A Simple, Accurate Approximation to the Sum of Gamma-Gamma variates and Applications in MIMO Free-Space Optical SystemsK
1 like637 views
The document proposes using an alpha-mu distribution to approximate the sum of independent gamma-gamma random variables, which model irradiance in free-space optical communications systems under atmospheric turbulence. This approximation allows accurate yet simple performance analysis of MIMO free-space optical systems in terms of outage probability and average bit error probability. Numerical results validate the proposed approximation method.
A Simple, Accurate Approximation to the Sum of Gamma-Gamma variates and Applications in MIMO Free-Space Optical SystemsK
- 1. A Simple, Accurate Approximation to the Sum of Gamma-Gamma variates and Applications in MIMO Free-Space Optical Systems K. Peppas, Ch. Tasiopoulos, A. Fotopoulos, A. Talavari
- 2. Free Space Optical Communications (1) Rapidly gaining popularity as effective means of transferring data at high rates over short distances These systems facilitate rapidly deployable, lightweight, high-capacity communication without licensing fees and tariff The performance of this technology depends strongly on the atmospheric conditions between the transmitter and the receiver and the parameters of the link such as the length and the operation wavelength. Effects of fog, rain, atmospheric gases, and aerosols result in beam attenuation due to photon absorption and scattering
- 3. Free Space Optical Communications (2) In clear weather conditions, the atmospheric turbulence results in fluctuations at the received signal, i.e., signal fading, also known as scintillation in optical communication terminology Turbulence is caused by inhomogeneities of both temperature and pressure in the atmosphere and can severely degrade the link performance, particularly over link distances of 1 km or longer Field measurements performed with a link of 2.4 km, in San Diego, on February 25, 1997, a sunny and warm day, showed that the scintillation was worst around 1:30 pm Pacific Standard Time and was very bad for several hours.
- 4. Free Space Optical Communications (3) Various statistical models have been proposed to describe the optical channel characteristics with respect to the atmospheric turbulence strength. For weak fluctuations, the distribution of received intensities is close to a log-normal. However, as the strength of turbulence increases, lognormal statistics exhibit large deviations compared with experimental data Recently, Al-Habash et al proposed the gamma-gamma distribution as a tractable mathematical model for atmospheric turbulence
- 5. The gamma-gamma distribution (1) This model is a two-parameter distribution that is based on a doubly stochastic theory of scintillation Assumes that small-scale irradiance fluctuations are modulated by large-scale irradiance fluctuations of the propagating wave, both governed by independent gamma distributions. The gamma-gamma distribution has been found to be suitable for modeling the irradiance of the optical channels for moderate-to-strong turbulence channels since it provides good agreement between theoretical and experimental data
- 6. The gamma-gamma distribution (2) A random variable I ≥ 0 follows a gamma-gamma distribution if its probability density function is given by Γ( . ) is the gamma function and K a ( . ) is the modified Bessel function of the second kind and order a The parameters a and b are directly related to atmospheric conditions
- 7. The gamma-gamma distribution (3) Assuming spherical wave propagation these parameters are given by In the above equations, σ 2 is the Rytov variance given by The parameter d is given by The parameter Cn is altitude dependent and given by
- 8. Fading mitigation techniques in FSO systems Powerful fading-mitigation techniques need to be deployed for FSO links particularly with transmission range of 1 km or longer Error control coding in conjunction with interleaving can be employed in FSO communications to combat fading Large size interleavers are required to achieve the theoretical coding gains Maximum Likelihood Sequence Detection (MLSD) is another solution Suffers from excessive computational complexity, requiring multi-dimensional integration Space Diversity and MIMO techniques
- 9. Space Diversity and MIMO techniques in FSO systems The employment of multiple transmit/receive apertures, provides an attractive alternative approach for fading compensation Besides its role as a fading-mitigation tool, multiple-aperture designs significantly reduce the potential for temporary blockage of the laser beam by obstructions Further justification for the employment of multiple apertures comes from limitations in transmit power density The allowable safe laser power depends on the wavelength and obviously a higher power at the receiver side allows the system to support longer distances and through heavier attenuation while achieving higher data rates. It has been demonstrated that ergodic capacity scales as the number of transmit apertures times the number of receive apertures for high signal-to-background noise ratio
- 10. The Considered MIMO system (1) We consider a MIMO FSO system where the information signal is transmitted via M apertures and received by N apertures The information bits are modulated using On-Off keying (OOK) and transmitted through the M apertures using repetition coding A high-energy FSO system whose performance is limited by background radiation and thermal noise is assumed. Under this assumption, the use of the AWGN model as a good approximation of the Poisson photon counting detection model is applicable The received signal at the n-th receive aperture is expressed as η is the optical-to-electrical conversion coefficient, s = {0, 1} are the information symbols, υ n is AWGN .
- 11. The Considered MIMO system ( 2 ) Furthermore, we assume that the I mn - s are independent random variables , a fact justified for link distances of the order of kilometers and for aperture separation distances of the order of centimeters The output of the receiver, assuming that EGC combining is employed, can be expressed as The received electrical SNR between the m transmit and n receive aperture is given by Assuming that I mn - s follow a G-G distribution what are the statistics of γ ?
- 12. The proposed solution (1) Finding the statistics of the sum of independent G-G random variables is a very difficult problem An exact closed-form solution is not available in the open technical literature Infinite series representations are proposed in 1. Accurate Approximations are proposed in 2. , 3. In this work we propose to use an α-μ distribution to approximate the sum of i.i.d G-G random variables E. Bayaki, R. Schober, and R. 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. N. D. Chatzidiamantis, G. K. Karagiannidis, and D. S. Michalopoulos, “On the distribution of the sum of gamma-gamma variates and application in MIMO optical wireless systems,” in IEEE Global Telecommunications Conference (GLOBECOM .09), Hawai, USA, 30 Nov. - 4Dec. 2009 S. Al-Ahmadi and H. Yanikomeroglu, “On the approximation of the generalized-K distribution by a gamma distribution for modeling composite fading channels,” IEEE Trans. Wireless Commun., vol. 9, no. 2, pp. 706–713, Feb. 2010
- 13. The proposed solution (2) The α-μ PDF and CDF can be expressed as Why using the α-μ distribution? In 3. it was shown that the gamma distribution can be used to approximate the sum of independent G-G variates. We feel that the use of a more generic distribution, which includes as special case the gamma distribution (in the α-μ case by setting α = 1) will produce more accurate results Τ he estimation of the parameters α, μ PDF requires the knowledge of the first , the second and the fourth moment of Z . Therefore, the resulting PDF incorporates information regarding the mean, the variance and the kurtosis of Z
- 14. The proposed solution ( 3 ) Moments-based estimation The moments of Z can be evaluated as
- 15. The application of the proposed method The proposed approximation performs well for a wide range of the involved parameters and summands
- 16. Performance analysis of MIMO FSO systems (1) Performance metrics: Outage Probability (OP) and Average Bit Error Probability (ABEP) Outage Probability: the probability that the instantaneous SNR of the combined signal at the receiver output, falls below a pre-defined threshold γ th It is obtained as Can be readily evaluate using the proposed method with low computational complexity
- 17. Performance analysis of MIMO FSO systems (2)
- 18. Performance analysis of MIMO FSO systems (3) Assuming perfect Channel State Information, the ABEP can be evaluated as This integral is impossible to evaluate in closed form One can however easily obtain a highly accurate approximate value by using the proposed method By approximating the PDF of I with the PDF of Z, ABEP can finally be obtained by performing numerical or symbolic integration
- 19. Performance analysis of MIMO FSO systems (4)
- 20. Conclusion We proposed a novel highly accurate closed-form approximation to the sum of i.i.d gamma-gamma random variables The α-μ distribution was used to solve this cumbersome statistical problem The proposed method can be efficiently applied in the performance evaluation of MIMO FSO systems in the presence of atmospheric turbulence Numerical results as well as Monte-Carlo simulations demonstrated the efficiency of the proposed methods and validated the accuracy of the proposed analysis
- 21. Thank you for your attention! Any questions?