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 | μ, σ) � � N� x | 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 (�r /2μ); hence, Xi,Yi ∼ SMN(0, (�r /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 � ��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 � t exp�− μt� � �W(t; θ)dt, r > 0, Γ(μ)�r 0 �r ](]t/2)(]/2)− 1 W(t; θ) � , (3) (5) 2Γ(]/2)exp(]t/2) �� where �r � 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 � (�rα/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, �μ X2 + Y2/(v(t))2|t � (Rα/(v(t))2) |t ∼ and the “t” case. i�1 i i χ2(2μ) since �μ X2/(v(t))2|t ∼ χ2(μ) and �μ Y2/ .is paper’s contribution can be summarised as follows: i�1 i i�1 i (v(t))2|t ∼ χ2(μ). Let K(t) � Rα/(v(t))2|t; 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 |t 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). Consider W(t; θ) as given by (2), and then (5) simpliﬁes 1 234 r to the well-known α − μ model [16]: W(t;θ) = δ(t – 1) αrαμ− 1μμ r α f (r) � exp�− μ� � �, r > 0, (8) W(t;θ) = Gam (30/2, 30/2) normal ( )�rαμ �r Γ μ W(t;θ) = Gam (15/2, 15/2) and if W(t; θ) as given by (3), then W(t;θ) = Gam (3/2, 3/2) αμ− 1 μ ∞ α (]/2)− 1 Figure − αr μ μ r ](]t/2) 1: Illustration of CDFs of the SMN (α μ) for diﬀerent ft(r) � αμ � t exp�− μt� � � dt mixing densities from (5). Γ(μ)�r 0 �r 2Γ(]/2)exp(]t/2)

αrαμ− 1μμ]]/2 Γ(μ + ]/2) previously unconsidered models, providing flexibility for � αμ ( +( )), Γ(μ)�r Γ(]/2)2]/2 μ(r/�r)α +(]/2)� μ ]/2 modelling that may yield improved fits to experimental data in practice when these data exhibit potential heavier ( ) 9 tail behaviour [2, 13] with μ + (]/2) > 0 and μ(r/�r)α + (]/2) > 0. □ 3. Reliability Analysis over an 2.2. Statistical Characteristics. In this section, some statis- Enriched Candidate tical characteristics necessary for the remainder of the paper of the SMN (α − μ) distribution are derived. .e results are In this section, the objective is to derive expressions for presented without proof. different performance metrics under SMN (α − μ) model as proposed in Section 2. Tractable expressions for the SNR, th AoF, and the Laplace transform are derived and analyzed for Theorem 2. From (5), the j moment of the SMN (α − μ) the SMN (α − μ). type model is μ ∞ ∞ α j αμ αμ+j− 1 μ r E�R � � αμ � r � t exp�− μt� � �W(t;θ)dtdr. Γ(μ)�r 0 0 �r 3.1. SNR. To measure and investigate the performance of the (10) SMN (α − μ) type model, the instantaneous SNR is needed. .e instantaneous SNR expressed in terms of the channel For W(t; θ) given by (2), it follows from (10) and envelope is c � c�(R/�r)2 � c�R2(1/E(R2)), where c� � E(�r2) Gradshteyn and Ryzhik ([15]; equation 3.326-2) that (Eb/No), and Eb is the energy per bit and N0 is the noise �rjΓ(μ +(j/α)) spectral density [17]. .e SNR is the building block to E �Rj � � . (11) normal μj/αΓ(μ) calculate the outage probability, AoF and ABER [18]. In this section, the density, CDF, AoF, and Laplace transform of the This will hold for α > 0, �rα > 0, and αμ + j > 1. SNR are derived. For W(t; θ) given by (3), it follows from (10) and From (5), the density of the SNR for the SMN (α − μ) Gradshteyn and Ryzhik ([15]; equation 3.241-4) that model is j/α �� ( +(j )) ((] ) − (j )) ]�rα j μ αμ ∞ α j Γ μ /α Γ /2 /α 2 αμ c − 1 μ c Et�R � � � � , ] > . f(c) � � � c � t exp�− μt� � �W(t; θ)dt. Γ(μ)Γ((]/2)) 2μ α 2Γ(μ) c� 0 c� (12) (13) Figure 1 illustrates the value which the SMN structure For the W(t; θ) (2) and (3), (13) simplifies to provides the practitioner with. .is graph illustrates the r μ �� α CDF, F(r) � � f(t)dt, using (5) for the two considered αμ (αμ/2)− 1 − (αμ/2) c 0 f (c) � c c� exp�− μ� � �, c > 0, mixing densities for certain arbitrary values of the pa- normal 2Γ(μ) c� rameters and highlights this potential heavier tail char- (14) acteristic. .e proposed SMN platform in this paper allows theoretical and resultant practical access to and 4 Mathematical Problems in Engineering

αμμ]]/2Γ(μ + (]/2)) 1 f (c) � c(αμ/2)− 1c�− (αμ/2) , c , t (] )+ �� ( +(] )) > 0 (15) Γ(μ)Γ(]/2)2 /2 1 μ( c/c� )α + (]/2)� μ /2 respectively. .e CDF of the SNR (13):

cOP FccOP � � � fc(c)dc 0 ( ) �� α 16 αμμ cOP ∞ c � � c(αμ/2)− 1 � tμ − t W(t; ) t c, αμ/2 exp�μ � � � θ d d 2Γ(μ)c� 0 0 c�

with threshold cOP > 0. .e CDF of the SNR’s for the normal where c(·) denotes the incomplete gamma function and t case will be obtained next. ([15]; equation 8.350-1). (i) From (2), (14), and (16) and using Gradshteyn and (ii) Similarly from (3), (15), (16), and Gradshteyn and Ryzhik ([15]; equation (3.381)-(8)), it follows Ryzhik ([15]; equation 3.194-1), it follows: α/2 c�μ, μcOP/c� � � F c � � , (17) normal OP Γ(μ)

] �� αμ cOP αμμΓ(μ + (]/2))] /2 c F c � � � c− 1� � c. t OP �� (μ+(]/2)) d (18) 0 Γ(μ)Γ(]/2)2(]/2)+1 μ( c/c� )α + (]/2)� c�

Let z � cα/2; then, the latter equals

μ μ− 1 αμ/2 ] 2 − (α/2) α/2 Γ(μ +(]/2))2 μ cOP FtcOP �� F �μ + , μ; 1 + μ, − μc� c � , (19) 2 1 2 ] OP Γ(μ)Γ(]/2)]μc�αμ/2

where 2F1(·) denotes the Gauss hypergeometric function and when AoF >1, the fading is considered to be severe. .e ([15]; equation 9.14-1). AoF for the normal case is given as Γ(μ +(4/α)) − [Γ(μ +(2/α))]2[Γ(μ)]− 1 AoF � , 3.2. AoF. .e AoF is a quantitative measure giving an in- normal [Γ(μ +(2/α))]2μ−(1/α)[Γ(μ)]− 1 dication of the severity or level of fading that is present in the (21) model [18, 19]. .e AoF is calculated as 2 2 � 2/α 2 var(c) Ec � − E(c) since Enormal(c) � cΓ(μ + (2/α))/μ Γ(μ) and Enormal(c ) � AoF � � , (20) c�2 Γ(μ + (4/α))/μ4/αΓ(μ). .e t case follows similarly as E(c)2 E(c)2

Γ(μ +(4/α))Γ((]/2) − 4) −� [Γ(μ +(2/α))Γ((]/2) − (α/2))]2/(Γ(μ)Γ(]/2))� AoFt � , (22) [Γ(μ +(2/α))Γ(]/2 − (α/2))]2/(Γ(μ)Γ(]/2)) since Mathematical Problems in Engineering 5

Table 1: AoF (22) for diﬀerent values of ] and μ, when α � 2. ] μ � 4 μ � 5 μ � 6 5 2.75 2.6 2.5 6 1.5 1.4 1.3333 7 1.083333 1 0.94444 8 0.875 0.8 0.75 9 0.75 0.68 0.633333 10 0.666667 0.6 0.555556 11 0.607143 0.542857 0.5 12 0.5625 0.5 0.458333 13 0.527778 0.466667 0.425926

2.5

2.0

1.5 AoF

1.0

0.5

6 8 10 12 v t case, μ = 4 Normal, μ = 4 t, μ = 5 Normal, μ = 5 t, μ = 6 Normal, μ = 6

Figure 2: AoF (21) and AoF (22) for diﬀerent values of ] and μ, when α � 2.

Γ(μ +(2/α))Γ((]/2) − (2/α)) ]2/αc� obtain numerical results for the ABER for this proposed Et(c) � , model (see (5)). From (13), the Laplace transform is Γ(μ)Γ(]/2) (2μ)2/α (23) αμμ ∞ ∞ (− c)k (− μ)u 4/α L(c) � E[exp(− cc)] � � � 2 Γ(μ +(4/α))Γ((]/2) − (4/α)) ] c� 2Γ(μ)c�αu/2 k! u!c�αu/2 Et�c � � . k�0 u�0 Γ(μ)Γ(]/2) (2μ)4/α ∞ ∞ (α(μ+u)/2)+k− 1 μ+u From Table 1 and Figure 2, we observe that the AoF is > 1 � c � t W(t; θ)dt dc. 0 0 for small values of ], corresponding to the observations of (24) Choi et al. [5] that the underlying t assumption is suitable for cases of severe fading. .e AoF for the underlying normal An alternative expression, using an approach similar to case is equal to 0.5, 0.447214, and 0.408248 (see (22)) for the that of Magableh and Matalgah [17], follows as � corresponding three diﬀerent μ values with α 2; it is further αμμ ∞ ∞ ] L(c) � � � G1,0 cc | − c(αμ/2)− 1 noted that as increases, the AoF decreases for the un- αμ/2 0,10 � 2Γ(μ)c� 0 0 derlying t model. (25) � − 1,0 α/μ − (α/2) � μ × G0,1�μtc c� � �W(t; θ)t dt dc, 3.3.LaplaceTransform. .e Laplace transform is useful as an μ integral transform in diﬀerent performance metrics. Relying where G(·) denotes Meijer’s G-function [22]. on the Laplace-based approach, the ABER can be evaluated .e expressions for the underlying normal and t case are P � (a ) �π/2 L(b2 2( )) � as aver m/π 0 m/2 sin θ dθ, where am 1 and 2 of special interest. bm � 2(Eb/N0) (see [20]). Analytical evaluation of ABER for fading channel distributions has been of interest lately [21]. (1) Using (2), (14), and Gradshteyn and Ryzhik ([15]; .erefore, alternative expressions for the Laplace transform equation 3.326-2), the Laplace transform for the are presented here, in order to enable the researcher to normal case is&ecmath; 6 Mathematical Problems in Engineering

∞ αμμc(αμ/2)− 1 c α/2 L (c) � � ⎡⎣− ⎤⎦ (− cc) c normal αμ/2 exp μ� � exp d 0 2Γ(μ)c� c�

1 ∞ (− c)k Γ(μ + (2k/α)) � � c�k. ( ) k! 2k/α Γ μ k�0 μ (26)

An alternative form is

�1 αμμ ∞ μ � L (c) � � c(αμ/2)− 1G1,0�cl/k� �G1,0�cc|1 �dc normal αμ/2 0,1 α/2 � 0,1 0 2Γ(μ)c� 0 c� 0 � � 2 − αμ 3 − αμ α − 1 − αμ � , , ... , (27) μ 1/2 αμ− 1/2 ⎜⎛ 2 α � 2α 2α 2α ⎟⎞ αμ 2 α 2,α⎜ μ α � ⎟ � G ⎜� � � ⎟. αμ/2 α/2 αμ/2 α,2⎜ α/2 cα 2 � ⎟ 2Γ(μ)c� (2π) c ⎝ c� 2 � 1 ⎠ � 0, � 2

(2) For the t case, applying Gradshteyn and Ryzhik ([15], equation 3.241-4) results in Meijer’s G-function for the integral considered can be found in ([17, 23], p. 346).

] αμμΓ(μ + (]/2))] /2 ∞ − (μ+(]/2)) L (c) � � cα/2c�− (α/2) + ( ) (− cc) c t αμ/2 �μ ]/2 � exp d Γ(μ)Γ(]/2)2(]/2)+1c� 0 (28) k αμμ2μ− 1 ∞ (− 1)k Γ(μ + (]/2) + k)Γ(α(μ + k)/2) 2μ � � � � , αμ/2 μ cα(μ+k)/2 k! α/2 Γ(μ)Γ(]/2)c� ] k�0 c� ] where 0 < μ + (2k/α) < μ + (]/2), (]/2) ≠ 0 and μc�− (α/2) ≠ 0. An alternative expression for (29) is obtained from ([22, 24], p. 81) and (15) as

1/2(αμ+1) μ 2μ+(]/2)− (1/2)(3+α) α μ 2 − (αμ/2) − (1/2)(1+α) Lt(c) � c (π) Γ(μ)Γ(]/2)(cc�)(αμ/2)]μ � � 1 − μ − (]/2) 2 − μ − (]/2) 1 − (αμ/2) 2 − (αμ/2) 1 − (αμ/2) + α − 1 � , , , , ... , (29) ⎜⎛ � ⎟⎞ ⎜ 4μ2αα � 2 2 α α α ⎟ 2,2+α⎜ � ⎟ × G2+α,2⎜ α � ⎟. ⎝⎜]2cαc� � ⎠⎟ � 1 � 0, 2

4. Analytical Illustration and Discussion Particular values are evaluated for the Laplace transform for the normal and t cases with these diﬀerent expressions An often considered performance measure for systems (see Table 2). .is table illustrates the accuracy of the dif- operating in fading environments is the outage probability, ferent expressions for the Laplace transform which the Pout, deﬁned as the probability that the output SNR falls practitioner may consider. below a given threshold cOP; thus, Pout(c) = F(cOP) (see (16)). Mathematical Problems in Engineering 7

Table 2: Values of Laplace transform computations for some arbitrary parameter values. Normal Equation (26) (27) L(0.1) 0.797194370359114 0.797194370359114 L(0.25) 0.619690325863945 0.619690325863945 L(0.5) 0.455537937967763 0.455537937967763 L(0.75) 0.36041670118177 0.36041670118177 L(1) 0.297676390851642 0.297676390851642 t (] � 3) Equation (28) (29) L(0.1) 0.624743148313397 0.624743148313397 L(0.25) 0.4840824983658209 0.4840824983658212 L(0.5) 0.373557082353699 0.373557082353697 L(0.75) 0.3113678073601211 0.3113678073601208 L(1) 0.269619410990287 0.269619410990287 t (] �15) Equation (28) (29) L(0.1) 0.753959031928999 0.753959031928984 L(0.25) 0.5805936227825265 0.5805936227824519 L(0.5) 0.431300920344367 0.431300920343910 L(0.75) 0.3463761840086145 0.3463761840072217 L(1) 0.290221789375107 0.290221789374477

1.0 1.0 0.8 0.9 0.6

0.8 0.4

0.7 Outage probability Outage probability

0.2 1.5 2.0 2.5 3.0 3.5 4.0 0.6 3.2 3.4 3.6 3.8 4.0 Threshold Threshold

W(t;θ) = δ(t – 1) W(t;θ) = δ(t – 1) W(t;θ) = Gam (30/2, 30/2) W(t;θ) = Gam (30/2, 30/2) W(t;θ) = Gam (15/2, 15/2) W(t;θ) = Gam (15/2, 15/2) W(t;θ) = Gam (3/2, 3/2) W(t;θ) = Gam (3/2, 3/2) (a) (b)

Figure 3: Outage probabilities of SNR of the SMN (α − μ) for diﬀerent mixing densities from 5, for α � 2, μ � 1, c� � 3.

By comparing the outage probability of the SMN (α − μ) with practitioner with engineering expertise in the communi- the mixing densities, i.e. for the underlying normal and t cations systems arena. .is unified scale mixture platform distributions, as depicted in Figure 3, it can be observed that allows theoretical and practical access to previously un- the assumption of an underlying t distribution results in a considered models that may yield improved fits to experi- lower outage probability than the α − μ distribution even for mental data. large degrees of freedom (the bottom graph in Figure 3 is a magnification of the top graph; for a smaller interval on the x-axis to provide the reader with ease of interpretation). As 5. Conclusion expected, the SMN (α − μ) approaches the α − μ distribution In this paper, we proposed a SMN generalization to Yacoub’s as the degrees of freedom of the t distribution increase, since α − μ distribution, to model process behaviour. .is provides a larger degree of freedom results in the tails being lighter a flexible tool for the practitioner to model observed ex- and closer to what is expected with the underlying normal perimental data when the underlying normality assumption distribution. .ese results and arguments highlight the does not seem feasible. Based on the performance evaluation contribution the SMN (α − μ) model which enables the for the two considered mixing densities, it is clear that there 8 Mathematical Problems in Engineering are instances in which the ABER and outage probability of Conference on Communications, pp. 4628–4633, IEEE, the α − μ distribution can be improved through the use of the Glasgow, UK, June 2007. proposed SMN (α − μ) model. .is reiterates the arguments [6] Y. He, F. R. Yu, N. Zhao, H. Yin, H. Yao, and R. C. Qiu, “Big presented in Choi et al. [5] and Ollila et al. [4] that fading data analytics in mobile cellular networks,” IEEE Access, vol. 4, models which do not exhibit normality from its genesis may pp. 1985–1996, 2016. be of distinct practical value. .is paper provides a [7] R. C. Qiu, “Large random matrices and big data analytics,” in framework within a computational reach utilising a SMN Big Data of Complex Networks, pp. 179–222, Chapman and Hall/CRC, London, UK, 2016. construction within the fading environment, which could [8] D. F. Andrews and C. L. Mallows, “Scale mixtures of normal now be implemented by the practitioner when field data distributions,” Journal of the Royal Statistical Society: Series B justify it. Further, as a potential future point of departure, (Methodological), vol. 36, no. 1, pp. 99–102, 1974. these scale mixture density representations situated within [9] S. T. B. Choy and J. S. K. Chan, “Scale mixtures distributions the fading environment may provide particular advances in statistical modelling,” Australian & New Zealand Journal of within Bayesian computing, especially in the case of the Statistics, vol. 50, no. 2, pp. 135–146, 2008. Gibbs sampler, where this hierarchical representation may [10] V. H. Lachos Davila,´ C. B. 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Wishart type distribution inspired by massive MIMO systems,” Journal of Statistical Distributions and Applications, Conflicts of Interest vol. 6, no. 1, p. 4, 2019. [14] K. C. Chu, “Estimation and decision for linear systems with .e authors declare that they have no conflicts of interest. elliptical random processes,” IEEE Transactions on Automatic Control, vol. 18, no. 5, pp. 499–505, 1973. Acknowledgments [15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, Cambridge, MA, USA, 2014. .e authors would like to hereby acknowledge the support of [16] M. D. Yacoub, “.e α − μ distribution: a general fading distri- the StatDisT group based at the University of Pretoria, bution,” in Proceedings of the 13th IEEE International Symposium Pretoria, South Africa. .is work was based upon research on Personal, Indoor and Mobile Radio Communications, vol. 2, supported by the National Research Foundation, South pp. 629–633, Lisboa, Portugal, September 2002. Africa (SRUG190308422768 nr. 120839), the South African [17] A. M. Magableh and M. M. Matalgah, “Moment generating NRF SARChI Research Chair in Computational and function of the generalized α − μ distribution with applica- Methodological Statistics (UID: 71199), the South African tions,” IEEE Communications Letters, vol. 13, no. 6, DST-NRF-MRC SARChI Research Chair in Biostatistics pp. 411–413, 2009. (Grant no. 114613), the Research Development Programme [18] M. K. Simon and M. S. Alouini, Digital Communication over at UP 296/2019, and the project UIDB/00297/2020 at UNL. 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