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Space-time trellis coding with transmit laser selection for FSO links over strong atmospheric turbulence channels Antonio Garc´ıa-Zambrana1 *, Carmen Castillo-V´azquez2 , and Beatriz Castillo-V´azquez1 1 Department 2 Department

of Communications Engineering, University of M´alaga, E-29071 M´alaga, Spain of Statistics and Operations Research, University of M´alaga, E-29071 M´alaga, Spain *[email protected]

Abstract: Atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the link performance. In this paper, a scheme combining transmit laser selection (TLS) and space-time trellis code (STTC) for multiple-input-single-output (MISO) free-space optical (FSO) communication systems with intensity modulation and direct detection (IM/DD) over strong atmospheric turbulence channels is analyzed. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas. Based on a pairwise error probability (PEP) analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2L − 1 when L transmit lasers are available and a simple two-state STTC with rate 1 bit/(s · Hz) is used. Simulation results are further demonstrated to confirm the analytical results. © 2010 Optical Society of America OCIS codes: (010.1330) Atmospheric turbulence; (060.2605) Free-space optical communication; (060.4510) Optical communications.

References and links 1. J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” Proc. IEEE 85, 265–298 (1997). 2. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009). 3. W. Lim, C. Yun, and K. Kim, “BER performance analysis of radio over free-space optical systems considering laser phase noise under Gamma-Gamma turbulence channels,” Opt. Express 17(6), 4479–4484 (2009). 4. K. Tsukamoto, A. Hashimoto, Y. Aburakawa, and M. Matsumoto, “The case for free space,” IEEE Microwave Mag. 10(5), 84–92 (2009). 5. L. Andrews, R. Phillips, and C. Hopen, Laser Beam Scintillation with Applications (SPIE Press, 2001). 6. X. Zhu and J. M. Kahn, “Free-Space Optical Communication through Atmospheric Turbulence Channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). 7. X. Zhu and J. M. Kahn, “Performance bounds for coded free-space optical communications through atmospheric turbulence channels,” IEEE Trans. Commun. 51(8), 1233–1239 (2003).

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8. J. Anguita, I. Djordjevic, M. Neifeld, and B. Vasic, “Shannon capacities and error-correction codes for optical atmospheric turbulent channels,” J. Opt. Netw. 4(9), 586–601 (2005). 9. M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over gammagamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 5(6), 1229–1233 (2006). 10. E. J. Shin and V. W. S. Chan, “Optical communication over the turbulent atmospheric channel using spatial diversity,” in Proc. IEEE GLOBECOM, pp. 2055–2060 (2002). 11. I. B. Djordjevic, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel using Q-ary pulse-position modulation,” Opt. Express 15(16), 10,026–10,032 (2007). 12. I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. Neifeld, “LDPC-Coded MIMO Optical Communication Over the Atmospheric Turbulence Channel,” J. Lightwave Technol. 26(5), 478–487 (2008). 13. F. Xu, A. Khalighi, P. Causs´e, and S. Bourennane, “Channel coding and time-diversity for optical wireless links,” Opt. Express 17(2), 872–887 (2009). 14. S. G. Wilson, M. Brandt-Pearce, Q. Cao, and I. Leveque, J. H., “Free-Space Optical MIMO Transmission With Q-ary PPM,” IEEE Trans. Commun. 53(8), 1402–1412 (2005). 15. S. M. Navidpour, M. Uysal, and M. Kavehrad, “BER Performance of Free-Space Optical Transmission with Spatial Diversity,” IEEE Trans. Wireless Commun. 6(8), 2813–2819 (2007). 16. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun. 8(2), 951–957 (2009). 17. M. Simon and V. Vilnrotter, “Alamouti-Type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wireless Commun. 4(1), 35–39 (2005). 18. A. Garc´ıa-Zambrana, “Error rate performance for STBC in free-space optical communications through strong atmospheric turbulence,” IEEE Commun. Lett. 11(5), 390–392 (2007). 19. C. Abou-Rjeily, “Orthogonal Space-Time Block Codes for Binary Pulse Position Modulation,” IEEE Trans. Commun. 57(3), 602–605 (2009). 20. M. Safari and M. Uysal, “Do We Really Need OSTBCs for Free-Space Optical Communication with Direct Detection?” IEEE Trans. Wireless Commun. 7(11), 4445–4448 (2008). 21. V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: performance criterion and code construction,” IEEE Trans. Information Theory 44(2), 744–765 (1998). 22. E. Bayaki and R. Schober, “On Space-Time Coding for Free-Space Optical Systems,” (2009). Accepted for future publication in IEEE Trans. Commun. 23. Z. Chen, B. Vucetic, and J. Yuan, “Space-time trellis codes with transmit antenna selection,” Electron. Lett. 39(11), 854–855 (2003). 24. D. A. Gore and A. J. Paulraj, “MIMO antenna subset selection with space-time coding,” IEEE Trans. Sig. Process. 50(10), 2580–2588 (2002). 25. A. F. Molisch and M. Z. Win, “MIMO systems with antenna selection,” IEEE Microwave Magazine 5(1), 46–56 (2004). 26. A. Garcia-Zambrana, C. Castillo-Vazquez, B. Castillo-Vazquez, and A. Hiniesta-Gomez, “Selection Transmit Diversity for FSO Links Over Strong Atmospheric Turbulence Channels,” IEEE Photon. Technol. Lett. 21(14), 1017–1019 (2009). 27. B. Castillo-Vazquez, A. Garcia-Zambrana, and C. Castillo-Vazquez, “Closed-form BER expression for FSO links with transmit laser selection over exponential atmospheric turbulence channels,” Electron. Lett. 45(23), 1185– 1187 (2009). 28. N. Letzepis and A. G. Fabregas, “Outage probability of the MIMO Gaussian free-space optical channel with PPM,” in Proc. IEEE International Symposium on Information Theory ISIT 2008, pp. 2649–2653 (2008). 29. S. Z. Denic, I. Djordjevic, J. Anguita, B. Vasic, and M. A. Neifeld, “Information Theoretic Limits for Free-Space Optical Channels With and Without Memory,” J. Lightwave Technol. 26(19), 3376–3384 (2008). 30. M. K. Simon and M.-S. Alouini, Digital Communications Over Fading Channels, 2nd ed. (Wiley-IEEE Press, New Jersey, 2005). 31. S. Hranilovic and F. R. Kschischang, “Optical intensity-modulated direct detection channels: signal space and lattice codes,” IEEE Trans. Information Theory 49(6), 1385–1399 (2003). 32. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. (Academic Press Inc., 2007). 33. H. Jafarkhani, Space-Time Coding: Theory and Practice (Cambridge University Press, New York, 2005). 34. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (John Wiley and Sons Inc., 2003). 35. M. Chiani, D. Dardari, and M. K. Simon, “New exponential bounds and approximations for the computation of error probability in fading channels,” IEEE Trans. Wireless Commun. 2(4), 840–845 (2003). 36. Wolfram Research, Inc., “The Wolfram functions site,” URL http://functions.wolfram.com. 37. Wolfram Research, Inc., Mathematica, version 7.0 ed. (Wolfram Research, Inc., Champaign, Illinois, 2008). 38. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth ed. (Dover, New York, 1970).

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1.

Introduction

Optical wireless communications using intensity modulation and direct detection (IM/DD) can provide high-speed links for a variety of applications [1], providing an unregulated spectral segment and high security. Recently, the use of atmospheric free-space optical (FSO) transmission is being specially interesting to solve the “last mile” problem, above all in densely populated urban areas, as well as a supplement to radio-frequency (RF) links [2] and the recent development of radio on free-space optical links (RoFSOLs) [3, 4]. However, atmospheric turbulence produces fluctuations in the irradiance of the transmitted optical beam, which is known as atmospheric scintillation, severely degrading the link performance [5, 6]. Error control coding as well as diversity techniques can be used over FSO links to mitigate turbulence-induced fading [7–13]. In particular, heuristic space-time code (STC) designs such as repetition codes (RCs) [14–16] and orthogonal space-time block codes (OSTBCs) [17–19] have been proposed for FSO systems with IM/DD. It must be emphasized that a simple translation of the analysis of STCs from RF systems is not plausible due to peculiarities proper to FSO scenario. It is well known from the vast literature on wireless RF systems that simply sending the same signal from different antennas (i.e., repetition coding) does not realize any transmit diversity advantage. However, in [20] it was shown that simple RCs not only are able to extract full diversity but also always outperform OSTBCs, because of the fact that the transmitted signal is an intensity and, hence, it is subject to a nonnegativity constraint. In this way, unlike in the RF case [21], performance bounds and systematic design guidelines for general FSO STCs are not available. In [22], a closedform expression has recently been derived for the asymptotic pairwise error probability (PEP) of general FSO STCs for two lasers and an arbitrary number of photodetectors for channels suffering from Gamma-Gamma fading, showing the quasi-optimality of STC designs based on repetition codes and their superiority compared to conventional orthogonal space-time block codes. Selection transmit diversity technique is well known for RF systems, presenting a vast amount literature on RF multiple-input-multiple-output (MIMO) schemes that pay attention to transmission using such scheme, being a promising approach for reducing complexity since one can employ a reduced number of RF chains [23–25]. This idea is proposed in [26, 27] 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 scintillation has shown to be able to extract full diversity as well as providing better performance compared to the STC designs previously commented, such as OSTBCs and RCs, implementing transmit diversity but not using channel state information (CSI) at the transmitter (CSIT). The knowledge of CSIT is feasible for FSO channels given what scintillation is a slow time varying process relative to the large symbol rate. This has been recently considered for FSO links from the point of view of information theory [28, 29]. In this paper, a scheme combining transmit laser selection (TLS) and space-time trellis code (STTC) for multiple-input-single-output (MISO) FSO communication systems with IM/DD over strong atmospheric turbulence channels is analyzed, where the turbulence-induced fading is described by the negative exponential and K distributions and the channel fade level is tracked by both the transmitter and receiver. Assuming CSI at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas, providing a better performance compared to a similar selection transmit diversity scheme investigated previously by the authors for uncoded links [26, 27]. Here, STTC is considered in order to assume a fast fading channel, where the channel coefficients fade independently from one symbol to the next. Such a channel is a #120032 - $15.00 USD

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suitable model for a fully interleaved flat fading channel where an interleaver of length longer than the coherence time of the channel is employed. This may be done for example to make sure that the consecutive symbols transmitted see almost independent fades in an attempt to improve the diversity order, taking advantage not only of the number of transmit lasers but also of the shortest error event length in the trellis code [30]. In relation to difficulty of STTC design and decoding complexity and the requirement of a large number of transmit antennas in order to achieve a full diversity order [21], the combination of transmit laser selection and existing STTCs designed for two transmit antennas is a good approach to achieving a high diversity order while maintaining low decoding complexity at the receiver. Based on a pairwise error probability analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2L − 1 when L transmit lasers are available and a simple two-state STTC with rate 1 bit/(s · Hz) is used. Simulation results are further demonstrated to confirm the analytical results. Here, not only rectangular pulses are considered but also on-off keying (OOK) formats with any pulse shape, corroborating the advantage of using pulses with high peak-to-average optical power ratio (PAOPR), such as Gaussian pulses with reduced duty cycle. 2.

Atmospheric turbulence channel model

The use of infrared technologies based on IM/DD links is considered; in this way, having a single-input-single-output (SISO) system as a reference, the instantaneous current in the receiving photodetector, y(t), can be written as y(t) = η i(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 source, and I  i(t) the scintillation at the optical path; 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 detector. In this case, the noise can usually be modeled to high accuracy as AWGN with zero mean and variance N0 /2, i.e. Z ∼ N(0, N0 /2), independent of the on/off state of the received bit [1]. 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. Although limits are placed on both the average and peak optical power transmitted, in the case of most practical modulated optical sources, it is the average optical power constraint that dominates [31]. The received electrical signal Y  y(t), however, can assume negative amplitude values. In this fashion, the atmospheric turbulence channel model consists of a multiplicative noise model, where the optical signal is multiplied by the channel irradiance. Considering strong turbulence conditions [5,16], negative exponential and K distribution for the i.i.d. channel irradiances can be assumed. The probability density function (PDF) corresponding to the K turbulence model is given by α +1  √  2α 2 α −1 i 2 Kα −1 2 α i , i ≥ 0 fI (i) = Γ(α )

(2)

where α is a channel parameter related to the effective number of discrete scatterers, Γ(·) is the well-known Gamma function, and Kν (·) is the ν th-order modified Bessel function of the second kind [32]. Since the mean value of this turbulence model is E[I] = 1 and the second moment is given by E[I 2 ] = (2 + 2/α ), the scintillation index (SI), a parameter of interest used to describe the strength of atmospheric fading, is defined as SI = #120032 - $15.00 USD

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E[I 2 ] 2 −1 = 1+ (E[I])2 α

(3)

Received 16 Nov 2009; revised 9 Jan 2010; accepted 21 Feb 2010; published 1 Mar 2010

15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5359

Together with this distribution and considering a limiting case of strong turbulence conditions [5, 18, 28], a negative exponential model with PDF given by fI (i) = exp (−i), i ≥ 0

(4)

is also adopted to describe turbulence-induced fading, leading to an easier mathematical treatment to evaluate error rate performance for any diversity order. This distribution can be seen as the K-distributed turbulence model in (2) when the channel parameter α → ∞. We consider OOK formats with any pulse shape and reduced duty cycle, allowing the increase of the PAOPR parameter [18, 26]. A new basis function φ (t) is defined as φ (t) = g(t) Eg where g(t) represents any normalized pulse shape satisfying the non∞ 2 g (t)dt negativity constraint, with 0 ≤ g(t) ≤ 1 in the bit period and 0 otherwise, and Eg = −∞ is the electrical energy. In this way, an expression for the optical intensity can be written as x(t) =





ak

k=−∞

2Tb P g (t − kTb ) G( f = 0)

(5)

where G( f = 0) represents the Fourier transform of g(t) evaluated at frequency f = 0, i.e. the area of the employed pulse shape. The random variable (RV) ak follows a Bernoulli distribution with parameter p = 1/2, taking the values of 0 for the bit “0” (off pulse) and 1 for the bit “1” (on pulse). From this expression, it is easy to deduce that the average optical power transmitted is P, defining a constellation of two equiprobable points in a one-dimensional space with an Euclidean distance of  (6) d = 2P Tb ξ where ξ = Tb Eg /G2 ( f = 0) represents the square of the increment in Euclidean distance due to the use of a pulse shape of high PAOPR, alternative to the classical rectangular pulse. The channel is assumed to be memoryless, stationary and ergodic, with independent and identically distributed intensity fast fading statistics. In spite of scintillation is a slow time varying process relative to typical symbol rates of an FSO system, having a coherence time on the order of milliseconds, this approach is valid because temporal correlation can in practice 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, 9, 11, 12]. 3.

Proposed transmit diversity scheme

The use of optical arrays, similar to the use of antenna-array technology for microwave systems, is considered as a means of combatting fading. In particular, we adopt a MISO array based on L laser sources, assumed to be intensity-modulated only 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, and the separation distance between the lasers is large enough so that the fading experienced between source-detector pairs I j (t) is assumed to be statistically independent. Assuming channel state information at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available L lasers corresponding to the optical paths with greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas [21]. To illustrate the proposed scheme, we adopt in this paper the example shown in Fig. 1, where a two-state STTC with rate 1 bit/(s · Hz) using OOK is displayed [33, Fig. 6.5.]. The incoming symbol stream is first encoded using the trellis structure and the encoded stream is then distributed among the two sources out of the available L lasers corresponding to greater values of scintillation, I(L) (t) and I(L−1) (t), where I(1) (t), I(2) (t), ..., I(L) (t) is a new sequence of #120032 - $15.00 USD

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L auxiliary random variables obtained by arranging the random sequence I1 (t), I2 (t), ..., IL (t) in an increasing order of magnitude. The labeling i/kk along each branch of the trellis refers to the input bit (i) and the corresponding pair of output symbols (kk) that result from the transition between the states at the beginning and end of the branch.

Fig. 1. Trellis diagram of two-state STTC, OOK, 1 bit/(s · Hz)

4.

Performance analysis

In this section, an optical array based on L = {1, 2, 4, 8} laser sources, all pointed towards a distant photodetector, is considered. We present aproximate closed-form expressions for the bit error rate (BER) using a pairwise error probability analysis when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turˆ bulence conditions. The PEP represents the probability of choosing the space-time sequence X when in fact the sequence X was transmitted [30, Chapter 16]. Assuming that the correct path is the all-zeros sequence, then for the shortest error event path of length N = 2 illustrated by shading in Fig. 1, we have     0 0 0 d ˆ X= , X= (7) 0 0 d 0 ˆ is associated with the two symbols transmitted from the two where each column of X and X lasers in a given symbol interval (time slot) and d is the Euclidean distance in (6) corresponding ˆ matrices assoto the OOK signaling (i.e., the OOK symbols are the elements of the X and X ciated with the trellis). In the proposed scheme, for example, we associate the first and second rows with the (L-1)th and Lth order statistics corresponding to the scintillation. Under the assumption of perfect CSI, the conditional PEP with respect to scintillation coefficients of greater value, I(L) and I(L−1) , is given as [30, Chapter 16] ⎛ ⎞ 2



ˆ I(L) , I(L−1) = Q ⎝ (d/2) i2 + i2 ⎠ (8) P X→X 1 2 2N0 where Q(·) is the Gaussian-Q function. Similar expressions to evaluate the pairwise error probability of coded FSO IM/DD links using OOK signaling can be found in [7,9]. Here, the division of d by 2 is considered so as to maintain the average optical power in the air at a constant level of P, being transmitted by each laser an average optical power of P/2. Substituting the value of d obtained in (6) gives  

γξ 2 2

ˆ P X → X I(L) , I(L−1) = Q (9) i +i 2 1 2 #120032 - $15.00 USD

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where γ = P2 Tb /N0 is the average receiver electrical signal-to-noise spectral density ratio (SNR) in the presence of the turbulence [6], knowing that PDF in (2) or (4) is normalized. Under the assumption of perfect interleaving, we can exploit independency among fading coˆ by averaging (9) as follows efficients to obtain the average PEP, P(X → X),    ∞ ∞ 2 2 γξ ˆ = i +i fI(L) (i1 ) fI(L−1) (i2 )di1 di2 Q (10) P(X → X) 2 1 2 0 0 According to order statistics [34], for i.i.d. RVs of {I j } j=1,2,···L , the PDF corresponding to I(L) and I(L−1) can be written as fI(L) (i) = L fI (i)[FI (i)]L−1 (11) fI(L−1) (i) = L(L − 1) fI (i)(1 − FI (i))(FI (i))L−2

(12)

being FI (i) the cumulative density function (CDF) corresponding to the turbulence model. An union bound on the average BER can be found as [30, eq. (13.44)] Pb (E) ≤

1 P(X) nc ∑ X

ˆ ˆ → X) ∑ n(X, X)P(X

(13)

ˆ X=X

ˆ is the numwhere P(X) is the probability that the coded sequence X is transmitted, n(X, X) ˆ ber of information bit errors in choosing another coded sequence X instead of X and nc is the number of information bits per transmission. Next, if we were to choose to approximate the average BER by considering only error event paths of minimum length (i.e., N = 2) ˆ To simplify the expres[30, Section 14.6.4], we can use (10) to obtain Pb (E) P(X → X). sion in (10), we use the approximation for the Q-function presented in [35, Eq. (14)] (i.e., Q(x) (1/12) exp (−x2 /2) + (1/4) exp (−2x2 /3)), finally obtaining       ∞ −γξ i21 −γξ i22 1 ∞ Pb (E) exp exp fI(L) (i1 )di1 fI(L−1) (i2 )di2 12 0 4 4 0 (14)       ∞ −γξ i21 −γξ i22 1 ∞ + exp exp fI(L) (i1 )di1 fI(L−1) (i2 )di2 4 0 3 3 0 To evaluate the improvement in performance, we compare the proposed scheme, which is referred to as the TLS/STTC scheme, with the transmit diversity technique TLS presented in [26] for uncoded FSO links, based on the selection of the optical path with a greater value of scintillation, where the average BER is given by  ∞   Pb (E) = Q 2γξ i2 fI(L) (i)di (15) 0

In the same way, we also include the performance corresponding to the STTC scheme when no transmit laser selection is used, where, following an approach as in (13), the average BER can be written as    ∞ ∞ 2 2 γξ ˆ = i +i Pb (E) P(X → X) fI (i1 ) fI (i2 )di1 di2 Q (16) 2 1 2 0 0

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4.1.

K atmospheric turbulence channel

Particularizing with the K distribution in (2) and using [36, Eq. (03.04.21.0013.01)] together with the fact that Kν (·) is a even function with respect to its parameter, the derived integral for the CDF of the K channel can be written as FI (i) = 1 −

2α α /2 α /2  √  i Kα 2 iα Γ(α )

(17)

The results corresponding to this FSO scenario are illustrated in the Fig. 2, when different levels of turbulence strength of α = 1 and α = 4 are assumed, corresponding to values of scintillation index of SI = 3 and SI = 1.5, respectively, and where rectangular pulse shapes with ξ = 1 are used. Additionally, a relevant improvement in performance must be noted as a consequence of pulse shape used, providing an increment in the average SNR of 10 log10 ξ dB. So, for instance, when a rectangular pulse shape of duration κ Tb , with 0 < κ ≤ 1, is adopted, √a value of ξ = 4/ κ π value of ξ = 1/κ can be easily shown. Nonetheless, a significantly higher

is obtained when a Gaussian pulse of duration κ Tb as g(t) = exp −t 2 /2σ 2 ∀|t| < κ Tb /2 is adopted, where σ = κ Tb /8 and the reduction of duty cycle is also here controlled by the parameter κ . In this fashion, 99.99% of the average optical power of a Gaussian pulse shape is being considered. Then, a Gaussian pulse shape with κ = 0.25 is also adopted when L = 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Numerical results for TLS/STTC in (14), TLS in (15) and STTC without laser selection in (16) are computed using a symbolic mathematics package [37]. BER simulation results are furthermore included as a reference. Due to the long simulation time involved, simulation results only up to BER=10−6 are included. Simulation results demonstrate an excellent agreement with the analytical results for L = {2, 4, 8}, as well as the greater diversity order for the transmit diversity technique here proposed if compared with TLS and STTC, being superior to the number of available transmit lasers L. 4.2.

Exponential atmospheric turbulence channel

In this subsection, considering a limiting case of strong turbulence conditions [5, 18, 28], a negative exponential model is adopted to describe turbulence-induced fading, leading to an easier mathematical treatment to evaluate error rate performance for any number of transmit lasers. Here, particularizing with the negative exponential distribution in (4), the derived integral for the CDF of the exponential turbulent channel can be written as FI (i) = 1 − exp (−i). Using the binomial theorem in (11) and (12), we obtain L

fI(L) (i) = L ∑

n=1

fI(L−1) (i) = L(L − 1)

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 L−1 (−1)n−1 exp (−ni) n−1

L



m=2



 L−2 (−1)m−2 exp (−mi) m−2

(18)

(19)

Received 16 Nov 2009; revised 9 Jan 2010; accepted 21 Feb 2010; published 1 Mar 2010

15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5363

Average bit error rate

10 10 10 10 10 10 10 10 10

No diversity α=1 STTC TLS; L={2,4,8} TLS/STTC; L={2,4,8} TLS/STTC−Gs; L=2 Simulation

−1 −2 −3 −4

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−5 −6 −7

L=4 −8

L=8 −9

0

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(a)

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−1

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0

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90

(b)

Fig. 2. Performance comparison of TLS/STTC and TLS in FSO IM/DD link over the K atmospheric turbulence channel when different levels of turbulence strength (a) (α = 1) and (b) (α = 4) are assumed, corresponding to values of scintillation index of SI = 3 and SI = 1.5, respectively. Next, substituting (18) and (19) in (14) and evaluating the integrals by using [38, eqn. (7.4.32)], a closed-form solution for the aproximate average BER yields as     L2 (L − 1) L L − 1 L m+n−3 L − 2 Pb (E) ∑ n − 1 ∑ (−1) m−2 48γξ n=1 m=2       2  m + n2 m n erfc  exp 4π erfc  (20) γξ γξ γξ   √   √

 3 m2 + n2 3m 3n   erfc exp + 9π erfc #120032 - $15.00 USD Received 16 Nov 2009; revised 92Janγξ 2010; accepted 214Feb γξ 2010; published 1 Mar 2010 2 γξ (C) 2010 OSA

15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5364

where erfc(·) is the complementary error function. The results corresponding to this FSO scenario are illustrated in the Fig. 3, where rectangular pulse shapes with κ = 1 are used.

Average bit error rate

10 10 10 10 10 10 10 10 10

No diversity STTC TLS; L={2,4,8} TLS/STTC; L={2,4,8} TLS/STTC−Gs; L=2 Simulation

−1 −2 −3 −4 −5 −6

L=2 L=4

−7 −8

L=8 −9

0

10

20

30

40

50

60

Average SNR, γ (dB)

70

80

90

Fig. 3. Performance comparison of TLS/STTC and TLS in FSO IM/DD link over the exponential atmospheric turbulence channel, corresponding to a value of scintillation index of SI = 1.

As before in Fig. 2, a Gaussian pulse shape with κ = 0.25 is also adopted when L = 2 in order to show the improvement in performance obtained with pulse shapes having a high PAOPR. Here, results for TLS/STTC from evaluating the expression in (20) are displayed together with numerical results for the TLS and STTC schemes computed as in previous subsection. BER simulation results are furthermore included as a reference, demonstrating an excellent agreement with the analytical results for L = {2, 4, 8}, as well as a better performance in terms of diversity gain, as previously concluded for the K channel. With the purpose of analyzing the diversity order achieved for the TLS/STTC scheme here proposed when L transmit lasers are available, we can use in (20) the series expansions corresponding to the exponential k function [38, eqn. (4.2.1)] exp (x) = ∑∞ k=0 x /k!) and the error function [38, eqn. (7.1.5)] √ (i.e., ∞ k 2k+1 /((2k + 1)k!)). In this way, it is straightforward to (i.e., erfc(x) = 1 − (2/ π ) ∑k=0 (−1) x show that the average BER behaves asymptotically as 1/γ (2L−1)/2 , corroborating a diversity gain of 2L−1 in relation to the absence of space-time trellis coding with laser selection, wherein the average BER varies as 1/γ 1/2 [17,18]. For example, in comparison to the selection transmit diversity scheme in [27], where a diversity order of L can be demonstrated in a similar way as before, the additional use of the simple two-state STTC in Fig. 1 provides a performance improvement of 20 dB at a target BER rate of 10−9 with only two transmit lasers. 5.

Conclusions

In this paper, a scheme combining transmit laser selection and space-time trellis code for MISO FSO communication systems with IM/DD over strong atmospheric turbulence channels is analyzed, where the turbulence-induced fading is described by the negative exponential and K distributions and the channel fade level is tracked by both the transmitter and receiver. Assuming CSI at the transmitter and receiver, we propose the transmit diversity technique based on the selection of two out of the available L lasers corresponding to the optical paths with #120032 - $15.00 USD

(C) 2010 OSA

Received 16 Nov 2009; revised 9 Jan 2010; accepted 21 Feb 2010; published 1 Mar 2010

15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5365

greater values of scintillation to transmit the baseline STTCs designed for two transmit antennas, providing a better performance compared to a similar selection transmit diversity scheme investigated previously by the authors for uncoded links [26, 27]. Based on a pairwise error probability analysis, results in terms of bit error rate are presented when the scintillation follows negative exponential and K distributions, which cover a wide range of strong atmospheric turbulence conditions. Obtained results show a diversity order of 2L − 1 when L transmit lasers are available and a simple two-state STTC with rate 1 bit/(s · Hz) is used. As revealed out by the results under no diversity assumption, a slow change in the slope of performance curve can be observed. This justifies the adoption of diversity techniques as here proposed since it is not practical for many applications to increase the power margin in the link budget to eliminate the deep fades observed under strong turbulence. Additionally, the use of pulse shapes having a high PAOPR has shown to be a key factor to achieve a significant improvement in performance. From the relevant results here obtained when a simple two-state STTC is used, investigating in the FSO case the impact on the diversity order of STTCs designs of greater minimum symbolwise Hamming distance, which is directly related to the shortest error event length in the trellis code, is an interesting topic for future research. Acknowledgments The authors are grateful for financial support from the Junta de Andaluc´ıa (research group “Communications Engineering (TIC-0102)”).

#120032 - $15.00 USD

(C) 2010 OSA

Received 16 Nov 2009; revised 9 Jan 2010; accepted 21 Feb 2010; published 1 Mar 2010

15 March 2010 / Vol. 18, No. 6 / OPTICS EXPRESS 5366

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