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On the Calculation of the Incomplete MGF with Applications to Wireless Communications

F. J. Lopez-Martinez, J. M. Romero-Jerez, J. F. Paris

TL;DR

A general method for calculating the IMGF of any arbitrary fading distribution of the very general shadowed fading model, which includes the popular Rician shadowed, and other classical models as particular cases.

Abstract

The incomplete moment generating function (IMGF) has paramount relevance in communication theory, since it appears in a plethora of scenarios when analyzing the performance of communication systems. We here present a general method for calculating the IMGF of any arbitrary fading distribution. Then, we provide exact closed-form expressions for the IMGF of the very general κ-μ shadowed fading model, which includes the popular κ-μ, η-μ, Rician shadowed and other classical models as particular cases. We illustrate the practical applicability of this result by analyzing several scenarios of interest in wireless communications: (1) Physical layer security in the presence of an eavesdropper, (2) Outage probability analysis with interference and background noise, (3) Channel capacity with side information at the transmitter and the receiver, and (4) Average bit-error rate with adaptive modulation, when the fading on the desired link can be modeled by any of the aforementioned distributions.

On the Calculation of the Incomplete MGF with Applications to Wireless Communications

TL;DR

A general method for calculating the IMGF of any arbitrary fading distribution of the very general shadowed fading model, which includes the popular Rician shadowed, and other classical models as particular cases.

Abstract

The incomplete moment generating function (IMGF) has paramount relevance in communication theory, since it appears in a plethora of scenarios when analyzing the performance of communication systems. We here present a general method for calculating the IMGF of any arbitrary fading distribution. Then, we provide exact closed-form expressions for the IMGF of the very general κ-μ shadowed fading model, which includes the popular κ-μ, η-μ, Rician shadowed and other classical models as particular cases. We illustrate the practical applicability of this result by analyzing several scenarios of interest in wireless communications: (1) Physical layer security in the presence of an eavesdropper, (2) Outage probability analysis with interference and background noise, (3) Channel capacity with side information at the transmitter and the receiver, and (4) Average bit-error rate with adaptive modulation, when the fading on the desired link can be modeled by any of the aforementioned distributions.

Paper Structure

This paper contains 13 sections, 7 theorems, 36 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Results
  3. Application to Physical Layer Security
  4. Single-antenna scenario
  5. Extension to the multi-antenna scenario
  6. Other Applications
  7. Outage probability analysis with interference and background noise
  8. Channel capacity with side information at the transmitter and the receiver
  9. Average bit-error rate with adaptive modulation
  10. Numerical Results
  11. Conclusions
  12. Proof of Lemma 2
  13. CDF of the $\kappa$-$\mu$ shadowed distribution for integer fading parameters

Key Result

Lemma 1

Let $X$ be a non-negative random variable with MGF ${\mathcal{M}} _{\gamma}(s)$. Its lower IMGFs can be computed by the inverse Laplace transform of the scaled-shifted MGF, i.e. where ${{\mathcal{L} }}\left\{ {h\left( z \right);z,p} \right\} \triangleq \int_0^\infty {e^{ - pz} h\left( z \right)dz}$ represents the Laplace transform from the $z$-domain to the $p$-domain, and ${{\mathcal{L} }}^{ -

Figures (8)

  • Figure 1: Outage probability of secrecy capacity under $\kappa$-$\mu$ shadowed fading as a function of $\bar{\gamma}_b$, for different values of $m$ and $\mu$. Parameter values: $\kappa=1.5$, $\bar{\gamma}_e=15$ dB and $R_S=0.1$.
  • Figure 2: Outage probability of secrecy capacity under $\kappa$-$\mu$ shadowed fading as a function of $\bar{\gamma}_b$, for different values of $m$ and $\mu$. Parameter values: $\kappa=10$, $\bar{\gamma}_e=15$ dB and $R_S=0.1$.
  • Figure 3: Outage probability of secrecy capacity under Rician shadowed fading as a function of $\bar{\gamma}_b$, for different values of $K$ and $m$. Parameter values: $\bar{\gamma}_e=15$ dB and $R_S=0.1$.
  • Figure 4: Outage probability of secrecy capacity under $\kappa$-$\mu$ fading as a function of $\bar{\gamma}_b$, for different values of $\kappa$ and $\mu$. Parameter values: $\bar{\gamma}_e=15$ dB and $R_S=0.1$.
  • Figure 5: Outage probability of secrecy capacity under $\eta$-$\mu$ fading as a function of $\bar{\gamma}_b$, for different values of $\eta$ and $\mu$. Parameter values: $\bar{\gamma}_e=15$ dB and $R_S=0.1$.
  • ...and 3 more figures

Theorems & Definitions (16)

  • Definition 1: Lower IMGF
  • Definition 2: Upper IMGF
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • ...and 6 more