On the Product of Two $\kappa$ – $\mu$ Random Variables and its Application to Double and Composite Fading Channels

In this paper, we perform a systematic investigation of the statistics associated with the product of two independent and non-identically distributed <inline-formula> <tex-math notation="LaTeX">$\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula> random variables. More specifically, we develop novel analytical formulations for many of the fundamental statistics of interest, namely, the probability density function, cumulative distribution function, and moment-generating function. Using these new results, closed-form expressions are obtained for the higher order moments, amount of fading and channel quality estimation index, while analytical formulations are obtained for the outage probability, average channel capacity, average symbol error probability, and average bit error probability. These general expressions can be reduced to a number of fading scenarios, such as the double Rayleigh, double Rice, double Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula>/Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, and Rice/Nakagami-<inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, which all occur as special cases. Additionally, as a byproduct of the work performed here, formulations for the <inline-formula> <tex-math notation="LaTeX">$\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu /\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula> composite fading model can also be deduced. To illustrate the efficacy of the novel expressions proposed here, we provide useful insights into the outage probability of a dual-hop system used in body area networks, and demonstrate the suitability of the <inline-formula> <tex-math notation="LaTeX">$\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu /\kappa$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula> composite fading for characterizing shadowed fading in device-to-device channels.

[1]  Ali Abdi,et al.  On the utility of gamma PDF in modeling shadow fading (slow fading) , 1999, 1999 IEEE 49th Vehicular Technology Conference (Cat. No.99CH36363).

[2]  E. Jakeman,et al.  Generalized K distribution: a statistical model for weak scattering , 1987 .

[3]  George K. Karagiannidis,et al.  Channel Quality Estimation Index (CQEI): An Improved Performance Criterion for Wireless Communications Systems Over Fading Channels , 2006 .

[4]  Mohamed-Slim Alouini,et al.  Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis , 2000 .

[5]  Simon L. Cotton,et al.  Secrecy Capacity Analysis Over κ-μ Fading Channels: Theory and Applications , 2015, IEEE Trans. Commun..

[6]  Ranjan K. Mallik,et al.  Performance analysis of MIMO free-space optical systems in gamma-gamma fading , 2009, IEEE Transactions on Communications.

[7]  Colby Boyer,et al.  — Invited Paper — Backscatter Communication and RFID: Coding, Energy, and MIMO Analysis , 2014, IEEE Transactions on Communications.

[8]  David B. Smith,et al.  An Experimental-Based Analysis of Inter-BAN Co-Channel Interference Using the κ - μ Fading Model , 2017 .

[9]  David W. Matolak,et al.  Worse-than-Rayleigh fading: Experimental results and theoretical models , 2011, IEEE Communications Magazine.

[10]  P. Vainikainen,et al.  Statistical Analysis of the Multiple Scattering Radio Channel , 2006, IEEE Transactions on Antennas and Propagation.

[11]  Marvin K. Simon,et al.  Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers, Scientists and Mathematicians , 2006 .

[12]  Jonathan Ling,et al.  Comparisons of a Computer-Based Propagation Prediction Tool with Experimental Data Collected in Urban Microcelluar Environments , 1997, IEEE J. Sel. Areas Commun..

[13]  Matthias Patzold,et al.  Channel Models for Mobile-to-Mobile Cooperative Communication Systems: A State of the Art Review , 2011, IEEE Vehicular Technology Magazine.

[14]  Aggelos Bletsas,et al.  Multistatic Scatter Radio Sensor Networks for Extended Coverage , 2017, IEEE Transactions on Wireless Communications.

[15]  Seong Ki Yoo,et al.  The Fisher-Snedecor F distribution: A Simple and Accurate Composite Fading Model , 2017 .

[16]  Saralees Nadarajah,et al.  Exact distribution of the product of m gamma and n Pareto random variables , 2011, J. Comput. Appl. Math..

[17]  David Tse,et al.  Fundamentals of Wireless Communication , 2005 .

[18]  Aggelos Bletsas,et al.  Coherent Detection and Channel Coding for Bistatic Scatter Radio Sensor Networking , 2015, IEEE Transactions on Communications.

[19]  Victor Adamchik,et al.  The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system , 1990, ISSAC '90.

[20]  Simon L. Cotton,et al.  An experimental investigation into the influence of user state and environment on fading characteristics in wireless body area networks at 2.45 GHz , 2009, IEEE Transactions on Wireless Communications.

[21]  F. Li,et al.  A new polynomial approximation for Jν Bessel functions , 2006, Appl. Math. Comput..

[22]  Reinaldo A. Valenzuela,et al.  Keyholes, correlations, and capacities of multielement transmit and receive antennas , 2002, IEEE Trans. Wirel. Commun..

[23]  Mohamed-Slim Alouini,et al.  Coded Communication over Fading Channels , 2005 .

[24]  Yoshio Karasawa,et al.  An analysis method of double fading MIMO channels including LOS environments , 2008, 2008 IEEE 19th International Symposium on Personal, Indoor and Mobile Radio Communications.

[25]  David B. Smith,et al.  An Experimental-Based Analysis of Inter-BAN Co-Channel Interference Using the $\kappa$ - $\mu$ Fading Model , 2017, IEEE Transactions on Antennas and Propagation.

[26]  M.D. Yacoub,et al.  The κ-μ distribution and the η-μ distribution , 2007, IEEE Antennas and Propagation Magazine.

[27]  Simon L. Cotton,et al.  Simultaneous channel measurements of the on-body and body-to-body channels , 2016, 2016 IEEE 27th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC).

[28]  Michail Matthaiou,et al.  The Fisher–Snedecor $\mathcal {F}$ Distribution: A Simple and Accurate Composite Fading Model , 2017, IEEE Communications Letters.

[29]  L. Andrews,et al.  Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media , 2001 .

[30]  Michel Daoud Yacoub,et al.  Product of Two Envelopes Taken From α-μ, κ-μ, and η-μ Distributions , 2018, IEEE Trans. Commun..

[31]  Michail Matthaiou,et al.  The κ-μ / Inverse Gamma and η-μ / Inverse Gamma Composite Fading Models: Fundamental Statistics and Empirical Validation , 2017, IEEE Transactions on Communications.

[32]  Hyundong Shin,et al.  Performance analysis of space-time block codes over keyhole Nakagami-m fading channels , 2004, IEEE Transactions on Vehicular Technology.

[33]  George N. Karystinos,et al.  Noncoherent Short Packet Detection and Decoding for Scatter Radio Sensor Networking , 2017, IEEE Transactions on Communications.

[34]  Daniel Benevides da Costa,et al.  Average channel capacity for generalized fading scenarios , 2007, IEEE Communications Letters.

[35]  Ali Abdi,et al.  K distribution: an appropriate substitute for Rayleigh-lognormal distribution in fading-shadowing wireless channels , 1998 .

[36]  Simon L. Cotton,et al.  Human Body Shadowing in Cellular Device-to-Device Communications: Channel Modeling Using the Shadowed $\kappa-\mu$ Fading Model , 2015, IEEE Journal on Selected Areas in Communications.

[37]  Michel Daoud Yacoub,et al.  Product of Two Envelopes Taken From $\alpha $ - $\mu $ , $\kappa $ - $\mu $ , and $\eta $ - $\mu $ Distributions , 2018, IEEE Transactions on Communications.

[38]  Michel Daoud Yacoub,et al.  The Product of Two α-μ Variates and the Composite α-μ Multipath–Shadowing Model , 2015 .

[39]  R. Valenzuela,et al.  Capacities of multi-element transmit and receive antennas: Correlations and keyholes , 2000 .

[40]  Mohamed-Slim Alouini,et al.  Dual diversity over correlated log-normal fading channels , 2002, IEEE Trans. Commun..