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Free-Space Optical Communication With SpatialModulation and Coherent Detection Over H-K

Atmospheric Turbulence ChannelsKostas P. Peppas, Senior Member, IEEE, and P. Takis Mathiopoulos, Senior Member, IEEE

Abstract—The use of optical spatial modulation (OSM) is pro-posed as a simple low-complexity means of achieving spatial diver-sity in coherent free-space optical (FSO) communication systems.In doing so, this paper presents a generic analytical framework forobtaining the average bit error probability (ABEP) of uncoded andcoded OSM with coherent detection in the presence of turbulence-induced fading. Although the framework is general enough to ac-commodate any type of channel models based on turbulence scat-tering, the focus of the analysis is on the H-K distribution as suchmodel is very general and valid over a wide range of atmosphericpropagation conditions. Using this framework, it is shown thatOSM can offer comparable performance with conventional coher-ent FSO schemes employing spatial diversity at the transmitteror the receiver only, while outperforming the latter in terms ofspectral efficiency and hardware complexity. Furthermore, vari-ous numerical performance evaluation results are also presentedand compared with equivalent results obtained by Monte Carlosimulations which verify the accuracy of the derived analyticalexpressions.

Index Terms—Atmospheric turbulence, average bit error prob-ability, coherent detection, free space optical communication sys-tems, H-K distribution, multiple-input multiple-output (MIMO)systems, optical spatial modulation.

I. INTRODUCTION

FREE-SPACE optical (FSO) communication systems haverecently attracted great attention within the research com-

munity as well as for commercial use. FSO systems can pro-vide ultra-high data rates (at the order of multiple gigabit persecond), immunity to electromagnetic interference, excellentsecurity and large unlicensed bandwidth, i.e., hundred or eventhousand times higher than radio-frequency (RF) systems, alongwith low installation and operational cost [1].

The challenge in employing such systems is that FSO links arehighly vulnerable due to the detrimental effects of attenuationunder adverse weather conditions (e.g., fog), pointing errorsand atmospheric turbulence [1]. One method to improve thereliability of the FSO link is to employ spatial diversity, i.e.multiple-lasers and multiple-apertures to create a multiple-inputmultiple-output (MIMO) optical channel. Because of its lowcomplexity, spatial diversity is a particularly attractive fading

Manuscript received March 23, 2015; revised June 21, 2015; accepted July14, 2015.

K. P. Peppas is with the Department of Informatics and Telecommunications,University of Peloponnese, Tripoli 22100, Greece (e-mail: peppas@uop.gr).

P. T. Mathiopoulos is with the Department of Informatics and Telecom-munications, National and Kapodistrian University of Athens, Athens 15784,Greece (e-mail: mathio@di.uoa.gr).

Digital Object Identifier 10.1109/JLT.2015.2465385

mitigation technique and performance enhancements have beenextensively studied in many past research works in the field ofFSO communications (e.g., see [2]–[5]).

In order to evaluate the impact of atmospheric turbulence onthe performance of OSM, accurate models for the fading dis-tribution are necessary. For example, the lognormal distributionis often used to model weak turbulence conditions whereas thenegative exponential and the K-distribution are used to modelstrong turbulence conditions [6]. Other more general statisti-cal models have also been proposed to model scintillation overall turbulence conditions, such as, for example, the Gamma–Gamma [7], the lognormal-Rice (or Beckmann) [8] the homo-dyned K distribution (H-K) [9] and the I-K [10]–[12] distribu-tions. The validity of these models is based on the argumentthat scintillation is a doubly stochastic random process mod-elling both small and large scale turbulence effects. Besides,they agree well with measurement data and simulations for awide range of turbulence conditions.

In this paper, the H-K distribution is adopted to modelturbulence-induced fading [9]. The main reason for this choiceis the fact that this distribution is based on a very general scat-tering model which is valid over a wide range of atmosphericconditions. It is also noted that the H-K distribution generalizesexisting models such as the K-distribution. The H-K distributionmodels the field of the optical wave as the sum of a determinis-tic component and a random component, the intensity of whichfollows the Rice (Nakagami-n) distribution. The average inten-sity of the random portion of the field is treated as a fluctuatingquantity [10]. It is important to underline that, to the best of ourknowledge, in the open technical literature there have been nopapers published analyzing and evaluating the performance ofFSO systems over such channels, because of the complicatedmathematical form of the required for the analysis probabilitydensity functions (PDF).

Depending on their detection, FSO systems can be classifiedinto two main categories, namely non-coherent (direct detec-tion) and coherent (heterodyne detection) systems [13]–[17].Non-coherent FSO systems employing intensity modulationand direct detection (IM/DD) have the optical power outputof a source varied in accordance with some characteristic ofthe modulating signal. A direct detection receiver only respondsto the intensity of the received field and is oblivious to thereceived signal phase. Coherent FSO systems encode the infor-mation bits directly onto the electric field of the optical beam.At the receiver, a local oscillator (LO) is employed to extractthe information encoded on the optical carrier electric field. On

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the one hand, coherent FSO systems can provide significantperformance enhancements due to spatial temporal selectivityand heterodyne gain in comparison to direct detection systems.Moreover, they are more versatile as any kind of amplitude,frequency, or phase modulation can be employed. On the otherhand, the implementation of coherent receivers is more complexas the LO field should be spatially and temporally coherent withthe received field. Throughout this work, coherent FSO systemsare only considered.

Recently, the so-called optical spatial modulation (OSM)has emerged as a power- and bandwidth-efficient single-carriertransmission technique for optical wireless communication sys-tems [18]–[20]. This spatial diversity scheme, initially proposedin [21] and further investigated in [22] and[23], employs a sim-ple modulation mechanism that foresees to activate just one outof several MIMO transmitters at any time instant and to use theindex of the activated transmitter as an additional dimension forconveying implicit information. It has been shown that OSMcan increase the data rate by base two logarithm of the numberof transmit units [18]. Also, OSM can increase the data rate bya factor of 2 and 4, as compared to on–off keying and pulse po-sition modulation, respectively [18], [19]. It is underlined thatsuch performance gains are obtained with a significant reductionin receiver complexity.

Because of the above mentioned advantages of OSM overother more conventional transmission schemes and given thewide applicability of FSO, it is important to investigate thepotential performance enhancements obtained by incorporatingOSM in FSO systems. However, in general, this research topichas not been thoroughly investigated. In fact, only recently,there have been papers published in the open technical litera-ture dealing with performance analysis studies of FSO systemsemploying spatial modulation and operating in the presence ofatmospheric turbulence, e.g., see [24] and [25]. Specifically, in[24], the combination of subcarrier IM and spatial modulationwith receiver diversity was proposed to enhance the perfor-mance of IM/DD FSO systems. In [25], another IM/DD basedsystem FSO system which combines antenna shift keying withjoint pulse position and amplitude modulations was considered.For this system, which was denoted as spatial pulse positionand amplitude modulation (SPPAM), the atmospheric turbu-lence channel was modeled as log-normal or Gamma–Gammadistributions and was evaluated, in terms of bounds, for un-coded and coded signals. Average bit error probability (ABEP)performance evaluation results have shown that SPPAM offers acompromise between spectral and power efficiencies as well asa certain degree of robustness against atmospheric turbulence.Despite these two papers which deal with non-coherent detec-tion schemes, the potential enhancements of OSM on improv-ing the performance of FSO systems with coherent detectionstill remains an open research topic which, to the best of ourknowledge, has not been addressed so far in the open technicalliterature.

Motivated by the above, in this paper we present for the firsttime a generic analytical framework which can be used to ac-curately obtain the performance of outdoor OSM with coherentdetection in the presence of turbulence-induced fading. More

TABLE ILIST OF MATHEMATICAL NOTATIONS

j2 = −1 denotes the imaginary unit|z | denotes the magnitude of the complex number z

�{z} denotes the real part of the complex number z

�{z} denotes the real part of the complex number z

f (x) = o[g(x)] as x → x0 if limx →x 0f (x )g (x ) = 0

‖ · ‖2F denotes the square Frobenius norm

(·)T denotes the matrix transpose∗ denotes convolutionE〈·〉 denotes expectationfX (·) denotes the PDF of the RV X

FX (·) denotes the cumulative distribution function of the RV X

MX (·) denotes the MGF of the RV X

Ia (·) is the modified Bessel function of the first kind and order a [26, eq. (8.431)]Ka (·) is the modified Bessel function of the second kind and order a [26, eq. (8.432)]Γ (x) =

∫ ∞0 exp(−t)tx −1 dt is the Gamma function [26, eq. (8.310/1)]

Q(x) = 1√2 π

∫ ∞x exp(−t2 /2)dt is the Gauss Q-function

Wp , q (·) is the Whittaker function [26, eq. (9.220)]Pr{·} denotes the probability operator· denotes estimated value at the receiver side

specifically and within this new analytical framework, the mainnovel research contributions of the paper are as follows:

1) New analytical expressions for the ABEP of coherent OSMunder turbulence conditions modeled by the H-K distribu-tion are derived. When the transmitter is equipped withtwo apertures the resulting analytical expressions are ex-act, whereas for an arbitrary number of transmit aperturestight upperbounds can be obtained.

2) ABEP bounds for coded OSM systems are derived andthe performance enhancements when channel coding isemployed are presented and analyzed.

The ABEP of OSM is also compared to that of conventionalFSO schemes with transmit or receive diversity only, i.e., whenmaximal ratio combining (MRC), selection combining (SC) orAlamouti-type space-time block codes (STBC) are employed.It is finally noted that the theoretical analysis is substantiated bycomparing the theoretical and equivalent simulated performanceevaluation results obtained by means of Monte Carlo techniques.

The paper is organized as follows. After this introduction,Section II outlines the system and channel models. In Section IIIanalytical expressions for the ABEP of uncoded OSM systemsare presented. Asymptotic ABEP expressions are also derived,wherefrom the diversity gain of coherent OSM can be readilydeduced. The performance of coded OSM systems is discussedin Section IV. In Section V an overview of the conventional di-versity techniques for coherent OSM is presented. In Section VIthe various performance evaluation results and their interpreta-tions as well as comparisons are presented. Finally, concludingremarks can be found in Section VII. Notations: A comprehen-sive list of all mathematical notations used in this paper can befound in Table I.

II. SYSTEM AND CHANNEL MODEL

In this section, a detailed description of the OSM FSO sys-tem model, i.e., transmitter, channel and receiver is provided.Moreover, the H-K distribution is introduced and analytical

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expressions for its parameters in terms of equivalent physical pa-rameters of the turbulence phenomenon, such as the refractive-index structure parameter, optical wave number, and propaga-tion path length, are derived.

A. Preliminaries

Let us consider an M × N MIMO FSO system with M trans-mit units (lasers) and N coherent receivers. It is assumed thatthe receiving apertures are separated by more than a coherencewavelength to ensure the independency of fading channels. Thebasic principle of OSM modulation is as follows [18], [20]:

1) The transmitter encodes blocks of log2(M) data bits intothe index of a single transmit unit. Such a block of bitsis hereafter referred to as “message” and is denoted bybm , ∀m = 1, 2, . . . ,M . It is assumed that the M mes-sages are transmitted by the encoder with equal probabil-ity and that the related transmitted signal is denoted byEm = Em exp(jφbm

). During each time slot, only onetransmitter �, where � = 1, 2, . . . ,M is active for datatransmission. The information bits are modulated on theelectric field of an optical signal beam through an externalmodulator. During each particular time slot, the remainingtransmit lasers are kept silent, i.e., they do not transmit.

2) At the receiver, the incoming optical field is mixed with aLO field and the combined wave is first converted by thephotodetector to an electrical one. A bandpass filter is thenemployed to extract the intermediate frequency (IF) com-ponent of the total output current. Finally, a N -hypothesisdetection problem is solved to retrieve the active transmitunit index, which results in the estimation of the uniquesequence of bits emitted by the transmitter.

B. Receiver Structure

The received electric field at the aperture plane of the nthreceiver after mixing with a LO beam, can be expressed as [16],[17]

en (t) =√

2PtZ0Em hm,n cos(ω0t + φm,n + φbm)

+√

2PLOZ0 cos(ωLOt). (1)

In the above equation, Pt is the transmit laser power, Z0 is thefree space impedance, hm,n and φm,n denote the magnitude andthe phase of the complex channel coefficient between the mthtransmit and the nth receive aperture, respectively. Furthermore,PLO denotes the power of the LO, ωLO = ω0 + ωIF where ω0and ωIF are the carrier and the intermediate radian frequencies,respectively.

The output current of the nth photodetector can be mathe-matically expressed as [16], [17]

in (t) =R

Z0[en (t)]2 (2)

where R = ηqe/(hν0) is the responsivity of the photodetectorwith qe = 1.6 × 10−19 Cb is the charge of an electron, h =6.6 × 10−34 J · s is the Planck constant, η is the photodetectorefficiency, and ν0 = ω0/(2π) is the optical center frequency.

Expanding (2) and ignoring the double-frequency terms that arefiltered out by the bandpass filter, the resulting photocurrent canbe expressed as

in (t) = RPtE2m h2

m,n + RPLO

+2R√

PtPLOEm hm,n cos(ωIFt − φm,n − φbm)

= iDC(t) + iAC(t). (3)

In (3), iDC(t) = RPtE2m h2

m,n + RPLO is the dc componentgenerated by the signal and LO fields, respectively, iAC(t) =2R

√PtPLO cos(ωIFt − φm,n − φbm

) is the ac component inthe received photocurrent which, unlike for direct detection,contains information about the frequency and phase of the re-ceived signal. It is assumed that for coherent detection the IFωIF is nonzero, so that the signal power can be expressed asPs = 2R2PtPLOE2

m h2m,n .

As in [13]–[17], we also consider that PLO � Ps and thus, thedc photocurrent can be approximated as iDC(t) ≈ RPLO. Thephotodetection process is impaired by shot noise with varianceσ2

shot,L = 2qeRPLOBe where Be is the electrical bandwidth ofthe photodetector. It is also noted that because of the large valueof RPLO the photocurrent due to thermal noise and the darkcurrent can be ignored [16].

Following [15] and[17], the sufficient statistics at the nthcoherent receiver can be expressed as

yn =√

μhm,nEm exp[j(φm,n + φbm)] + zn (4)

where μ = RPt/(qeBe) is the average signal-to-noise ratio(SNR) and zn is the noise at the nth receiver. Assuming thatthe LO power is large and the receiver noise is dominated bythe LO related noise terms, the additive white Gaussian noisemodel can be employed as an accurate approximation of thePoisson photon-counting detection model [16], [17]. Thus, zn

can be modeled as a zero-mean unit variance complex Gaussianrandom variable (RV) [17].

Similar to [27], it is assumed that the receiver has knowledgeof the actual fading gains and that the fading process remainsconstant over the bit interval and changes from bit to bit in anindependent manner. At the receiver, the optimal spatial mod-ulation detector estimates the active transmitter index, �, at agiven time slot according to [28]

� = argmax�

py (y|x,H)

= argmin�

{√μ ‖ h�x� ‖2

F −2(yT h�x�

)}(5)

where- x is an M -dimensional vector with elements corresponding

to the electrical field Em exp(jφbm) transmitted over the

optical MIMO channel;

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- H(t) is an N × M optical MIMO channel defined as

H(t) = [h1 ,h2 , . . . ,hM ]

�

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

h11 (t) exp(jφ11 ) . . . h1M (t) exp(jφ1M )

h21 (t) exp(jφ21 ) . . . h2M (t) exp(jφ2M )

.... . .

...

hN 1 (t) exp(jφN 1 ) . . . hN M (t) exp(jφN M )

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(6)

- z is the N -dimensional noise vector;- py (y|x,H) is the PDF of y conditioned on the transmitted

vector x and the channel H.

C. Channel Model

A discrete scattering channel model is considered, where theradiation field of an optical wave at a particular point is assumedto be composed of a number of scattered components that havepropagated through different paths. Under the Ricean assump-tion [10], the complex channel path gains hij (t) between theith transmitter and the jth photodetector can be expressed ashij (t) = hij (t) exp(jωt) where ω is the radian frequency of theoptical wave and

hij (t) = �{hij (t)} + j�{hij (t)}= Aij exp[jθij (t)] + Rij (t) exp[jΦij (t)] (7)

where the term Aij exp[jθij (t)] is a deterministic componentand Rij (t) exp[jΦij (t)] is a circular complex Gaussian RV.Hence, the amplitude Rij is Rayleigh distributed with scaleparameter σ2

ij = bij /2 [10, Eq. (13)] and the phase Φij is uni-formly distributed over [0, 2π). Under the assumption of a dou-bly stochastic scintillation model [10], the effect of randomfluctuations in the turbulence parameters is modeled by allow-ing random variations in the parameter bij of the Rayleigh com-ponent. Following [10], it is further assumed that bij follows aGamma distribution with PDF given by

fbi j(b) =

(αij

b0

)αi j bαi j −1

Γ(αij )exp

(

− αb

b0i j

)

(8)

where α is the shaping parameter which represents the effectivenumber of scatters and b0i j

= E{bij}. Then, the PDF of theirradiance Iij = |hij (t)|2 , fIi j

(I), can be expressed as [10, Eq.(8)]

fIi j(I) =

(αij /b0i j

)αi j

Γ(αij )

×∫ ∞

0bαi j −2 exp

(

−αij b

b0i j

− I + Aij2

b

)

×I0

(2Aij

√I

b

)

db (9)

which is actually the integral representation of the H-K distri-bution [10]. It is noted that, with the exception of the specialcases Aij = 0 or α = 1, fIi j

(I) cannot be expressed in closed

form. Specifically, for Aij = 0 (9) reduces to the K-distributionwhereas for α = 1, (9) reduces to a special case of the I-Kdistribution [10, Eq. (10)].

The νth normalized moment of Iij is given by [10, Eq. (23)]as

E{Iνij}E{Iij}ν

=ν!

ανij (1 + ρij )ν

ν∑

k=0

(ν

k

)Γ(αij + ν − k)

Γ(αij )(αij ρij )ν

ν!(10)

where ρij � A2ij /b0i j

is the coherence parameter, defined asthe power ratio of mean intensities of the constant-amplitudecomponent and random component of the field in (7) [10], [11].Using (10), the scintillation index can be readily calculated as

σ2Ii j

�E{I2ij}E{Iij}2 − 1 =

αij + 2αij ρij + 2αij (1 + ρij )2 . (11)

Under the assumption of plane wave propagation, σ2Ii j

can bedirectly related to atmospheric conditions as [11, Eq. (7), Eq.(9)]

σ2Ii j

≈{

σ21 (1 + 0.5σ2

1 ), σ1 � 1

1 + 0.86/σ4/51 , σ1 � 1

(12)

where σ21 = 1.23C2

ni jk7/6L

11/6ij is the Rytov variance, k =

2π/λ is the optical wave number with λ being the wave-length, Lij is the link distance and Cni j

denotes the index ofrefraction structure parameter. For FSO links which operatenear the ground plane, C2

ni j≈ 1.7 × 10−14 m−2/3 and 8.4 ×

10−15 m−2/3 for the daytime and night time periods, respec-tively [29]. Moreover, σ1 � 1 and σ1 � 1 correspond to weakand strong turbulence conditions, respectively.

Using (12), the parameters of the H-K distribution, α and ρ,can be directly related to physical parameters of the turbulenceby following a similar line of arguments as in [11], where similarresults have been derived for the I-K distribution. In particular,on the one hand, weak turbulence conditions are characterizedin the H-K distribution by large values of ρij . In this case thescintillation index given by (11) can be approximated as

σ2Ii j

≈ 2ρij

, with ρij � 1. (13)

On the other hand assuming strong turbulence conditions, inwhich case ρij goes towards zero, (11) can be approximated as

σ2Ii j

≈ 1 +2

αij, with ρij � 1. (14)

By comparing (13) and (14) with the first and second branchesof (12), respectively, αij and ρij can be obtained as

αij = 2.33σ4/51i j

(15)

ρij =2

σ21i j

(1 + 0.5σ21i j

). (16)

To the best of our knowledge, the relationship of αij and ρij withσ1i j

given by (15) and (16) has not been previously published.

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III. PERFORMANCE ANALYSIS OF UNCODED OSM

In this section, by employing the well-known moment gen-erating function (MGF)-based approach for the performanceanalysis of digital communications over fading channels [30],analytical expressions for the ABEP of uncoded OSM systemswill be derived. In addition, expressions for the diversity andcoding gains of OSM systems are also presented, thus provid-ing useful insight as to how these parameters affect the overallsystem performance.

A. Preliminaries

For M = 2, the conditional bit error probability (CBEP) ofOSM systems when no turbulence induced fading is consideredcan be obtained in closed form as [27]

PE (h1 ,h2) = Q

(√μ

4‖ h1 − h2 ‖2

F

)

. (17)

The squared Frobenius norm in (17) can be expressed as

‖ h1 − h2 ‖2F =

N∑

n=0

|h1,n − h2,n |2 (18)

where hi,n is the nth element of hi , ∀i ∈ {1, 2}. When M > 2transmitters are considered, a tight upper bound for the CBEPof the above system can be obtained as [18, Eq. (7)]

PE (H) ≤ M−1

log2(M)

×M∑

m 1 =1

M∑

m 2 �=m 1 =1

Nb(m1 ,m2)PEP(m1 → m2) (19)

where PEP(m1 → m2) denotes the pairwise error probability(PEP) related to the pair of transmitters m1 and m2 , where m1and m2 ∈ 1, 2, . . . ,M , and Nb(m1 ,m2) is the number of bits inerror which have occurred when the receiver decides incorrectlythat m2 instead of m1 has been active. The PEP(m1 → m2)can be evaluated as [18, Eq. (8)]

PEP(m1 → m2) = Q

(√μ

4‖ hm 1 − hm 2 ‖2

F

)

. (20)

B. MGF-Based Approach

When atmospheric turbulence is included in the analysis theconditional error probabilities in (17) and (19) need to be av-eraged over the elements of the channel matrix H in order toevaluate the ABEP. Without loss of generality, let us consider thecase of a 2 × N MIMO system. Since hi,n are complex Gaus-sian RVs, the difference Δn = h1,n − h2,n is also a complexGaussian RV having mean equal to the difference of the meansof hi,n and variance equal to the sum of variances of hi,n . In or-der to deduce a closed form expression for the ABEP, it is furtherassumed that hi,n have uncorrelated real and imaginary com-ponents with the same variance σ2

n = bn/2. It is noted that thisassumption has been experimentally verified in the past for linkdistances of the order of kilometer and for aperture separationdistances of the order of centimeter [31], [32]. Consequently,

Δn has uncorrelated components too and its squared envelope,|Δn |2 , is characterized by a non-central chi-square PDF

f|Δn |2 (x|bn ) =1

2bnexp

(

−x + A2n

2bn

)

I0

(An

√x

bn

)

(21)

where An = |A2,n exp(jθ2,n ) − A1,n exp(jθ1,n )|. Assumingthat bn follows the Gamma with parameters αn and b0,n , thePDF of |Δn |2 is obtained by averaging (21) with respect to bn ,i.e.,

f|Δn |2 (x) =(αn/b0,n )αn

2Γ(αn )

×∫ ∞

0bαn −2n exp

(

−αnbn

b0,n− x + A2

n

2bn

)

I0

(An

√x

bn

)

dbn . (22)

As was pointed out in [10], f|Δn |2 (x) can’t be expressed inclosed form. Nevertheless, for the special case of αn = 1, i.e.,when one scatterer per branch is considered, and by employing[10, Eq. (10)], this integral can be evaluated in closed form as

f|Δn |2 (x)

=

⎧⎪⎪⎨

⎪⎪⎩

1b0,n

K0

(√2An/b0,n

)

I0

(√2x/b0,n

), x < A2

n

1b0,n

I0

(√2An/b0,n

)

K0

(√2x/b0,n

), x > A2

n .

(23)

Moreover, for the special case where h1,n and h2,n have identi-cal mean value, i.e., when An = 0, (22) yields the well-knownK-distribution with PDF given by

f|Δn |2 (x) = 2(1−αn )/2Γ(αn )(

αnx

b0,n

)(αn −1)/2

×Kαn −1

(√2αnx

b0,n

)

. (24)

By employing the MGF-based approach for the performanceanalysis of digital communications over fading channels, theaverage PEP (APEP) can be obtained as

APEP =1π

∫ π/2

0

N∏

n=1

[M|Δn |2

( μ

8 sin2 θ

)]dθ. (25)

Moreover, using the tight approximation for the Gaussian Q-function presented in [33, Eq. (14)] (i.e., Q(x) ≈ 1/12 exp(−x2) + 1/4 exp(−2x2/3)), an expression accurately approx-imating APEP can be deduced as

APEP ≈ 112

N∏

n=1

[M|Δn |2

(μ

8

)]+

14

N∏

n=1

[M|Δn |2

(μ

6

)].

(26)In the following, analytical expressions for the MGF of |Δn |2will be obtained.

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Proposition 1: An integral representation for the MGF of|Δn |2 can be deduced as

M|Δn |2 (s)=(αn/b0,n )αn

Γ(αn )

×∫ ∞

0

bαn −1

2bs + 1exp

(

− Ans

2bs + 1− αnb

b0,n

)

db.

(27)

Proof: By employing the definition of the MGF, M|Δn |2 (s)can be obtained as

M|Δn |2 (s) =∫ ∞

0exp(−sx)f|Δn |2 (x)dx

=(αn/b0,n )αn

2Γ(αn )

×∫ ∞

0

∫ ∞

0exp

(

−sx − αnb

b0,n− x + A2

n

2b

)

×I0

(An

√x

b

)

bαn −2 dbdx. (28)

By changing the order of integration, the above equation can beexpressed as

M|Δ n |2 (s) =(αn /b0 ,n )α n

2Γ(αn )

∫ ∞

0bα n −2 exp

(

−αn b

b0 ,n

)

[∫ ∞

0exp

(

−sx − x + A2n

2b

)

I0

(An

√x

b

)

dx

]

db.

(29)

The inner integral, i.e., with respect to x can be evaluated byemploying [34, Eq. (3.15.2.2)] as

∫ ∞

0exp

(

−sx − x + A2n

2b

)

I0

(An

√x

b

)

dx

=2b

2sb + 1exp

[1

2An b(2sb + 1)

]

. (30)

Substituting (30) into (29) and after some straightforward math-ematical manipulations, (27) is readily deduced thus completingthe proof. �

The integral in (27) can be accurately approximated by em-ploying a Gauss–Chebyshev quadrature (GCQ) technique as[35]

M|Δn |2 (s)≈(αn/b0,n )αn

Γ(αn )

×J∑

j=0

wjtj

αn −1

2tj s + 1exp

(

− Ans

2tj s + 1− αntj

b0,n

)

(31)

where J is the number of integration points, tj are the abscissasand wj the corresponding weights. In [36, eqs. (22) and (23)],

tj and wj are given as

tj = tan[π

4cos

(2j − 1

2Jπ

)

+π

4

]

(32a)

wj =π2 sin

( 2j−12J π

)

4J cos2[

π4 cos

( 2j−12J π

)+ π

4

] . (32b)

For the special case of An = 0, (27) can be evaluated in closedform, as it will be shown next.

Corrolary 1: For the special case of A = 0 the MGF of|Δn |2 can be deduced in closed form as

M|Δn |2 (s) =(

αn

2sb0,n

)αn /2

exp(

αn

4sb0,n

)

×W−αn /2,(αn −1)/2

(αn

2sb0,n

)

. (33)

This result can be readily deduced by employing the integralrepresentation of the Whittaker W -function given in [26, Eq.(9.222)]. It is noted that (33) is in agreement with a previouslyknown result, namely the analytical expression for the MGF ofthe K-distribution [37, Eq. (4)].

C. Diversity Gain Analysis

The diversity gain of the considered OSM MIMO system canbe obtained by using the approach presented in [38]. In partic-ular, a generic analytical expression, which becomes asymptot-ically tight at high SNR values, will be derived for the APEPappearing in (25), as follows:

Proposition 2: For high SNR values, (25) can be approxi-mated by

APEPμ�1≈

2N −1Γ(N + 1

2

)

√πΓ (N + 1)

[N∏

n=1

c�

](μ

4

)−N

(34)

where

cn =

(An

2

)(αn −1)/2(αn/b0,n )(αn +1)/2

Γ(αn )

×Kαn −1

⎛

⎝

√2Anαn

b0,n

⎞

⎠. (35)

Proof: According to [38, Proposition 3], the asymptotic errorperformance of the OSM system depends on the behavior ofM|Δn |2 (s), as s → ∞. To determine an analytical asymptoticexpression for APEP a Taylor series expansion is employed toapproximate M|Δn |2 (s) as

|M|Δn |2 (s)| = cn |s|−dn + o(|s|−dn ), s → ∞ (36)

where cn and dn are parameters that determine the diversity andcoding gains of the nth diversity branch, respectively. Observe

that since As/(2sb + 1)s→∞≈ A/(2b) and 1/(2sb + 1)

s→∞≈

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1/(2bs), (27) yields

M|Δn |2 (s) ≈(αn/b0,n )αn

2sΓ(αn )

×∫ ∞

0bαn −2 exp

(

− An

2b− αnb

b0,n

)

db. (37)

By employing [34, Eq. (2.2.2.1)], (37) can be solved in closedform yielding

M|Δn |2 (s) ≈(

An

2

)(αn −1)/2(αn/b0,n )(αn +1)/2

sΓ(αn )

×Kαn −1

⎛

⎝

√2Anαn

b0,n

⎞

⎠ . (38)

By comparing (38) and (36) it is readily deduced that dn = 1and cn is given by (35). Thus, by substituting (36) into (25),the asymptotic PEP expression can be obtained as in (34) whichconcludes the proof. �

From (34) it is clear that the diversity gain achieved by theconsidered system is equal to N . It is also evident that the diver-sity gain depends only on the number of the receive aperturesand is independent of the fading severity. This finding is in agree-ment with other equivalent observations reported in [27], [39]and[40], which have been obtained for the case of RF MIMOwireless systems.

It is noted that for the special case An = 0, i.e., when |Δn |2follows the K-distribution, by employing the asymptotic result

Kt(x)x→0≈ (Γ(t)/2) (2/x)t [35], cn can be further simplified as

cn =αn

2b0,n (αn − 1). (39)

IV. PERFORMANCE ANALYSIS OF CODED OSM OVER

TURBULENCE CHANNELS

When coded OSM is employed, the input signal s(t) is firstencoded by a convolutional encoder. The encoded data are inter-leaved by a random block interleaver and transmitted through theoptical wireless channels using spatial modulation. It is assumedthat perfect interleaving at the transmitter and de-interleavingat the receiver is used. Considering maximum likelihood softdecision decoding, the log likelihood ratios (LLRs) for the ithconstellation bit when the �th transmitting antenna is active arecomputed as [18, Eq. (6)]

LLR = logPr{�i = 1|y}Pr{�i = 0|y}

= log

∑�∈Li

1exp

(−‖ y − h� s� ‖2/N0

)

∑�∈Li

0exp

(−‖ y − h� s� ‖2/N0

) (40)

where L ∈ {1 : M} is the set of spatial constellation points, Li1

and Li0 are subsets from L containing the transmitter indices

having “1” and “0” at the ith bit, respectively. The resulting dataare finally decoded by a Viterbi decoder.

A union bound on the ABEP of a coded communicationsystem can be obtained as [30]

Pub ≤ 1n

∑

X

P (X)∑

X �=X ′

q(X,X′)PEP(X,X′) (41)

where P (X) is the probability that the coded sequence X istransmitted, q(X,X′) is the number of information bit errorsin choosing another coded sequence X′ instead of X n is thenumber of information bits per transmission and PEP(X,X′)is the PEP, i.e., the probability of selecting X′ when X wasactually transmitted.

By employing [30, p. 510], (41) can be efficiently evaluatedas

Pub ≤ 1n

∑

X

P (X)∫ π/2

0

[∂

∂NT [D(θ), N ]

∣∣∣∣N =1

]

(42)

where T [D(θ), N ] is the transfer function of the employed con-volutional code, N is an indicator variable taking into accountthe number of the erroneous bits and D(θ) depends on the un-derlying PEP expression. Furthermore, assuming that uniformerror probability (UEP) codes are considered and taking intoaccount the symmetry property this code family exhibits, thusmaking the distance structure of a UEP code independent of thetransmitted sequence, (42) can be further simplified as [30]

Pub ≤ 1π

∫ π/2

0

[1n

∂

∂NT [D(θ), N ]

∣∣∣∣N =1

]

. (43)

For M = 2, using (17), (18) and Craig’s formula for the Q-function, i.e., Q(x) = 1/π

∫ π/20 exp(−x2/2 sin2 θ)dθ, D(θ)

can be expressed as

D(θ) =N∏

n=1

M|Δn |2( μ

8 sin2 θ

)(44)

where M|Δn |2 can be obtained from (27). When M > 2, byemploying [18, Eq. (13)], and using a similar line of argumentsas for the case of M = 2, D(θ) can be written as

M∏

m 1 =1

M∏

m 2 �=m 1 =1

M|Δm 1 , m 2 |2( μ

8 sin2 θ

)(45)

where |Δm 1 ,m 2 |2 =‖ hm 1 − hm 2 ‖2 . The last MGF can be an-alytically computed with the help of (27) in a straight-forwardway.

V. DIVERSITY TECHNIQUES FOR COHERENT FSO SYSTEMS

Diversity at the transmitter and/or the receiver is a commonlyused technique to mitigate the deleterious impact of atmosphericturbulence on coherent FSO system performance. In this sec-tion, firstly the most popular diversity techniques for coherentFSO systems will be presented [41]. Such diversity schemeswill be also considered for the FSO communication systems in-vestigated in this paper as follows: Receive diversity by employ-ing MRC or SC and transmit diversity based on the Alamoutischeme.

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The instantaneous SNR at the output of the coherent MRCreceiver assuming equal average SNR per receiving aperture, μcan be expressed as [41, Eq. (11)]

γMRC = μ

N∑

n=1

In (46)

where In denotes the optical signal irradiance at the nth branch.The ABEP of coherent MRC receivers with differential phaseshift keying (DPSK) can be deduced as [13, Eq. (14)]

PE =12

N∏

n=1

MIn(μ) (47)

whereas for binary phase shift keying (BPSK) as [41, Eq. (18)]

PE =1π

∫ π/2

0

N∏

n=1

MIn

( μ

sin2 θ

)dθ. (48)

The instantaneous SNR at the output of the coherent SCreceiver is given by [41, Eq. (15)]

γSC = max{μI1 , μI2}. (49)

For the SC case, an analytical expression for the ABEP is muchmore difficult to be deduced and, thus for this case the ABEP willbe evaluated exclusively by means of Monte Carlo simulations.

In [16], transmit diversity techniques, based on the Alam-outi scheme, were explored for coherent OWC applications byemploying STBC. The instantaneous SNR at the input of thedemodulator of the optical receiver has a similar form as (46)[16], and therefore the ABEP of BPSK can be readily evaluatedby employing (48).

VI. PERFORMANCE EVALUATION RESULTS AND DISCUSSION

In this section the various performance evaluation resultswhich have been obtained by numerically evaluating the math-ematical expressions presented in Sections III and IV for un-coded and coded OSM systems operating over H-K turbulentchannels will be presented. In particular, for uncoded OSM sys-tems, ABEP vs. SNR for 2 × N OSM systems (obtained using(26) with (27), and (34)) the results are presented in Figs. 1–4.For the uncoded schemes, in order to validate the accuracy ofthe previously mentioned expressions, comparisons with com-plementary Monte Carlo simulated performance results are alsoincluded in these figures. As far as the performance of codedOSM systems is concerned, ABEP upper bounds vs. SNR havebeen obtained using (43) with (27) and the results are illustratedin Fig. 5. Table VI summarizes the different values of the systemparameters used to obtain the simulated performance evaluationresults. Note that in obtaining the various performance resultsa system with M = 2 transmitting apertures was considered.The motivation behind this choice is the inherent design com-plexities and cost of adopting multiple transmitters in currentstate-of-the-art coherent optical wireless links [16], as well asthe fact that the proposed framework yields exact results forM = 2 transmitters.

Fig. 1. ABEP of uncoded OSM for 2 × N MIMO H-K turbulent channels asa function of the average SNR, μ, for various number of receiving apertures,N . Simulation Parameters: A1 ,n = 2, A2 ,n = 1, θ1 ,n = π/3, θ2 ,n = π/4,αn = 2, b0 ,n = 2.

Fig. 2. ABEP of uncoded OSM for 2 × 2 and 2 × 4 MIMO H-K turbulentchannels as a function of the average SNR, μ, for various values of link distances,L. Simulation Parameters: λ = 1550 nm, C 2

n = 1.7 × 10−14 m−2/3 , θ1 ,n =π/3, θ2 ,n = π/4, b0 ,n = 1.

In order to demonstrate the impact of the number of the re-ceiving apertures on the system performance, Fig. 1, presentsthe ABEP performance as a function of the average SNR, μ,of 2 × N MIMO OSM systems with N ∈ {1, 2, 3, 4}. Inde-pendent and identically distributed branches are consideredwith A1,n = 2, A2,n = 1, θ1,n = π/3, θ2,n = π/4, αn = 2,b0,n = 2. The obtained results clearly indicate that the ABEP

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Fig. 3. ABEP of uncoded OSM for 2 × 3 MIMO H-K turbulent channels asa function of the average SNR, μ, for weak (C 2

n = 10−15 m−2/3 ) and strongC 2

n = 3 × 10−14 m−2/3 turbulence and for various values of b0 ,n . SimulationParameters: λ = 1550 nm, θ1 ,n = π/3, θ2 ,n = π/4.

Fig. 4. ABEP Comparison of 2 × 2 OSM with 1 × 2 coherent MRC systemsemploying DPSK, as a function of the average SNR, μ, for various valuesof A1 ,n . Simulation Parameters: A2 ,n = 0, θ1 ,n = 0, θ2 ,n = 0, αn = 1.5,b0 ,n = 1.5.

curves, obtained using (26), are in close agreement with theperformance obtained via simulations, thus verifying the cor-rectness of the proposed analysis. Moreover, it is evident thatthe asymptotic ABEP curves accurately predict the diversitygain achieved by the considered system for all test cases.

In Fig. 2, the dependence on the link distance, L, of the ABEPof a 2 × N MIMO OSM system is illustrated. The considered

Fig. 5. ABEP upper bounds of convolutional coded OSM for 2 × 2 and 2 ×1 H-K turbulent channels as a function of the average SNR, μ, for variousvalues of link distances, L. Simulation Parameters: λ = 1550 nm, C2

n = 1.7 ×10−14 m−2/3 , θ1 ,n = π/3, θ2 ,n = π/4, b0 ,n = 1.

TABLE IISYSTEM PARAMETERS FOR SIMULATION

Parameter Value

Operating wavelength, λ (nm) 1500Refractive index parameter, C 2

n (m−2 / 3 ) {3, 1.7, 0.1} × 10−1 4 ,Link distance, L (m) {500, 1000, 1500, 2000}θ1 , n π/3θ2 , n π/4A 1 , n {0, 1, 2, 3}A 2 , n {0, 1}b0 , n {0.25, 0.5, 1, 1.5, 2}Number of Transmitting apertures, M 2Number of Receiving apertures, N {1, 2, 3, 4}

system is again equipped with either N = 2 or N = 4 receivingapertures and identically distributed branches are assumed. Theparameters of the H-K distribution are calculated from (15)and (16) assuming plane wave propagation. Following [42], itis further assumed that the operating wavelength is λ = 1550nm and C2

n = 1.7 × 10−14 m−2/3 , which, as it was pointed outin Section II, is a typical value of refractive index for FSOlinks operating near the ground plane during daytime [29]. Asexpected, the error performance deteriorates as L increases fromL = 500 m to L = 1500 m. Moreover, it is evident that anincrease in L from 500 to 1000 m leading to a more severeperformance deterioration than in the case where L increasesfrom 1000 to 1500 m. For all cases considered, the analyticalresults obtained using (26) are compared with the equivalentresults obtained by means of Monte-Carlo computer simulationsand again they match very well.

In Fig. 3 the impact of the turbulence strength as well asof the parameter b0 on system performance is illustrated. In

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particular, Fig. 3 presents the ABEP performance as a functionof the average SNR, μ, of 2 × N MIMO OSM systems withN = 3 receiving apertures, link distance L = 2000 m, and var-ious values of b0 . Two different values of the refractive indexparameter C2

n are considered, i.e., C2n = 3 × 10−14 m−2/3 and

C2n = 10−15 m−2/3 to account for strong and weak turbulence,

respectively [29], [43]. The obtained results clearly show thatthe error performance improves as b0 increases and/or C2

n de-creases. Moreover, the analytical results obtained using (26)agree well with the equivalent results obtained by means ofMonte–Carlo computer simulations.

Next we compare the proposed OSM system with two al-ternative coherent FSO systems that can provide performanceenhancements by means of transmit, i.e., multiple-input single-output (MISO) or receive diversity i.e. single-input multiple-output (SIMO). It is noted that for similar aperture configura-tions, a fair comparison between coherent and IM/DD systemsseems difficult as the same received laser power leads to differentSNRs for each of these schemes [17]. On the other hand, in orderto perform a fair comparison between OSM and the alternativeMISO or SIMO systems under the same propagation channelconditions, the aperture configuration of the FSO systems undercomparison should be carefully selected. Specifically, becauseof the fact that the diversity gain of OSM equals to only thenumber of the receive apertures, i.e., no transmit diversity gaincan be achieved, the number of transmit or receive apertures ofthe alternative systems must be hence selected to be equal to thenumber of receive apertures of the OSM system. To this end andfor a fair comparison in our paper a 2 × 2 OSM system is com-pared with the following two alternative FSO communicationsystems which also employ coherent detection:

1) A 1 × 2 heterodyne FSO communication system whichemploys DPSK [13] and MRC or SC;

2) A 2 × 1 coherent FSO system employing the Alamoutischeme [16] and BPSK.

In order to simplify the underlying mathematical analysis,it is assumed that the PDF of In is given by (9) with the pa-rameters An being all zero, i.e., the PDF considered here is theK-distribution. Thus, MIn

(μ) can be readily obtained in closedform from (33) by replacing b0,n with b0,n /2. In Fig. 4, theABEP of 2 × 2 MIMO OSM links is compared with the ABEPof 1 × 2 coherent FSO systems with DPSK considering identi-cally distributed links. In order to compare these systems underthe same propagation conditions, it is assumed that αn = 1, 5,b0,n = 1.5, A2,n = 0 and A1,n = {0, 1, 2, 3}. As it can be ob-served, when either MRC or SC are employed, although coher-ent DPSK performs worse than the OSM for values of A1,n upto approximately 1, it outperforms OSM at lower values of A1,n .Moreover, although the OSM outperforms the Alamouti schemefor A1,n = 2 and 3, it performs similarly for high SNR valueswhen A1,n = 1. It is noted that for A1,n = 1 and lower valuesof A1,n the Alamouti scheme yields the best performance of theconsidered schemes. However, when more transmit aperturesare employed, this advantage is compensated by the superiorspectral efficiency of OSM and its lower hardware complex-ity as compared to coherent MRC. Specifically, as pointed out

in [18], OSM offers increased spectral efficiency by a factorlog2(M). Moreover, as only one transmitting aperture is acti-vated at any bit duration, OSM has a lower decoding complexityas compared to conventional MRC and Alamouti schemes.

In Fig. 5, upper bounds on the ABEP of convolutional coded2 × 1 and 2 × 1 OSM systems are depicted, assuming similarpropagation conditions to those considered in Fig. 2. Consider-ing a convolutional code with rate 1/3 and constraint length of3, its transfer function is given as [44, Eq. (8.2.6)]

T [D(θ), N ] =D(θ)6N

1 − 2ND(θ)2 . (50)

Substituting (50) into (43), a union bound on the ABEP can beobtained as

Pub ≤ 1π log2(M)

∫ π/2

0

D(θ)6

(1 − 2D(θ)2)2 dθ. (51)

The performance results of Fig. 5 clearly show that, as expected,the incorporation of convolutional coding significantly enhancesthe performance of OSM systems, even when a small numberof receive apertures is employed.

VII. CONCLUSION

In this paper, the use of spatial modulation technique forcoherent FSO communication systems has been proposed. Wehave provided a comprehensive analytical framework for errorperformance analysis which is valid for a great deal of atmo-spheric turbulence scattering channel models but focusing onthe H-K distribution. The proposed framework has revealed im-portant information about the performance of OSM over suchturbulent channels, including the effect of fading severity andthe achievable diversity gain. In particular, it has been shownthat the diversity gain depends on the number of receive aper-tures only while channel parameters affect the coding gain only.Upper bounds for the ABEP performance of coded OSM sys-tems have also been derived, demonstrating that coding tech-niques can greatly enhance the performance of OSM. Extensivecomputer simulation performance evaluation results have beenalso obtained which have verified the accuracy of the analyti-cal approach. It was shown that OSM can provide significantperformance enhancements in the presence of H-K atmosphericturbulence. The improvements are comparable to the ones of-fered by conventional coherent systems with spatial diversity,while outperforming the latter in terms of spectral efficiencyand hardware complexity. Specifically, OSM offers increasedspectral efficiency by a factor log2(M) when multiple transmit-ting apertures are employed while it completely avoids multiplecommunication chains in a similar fashion as in RF systems.Besides, under specific propagation conditions, OSM can yieldbetter performance than conventional SIMO systems employingMRC or SC. The proposed framework, apart from enabling thecomputation of accurate performance evaluation results in anefficient manner, it also provides a useful tool for understand-ing the performance trend, important properties and tradeoffs of

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outdoor OSM operating in the presence of atmospheric turbu-lence.

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[33] M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds andapproximations for the computation of error probability in fading chan-nels,” IEEE Trans. Wireless Commun., vol. 2, no. 4, pp. 840–845, Jul.2003.

[34] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and SeriesVolume 4: Direct Laplace Transforms, 1st ed. Boca Raton, FL, USA: CRC,1992.

[35] M. Abramovitz and I. Stegun, Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables. New York, NY, USA: Dover,1964.

[36] F. Yilmaz and M.-S. Alouini, “An MGF-based capacity analysis of equalgain combining over fading channels,” in Proc. IEEE 21st Int. Symp. Pers.Indoor Mobile Radio Commun., Sep. 2010, pp. 945–950.

[37] P. Theofilakos, A. G. Kanatas, and G. P. Efthymoglou, “Performance ofgeneralized selection combining receivers in K fading channels,” IEEECommun. Lett., vol. 12, no. 11, pp. 816–818, Nov. 2008.

[38] Z. Wang and G. Giannakis, “A simple and general parametrization quan-tifying performance in fading channels,” IEEE Trans. Commun., vol. 51,no. 8, pp. 1389–1398, Aug. 2003.

[39] M. D. Renzo and H. Haas, “Bit error probability of space modulation overNakagami-m fading: Asymptotic analysis,” IEEE Commun.Lett., vol. 15,no. 10, pp. 1026–1028, Oct. 2011.

[40] K. P. Peppas, M. Zamkotsian, F. Lazarakis, and P. G. Cottis, “Asymp-totic error performance analysis of spatial modulation under generalizedfading,” IEEE Wireless Commun. Lett., vol. 3, no. 4, pp. 421–424, Aug.2014.

[41] M. Niu, J. Cheng, and J. F. Holzman, “Error rate performance com-parison of coherent and subcarrier intensity modulated optical wirelesscommunications,” J. Opt. Commun. Netw., vol. 5, no. 6, pp. 554–564, Jun.2013.

[42] M. Uysal, J. T. Li, and M. Yu, “Error rate performance analysis of codedfree-space optical links over gamma-gamma atmospheric turbulence chan-nels,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1229–1233, Jun.2006.

[43] H. E. Nistazakis, E. A. Karagianni, A. D. Tsigopoulos, M. E. Fafalios,and G. S. Tombras, “Average capacity of optical wireless communica-tion systems over atmospheric turbulence channels,” J. Lightw. Technol.,vol. 27, no. 8, pp. 974–979, Apr. 2009.

[44] J. G. Proakis, Digital Communications, 3rd ed. New York, NY, USA: McGraw Hill, 1995.

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Kostas P. Peppas was born in Athens, Greece, in 1975. He received the Diplomadegree in electrical and computer engineering and the Ph.D. degree in wirelesscommunications from the National Technical University of Athens, Athens, in1997 and 2004, respectively. From 2004 to 2007, he was with the Department ofComputer Science, University of Peloponnese, Tripolis, Greece, and from 2008to 2014 with the National Center for Scientific Research–“Demokritos,” Instituteof Informatics and Telecommunications as a Researcher. In 2014, he joinedthe Department of Telecommunication Science and Technology, University ofPeloponnese, where he is currently a Lecturer. His current research interestsinclude digital communications over fading channels, MIMO systems, wirelessand personal communication networks, and system level analysis and design.He has authored more than 70 journal and conference papers.

P. Takis Mathiopoulos (SM’ 94) received the Ph.D. degree in digital commu-nications from the University of Ottawa, Ottawa, ON, Canada, in 1989.

From 1982 to 1986, he was with Raytheon Canada, Ltd., working in the areasof air navigational and satellite communications. In 1988, he joined the Depart-ment of Electrical and Computer Engineering (ECE), University of BritishColumbia (UBC), Vancouver, BC, Canada, where he was a Faculty Member asa Professor from 2000 to 2003. From 2000 to 2014, he was with the Institutefor Space Applications and Remote Sensing (ISARS), National Observatory ofAthens (NOA), Athens, Greece, first as the Director and then as the Director ofresearch and established the Wireless Communications Research Group. As IS-ARS’ Director during 2000–2004, he led the Institute to a significant expansionR&D growth, and international scientific recognition. For these achievements,ISARS has been selected as a National Center of Excellence for the years 2005–2008. Since 2014, he has been an Adjunct Researcher at the Institute of Astron-omy, Astrophysics, Space Applications and Remote Sensing, NOA. Since 2003,he worked part time at the Department of Informatics and Telecommunications,University of Athens, Athens, where since 2014, he has been a Professor ofdigital communications. From 2008 to 2013, he was a Guest Professor with theSouthwest Jiaotong University, China. He is also appointed as a Guest Profes-sor at the School of Information Engineering, Yangzhou University, Yangzhou,China, for 2014–2015 under the Senior Foreign Expert Program of the Govern-ment of PR of China. Furthermore he is appointed by Keio University, Tokyo,Japan, as a Visiting Professor at the Department of Information and ComputerScience for 2015–2016 under the Top Global University Project of the Ministryof Education, Culture, Sports, Science and Technology Government of Japan.For the last 25 years, he has been conducting research mainly on the physicallayer of digital communication systems for terrestrial and satellite applications,including digital communications over fading and interference environments.He coauthored a paper in GLOBECOM’89 establishing for the first time in theopen technical literature the link between MLSE and multiple (or multisymbol)differential detection for the AWGN and fading channels. He is also interested inchannel characterization and measurements, modulation and coding techniques,synchronization, SIMO/MIMO, UWB, OFDM, software/cognitive radios, andgreen communications. In addition, since 2010, he has been actively involvedwith research activities in the fields of remote sensing, LiDAR systems, and pho-togrammetry. In these areas, he has coauthored more than 100 journal papers,mainly published in various IEEE and IET journals, four book chapters, andmore than 120 conference papers. He has been the PI for more than 40 researchgrants and has supervised the thesis of 11 Ph.D. and 23 Master students.

Dr. Mathiopoulos has been or currently serves on the editorial board ofseveral archival journals, including the IET Communications, and the IEEETRANSACTIONS ON COMMUNICATIONS from 1993 to 2005. He has regularlyacted as a consultant for various governmental and private organizations. Since1993, he has served on a regular basis as a Scientific Advisor and a TechnicalExpert for the European Commission (EC). In addition, since 2001, he has beenthe Greek Representative to high-level committees in the EC and the EuropeanSpace Agency. He has been a Member of the TPC of more than 70 internationalIEEE conferences, as well as the TPC Vice-Chair for the 2006-S IEEE VTC andthe 2008-F IEEE VTC, as well as the Cochair of the FITCE2011. He has de-livered numerous invited presentations, including plenary and keynote lectures,and has taught many short courses all over the world. As a Faculty Memberat the ECE, UBC, he was elected as an ASI Fellow and a Killam ResearchFellow. He received the two best paper awards for papers published in the 2ndInternational Symposium on Communication, Control, and Signal Processingin 2008), and the 3rd International Conference on Advances in Satellite andSpace Communications in 2011.

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Free-Space Optical Communication With SpatialModulation and Coherent Detection Over H-K

Atmospheric Turbulence ChannelsKostas P. Peppas, Senior Member, IEEE, and P. Takis Mathiopoulos, Senior Member, IEEE

Abstract—The use of optical spatial modulation (OSM) is pro-posed as a simple low-complexity means of achieving spatial diver-sity in coherent free-space optical (FSO) communication systems.In doing so, this paper presents a generic analytical framework forobtaining the average bit error probability (ABEP) of uncoded andcoded OSM with coherent detection in the presence of turbulence-induced fading. Although the framework is general enough to ac-commodate any type of channel models based on turbulence scat-tering, the focus of the analysis is on the H-K distribution as suchmodel is very general and valid over a wide range of atmosphericpropagation conditions. Using this framework, it is shown thatOSM can offer comparable performance with conventional coher-ent FSO schemes employing spatial diversity at the transmitteror the receiver only, while outperforming the latter in terms ofspectral efficiency and hardware complexity. Furthermore, vari-ous numerical performance evaluation results are also presentedand compared with equivalent results obtained by Monte Carlosimulations which verify the accuracy of the derived analyticalexpressions.

Index Terms—Atmospheric turbulence, average bit error prob-ability, coherent detection, free space optical communication sys-tems, H-K distribution, multiple-input multiple-output (MIMO)systems, optical spatial modulation.

I. INTRODUCTION

FREE-SPACE optical (FSO) communication systems haverecently attracted great attention within the research com-

munity as well as for commercial use. FSO systems can pro-vide ultra-high data rates (at the order of multiple gigabit persecond), immunity to electromagnetic interference, excellentsecurity and large unlicensed bandwidth, i.e., hundred or eventhousand times higher than radio-frequency (RF) systems, alongwith low installation and operational cost [1].

The challenge in employing such systems is that FSO links arehighly vulnerable due to the detrimental effects of attenuationunder adverse weather conditions (e.g., fog), pointing errorsand atmospheric turbulence [1]. One method to improve thereliability of the FSO link is to employ spatial diversity, i.e.multiple-lasers and multiple-apertures to create a multiple-inputmultiple-output (MIMO) optical channel. Because of its lowcomplexity, spatial diversity is a particularly attractive fading

Manuscript received March 23, 2015; revised June 21, 2015; accepted July14, 2015.

K. P. Peppas is with the Department of Informatics and Telecommunications,University of Peloponnese, Tripoli 22100, Greece (e-mail: peppas@uop.gr).

P. T. Mathiopoulos is with the Department of Informatics and Telecom-munications, National and Kapodistrian University of Athens, Athens 15784,Greece (e-mail: mathio@di.uoa.gr).

Digital Object Identifier 10.1109/JLT.2015.2465385

mitigation technique and performance enhancements have beenextensively studied in many past research works in the field ofFSO communications (e.g., see [2]–[5]).

In order to evaluate the impact of atmospheric turbulence onthe performance of OSM, accurate models for the fading dis-tribution are necessary. For example, the lognormal distributionis often used to model weak turbulence conditions whereas thenegative exponential and the K-distribution are used to modelstrong turbulence conditions [6]. Other more general statisti-cal models have also been proposed to model scintillation overall turbulence conditions, such as, for example, the Gamma–Gamma [7], the lognormal-Rice (or Beckmann) [8] the homo-dyned K distribution (H-K) [9] and the I-K [10]–[12] distribu-tions. The validity of these models is based on the argumentthat scintillation is a doubly stochastic random process mod-elling both small and large scale turbulence effects. Besides,they agree well with measurement data and simulations for awide range of turbulence conditions.

In this paper, the H-K distribution is adopted to modelturbulence-induced fading [9]. The main reason for this choiceis the fact that this distribution is based on a very general scat-tering model which is valid over a wide range of atmosphericconditions. It is also noted that the H-K distribution generalizesexisting models such as the K-distribution. The H-K distributionmodels the field of the optical wave as the sum of a determinis-tic component and a random component, the intensity of whichfollows the Rice (Nakagami-n) distribution. The average inten-sity of the random portion of the field is treated as a fluctuatingquantity [10]. It is important to underline that, to the best of ourknowledge, in the open technical literature there have been nopapers published analyzing and evaluating the performance ofFSO systems over such channels, because of the complicatedmathematical form of the required for the analysis probabilitydensity functions (PDF).

Depending on their detection, FSO systems can be classifiedinto two main categories, namely non-coherent (direct detec-tion) and coherent (heterodyne detection) systems [13]–[17].Non-coherent FSO systems employing intensity modulationand direct detection (IM/DD) have the optical power outputof a source varied in accordance with some characteristic ofthe modulating signal. A direct detection receiver only respondsto the intensity of the received field and is oblivious to thereceived signal phase. Coherent FSO systems encode the infor-mation bits directly onto the electric field of the optical beam.At the receiver, a local oscillator (LO) is employed to extractthe information encoded on the optical carrier electric field. On

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2 JOURNAL OF LIGHTWAVE TECHNOLOGY

the one hand, coherent FSO systems can provide significantperformance enhancements due to spatial temporal selectivityand heterodyne gain in comparison to direct detection systems.Moreover, they are more versatile as any kind of amplitude,frequency, or phase modulation can be employed. On the otherhand, the implementation of coherent receivers is more complexas the LO field should be spatially and temporally coherent withthe received field. Throughout this work, coherent FSO systemsare only considered.

Recently, the so-called optical spatial modulation (OSM)has emerged as a power- and bandwidth-efficient single-carriertransmission technique for optical wireless communication sys-tems [18]–[20]. This spatial diversity scheme, initially proposedin [21] and further investigated in [22] and[23], employs a sim-ple modulation mechanism that foresees to activate just one outof several MIMO transmitters at any time instant and to use theindex of the activated transmitter as an additional dimension forconveying implicit information. It has been shown that OSMcan increase the data rate by base two logarithm of the numberof transmit units [18]. Also, OSM can increase the data rate bya factor of 2 and 4, as compared to on–off keying and pulse po-sition modulation, respectively [18], [19]. It is underlined thatsuch performance gains are obtained with a significant reductionin receiver complexity.

Because of the above mentioned advantages of OSM overother more conventional transmission schemes and given thewide applicability of FSO, it is important to investigate thepotential performance enhancements obtained by incorporatingOSM in FSO systems. However, in general, this research topichas not been thoroughly investigated. In fact, only recently,there have been papers published in the open technical litera-ture dealing with performance analysis studies of FSO systemsemploying spatial modulation and operating in the presence ofatmospheric turbulence, e.g., see [24] and [25]. Specifically, in[24], the combination of subcarrier IM and spatial modulationwith receiver diversity was proposed to enhance the perfor-mance of IM/DD FSO systems. In [25], another IM/DD basedsystem FSO system which combines antenna shift keying withjoint pulse position and amplitude modulations was considered.For this system, which was denoted as spatial pulse positionand amplitude modulation (SPPAM), the atmospheric turbu-lence channel was modeled as log-normal or Gamma–Gammadistributions and was evaluated, in terms of bounds, for un-coded and coded signals. Average bit error probability (ABEP)performance evaluation results have shown that SPPAM offers acompromise between spectral and power efficiencies as well asa certain degree of robustness against atmospheric turbulence.Despite these two papers which deal with non-coherent detec-tion schemes, the potential enhancements of OSM on improv-ing the performance of FSO systems with coherent detectionstill remains an open research topic which, to the best of ourknowledge, has not been addressed so far in the open technicalliterature.

Motivated by the above, in this paper we present for the firsttime a generic analytical framework which can be used to ac-curately obtain the performance of outdoor OSM with coherentdetection in the presence of turbulence-induced fading. More

TABLE ILIST OF MATHEMATICAL NOTATIONS

j2 = −1 denotes the imaginary unit|z | denotes the magnitude of the complex number z

�{z} denotes the real part of the complex number z

�{z} denotes the real part of the complex number z

f (x) = o[g(x)] as x → x0 if limx →x 0f (x )g (x ) = 0

‖ · ‖2F denotes the square Frobenius norm

(·)T denotes the matrix transpose∗ denotes convolutionE〈·〉 denotes expectationfX (·) denotes the PDF of the RV X

FX (·) denotes the cumulative distribution function of the RV X

MX (·) denotes the MGF of the RV X

Ia (·) is the modified Bessel function of the first kind and order a [26, eq. (8.431)]Ka (·) is the modified Bessel function of the second kind and order a [26, eq. (8.432)]Γ (x) =

∫ ∞0 exp(−t)tx −1 dt is the Gamma function [26, eq. (8.310/1)]

Q(x) = 1√2 π

∫ ∞x exp(−t2 /2)dt is the Gauss Q-function

Wp , q (·) is the Whittaker function [26, eq. (9.220)]Pr{·} denotes the probability operator· denotes estimated value at the receiver side

specifically and within this new analytical framework, the mainnovel research contributions of the paper are as follows:

1) New analytical expressions for the ABEP of coherent OSMunder turbulence conditions modeled by the H-K distribu-tion are derived. When the transmitter is equipped withtwo apertures the resulting analytical expressions are ex-act, whereas for an arbitrary number of transmit aperturestight upperbounds can be obtained.

2) ABEP bounds for coded OSM systems are derived andthe performance enhancements when channel coding isemployed are presented and analyzed.

The ABEP of OSM is also compared to that of conventionalFSO schemes with transmit or receive diversity only, i.e., whenmaximal ratio combining (MRC), selection combining (SC) orAlamouti-type space-time block codes (STBC) are employed.It is finally noted that the theoretical analysis is substantiated bycomparing the theoretical and equivalent simulated performanceevaluation results obtained by means of Monte Carlo techniques.

The paper is organized as follows. After this introduction,Section II outlines the system and channel models. In Section IIIanalytical expressions for the ABEP of uncoded OSM systemsare presented. Asymptotic ABEP expressions are also derived,wherefrom the diversity gain of coherent OSM can be readilydeduced. The performance of coded OSM systems is discussedin Section IV. In Section V an overview of the conventional di-versity techniques for coherent OSM is presented. In Section VIthe various performance evaluation results and their interpreta-tions as well as comparisons are presented. Finally, concludingremarks can be found in Section VII. Notations: A comprehen-sive list of all mathematical notations used in this paper can befound in Table I.

II. SYSTEM AND CHANNEL MODEL

In this section, a detailed description of the OSM FSO sys-tem model, i.e., transmitter, channel and receiver is provided.Moreover, the H-K distribution is introduced and analytical

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PEPPAS AND MATHIOPOULOS: FREE-SPACE OPTICAL COMMUNICATION WITH SPATIAL MODULATION AND COHERENT DETECTION 3

expressions for its parameters in terms of equivalent physical pa-rameters of the turbulence phenomenon, such as the refractive-index structure parameter, optical wave number, and propaga-tion path length, are derived.

A. Preliminaries

Let us consider an M × N MIMO FSO system with M trans-mit units (lasers) and N coherent receivers. It is assumed thatthe receiving apertures are separated by more than a coherencewavelength to ensure the independency of fading channels. Thebasic principle of OSM modulation is as follows [18], [20]:

1) The transmitter encodes blocks of log2(M) data bits intothe index of a single transmit unit. Such a block of bitsis hereafter referred to as “message” and is denoted bybm , ∀m = 1, 2, . . . ,M . It is assumed that the M mes-sages are transmitted by the encoder with equal probabil-ity and that the related transmitted signal is denoted byEm = Em exp(jφbm

). During each time slot, only onetransmitter �, where � = 1, 2, . . . ,M is active for datatransmission. The information bits are modulated on theelectric field of an optical signal beam through an externalmodulator. During each particular time slot, the remainingtransmit lasers are kept silent, i.e., they do not transmit.

2) At the receiver, the incoming optical field is mixed with aLO field and the combined wave is first converted by thephotodetector to an electrical one. A bandpass filter is thenemployed to extract the intermediate frequency (IF) com-ponent of the total output current. Finally, a N -hypothesisdetection problem is solved to retrieve the active transmitunit index, which results in the estimation of the uniquesequence of bits emitted by the transmitter.

B. Receiver Structure

The received electric field at the aperture plane of the nthreceiver after mixing with a LO beam, can be expressed as [16],[17]

en (t) =√

2PtZ0Em hm,n cos(ω0t + φm,n + φbm)

+√

2PLOZ0 cos(ωLOt). (1)

In the above equation, Pt is the transmit laser power, Z0 is thefree space impedance, hm,n and φm,n denote the magnitude andthe phase of the complex channel coefficient between the mthtransmit and the nth receive aperture, respectively. Furthermore,PLO denotes the power of the LO, ωLO = ω0 + ωIF where ω0and ωIF are the carrier and the intermediate radian frequencies,respectively.

The output current of the nth photodetector can be mathe-matically expressed as [16], [17]

in (t) =R

Z0[en (t)]2 (2)

where R = ηqe/(hν0) is the responsivity of the photodetectorwith qe = 1.6 × 10−19 Cb is the charge of an electron, h =6.6 × 10−34 J · s is the Planck constant, η is the photodetectorefficiency, and ν0 = ω0/(2π) is the optical center frequency.

Expanding (2) and ignoring the double-frequency terms that arefiltered out by the bandpass filter, the resulting photocurrent canbe expressed as

in (t) = RPtE2m h2

m,n + RPLO

+2R√

PtPLOEm hm,n cos(ωIFt − φm,n − φbm)

= iDC(t) + iAC(t). (3)

In (3), iDC(t) = RPtE2m h2

m,n + RPLO is the dc componentgenerated by the signal and LO fields, respectively, iAC(t) =2R

√PtPLO cos(ωIFt − φm,n − φbm

) is the ac component inthe received photocurrent which, unlike for direct detection,contains information about the frequency and phase of the re-ceived signal. It is assumed that for coherent detection the IFωIF is nonzero, so that the signal power can be expressed asPs = 2R2PtPLOE2

m h2m,n .

As in [13]–[17], we also consider that PLO � Ps and thus, thedc photocurrent can be approximated as iDC(t) ≈ RPLO. Thephotodetection process is impaired by shot noise with varianceσ2

shot,L = 2qeRPLOBe where Be is the electrical bandwidth ofthe photodetector. It is also noted that because of the large valueof RPLO the photocurrent due to thermal noise and the darkcurrent can be ignored [16].

Following [15] and[17], the sufficient statistics at the nthcoherent receiver can be expressed as

yn =√

μhm,nEm exp[j(φm,n + φbm)] + zn (4)

where μ = RPt/(qeBe) is the average signal-to-noise ratio(SNR) and zn is the noise at the nth receiver. Assuming thatthe LO power is large and the receiver noise is dominated bythe LO related noise terms, the additive white Gaussian noisemodel can be employed as an accurate approximation of thePoisson photon-counting detection model [16], [17]. Thus, zn

can be modeled as a zero-mean unit variance complex Gaussianrandom variable (RV) [17].

Similar to [27], it is assumed that the receiver has knowledgeof the actual fading gains and that the fading process remainsconstant over the bit interval and changes from bit to bit in anindependent manner. At the receiver, the optimal spatial mod-ulation detector estimates the active transmitter index, �, at agiven time slot according to [28]

� = argmax�

py (y|x,H)

= argmin�

{√μ ‖ h�x� ‖2

F −2(yT h�x�

)}(5)

where- x is an M -dimensional vector with elements corresponding

to the electrical field Em exp(jφbm) transmitted over the

optical MIMO channel;

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4 JOURNAL OF LIGHTWAVE TECHNOLOGY

- H(t) is an N × M optical MIMO channel defined as

H(t) = [h1 ,h2 , . . . ,hM ]

�

⎡

⎢⎢⎢⎢⎢⎢⎢⎣

h11 (t) exp(jφ11 ) . . . h1M (t) exp(jφ1M )

h21 (t) exp(jφ21 ) . . . h2M (t) exp(jφ2M )

.... . .

...

hN 1 (t) exp(jφN 1 ) . . . hN M (t) exp(jφN M )

⎤

⎥⎥⎥⎥⎥⎥⎥⎦

(6)

- z is the N -dimensional noise vector;- py (y|x,H) is the PDF of y conditioned on the transmitted

vector x and the channel H.

C. Channel Model

A discrete scattering channel model is considered, where theradiation field of an optical wave at a particular point is assumedto be composed of a number of scattered components that havepropagated through different paths. Under the Ricean assump-tion [10], the complex channel path gains hij (t) between theith transmitter and the jth photodetector can be expressed ashij (t) = hij (t) exp(jωt) where ω is the radian frequency of theoptical wave and

hij (t) = �{hij (t)} + j�{hij (t)}= Aij exp[jθij (t)] + Rij (t) exp[jΦij (t)] (7)

where the term Aij exp[jθij (t)] is a deterministic componentand Rij (t) exp[jΦij (t)] is a circular complex Gaussian RV.Hence, the amplitude Rij is Rayleigh distributed with scaleparameter σ2

ij = bij /2 [10, Eq. (13)] and the phase Φij is uni-formly distributed over [0, 2π). Under the assumption of a dou-bly stochastic scintillation model [10], the effect of randomfluctuations in the turbulence parameters is modeled by allow-ing random variations in the parameter bij of the Rayleigh com-ponent. Following [10], it is further assumed that bij follows aGamma distribution with PDF given by

fbi j(b) =

(αij

b0

)αi j bαi j −1

Γ(αij )exp

(

− αb

b0i j

)

(8)

where α is the shaping parameter which represents the effectivenumber of scatters and b0i j

= E{bij}. Then, the PDF of theirradiance Iij = |hij (t)|2 , fIi j

(I), can be expressed as [10, Eq.(8)]

fIi j(I) =

(αij /b0i j

)αi j

Γ(αij )

×∫ ∞

0bαi j −2 exp

(

−αij b

b0i j

− I + Aij2

b

)

×I0

(2Aij

√I

b

)

db (9)

which is actually the integral representation of the H-K distri-bution [10]. It is noted that, with the exception of the specialcases Aij = 0 or α = 1, fIi j

(I) cannot be expressed in closed

form. Specifically, for Aij = 0 (9) reduces to the K-distributionwhereas for α = 1, (9) reduces to a special case of the I-Kdistribution [10, Eq. (10)].

The νth normalized moment of Iij is given by [10, Eq. (23)]as

E{Iνij}E{Iij}ν

=ν!

ανij (1 + ρij )ν

ν∑

k=0

(ν

k

)Γ(αij + ν − k)

Γ(αij )(αij ρij )ν

ν!(10)

where ρij � A2ij /b0i j

is the coherence parameter, defined asthe power ratio of mean intensities of the constant-amplitudecomponent and random component of the field in (7) [10], [11].Using (10), the scintillation index can be readily calculated as

σ2Ii j

�E{I2ij}E{Iij}2 − 1 =

αij + 2αij ρij + 2αij (1 + ρij )2 . (11)

Under the assumption of plane wave propagation, σ2Ii j

can bedirectly related to atmospheric conditions as [11, Eq. (7), Eq.(9)]

σ2Ii j

≈{

σ21 (1 + 0.5σ2

1 ), σ1 � 1

1 + 0.86/σ4/51 , σ1 � 1

(12)

where σ21 = 1.23C2

ni jk7/6L

11/6ij is the Rytov variance, k =

2π/λ is the optical wave number with λ being the wave-length, Lij is the link distance and Cni j

denotes the index ofrefraction structure parameter. For FSO links which operatenear the ground plane, C2

ni j≈ 1.7 × 10−14 m−2/3 and 8.4 ×

10−15 m−2/3 for the daytime and night time periods, respec-tively [29]. Moreover, σ1 � 1 and σ1 � 1 correspond to weakand strong turbulence conditions, respectively.

Using (12), the parameters of the H-K distribution, α and ρ,can be directly related to physical parameters of the turbulenceby following a similar line of arguments as in [11], where similarresults have been derived for the I-K distribution. In particular,on the one hand, weak turbulence conditions are characterizedin the H-K distribution by large values of ρij . In this case thescintillation index given by (11) can be approximated as

σ2Ii j

≈ 2ρij

, with ρij � 1. (13)

On the other hand assuming strong turbulence conditions, inwhich case ρij goes towards zero, (11) can be approximated as

σ2Ii j

≈ 1 +2

αij, with ρij � 1. (14)

By comparing (13) and (14) with the first and second branchesof (12), respectively, αij and ρij can be obtained as

αij = 2.33σ4/51i j

(15)

ρij =2

σ21i j

(1 + 0.5σ21i j

). (16)

To the best of our knowledge, the relationship of αij and ρij withσ1i j

given by (15) and (16) has not been previously published.

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III. PERFORMANCE ANALYSIS OF UNCODED OSM

In this section, by employing the well-known moment gen-erating function (MGF)-based approach for the performanceanalysis of digital communications over fading channels [30],analytical expressions for the ABEP of uncoded OSM systemswill be derived. In addition, expressions for the diversity andcoding gains of OSM systems are also presented, thus provid-ing useful insight as to how these parameters affect the overallsystem performance.

A. Preliminaries

For M = 2, the conditional bit error probability (CBEP) ofOSM systems when no turbulence induced fading is consideredcan be obtained in closed form as [27]

PE (h1 ,h2) = Q

(√μ

4‖ h1 − h2 ‖2

F

)

. (17)

The squared Frobenius norm in (17) can be expressed as

‖ h1 − h2 ‖2F =

N∑

n=0

|h1,n − h2,n |2 (18)

where hi,n is the nth element of hi , ∀i ∈ {1, 2}. When M > 2transmitters are considered, a tight upper bound for the CBEPof the above system can be obtained as [18, Eq. (7)]

PE (H) ≤ M−1

log2(M)

×M∑

m 1 =1

M∑

m 2 �=m 1 =1

Nb(m1 ,m2)PEP(m1 → m2) (19)

where PEP(m1 → m2) denotes the pairwise error probability(PEP) related to the pair of transmitters m1 and m2 , where m1and m2 ∈ 1, 2, . . . ,M , and Nb(m1 ,m2) is the number of bits inerror which have occurred when the receiver decides incorrectlythat m2 instead of m1 has been active. The PEP(m1 → m2)can be evaluated as [18, Eq. (8)]

PEP(m1 → m2) = Q

(√μ

4‖ hm 1 − hm 2 ‖2

F

)

. (20)

B. MGF-Based Approach

When atmospheric turbulence is included in the analysis theconditional error probabilities in (17) and (19) need to be av-eraged over the elements of the channel matrix H in order toevaluate the ABEP. Without loss of generality, let us consider thecase of a 2 × N MIMO system. Since hi,n are complex Gaus-sian RVs, the difference Δn = h1,n − h2,n is also a complexGaussian RV having mean equal to the difference of the meansof hi,n and variance equal to the sum of variances of hi,n . In or-der to deduce a closed form expression for the ABEP, it is furtherassumed that hi,n have uncorrelated real and imaginary com-ponents with the same variance σ2

n = bn/2. It is noted that thisassumption has been experimentally verified in the past for linkdistances of the order of kilometer and for aperture separationdistances of the order of centimeter [31], [32]. Consequently,

Δn has uncorrelated components too and its squared envelope,|Δn |2 , is characterized by a non-central chi-square PDF

f|Δn |2 (x|bn ) =1

2bnexp

(

−x + A2n

2bn

)

I0

(An

√x

bn

)

(21)

where An = |A2,n exp(jθ2,n ) − A1,n exp(jθ1,n )|. Assumingthat bn follows the Gamma with parameters αn and b0,n , thePDF of |Δn |2 is obtained by averaging (21) with respect to bn ,i.e.,

f|Δn |2 (x) =(αn/b0,n )αn

2Γ(αn )

×∫ ∞

0bαn −2n exp

(

−αnbn

b0,n− x + A2

n

2bn

)

I0

(An

√x

bn

)

dbn . (22)

As was pointed out in [10], f|Δn |2 (x) can’t be expressed inclosed form. Nevertheless, for the special case of αn = 1, i.e.,when one scatterer per branch is considered, and by employing[10, Eq. (10)], this integral can be evaluated in closed form as

f|Δn |2 (x)

=

⎧⎪⎪⎨

⎪⎪⎩

1b0,n

K0

(√2An/b0,n

)

I0

(√2x/b0,n

), x < A2

n

1b0,n

I0

(√2An/b0,n

)

K0

(√2x/b0,n

), x > A2

n .

(23)

Moreover, for the special case where h1,n and h2,n have identi-cal mean value, i.e., when An = 0, (22) yields the well-knownK-distribution with PDF given by

f|Δn |2 (x) = 2(1−αn )/2Γ(αn )(

αnx

b0,n

)(αn −1)/2

×Kαn −1

(√2αnx

b0,n

)

. (24)

By employing the MGF-based approach for the performanceanalysis of digital communications over fading channels, theaverage PEP (APEP) can be obtained as

APEP =1π

∫ π/2

0

N∏

n=1

[M|Δn |2

( μ

8 sin2 θ

)]dθ. (25)

Moreover, using the tight approximation for the Gaussian Q-function presented in [33, Eq. (14)] (i.e., Q(x) ≈ 1/12 exp(−x2) + 1/4 exp(−2x2/3)), an expression accurately approx-imating APEP can be deduced as

APEP ≈ 112

N∏

n=1

[M|Δn |2

(μ

8

)]+

14

N∏

n=1

[M|Δn |2

(μ

6

)].

(26)In the following, analytical expressions for the MGF of |Δn |2will be obtained.

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Proposition 1: An integral representation for the MGF of|Δn |2 can be deduced as

M|Δn |2 (s)=(αn/b0,n )αn

Γ(αn )

×∫ ∞

0

bαn −1

2bs + 1exp

(

− Ans

2bs + 1− αnb

b0,n

)

db.

(27)

Proof: By employing the definition of the MGF, M|Δn |2 (s)can be obtained as

M|Δn |2 (s) =∫ ∞

0exp(−sx)f|Δn |2 (x)dx

=(αn/b0,n )αn

2Γ(αn )

×∫ ∞

0

∫ ∞

0exp

(

−sx − αnb

b0,n− x + A2

n

2b

)

×I0

(An

√x

b

)

bαn −2 dbdx. (28)

By changing the order of integration, the above equation can beexpressed as

M|Δ n |2 (s) =(αn /b0 ,n )α n

2Γ(αn )

∫ ∞

0bα n −2 exp

(

−αn b

b0 ,n

)

[∫ ∞

0exp

(

−sx − x + A2n

2b

)

I0

(An

√x

b

)

dx

]

db.

(29)

The inner integral, i.e., with respect to x can be evaluated byemploying [34, Eq. (3.15.2.2)] as

∫ ∞

0exp

(

−sx − x + A2n

2b

)

I0

(An

√x

b

)

dx

=2b

2sb + 1exp

[1

2An b(2sb + 1)

]

. (30)

Substituting (30) into (29) and after some straightforward math-ematical manipulations, (27) is readily deduced thus completingthe proof. �

The integral in (27) can be accurately approximated by em-ploying a Gauss–Chebyshev quadrature (GCQ) technique as[35]

M|Δn |2 (s)≈(αn/b0,n )αn

Γ(αn )

×J∑

j=0

wjtj

αn −1

2tj s + 1exp

(

− Ans

2tj s + 1− αntj

b0,n

)

(31)

where J is the number of integration points, tj are the abscissasand wj the corresponding weights. In [36, eqs. (22) and (23)],

tj and wj are given as

tj = tan[π

4cos

(2j − 1

2Jπ

)

+π

4

]

(32a)

wj =π2 sin

( 2j−12J π

)

4J cos2[

π4 cos

( 2j−12J π

)+ π

4

] . (32b)

For the special case of An = 0, (27) can be evaluated in closedform, as it will be shown next.

Corrolary 1: For the special case of A = 0 the MGF of|Δn |2 can be deduced in closed form as

M|Δn |2 (s) =(

αn

2sb0,n

)αn /2

exp(

αn

4sb0,n

)

×W−αn /2,(αn −1)/2

(αn

2sb0,n

)

. (33)

This result can be readily deduced by employing the integralrepresentation of the Whittaker W -function given in [26, Eq.(9.222)]. It is noted that (33) is in agreement with a previouslyknown result, namely the analytical expression for the MGF ofthe K-distribution [37, Eq. (4)].

C. Diversity Gain Analysis

The diversity gain of the considered OSM MIMO system canbe obtained by using the approach presented in [38]. In partic-ular, a generic analytical expression, which becomes asymptot-ically tight at high SNR values, will be derived for the APEPappearing in (25), as follows:

Proposition 2: For high SNR values, (25) can be approxi-mated by

APEPμ�1≈

2N −1Γ(N + 1

2

)

√πΓ (N + 1)

[N∏

n=1

c�

](μ

4

)−N

(34)

where

cn =

(An

2

)(αn −1)/2(αn/b0,n )(αn +1)/2

Γ(αn )

×Kαn −1

⎛

⎝

√2Anαn

b0,n

⎞

⎠. (35)

Proof: According to [38, Proposition 3], the asymptotic errorperformance of the OSM system depends on the behavior ofM|Δn |2 (s), as s → ∞. To determine an analytical asymptoticexpression for APEP a Taylor series expansion is employed toapproximate M|Δn |2 (s) as

|M|Δn |2 (s)| = cn |s|−dn + o(|s|−dn ), s → ∞ (36)

where cn and dn are parameters that determine the diversity andcoding gains of the nth diversity branch, respectively. Observe

that since As/(2sb + 1)s→∞≈ A/(2b) and 1/(2sb + 1)

s→∞≈

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1/(2bs), (27) yields

M|Δn |2 (s) ≈(αn/b0,n )αn

2sΓ(αn )

×∫ ∞

0bαn −2 exp

(

− An

2b− αnb

b0,n

)

db. (37)

By employing [34, Eq. (2.2.2.1)], (37) can be solved in closedform yielding

M|Δn |2 (s) ≈(

An

2

)(αn −1)/2(αn/b0,n )(αn +1)/2

sΓ(αn )

×Kαn −1

⎛

⎝

√2Anαn

b0,n

⎞

⎠ . (38)

By comparing (38) and (36) it is readily deduced that dn = 1and cn is given by (35). Thus, by substituting (36) into (25),the asymptotic PEP expression can be obtained as in (34) whichconcludes the proof. �

From (34) it is clear that the diversity gain achieved by theconsidered system is equal to N . It is also evident that the diver-sity gain depends only on the number of the receive aperturesand is independent of the fading severity. This finding is in agree-ment with other equivalent observations reported in [27], [39]and[40], which have been obtained for the case of RF MIMOwireless systems.

It is noted that for the special case An = 0, i.e., when |Δn |2follows the K-distribution, by employing the asymptotic result

Kt(x)x→0≈ (Γ(t)/2) (2/x)t [35], cn can be further simplified as

cn =αn

2b0,n (αn − 1). (39)

IV. PERFORMANCE ANALYSIS OF CODED OSM OVER

TURBULENCE CHANNELS

When coded OSM is employed, the input signal s(t) is firstencoded by a convolutional encoder. The encoded data are inter-leaved by a random block interleaver and transmitted through theoptical wireless channels using spatial modulation. It is assumedthat perfect interleaving at the transmitter and de-interleavingat the receiver is used. Considering maximum likelihood softdecision decoding, the log likelihood ratios (LLRs) for the ithconstellation bit when the �th transmitting antenna is active arecomputed as [18, Eq. (6)]

LLR = logPr{�i = 1|y}Pr{�i = 0|y}

= log

∑�∈Li

1exp

(−‖ y − h� s� ‖2/N0

)

∑�∈Li

0exp

(−‖ y − h� s� ‖2/N0

) (40)

where L ∈ {1 : M} is the set of spatial constellation points, Li1

and Li0 are subsets from L containing the transmitter indices

having “1” and “0” at the ith bit, respectively. The resulting dataare finally decoded by a Viterbi decoder.

A union bound on the ABEP of a coded communicationsystem can be obtained as [30]

Pub ≤ 1n

∑

X

P (X)∑

X �=X ′

q(X,X′)PEP(X,X′) (41)

where P (X) is the probability that the coded sequence X istransmitted, q(X,X′) is the number of information bit errorsin choosing another coded sequence X′ instead of X n is thenumber of information bits per transmission and PEP(X,X′)is the PEP, i.e., the probability of selecting X′ when X wasactually transmitted.

By employing [30, p. 510], (41) can be efficiently evaluatedas

Pub ≤ 1n

∑

X

P (X)∫ π/2

0

[∂

∂NT [D(θ), N ]

∣∣∣∣N =1

]

(42)

where T [D(θ), N ] is the transfer function of the employed con-volutional code, N is an indicator variable taking into accountthe number of the erroneous bits and D(θ) depends on the un-derlying PEP expression. Furthermore, assuming that uniformerror probability (UEP) codes are considered and taking intoaccount the symmetry property this code family exhibits, thusmaking the distance structure of a UEP code independent of thetransmitted sequence, (42) can be further simplified as [30]

Pub ≤ 1π

∫ π/2

0

[1n

∂

∂NT [D(θ), N ]

∣∣∣∣N =1

]

. (43)

For M = 2, using (17), (18) and Craig’s formula for the Q-function, i.e., Q(x) = 1/π

∫ π/20 exp(−x2/2 sin2 θ)dθ, D(θ)

can be expressed as

D(θ) =N∏

n=1

M|Δn |2( μ

8 sin2 θ

)(44)

where M|Δn |2 can be obtained from (27). When M > 2, byemploying [18, Eq. (13)], and using a similar line of argumentsas for the case of M = 2, D(θ) can be written as

M∏

m 1 =1

M∏

m 2 �=m 1 =1

M|Δm 1 , m 2 |2( μ

8 sin2 θ

)(45)

where |Δm 1 ,m 2 |2 =‖ hm 1 − hm 2 ‖2 . The last MGF can be an-alytically computed with the help of (27) in a straight-forwardway.

V. DIVERSITY TECHNIQUES FOR COHERENT FSO SYSTEMS

Diversity at the transmitter and/or the receiver is a commonlyused technique to mitigate the deleterious impact of atmosphericturbulence on coherent FSO system performance. In this sec-tion, firstly the most popular diversity techniques for coherentFSO systems will be presented [41]. Such diversity schemeswill be also considered for the FSO communication systems in-vestigated in this paper as follows: Receive diversity by employ-ing MRC or SC and transmit diversity based on the Alamoutischeme.

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The instantaneous SNR at the output of the coherent MRCreceiver assuming equal average SNR per receiving aperture, μcan be expressed as [41, Eq. (11)]

γMRC = μ

N∑

n=1

In (46)

where In denotes the optical signal irradiance at the nth branch.The ABEP of coherent MRC receivers with differential phaseshift keying (DPSK) can be deduced as [13, Eq. (14)]

PE =12

N∏

n=1

MIn(μ) (47)

whereas for binary phase shift keying (BPSK) as [41, Eq. (18)]

PE =1π

∫ π/2

0

N∏

n=1

MIn

( μ

sin2 θ

)dθ. (48)

The instantaneous SNR at the output of the coherent SCreceiver is given by [41, Eq. (15)]

γSC = max{μI1 , μI2}. (49)

For the SC case, an analytical expression for the ABEP is muchmore difficult to be deduced and, thus for this case the ABEP willbe evaluated exclusively by means of Monte Carlo simulations.

In [16], transmit diversity techniques, based on the Alam-outi scheme, were explored for coherent OWC applications byemploying STBC. The instantaneous SNR at the input of thedemodulator of the optical receiver has a similar form as (46)[16], and therefore the ABEP of BPSK can be readily evaluatedby employing (48).

VI. PERFORMANCE EVALUATION RESULTS AND DISCUSSION

In this section the various performance evaluation resultswhich have been obtained by numerically evaluating the math-ematical expressions presented in Sections III and IV for un-coded and coded OSM systems operating over H-K turbulentchannels will be presented. In particular, for uncoded OSM sys-tems, ABEP vs. SNR for 2 × N OSM systems (obtained using(26) with (27), and (34)) the results are presented in Figs. 1–4.For the uncoded schemes, in order to validate the accuracy ofthe previously mentioned expressions, comparisons with com-plementary Monte Carlo simulated performance results are alsoincluded in these figures. As far as the performance of codedOSM systems is concerned, ABEP upper bounds vs. SNR havebeen obtained using (43) with (27) and the results are illustratedin Fig. 5. Table VI summarizes the different values of the systemparameters used to obtain the simulated performance evaluationresults. Note that in obtaining the various performance resultsa system with M = 2 transmitting apertures was considered.The motivation behind this choice is the inherent design com-plexities and cost of adopting multiple transmitters in currentstate-of-the-art coherent optical wireless links [16], as well asthe fact that the proposed framework yields exact results forM = 2 transmitters.

Fig. 1. ABEP of uncoded OSM for 2 × N MIMO H-K turbulent channels asa function of the average SNR, μ, for various number of receiving apertures,N . Simulation Parameters: A1 ,n = 2, A2 ,n = 1, θ1 ,n = π/3, θ2 ,n = π/4,αn = 2, b0 ,n = 2.

Fig. 2. ABEP of uncoded OSM for 2 × 2 and 2 × 4 MIMO H-K turbulentchannels as a function of the average SNR, μ, for various values of link distances,L. Simulation Parameters: λ = 1550 nm, C 2

n = 1.7 × 10−14 m−2/3 , θ1 ,n =π/3, θ2 ,n = π/4, b0 ,n = 1.

In order to demonstrate the impact of the number of the re-ceiving apertures on the system performance, Fig. 1, presentsthe ABEP performance as a function of the average SNR, μ,of 2 × N MIMO OSM systems with N ∈ {1, 2, 3, 4}. Inde-pendent and identically distributed branches are consideredwith A1,n = 2, A2,n = 1, θ1,n = π/3, θ2,n = π/4, αn = 2,b0,n = 2. The obtained results clearly indicate that the ABEP

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Fig. 3. ABEP of uncoded OSM for 2 × 3 MIMO H-K turbulent channels asa function of the average SNR, μ, for weak (C 2

n = 10−15 m−2/3 ) and strongC 2

n = 3 × 10−14 m−2/3 turbulence and for various values of b0 ,n . SimulationParameters: λ = 1550 nm, θ1 ,n = π/3, θ2 ,n = π/4.

Fig. 4. ABEP Comparison of 2 × 2 OSM with 1 × 2 coherent MRC systemsemploying DPSK, as a function of the average SNR, μ, for various valuesof A1 ,n . Simulation Parameters: A2 ,n = 0, θ1 ,n = 0, θ2 ,n = 0, αn = 1.5,b0 ,n = 1.5.

curves, obtained using (26), are in close agreement with theperformance obtained via simulations, thus verifying the cor-rectness of the proposed analysis. Moreover, it is evident thatthe asymptotic ABEP curves accurately predict the diversitygain achieved by the considered system for all test cases.

In Fig. 2, the dependence on the link distance, L, of the ABEPof a 2 × N MIMO OSM system is illustrated. The considered

Fig. 5. ABEP upper bounds of convolutional coded OSM for 2 × 2 and 2 ×1 H-K turbulent channels as a function of the average SNR, μ, for variousvalues of link distances, L. Simulation Parameters: λ = 1550 nm, C 2

n = 1.7 ×10−14 m−2/3 , θ1 ,n = π/3, θ2 ,n = π/4, b0 ,n = 1.

TABLE IISYSTEM PARAMETERS FOR SIMULATION

Parameter Value

Operating wavelength, λ (nm) 1500Refractive index parameter, C 2

n (m−2 / 3 ) {3, 1.7, 0.1} × 10−1 4 ,Link distance, L (m) {500, 1000, 1500, 2000}θ1 , n π/3θ2 , n π/4A 1 , n {0, 1, 2, 3}A 2 , n {0, 1}b0 , n {0.25, 0.5, 1, 1.5, 2}Number of Transmitting apertures, M 2Number of Receiving apertures, N {1, 2, 3, 4}

system is again equipped with either N = 2 or N = 4 receivingapertures and identically distributed branches are assumed. Theparameters of the H-K distribution are calculated from (15)and (16) assuming plane wave propagation. Following [42], itis further assumed that the operating wavelength is λ = 1550nm and C2

n = 1.7 × 10−14 m−2/3 , which, as it was pointed outin Section II, is a typical value of refractive index for FSOlinks operating near the ground plane during daytime [29]. Asexpected, the error performance deteriorates as L increases fromL = 500 m to L = 1500 m. Moreover, it is evident that anincrease in L from 500 to 1000 m leading to a more severeperformance deterioration than in the case where L increasesfrom 1000 to 1500 m. For all cases considered, the analyticalresults obtained using (26) are compared with the equivalentresults obtained by means of Monte-Carlo computer simulationsand again they match very well.

In Fig. 3 the impact of the turbulence strength as well asof the parameter b0 on system performance is illustrated. In

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particular, Fig. 3 presents the ABEP performance as a functionof the average SNR, μ, of 2 × N MIMO OSM systems withN = 3 receiving apertures, link distance L = 2000 m, and var-ious values of b0 . Two different values of the refractive indexparameter C2

n are considered, i.e., C2n = 3 × 10−14 m−2/3 and

C2n = 10−15 m−2/3 to account for strong and weak turbulence,

respectively [29], [43]. The obtained results clearly show thatthe error performance improves as b0 increases and/or C2

n de-creases. Moreover, the analytical results obtained using (26)agree well with the equivalent results obtained by means ofMonte–Carlo computer simulations.

Next we compare the proposed OSM system with two al-ternative coherent FSO systems that can provide performanceenhancements by means of transmit, i.e., multiple-input single-output (MISO) or receive diversity i.e. single-input multiple-output (SIMO). It is noted that for similar aperture configura-tions, a fair comparison between coherent and IM/DD systemsseems difficult as the same received laser power leads to differentSNRs for each of these schemes [17]. On the other hand, in orderto perform a fair comparison between OSM and the alternativeMISO or SIMO systems under the same propagation channelconditions, the aperture configuration of the FSO systems undercomparison should be carefully selected. Specifically, becauseof the fact that the diversity gain of OSM equals to only thenumber of the receive apertures, i.e., no transmit diversity gaincan be achieved, the number of transmit or receive apertures ofthe alternative systems must be hence selected to be equal to thenumber of receive apertures of the OSM system. To this end andfor a fair comparison in our paper a 2 × 2 OSM system is com-pared with the following two alternative FSO communicationsystems which also employ coherent detection:

1) A 1 × 2 heterodyne FSO communication system whichemploys DPSK [13] and MRC or SC;

2) A 2 × 1 coherent FSO system employing the Alamoutischeme [16] and BPSK.

In order to simplify the underlying mathematical analysis,it is assumed that the PDF of In is given by (9) with the pa-rameters An being all zero, i.e., the PDF considered here is theK-distribution. Thus, MIn

(μ) can be readily obtained in closedform from (33) by replacing b0,n with b0,n /2. In Fig. 4, theABEP of 2 × 2 MIMO OSM links is compared with the ABEPof 1 × 2 coherent FSO systems with DPSK considering identi-cally distributed links. In order to compare these systems underthe same propagation conditions, it is assumed that αn = 1, 5,b0,n = 1.5, A2,n = 0 and A1,n = {0, 1, 2, 3}. As it can be ob-served, when either MRC or SC are employed, although coher-ent DPSK performs worse than the OSM for values of A1,n upto approximately 1, it outperforms OSM at lower values of A1,n .Moreover, although the OSM outperforms the Alamouti schemefor A1,n = 2 and 3, it performs similarly for high SNR valueswhen A1,n = 1. It is noted that for A1,n = 1 and lower valuesof A1,n the Alamouti scheme yields the best performance of theconsidered schemes. However, when more transmit aperturesare employed, this advantage is compensated by the superiorspectral efficiency of OSM and its lower hardware complex-ity as compared to coherent MRC. Specifically, as pointed out

in [18], OSM offers increased spectral efficiency by a factorlog2(M). Moreover, as only one transmitting aperture is acti-vated at any bit duration, OSM has a lower decoding complexityas compared to conventional MRC and Alamouti schemes.

In Fig. 5, upper bounds on the ABEP of convolutional coded2 × 1 and 2 × 1 OSM systems are depicted, assuming similarpropagation conditions to those considered in Fig. 2. Consider-ing a convolutional code with rate 1/3 and constraint length of3, its transfer function is given as [44, Eq. (8.2.6)]

T [D(θ), N ] =D(θ)6N

1 − 2ND(θ)2 . (50)

Substituting (50) into (43), a union bound on the ABEP can beobtained as

Pub ≤ 1π log2(M)

∫ π/2

0

D(θ)6

(1 − 2D(θ)2)2 dθ. (51)

The performance results of Fig. 5 clearly show that, as expected,the incorporation of convolutional coding significantly enhancesthe performance of OSM systems, even when a small numberof receive apertures is employed.

VII. CONCLUSION

In this paper, the use of spatial modulation technique forcoherent FSO communication systems has been proposed. Wehave provided a comprehensive analytical framework for errorperformance analysis which is valid for a great deal of atmo-spheric turbulence scattering channel models but focusing onthe H-K distribution. The proposed framework has revealed im-portant information about the performance of OSM over suchturbulent channels, including the effect of fading severity andthe achievable diversity gain. In particular, it has been shownthat the diversity gain depends on the number of receive aper-tures only while channel parameters affect the coding gain only.Upper bounds for the ABEP performance of coded OSM sys-tems have also been derived, demonstrating that coding tech-niques can greatly enhance the performance of OSM. Extensivecomputer simulation performance evaluation results have beenalso obtained which have verified the accuracy of the analyti-cal approach. It was shown that OSM can provide significantperformance enhancements in the presence of H-K atmosphericturbulence. The improvements are comparable to the ones of-fered by conventional coherent systems with spatial diversity,while outperforming the latter in terms of spectral efficiencyand hardware complexity. Specifically, OSM offers increasedspectral efficiency by a factor log2(M) when multiple transmit-ting apertures are employed while it completely avoids multiplecommunication chains in a similar fashion as in RF systems.Besides, under specific propagation conditions, OSM can yieldbetter performance than conventional SIMO systems employingMRC or SC. The proposed framework, apart from enabling thecomputation of accurate performance evaluation results in anefficient manner, it also provides a useful tool for understand-ing the performance trend, important properties and tradeoffs of

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outdoor OSM operating in the presence of atmospheric turbu-lence.

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12 JOURNAL OF LIGHTWAVE TECHNOLOGY

Kostas P. Peppas was born in Athens, Greece, in 1975. He received the Diplomadegree in electrical and computer engineering and the Ph.D. degree in wirelesscommunications from the National Technical University of Athens, Athens, in1997 and 2004, respectively. From 2004 to 2007, he was with the Department ofComputer Science, University of Peloponnese, Tripolis, Greece, and from 2008to 2014 with the National Center for Scientific Research–“Demokritos,” Instituteof Informatics and Telecommunications as a Researcher. In 2014, he joinedthe Department of Telecommunication Science and Technology, University ofPeloponnese, where he is currently a Lecturer. His current research interestsinclude digital communications over fading channels, MIMO systems, wirelessand personal communication networks, and system level analysis and design.He has authored more than 70 journal and conference papers.

P. Takis Mathiopoulos (SM’ 94) received the Ph.D. degree in digital commu-nications from the University of Ottawa, Ottawa, ON, Canada, in 1989.

From 1982 to 1986, he was with Raytheon Canada, Ltd., working in the areasof air navigational and satellite communications. In 1988, he joined the Depart-ment of Electrical and Computer Engineering (ECE), University of BritishColumbia (UBC), Vancouver, BC, Canada, where he was a Faculty Member asa Professor from 2000 to 2003. From 2000 to 2014, he was with the Institutefor Space Applications and Remote Sensing (ISARS), National Observatory ofAthens (NOA), Athens, Greece, first as the Director and then as the Director ofresearch and established the Wireless Communications Research Group. As IS-ARS’ Director during 2000–2004, he led the Institute to a significant expansionR&D growth, and international scientific recognition. For these achievements,ISARS has been selected as a National Center of Excellence for the years 2005–2008. Since 2014, he has been an Adjunct Researcher at the Institute of Astron-omy, Astrophysics, Space Applications and Remote Sensing, NOA. Since 2003,he worked part time at the Department of Informatics and Telecommunications,University of Athens, Athens, where since 2014, he has been a Professor ofdigital communications. From 2008 to 2013, he was a Guest Professor with theSouthwest Jiaotong University, China. He is also appointed as a Guest Profes-sor at the School of Information Engineering, Yangzhou University, Yangzhou,China, for 2014–2015 under the Senior Foreign Expert Program of the Govern-ment of PR of China. Furthermore he is appointed by Keio University, Tokyo,Japan, as a Visiting Professor at the Department of Information and ComputerScience for 2015–2016 under the Top Global University Project of the Ministryof Education, Culture, Sports, Science and Technology Government of Japan.For the last 25 years, he has been conducting research mainly on the physicallayer of digital communication systems for terrestrial and satellite applications,including digital communications over fading and interference environments.He coauthored a paper in GLOBECOM’89 establishing for the first time in theopen technical literature the link between MLSE and multiple (or multisymbol)differential detection for the AWGN and fading channels. He is also interested inchannel characterization and measurements, modulation and coding techniques,synchronization, SIMO/MIMO, UWB, OFDM, software/cognitive radios, andgreen communications. In addition, since 2010, he has been actively involvedwith research activities in the fields of remote sensing, LiDAR systems, and pho-togrammetry. In these areas, he has coauthored more than 100 journal papers,mainly published in various IEEE and IET journals, four book chapters, andmore than 120 conference papers. He has been the PI for more than 40 researchgrants and has supervised the thesis of 11 Ph.D. and 23 Master students.

Dr. Mathiopoulos has been or currently serves on the editorial board ofseveral archival journals, including the IET Communications, and the IEEETRANSACTIONS ON COMMUNICATIONS from 1993 to 2005. He has regularlyacted as a consultant for various governmental and private organizations. Since1993, he has served on a regular basis as a Scientific Advisor and a TechnicalExpert for the European Commission (EC). In addition, since 2001, he has beenthe Greek Representative to high-level committees in the EC and the EuropeanSpace Agency. He has been a Member of the TPC of more than 70 internationalIEEE conferences, as well as the TPC Vice-Chair for the 2006-S IEEE VTC andthe 2008-F IEEE VTC, as well as the Cochair of the FITCE2011. He has de-livered numerous invited presentations, including plenary and keynote lectures,and has taught many short courses all over the world. As a Faculty Memberat the ECE, UBC, he was elected as an ASI Fellow and a Killam ResearchFellow. He received the two best paper awards for papers published in the 2ndInternational Symposium on Communication, Control, and Signal Processingin 2008), and the 3rd International Conference on Advances in Satellite andSpace Communications in 2011.