Skip to main content
Top
Published in: Wireless Personal Communications 3/2022

01-11-2021

LCR and AFD of the Products of Nakagami-m and Nakagami-m Squared Random Variables: Application to Wireless Communications Through Relays

Authors: Caslav Stefanovic, Ivan Milovanovic, Stefan Panic, Mihajlo Stefanovic

Published in: Wireless Personal Communications | Issue 3/2022

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

The paper considers level crossing rate (LCR) and average fade duration (AFD) of the product of independent and identically distributed (i.i.d) Nakagami-m (NM) and double NM squared (also known as gamma-gamma) random variables (RVs). The derived statistics are then directly applied to the radio-frequency (RF) - free space optical (FSO), dual-hop (DH), amplify-and-forward (AF) relaying system over non turbulent-induced-fading channels (nTIFCs) and turbulent-induced-fading channels (TIFCs). The obtained results for LCR and AFD of DH-AF, RF-FSO system over TIFCs and nTIFCs are numerically evaluated and graphically presented for various system model parameters.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Appendix
Available only for authorised users
Literature
1.
go back to reference Talha, B., & Pätzold, M. (2011). Channel models for mobile-to-mobile cooperative communication systems: A state of the art review. IEEE Vehicular Technology Magazine, 6(2), 33–43.CrossRef Talha, B., & Pätzold, M. (2011). Channel models for mobile-to-mobile cooperative communication systems: A state of the art review. IEEE Vehicular Technology Magazine, 6(2), 33–43.CrossRef
2.
go back to reference Park, J., Chae, C. B., & Yoon, G. (2017). Amplify-and-forward two-way relaying system over free-space optics channels. Journal of Communications and Networks, 19(5), 481–492.CrossRef Park, J., Chae, C. B., & Yoon, G. (2017). Amplify-and-forward two-way relaying system over free-space optics channels. Journal of Communications and Networks, 19(5), 481–492.CrossRef
3.
go back to reference Zhang, J., Pan, X., Pan, G., & Xie, Y. (2020). Secrecy analysis for multi-relaying RF-FSO systems with a multi-aperture destination. IEEE Photonics Journal, 12(2), 1–11. Zhang, J., Pan, X., Pan, G., & Xie, Y. (2020). Secrecy analysis for multi-relaying RF-FSO systems with a multi-aperture destination. IEEE Photonics Journal, 12(2), 1–11.
4.
go back to reference Zhang, Y., Zhang, J., Yang, L., Ai, B., & Alouini, M. S. (2020). On the performance of dual-hop systems over mixed FSO/mmWave fading channels. IEEE Open Journal of the Communications Society, 1, 477–489.CrossRef Zhang, Y., Zhang, J., Yang, L., Ai, B., & Alouini, M. S. (2020). On the performance of dual-hop systems over mixed FSO/mmWave fading channels. IEEE Open Journal of the Communications Society, 1, 477–489.CrossRef
5.
go back to reference Douik, A., Dahrouj, H., Al-Naffouri, T. Y., & Alouini, M. S. (2016). Hybrid radio/free-space optical design for next generation backhaul systems. IEEE Transactions on Communications, 64(6), 2563–2577.CrossRef Douik, A., Dahrouj, H., Al-Naffouri, T. Y., & Alouini, M. S. (2016). Hybrid radio/free-space optical design for next generation backhaul systems. IEEE Transactions on Communications, 64(6), 2563–2577.CrossRef
6.
go back to reference Stefanovic, C., Pratesi, M., & Santucci, F. (2019). Second order statistics of mixed RF-FSO relay systems and its application to vehicular networks. In: IEEE ICC’19 ONF Symposium Stefanovic, C., Pratesi, M., & Santucci, F. (2019). Second order statistics of mixed RF-FSO relay systems and its application to vehicular networks. In: IEEE ICC’19 ONF Symposium
7.
go back to reference Stefanovic, C., Pratesi, M., & Santucci, F. (2018). Second Performance evaluation of cooperative communications over fading channels in vehicular networks. In: Second URSI Atalntic Science Radio Meeting Stefanovic, C., Pratesi, M., & Santucci, F. (2018). Second Performance evaluation of cooperative communications over fading channels in vehicular networks. In: Second URSI Atalntic Science Radio Meeting
8.
go back to reference Fawaz, W., Abou-Rjeily, C., & Assi, C. (2017). UAV-aided cooperation for FSO communication systems. IEEE Communications Magazine, 56(1), 70–75.CrossRef Fawaz, W., Abou-Rjeily, C., & Assi, C. (2017). UAV-aided cooperation for FSO communication systems. IEEE Communications Magazine, 56(1), 70–75.CrossRef
9.
go back to reference Bhargav, N., da Silva, C. R. N., Chun, Y. J., Leonardo, E. J., Cotton, S. L., & Yacoub, M. D. (2018). On the product of two \(\kappa\)-\(\mu\) random variables and its application to double and composite fading channels. IEEE Transactions on Wireless Communications, 17(4), 2457–2470.CrossRef Bhargav, N., da Silva, C. R. N., Chun, Y. J., Leonardo, E. J., Cotton, S. L., & Yacoub, M. D. (2018). On the product of two \(\kappa\)-\(\mu\) random variables and its application to double and composite fading channels. IEEE Transactions on Wireless Communications, 17(4), 2457–2470.CrossRef
10.
go back to reference Milosevic, N., Stefanovic, M., Nikolic, Z., Spalevic, P., & Stefanovic, C. (2018). Performance analysis of interference-limited mobile-to-mobile \(\kappa\)-\(\mu\) fading channel. Wireless Personal Communications, 101(3), 1685–1701.CrossRef Milosevic, N., Stefanovic, M., Nikolic, Z., Spalevic, P., & Stefanovic, C. (2018). Performance analysis of interference-limited mobile-to-mobile \(\kappa\)-\(\mu\) fading channel. Wireless Personal Communications, 101(3), 1685–1701.CrossRef
11.
go back to reference Milosevic, N., Stefanovic, C., Nikolic, Z., Bandjur, M., & Stefanovic, M. (2018). First-and second-order statistics of interference-limited mobile-to- mobile Weibull fading channel. Journal of Circuits, Systems and Computers, 27(11), 1685–1701.CrossRef Milosevic, N., Stefanovic, C., Nikolic, Z., Bandjur, M., & Stefanovic, M. (2018). First-and second-order statistics of interference-limited mobile-to- mobile Weibull fading channel. Journal of Circuits, Systems and Computers, 27(11), 1685–1701.CrossRef
12.
go back to reference Stefanovic, C., Panic, S., Mladenovic, V., Jovkovic, S., & Stefanovic, M. (2020). Higher order statistics of cooperative mobile-to-mobile relay communications over composite fading channels. International Journal of Ad Hoc and Ubiquitous Computing, 35(2), 61–70.CrossRef Stefanovic, C., Panic, S., Mladenovic, V., Jovkovic, S., & Stefanovic, M. (2020). Higher order statistics of cooperative mobile-to-mobile relay communications over composite fading channels. International Journal of Ad Hoc and Ubiquitous Computing, 35(2), 61–70.CrossRef
13.
go back to reference Hajri, N., Khedhiri, R., & Youssef, N. (2020). On selection combining diversity in dual-hop relaying systems over double rice channels: Fade statistics and performance analysis. IEEE Access, 8, 72188–72203.CrossRef Hajri, N., Khedhiri, R., & Youssef, N. (2020). On selection combining diversity in dual-hop relaying systems over double rice channels: Fade statistics and performance analysis. IEEE Access, 8, 72188–72203.CrossRef
14.
go back to reference Li, S., Yang, L., Da Costa, D. B., Zhang, J., & Alouini, M. S. (2020). Performance analysis of mixed RF-UWOC dual-hop transmission systems. IEEE Transactions on Vehicular Technology, 69(11), 14043–14048.CrossRef Li, S., Yang, L., Da Costa, D. B., Zhang, J., & Alouini, M. S. (2020). Performance analysis of mixed RF-UWOC dual-hop transmission systems. IEEE Transactions on Vehicular Technology, 69(11), 14043–14048.CrossRef
15.
go back to reference Liu, W., Ding, J., Zheng, J., Chen, X., & Chih-Lin, X. I. (2020). Relay-assisted technology in optical wireless communications: A survey. IEEE Access, 8, 194384–194409.CrossRef Liu, W., Ding, J., Zheng, J., Chen, X., & Chih-Lin, X. I. (2020). Relay-assisted technology in optical wireless communications: A survey. IEEE Access, 8, 194384–194409.CrossRef
16.
go back to reference Liang, H., Gao, C., Li, Y., Miao, M., & Li, X. (2020). Performance analysis of mixed MISO RF/SIMO FSO relaying systems. Optics Communications, 478, 126344.CrossRef Liang, H., Gao, C., Li, Y., Miao, M., & Li, X. (2020). Performance analysis of mixed MISO RF/SIMO FSO relaying systems. Optics Communications, 478, 126344.CrossRef
17.
go back to reference Sharma, S., Madhukumar, A. S., & Swaminathan, R. (2020). Performance of dual-hop hybrid FSO/RF system with pointing errors optimization. In: Proceeding of the 2020 IEEE 91st vehicular technology conference (VTC2020-Spring), pp. 1–5 Sharma, S., Madhukumar, A. S., & Swaminathan, R. (2020). Performance of dual-hop hybrid FSO/RF system with pointing errors optimization. In: Proceeding of the 2020 IEEE 91st vehicular technology conference (VTC2020-Spring), pp. 1–5
18.
go back to reference Panic, S., Mohanchenko, S., Stefanovic, C., & Stefanovic, M. (2020). First order outage statistics of asymmetrical RF-OW dual-hop relay communications. The University Thought-Publication in Natural Sciences, 10(1), 57–62.CrossRef Panic, S., Mohanchenko, S., Stefanovic, C., & Stefanovic, M. (2020). First order outage statistics of asymmetrical RF-OW dual-hop relay communications. The University Thought-Publication in Natural Sciences, 10(1), 57–62.CrossRef
19.
go back to reference Nor, N. A. M., Ghassemlooy, Z., Bohata, J., Saxena, P., Komanec, M., Zvanovec, S., et al. (2016). Experimental investigation of all-optical relay-assisted 10 Gb/s FSO link over the atmospheric turbulence channel. Journal of Lightwave Technology, 35(1), 45–53.CrossRef Nor, N. A. M., Ghassemlooy, Z., Bohata, J., Saxena, P., Komanec, M., Zvanovec, S., et al. (2016). Experimental investigation of all-optical relay-assisted 10 Gb/s FSO link over the atmospheric turbulence channel. Journal of Lightwave Technology, 35(1), 45–53.CrossRef
20.
go back to reference Juel, N. H. (2021). Secrecy performance analysis of mixed and exponentiated Weibull RF-FSO cooperative relaying system. IEEE Access, 9, 72342–72356.CrossRef Juel, N. H. (2021). Secrecy performance analysis of mixed and exponentiated Weibull RF-FSO cooperative relaying system. IEEE Access, 9, 72342–72356.CrossRef
21.
go back to reference Salhab, M. A., & Yang, L. (2021). Mixed RF/FSO relay networks: RIS-equipped RF source vs RIS-aided RF source. IEEE Wireless Communications Letters, 10(8), 1712–1716.CrossRef Salhab, M. A., & Yang, L. (2021). Mixed RF/FSO relay networks: RIS-equipped RF source vs RIS-aided RF source. IEEE Wireless Communications Letters, 10(8), 1712–1716.CrossRef
22.
go back to reference Liu, X., Gu, C., Guo, K., Cheng, M., Lin, M., & Zhu, W.-P. (2021). Robust beamforming and outage performance of uplink multiuser satellite-aerial-terrestrial networks with mixed RF-FSO channels. IEEE Photonics Journal, 13(4), 1–8. Liu, X., Gu, C., Guo, K., Cheng, M., Lin, M., & Zhu, W.-P. (2021). Robust beamforming and outage performance of uplink multiuser satellite-aerial-terrestrial networks with mixed RF-FSO channels. IEEE Photonics Journal, 13(4), 1–8.
23.
go back to reference Yura, H. T., & Hanson, S. G. (2010). Mean level signal crossing rate for an arbitrary stochastic process. Journal of the Optical Society of America A, 27(4), 797–807.CrossRef Yura, H. T., & Hanson, S. G. (2010). Mean level signal crossing rate for an arbitrary stochastic process. Journal of the Optical Society of America A, 27(4), 797–807.CrossRef
25.
go back to reference Stefanovic, D., Stefanovic, C., Djosic, D., Milic, D., Rancic, D., & Stefanovic, M. (2019). LCR of the ratio of the product of two squared Nakagami-m random processes and its application to wireless. In: Proceedings of the 2019 18th IEEE international symposium INFOTEH-JAHORINA (INFOTEH), pp. 1–4 Stefanovic, D., Stefanovic, C., Djosic, D., Milic, D., Rancic, D., & Stefanovic, M. (2019). LCR of the ratio of the product of two squared Nakagami-m random processes and its application to wireless. In: Proceedings of the 2019 18th IEEE international symposium INFOTEH-JAHORINA (INFOTEH), pp. 1–4
26.
go back to reference Issaid, C. B., & Alouini, M. S. (2019). Level crossing rate and average outage duration of free space optical links. IEEE Transactions on Communications, 67(9), 6234–6242.CrossRef Issaid, C. B., & Alouini, M. S. (2019). Level crossing rate and average outage duration of free space optical links. IEEE Transactions on Communications, 67(9), 6234–6242.CrossRef
27.
go back to reference Stefanovic, C., Panic, S., Djosic, D., Milic, D., & Stefanovic, M. (2021). On the second order statistics of N-hop FSO communications over N-gamma-gamma turbulence induced fading channels. Physical Communication, p. 101289 Stefanovic, C., Panic, S., Djosic, D., Milic, D., & Stefanovic, M. (2021). On the second order statistics of N-hop FSO communications over N-gamma-gamma turbulence induced fading channels. Physical Communication, p. 101289
28.
go back to reference Vetelino, F. S., Young, C., & Andrews, L. (2007). Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence. Applied Optics, 46(18), 3780–3789.CrossRef Vetelino, F. S., Young, C., & Andrews, L. (2007). Fade statistics and aperture averaging for Gaussian beam waves in moderate-to-strong turbulence. Applied Optics, 46(18), 3780–3789.CrossRef
29.
go back to reference Kim, H. K., Higashino, T., Tsukamoto, K., & Komaki, S. (2011). Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel. IEEE International Conference on Space Optical Systems and Applications, pp. 58–66 Kim, H. K., Higashino, T., Tsukamoto, K., & Komaki, S. (2011). Optical fading analysis considering spectrum of optical scintillation in terrestrial free-space optical channel. IEEE International Conference on Space Optical Systems and Applications, pp. 58–66
30.
go back to reference Le, H. D., & Pham, A. T. (2021). Level crossing rate and average fade duration of satellite-to-UAV FSO channels. IEEE Photonics Journal, 13(1), 1–14.MathSciNetCrossRef Le, H. D., & Pham, A. T. (2021). Level crossing rate and average fade duration of satellite-to-UAV FSO channels. IEEE Photonics Journal, 13(1), 1–14.MathSciNetCrossRef
31.
go back to reference Stüber, L. G. (1996). Principles of mobile communication. Mass, USA: Kluwer Academic.CrossRef Stüber, L. G. (1996). Principles of mobile communication. Mass, USA: Kluwer Academic.CrossRef
33.
go back to reference Stefanovic, C., Jaksic, B., Spalevic, P., Panic, S., & Trajcevski, Z. (2013). Performance analysis of selection combining over correlated Nakagami-m fading channels with constant correlation model for desired signal and cochannel interference. Radioengineering, 22(4), 1176–1181. Stefanovic, C., Jaksic, B., Spalevic, P., Panic, S., & Trajcevski, Z. (2013). Performance analysis of selection combining over correlated Nakagami-m fading channels with constant correlation model for desired signal and cochannel interference. Radioengineering, 22(4), 1176–1181.
34.
go back to reference Al-Ahmadi, S. (2014). The gamma-gamma signal fading model: A survey. IEEE Antennas and Propagation Magazine, 56(5), 245–260.CrossRef Al-Ahmadi, S. (2014). The gamma-gamma signal fading model: A survey. IEEE Antennas and Propagation Magazine, 56(5), 245–260.CrossRef
35.
go back to reference Petkovic, M., Zdravkovic, N., Stefanovic, C., & Djordjevic, G. (2014). Performance analysis of SIM-FSO system over Gamma–Gamma atmospheric channel. In: XLIX International Scientific Conference on Information, Communication and Energy Systems and Technologies—ICEST 2014, vol. 1, pp. 19–22 Petkovic, M., Zdravkovic, N., Stefanovic, C., & Djordjevic, G. (2014). Performance analysis of SIM-FSO system over Gamma–Gamma atmospheric channel. In: XLIX International Scientific Conference on Information, Communication and Energy Systems and Technologies—ICEST 2014, vol. 1, pp. 19–22
36.
go back to reference Datsikas, C. K., Peppas, K. P., Sagias, N. C., & Tombras, G. S. (2010). Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels. IEEE/OSA Journal of Optical Communications and Networking, 2(8), 576–586.CrossRef Datsikas, C. K., Peppas, K. P., Sagias, N. C., & Tombras, G. S. (2010). Serial free-space optical relaying communications over gamma-gamma atmospheric turbulence channels. IEEE/OSA Journal of Optical Communications and Networking, 2(8), 576–586.CrossRef
37.
go back to reference Badarneh, O. S., Muhaidat, S., Sofotasios, P. C., Cotton, S. L., Rabie, K., & da Costa, D. B. (2018). The N\(*\) Fisher–Snedecor F cascaded fading model. In: Proceedings of the 14th international conference on wireless and mobile computing, networking and communications (WiMob), p. 1–7 Badarneh, O. S., Muhaidat, S., Sofotasios, P. C., Cotton, S. L., Rabie, K., & da Costa, D. B. (2018). The N\(*\) Fisher–Snedecor F cascaded fading model. In: Proceedings of the 14th international conference on wireless and mobile computing, networking and communications (WiMob), p. 1–7
38.
go back to reference Karagiannidis, G. K., Sagias, N. C., & Mathiopoulos, P. T. (2007). N* Nakagami: A novel stochastic model for cascaded fading channels. IEEE Transactions on Communications, 55(8), 1453–1458.CrossRef Karagiannidis, G. K., Sagias, N. C., & Mathiopoulos, P. T. (2007). N* Nakagami: A novel stochastic model for cascaded fading channels. IEEE Transactions on Communications, 55(8), 1453–1458.CrossRef
39.
go back to reference Bithas, P. S., Kanatas, A. G., da Costa, D. B., Upadhyay, P. K., & Dias, U. S. (2018). On the double-generalized gamma statistics and their application to the performance analysis of V2V communications. IEEE Transactions on Communications, 66(1), 448–460.CrossRef Bithas, P. S., Kanatas, A. G., da Costa, D. B., Upadhyay, P. K., & Dias, U. S. (2018). On the double-generalized gamma statistics and their application to the performance analysis of V2V communications. IEEE Transactions on Communications, 66(1), 448–460.CrossRef
42.
go back to reference Al-Hmood, H., Abbas, R. S., & Al-Raweshidy,H. S. (2020). Ratio of products of mixture gamma variates with applications to wireless communications. IEEE Transactions on Communications. arXiv preprint arXiv:2007.10826 Al-Hmood, H., Abbas, R. S., & Al-Raweshidy,H. S. (2020). Ratio of products of mixture gamma variates with applications to wireless communications. IEEE Transactions on Communications. arXiv preprint arXiv:​2007.​10826
43.
go back to reference Stankovic, A., Stefanovic, C., Sekulovic, N., Popovic, Z., & Stefanovic, M. (2012). The distribution of minimum of ratios of two random variables and its application in analysis of multi-hop systems. Radioengineering, 21(4), 1156–1162. Stankovic, A., Stefanovic, C., Sekulovic, N., Popovic, Z., & Stefanovic, M. (2012). The distribution of minimum of ratios of two random variables and its application in analysis of multi-hop systems. Radioengineering, 21(4), 1156–1162.
44.
45.
go back to reference Hadzi-Velkov, Z., Zlatanov, N., & Karagiannidis, G. K. (2009). On the second order statistics of the multihop rayleigh fading channel. IEEE Transactions on Communications, 57(6), 1815–1823.CrossRef Hadzi-Velkov, Z., Zlatanov, N., & Karagiannidis, G. K. (2009). On the second order statistics of the multihop rayleigh fading channel. IEEE Transactions on Communications, 57(6), 1815–1823.CrossRef
46.
go back to reference Al-Habash, A., Andrews, L. C., & Phillips, R. L. (2001). Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media. Optical Engineering, 40(8), 1554–1562.CrossRef Al-Habash, A., Andrews, L. C., & Phillips, R. L. (2001). Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media. Optical Engineering, 40(8), 1554–1562.CrossRef
47.
go back to reference Yacoub, M. D., Bautista, J. V., & Guedes, L. G. R. (1999). On higher order statistics of the Nakagami-m distribution. IEEE Transactions on Vehicular Technology, 48(3), 790–794.CrossRef Yacoub, M. D., Bautista, J. V., & Guedes, L. G. R. (1999). On higher order statistics of the Nakagami-m distribution. IEEE Transactions on Vehicular Technology, 48(3), 790–794.CrossRef
48.
go back to reference Stefanovic, C., Djosic, D., Panic, S., Milic, D., & Stefanovic, M. (2020). A framework for statistical channel modeling in 5G wireless communication systems. 5G Multimedia communications: Technology, multiservices, deployment, CRC Press 2020, pp. 31–54 Stefanovic, C., Djosic, D., Panic, S., Milic, D., & Stefanovic, M. (2020). A framework for statistical channel modeling in 5G wireless communication systems. 5G Multimedia communications: Technology, multiservices, deployment, CRC Press 2020, pp. 31–54
49.
go back to reference Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of Integrals, Series, and Products (6th ed.). New York: Academic.MATH Gradshteyn, I. S., & Ryzhik, I. M. (2000). Table of Integrals, Series, and Products (6th ed.). New York: Academic.MATH
Metadata
Title
LCR and AFD of the Products of Nakagami-m and Nakagami-m Squared Random Variables: Application to Wireless Communications Through Relays
Authors
Caslav Stefanovic
Ivan Milovanovic
Stefan Panic
Mihajlo Stefanovic
Publication date
01-11-2021
Publisher
Springer US
Published in
Wireless Personal Communications / Issue 3/2022
Print ISSN: 0929-6212
Electronic ISSN: 1572-834X
DOI
https://doi.org/10.1007/s11277-021-09258-6

Other articles of this Issue 3/2022

Wireless Personal Communications 3/2022 Go to the issue