Letter
Vol. 40, No. 13 / July 1 2015 / Optics Letters
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General closed-form bit-error rate expressions for coded M-distributed atmospheric optical communications JOSÉ M. GARRIDO BALSELLS,* FRANCISCO J. LÓPEZ-GONZÁLEZ, ANTONIO JURADO-NAVAS, MIGUEL CASTILLO-VÁZQUEZ, AND ANTONIO PUERTA NOTARIO Department of Communications Engineering, University of Málaga, Campus de Teatinos s/n, E-29071 Málaga, Spain *Corresponding author:
[email protected] Received 9 April 2015; revised 27 May 2015; accepted 27 May 2015; posted 28 May 2015 (Doc. ID 237860); published 16 June 2015
In this Letter, general closed-form expressions for the average bit error rate in atmospheric optical links employing rate-adaptive channel coding are derived. To characterize the irradiance fluctuations caused by atmospheric turbulence, the Málaga or M distribution is employed. The proposed expressions allow us to evaluate the performance of atmospheric optical links employing channel coding schemes such as OOK-GSc, OOK-GScc, HHH(1,13), or vw-MPPM with different coding rates and under all regimes of turbulence strength. A hyper-exponential fitting technique applied to the conditional bit error rate is used in all cases. The proposed closed-form expressions are validated by Monte-Carlo simulations. © 2015 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence; (060.2605) Free-space optical communication; (290.5930) Scintillation. http://dx.doi.org/10.1364/OL.40.002937
Atmospheric optical communications (AOC) can provide secure and high-speed transmissions for multiple applications, especially in the deployment of broadband wireless links in next-generation networks (NGN) [1,2]. Nevertheless, AOC links are not without drawbacks and may experience fading due to the turbulence atmosphere, causing fluctuations in the irradiance of the optical beam propagating through this medium [2–4]. Such irradiance fluctuations, known as scintillation, lead to an increase of the bit-error rate (BER), limiting the performance of AOC systems. Extensive research has been performed by the scientific community over the years in the search of a tractable probability density function (pdf) capable to model these irradiance fluctuations. As a result, different distributions have been proposed so far, the Log-normal [2] and the Gamma–Gamma [5] distributions being the most accepted ones. Recently, a promising statistical model called Málaga, or M model, was proposed [6]. This new M model not only offers excellent results under 0146-9592/15/132937-04$15/0$15.00 © 2015 Optical Society of America
all irradiance fluctuation regimes [6–8], but also unifies in a closed-form expression most of the statistical models for the irradiance fluctuations proposed in the bibliography. Thus, by using the M model, general closed-form expressions for the BER of AOC links under all regimes of turbulence strength can be derived, as reported in [7]. In this Letter, the analysis presented in [7] is extended to include the effect of channel coding in the closed-form BER expressions for AOC links. In particular, several rate-adaptive channel-coding schemes, employed to mitigate the irradiance fluctuations, are analyzed. First, two schemes based on the variable silence periods insertion and on–off keying (OOK) are examined. These schemes, named OOK-GSc and OOKGScc, were originally developed for indoor optical links [9], although they were later evaluated in AOC links with promising results [10]. OOK-GSc and OOK-GScc provide native coding rates, i.e., the transmission rates reduction due to the coding process, of R 1∕2 and R 1∕4, respectively, although these rates can be reduced including additional silence periods [9]. In addition, as described in [9,11], these two coding schemes can be considered as runlength-limited (RLL) sequences, as well as the HHH(1,13) technique [12] also analyzed here, providing a coding rate R 2∕3. Moreover, the variable-weight multi-pulse position modulation (vw-MPPM) is also examined [13]. In this case, it is worth noting that a similar analysis to obtain the BER for AOC systems with vw-MPPM coding under Gamma–Gamma atmospheric turbulence was reported in [14]. Here, a point-to-point atmospheric optical link using intensity modulation with direct detection (IM/DD) and OOK is considered. The receiver noise, mainly caused by ambient light [15], is modeled as an additive white Gaussian noise, nt, with zero mean and variance σ 2n . In this way, the time-dependent photocurrent at the detector output is expressed as yt η · I rx t nt;
(1)
where η denotes the detector responsivity, and I rx is the received optical irradiance, which can be written as the product of the received irradiance in absence of turbulence, I 0 , and the normalized scintillation induced irradiance, I , with EI 1.
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Vol. 40, No. 13 / July 1 2015 / Optics Letters
Letter
Then, I rx I 0 I, and the instantaneous electrical signal-tonoise ratio (SNR) at the receiver can be expressed as γ
ηI 0 I2 γ0I 2; σ 2n
(2)
being γ 0 the received electrical SNR in absence of atmospheric turbulence, being Eγ γ 0 . To model the turbulence-induced scintillation, the M statistical distribution is assumed here. In this respect, as reported in [6,7], the pdf of the normalized irradiance for this distribution is given by the following expression: sffiffiffiffiffi ∞ X αI αk G ak I 2 −1 K α−k 2 f I I AG ; (3) ξg k1 where 8 G
