IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 23, NO. 4, FEBRUARY 15, 2011
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Average Bit-Error Rate of the Alamouti Scheme in Gamma-Gamma Fading Channels Jaedon Park, Eunju Lee, and Giwan Yoon
Abstract—Multi-input multi-output (MIMO) in free-space optics communications with subcarrier intensity modulation has been a big research issue due to the diversity gain of MIMO to mitigate the signal scintillation caused by the atmospheric turbulence. In this work, we have analyzed the performance of the Alamouti scheme in FSO links. Particularly, a power series expression of the average bit-error rate (BER) of the Alamouti scheme is derived in gamma-gamma fading channels. As a result, the Alamouti scheme could achieve a high signal-to-noise ratio (SNR) gain of 37 dB in a strong turbulence regime, and also a high SNR gain of 27 dB in a moderate turbulence regime over the no diversity at the BER of 10 6 .
expressions for FSO links with a spatial diversity were derived in IM/DD and K-distributed channels. In this work, we, for the first time, investigate the performance analysis of the 2 1 Alamouti scheme [13] in FSO communications. We adopt the SIM scheme with the binary phase shift keying (BPSK) modulation. First, we derive the probability density function (PDF) of the received SNR of the Alamouti scheme as power series in gamma-gamma fading channels. Then, with the PDF, we present a power series expression of the average BER of the Alamouti scheme.
Index Terms—Alamouti scheme, free-space optics (FSO) communications, gamma-gamma fading.
II. SYSTEM AND CHANNEL MODEL A. System Model
I. INTRODUCTION
F
REE-SPACE optics (FSO) communications technology has attracted an enormous attention because of its ability to provide high security and high data rates with license-free bands [1]–[4]. In the implementation of the FSO communications systems, the intensity modulation and direct detection (IM/DD) with on/off keying (OOK) systems have been originally used for its simplicity. Since the OOK systems, however, have a big issue in the atmospheric turbulence mainly due to the fixed threshold, the subcarrier intensity modulation (SIM) has been considered as one of promising candidates to further enhance the performance of the FSO communications [5]–[7]. In addition, for the analysis of the atmospheric turbulence of the FSO communications channels, the gamma-gamma distribution model has been widely used because it fits very well to the experimental results [2], [6]–[9]. Recently, with the research on the MIMO in radio frequency (RF) links [10], [11], extensive studies on the MIMO in FSO communications have been reported [8], [9], [12]. In [8], the pairwise error probability (PEP) of the multi-input multi-output (MIMO) FSO systems with IM/DD was presented by assuming the repetition coding at the transmitter as well as the equal gain combining (EGC) and maximal ratio combining (MRC) at the receiver. In [9], the outage probability of the MIMO FSO channel for pulse position modulation (PPM) and EGC was studied. In [12], the closed-form bit-error-rate (BER) Manuscript received August 16, 2010; revised November 11, 2010; accepted December 11, 2010. Date of publication December 20, 2010; date of current version February 04, 2011. J. Park is with Agency for Defense Development, Daejeon 305-152, Korea (e-mail:
[email protected]). E. Lee and G. Yoon are with Korea Advanced Institute of Science and Technology, Daejeon 305-732, Korea (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/LPT.2010.2100815
We consider the conventional Alamouti scheme with two transmit apertures and one receive aperture [13] employing SIM [5]–[7] in gamma-gamma fading channels [2], [6]–[9]. During the first symbol period, and signals are transmitted from aperture one and aperture two, respectively, and during and signals are transmitted the second symbol period, from aperture one and aperture two, respectively [13]. Here and for the rest of this letter, the superscripts and denote the complex conjugate and the matrix transpose, respectively. In this scenario, the received signal vector , where and are the received signals during the first and second symbol periods, can be expressed as (1) where , , and are the AWGN where . Also, with zero mean and variance ( , 2) 0 with is the real-valued irradiance from the th transmitter to the receiver, following a gammagamma distribution, and is the optical-to-electrical conversion coefficient. Here, it is assumed that the average transmit energy for each antenna. per symbol is After the Alamouti decoding at the receiver which is , we get (2) , and
where Since symbol is given by
and
. Here . , the received SNR per
(3) where
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and
.
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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 23, NO. 4, FEBRUARY 15, 2011
B. Channel Model
Therefore, the PDF of the received SNR can be
In this section, we describe the gamma-gamma distribution follows a gamma-gamma disin brief. The random variable tribution and the PDF of the variable is given by [2], [6]–[9]
(12)
(4) where the is the modified Bessel function of the second kind of order . The parameters and are related to the atmospheric turbulence conditions [2], [6]–[9]. Using the generalized power series representation method, the distribution in (4) can be expressed as [8] (5)
IV. AVERAGE BER ANALYSIS denote the conditional error probability If we let in an AWGN channel, the average error probability can be obtained as (13) If we adopt the BPSK modulation, substitute (12) to (13), the average BER will be
, and
where
(6) (14) III. PDF ANALYSIS OF THE RECEIVED SNR With the relation ment generating function (MGF) of
, the mocan be given by
If we apply the Gaussian Q-function [15] (15) and the gamma function [14, 3.381]
(7) (16) where, to (14), the average BER is derived as (8) The MGF of
(17)
can be expressed as
where the
is the beta function defined by [14, 3.621]
(9) (18) where (10) means that is convolved times with itself, Here, e.g. , , . The variable in (10) can be derived with the help in [14, 0.316]. Also, . Applying the inverse Laplace transform to (9), the PDF of can be derived as
(11)
V. NUMERICAL RESULTS Fig. 1 shows the numerical results of the average BER as derived in (17) for the 2 1 Alamouti scheme employing SIM with BPSK through gamma-gamma fading channels. We consider the atmospheric turbulence parameters and as 4.2 and 1.4, respectively for a strong turbulence regime, and 4.0 and 1.9, respectively for a moderate turbulence regime [6]. Here, the upper limit of the infinite sum in (17) is truncated to 50 for practical analysis. According to the analysis results, it is found that the required SNR is 82 dB for no diversity (one transmit and one receive in a strong turbulence regime apertures) to get a BER of
PARK et al.: AVERAGE BER OF THE ALAMOUTI SCHEME IN GAMMA-GAMMA FADING CHANNELS
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1 that, in gamma-gamma fading channels with SIM, the 2 Alamouti scheme could achieve a significantly large SNR gain of 37 dB over the no diversity for a strong turbulence regime at . For a moderate turbulence regime, the Alamthe BER of outi scheme could achieve also a high SNR gain of 27 dB at the same condition. REFERENCES
Fig. 1. Average BER of Alamouti scheme with SIM BPSK in gamma-gamma fading channels.
TABLE I NUMBER OF REQUIRED SUM INSTEAD OF THE INFINITE SUM FOR THE CONVERGENCE OF THE AVERAGE BER IN (17)
while it is only 45 dB for the 2 1 Alamouti scheme to get the same BER. In the case of a moderate turbulence regime, it is found that the required SNR is 64 dB for no diversity while it is only 37 dB for the 2 1 Alamouti scheme to get the same BER. As shown in the figure, the simulation results (given up to due to the long simulation time) correspond exactly to the analysis results. Table I summarizes the number of required sum instead of the infinite sum for the convergence of the average BER in (17), where the convergence is defined when the error value between the simulation and analysis results is less than . As shown in the table, as the SNR increases, the number of the required terms decreases. VI. CONCLUSION In this letter, we, for the first time, report a power series expression of the average BER of Alamouti scheme for SIM FSO communications. The gamma-gamma fading channel is used for the FSO atmospheric turbulence model. Analysis results show
[1] L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation With Applications. Bellingham, WA: SPIE Press, 2001. [2] M. Uysal, J. Li, and M. Yu, “Error rate performance analysis of coded free-space optical links over Gamma-Gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 5, no. 6, pp. 1229–1233, Jun. 2006. [3] H. E. Nistazakis, D. Marinos, M. Hanias, A. D. Tsigopoulos, and M. E. Fafalios, “Estimation of capacity bounds of free space optical channels under strong turbulence conditions,” in Proc. 18th Int. Conf. Microwave Radar and Wireless Communications, Vilnius, Lithuania, 2010, pp. 1–3. [4] W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of PPM on terrestrial FSO links with turbulence and pointing errors,” IEEE Commun. Lett., vol. 14, no. 5, pp. 468–470, May 2010. [5] J. Li, J. Q. Liu, and D. P. Taylor, “Optical communication using subcarrier PSK intensity modulation through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 55, no. 8, pp. 1598–1606, Aug. 2007. [6] W. O. Popoola and Z. Ghassemlooy, “BPSK subcarrier intensity modulated free-space optical communications in atmospheric turbulence,” J. Lightw. Technol., vol. 27, no. 8, pp. 967–973, Apr. 2009. [7] K. P. Peppas and C. K. Datsikas, “Average symbol error probability of general-order rectangular quadrature amplitude modulation of optical wireless communication systems over atmospheric turbulence channels,” J. Opt. Commun. Netw., vol. 2, no. 2, pp. 102–110, Feb. 2010. [8] E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in Gamma-Gamma fading,” IEEE Trans. Commun., vol. 57, no. 11, pp. 3415–3424, Nov. 2009. [9] N. Letzepis and A. G. Fabregas, “Outage probability of the Gaussian MIMO free-space optical channel with PPM,” IEEE Trans. Commun., vol. 57, no. 12, pp. 3682–3690, Dec. 2009. [10] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1456–1467, Jul. 1999. [11] A. Vielmon, Y. Li, and J. R. Barry, “Performance of Alamouti transmit diversity over time-varying rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 3, no. 5, pp. 1369–1373, Sep. 2004. [12] 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., vol. 8, no. 2, pp. 951–957, Feb. 2009. [13] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [14] I. S. Gradshteyn, I. M. Ryzhik, and A. Jeffrey, Table of Integrals, Series, and Products, 5th ed. New York: Academic, 1994. [15] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channels. Hoboken, NJ: Wiley, 2000.
