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On the Bivariate Nakagami-$m$ Cumulative Distribution Function: Closed-form Expression and Applications

F. J. Lopez-Martinez, D. Morales-Jimenez, E. Martos-Naya, J. F. Paris

TL;DR

This paper derives exact closed-form expressions for the bivariate Nakagami-m cumulative distribution function (CDF) with positive integer fading severity index m in terms of a class of hypergeometric functions and shows that it can be expressed as a finite sum of elementary functions and bivariate confluenthypergeometric Φ3 functions.

Abstract

In this paper, we derive exact closed-form expressions for the bivariate Nakagami-$m$ cumulative distribution function (CDF) with positive integer fading severity index $m$ in terms of a class of hypergeometric functions. Particularly, we show that the bivariate Nakagami-$m$ CDF can be expressed as a finite sum of elementary functions and bivariate confluent hypergeometric $Φ_3$ functions. Direct applications which arise from the proposed closed-form expression are the outage probability (OP) analysis of a dual-branch selection combiner in correlated Nakagami-$m$ fading, or the calculation of the level crossing rate (LCR) and average fade duration (AFD) of a sampled Nakagami-$m$ fading envelope.

On the Bivariate Nakagami-$m$ Cumulative Distribution Function: Closed-form Expression and Applications

TL;DR

This paper derives exact closed-form expressions for the bivariate Nakagami-m cumulative distribution function (CDF) with positive integer fading severity index m in terms of a class of hypergeometric functions and shows that it can be expressed as a finite sum of elementary functions and bivariate confluenthypergeometric Φ3 functions.

Abstract

In this paper, we derive exact closed-form expressions for the bivariate Nakagami- cumulative distribution function (CDF) with positive integer fading severity index in terms of a class of hypergeometric functions. Particularly, we show that the bivariate Nakagami- CDF can be expressed as a finite sum of elementary functions and bivariate confluent hypergeometric functions. Direct applications which arise from the proposed closed-form expression are the outage probability (OP) analysis of a dual-branch selection combiner in correlated Nakagami- fading, or the calculation of the level crossing rate (LCR) and average fade duration (AFD) of a sampled Nakagami- fading envelope.

Paper Structure

This paper contains 14 sections, 7 theorems, 69 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statistical Analysis
  3. Notation and preliminaries
  4. Results
  5. Applications
  6. Outage probability of dual-branch SC
  7. Higher order statistics of sampled Nakagami-m fading channels
  8. Numerical Results
  9. Conclusion
  10. Proof of proposition \ref{['prop3']}
  11. Proof of proposition \ref{['prop1']}
  12. Proof of proposition \ref{['prop2']}
  13. Proof of lemma \ref{['lemma1']}
  14. Mathematica$\texttrademark$ program for the computation of $\Phi_3$ function

Key Result

Proposition 1

The integral ${\mathcal{H} }_m\left(u,\gamma,\delta\right)$ can be expressed as where $\alpha=\gamma^2+2m$ and

Figures (6)

  • Figure 1: Outage probability versus threshold level $\gamma_{th}$ (normalized to $\bar{\gamma}_1$), for balanced reception $\bar{\gamma}_1=\bar{\gamma}_2$ and different values of $m$ and $\rho$. Markers correspond to Monte Carlo simulations.
  • Figure 2: Outage probability versus threshold level $\gamma_{th}$ (normalized to $\bar{\gamma}_1$), for unbalanced reception $\bar{\gamma}_1=5\bar{\gamma}_2$ and different values of $m$ and $\rho$. Markers correspond to Monte Carlo simulations.
  • Figure 3: Normalized level crossing rate $N_R(u)/f_d$ of a sampled Nakagami-$m$ fading channel, for different values of $f_d · T$ and $m$. Markers correspond to Monte Carlo simulations.
  • Figure 4: Normalized average fade duration $A_R(u)·f_d$ of a sampled Nakagami-$m$ fading channel, for different values of $f_d · T$ and $m$.
  • Figure 5: Contour integration for integral $\mathcal{I}_{k}$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 1
  • Corollary 1
  • Corollary 2
  • Corollary 3