The {\alpha}-{\kappa}-{\mu} Shadowed Fading Distribution: Statistical Characterization and Applications

TL;DR

This paper introduces the { extalpha}-{ extkappa}-{ extmu} shadowed ({ extalpha}-KMS) fading distribution, a comprehensive model unifying many classical fading distributions, and analyzes key performance metrics for wireless channels.

Contribution

It proposes the { extalpha}-KMS distribution as a unifying fading model and derives analytical expressions for performance metrics like outage probability and capacity.

Findings

{ extalpha}-KMS includes many classical fading models as special cases.

Performance metrics can be derived using existing { extalpha}-{ extmu} results.

Finite mixture representation simplifies analysis for integer parameters.

Abstract

We introduce the {\alpha}-{\kappa}-{\mu} shadowed ({\alpha}-KMS) fading distribution as a natural generalization of the versatile {\alpha}-{\kappa}-{\mu} and {\alpha}-{\eta}-{\mu} distributions. The {\alpha}-KMS fading distribution unifies a wide set of fading distributions, as it includes the {\alpha}-{\kappa}-{\mu}, {\alpha}- {\eta}-{\mu}, {\alpha}-{\mu}, Weibull, {\kappa}-{\mu} shadowed, Rician shadowed, {\kappa}-{\mu} and {\eta}- {\mu} distributions as special cases, together with classical models like Rice, Nakagami-m, Hoyt, Rayleigh and one-sided Gaussian. Notably, the {\alpha}-KMS distribution reduces to a finite mixture of {\alpha}-{\mu} distributions when the fading parameters {\mu} and m take positive integer values, so that performance analysis over {\alpha}-KMS fading channels can be tackled by leveraging previous (existing) results in the literature for the simpler {\alpha}-{\mu} case. As application examples, important performance metrics like the outage probability and average channel capacity are analyzed.

Equations

$$R_{\mathcal{S}}^2=\sum _{i=1}^{\mu }{\left(X_i+\xi p_i\right)}^2+\sum _{i=1}^{\mu }{\left(Y_i+\xi q_i\right)}^2.$$

$$R^{\alpha }=\sum _{i=1}^{\mu }{\left(X_i+\xi p_i\right)}^2+\sum _{i=1}^{\mu }{\left(Y_i+\xi q_i\right)}^2$$

$$f_{\alpha \mu }(\overline{\gamma };\alpha ,\mu ;\gamma )=\frac{\alpha }{2\Gamma (\mu )}{D}^{\mu \alpha /2}{\gamma }^{\alpha \mu /2-1}{e}^{-{\gamma }^{\alpha }{D}^{\alpha /2}},$$

$$f_{\gamma }(\gamma )=$$

$$\times {}_1{F}_1\left(m,\mu ;\frac{\mu \kappa }{c(\mu \kappa +m)}{\left(\frac{\gamma }{\overline{\gamma }}\right)}^{\frac{\alpha }{2}}\right),$$

$$c={\left(\frac{(\mu \kappa +m{)}^m\Gamma (\mu )}{m^m\Gamma \left(\mu +\frac{2}{\alpha }\right){}_2{F}_1\left(m,\mu +\frac{2}{\alpha };\mu ;\frac{\mu \kappa }{\mu \kappa +m}\right)}\right)}^{\alpha /2},$$

$$f_R(r)=$$

$$\times {}_1{F}_1\left(m,\mu ;\frac{\mu \kappa r^{\alpha }}{{\Omega }^{\alpha /2}c(\mu \kappa +m)}\right),$$

Full text

We introduce the $\alpha$-$\kappa$-$\mu$ shadowed ($\alpha$-KMS) fading distribution as a natural generalization of the versatile $\alpha$-$\kappa$-$\mu$ and $\alpha$-$\eta$-$\mu$ distributions. The $\alpha$-KMS fading distribution unifies a wide set of fading distributions, as it includes the $\alpha$-$\kappa$-$\mu$, $\alpha$-$\eta$-$\mu$, $\alpha$-$\mu$, Weibull, $\kappa$-$\mu$ shadowed, Rician shadowed, $\kappa$-$\mu$ and $\eta$-$\mu$ distributions as special cases, together with classical models like Rice, Nakagami-$m$, Hoyt, Rayleigh and one-sided Gaussian. Notably, the $\alpha$-KMS distribution reduces to a finite mixture of $\alpha$-$\mu$ distributions when the fading parameters $\mu$ and $m$ take positive integer values, so that performance analysis over $\alpha$-KMS fading channels can be tackled by leveraging previous (existing) results in the literature for the simpler $\alpha$-$\mu$ case. As application examples, important performance metrics like the outage probability and average channel capacity are analyzed.

I Introduction

The $\alpha$-$\mu$ distribution originally proposed by Yacoub [Yacoub2002, Yacoub2007] has become very popular in the context of wireless channel modelling, due to not only its reasonably simple analytical form but also because of its improved fit to field measurements. Its original formulation was motivated by the interest on exploring and eventually characterizing the non-linearity of the propagation medium. Interestingly, the $\alpha$-$\mu$ distribution arises as a power transformation over the received signal envelope in a Nakagami-$m$ set-up, in a similar way as the Weibull distribution is obtained from an underlying Rayleigh distribution.

The $\alpha$-$\eta$-$\mu$ and $\alpha$-$\kappa$-$\mu$ distributions [Fraidenraich2006] arose as generalizations of the equally popular and versatile $\kappa$-$\mu$ and $\eta$-$\mu$ distributions [Yacoub2000, Yacoub2001, Yacoub2007b]. Despite their versatility on extending the range of propagation conditions which they are able to model, their applicability for performance analysis purposes is hindered by the complicated form of their chief probability functions, and the scarce results available in the literature often have complicated form [Aldalgamouni2017, Moualeu2018, Magableh2018], compared to their $\alpha$-$\mu$ fading counterparts.

In [Paris2014, Cotton2015], the $\kappa$-$\mu$ shadowed fading distribution was introduced with the aim of generalizing the $\kappa$-$\mu$ distribution by allowing the dominant specular components to randomly fluctuate. This effect was introduced through an additional parameter $m$ similar to that in the Nakagami-$m$ distribution, and provided additional flexibility to the model in order to better accommodate to field measurements. It was later demonstrated in [Moreno2016] that, somehow counterintuitively, the $\kappa$-$\mu$ shadowed fading model included as special cases both the $\kappa$-$\mu$ and $\eta$-$\mu$ distributions. This allows to unify both fading models under a common, more general model with a similar mathematical complexity. Besides, in the special cases of considering integer values for the fading parameters $\mu$ and $m$, the probability density function (PDF) and the cumulative distribution function (CDF) of the $\kappa$-$\mu$ shadowed fading model are expressed in terms of a finite sum of powers and exponentials [Lopez2017]. Hence, in these circumstances, its tractability is as simple as if the simple Nakagami-$m$ model was assumed.

As we can see from the previous works, the consideration of channel propagation effects like non-linearity of the propagation medium and the fluctuation of dominant specular components have been tackled separately to the best of our knowledge. In order to fill this gap, and therefore to jointly consider both effects, we formulate the $\alpha$-$\kappa$-$\mu$ shadowed fading model ($\alpha$-KMS) as a natural generalization of the $\kappa$-$\mu$ shadowed fading model, which inherits its main key features: (i) it includes as special cases both the $\alpha$-$\kappa$-$\mu$ and the $\alpha$-$\eta$-$\mu$ fading models; (ii) it also includes as special cases virtually all classical models (Nakagami, Hoyt, Rice, $\kappa$-$\mu$, $\eta$-$\mu$, Weibull, $\alpha$-$\mu$, Rician Shadowed); (iii) for integer values of $\mu$ (corresponding to the physical model based on clusters in [Yacoub2007b]) and $m$, the $\alpha$-KMS model can be expressed as a finite mixture of $\alpha$-$\mu$ distributions, which implies that performance analysis over $\alpha$-KMS fading channels is as tractable as the $\alpha$-$\mu$ case. Hence, the $\alpha$-KMS fading model not only generalizes but also simplifies the analysis of the baseline fading models from which it originates. We exemplify the tractability of the $\alpha$-KMS fading channel model by analyzing two key performance metrics, i.e., the outage probability and the average channel capacity. In both cases, we provide analytical expressions for these performance metrics, together with simple but tight approximations in the high signal-to-noise ratio (SNR) regime.

The remainder of this paper is structured as follows: the physical model for the $\alpha$-KMS fading distribution is described in Section II, whereas the statistical characterization of this distribution is carried out in Section III. The application to system performance analysis is introduced in Section LABEL:S4, covering outage probability and average channel capacity. Numerical results are presented in Section LABEL:S5, and conclusions are discussed in Section LABEL:S6.

II Physical Model – $\alpha$-KMS Fading Distribution

According to the physical model of the $\kappa$-$\mu$ shadowed fading distribution originally given in [Paris2014], the received signal envelope $R_{\mathcal{S}}$ can be expressed as a superposition of $\mu$ multipath clusters in the following form:

[TABLE]

Within each cluster, the scattered components are modeled as zero-mean independent Gaussian random variables (RVs), i.e. $X_{i},Y_{i}\sim\mathcal{N}(0,\sigma^{2})$ $\forall$ $i$, whereas the dominant components are defined by the real parameters $p_{i}$ and $q_{i}$. The RV $\xi$ is defined so that its square value is gamma distributed with unit mean and shape parameter $m$, i.e. $\xi^{2}\sim\Gamma(m,1/m)$, which ultimately encapsulates any sort of amplitude fluctuation suffered by the line-of-sight (LoS) component.

The physical models of the $\alpha$-$\mu$, $\alpha$-$\kappa$-$\mu$ and $\alpha$-$\eta$-$\mu$ fading distributions are formulated by applying a power transformation over the received signal envelope, in a similar way as the Weibull distribution is derived from the Rayleigh model. Using this approach [Yacoub2007], the received signal envelope $R$ in an $\alpha$-KMS fading environment is given by

[TABLE]

where $\alpha>0$ is a real number characterizing the non-linearity of the propagation medium. Note that, if $\alpha=2$ then (2) reduces to (1). The $\alpha$-KMS model is completely defined by the set of parameters $\alpha$, $\mu$, $m$ and $\kappa=\sum_{i}^{\mu}(p_{i}^{2}+q_{i}^{2})/(2\sigma^{2})$. In the following derivations, we will pursue the statistical characterization of the RVs arising from this physical model. We will focus on the characterization of the instantaneous SNR under $\alpha$-KMS fading, defined as $\gamma=\overline{\gamma}R^{2}/\,\mathbb{E}\left[R^{2}\right]$, where $\overline{\gamma}$ is the average SNR. From the formulations for $\gamma$, those of the received signal envelope $R$ can be easily obtained by using a simple change of variables.

III Statistical Characterization

III-A Definitions

Definition 1** ($\alpha$-$\mu$ distribution)**

Let $\gamma$ be a real RV characterizing the instantaneous SNR under $\alpha$-$\mu$ fading [Yacoub2007b], with mean $\overline{\gamma}=\mathbb{E}\left[\gamma\right]$ and real shape parameters $\alpha$ and $\mu$. Then, the PDF of $\gamma$ is given by

[TABLE]

with $D=\frac{\Gamma\left(\mu+\frac{2}{\alpha}\right)}{\overline{\gamma}\Gamma(\mu)}$, and $\Gamma(\cdot)$ being the gamma function [Gradshteyn07, 8.310].

III-B Chief Probability Functions

Lemma 1** (The $\alpha$-KMS distribution: PDF)**

Let $\gamma$ be a real RV characterizing the instantaneous SNR under $\alpha$-KMS fading, with $\overline{\gamma}=\mathbb{E}[\gamma]$ and non-negative real shape parameters $\alpha$, $\kappa$, $\mu$ and $m$, i.e. $\gamma\sim\mathcal{A}_{kms}\left(\overline{\gamma};\alpha,\kappa,\mu,m\right)$. Then, its PDF is given by

[TABLE]

where ${}_{1}{F}_{1}(\cdot)$ is the confluent hypergeometric function [Abra72, Eq. (13.1.2)] and

[TABLE]

with ${}_{2}F_{1}(\cdot)$ denoting the Gauss hypergeometric function [Abra72, Eq. (15.1.1)].

Proof:

Using standard techniques of transformation of random variables, and starting from the original expressions given in [Paris2014, Cotton2015] for the $\kappa$-$\mu$* shadowed fading distribution, the PDF of the signal amplitude $R$ can be deduced as*

[TABLE]

where $\Omega=\mathbb{E}\left[R^{2}\right]$. Finally, since $\gamma=\overline{\gamma}\frac{R^{2}}{\Omega}$ we can obtain the desired PDF as $f_{\gamma}(\gamma)=\frac{1}{2}\sqrt{\frac{\Omega}{\overline{\gamma}\gamma}}f_{R}\left(\sqrt{\frac{\Omega\gamma}{\overline{\gamma}}}\right)$. ∎