Composite Fading Models based on Inverse Gamma Shadowing: Theory and Validation

TL;DR

It is proved that the probability density function and cumulative density function of any IG-based composite fading model are directly expressed in terms of a Laplace-domain statistic of the underlying fast fading model, and in some relevant cases, as a mixture of well-known state-of-the-art distributions.

Abstract

We introduce a general approach to characterize composite fading models based on inverse gamma (IG) shadowing. We first determine to what extent the IG distribution is an adequate choice for modeling shadow fading, by means of a comprehensive test with field measurements and other distributions conventionally used for this purpose. Then, we prove that the probability density function and cumulative distribution function of any IG-based composite fading model are directly expressed in terms of a Laplace-domain statistic of the underlying fast fading model and, in some relevant cases, as a mixture of wellknown state-of-the-art distributions. Also, exact and asymptotic expressions for the outage probability are provided, which are valid for any choice of baseline fading distribution. Finally, we exemplify our approach by presenting several application examples for IG-based composite fading models, for which their statistical characterization is directly obtained in a simple form.

Figures (9)

• Figure 1: Shadowing CDF for data in Babiroli2017, corresponding to urban and suburban scenarios at $169$ MHz. Parameters for urban scenario: lognormal ($\mu=0.05$, $\sigma=1.08$), gamma ($k = 1.15$, $\Omega = 1.57$), inverse gamma ($m = 1.18$, $\Omega_i = 4.60$) and inverse Gaussian ($\mu_i = 1.86$, $\lambda = 1.04$). Parameters for suburban scenario: lognormal ($\mu=0$, $\sigma=0.33$), gamma ($k = 10.15$, $\Omega = 1.02$), inverse gamma ($m = 9.82$, $\Omega_i = 1.05$) and inverse Gaussian ($\mu_i = 1.04$, $\lambda = 9.58$).
• Figure 2: Shadowing CDF data in Abdi1998, corresponding to suburban scenarios at $910.25$ MHz. Parameters for case $(a)$: lognormal ($\mu=7.04$, $\sigma=0.57$), gamma ($k = 3.45$, $\Omega = 1293$), inverse gamma ($m = 3.32$, $\Omega_i = 1433$) and inverse Gaussian ($\mu_i = 1345$, $\lambda = 3607$). Parameters for case $(b)$: lognormal ($\mu=6.08$, $\sigma=0.40$), gamma ($k = 6.80$, $\Omega = 465.4$), inverse gamma ($m = 6.54$, $\Omega_i=485.7$) and inverse Gaussian ($\mu_i = 473.4$, $\lambda = 2848$).
• Figure 3: Shadowing CDF for data in Ai2017, corresponding to an indoor scenario at $26$ GHz. Parameters: lognormal ($\mu=0$, $\sigma=0.011$), gamma ($k = 8319$, $\Omega = 0.998$), inverse gamma ($m = 8316$, $\Omega_i=1$) and inverse Gaussian ($\mu_i = 1$, $\lambda = 8310$).
• Figure 4: Shadowing CDF for data in Zhu2015, corresponding to an indoor scenario at $45$ GHz with two distinct types of antennas: horn antenna (case $(a)$) and open ended guide antenna (case $(b)$). Parameters for case $(a)$: lognormal ($\mu=0$, $\sigma=0.20$), gamma ($k = 26.49$, $\Omega = 1.01$), inverse gamma ($m = 26.17$, $\Omega_i=1.02$) and inverse Gaussian ($\mu_i = 1$, $\lambda = 25.74$). Parameters for case $(b)$: lognormal ($\mu=0.4$, $\sigma=0.49$), gamma ($k = 4.52$, $\Omega = 1.64$), inverse gamma ($m = 4.69$, $\Omega_i=1.72$) and inverse Gaussian ($\mu_i = 1.67$, $\lambda = 6.51$).
• Figure 5: Impact of assuming $m \in \mathbb{N}^+$ in the inverse gamma distribution for data in Babiroli2017, corresponding to urban and suburban scenarios at $169$ MHz. Parameters for urban scenario: inverse gamma ($m = 1.18$, $\Omega_i = 4.60$) and inverse gamma with integer $m$ ($m = 1$, $\Omega_i \to \infty$). Parameters for suburban scenario: inverse gamma ($m = 9.82$, $\Omega_i = 1.05$) inverse gamma with integer $m$ ($m = 10$, $\Omega_i = 1.05$).

Theorems & Definitions (11)

• Definition 1: Lognormal distribution
• Definition 2: Gamma distribution
• Definition 3: Inverse Gaussian distribution
• Definition 4: Inverse gamma distribution
• Definition 5: GMGF
• Lemma 1
• Lemma 2
• Lemma 3
• Corollary 1
• Lemma 4