Advanced
BER Analysis of Coherent Free Space Optical Systems with BPSK over Gamma-Gamma Channels
BER Analysis of Coherent Free Space Optical Systems with BPSK over Gamma-Gamma Channels
Journal of the Optical Society of Korea. 2015. Jun, 19(3): 237-240
Copyright © 2015, Optical Society of Korea
  • Received : February 02, 2015
  • Accepted : June 06, 2015
  • Published : June 25, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
Wansu Lim
wansu.lim@kumoh.ac.kr
Abstract
We derived the average bit error rate (BER) of coherent free-space optical (FSO) systems with digital binary phase shift keying (BPSK) modulations over atmospheric turbulence channels with a gamma–gamma distribution. To obtain a generalized derivation in a closed-form expression, we used special integrals and transformations of the Meijer G function. Furthermore, we numerically analyzed and simulated the average BER behavior according to the average SNR for different turbulence strengths. Simulation results are demonstrated to confirm the analytical results.
Keywords
I. INTRODUCTION
Free-space optical (FSO) communication systems [1 - 5] are commonly used to provide an attractive and cost-effective link for high-data-rate wireless transmission. FSO systems support diverse applications ranging from highly directive point-to-point links for terrestrial last-mile and long-haul intersatellite solutions to quick and efficient deployment in densely populated urban areas or in unstructured environments such as disaster-prone areas.
FSO systems utilize a free-space medium for transmission, but they are inherently affected by atmospheric conditions, among which turbulence has the most significant effect, especially for high-data-rate point-to-point links. As such, it is an interesting problem to analyze the degradation of signal strength due to scintillation of the optical signal as well as link performance against atmospheric turbulence channels. In atmospheric turbulence channels, the coherence time of the channels is on the order of milliseconds, which is typically much larger than the one-bit time interval of gigabit-per-second (Gbps) FSO signals [6 - 11] . Hence, for a one-bit time interval, the FSO channels are modeled as a constant and random variable that is governed by a log-normal, K, or gamma-gamma distribution. The gamma-gamma distribution is a tractable mathematical model with a multiplication of two parameters of small-scale and large-scale irradiance fluctuations, the probability density functions (PDFs) of which are independent gamma distributions, and it provides excellent agreement between theoretical and simulation results [6 - 8] .
Many authors have intensively researched and analyzed several implementation techniques for FSO systems under turbulence. In Ref. [8] , a performance analysis for intensity modulation-direct detection (IM-DD) FSO systems over gamma-gamma turbulence channels was presented. For coherent FSO systems, Refs. [12] and [13] proposed alternative implementations enabling a higher receiver sensitivity than that of IM-DD, especially when the power of the local oscillator laser is sufficiently high; Refs. [6] and [14] presented analyses of coherent heterodyne DPSK systems over K and gamma-gamma turbulence channels, respectively, considering thermal noise caused by the high operating temperature of FSO systems. Coherent PSK requires the proper control of laser coherence, which is challenging in FSO systems because of difficulties associated with the phase-locking of the local oscillators. However, coherent PSK is expected to provide performance benefits over DPSK. Furthermore, there are known results indicating that the performance of coherent FSO systems is limited by the shot noise of the receiver, which heterodynes with a local oscillator laser having sufficiently high power; this needs to be analyzed further under turbulence channels.
In this study, we derived a generalized closed-form expression for the average BER performance of coherent FSO systems by using binary phase shift keying (BPSK) over atmospheric turbulence channels, in which the turbulence-induced fading of the signal intensity is described by a gamma-gamma distribution. Moreover, theoretical results are provided to understand the degradation of performance as a function of scintillation depth. Analytical results are further confirmed through Monte Carlo simulations and VPI transmission Maker results.
II. SYSTEM AND CHANNEL MODEL
The overall architecture of coherent FSO systems is shown in Fig. 1 (a) . The optical modulator processes data by using a laser at a transmitter. The output signal of the optical modulator is transmitted via atmospheric turbulence channels between telescopes. In Fig. 1 (b) , the received signal is combined with a local oscillator laser through a half-mirror at a receiver. Then, a photodetector detects the compound optical signals to generate the photocurrent. Finally, a decision module extracts data. In this section, we explain the coherent PSK receiver in detail because it is the basic model for coherent receivers and can be adapted to other modulation receivers. In Fig. 1 (b) , the received optical signal and the local oscillator laser in scalar form can be expressed, respectively, as [12 , 13]
PPT Slide
Lager Image
(a) Overall architecture of coherent FSO systems. (b) Structure of a dual-photodiode balanced receiver of coherent FSO systems.
PPT Slide
Lager Image
where E S is the electrical field of the received signal, I is the intensity-fading coefficient, a = ± 1 is the information, f c is the optical carrier frequency, and E LO is the electric field of the local oscillator laser. The total power of the received signal and local oscillator laser is
PPT Slide
Lager Image
where P S is the power of the received signal and P LO is the power of the local oscillator laser. Additionally, the output current of the photodetector is expressed as
PPT Slide
Lager Image
where R is the responsivity of the receiver; the short noise ( iSH ( t )) and thermal noise ( iTH ( t )) have power spectral densities (PSDs) of G SH = RP LOq and
PPT Slide
Lager Image
, respectively; q is the electron charge; k is the Boltzmann constant; B is the noise-equivalent bandwidth of the filter; and R L is the load resistance. When iPD ( t ) is low-pass filtered to limit the noise power, the instantaneous signal-to-noise ratio (SNR) is given as
PPT Slide
Lager Image
In coherent FSO systems, if the power of the local oscillator laser were sufficiently high, the second term in the denominator of Eq. (4) would vanish. Thus, Eq. (4) is reduced to [5]
PPT Slide
Lager Image
As in [7] and [8] , the PDF of a gamma-gamma distribution is represented by the product of small-scale and large-scale irradiance fluctuations, both of which have gamma distributions. The gamma-gamma distribution is
PPT Slide
Lager Image
where I >0,
PPT Slide
Lager Image
is the average irradiance of the channel, α and β are the scintillation parameters, K s (ㆍ) is the modified Bessel function of the second kind of order ε , and Γ(ㆍ) is the Gamma function. Here, α and β are defined based on the atmospheric conditions, as in [8] .
III. DERIVATION OF THE AVERAGE BER
In this section, we derive the average BER of coherent FSO systems according to BPSK modulation. We first calculate the average SNR (µ) using [15 , Eq. 07.34.21.0009.01] as follows:
PPT Slide
Lager Image
The conditional BER, P b ( I ), for coherent systems is represented as
PPT Slide
Lager Image
where erfc is the complementary error function. To obtain a closed-form expression, we used the following Meijer G functions that were reported in [15 , Eq. 07.34.03.0619.01 and Eq. 07.34.03.0605.01], which are, respectively, expressed as
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Thus, the conditional BER (Eq. 8) and gamma-gamma distribution (Eq. 6) can be represented by the Meijier G function using Eqs. 9 and 10. Thus, the average BER (
PPT Slide
Lager Image
) can be obtained using the following integral:
PPT Slide
Lager Image
By substituting Eqs. (6) and (8) into Eq. (11), we can obtain the following equation:
PPT Slide
Lager Image
Finally, by using the classic Meijer integral of the two G functions [15 , Eq. 07.34.21.0011.01], Eq. 12 is simplified as follows:
PPT Slide
Lager Image
IV. NUMERICAL RESULTS
Figure 2 represents the results for a closed-form expression of the average BER,
PPT Slide
Lager Image
, as a function of the average SNR at different turbulence strengths. For instance, we considered the following turbulence strengths: ( α : β )∈ {(4,4),(6,6),(8,8),(10,10)}. Monte Carlo simulation results are included as a reference to validate our theoretical analysis. For creating gamma-gamma turbulence channels in the simulation, we used the multiplication of two random variables with a gamma distribution [3] . Then, we confirm that the PDF from a gamma-gamma random variable using the histogram method in MATLAB is the same as Eq. (6). Owing to the long simulation time, only simulation results up to BER = 10 −5 are included. The simulation results demonstrate an excellent agreement with the results of theoretical analysis. Considering that BER = 10 −9 is a practical performance target for an FSO system, our analytical results can serve as a simple and reliable method to estimate BER performance without resorting to lengthy simulations. In addition, to clarify our analysis further, we confirm its accuracy through system-level simulations using VPI transmission Maker. Because the validity of the FSO toolboxes of VPI transmission Maker has been established by verified FSO experimental data, we consider the results from VPI transmission Maker as an alternative experimental approach. The outputs of VPI transmission Maker are completely consistent with that of our analysis, as well as that of our MATLAB simulation.
PPT Slide
Lager Image
Comparison of the average BER performance as a function of the average SNR for (α ,β )∈{(4,4), (6,6), (8,8), (10,10)}.
Figure 2 plots the average BER as a function of the average SNR for different turbulence channel strengths. Recall that the channel strength depends on the scintillation parameters, α and β , such that the turbulence effects become stronger as α and β decrease. We observe that the BER of the coherent FSO system under a strong turbulence effect of (α,β) = (4,4) is significantly higher than that under a weak turbulence effect of (α,β) = (10,10). More specifically, in the weak turbulence case, the average BER with an SNR of 20 dB is less than 10 −8 . In the strong turbulence case, the average BER severely increases to 10 −3 at 20-dB SNR.
V. CONCLUSION
We obtained a closed-form expression for the average BER of coherent FSO systems over atmospheric turbulence channels with a gamma-gamma distribution by using special integrals and transformations of the Meijer G function. Furthermore, we simulated the average BER performance using the Monte Carlo method to confirm the theoretical results. Simulation results show an excellent agreement with the analytical results. Therefore, we can more easily predict BER performance by using a simple closed-form expression without any complicated calculation. In practical terms, when we establish coherent FSO systems, we can create an engineering table by using the derived BER.
Acknowledgements
This paper was supported by Research Fund, Kumoh National Institute of Technology.
References
Zedini E. , Ansari I. S. , Alouini M. 2015 “Performance analysis of mixed Nakagami- m and Gamma-Gamma dual-Hop FSO transmission systems,” IEEE Photon. J. 7 7900120 -
Kiasaleh K. 2015 “Receiver architecture for channel-aided, OOK, APD-based FSO communications through turbulent atmosphere,” IEEE Trans. Commun. 63 186 - 194
Song T. , Kam P.-Y. 2014 “A robust GLRT receiver with implicit channel estimation and automatic threshold adjustment for the free space optical channel with IM/DD,” IEEE J. Lightwave Technol. 32 369 - 383
Feng F. , White I. H. , Wilkinson T. D. 2014 “Aberration correction for free space optical communications using rectangular zernike modal wavefront sensing,” IEEE J. Lightwave Technol. 32 1239 - 1245
Tang Y. , Brandt-Pearce M. 2014 “Link allocation, routing, and scheduling for hybrid FSO/RF wireless mesh networks,” IEEE J. Opt. Commun. Netw. Opt. 6 86 - 95
Kiasaleh K. 2006 “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54 604 - 607
Al-Habash M. A. , Andrews L. C. , Phillips R. L. 2001 “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40 1554 - 1562
Uysal M. , Li J. , Yu M. 2006 “Error rate performance analysis of coded free-space optical links over gamma-gamma atmospheric turbulence channels,” IEEE Trans. Wireless Commun 5 1229 - 1233
Nistazakis H. E. , Stassinakis A. N. , Sandalidis H. G. , Tombras G. S. 2015 “QAM and PSK OFDM RoFSO over M-turbulence induced fading channels,” IEEE Photon. J. 7 7900411 -
Rajbhandari S. , Ghassemlooy Z. , Haigh P. A. , Kanesan T. , Tang X. 2015 “Experimental error performance of modulation schemes under a controlled laboratory turbulence FSO channel,” IEEE J. Lightwave Technol. 33 244 - 250
Yang L. , Gao X. , Alouini M.-S. 2014 P“Performance analysis of free-space optical communication systems with multiuser diversity over atmospheric turbulence channels,” IEEE Photon. J. 6 7901217 -
Kazovsky L. , Benedetto S. , Willner A. 1996 Optical Fiber Communication Systems Artech House Norwood, MA, USA
Kim I. , Goldfarb G. , Li G. 2007 “Electronic wavefront correction for PSK free-space optical communications,” Electron. Lett 43 1108 - 1109
Tsiftsis T. A. 2008 “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44 372 - 373
2004 The Wolfram function site Internet [online]. Available: