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An Enriched Α− Μ Model As Fading Candidate Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 5879413, 9 pages https://doi.org/10.1155/2020/5879413 Research Article An Enriched α − μ Model as Fading Candidate J. T. Ferreira ,1 A. Bekker,1 F. Marques ,2 and M. Laidlaw1 1Department of Statistics, University of Pretoria, Pretoria, South Africa 2Centro de Matema´tica e Aplicações (CMA) and Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universidade Nova de Lisboa, Lisbon, Portugal Correspondence should be addressed to J. T. Ferreira; [email protected] Received 29 September 2019; Revised 14 January 2020; Accepted 7 March 2020; Published 25 April 2020 Academic Editor: Rafał Stanisławski Copyright © 2020 J. T. Ferreira et al. .is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. .is paper introduces an enriched α − μ distribution which may act as fading model with its origins via the scale mixture construction. .e distribution’s characteristics are visited and its feasibility as a fading candidate in wireless communications systems is investigated. .e analysis of the system reliability and some performance measures of wireless communications systems over this enriched α − μ fading candidate are illustrated. Computable representations of the Laplace transform for this scale mixture construction are also provided. .e derived expressions are explored via numerical investigations. Tractable results are computed in terms of the Meijer G-function. .is unified scale mixture approach allows access to previously unconsidered underlying models that may yield improved fits to experimental data in practice. 1. Introduction the modelling of on-body communications networks as well as vehicle communication (see [3]). In the field of communication systems, fading channels are .e underlying assumption of normality has received characterized with statistical distributions to describe the some criticism. Increasing evidence in real applications has signal degradation from the transmitter to receiver of wireless illustrated that the normal distribution is not always an signals. Commonly used fading models such as the Rayleigh appropriate choice, as many experimental measurements distribution assume received signals can be found by the show a lepto- or platykurtic shape. Ollila et al. [4] noted that addition of vector sums representing scattering, diffraction, a more general assumption than that of the normal may not and reflection from different objects. .ese models have a be far from reality, with an underlying t assumption being drawback in that they can only model the fading accurately deployed within communication systems to account for when the scattering is homogeneous and require a large severe fading by Choi et al. [5]. .e t distribution is useful number of interfering signals to apply the central limit the- and well known in statistics, but there is a lack of results in orem (CLT) and so are built upon an underlying assumption communication systems pertaining to real life modelling of normality [1]. .e α − μ distribution was introduced by when compared to its normal counterpart. .is raises the Yacoub [2] such that the parameters have physical meaning. challenge of modelling data which emanate from inhomo- Fading is modelled in nonlinear environments where there geneous environments. Indeed, He et al. [6] and Qiu [7] exists a spatial correlation between the surfaces which causes explicitly asked what the consequences of analyses are when the diffusion and scattering and introduces the α − μ model underlying normality does not seem plausible. .e contri- with the normal assumption for the underlying process be- bution of the work in this paper aims to assist answering this haviour. .e α − μ model is statistically complex in its ex- question. A more pliable approach is to model these data by planation of fading while also being diverse and contains taking into account a stochastic element which is embedded certain well-known statistical distributions which are of in- in the variance component of the underlying model; this terest within the statistics and wireless communications leads to the consideration of scale mixtures of normal (SMN) paradigm. .is fading candidate has been found valuable in distribution alternatives. For the interested reader, the SMN 2 Mathematical Problems in Engineering approach and the value thereof can be consulted in, for platform and particular statistical characteristics of example, Andrews and Mallows [8]; Choy and Chan [9]; and this model are derived Lachos Davila´ et al. [10]. (2) Expressions for system reliability are derived and .e continuous demand for reliable communications investigated systems has caused to the resurgence of research in various (3) Computable representations of the Laplace trans- channel modelling approaches in system design and eval- form of special cases of this SMN (α − μ) are given uation to improve model performance. .erefore, in this paper, we propose a SMN alternative of the well-known α − μ (4) Speculative comparison between the different con- distribution. .is paper generalizes Yacoub’s α − μ model to sidered mixing densities in terms of outage proba- one that emanates from an underlying SMN assumption for bility is included the process behaviour. In this way, an enriched α − μ dis- .e departure point of this paper is the systematic tribution, a powerful umbrella model, is derived which al- construction of an enriched α − μ distribution with genesis as lows the practitioner to amend the underlying distribution a fading model in communication systems; this is given in in the case that observed experimental data exhibit greater Section 2. Subsequently, in Section 3, the system reliability is heterogeneity than the usual α − μ distribution would ac- analyzed via the signal to noise ratio (SNR), amount of commodate. To the authors’ knowledge, this representation fading (AoF), and the Laplace transform which is of value in of the α − μ model within communications systems has not practice for evaluation the average bit-error-rate (ABER). yet been considered. Some performance measures of a fading .e direct applications which may arise from the proposed channel subject to this enriched α − μ model are compara- model are comparatively investigated via the outage prob- tively investigated against that of the well-studied normal ability in Section 4 with final thoughts given in Section 5. (for recent contributions in this domain, see [11–13]). .e SMN class uses the Laplace transform technique to 2. System Model and Statistical Characteristics illustrate that a variable X with location μ and scale σ has a SMN representation if it can be expressed as In this section, some theoretical properties such as the ∞ − 1 2 density of the SMN (α − μ) distribution will be proposed and fX(x j μ; σ) � Z N� x j μ; t σ �W(t; θ)dt; (1) 0 derived. where N(x|·, ·) is the normal density function and W(t; θ) is a density function of arbitrary variable T defined on R+ in 2.1. System Model. .e following theorem introduces the which θ is a scalar or vector parameter indexing the dis- SMN(α − μ) model and derives the corresponding density tribution of T (see [9]). In this case, W(t; θ) is called the function. mixing density of this SMN representation, and the distri- 2 bution of X is denoted by SMN(μ, σ , W(·)). In general, Theorem 1. Let Xi and Yi be mutually independent SMN W(t; θ) may be posed as an arbitrary random variable. .e processes with E(Xi) � E(Yi) � 0 and var(Xi) � var(Yi) � α α advantage of the representation in (1) is gaining access to (br /2μ); hence, Xi;Yi ∼ SMN(0; (br /2μ);W(t; θ)). 7e class of previously unexplored distributions to accom- α− power envelope (amplitude of the process stemming from Xi modate experimental data better in practice. and Yi) emanating from the SMN construction is defined by In this paper, two mixing densities are of particular (see [2]) interest. .e usual α − μ model is obtained when the random μ α 2 2 variable T is degenerate at 0; that is, W(t; θ) is given by R � X�Xi + Yi �; (4) i� W(t; θ) � δ(t − 1); (2) 1 where α and μ are positive integers. 7e density of the envelope where δ(·) denotes the Dirac delta function. In the case R is given by where an enriched α − μ model emanates from an underlying rαμ− 1 μ ∞ r α t distribution, the mixing density is given by (see [14]) α μ μ fR(r) � αμ Z t exp�− μt� � �W(t; θ)dt; r > 0; Γ(μ)br 0 br ](]t/2)(]/2)− 1 W(t; θ) � ; (3) (5) 2Γ(]/2)exp(]t/2) p������ where br � α E(Rα). 7e distribution with density (5) is called � ] (·) where θ > 0 degrees of freedom and Γ denotes the a SMN (α − μ) distribution. gamma function ([15], equation 8.310.1). Note that W(t; θ) ≡ Gam((]/2); (]/2)), where Gam((]/2); (]/2)) de- Proof. Note that (v(t))2 � (brα/2μ) (χ2(]) denotes a chi- notes the density of a gamma random variable with both square distributed random variable with ] > 0 degrees of parameters equal to ]/2. In the literature, these two mixing freedom); then (X /v(t))|t ∼ N(0; 1) and (Y /v(t))|t ∼ N densities, respectively, lead to what is called the “normal” i i (0; 1); therefore, Pμ X2 + Y2/(v(t))2jt � (Rα/(v(t))2) jt ∼ and the “t” case. i�1 i i χ2(2μ) since Pμ X2/(v(t))2jt ∼ χ2(μ) and Pμ Y2/ .is paper’s contribution can be summarised as follows: i�1 i i�1 i (v(t))2jt ∼ χ2(μ). Let K(t) � Rα/(v(t))2jt; thus, K(t) ∼ χ2 (1) A SMN (α − μ) model is systematically developed (2μ) with the conditional density of the α− power envelope with genesis from the communication system as Mathematical Problems in Engineering 3 α μ− 1 α 1.0 α 1 r r f α r � � ! exp − !: R jt μ ( )(v(t))2 (v(t))2 (v(t))2 2 Γ μ 2 0.8 (6) 0.6 Subsequently, the conditional density of the envelope, R, is CDF 0.4 α α αμ− 1 r fR t(r) � !r exp − !; (7) ∣ 2μΓ(μ)(v(t))2μ 2(v(t))2 0.2 with the unconditional distribution given by (5).
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