bps/hz, 7 bps/hz and 8 bps/hz for 8 8, 6 6, 4 4 and O systems respetively at.9. in the number of transmit and reeive antennas. But this is greatly notied when there is an inrease in the reeive antennas, suggesting that the O and SO FSO systems provide a better hannel performane than the SSO and SO FSO systems onsidered. However, from the analysis, it is learly notied that the A depends strongly on the strength of the atmospheri turbulene whih determines the log-irradiane variane value. As the log irradiane variane values inreases, the A of the onsidered systems redues. Fig. 7. Ergodi hannel apaity analysis for FSO-SSO and FSOSpatial diversity systems over a log-normal hannel with. and.9. Fig. 8. Ergodi hannel apaity analysis for transmit and reeive FSOSpatial diversity systems over a log-normal hannel with.9. Fig. 8 shows the A of diverse FSO spatial diversity hannels with the different number of transmit and reeive antennas (i.e. ) as a funtion of the S, for atmospheri turbulene strength.9 representing the moderately weak turbulene regime. his explains the disparity in the Average (Ergodi) hannel apaity of the transmit diversity against the reeive diversity with tehnique. Also in this figure, using the "non-diversity" system as the benhmark, at.9, we dedued that the bandwidth effiieny provided by the reeive diversity systems with tehnique is far greater than that obtained from the transmit diversity. As the order of the spatial diversity inreases, a tremendous inrease in A was notied. his suggests that the higher-order SO spatial diversity systems provide signifiant improvement to the A. herefore, using the "non-diversity" FSO system as the benhmark, at S of 3 db, we notied that the A gain was approximately bps/hz for 8 transmit diversity FSO system, while it extends up to 4 bps/hz, 6 bps/hz, 8 bps/hz and 6.5 bps/hz for 8, 6, 4 and reeive diversity FSO systems respetively. herefore, from Figs. 7 and 8, it an be notied that hannel performane improves only if there is an inrease 7 Journal of ommuniations 39. Outage Probability esults he DFs of the hannel apaities for the spatial diversity FSO systems under the Log-normal fading are illustrated in Figs. 9 and when the S is db and db respetively, at optial link distane L = 6 km, optial wavelength 85 nm and the atmospheri turbulene strength n 9 6 m / 3. Fig. 9 illustrates the DF of some spatial diversity FSO systems as a funtion of the transmission rate, at S of db, optial link distane L = 6 km and under a weak turbulene ondition of atmospheri turbulene strength / 3. At the indiated O of n 9 6 m., it an be observed that the higher-order spatial diversity systems yields signifiant data rate of 5.5 b/s/hz, 7 b/s/hz,.3 b/s/hz,.6 b/s/hz and 5.5 b/s/hz for 4, 4 4, 6 6, 4 8 and 8 8 O FSO systems. However, it an also be notied that the 4 8 and 6 6 O FSO systems have almost the same data rate at. when the S is db, despite the differene in their antenna order onfigurations. n ontrast, as expeted, the non-diversity FSO system (SSO) shows no signifiant improvement in its hannel apaity unlike the spatial diversity FSO systems.
Journal of ommuniations Vol., o. 7, July 7 S = db. he 6 6 O system has a data rate gain of about b/s/hz over the 4 8 O system when the S inreases from db to db. his an also be as a result of the fat that the number of hannels of the 6 6 O system is greater than that of the 4 8 O system. hus, these two figures (9 and ) have shown that hannel apaities of the higher-order spatial diversity FSO systems improve optimally as the number of the transmit antennas and reeive antennas inreases. V. he optial SSO and spatial diversity systems employing the BPSK-S aross optial soures with oherent Detetion over the Log-normal model for weak atmospheri turbulene ondition, with two hosen sintillation indies have been analyzed. he two sintillation indies for this analysis were given in terms of the log-irradiane variane values of. and Fig. 9. umulative Distribution Funtion of the FSO-SSO and FSOSpatial diversity hannel apaities as a funtion of the transmission rate over a Log-normal hannel with n 9 6 m / 3, at S = db and distane L = 6 km..9, respetively representing mildly weak and moderately weak atmospheri turbulene onditions. he analyses made in this report were neessary for observing the best performane improvement that an ever be obtained from higher-order spatial diversity systems for Free Spae Optial ommuniations. his knowledge an be used to avoid redution in transmission apaity as well as any possible outages. However, aording to the results of the BE performane analysis for all the senarios onsidered, it was observed that the S (db) needed to obtain an optimal BE inreases along with the inrease in the atmospheri turbulene level whih is indiated by the sintillation indies. Among the higherorder spatial diversity systems that were onsidered under the weak atmospheri turbulene regime, 8 8 O system gave the optimum BE and hannel apaity performanes. herefore, the effiieny and power of the higher-order spatial diversity system espeially the 8 8 O system was established in the presene of weak atmospheri turbulene and onsidered for average BE Fig. illustrates the DF of some spatial diversity FSO systems as a funtion of the transmission rate, at S of db, optial link distane L = 6 km and under atmospheri turbulene strength n 9 6 m / 3. At the indiated O of., the data transmission rate of the higher-order spatial diversity systems are b/s/hz, 6.5 b/s/hz, 4 b/s/hz, 6 b/s/hz and 35.5 b/s/hz for 4, 4 4, 4 8, 6 6 and 8 8 O FSO systems, while the non-diversity FSO system yields a data rate of.5 b/s/hz. 9 of. As a matter of fat, it is important to note that as the spatial diversity antenna order onfiguration inreases, there exist a pratial omplexity limitation introdued into the system. his pratial limitation is as a result of the omplexities of the higher-order spatial diversity tehniques with higher number of transmit and reeive antennas involved in the build-up of the FSO ommuniation system design. his makes it ompliated in the pratial sense. his form of advane spatial diversity tehniques has been suggested to provide improvement on the system performane in FSO ommuniations when onsidering relatively severe sintillation hannels. hus, when implementation is not foused on the omplexity of the design, higher-order spatial diversity tehniques and BPSK-S an be employed in FSO Fig.. umulative Distribution Funtion of the FSO-SSO and FSOSpatial diversity hannel apaities as a funtion of the transmission rate over a Log-normal hannel with n 9 6 m / 3, at S = db and distane L = 6 km. However, when the S is db, one of the notable fats is that at., the DF of the 4 8 and 6 6 O systems are not quite lose unlike the DF at 7 Journal of ommuniations OLUSO 39