Sum of arbitrarily correlated Gamma random variables with unequal parameters and its application in wireless communications

Characterizing the distribution for the sum of correlated Gamma random variables (RVs), especially for the sum of those with unequal fading and power parameters, is still an open issue. In this paper, based on the Cholesky factorization on the covariance matrix and moments matching method, we propose an approximate expression for the probability density function (PDF) of the sum of correlated Gamma RVs with unequal fading and power parameters and arbitrary correlation matrix. The proposed PDF expression is simple, accurate, and closed-form and thus can be conveniently used for general performance analysis in wireless communications. Simulation results are used to confirm the validity of the proposed PDF expression. The performance analysis of maximal-ratio combining (MRC) diversity system and cellular mobile radio system in wireless communications using the proposed PDF expression is also presented.

[1]  D. Rajan.  Probability, Random Variables, and Stochastic Processes , 2017 .

[2]  M. Nakagami.  The m-Distribution—A General Formula of Intensity Distribution of Rapid Fading , 1960 .

[3]  Cheol Mun,et al.  Approximation of SNR statistics for MRC diversity systems in arbitrarily correlated Nakagami fading channels , 1999 .

[4]  Moe Z. Win,et al.  MRC performance for M-ary modulation in arbitrarily correlated Nakagami fading channels , 2000, IEEE Communications Letters.

[5]  Nikos C. Sagias,et al.  New results for the multivariate Nakagami-m fading model with arbitrary correlation matrix and applications , 2009, IEEE Transactions on Wireless Communications.

[6]  Keith Q. T. Zhang.  A generic correlated Nakagami fading model for wireless communications , 2003, IEEE Trans. Commun..

[7]  John G. Proakis,et al.  Probability, random variables and stochastic processes , 1985, IEEE Trans. Acoust. Speech Signal Process..

[8]  George K. Karagiannidis,et al.  An efficient approach to multivariate Nakagami-m distribution using Green's matrix approximation , 2003, IEEE Trans. Wirel. Commun..

[9]  Yong Liang Guan,et al.  Simulation of Nakagami fading channels with arbitrary cross-correlation and fading parameters , 2004, IEEE Transactions on Wireless Communications.

[10]  James R. Zeidler,et al.  Performance analysis of compact antenna arrays with MRC in correlated Nakagami fading channels , 2001, IEEE Trans. Veh. Technol..

[11]  Q. T. Zhang.  Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix , 1999 .

[12]  Mohamed-Slim Alouini,et al.  On the sum of gamma random variates with application to the performance of maximal ratio combining over Nakagami-m fading channels , 2012, 2012 IEEE 13th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC).

[13]  S. Nadarajah.  A Review of Results on Sums of Random Variables , 2008 .

[14]  George K. Karagiannidis,et al.  Closed-form statistics for the sum of squared Nakagami-m variates and its applications , 2006, IEEE Transactions on Communications.

[15]  Chengwen Xing,et al.  Exact Performance Analysis of Dual-Hop Semi-Blind AF Relaying over Arbitrary Nakagami-m Fading Channels , 2011, IEEE Transactions on Wireless Communications.

[16]  Ali Abdi,et al.  Sum of gamma variates and performance of wireless communication systems over Nakagami-fading channels , 2001, IEEE Trans. Veh. Technol..

[17]  Valentine A. Aalo,et al.  Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment , 1995, IEEE Trans. Commun..

[18]  Chintha Tellambura,et al.  Error rates for Nakagami-m fading multichannel reception of binary and M-ary signals , 2001, IEEE Trans. Commun..

[19]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[20]  Tuan A. Tran,et al.  Sum of arbitrarily correlated Gamma variates and performance of wireless communication systems over Nakagami-m fading channels , 2007, IET Commun..