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Alternative Formulations for the Fluctuating Two-Ray Fading Model

Maryam Olyaee, Juan M. Romero-Jerez, F. Javier Lopez-Martinez, Andrea J. Goldsmith

TL;DR

Alternative expressions for the probability density function and cumulative distribution function of the FTR model are obtained, as well as new expressions for some Laplace-domain statistics of interest to exemplify the practical relevance of this new formulation for performance analysis.

Abstract

We present two alternative formulations for the distribution of the fluctuating two-ray (FTR) fading model, which simplify its statistical characterization and subsequent use for performance evaluation. New expressions for the probability density function (PDF) and cumulative distribution function of the FTR model are obtained based on the observation that the FTR fading distribution is described, for arbitrary $m$, as an underlying Rician Shadowed (RS) distribution with continuously varying parameter $K$, while for the special case of $m$ being an integer, the FTR fading model is described in terms of a finite number of underlying squared Nakagami-$m$ distributions. It is shown that the chief statistics and any performance metric that are computed by averaging over the PDF of the FTR fading model can be expressed in terms of a finite-range integral over the corresponding statistic or performance metric for the RS (for arbitrary $m$) or the Nakagami-$m$ (for integer $m$) fading models, which have a simpler analytical characterization than the FTR model and for which many results are available in closed-form. New expressions for some Laplace-domain statistics of interest are also obtained; these are used to exemplify the practical relevance of this new formulation for performance analysis.

Alternative Formulations for the Fluctuating Two-Ray Fading Model

TL;DR

Alternative expressions for the probability density function and cumulative distribution function of the FTR model are obtained, as well as new expressions for some Laplace-domain statistics of interest to exemplify the practical relevance of this new formulation for performance analysis.

Abstract

We present two alternative formulations for the distribution of the fluctuating two-ray (FTR) fading model, which simplify its statistical characterization and subsequent use for performance evaluation. New expressions for the probability density function (PDF) and cumulative distribution function of the FTR model are obtained based on the observation that the FTR fading distribution is described, for arbitrary , as an underlying Rician Shadowed (RS) distribution with continuously varying parameter , while for the special case of being an integer, the FTR fading model is described in terms of a finite number of underlying squared Nakagami- distributions. It is shown that the chief statistics and any performance metric that are computed by averaging over the PDF of the FTR fading model can be expressed in terms of a finite-range integral over the corresponding statistic or performance metric for the RS (for arbitrary ) or the Nakagami- (for integer ) fading models, which have a simpler analytical characterization than the FTR model and for which many results are available in closed-form. New expressions for some Laplace-domain statistics of interest are also obtained; these are used to exemplify the practical relevance of this new formulation for performance analysis.

Paper Structure

This paper contains 14 sections, 10 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminary Definitions
  3. FTR Formulation as a Continuous Mixture of RS Variates
  4. PDF and CDF of the FTR fading
  5. MGF of the FTR fading
  6. Generalized MGF and moments
  7. Incomplete MGF of the FTR model
  8. FTR Formulation as a Continuous Mixture of Nakagami-$m$ Variates
  9. Incomplete generalized MGF of FTR
  10. Application Example: Outage Probability of Multi-antenna receiver with CCI
  11. CCI with background noise case with integer $m$
  12. Interference limited case with arbitrary $m$
  13. Numerical results
  14. Conclusion

Key Result

Lemma 1

Let $\gamma$ be a random variable such that $\gamma \sim \mathcal{FTR}(\overline{\gamma};m,K,\Delta)$, then $\gamma$ is a continuous mixture of squared RS variates, whose PDF is given as

Figures (5)

  • Figure 1: PDF of the SNR ($\gamma$) under FTR fading obtained using the FTR-RS connection ($m=1.5,3$) and the FTR-Nakagami-$m$ connection ($m=3$). Monte-Carlo simulation are also plotted.
  • Figure 2: Analytical and simulation results for the outage probability in the presence of CCI and background noise versus normalized SINR (dB) for different $m$, $K$ and $\Delta$ with $L=2$, $P_I=0.01$ and $R_{th} =1$.
  • Figure 3: Analytical and simulation results for the outage probability in the presence of CCI and background noise versus normalized SINR (dB) for different values of the SINR threshold ($R_{th}$) with $m=2$, $\Delta = 0.6$ and $K=10$.
  • Figure 4: Analytical and simulation results for the outage probability versus SIR threshold (dB) in a interference-limited system for different values of $m$$(0.5,1,1.5,2.5)$ with $K=10$, $\Delta = 0.6$, $N = 2$, $P_I =1$ and $L= 1$.
  • Figure 5: Analytical and simulation results for the outage probability versus SIR threshold (dB) for different numbers of antennas ($N$) and interferers ($L$) with $K=10$, $\Delta = 0.6$, $m = 2.5$, and $P_I=1$.

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 12 more