On the Bivariate Nakagami-Lognormal Distribution and Its Correlation Properties

The bivariate Nakagami-lognormal distribution used to model the composite fast fading and shadowing has been examined exhaustively. In particular, we have derived the joint probability density function, the cross-moments, and the correlation coefficient in power terms. Also, two procedures to generate two correlated Nakagami-lognormal random variables are described. These procedures can be used to evaluate the robustness of the sample correlation coefficient distribution in both macro- and microdiversity scenarios. It is shown that the bias and the standard deviation of this sample correlation coefficient are substantially high for large shadowing standard deviations found in wireless communication measurements, even if the number of observations is considerable.This work was supported by the Spanish Ministerio de Ciencia e Innovacion TEC-2010-20841-C04-1.Reig, J.; Rubio Arjona, L.; Rodrigo Peñarrocha, VM. (2014). On the Bivariate Nakagami-Lognormal Distribution and Its Correlation Properties. International Journal of Antennas and Propagation. 2014:1-8. https://doi.org/10.1155/2014/328732S182014Rubio, L., Reig, J., & Cardona, N. (2007). Evaluation of Nakagami fading behaviour based on measurements in urban scenarios. AEU - International Journal of Electronics and Communications, 61(2), 135-138. doi:10.1016/j.aeue.2006.03.004Suzuki, H. (1977). A Statistical Model for Urban Radio Propogation. IEEE Transactions on Communications, 25(7), 673-680. doi:10.1109/tcom.1977.1093888Abu-Dayya, A. A., & Beaulieu, N. C. (1994). Micro- and macrodiversity NCFSK (DPSK) on shadowed Nakagami-fading channels. IEEE Transactions on Communications, 42(9), 2693-2702. doi:10.1109/26.317410Tjhung, T. T., & Chai, C. C. (1999). Fade statistics in Nakagami-lognormal channels. IEEE Transactions on Communications, 47(12), 1769-1772. doi:10.1109/26.809692Shankar, P. M. (2004). Error Rates in Generalized Shadowed Fading Channels. Wireless Personal Communications, 28(3), 233-238. doi:10.1023/b:wire.0000032253.68423.86Atapattu, S., Tellambura, C., & Jiang, H. (2011). A Mixture Gamma Distribution to Model the SNR of Wireless Channels. IEEE Transactions on Wireless Communications, 10(12), 4193-4203. doi:10.1109/twc.2011.111210.102115Reig, J., & Rubio, L. (2013). Estimation of the Composite Fast Fading and Shadowing Distribution Using the Log-Moments in Wireless Communications. IEEE Transactions on Wireless Communications, 12(8), 3672-3681. doi:10.1109/twc.2013.050713.120054Mukherjee, S., & Avidor, D. (2003). Effect of microdiversity and correlated macrodiversity on outages in a cellular system. IEEE Transactions on Wireless Communications, 2(1), 50-58. doi:10.1109/twc.2002.806363Zhang, R., Wei, J., Michelson, D. G., & Leung, V. C. M. (2012). Outage Probability of MRC Diversity over Correlated Shadowed Fading Channels. IEEE Wireless Communications Letters, 1(5), 516-519. doi:10.1109/wcl.2012.072012.120452Rui, Z., Jibo, W., & Leung, V. C. M. (2013). Outage probability of composite microscopic and macroscopic diversity over correlated shadowed fading channels. China Communications, 10(11), 129-142. doi:10.1109/cc.2013.6674217Abdel-Hafez, M., & Safak, M. (1999). Performance analysis of digital cellular radio systems in Nakagami fading and correlated shadowing environment. IEEE Transactions on Vehicular Technology, 48(5), 1381-1391. doi:10.1109/25.790511Shankar, P. M. (2009). Macrodiversity and Microdiversity in Correlated Shadowed Fading Channels. IEEE Transactions on Vehicular Technology, 58(2), 727-732. doi:10.1109/tvt.2008.926622MOSTAFA, M. D., & MAHMOUD, M. W. (1964). On the problem of estimation for the bivariate lognormal distribution. Biometrika, 51(3-4), 522-527. doi:10.1093/biomet/51.3-4.522Reig, J., Rubio, L., & Cardona, N. (2002). Bivariate Nakagami-m distribution with arbitrary fading parameters. Electronics Letters, 38(25), 1715. doi:10.1049/el:20021124Tan, C. C., & Beaulieu, N. C. (1997). Infinite series representations of the bivariate Rayleigh and Nakagami-m distributions. IEEE Transactions on Communications, 45(10), 1159-1161. doi:10.1109/26.634675Lien, D., & Balakrishnan, N. (2006). Moments and properties of multiplicatively constrained bivariate lognormal distribution with applications to futures hedging. Journal of Statistical Planning and Inference, 136(4), 1349-1359. doi:10.1016/j.jspi.2004.10.004Sørensen, T. B. (1999). Slow fading cross-correlation against azimuth separation of base stations. Electronics Letters, 35(2), 127. doi:10.1049/el:19990085Reig, J., Martinez-Amoraga, M. A., & Rubio, L. (2007). Generation of bivariate Nakagami-m fading envelopes with arbitrary not necessary identical fading parameters. Wireless Communications and Mobile Computing, 7(4), 531-537. doi:10.1002/wcm.386Lai, C. D., Rayner, J. C. W., & Hutchinson, T. P. (1999). Robustness of the sample correlation - the bivariate lognormal case. Journal of Applied Mathematics and Decision Sciences, 3(1), 7-19. doi:10.1155/s117391269900001