Transmit alternate laser selection with time diversity ... - OSA Publishing

Transmit alternate laser selection with time diversity for FSO communications Antonio Garc´ıa-Zambrana,1,∗ Rubén Boluda-Ruiz,1 Carmen Castillo-Vázquez,2 and Beatriz Castillo-Vázquez1 1 Department

2 Department

of Communications Engineering, University of Málaga, E-29071 Málaga, Spain of Statistics and Operations Research, University of Málaga, E-29071 Málaga, Spain ∗ [email protected]

Abstract: In this paper, a new transmit alternate laser selection (TALS) scheme for FSO communication systems using intensity modulation and direct detection (IM/DD) over atmospheric turbulence and misalignment fading channels is presented when limited time diversity is available in the turbulent channel. Assuming channel state information (CSI) at the transmitter and receiver and a time diversity order (TDO) limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. Closed-form asymptotic bit error-rate (BER) expressions are derived when the irradiance of the transmitted optical beam is susceptible to moderate-tostrong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO available in the turbulent channel, a significant diversity gain is achieved, providing a diversity order of (2L + 1 − TDO)TDO/2. © 2014 Optical Society of America

OCIS codes: (010.1330) Atmospheric turbulence; (060.2605) Free-space optical communication; (060.4510) Optical communications.

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#213237 - $15.00 USD Received 2 Jun 2014; revised 10 Sep 2014; accepted 16 Sep 2014; published 23 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023861 | OPTICS EXPRESS 23861

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1.

Introduction

Atmospheric free-space optical (FSO) transmission using intensity modulation and direct detection (IM/DD) can be considered as an important alternative to consider for next generation broadband in order to support large bandwidth, unlicensed spectrum, excellent security, and quick and inexpensive setup [1]. Nonetheless, this technology is not without drawbacks, being the atmospheric turbulence one of the most impairments, producing fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely reducing the link performance [2]. Additionally, an unsuitable alignment between transmitter and receiver due to vibrations in the transmitted beam can produce a greater performance degradation. In [3], the effects of atmospheric turbulence and misalignment considering aperture average effect were considered to study the outage capacity for single-input/single-output (SISO) links. In [4, 5], a wide range of turbulence conditions with gamma-gamma atmospheric turbulence and pointing errors is also considered on terrestrial FSO links, deriving closed-form expressions for the error-rate performance in terms of Meijer’s G-functions. Error control coding as well as diversity schemes, including both spatial-domain and temporal-domain techniques, can be used over FSO links to mitigate turbulence-induced fading [6–13]. In case of temporal-domain techniques, interleaving is usually adopted to improve the channel coding performance [12]. In [14–16], selection transmit diversity is proposed for FSO links over strong turbulence channels, where the transmit diversity technique based on the selection of the optical path with a greater value of irradiance has shown to be able to extract full diversity as well as providing better performance compared to general FSO spacetime codes (STCs) designs, such as conventional orthogonal space-time block codes (OSTBCs) and repetition codes (RCs). The optimality of selection transmit diversity as an optimal power allocation strategy for shot noise limited FSO systems has been proved in [17], proposing an extension of this scheme to systems with limited feedback. In [18], a novel approximate closed-form bit error-rate (BER) expression is derived for FSO links with transmit laser selection over K-distributed atmospheric turbulence channels. In [19, 20], comparing different diversity techniques, a significant improvement in terms of outage and error-rate performance is demonstrated when multiple-input/multiple-output (MIMO) FSO links based on transmit laser selection are adopted over misalignment fading channels in the context of wide range of turbulence conditions, showing that the diversity order is independent of the pointing error effects when the equivalent beam radius at the receiver is at least twice the value of the pointing error displacement standard deviation at the receiver. Another remarkable conclusion is the fact that better performance is achieved when increasing the number of transmit apertures instead of the number of receive apertures in order to guarantee a same diversity order. In this paper, a new transmit alternate laser selection (TALS) scheme with repetition coding for IM/DD FSO communications over atmospheric turbulence and misalignment fading channels is presented. Assuming channel state information (CSI) at the transmitter and receiver and a time diversity order (TDO) limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statis#213237 - $15.00 USD Received 2 Jun 2014; revised 10 Sep 2014; accepted 16 Sep 2014; published 23 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023861 | OPTICS EXPRESS 23863

tics in an alternating way. Taking advantage of the temporal diversity limited to TDO, we can assume that the channel coefficients fade independently from one block to the next repetition of the same block. Closed-form asymptotic bit error-rate (BER) expressions are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma (GG) distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO available in the turbulent channel, a significant diversity gain is achieved, obtaining a diversity order of (2L + 1 − TDO)TDO/2. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. However, in spite of obtaining a higher diversity order as TDO increases, the adoption of a time diversity order superior to 2 translates to a remarkable coding gain disadvantage, not providing a significant better performance in the context of moderate turbulence when TDO ≥ 3 for FSO systems where BER targets as low as 10−9 are typically aimed to achieve. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results, showing that asymptotic expressions here obtained lead to simple bounds on the bit error probability in the range from low to high signal-to-noise ratio (SNR). 2.

System and channel model

We adopt a multiple-input/single-output (MISO) array based on L laser sources, assumed to be intensity-modulated only and and all pointed towards a distant photodetector, assumed to be ideal noncoherent (direct-detection) receiver. The sources and the detector are physically situated so that all transmitters are simultaneously observed by the receiver. The use of infrared technologies based on IM/DD links is considered, where the instantaneous current ym (t) in the receiving photodetector corresponding to the information signal transmitted from the mth laser can be written as ym (t) = ηim (t)x(t) + z(t) (1) where η is the detector responsivity, assumed hereinafter to be the unity, X , x(t) represents the optical power supplied by the mth source and Im , im (t) the equivalent real-valued fading gain (irradiance) through the optical channel between the mth laser and the receive aperture. Additionally, the fading experienced between source-detector pairs Im is assumed to be statistically independent. Z , z(t) is assumed to include any front-end receiver thermal noise as well as shot noise caused by ambient light much stronger than the desired signal at the detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance σ 2 = N0 /2, i.e. Z ∼ N(0, N0 /2), independent of the on/off state of the received bit. Since the transmitted signal is an intensity, X must satisfy ∀t x(t) ≥ 0. Due to eye and skin safety regulations, the average optical power is limited and, hence, the average amplitude of X is limited. The received electrical signal Ym , ym (t), however, can assume negative amplitude values. We use Ym , X, Im and Z to denote random variables and ym (t), x(t), im (t) and z(t) their corresponding realizations. (a) (p) The irradiance is considered to be a product of three factors i.e., Im = ζm Im Im where ζm is (a) (p) the deterministic propagation loss, Im is the attenuation due to atmospheric turbulence and Im the attenuation due to geometric spread and pointing errors. ζm is determined by the exponential Beers-Lambert law as ζm = e−Φd , where d is the link distance and Φ is the atmospheric attenuation coefficient. It is given by Φ = (3.91/V (km)) (λ (nm)/550)−q where V is the visibility in kilometers, λ is the wavelength in nanometers and q is the size distribution of the scattering particles, being q = 1.3 for average visibility (6 km < V < 50 km), and q = 0.16V + 0.34 for haze visibility (1 km < V < 6 km) [21]. To consider a wide range of turbulence conditions, the gamma-gamma turbulence model proposed in [2] is here assumed. Regarding to the impact of


pointing errors, we use the general model of misalignment fading given in [3] by Farid and Hranilovic, wherein the effect of beam width, detector size and jitter variance is considered. A closed-form expression of the combined probability density function (PDF) of Im was derived in [4] as αm βm ϕm2 αm βm ϕm2 , i≥0 (2) G3,0 i fIm (i) = A0 ζm Γ(αm )Γ(βm ) 1,3 A0 ζm ϕm2 − 1, αm − 1, βm − 1 where Gm,n p,q [·] is the Meijer’s G-function [22, eqn. (9.301)] and Γ(·) is the well-known Gamma function. Assuming plane wave propagation, α and β can be directly linked to physical parameters through the following expresions [23]: h i−1 12/5 α = exp 0.49σR2 /(1 + 1.11σR )7/6 − 1 h i−1 12/5 β = exp 0.51σR2 /(1 + 0.69σR )5/6 − 1

(3a) (3b)

where σR2 = 1.23Cn2 κ 7/6 d 11/6 is the Rytov variance, which is a measure of optical turbulence strength. Here, κ = 2π/λ is the optical wave number and d is the link distance in meters. Cn2 stands for the altitude-dependent index of the refractive structure parameter and varies from 10−13 m−2/3 for strong turbulence to 10−17 m−2/3 for weak turbulence [2]. It must be emphasized that parameters α and β cannot be arbitrarily chosen in FSO applications, being related through the Rytov variance. It can be shown that the relationship α > β always holds, and the parameter β is lower bounded above 1 as the Rytov variance approaches ∞ [24]. In relation to the impact of pointing errors [3], assuming a Gaussian spatial intensity profile of beam waist radius, ωz , on the receiver plane at distance z from the transmitter and a circular receive aperture of radius r, ϕ = ωzeq /2σs is the ratio between the equivalent beam radius at the receiver displacement standard deviation (jitter) at the receiver, √ √ and the pointing error √ ωz2eq = ωz2 πerf(v)/2v exp(−v2 ), v = πr/ 2ωz , A0 = [erf(v)]2 and erf(·) is the error function [22, eqn. (8.250)]. Nonetheless, the PDF in Eq. (2) appears to be cumbersome to use in order to obtain simple closed-form expressions in the analysis of FSO communication systems. To overcome this inconvenience, the PDF is approximated by using the first two terms of the Taylor expansion at i = 0 as fIm (i) = am ibm −1 +cm ibm +O(ibm +1 ). As proposed in [25], we adopt the approximation fIm (i) ≈ am ibm −1 exp(icm /am ). Different expressions for fIm (i), depending on the relation between the values of ϕ 2 and β , can be written as ϕm2 (αm βm )βm Γ(αm − βm )

βm −1

i

2 −β αm βm ϕm m

i e (A0 ζm )βm Γ(αm )Γ(βm ) (ϕm2 − βm ) 2 ϕm2 (αm βm )ϕm Γ αm − ϕm2 Γ βm − ϕm2 ϕ 2 −1 fIm (i) ≈ i m , 2 (A0 ζm )ϕm Γ(αm )Γ(βm )

fIm (i) ≈

(

)

2 +1 A0 ζ (αm −βm −1) βm −ϕm

(

),

ϕm2 > βm

ϕm2 < βm

(4a) (4b)

Therefore, corresponding expressions for the coefficients am , bm and cm are given by  ϕ 2 (αm βm )βm Γ(αm −βm )  ϕm2 > βm  (A ζ m)βm Γ(α )Γ(β )(ϕ 2 −β ) , m m m 0 m m 2 am = 2 ϕm 2 2   ϕm (αm βm ) Γ(2αm −ϕm )Γ(βm −ϕm ) , ϕm2 < βm (A0 ζm )ϕm Γ(αm )Γ(βm )  2 −β αm βm (ϕm  m) βm − 1, ϕm2 > βm ϕm2 > βm 2 +1) , A ζ (α −β −1) β −ϕm ( m m m bm = c = 0 m 2 2 ϕm − 1, ϕm < βm  0, ϕ2 < β m

(5)

m


It can be noted that the second term of the Taylor expansion is equal to 0 when the diversity order is not independent of the pointing error effects, i.e. ϕm2 < βm . Assuming CSI at the transmitter and receiver and temporal diversity order equal to TDO, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit a block of TDO information bits. Implementing repetition coding with these blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. In this way, information bits in each block will be distributed among the TDO sources out of the available L lasers corresponding to greater values of scintillation, I(L) , I(L−1) and I(L−TDO+1) , where I(1) , I(2) , ..., I(L) is a new sequence of L auxiliary random variables obtained by arranging the random sequence I1 , I2 , ..., IL in an increasing order of magnitude. Taking advantage of the temporal diversity limited to TDO, we can assume that the channel coefficients fade independently from one block to the next repetition of the same block. As a result of this, different TDO order statistics affecting to the same information bit can be considered independent since they are selected from the different groups of random variables and, hence, they do not influence each other. This fact is key to achieve a relevant improvement in performance. Since atmospheric scintillation is a slow time varying process relative to typical symbol rates of an FSO system, wherein the coherence time ranges from a few milliseconds to tens of milliseconds [13], we consider the time variations according to the theoretical block-fading model due to the frozen-atmosphere characteristics of optical turbulence, where the channel fade remains constant during a block (corresponding to the channel coherence interval τc ) and changes to a new independent value from one block to next. Hence, the channel may be assumed to be constant for hundreds of thousands of bits for gigabits per second (Gbps) signaling rates [2]. In other words, channel fades are assumed to be independent and identically distributed (i.i.d.). This temporal correlation can be overcome by means of long interleavers, being usually assumed both in the analysis from the point of view of information theory and error rate performance analysis of coded FSO links [8, 26, 27]. In [13] turbo product code (TPC) as the channel coding scheme is applied to FSO communications, investigating the efficiency of interleaving for different interleaving depths. However, as in [12, 28], we here assume that the interleaver depth can not be infinite and, hence, we can potentially benefit from a degree of time diversity limited equal to TDO. This consideration is justified from the fact that the latency introduced by the interleaver is not an inconvenience for the required application. For example, for a time diversity order available of TDO = 2, i.e. two channel fades i1 and i2 per frame, perfect interleaving can be done by simply sending the same information delayed at least the expected fade duration τc , as shown experimentally in [29]. In this case, a minimum required buffer size corresponding to (TDO − 1)Rb τ symbols must be assumed, being τ > τc and Rb the signaling rate. Based on the concept of temporal-domain diversity reception (TDDR), this idea has been applied for FSO links in [29–31], where two separate channels over the same transmit and receive path are implemented. Both channels carry the same data, but one of the channels is delayed by the expected fade duration. In [32] subcarrier time delay diversity (STDD) is proposed as an alternative means of mitigating the channel fading. In [33] the use of wavelength and time diversity in wireless optical communication systems that operate under different atmospheric turbulence conditions is analyzed, considering BER and outage probability as performance metrics. Decode-and-forward relay-assisted FSO communications using time-diversity are proposed in [34].


3.

Error-rate performance analysis

In this section, an optical array based on L laser sources, all pointed towards a distant photodetector, is considered. We present simple bounds on the bit error rate using a pairwise error probability (PEP) analysis in the range from low to high SNR, taking advantage of the simpler expressions in Eq. (4). Here, it is assumed that the average optical power transmitted is Popt , being adopted an on-off keying (OOK) signaling based on a constellation of two p equiprobable points in a one-dimensional space with an Euclidean distance of dE = 2Popt Tb ξ , where the parameter Tb is the bit period and ξ represents the square of the increment in Euclidean distance due to the use of a pulse shape of high peak-to-average optical power ratio (PAOPR), as explained in a greater detail in [19, appendix]. In the analysis, a perfect Gray code is also assumed and all symbols are transmitted with equal probability. The PEP represents the probˆ when in fact the codeword X was transmitted. ability of choosing the space-time codeword X According to the novel transmit diversity transmit squeme here proposed, based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit a block of TDO information bits, the encoder takes a block of TDO bits x1 , x2 ,...,xTDO in each encoding operation and maps them to the transmit lasers according to a code matrix given by     x1 x4 x3 x2 x1 x3 x2  x2 x1 x4 x3  x1 x2  , X3 =  x2 x1 x3  , X4 =  X2 =  x3 x2 x1 x4  (6) x2 x1 x3 x2 x1 x4 x3 x2 x1 for values of TDO of 2, 3 and 4, respectively. Each row is associated to the Lth, (L − 1)th, (L − 2)th,... order statistics corresponding to the scintillation and each column is related to the repetition of the block of TDO information bits using the TDO largest order statistics in an alternating way, and assuming that channel coefficients fade independently as a consequence of the temporal diversity. In order to suit space-time coded OOK formats using any pulse shape to op∆ tical wireless communications, the complement of a signal xi can be redefined as xi = −xi + dE , where by complement we mean that the signal waveform is obtained by reversing the roles of ”on” and ”off”, following the approach presented by Simon in [35]. For instance, according to Eq. (1) and for a time diversity order of 3, the received signals in the first, second and third bit intervals, respectively, are given by r = [ r1 r2 r3 ]T so that          I(L),(1) I(L−1),(1) I(L−2),(1) 0 r1 x1 z1  r2  −  I(L),(2)  · dE =  I(L−1),(2) I(L−2),(2) −I(L),(2)   x2  +  z2  (7) r3 I(L),(3) x3 z3 I(L−2),(3) −I(L),(3) I(L−1),(3) which can be rewritten in a matrix form as r − ro = H · x + z. Here, I(1),(n) , I(2),(n) , ..., I(L),(n) represent the order statistics corresponding to the nth bit interval. Taking advantage of the temporal diversity, it must be noted that the different order statistics I(L) , I(L−1) and I(L−2) , affecting to the same information bit, can be considered independent since they are selected from the different groups of random variables and, hence, they do not influence each other. It must be noted that this approach cannot only be viewed as a two-stage transmit laser selection scheme where a space-time block coding scheme is applied to the system using the selected transmit lasers as proposed in antenna-subset selection, technique well known for RF systems [36, 37]. Here, according to the code matrix in (6), the available temporal diversity is fully exploited in order to achieve different channel gains corresponding to each column. In this way, the space-time coding takes into account the repetition of the block of TDO information bits using different laser-subset selections and, hence, different order statistics affecting to the same #213237 - $15.00 USD Received 2 Jun 2014; revised 10 Sep 2014; accepted 16 Sep 2014; published 23 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023861 | OPTICS EXPRESS 23867

information bit will not mutually influence each other, since they are selected from different subsets of random variables. For the sake of clarity, it must be emphasized that the space-time codeword in (6) is here defined by the repetition of the block of TDO information bits using the TDO largest order statistics from different TDO subsets in an alternating way. Assuming perfect CSI and a receiver that implements a maximum-likelihood metric, i.e., one that minimizes the metric m(r, x) = kr − ro −H · xk2 (8) where the squared norm of a matrix is defined by the sum of the magnitude squared of all its elements, the conditional PEP with respect to scintillation coefficients of greater value, I(L) , I(L−1) ,... and I(L−TDO+1) , for the shortest error event is given as [38]  v u 2 u dE u TDO TDO 2  t ˆ I(r) P X→X = Q ∑ ik    L−TDO+1≤r≤L 2N0 k=1

(9)

where Q(·) is the Gaussian-Q function. Here, the division of dE by TDO is considered so as to maintain the average optical power in the air at a constant level of Popt , being transmitted by each laser an average optical power of Popt /TDO. Substituting the value of dE gives v  u u 2γξ TDO ˆ I(r) = Q t P X→X (10) ∑ i2  L−TDO+1≤r≤L TDO2 k=1 k 2 T /N is the average receiver electrical SNR in the presence of the turbulence, where γ = Popt 0 b knowing that PDF in (2) is normalized. Under the assumption of time diversity, we can exploit ˆ by averaging independency among fading coefficients to obtain the average PEP, P(X → X), (10) as follows  v u Z ∞Z ∞ Z ∞ L u 2γξ TDO ˆ = P(X → X) ... Q t i2  fI(r) (ir−L+TDO )dir−L+TDO (11) ∏ 2 ∑ k TDO k=1 r=L−TDO+1 | 0 0{z 0 }

TDO-fold

According to order statistics [39], the PDF corresponding to I(r) can be written as fI(r) (i) =

Γ(L + 1) fI (i)(1 − FI (i))L−r (FI (i))r−1 Γ(r)Γ(L − r + 1)

(12)

being FI (i) the cumulative density function (CDF) corresponding to the turbulence model. Using the binomial expansion to approximate the term (1 − FI (i))L−r and taking advantage of the simpler expressions in Eq. (4) wherein fI (i) ≈ aib−1 exp(ic/a), the PDF in Eq. (12) can be simplified as Γ(L + 1) Γ(L + 1)ar b1−r c(br + 1) br−1 r−1 fI(r) (i) ' fI (i)(FI (i)) = exp i i (13) Γ(r)Γ(L − r + 1) Γ(r)Γ(L − r + 1) a(b + 1) An union bound on the average BER can be found as [38] Pb (E) ≤

1 P(X) nc ∑ X

ˆ ˆ → X) ∑ n(X, X)P(X

(14)

ˆ X6=X


ˆ is the number of where P(X) is the probability that the codeword X is transmitted, n(X, X) ˆ instead of X and nc is the number of information bit errors in choosing another codeword X information bits per transmission. Next, if we were to choose to approximate the average BER by considering only codeword errors of minimum distance and knowing that Gray coding is adopted, we can use (11) to obtain Pb (E) '

TDO + 1 ˆ P(X → X). 2

(15)

To simplify the expression in (11), we use the approximation for the Q-function presented in [40, eq. (14)] (i.e., Q(x) ' (1/12) exp (−x2 /2) + (1/4) exp (−2x2 /3)), finally obtaining L TDO + 1 ∏ 24 r=L−TDO+1

γξ 2 exp − i r−L+TDO f I(r) (ir−L+TDO )dir−L+TDO TDO2 0 (16) Z ∞ L 4γξ 2 TDO + 1 exp − ir−L+TDO fI(r) (ir−L+TDO )dir−L+TDO + ∏ 8 3TDO2 r=L−TDO+1 0

Pb (E) '

Z ∞

This expression can be simplified as TDO + 1 Pb (E) ' 8

! L L γξ 1 4γξ ∏ Tr ( TDO2 ) + ∏ Tr ( 3TDO2 ) 3 r=L−TDO+1 r=L−TDO+1

(17)

where Tr (ρ) is defined as follows Tr (ρ) =

Z ∞ 0

exp −ρi2 fI(r) (i)di

(18)

Taking advantage of the simpler expression in Eq. (13) for the PDF corresponding to the rth order statistic, this integral can be solved in terms of Hermite polynomials Hn (x) [22, eqn. (8.950.1)]. By using the integral representation of the parabolic cylinder funcR tion D p (z) = (1/Γ(−p)) exp(−z2 /4) 0∞ x−p−1 exp(−x2 /2 − xz)dx [22, eqn. (9.241.2)], and the fact that the parabolic cylinder √ function is connected to the Hermite polynomials as Dn (z) = 2−n/2 exp(−z2 /4)Hn (z/ 2) [22, eqn. (9.253)], the function Tr (ρ) can be written as Γ(L + 1)ar b1−r c(br + 1) Tr (ρ) = Γ(br)ρ −br/2 H−br − (19) √ Γ(r)Γ(L − r + 1) 2a(b + 1) ρ Substituting this expression in Eq. (17) and after some algebraic manipulations, the approximation of average BER is given by ! TDO (L+L0 ) ! 2 b L TDO + 1 aTDO Γ(bk) (bΓ(L + 1))TDO Pb (E) ' ∏ Γ(k)Γ(L − k + 1) 8 b(γξ )b/2 k=L0  ! TDO (L+L0 ) ! √ 2 L L b/2 1 3 3 ×  ∏ H−bk (A(bk + 1)) + ∏0 H−bk 2 A(bk + 1)  3 k=L0 2b k=L

(20)

cTDO where L0 = L − TDO + 1 and A = − 2a(b+1)(γξ . Considering now that the PDF in Eq. (2) )1/2

is approximated by using the first term of the Taylor expansion, i.e. assuming in Eq. (20) a value of c = 0, it is straightforward to show that the average BER behaves asymptotically as (Λc γξ )−Λd , where Λd and Λc denote diversity order and coding gain, respectively. At high #213237 - $15.00 USD Received 2 Jun 2014; revised 10 Sep 2014; accepted 16 Sep 2014; published 23 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023861 | OPTICS EXPRESS 23869

SNR, if asymptotically the error probability behaves as (Λc γξ )−Λd , the diversity order Λd determines the slope of the BER versus average SNR curve in a log-log scale and the coding gain Λc (in decibels) determines the shift of the curve in SNR. With the purpose of analyzing the diversity order achieved for the TALS scheme here proposed when L transmit lasers are available and since c = 0 implies that A = 0 and, √ hence, theargument of the Hermite polynomials is 0, [41, eqn. (07.01.03.0001.01)]. It is easy to we can use in Eq. (20) that Hν (0) = π2ν /Γ 1−ν 2 deduce that the average BER behaves asymptotically as 1/γ bTDO(2L−TDO−1)/4 , corroborating a diversity gain of (2L − TDO − 1)TDO/2 in relation to the absence of TALS scheme, wherein the average BER varies as 1/γ b/2 [20]. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. The results corresponding to this FSO scenario with rectangular pulse shapes and ξ = 1 are illustrated in Fig. 1, corroborating previous conclusions. Different weather conditions are considered: haze visibility of 4 km with Cn2 = 1.7 × 10−14 m−2/3 and average visibility of 16 km with Cn2 = 8 × 10−14 m−2/3 , corresponding to moderate and strong turbulence, respectively. Together with λ = 1550 nm, a link distance of d = 3 km and values of normalized beamwidth and jitter of (ωz /r, σs /r) = (5, 1), α and β are calculated from Eq. (3). Monte Carlo simulation results are furthermore included as a reference, confirming the accuracy and usefulness of the analytical expressions here obtained. Due to the long simulation time involved, simulation results only up to BER=10−8 are included. Simulation results corroborate that asymptotic expressions lead to simple bounds on the bit error probability in the range from low to high SNR. Here, it must be emphasized that a value of TDO = 1 corresponds to the case of transmit laser selection (TLS), obtaining expressions with better accuracy than previously reported in the literature in this general FSO scenario wherein GG fading model with pointing errors has been considered. At this point, it must be noted that the TLS scheme based on the selection of the optical path with a greater value of irradiance has shown to be able to provide better performance compared to general FSO STCs designs, such as conventional OSTBCs and repetition codes [14]. In this way, the TLS scheme, i.e. TALS with TDO = 1, can serve us as a benchmark to show the improvement in performance with the TALS scheme here proposed. In spite of obtaining a higher diversity order as TDO increases, the adoption of a time diversity order superior to 2 translates to a remarkable coding gain disadvantage, not providing a significant better performance in the context of moderate turbulence when TDO ≥ 3 for FSO systems where BER targets as low as 10−9 are typically aimed to achieve. This can be justified from Eq. (20) wherein the value of TDO is related to the SNR as (γ/TDO2 )−bTDO(2L−TDO−1)/4 , as a consequence of maintaining the average optical power in the air at a constant level of Popt , being transmitted by each laser an average optical power of Popt /TDO, as previously considered in Eq. (9). Taking a more in-depth look at the impact of pointing errors and the relation between the values of ϕ 2 and β , BER performance in FSO links using TALS with/without pointing errors in the context of moderate turbulence is displayed in Fig. 2, assuming L = {1, 3}, a link distance of d = 5 km and values of normalized beamwidth of ωz /r = 5 in Fig. 2(a) and ωz /r = 10 in Fig. 2(b) and normalized jitter of σs /r = {1, 3, 6}. Additionally, the numerical simulation of Eq. (15) has been also included for L = 3 with TDO = {1, 2} and different pointing errors in order to contrast the accuracy of the average BER obtained through accounting for error event paths of minimum distance. To analyze the BER performance obtained in a similar context when misalignment fading is not present and knowing that the impact of pointing errors in our analysis can be suppressed by assuming A0 → 1 and ϕ 2 → ∞ [3], the corresponding approximate BER


100

L=1 L=2 L=3 bound

10−1 Average bit-error probability

10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 25

d = 3 km TDO = 1 TDO = 2 TDO = 3 Cn2 = 1.7 × 10−14 m−2/3 35

45

55 65 Average SNR, γ (dB)

75

85

95

(a)

100

L=1 L=2 L=3 bound

10−1 Average bit-error probability

10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12 25

d = 3 km TDO = 1 TDO = 2 TDO = 3 Cn2 = 8 × 10−14 m−2/3 35

45

55 65 Average SNR, γ (dB)

75

85

95

(b)

Fig. 1. BER performance using TALS over atmospheric turbulence and misalignment

fading channels, when different weather conditions (a) Cn2 = 1.7 × 10−14 m−2/3 and (b) Cn2 = 8 × 10−14 m−2/3 are assumed for a link distance of d = 3 km and values of normalized beamwidth and jitter of (ωz /r, σs /r) = (5, 1).


L=1 L = 3 - TDO = 1 L = 3 - TDO = 2 bound

100

Average bit-error probability

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12

d = 5 km Cn2 = 1.7 × 10−14 m−2/3 (ωz /r, σs /r) = (5, 1) (ωz /r, σs /r) = (5, 3) no pointing errors Eq. (15) 25

35

45

55 65 75 85 Average SNR, γ (dB)

95

105

115

125

(a)

L=1 L = 3 - TDO = 1 L = 3 - TDO = 2 bound

100

Average bit-error probability

10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10 10−11 10−12

d = 5 km Cn2 = 1.7 × 10−14 m−2/3 (ωz /r, σs /r) = (10, 1) (ωz /r, σs /r) = (10, 6) no pointing errors Eq. (15) 25

35

45

55 65 75 85 Average SNR, γ (dB)

95

105

115

125

(b)

Fig. 2. BER performance using TALS over atmospheric turbulence and misalignment fading channels when a link distance of d = 5 km is assumed in the context of moderate turbulence together with values of normalized beamwidth of (a) ωz /r = 5 and (b) ωz /r = 10 and normalized jitter of σs /r = {1, 3, 6}.


expression can be easily derived from (20) as follows 0 ! !TDO 2 2 (L+L ) L TDO + 1 (β Γ(L + 1)) L+L0 Γ(α − β )(αTDO)β Γ(β k) Pb (E) ' ∏ Γ(k)Γ(L − k + 1) 8 β −β +1 Γ(α)Γ(β )(γξ )β /2 k=L0  ! ! TDO (L+L0 ) √ 2 L L β /2 1 3 3 ×  ∏ H−β k (A(β k + 1)) + ∏ H−β k 2 A(β k + 1)  3 k=L0 2β k=L0

(21) −β +1

(αβ ) TDO where L0 = L − TDO + 1 and A = 2(β +1)(α−β . −1)(γξ )1/2 As was also concluded in [20], it can be seen that pointing errors can in fact impact the diversity order under assumption of ϕ 2 < β , as corroborated for the cases (ωz /r, σs /r) = (5, 3) in Fig. 2(a) and (ωz /r, σs /r) = (10, 6) in Fig. 2(b). Therefore, the main aspect to consider in order to optimize the error-rate performance is the relation between the values of ϕ 2 and β , being the adoption of transmitters with accurate control of their beamwidth especially important to satisfy the condition ϕ 2 > β in order to maximize the diversity order gain. Once this condition is satisfied, the impact of the pointing error effects translates into a coding gain disadvantage, D[dB], relative to GG channel model without misalignment fading given by ! 20 ϕ2 . (22) D[dB] , log10 β β A (ϕ 2 − β ) 0

As in [19, 20], it is also here corroborated that the impact of pointing errors is not related to the diversity order, obtaining the same coding gain disadvantage regardless the number of transmit lasers. As a result, an identical coding gain of 10.6 decibels between values of normalized beamwidth and jitter of (ωz /r, σs /r) = (5, 1) and (ωz /r, σs /r) = (10, 1) can be contrasted comparing Fig. 2(a) and Fig. 2(b) for L = 1 as well as for L = 3, in excellent agreement with the result obtained from Eq. (22). Nonetheless, when the adoption of transmitters with precise control of their beamwidth is not possible the adoption of the TALS scheme here proposed is an attractive alternative as can be seen in Fig. 2 for values of normalized beamwidth and jitter of (ωz /r, σs /r) = (5, 3) and (ωz /r, σs /r) = (10, 6) when L = 3 and a time diversity order of 2 is available. For instance, an improvement in average SNR above 20 decibels can be achieved for values of normalized beamwidth and jitter of (ωz /r, σs /r) = (5, 3) when considering BER=10−9 as a practical performance target. 4.

Conclusions

In this paper, a new transmit alternate laser selection scheme with repetition coding for IM/DD FSO communications over atmospheric turbulence and misalignment fading channels is analyzed. Assuming CSI at the transmitter and receiver and a time diversity order limited, we propose the transmit diversity technique based on the rotating selection of TDO out of the available L lasers corresponding to the optical paths with greater values of scintillation. Implementing repetition coding with blocks of TDO information bits, each information bit will be retransmitted TDO times using the TDO largest order statistics in an alternating way. Closedform asymptotic BER expressions using a pairwise error probability analysis in the range from low to high SNR are derived when the irradiance of the transmitted optical beam is susceptible to moderate-to-strong turbulence conditions, following a gamma-gamma distribution, and pointing error effects, following a misalignment fading model where the effect of beam width, detector size and jitter variance is considered. Fully exploiting the potential time-diversity TDO #213237 - $15.00 USD Received 2 Jun 2014; revised 10 Sep 2014; accepted 16 Sep 2014; published 23 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023861 | OPTICS EXPRESS 23873

available in the turbulent channel, a significant diversity gain is achieved, obtaining a diversity order of (2L + 1 − TDO)TDO/2. In this way, diversity orders of 2L − 1 and 3L − 3 are achieved for time-diversity orders of 2 and 3, respectively. Simulation results are further demonstrated to confirm the accuracy and usefulness of the derived results. In relation to the impact of pointing errors, TALS scheme can be adopted as an alternative to achieve a relevant improvement in performance when the optimization of the transmit laser beamwidth as presented in [20] is not possible. Additionally, it must be emphasized that expressions with better accuracy than previously reported in the literature have been here presented for the transmit laser selection scheme (i.e., for the case TDO = 1) in this general FSO scenario wherein GG fading model with pointing errors has been considered. Acknowledgment The authors wish to acknowledge the financial support given by Spanish MINECO Project TEC2012-32606.
