Product of Three Random Variables and its Application in Relay Telecommunication Systems in the Presence of Multipath Fading

• Autorzy: Krstic Dragana , Nikolic Petar , Aleksic Danijela , Minic Sinisa , Vuckovic Dragan , Stefanovic Mihajlo

Identyfikatory

DOI

10.26636/jtit.2019.130018

• Języki publikacji: EN

Abstrakty

EN

In this paper, the product of three random variables (RVs) will be considered. Distribution of the product of independent random variables is very important in many applied problems, including wireless relay telecommunication systems. A few of such products of three random variables are observed in this work: the level crossing rate (LCR) of the product of a Nakagami-m random variable, a Rician random variable and a Rayleigh random variable, and of the products of two Rician RVs and one Nakagami-m RV is calculated in closed forms and presented graphically. The LCR formula may be later used for derivation of average fade duration (AFD) of a wireless relay communication radio system with three sections, working in the multipath fading channel. The impact of fading parameters and multipath fading power on the LCR is analyzed based on the graphs presented.

Słowa kluczowe

EN

level crossing rate; Nakagami-m fading; Rayleigh fading; relay telecommunication systems; Rician fading

• Wydawca: Instytut Łączności - Państwowy Instytut Badawczy
• Czasopismo: Journal of Telecommunications and Information Technology
• Rocznik: 2019
• Tom: nr 1
• Strony: 83--92
• Opis fizyczny: Bibliogr. 39 poz., rys.

Twórcy

autor

Krstic Dragana

• Faculty of Electronic Engineering, University of Niš, Niš, Serbia

autor

Nikolic Petar

• TigarTyres, Pirot, Serbia

autor

Aleksic Danijela

• College of Applied Technical Sciences Niš, Serbia

autor

Minic Sinisa

• Teachers' Training Faculty, Prizren-Leposavic, University of Pristina, Kosovska Mitrovica, Serbia

autor

Vuckovic Dragan

• Faculty of Economics and Engineering Management, University Business Academy, Novi Sad, Serbia

autor

Stefanovic Mihajlo

• Faculty of Electronic Engineering, University of Niš, Niš, Serbia

Bibliografia

• [1] Y. Chen, G. K. Karagiannidis, Hao Lu, and Ning Cao, „Novel approximations to the statistics of products of independent random variables and their applications in wireless communications", IEEE Trans. on Veh. Technol., vol. 61, no. 2, 2012, pp. 443-454 (doi: 10.1109/TVT.2011.2178441).
• [2] S. Nadarajaha and D. K. Dey, „On the product and ratio of t random variables", Appl. Mathem. Lett., vol. 19, no. 1, pp. 45-55, 2006 (doi: 10.1016/j.aml.2005.01.004).
• [3] E. Mekić, N. Sekulović, M. Bandjur, M. Stefanović, and P. Spalević, „The distribution of ratio of random variable and product of two random variables and its application in performance analysis of multi-hop relaying communications over fading channels", Przegląd Elektrotechniczny (Electrical Review), vol. 88, no. 7a, pp. 133-137, 2012.
• [4] D. Krstic, M. Stefanovic, V. Milenkovic, and Dj. Bandjur, „Level crossing rate of ratio of product of two a-k- m random variables and a-k- m random variable", WSEAS Trans. on Commun., vol. 13, no. 1, pp. 622-630, 2014.
• [5] D. Krstic, I. Romdhani, M. B. Y. Masadeh, S. Minic, G. Petkovic, and P. Milacic, „Level crossing rate of ratio of product of two ku random variables and Nakagami-m random variable", IEEE Int. Conf. on Comp. and Inform. Technol.; Ubiquitous Comput. and Commun.; Depend., Autonom. and Sec. Computi.; Perv. Intellig. and Comput., Liverpool, UK, 2015 (doi: 10.1109/CIT/IUCC/DASC/PICOM.2015.244).
• [6] N. Zlatanov, Z. Hadzi-Velkov, and G. K. Karagiannidis, „Level crossing rate and average fade duration of the double Nakagami-m random process and application in MIMO keyhole fading channels", IEEE Commun. Lett., vol. 12, no. 11, pp. 822-824, 2008 (doi: 10.1109/LCOMM.2008.081058).
• [7] J. D. Donahue, „Products and quotients of random variables and their applications", ARL 64-115, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio, The Martin Company, Denver, Colorado, July 1964 [Online]. Available: https://apps.dtic.mil/dtic/tr/fulltext/u2/603667.pdf
• [8] M. D. Springer and W. E. Thompson, „The distribution of products of independent random variables", SIAM J. on Appl. Mathem., vol. 14, no. 3, pp. 511-526, 1966 (doi: 10.1137/0114046).
• [9] M. D. Springer and W. E. Thompson, „The distribution of products of beta, gamma and Gaussian random variables", SIAM J. on Appl. Mathem., vol. 18, no 4, pp. 721-737, 1970 (doi: 10.1137/0118065).
• [10] Z. A. Lomnicki, „On the distribution of products of random variables", J. of the Royal Statist. Soc. Series B (Methodological), vol. 29, no. 3, pp. 513-524, 1967.
• [11] A. G. Glen, L. M. Leemis, and J. H. Drew, „Computing the distribution of the product of two continuous random variables", Comput. Statist. and Data Anal., vol. 44, no. 3, pp. 451-464, 2004 (doi: 10.1016/S0167-9473(02)00234-7).
• [12] T. S. Glickman and F. Xu, „The distribution of the product of two triangular random variables", Statist. & Probab. Lett., vol. 78, no. 16, pp. 2821-2826, 2008 (doi: 0.1016/j.spl.2008.03.031).
• [13] G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, „N*Nakagami: a novel stochastic model for cascaded fading channels", IEEE Trans. Commun., vol. 55, no. 8, pp. 1453-1458, 2007 (doi: 10.1109/TCOMM.2007.902497).
• [14] Z. Zheng, L. Wei, J. Hamalainen, and O. Tirkkonen, „Approximation to distribution of product of random variables using orthogonal polynomials for lognormal density", IEEE Commun. Lett., vol. 16, no. 12, pp. 2028-2031, 2012 (doi: 10.1109/LCOMM.2012.101712.122141).
• [15] E. J. Leonardo and M. D. Yacoub, „Statistics of the product of arbitrary a- m variates with applications", in Proc. 25th Int. Symp. on Pers., Indoor and Mob. Radio Commun. PIMRC 2014, Washington, DC, USA, 2014, pp. 73-76 (doi: 10.1109/PIMRC.2014.7136135).
• [16] Z. Stojanac, D. Suess, and M. Kliesch, „On products of Gaussian random variables", arXiv:1711.10516 [math.PR], 2018.
• [17] N. Bhargav et al., „On the product of two k- m random variables and its application to double and composite fading channels", IEEE Trans. on Wirel. Commun., vol. 17, no. 4, pp. 2457-2470, 2018 (doi: 10.1109/TWC.2018.2796562).
• [18] D. H. Pavlovic et al., „Statistics for ratios of Rayleigh, Rician, Nakagami-m, and Weibull distributed random variables", Mathem. Problems in Engin., vol. 2013, Article ID 252804 (doi: 10.1155/2013/252804).
• [19] M. Shakil and B. M. Golam Kibria, „On the product of Maxwell and Rice random variables", J. of Modern Appl. Statist. Meth., vol. 6, no. 1, Article 19, pp. 212-218, 2007 (doi: 10.22237/jmasm/1177993080) [Online]. Available: http://digitalcommons.wayne.edu/jmasm/vol6/iss1/19
• [20] K. Pearson, „The problem of the random walk", Nature, vol. 72, p. 318, 1905 (doi: 10.1038/072294b0).
• [21] M. Nakagami, „The m-distribution: A general formula of intensity distribution of rapid fading", in Statistical Methods in Radio Wave Propagation: Proceedings of a Symposium held June 18-20, 1958, W. C. Hoffman, Ed. New York: Pergamon Press, 1960, pp. 3-36 (doi: 10.1016/b978-0-08-009306-2.50005-4).
• [22] S. O. Rice, „Mathematical analysis of random noise", Bell Syst. Technic. J., vol. 24, no. 1, pp. 46-156, 1945 (doi: 10.1002/j.1538-7305.1945.tb00453.x).
• [23] D. Chizhik, G. J. Foschini, M. J. Gans, and R. A. Valenzuela, „Keyholes, correlations, and capacities of multielement transmit and receive antennas", IEEE Trans. on Wirel. Commun., vol. 1, no. 2, pp. 361-368, 2002 (doi: 10.1109/7693.994830).
• [24] T. Taniguchi, Y. Karasawa, and M. Tsuruta, „An analysis method of double fading MIMO channels including LOS environments", in Proc. of the IEEE 19th Int. Symp. Pers., Indoor Mob. Radio Commun., Cannes, France, 2008, pp. 1-5 (doi: 10.1109/PIMRC.2008.4699512).
• [25] D. Krstic, M. Stefanovic, and P. Nikolić, „Level crossing rate of product of Nakagami-m random variable, Rician random variable and Rayleigh random variable", ICTF 2018, IEICE Information and Communication Technology Forum, July 2018, Graz, Austria.
• [26] D. Krstic, M. Stefanovic, M. M. B. Yaseen, S. Aljawarneh, and P. Nikolić, „Statistics of the product of three Rician random processes with application", in Proc. 1st Int. Conf. on Data Sci., Elearn. and Inform. Syst. DATA'18, Madrid, Spain, 2018 (doi: 10.1145/3279996.3280015).
• [27] G. L. Stüber, Principles of Mobile Communication, 2nd ed. Norwell, MA, USA: Kluwer, 2001 (ISBN: 0792379985).
• [28] P. M. Shankar, Fading and Shadowing in Wireless Systems. New York Dordrecht Heidelberg London: Springer, 2012 (doi: 10.1007/978-1-4614-0367-8).
• [29] S. Panic, M. Stefanovic, J. Anastasov, and P. Spalevic Fading and Interference Mitigation in Wireless Communications. Boca Raton, USA: CRC Press, 2013 (ISBN: 9781466508415).
• [30] D. Shen, Y. Cui, A. Zhang, Y. Yang, and K. Wu, „A simple simulation method for Nakagami fading channel", in Proc. Int. Conf. on Microw. and Millim. Wave Technol., Chengdu, China, 2010 (doi:10.1109/icmmt.2010.5525262).
• [31] A. Goldsmith, Wireless Communications. Stanford University, 2004 (ISBN: 9780521837163).
• [32] S. O. Rice, „Statistical properties of a sine wave plus random noise", Bell Syst. Tech. J., vol. 27, no. 1, pp. 109-157, 1948 (doi:10.1002/j.1538-7305.1948.tb01334.x).
• [33] A. Mitić, D. Milović, M. Jakovljević, A. Panajotović, „Second order statistics of signal in Nakagami - lognormal fading channels with selection combining", in XIII Telekomunikacioni forum TELFOR 2005, Beograd, Serbia, 2005 [in Serbian].
• [34] M. D. Yacoub, J. E. V. Bautista, and L. Guerra de Rezende Guedes, „On higher order statistics of the Nakagami-m distribution", IEEE Trans. on Veh. Technol., vol. 48, no. 3, pp. 790-794, 1999 (doi: 10.1109/25.764995).
• [35] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 2002 (ISBN: 0130422320).
• [36] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. National Bureau of Standards, 1964, reprinted Dover Publications, 1965 (ISBN: 9780486612720).
• [37] J. L. Lopez and P. J. Pagola, „A simplification of the Laplace method for double integrals. Application to the second Appell function*", Electron. Trans. on Num. Analysis, vol. 30, pp. 224-236, 2008 (ISSN: 1068-9613).
• [38] N. C. Karmakar, Ed., Handbook of Smart Antennas for RFID Systems. Wiley, 2010 (doi: 10.1002/9780470872178, ISSN: 1068-9613).
• [39] Z. Cao and Y. D. Yao, „Definition and derivation of level crossing rate and average fade duration in an interference-limited environment", in Proc. IEEE 54th Veh. Technol. Conf. VTC Fall 2001, Atlantic City, N.J., USA, 2001 (doi: 10.1109/VTC.2001.956470).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).