Ratio of Two Envelopes Taken from Variates and Some Practical Applications

Carlos Rafael Nogueira Da Silva*, Nidhi Bhargav, Elvio J. Leonardo, Simon L. Cotton, Michel Daoud Yacoub

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)
67 Downloads (Pure)


In this paper, new, exact expressions for the probability density functions and the cumulative distribution functions of the ratio of random envelopes involving the fading distributions are derived. The expressions are obtained in terms of easily computable infinite series and also in terms of the multivariable Fox H-function. Some special cases of these ratios, namely, Hoyt/Hoyt, Nakagami-m/Nakagami-m, with an integer for the variate, with an integer for only one of the and their reciprocals are found in novel exact closed-form expressions. In addition, simple closed-form expressions for the asymptotes of the probability density functions and cumulative distribution functions of all ratios, both for the lower and upper tails of the distributions are derived. In the same way, asymptotes for the bit error rate on a binary signaling channel are obtained in closed-form expressions. To demonstrate the practical utility of these new formulations, an application example is provided. In particular, the secrecy capacity of a Gaussian wire-Tap channel used for device-To-device and vehicle-To-vehicle communications is characterized using data obtained from field measurements conducted at 5.8 GHz.

Original languageEnglish
Article number8703806
Pages (from-to)54449-54463
Number of pages15
JournalIEEE Access
Publication statusPublished - 01 May 2019


  • D2D
  • multihop systems
  • spectrum sensing
  • spectrum sharing
  • V2V
  • α-μ distribution
  • η-μ distribution
  • κ-μ distribution

ASJC Scopus subject areas

  • Computer Science(all)
  • Materials Science(all)
  • Engineering(all)


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