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Advanced Optical Spatial Modulation Techniquesfor FSO Communication

Anirban Bhowal, Member, IEEE and Rakhesh Singh Kshetrimayum, Senior Member, IEEEDepartment of Electronics and Electrical Engineering

Indian Institute of Technology GuwahatiGuwahati 781039, India

Email: a.bhowal@alumni.iitg.ac.in

Abstract—In this paper, spectral efficiency of free-space op-tical (FSO) communication is enhanced by the utilization ofadvanced schemes of optical spatial modulation (OSM). Opticalgeneralized spatial modulation (OGSM), optical enhanced spatialmodulation (OESM), and optical improved quadrature spatialmodulation (OIQSM) are proposed for performance improve-ment of FSO systems over Gamma-Gamma (G-G) channelincorporating pointing errors and path loss effect. These methodsare investigated in terms of bit error rate (BER) and theanalytical BER upper bound results are corroborated by MonteCarlo simulations. Both spatial domain and symbol domainerrors are considered while calculating BER upper bounds.The specifications required to implement the schemes are alsopresented. The proposed methods offer benefits of reduced costand power consumption, and BER performance improvementover OSM for higher spectral efficiencies. The upper boundsof BER are tight for all values of transmit power for variousconfigurations of the proposed schemes.

Index Terms—FSO, OGSM, OESM, OIQSM

I. INTRODUCTION

Free-space optical (FSO) communication has the potentialto overcome the limitations of the existing radio frequency(RF) communication because of its higher data rates, band-width, lower installation and maintenance costs, and inherentsecurity solutions [1]. Optical spatial modulation (OSM) iscommonly deployed in FSO communication to improve thespectral efficiency. In OSM [2], a single optical chain andmultiple lasers with its corresponding apertures are present.According to the message bits, the corresponding laser isconnected to the optical chain and the modulated symbol issent.

The channel model needs to incorporate the rapidly chang-ing atmospheric conditions in FSO communication. Opticalsignals undergo directional and energy changes due to randomvariations of refractive index, temperature, and humidity ofatmospheric regions. Atmospheric turbulence (AT) [3] causesmodulation of small scale turbulent eddies with the large scaleturbulent eddies resulting in variations of the received opticalintensity and this phenomenon is known as scintillation. SuchAT induced fading can be properly represented by Gamma-Gamma (G-G) channel model as reported in literature [1], [4].The FSO performance may be influenced by building sways,wind speeds which may cause misalignment of laser sourcesand receivers. This leads to pointing error [4], so our channel

model also incorporates pointing error to consider all thesefactors.

There are different modulation schemes which can beused for FSO communication. On-off keying (OOK) requiresadaptive threshold for adjustment in accordance with thevarying atmospheric turbulences. M-ary quadrature amplitudemodulation (M-QAM) is the preferred modulation scheme forFSO communication as its threshold is fixed irrespective ofthe signal distribution [5], [6]. In literature, error analysis hasbeen done for multiple-input-multiple-output (MIMO) [7] andconventional FSO links [8]. Cooperative FSO communicationhas been analyzed in terms of outage probability for channelsincorporating pointing errors in [9]. Optical MIMO systemshave been proposed in [10] to enhance system performance.The performance of QAM with spatial diversity has beenanalyzed in terms of outage probability and bit error rate fordifferent degrees of turbulence in FSO systems [11]. Spectralefficiency of a single-input-multiple-output (SIMO) FSO sys-tem has been analyzed for different turbulence effects alongwith pointing errors in [12] and depending on the bit error rateand channel conditions, the order of the phase-shift keyingmodulation scheme is varied. To improve the performanceof SIMO FSO systems under atmospheric turbulence andpointing errors, multi-pulse pulse position modulation schemehas been proposed in [13]. Space shift keying (SSK) for FSOcommunication has been explored in [14]. Diversity analysisof optical SSK in MIMO FSO systems has been done inthe presence of atmospheric turbulence and pointing errorsin [15]. OSM for lognormal and G-G channels has beeninvestigated in [16]. OSM with photon counting receivers andtransmit diversity analysis has been reported in [17]. Spatialmodulation based subcarrier intensity-modulation (SIM) FSOsystems have been analyzed in terms of error probabilityover lognormal channels in [18]. OSM over H-K turbulencechannels in FSO system has been reported in [19], where theanalytical average bit error probability for both coded anduncoded OSM have been derived. OSM with spatial diversityhas been reported for FSO systems over G-G channel in [20].An upper bound for average bit error rate using maximum ratiocombining and equal gain combining has been analyzed. Asimilar analysis of OSM based FSO system has been done overG-G channel in presence of pointing errors in [21]. Transmitdiversity and pulse position modulation have been utilized toimprove the performance of OSM based FSO systems in [22].

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Differential OSM has been explored in [23]. Enhancedspatial modulation (ESM), the RF version of optical ESM(OESM), has been also extensively investigated in [24]. Im-proved quadrature spatial modulation (IQSM), the RF versionof optical IQSM (OIQSM), has been explored in [25]. Gen-eralized spatial modulation (GSM) in RF domain has beenreported in [26]. Optical MIMO enhances spectral efficiency,but it comes at the cost of increased hardware cost and powerconsumption, which will be discussed in the forthcomingsections. Coding techniques can be used to improve the FSOsystem performance [27]–[29], but it also requires overheadbits adding to the system complexity. These drawbacks canbe mitigated by the use of OSM, but modern FSO systemsrequire much higher spectral efficiencies. The research gapsfrom the literature survey include absence of performanceanalysis of any advanced OSM schemes for outdoor FSOcommunication. To the best of the author’s knowledge, costand power consumption analysis for such systems has not beenreported. Spatial domain and modulation domain BER analysisfor such advanced OSM schemes are not available for FSOsystems. All these factors motivate us to propose advancedOSM schemes like OESM, optical GSM (OGSM) and OIQSMfor FSO communication and analyze its BER, cost, and powerconsumption.

In OESM, single or double laser activation takes place,thereby requiring 2 optical chains. In OGSM, a certain numberof lasers are activated and the number of optical chains is lessthan the number of lasers. Thus OGSM requires Nopt opticalchains to activate Nopt lasers, where Nopt < NL . Note thatNL denotes the total number of source lasers. Meanwhile inOIQSM, double layer laser activation occurs for transmissionof real and imaginary parts of two symbols, and hence itrequires 4 optical chains. To increase the spectral efficiency,multiple modulated symbols can be transmitted by a precodingbased scheme, as proposed in RF domain [30]. The transmis-sion of multiple modulated symbols is basically implementedin OGSM, thus OIQSM is essentially a simplified version ofOGSM. There is a trade-off between spectral efficiency andenergy efficiency. Hence, we are limiting it to transmission oftwo modulated symbols at a time only in OIQSM, but it canbe extended for any number of modulated symbols.

The novelty of our paper lies in the fact that performanceanalysis of these advanced OSM schemes are investigatedin terms of BER for FSO communication for the first time.The upper bound of BER expressions are derived consider-ing both spatial and symbol modulation error. Even thoughthese schemes already exist in RF domain, they cannot beimplemented directly in optical domain due to the presence ofpointing errors, atmospheric turbulences, and different natureof optical transmitter and receiver than their RF counterparts.The system parameters required for the optical transmitter andreceiver are specified. The pointing error effect is incorporatedinto the channel model and cost and power consumptionanalysis of the proposed FSO systems is reported.

Section II describes the channel model while Section IIIelaborates the system model for the various schemes. Perfor-mance analysis is carried out in Section IV, while comparisonof the proposed schemes in terms of different performance

metrics is done in Section V. Section VI provides the detailedresults and Section VII concludes the paper.

II. CHANNEL MODEL

AT induced fading can be suitably described by G-Gchannel as described earlier. The effective number of largescale and small scale cells of the scattering process aredenoted by α and β respectively. These values depend on theturbulence conditions and have been defined in [31]. Rytovvariance σ2

lis defined in [32, eq. (4)]. The AT induced fading

coefficient is represented by h. The attenuation of laser poweron atmospheric propagation follows Beer Lambert Law and thelaser power at a distance of z is given by hl (z). For a longerobservation period, the term hl is considered to be fixed. Forpointing error model, it has been assumed that the detectorcan collect fraction hp of the power and its probability densityfunction (PDF) can be written as f (hp ) = ζ 2

Aζ 20

hζ2−1

p , where

hpε[0, A0] and ζ = wzeq /2σs . The ratio between equivalentbeam radius at the receiver (wzeq ) and twice the pointingerror displacement standard deviation at the receiver (σs) isrepresented by ζ . The fraction of collected power in absenceof pointing error is denoted by A0. The above mentioned termslike ζ , A0, wzeq and σs are already defined elaborately in [4].The overall PDF of G-G channel incorporating pointing errors,where heq = hphl h, can be written as in [4], [33]:

f (heq ) =ζ2(αβ)

A0hlΓ(α)Γ(β)G 3,0

1,3

(ζ 2

ζ 2−1,α−1,β−1

∣∣∣∣∣∣ αβheq

A0hl

), (1)

where Γ( ) denotes the Gamma function, Gm ,np ,q (:| z) is the

Meijer G function [34], [35].

III. SYSTEM MODEL

For all the three proposed schemes, the distance of separa-tion between source node (SN) and destination node (DN) isL. Source node has NL lasers with its corresponding apertureswhile DN has ND photodetectors with its correspondingapertures. The photodetector has a responsivity of Re . Inan optical transmit chain, source data initially modulates ahigh frequency RF sinusoidal signal. The modulated sinusoidalsignal contains negative data, which needs to be eliminated inorder to drive the laser as laser can only be driven by a positivesignal. Hence, a DC bias is added to make the RF modulatedsignal positive by shifting the level of the signal above zeromark. In this manner, the negative part of the modulatedsignal will now be raised above zero, whereas the positiveportion will be raised to a more positive value. Thus the signalafter DC bias addition is used to modulate the laser intensity.If the signal is zero, laser will be switched off, whereasaccording to various levels of signal value, laser intensitycan be varied. The laser output is transmitted into free-spacethrough an aperture. At the receiver, an aperture collects theoptical wave and focuses the optical wave to the photodetectorwhich performs direct detection in absence of local oscillator.The information content is directly proportional to the intensityof the transmitted optical signal. This optical wave is convertedto electrical signal, subsequently followed by demodulation inRF domain.

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The proposed schemes when used in the optical domainhave to take care of several parameters like jitter, pointing errordeviation, transmit and receiver aperture separation. The valuesof the different parameters for optical domain are mentionedin Results section. In RF domain, signal propagation mainlytakes place by reflection and multipath propagation. But inFSO communication, signal propagation is primarily line-of-sight based and undergoes atmospheric turbulence inducedfading and jitters due to environmental conditions. Hence, thechannel model and parameter values are quite different fromthe existing RF schemes. The proposed schemes are elaboratedin the following subsections.

A. Optical Enhanced Spatial Modulation

In OESM, two optical transmit chains are used and singleor double laser activation takes place depending upon themessage bits. A primary constellation and two secondary con-stellation schemes are required for this technique. The size ofthe secondary constellation is half of the primary constellationscheme. The total message bits are partitioned into two parts-control bits and modulation bits. The value of the control bitsdetermines the lasers to be activated and these space modu-lation bits determine the laser(s) to be used for transmissionaccording to a mapping table. The modulation bits are used forsymbol mapping. When a single laser is activated, the symbolis mapped according to the primary constellation, while fordouble laser activation, two symbols are mapped to eitherof the two secondary constellation schemes. For example ifprimary constellation size is m where m = log2(M) then sizeof secondary constellation is m/2. Hence either one symbolof m bits is transmitted from a primary constellation or twosymbols each of m/2 bits are transmitted from a secondaryconstellation. The total number of possible combination ofsymbol transmission is N2

Lwhere NL denotes the number of

source lasers. Hence spectral efficiency of OESM is given byηOESM = log2(N2

L) + log2(M).

For example if NL = 4 and M = 4, then primary constella-tion will be 4-QAM also equivalent to quadrature phase shiftkeying (QPSK), and secondary constellation will be binaryphase shift keying (BPSK). Since 4-QAM and QPSK are same,hence the constellation schemes can take symbols from the set{±1 ± j}. BPSK0 and BPSK1 are the secondary constellationschemes given by BPSK0 = {±1} and BPSK1 = {± j}. Inthis case, OESM can transmit 6 bpcu (bits per channel use)while normal OSM can transmit only 4 bpcu with NL = 4and M = 4. Thus this OESM scheme has been named as 4t6b.Note that atyb means a number of source lasers will be usedto achieve spectral efficiency of y. For this particular OESMscheme, 16 total combinations are possible out of which thefirst 4 combinations are same as that of OSM with singleoptical source being active. Double source activation is donefor the next 6 cases where symbol mapping is done accordingto one of the secondary constellation scheme BPSK0, whilethe secondary constellation scheme BPSK1 is used for the next6 cases. The active laser combination for space modulationfor this scheme is tabulated in spatial mapping Table I. Thefour laser sources are denoted by L1, L2, L3 and L4, and the

various combinations of laser sources are denoted by Cn wheren = 1, 2, ..., N2

L.

Example I: An example of OESM is illustrated in Fig. 1where the information stream is given by [1 0 0 1 1 1]. Thusthe first four bits [1 0 0 1] are used as control bits, whichcorresponds to the combination labelled C10 in Table I, wherethe third and fourth lasers will be active and BPSK0 schemewill be used. Optical chain (OC) 1 will be connected to laser3 and OC 2 will be connected to laser 4. BPSK0 symbol willbe sent from laser 3 and 4. All other lasers will be sitting idle.According to the secondary constellation scheme BPSK0, themodulated bits would be [1 1] for the input bits [1 1]. Hencethe transmitted symbol vector will be xxx = [0 0 1 1]T . Here [ ]T

denotes the transpose of a matrix. The other OESM schemescan be similarly implemented for higher modulation schemesand transmit sources.

TABLE I: OESM 4t6b

L1 L2 L3 L4C1 = 0000 QPSK 0 0 0C2 = 0001 0 QPSK 0 0C3 = 0010 0 0 QPSK 0C4 = 0011 0 0 0 QPSKC5 = 0100 BPSK0 BPSK0 0 0C6 = 0101 BPSK0 0 BPSK0 0C7 = 0110 BPSK0 0 0 BPSK0C8 = 0111 0 BPSK0 BPSK0 0C9 = 1000 0 BPSK0 0 BPSK0C10 = 1001 0 0 BPSK0 BPSK0C11 = 1010 BPSK1 BPSK1 0 0C12 = 1011 BPSK1 0 BPSK1 0C13 = 1100 BPSK1 0 0 BPSK1C14 = 1101 0 BPSK1 BPSK1 0C15 = 1110 0 BPSK1 0 BPSK1C16 = 1111 0 0 BPSK1 BPSK1

Fig. 1: Proposed system model for OESM (In Example I, NL =

4, for input bits [1 0 0 1 1 1], C10 combination in Table I willbe implemented in which lasers 3 and 4 will send symbol 1).

B. Optical Generalized Spatial Modulation

This technique combines the benefit of optical chain andlaser selection to achieve improved spectral efficiencies than

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OSM. In OGSM, multiple lasers can be activated to improvethe spectral efficiency. However, decoding complexity and costalso increase with use of multiple laser sources. Therefore,it requires an optimum combination of optical chains andlaser sources. OGSM system uses Nopt optical chains andNL laser sources. A multiple pole multiple throw (Nopt , NL )switch is used for connecting the optical chains to the lasers.Note that Nopt out of NL lasers are chosen to be activated.In OGSM, the message bits are partitioned into control bitsand modulation bits. C is defined as the control bits usedto choose the set of active laser sources and is defined byC = blog2

(NL

Nopt

)c. The set of valid laser patterns to be

activated is given by K having dimension of 2C . Out of NL

lasers, Nopt number of lasers can be chosen in a possible(NL

Nopt

)manner. Since the data will be transmitted through

Nopt lasers, hence the laser indices of such Nopt lasers arerequired. This multiple laser indices selection can be done byblog2

(NL

Nopt

)c number of bits. The value of the control bits

determines which numbered laser activation pattern is to beselected from set K . K set is a predefined spatial mappingtable fixed before transmission. If bit value in the selected laseractivation pattern is 1, then that particular transmit laser willbe activated. The modulated symbol is partitioned into Nopt

groups and each group comprising of m bits (where M = 2m)is transmitted from each laser. Symbol mapping is performedaccording to the M-ary modulation scheme. Hence the spectralefficiency can be defined as ηOGSM =

⌊log2

(NL

Nopt

)⌋+ mNopt .

The optimum selection of number of optical chains Nopt isa matter of concern in OGSM systems. The selection has to bedone in such a way that it maximizes the spectral efficiency.The first part of the spectral efficiency expression increasesfrom Nopt = 0 to Nopt = bNL/2c and then decreases.However, the second part of the expression increases linearlywith Nopt keeping the modulation term constant. Thus theoptimum value of Nopt for which the spectral efficiencyis maximum should be between bNL/2c and NL . To betterunderstand the trade-off of spectral efficiency and BER, letus discuss a more generalized version of OGSM where thenumber of optical chains can be varied. The RF version of sucha scheme named orthogonal frequency division multiplexingwith generalized index modulation (OFDM-GIM) is given in[36]. Let there be g groups of optical transmit chains andlasers. If total incoming bits is b, then it will be split into ggroups of p bits each. The p bits are further subdivided into p1and p2 bits where p1 bits are used for activating Nopt out ofNL lasers. The p2 bits are used for symbol mapping accordingto a M-ary modulation scheme. Hence a total number of g×NL

number of lasers and optical chains are present. In this casethe Nopt value (i.e. number of activated chains and lasers)is not constant for the entire system. It can vary in differentgroups depending on the input binary stream. Such a systemcan achieve a spectral efficiency of blog2(MNopt

(NL

Nopt

)cg. By

allowing variable values of activated lasers for each group,spectral efficiency can be increased as compared to thatof a fixed Nopt value [36]. It has been shown that suchOFDM-GIM based scheme can offer much improved spectralefficiency value at the cost of marginal BER performance loss.

But as the BER value gets lower for high SNR values, the BERperformance loss is very negligible for high spectral efficiencyvalues. Thus high spectral efficiency values can be obtainedfor OGSM also by adopting such schemes and the optimumvalue of number of activated lasers can be considered similarto what has been reported in [36] for OFDM based RF system.

Example II: Let us consider an OGSM system with NL =

4, Nopt = 2 and M = 4. It gives C = 2 and ηOGSM = 6bpcu (bits per channel use). The laser pattern set is given byK = [1, 1, 0, 0], [1, 0, 1, 0], [0, 1, 0, 1], [0, 0, 1, 1]. The patterncan be different, but should be fixed before transmission andshould have a cardinality of 2C . Let us assume the incomingmessage bit sequence is [1 0 0 1 1 1]. The first 2 bits willdefine the control bits (in this case it is [1 0] or decimal valueof 3) indicating that the third pattern [0 1 0 1] from the setK will be used for activating the lasers. In the third pattern,the second fourth bit values are one signifying that secondand fourth lasers will be activated. It means OC 1 will beconnected to laser 2 and OC 2 will be connected to laser 4.All other lasers are not connected to OC and sitting idle. Theremaining bits in the incoming bit sequence, [0 1 1 1] willbe used for symbol mapping. Thus two 4-QAM symbols willbe sent. The symbols are 1 − j and −1 − j for bits [0 1] and[1 1] respectively. Thus, the transmitted signal vector is givenby xxx = [0, 1− j, 0, −1− j]T . The example is pictorially shownin Fig. 2.

Fig. 2: Proposed OGSM system model: OC-Optical chain, PD-Photodetector (In Example II, NL = 4, Nopt = 2, M = 4, forinput bits [1 0 0 1 1 1], third pattern from the set K will beselected for the control bits [1 0], in which second and fourthlasers will send 4-QAM symbols 1 − j and −1 − j for themodulation bits [0 1] and [1 1] respectively).

C. Optical Improved Quadrature Spatial Modulation

A new scheme called optical improved quadrature spatialmodulation (OIQSM) is proposed for FSO system to enhancethe spectral efficiency. OIQSM can transmit double modulatedsymbols in a single time slot by using two different layersof laser sources. The original data stream is partitioned intotwo parts- one is used for laser index mapping and the otherfor symbol mapping. Again the laser index mapping bits aresplit into two parts, which denote the indices of in-phase andquadrature phase activated lasers. The mapped symbols arepartitioned into real and imaginary parts of the symbols. Now

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since there are two modulated symbols, hence two in-phaseand two quadrature phase lasers are activated for transmittingthe two real and two imaginary parts respectively. The laserset (1,2), (1,3), (1,4) and (2,3) are activated for the bit patterns(00), (01), (10) and (11) respectively for NL = 4. This laser setactivation pattern can be changed but should be fixed beforetransmission. The spectral efficiency of OIQSM [25] is givenby 2

∣∣∣∣log2

(NL

2

)∣∣∣∣ + 2 log2 M .Example III: For our proposed OIQSM based FSO system,

let us consider input data stream is [11010110]. The partof the data stream which will be used for activating thecorresponding laser source is [1101], while the remaining part[0110] will be used for symbol mapping. Hence, the laserswhich are activated for transmitting real and imaginary partsare {2, 3} and {1, 3} for bits [1 1] and [0 1] respectively. 4-QAM modulation scheme is used in this case and accordinglythe symbols will be

[1 − j, −1 + j

]for the symbol mapping

bits [01, 10]. The real and imaginary parts will be separated.Note that +1 and -1 are the real parts of the above two 4-QAM symbols, and they are to be sent from lasers 2 and 3respectively. Similarly, -1 and +1 are the imaginary parts of theabove two 4-QAM symbols, and they are to be sent from lasers1 and 3 respectively. Hence, we can write xrxrxr = [0, +1, −1, 0]T

and xixixi = [−1, 0, +1, 0]T . Thus, overall signal vector which istransmitted is xxx = xrxrxr + jxixixi =

[− j, 1, −1 + j, 0

]T . The in-phaseand quadrature phase laser activation are shown in Fig. 3(a)and 3(b) respectively.

IV. PERFORMANCE ANALYSIS

Message bits are transmitted by source node to destinationnode in a single time slot. At the photodetector, thermal, shotand dark current noise are the dominant sources of noise. Thetotal noise variance is computed by adding up the variancesof the three types of noises. Ambient light produces thermalnoise which is represented by additive white Gaussian noiseof zero mean and variance N0. Shot noise arises due tothe random arrival of photons. Dark current noise arisesin the absence of light. x is the transmitted source vector.s is the transmitted signal vector and can be expressed ass =

√Oe f f E x, where E denotes the average symbol energy,

Oe f f is the optical efficiency of the system. Assume y isthe receiver output. Decoding at the receiver takes placeby maximum likelihood (ML) decoding which is given bys = arg min

s∈S‖y − HHHs‖2, where S is the set of normalized

transmit symbol vectors. Each scheme - OSM, OESM,OIQSM and OGSM - has its own predefined set S, containingall the symbol vectors for that particular scheme. PairwiseError Probability (PEP) is the probability of symbol vector sbeing decoded as s′. For detection of symbol mapping bits,the average PEP can be computed as:

APEP(s → s′)Symbol =1|S|EH

Q 1

ND

√γe f f | |HHHs − HHHs′ | |2

2

=1|S|

∫ ∞0Q

1ND

√γe f f

∑ND

j=1∑NL

i=1 | |hi j s − hi j s′ | |2

2

fγSM(h)dh ,

(2)

(a)

(b)

Fig. 3: Proposed OIQSM system model for (a) in-phase laseractivation (b) quadrature phase laser activation (In ExampleIII, NL = 4, M = 4, for input bits [11010110], the controlbits are [1 1] and [0 1] for which lasers {2, 3} and {1, 3} willbe activated for real and imaginary parts of the two 4-QAMsymbols (1− j) and (−1+ j) for the symbol mapping bits [0 1]and [1 0] respectively).

where average received signal-to-noise ratio (SNR) is γ = EN0

,

whereas γe f f =ReOe f f E

N0, γSM =

∑ND

j=1 (∑NL

i=1 hi j )2. Notethat HHH is the channel matrix having dimension of ND × NL

and hi j is an element of the channel matrix which followsG-G distribution with pointing errors as in Eq. (1). Theintegration involves multi-dimensional integration over thechannel gains from each laser to photodetector. Hence, exactclosed-form expressions cannot be derived and the integralsare numerically evaluated with the help of Mathematica.The average BER (ABER) (denoted by Ps) occurring duringdetection of the symbol mapping bits is bounded by [24]:

Ps ≤1

|S|log2(|S|)

∑s∈S

∑s′∈S

n(s→ s′) APEP(s→ s′)Symbol ,

(3)where n(s → s′) is the number of bit errors that occur whens is decoded as s′. During detection of the laser index, theasymptotic APEP (where the laser index j can be incorrectlydetected as i) for G-G channel with pointing errors can be

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computed as:

APEP( j → i) =1|S|

∑j∈S

∑i∈S

C ∞∑k=0

(−αβ/(A0hl ))k

γ (k+1)/2e f f

k!

×G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1))ND

×(2ND )kND+NDΓ( kND+ND+1

2 )

γ (kND+ND )/2e f f

√π(k ND + ND )Γ((k ND + ND )/2)

, (4)

where C = αβζ 4

A0hl (Γ(α)Γ(β))2 . The details of this derivation can befound in Appendix A. This expression contains summation ofk from 0 to ∞. To truncate the infinite sum series in Eq. (4),we are going to explore the value of k for which the truncatedfinite series is giving insignificant errors. The Cauchy ratio testcan be used to check the convergence of the infinite series first.The series is convergent if:

limk→∞

|ak+1

ak

| ≤ 1 . (5)

The series coefficient of k + 1th and k th terms are obtainedonly from Eq. (4). The ratio of two Meijer G functions willalways be a nonzero real number for all values of k. It canbe derived in a similar manner as shown in [14]. However,after solving the ratio of k + 1th and k th term, it can beobserved that the degree of k in the denominator exceedsthat of the numerator by one. Hence on applying the limitcondition, the ratio equation will result in zero indicatingthat the infinite series is convergent. The values of k whichgive insignificant numerical error due to truncation of theinfinite series to a finite series is investigated [37, pp. 135]in Mathematica. We have expressed PEP values in the formof scientific notation a × 10b with accuracy upto 5th placeof decimal digit. The numerical error due to truncation of theinfinite sum is considered upto 10−7. Numerical error of anyvalue less than 1 × 10−7 is considered to be insignificant. Wewill explore the values of k for which we can get accurateresults upto 5 places of decimal by truncating the infinitesum to first k terms, i.e. the first five digits of decimalremain unaltered even on increasing the value of k further.It is observed that at higher SNR values (or higher values oftransmit power), lower k values can ensure that the truncationof terms produces insignificant error. For example at transmitpower value of 10 dBm, PEP for k = 0 is 4.44444×10−2,for k = 2 value is 7.42234×10−2, for k = 4 value is8.88875×10−2, while for k ≥ 6 value is 9.50011×10−2.Thus at 10 dBm transmit power, k = 6 is the limit at whichnumerical error due to truncation is insignificant. Similarly,for 15 dBm transmit power, PEP for k = 0 is 1.39186×10−2,for k = 2 PEP value is 1.49666×10−2, while for k ≥ 4 PEPvalue is 1.52222×10−2. Hence, the limit of k is 4 at 15 dBmtransmit power. Now for 20 dBm transmit power, PEP valuefor k ≥ 0 is 4.44222×10−3, while for 30 dBm transmit power,PEP value for k ≥ 0 is 4.44444×10−4. Thus, the truncatingof terms generates insignificant numerical error for k ≥ 0at high values of transmit power exceeding 20 dBm. It is tobe noted that as the value of k increases, the computationtime will also increase. The study of k values for which thetruncating of terms of the series produces insignificant error is

also done for moderate and weak turbulence conditions. Formoderate turbulence conditions, k ≥ 0 generates negligibletruncation error for more higher transmit power values of 30dBm or more. At transmit power value of 20 dBm, PEP undermoderate turbulence conditions gives almost the same valuefor k ≥ 4. Similarly under weak turbulence conditions, k ≥ 3produces negligible truncation error for the series at transmitpower value of 30 dBm, while for transmit power value of20 dBm, k ≥ 5 yields insignificant truncation error. Hence, itis concluded that as the turbulence conditions decrease, the kvalue required to generate negligible truncation error is largerat a particular value of transmit power. It is pertinent to notethat for more number of photodetectors, the limiting valueof k to produce insignificant truncation error is greater fora particular value of transmit power. For ND = 2, k ≥ 4generates insignificant numerical error at transmit power valueof 30 dBm. The upper limit of summation variable k togenerate accuracy till 5th place of decimal digit for differentcases is shown in Table II. The different scenarios are allotteda particular case number in the table. For Case 1 and 2,the analysis of k values for which the truncation of termsgenerates insignificant numerical errors, is illustrated in Fig.4, while a similar analysis for Case 3 and 4 is shown in Fig.5. The values of PEP at different k values are shown in thefigures, which helps the readers to easily understand the kvalues required to generate insignificant truncation error atdifferent transmit power values. In this paper, we have consid-ered strong turbulence conditions and a single photodetectorscenario. Hence considering all values of transmit power, wehave considered the limiting value of k as 10 for which thenumerical error induced due to truncating terms of the seriesis insignificant. Thus, the upper bound results will be tightfor the considered range of transmit power values for strongturbulence conditions.

Value of summation variable (k)

0 1 2 3 4 5 6 7 8 9 10

PE

P

10-4

10-3

10-2

10-1

Case 1, 10 dBm power

Case 1, 15 dBm power

Case 1, 20 dBm power

Case 1, 30 dBm power

Case 2, 10 dBm power

Case 2, 15 dBm power

Case 2, 20 dBm power

Case 2, 30 dBm power

X: 8Y: 0.095

X: 6Y: 0.095

X: 4Y: 0.08811

X: 10Y: 0.05544

X: 6Y: 0.05222

X: 8Y: 0.05544

X: 10Y: 0.007244

X: 4Y: 0.01522

X: 6Y: 0.01522

X: 2Y: 0.01497

X: 6Y: 0.007244

X: 4Y: 0.007

X: 8Y: 0.007244

X: 0Y: 0.004422

X: 2Y: 0.004422

X: 4Y: 0.003122

X: 6Y: 0.003121

X: 8Y: 0.003121

X: 0Y: 0.0004444

X: 2Y: 0.0004444

X: 0Y: 0.0002444

X: 2Y: 0.0002444

Fig. 4: Analysis of k values for Case 1 and 2 due to truncationof the infinite series to a finite series for different power values.

The average bit error rate for laser index detection (denotedby Plide) is given by:

Plide ≤1

|S|log2(|S|)

∑j∈S

∑i∈S

n( j → i) APEP( j → i) . (6)

The total bit error rate of the system (Pe) can be upper

7

TABLE II: k values required for accuracy till 5th place of decimal digit in scientific notation

Parameters 10dBm power 15dBm power 20dBm power 30dBm powerk PEP k PEP k PEP k PEP

α = 4.2, β = 1.4, ND = 1 (Case 1) 6 9.50011×10−2 4 1.52222×10−2 0 4.44222×10−3 0 4.44444×10−4

α = 4.0, β = 1.9, ND = 1 (Case 2) 8 5.54444×10−2 6 7.24444×10−3 4 3.12211×10−3 0 2.44411×10−4

α = 11.6, β = 10.1, ND = 1 (Case 3) 10 5.54444×10−3 8 8.42213×10−4 5 3.00001×10−4 3 2.00112×10−5

α = 4.2, β = 1.4, ND = 2 (Case 4) 10 8.44431×10−3 8 1.12113×10−3 6 1.81112×10−4 4 1.21100×10−6

Value of summation variable (k)0 2 4 6 8 10 12

PE

P

10-7

10-6

10-5

10-4

10-3

10-2

Case 3, 10 dBm power

Case 3, 15 dBm power

Case 3, 20 dBm power

Case 3, 30 dBm power

Case 4, 10 dBm power

Case 4, 15 dBm power

Case 4, 20 dBm power

Case 4, 30 dBm power

X: 10Y: 0.008444

X: 12Y: 0.008443

X: 6Y: 0.001111

X: 8Y: 0.001121

X: 10Y: 0.001121

X: 7Y: 0.0003

X: 4Y: 0.0002712

X: 5Y: 0.0003

X: 8Y: 0.0008422

X: 10Y: 0.0008422

X: 6Y: 0.0001811

X: 8Y: 0.0001811

X: 4Y: 0.0001754

X: 3Y: 2.001e-05

X: 5Y: 2.001e-05

X: 2Y: 1.801e-05

X: 4Y: 1.211e-06

X: 6Y: 1.211e-06X: 2

Y: 1e-06

Fig. 5: Analysis of k values for Case 3 and 4 due to truncationof the infinite series to a finite series for different power values.

bounded as:Pe ≤ Ps + Plide − PsPlide . (7)

The BER bounds depend on the different OSM schemes.Depending upon the number of lasers activated and whichtransmit lasers are active, the corresponding channel gains willvary and accordingly the integration over those channel gainswill be done. The predefined set of possible symbol vectorsalso vary according to the schemes, thereby changing the BERvalues.

V. COMPARISON OF PERFORMANCE METRICS

Power consumption, cost and complexity analysis of theproposed methods are carried out in this section. Powerconsumption of optical chain and optical switch are consideredas Poc and Psw [38] respectively. Note that αtr is the slopedependent load factor and Pt is the total transmitted opticalpower. Power consumption of OSM, OESM, OGSM andOIQSM can be explained as Eq. (8), (9), (10), and (11)respectively.

POSMtot = Poc + αtr Pt + Psw NOSM

sw , (8)

POESMtot = 2Poc + αtr Pt + Psw NOESM

sw , (9)

POGSMtot = Poc Nopt + αtr Pt + Psw NOGSM

sw , (10)

POIQSMtot = 4Poc + αtr Pt + Psw NOIQSM

sw . (11)

Power consumption of optical MIMO is given by PMIMOtot =

Poc NL + αtr Pt . Number of switches are different for eachscheme and number of switch requirement for OSM, OESM,OGSM, and OIQSM to achieve a spectral efficiency of η, aregiven by Eq. (12), (13), (14) and (15) respectively.

NOSMsw = 2(η−log2 (M )−1) , (12)

NOESMsw = 2(η−log2 (M ))/2−1 , (13)

NOGSMsw = 1/4

(1 +

√1 + 4 × 2((η−2log2 (M ))+1)

), (14)

NOIQSMsw = 1/4

(1 +

√1 + 4 × 2((η−2log2 (M ))/2+1)

). (15)

The derivation of number of switches required for all theseschemes is provided in Appendix B. The hardware cost iscalculated for all the schemes. Let us assume that Co , CS/P

and Csw are the costs of optical chain, serial to parallelconverter, and optical switch respectively. Total hardware costsof OSM, OESM, OGSM and OIQSM are given by Eq. (16),(17), (18), and (19) respectively.

COSM = Co + CS/P + Csw NOSMsw , (16)

COESM = 2Co + CS/P + Csw NOESMsw , (17)

COGSM = Co Nopt + CS/P + Csw NOGSMsw , (18)

COIQSM = 4Co + CS/P + Csw NOIQSMsw . (19)

Total cost of optical spatial multiplexing based MIMO systemis given by CMIMO

tot = Co NL . The number of switches requiredfor OGSM, OESM, and OIQSM are much lesser than therequirement in OSM for a particular spectral efficiency andM-ary modulation scheme. Hence the switch cost and powerconsumption of switches are much lesser for higher spectralefficiencies in OESM, OGSM and OIQSM. An example isconsidered for η = 8 bpcu, NL = 6 and M = 4. ForOESM 4 switches are required, for OGSM 3 switches arerequired (assuming Nopt = 2), for OIQSM switch requirementis 2, while the switch requirement is 32 for OSM. Theexample is pictorially illustrated in Fig. 6 where the overallpower consumption and cost of the proposed methods arecompared with that of OSM and optical MIMO (Optical spatialmultiplexing (SMX)) techniques. The parameters consideredare Poc = 53W, αtr = 3.1, Pt = 6.3W, Psw = 0.2W,Co =

1000$,CS/P = 2$,Csw = 220$. Please note that these aretypical values obtained through quotations and websites [39]–[48]. The cost may vary according to location, however theproportion of the values with respect to each other is nearlycorrect to the best of the author’s knowledge. An opticalchain comprises of intensity modulator, RF modulator, DCbias adder, pulse shaping and I/Q modulator blocks. Hence thecost is calculated accordingly. For the particular example asshown in the figure (for η = 8 bpcu), the power consumptionof OSM is relatively lesser than the other schemes, howeverthis trend is not fixed for all values of spectral efficiencyand modulation scheme. As the spectral efficiency increases,the number of switches in OSM increase by a power of 2whereas for other schemes, the growth in switches is linear.

8

For example if η = 20 bpcu and M = 4, number of switchesin OSM is 131072 while number of switches in OESM is256, number of switches in OGSM is 362, and number ofswitches in OIQSM is 11. In this case, we can clearly see thatthe total power consumption of OSM will exceed that of theother advanced OSM schemes due to excessive requirementof switches. The slight increase in the power consumption ofmultiple optical chains in advanced OSM schemes is negatedby the excessive power consumption of switches in OSM forvery high spectral efficiency values. The total cost of OSMalso exceeds the cost of other advanced OSM schemes forsuch high spectral efficiency values. Thus, we can concludefrom this detailed analysis that as spectral efficiency valueincreases, advanced OSM schemes perform better than OSMboth in terms of power consumption and cost.

OSM OESM OGSM OIQSM Optical SMX0

500

1000

1500

2000

2500

3000

3500

Total power consumption (Watts)

OSM OESM OGSM OIQSM Optical SMX

×104

0

1

2

3

4

5

6

7

Total cost (dollars)

78.9

3411

8042

126.3 126.13231.9

(a) (b)

4442

2662

64002

2882

Fig. 6: (a) Power consumption and (b) cost comparison of theproposed methods.

VI. RESULTS

Strong turbulence conditions (α = 4.2, β = 1.4) are consid-ered for all simulations. The separation distance between twonodes is considered to be 2 Km. Single photodetector, ND = 1is considered for analysis. Aperture size of the transmitteraperture is 7.5 cm, separation between two transmitter apertureis 40 mm, diameter of receiver aperture is 20 cm, 2 mradis considered as the divergence angle, responsivity (Re) is0.5, 0.8 is the optical efficiency of the system (Oe f f ), 30 cmis considered as the corresponding jitter standard deviation,corresponding beam radius at 1 Km is 2.5 m, and standarddeviation of noise is 10−7. In FSO systems, transmissionpower is lost as the beam becomes more wider as it travelsthrough free space. Hence, the narrow receiving aperture failsto receive wide beam signals and much of the optical signalis also lost due to misalignment errors. 2 mrad divergenceangle transmitter is chosen such that it can allow enoughmargin of error in pointing angle to maintain the target ofthe circular beam. Such a 2 mrad divergence angle will give abeam diameter of 2.5 m at 1 Km. A wider beam angle at thetransmit laser can be considered for a wider aperture diameterat the receiver, but it will also mean more transmission power

loss. The transmit and receive aperture diameters are relatedby the following equation [49]:

AR

AB

=

(DR

DT + 100 × d × θ

)2

, (20)

where DR and DT are the diameters of receiving and trans-mitting apertures, θ is the divergence angle, d is the distanceat which beam radius is calculated, AR and AB are the area ofreceiver and area of transmitted beam respectively. From thisequation, the ratio of areas is kept at -20 dB and the values oftransmit and receive aperture diameters are calculated as 20cm and 7.5 cm respectively. These values basically determinethe value of ζ2 which indicate the effect of pointing error.Since we wanted to analyze advanced OSM schemes in theworst atmospheric conditions, we have chosen such a high ζ2

value and accordingly generated the required aperture diametervalues. For other turbulence conditions, the parameters canbe varied and investigations can be done for the performanceanalysis of OSM schemes with different parameters. Thedistance between two transmit apertures are chosen such thatit exceeds the spatial coherence distance in order to ensurethat the channel gains are independent of each other.

For Monte Carlo simulations, 106 G-G channel realizationsare generated. The source bits are randomly generated and thesymbol mapping is done according to the particular scheme.Thus for 106 bits, number of erroneous bits are computed byactivating the corresponding lasers and transmitting the bitsacross the particular channel. At the receiver, symbol detectionis done in two steps- first the laser activation bits are estimatedto know the particular laser index and in the next step thesymbol mapping bits are decoded.

The performance of the proposed methods- OESM andOGSM are compared with that of OSM and other techniquesavailable in the open literature, in terms of BER as shown inFig. 7. The results from [14], [50], [51] are used for perfor-mance comparison. In [50], single-input-single-output (SISO)optical system has been implemented over G-G channel em-ploying OOK and assuming strong turbulence conditions. BERhas been computed over G-G channel assuming light fogand strong turbulence conditions, and 4-QAM modulation in[51]. In [14], BER has been computed for SSK based FSOsystem under strong turbulence (for NL = 4, ND = 1). Theresults of all these methods are obtained for our values oftransmit optical power and plotted in the figure. We haveconsidered NL = 4 and spectral efficiency of 6 bpcu forour analysis of the proposed methods. We have consideredNopt = 2, and 4-QAM modulation is used for OGSM andOESM, while OSM uses 16-QAM to achieve the same spectralefficiency. OESM and OGSM easily outperform OSM as OSMuses a higher modulation scheme which has closer spacingbetween constellation points, thus causing more error. OESMoutperforms OGSM as single laser activation also occurssometimes in OESM, while OGSM always activates 2 lasersfor the above scenario, hence OESM suffers from less opticalinterference as compared to OGSM. It is to be noted thatwhen more lasers are activated, more number of laser indiceshave to be estimated correctly at the receiver leading to morechances of error. Thus, increase in spatial domain error leads to

9

overall increment in BER. It is evident from the figure that ourproposed methods perform better than the existing methods.

Transmit optical power (dBm)0 10 20 30 40

BE

R

10-6

10-5

10-4

10-3

10-2

10-1

OSM upper boundOSM simulationOGSM upper boundOGSM simulationOESM upper boundOESM simulationSSK upper bound4-QAM FSO upper boundSISO upper bound

Fig. 7: Performance comparison of the proposed methods forspectral efficiency of 6 bpcu.

The comparison of OGSM, OESM, OIQSM, and OSM iscarried out in terms of BER in Fig. 8 for NL = 4 and a spectralefficiency of 8 bpcu. OGSM uses Nopt = 3 and 4-QAM modu-lation while OESM utilize 16-QAM modulation. OIQSM uses4-QAM modulation while OSM uses 64-QAM modulation.OSM uses the highest modulation scheme, thereby yieldingthe most inferior performance. OESM uses single or doublelaser activation, but requires a higher modulation scheme thanOGSM and OIQSM, thereby leading to inferior performancein OESM. OIQSM and OGSM uses the same modulationscheme, but OIQSM uses only 2 optical chains while OGSMuses 3 optical chains, leading to improved performance ofOIQSM over OGSM. It can be observed that BER upper boundis tight in comparison to the simulation results.

Transmit optical power (dBm)

0 5 10 15 20 25 30 35 40 45

BE

R

10-6

10-5

10-4

10-3

10-2

10-1

OESM upper bound

OESM simulation

OIQSM upper bound

OIQSM simulation

OGSM upper bound

OGSM simulation

OSM upper bound

OSM simulation

Fig. 8: Performance comparison of the proposed methods forspectral efficiency of 8 bpcu.

The performance of OESM and OGSM for different pa-rameter variations are compared in Fig. 9. It is observedthat the performance of OGSM and OESM degrades with anincrease in spectral efficiency and hence higher target datarates can be achieved at the cost of higher BER. With increasein modulation order, the BER performance also degrades.

For example, OESM uses 16-QAM for NL = 4, ηSE = 8bpcu, while OESM uses 4-QAM for NL = 4, ηSE = 6 bpcu.Similarly, OGSM with ηSE = 6 bpcu and OGSM with ηSE = 8bpcu uses 4-QAM but have different number of optical chains.As Nopt value increases in OGSM, more lasers are activeand the performance degrades due to increase in opticalinterference. It is pertinent to note that OGSM having 3 opticalchains performs better than OESM with 2 optical chains at thesame spectral efficiency of 8 bpcu. This is because OGSMuses 4-QAM and OESM uses 16-QAM to attain the samespectral efficiency. As constellation size increases, the symbolvector spacing decreases leading to more chances of error.Again for same spectral efficiency of 6 bpcu, OESM having 2optical chains performs better than OGSM having 2 opticalchains. This is because of the fact that OESM sometimesactivates single laser and optical chain also depending onthe message bits, which reduces the optical interference ascompared to OGSM with 2 optical chains, where 2 lasersand 2 optical chains are always activated. In all the figures,the analytical upper bounds are tight in comparison to MonteCarlo simulations.

Transmit optical power (dBm)0 5 10 15 20 25 30 35 40 45

BE

R

10-6

10-5

10-4

10-3

10-2

OGSM NL

=4 Nopt

=2, η=6 upper bound

OGSM NL

=4 Nopt

=2, η=6 simulation

OGSM NL

=4 Nopt

=3, η=8 upper bound

OGSM NL

=4 Nopt

=3, η=8 simulation

OESM NL

=4, η=6 upper bound

OESM NL

=4, η=6 simulation

OESM NL

=4, η=8 upper bound

OESM NL

=4, η=8 simulation

Fig. 9: Effect of parameter variations on system performance.

VII. CONCLUSION

Three advanced schemes of OSM named OESM, OGSM,and OIQSM have been proposed for FSO communication overG-G channel with pointing errors. It has been inferred that theproposed methods perform better than OSM in terms of BER,cost, and power consumption for higher spectral efficiencies.The proposed methods can attain the same spectral efficiencyas that of OSM by using a lower modulation scheme whichgives improved BER performance. The less number of opticalswitches required in these advanced OSM schemes offerbenefits in terms of total cost and power consumption for highspectral efficiencies. The derived BER upper bounds are tightfor any SNR range. Such analysis along with investigationof various system parameters can be useful to FSO systemdesigners who want to incorporate these technologies in ap-plications of fifth generation (5G) networks like smart health-care, smart transport, smart sensors for surveillance and airquality control, etc. These works can be extended in future byapplying these advanced OSM schemes for FSO cooperativecommunication. Outage probability and asymptotic analysis

10

can be carried out as well. The effect of receiver diversity onsystem performance can be another scope of future work.

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APPENDIX ACALCULATION OF SPATIAL DOMAIN PEP

The PEP calculation of the spatial domain in all OSM basedschemes is done in the following manner. The PEP for spatialdomain part (where laser index j is estimated as i) is givenby:

PEP( j → i) = Q(

1ND

√√√γe f f

2

ND∑l=1

|hl j − hli |2). (21)

To calculate the PEP for spatial domain part, the PDF of theabsolute value of the difference between two G-G randomvariables need to be calculated. Let Yl = |hl j − hli |, Xl = hl j −hli and γl be a new random variable such that γl = Y 2

lγe f f and

γz =∑γl . hli and hl j are two independent random variables

following non negative G-G distribution (as in Eq. (1)), so thePDF of Xl is given by:

fXl(x) =

∫ ∞

0 fhl(x + hi ) fhi

(hi )dhi , x ≥ 0∫ ∞−x

fhl(x + hi ) fhi

(hi )dhi , x < 0 .(22)

Now the PDF of Yl can be written as:

fYl(y) = fXl

(y) + fXl(−y) , (23)

where fXl(y) = fXl

(x) for x > 0 and fXl(−y) = fXl

(x)for x < 0. Using the PDF of G-G distribution, the PDF offXl

(x) can be written as:

fXl(x) =

(αβζ2)2

(A0hlΓ(α)Γ(β))2

∫ ∞0

G 3,01,3

(ζ 2

ζ 2−1,α−1,β−1

∣∣∣∣∣ αβ(x + hi )A0hl

)× G 3,0

1,3

(ζ 2

ζ 2−1,α−1,β−1

∣∣∣∣∣ αβhi

A0hl

)dhi . (24)

We can evaluate fXl(x) for x ≥ 0 (using [52] Eq. (2.24.1.3))

[53]:

fXl(x) = fXl

(y) =αβξ4

A0hl (Γ(α)Γ(β))2

∞∑k=0

(−αβy/(A0hl ))k

k!

× G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1) . (25)

Similarly, PDF of fXl(−y) can be calculated. Finally, the PDF

of Yl (from Eq. (23)) can be written as:

fYl(y) =

2αβξ4

A0hl (Γ(α)Γ(β))2

∞∑k=0

(−αβy/(A0hl ))k

k!

×G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1)] . (26)

The PDF of γl (considering change in random variables) iscomputed as:

fγl (γl ) =αβξ4

A0hl (Γ(α)Γ(β))2√γl

∞∑k=0

(−αβ√γl/(A0hl ))k

γ (k+1)/2e f f

k!

× G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1) . (27)

Now the moment generating function (MGF) of γl is writtenas:

Mγl (s) =∫ ∞

0e−sx fγl (x)dx . (28)

The MGF of γl can be obtained with a few mathematicalcomputations and can be written as:

Mγl (s) =αβξ4

A0hl (Γ(α)Γ(β))2s(k+1)/2

∞∑k=0

(−αβ/(A0hl ))k

γ (k+1)/2e f f

k!

× G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1) . (29)

The channels are independent of each other, so the MGF ofγz =

∑γl can be evaluated as:

Mγz(s) = CND s−

kND+ND2

( ∞∑k=0

(−αβ/(A0hl ))k

γ (k+1)/2e f f

k!

× G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1))ND

, (30)

where C =αβξ4

A0hl (Γ(α)Γ(β))2 . The PDF of fγz(γz ) is now

evaluated by computing the inverse Laplace transform of MGFand is written as:

fγz(γz ) =

(C∞∑k=0

(−αβ/(A0hl ))k

γ (k+1)/2e f f

k!

× G 3,45,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1))ND γkND+ND−2

2z

( kND+ND−22 )!

.

(31)

Now by using Eq. (31), the PEP can be evaluated as:

PEP( j → i) =∫ ∞

0Q

( 1ND

√γzγe f f

2

)fγz

(γz )dγz

=

(C∞∑k=0

(−αβ/(A0hl ))k

γ (k+1)/2e f f

k!G 3,4

5,5

(0,k+1−ζ 2 ,k+1−α,k+1−β,ζ 2

ζ 2−1,α−1,β−1,k−ζ 2 ,k

∣∣∣∣ 1))ND

×1

2( kND+ND−22 )!

∫ ∞0

γkND+ND−2

2z er f c

( 12ND

√γzγe f f

)dγz .

(32)

12

By using the relation,∫ ∞

0 xa−1

2 er f c(b√

x)dx = 2 Γ( a+22 )

(a+1)ba+1√π,

the APEP of the spatial domain part is obtained as in Eq. (4).

APPENDIX BNUMBER OF SWITCHES CALCULATION FOR ADVANCED

OSM SCHEMES

For calculation of number of switches, first the number oflasers for each scheme is to be determined. One switch canconnect two lasers, hence number of switches will be half ofthe number of lasers. For OSM the spectral efficiency is η =log2(NL ) + log2 M . Number of lasers required can be writtenas NL = 2η−log2 M . Hence number of switches for OSM isNOSMsw = NL/2 = 2η−log2 M−1. For OESM, the number of

switches can be derived in a similar manner:

η = log2 N2L + log2 M

2 log2 NL = η − log2 M

NL = 2(η−log2 M )/2

NOESMsw = 2(η−log2 M )/2−1 . (33)

For OGSM, it is considered that Nopt = 2. For other values ofNopt number of switches can also be done in a similar manner.Thus the derivation of number of switches for Nopt = 2 isshown below:

η = blog2

(NL

Nopt

)c + Nopt log2 M

2η−Nopt log2 M =NL!

Nopt !(NL − Nopt )!

Considering Nopt = 2, 2η−2 log2 M =NL (NL − 1)

2N2L − NL − 2η−2 log2 M+1 = 0

NL =1 +√

1 + 4 × 2(η−log2 M+1)

2

NOGSMsw =

1 +√

1 + 4 × 2(η−log2 M+1)

4. (34)

Similarly for OIQSM, number of switches required is calcu-lated as shown below:

η = 2 log2 M + 2blog2

(NL

2

)c

log2

(NL

2

)=η − 2 log2 M

2NL (NL − 1)

2= 2(η−2 log2 M )/2

N2L − NL − 2((η−2 log2 M )/2+1) = 0

NL =1 +√

1 + 4 × 2(η−2 log2 M )/2+1

2

NOIQSMsw =

1 +√

1 + 4 × 2(η−2 log2 M )/2+1

4. (35)

Please note that only the positive square root is consideredfor switch calculation in OGSM and OIQSM as number ofswitches cannot be less than 1 or negative. The exact numberof switches are rounded off to the nearest integer value.

Anirban Bhowal obtained his Bachelor of Tech-nology degree in Electronics and CommunicationEngineering from Heritage Institute of Technology,Kolkata in 2012. He received his Master of Technol-ogy degree in Communication Engineering from In-dian Institute of Information Technology, Allahabadin 2015. He graduated with a PhD degree fromthe Department of Electronics and Electrical Engi-neering, Indian Institute of Technology, Guwahati in2020. He was also the recipient of best PhD thesisaward from IIT Guwahati. His research interests in-

clude spatial modulation in RF and FSO systems, UOWC and FSO cooperativesystems, hybrid FSO/RF communication and BAN communication. He hasserved as the referee of several reputed journals including IEEE Comm.Letters, IEEE Wireless COmm. Letters, IEEE Transactions on Communica-tions, IEEE Internet of Things, IET Radar, Sonar and Navigation, ElectronicsLetters, IET Journal of Engineering, Journal of Physical CommunicationElsevier, OSA Applied Optics etc.

Rakhesh Singh Kshetrimayum (S’01-M’05-SM’14) received the Ph.D. degree from the Schoolof Electrical and Electronics Engineering (EEE),Nanyang Technological University, Singapore,and the B.Tech. degree in Electrical Engineering(EE) from the Indian Institute of Technology (IIT)Bombay, India. Since Sept. 2005, he has beena faculty member with the Department of EEE,IIT Guwahati, presently working as a Professorand was the former Head of Centre for CareerDevelopment. Before joining IIT Guwahati, he

did postdoctoral research at the Department of EE, Pennsylvania StateUniversity, USA and at the Department of Electrical CommunicationEngineering, Indian Institute of Science, Bangalore and worked as a SoftwareEngineer with the Mphasis, Pune. His research interests are in the broadareas of printed antennas, passive microwave devices, spatial modulation,cooperative communications, and optical wireless communications. Hehas authored the book Fundamentals of MIMO Wireless Communications(Cambridge University Press, 2017), co-authored the book MIMO WirelessCommunications Over Generalized Fading Channels (CRC Press, 2017) andpublished numerous journal and conference articles in the area of his researchinterests. Prof. Kshetrimayum is on the editorial board of AEU InternationalJournal of Electronics and Communications, International Journal of RFand Microwave Computer-Aided Engineering and Physical Communication.He has served on the committees of several IEEE conferences in variouscapacities such as TPC Chair (Communications track) of the NationalConference on Communications (NCC) 2016. He is a Fellow of the IET, UKand a Senior Member of the IEEE, USA.