Advances in the Statistical Characterization of Fading: From 2005 to Present

Abstract

This review paper presents some of the most significant advances in statistical characterization of wireless communication fading. We will focus on the fundamental statistical characterizations of the fading distributions in terms of the probability density function (PDF), the cumulative density function (CDF), and the moment generating function (MGF). The period from 2005 to present will be considered in order to complement the information which can be found in the specialized literature like Simon and Alouini’s book on digital communications over fading channels. We present both the advances on classical fading models and a brief summary of some selected new fading models. Some open problems in the field will be also stated.

1. Introduction

Fading is an important peculiarity of wireless communication channels. Statistical characterization of fading is a key aspect for the analysis and simulation of wireless communication systems. In the last years, there has been a huge amount of research work on statistical characterization of wireless communication fading. The most comprehensive account of this research topic is Simon and Alouini’s book on digital communications over fading channels [1]. This book presents the statistical characterization of the most popular fading models by 2005.

In this review paper, in order to complement the information available in [1], a selection of the new results on fading statistics in the period from 2005 to present will be considered. Only single-input single-output (SISO) flat fading channels are considered; for an introduction to multiple-input multiple-output (MIMO) fading channels the reader should consider [2, 3]. We will focus on the fundamental statistical characterizations of the fading distributions in terms of the probability density function (PDF), the cumulative density function (CDF), and the moment generating function (MGF). We will also pay special attention to the exact closed-form results concerning the PDF, CDF, and MGF of the fading distributions. The advances on classical fading models and some results on new fading models will be presented. Along the paper, some open problems on fading statistics will be stated.

The remainder of this paper is organized as follows. In Section 2 the precise mathematical definitions of the fundamental statistics are provided. The advances concerning the classical fading distributions are presented in Section 3 and some selected results on new fading models are presented in Section 4. Finally, some conclusions are given in Section 5.

2. Statistical Preliminaries

Let us denote 𝑅 which is the envelope of the fading channel. In this review paper we will focus on the fundamental statistics of the instantaneous signal-to-noise ratio (SNR) 𝛾. Let 𝐸[⋅] be the expectation operator. The instantaneous SNR can be expressed in terms of the channel envelope 𝑅 and the average SNR 𝛾≜𝐸[𝛾]: 𝑅𝛾=²𝐸𝑅²𝛾. (1)

The CDF of the instantaneous SNR 𝛾 can be obtained from the PDF 𝑓_𝛾(𝛾) as follows: 𝐹_𝛾𝛾=^𝛾₀𝑓_𝛾(𝜉)𝑑𝜉, (2) while the MGF is computed as ℳ_𝛾(𝑠)=𝐸𝑒^𝑠𝛾=^∞₀𝑓_𝛾(𝜉)𝑒^𝑠𝜉𝑑𝜉. (3) Note that the definition of the MGF given in (3) differs from the unilateral Laplace transform definition in the sign of the variable 𝑠.

From a pure statistical point of view, the three characterizations of the instantaneous SNR 𝛾 (the PDF, the CDF, and the MGF) are equivalent; however, from the point of view of the performance analysis of fading channels, the MGF and the CDF play an important role in the computation of key performance metrics such as error probability, outage probability, or outage capacity [1].

3. Advances on Classical Fading Models

In this section we will consider as classical fading models the following: Rayleigh, Nakagami-𝑞 (Hoyt), Nakagami-𝑛 (Rice), Nakagami-𝑚, Weibull, Beckmann, Lognormal, 𝐾, and Rician shadowed. All these fading models are well described in [1].

The PDF, CDF, and MGF of the Rayleigh, Nakagami-𝑚, Nakagami-𝑛 (Rice), and Weibull distributions are expressed in closed-form, and the corresponding expressions are found in [1].

Only the PDF and MGF of the Nakagami-𝑞 (Hoyt) distribution are given in [1]. A closed-form expression for its CDF was derived in 2009 [4, 5]: 𝐹_𝛾𝛾=𝑄𝑢𝑞𝛾𝛾,𝑣𝑞𝛾𝛾−𝑄𝑣𝑞𝛾𝛾,𝑢𝑞𝛾𝛾,𝛾≥0, (4) where 𝛾≜𝐸[𝛾], 𝑞 is the Hoyt shape parameter, 𝑢(𝑞)≐(1−𝑞⁴/2𝑞)(1+𝑞)/(1−𝑞), 𝑣(𝑞)≐(1−𝑞⁴/2𝑞)(1−𝑞)/(1+𝑞), and 𝑄(⋅) is the Marcum Q function [1].

Interestingly, the Beckmann fading distribution includes the Rayleigh, Nakagami-𝑞 (Hoyt), Nakagami-𝑛 (Rice), and Nakagami-𝑚 distributions as particular cases by a suitable combination of its parameters. Its MGF is expressed in elementary terms [1], but closed-form expressions for both its PDF and its CDF in terms of classical or well-studied special functions are challenging open problems.

Beyond channel modeling, the Lognormal distribution is widely used; its PDF and CDF are easily expressed in closed-form; however, no closed-form expression is known for its MGF. A recent paper on this particular problem which derives an approximate expression is [6].

The PDF and MGF of the 𝐾 distribution are expressed in closed-form in [1]. Its CDF was derived in closed-form in 2006 [7]: 𝐹_𝛾𝛾(𝜋(×=𝜋csc𝜈−1))𝜈𝛾/𝛾₁𝐹₂1;2−𝜈,2;𝜈𝛾/𝛾Γ(𝜈)Γ()−2−𝜈𝜈𝛾/𝛾^𝜈₁𝐹₂𝜈;𝜈,1+𝜈;𝜈𝛾/𝛾Γ(𝜈)Γ(),1+𝜈 (5) where 𝛾≜𝐸[𝛾], 𝜈 is the distribution shaping parameter, Γ(⋅) is the gamma function, and _𝑝𝐹_𝑞(⋅) is the generalized hypergeometric function [8].

As in the previous model, the PDF and MGF of the Rician shadowed distribution are expressed in closed-form in [1]. Its CDF was derived in closed-form in 2010 [9]: 𝐹_𝛾𝛾=𝑚^𝑚(1+𝐾)(𝑚+𝐾)^𝑚𝛾𝛾×Φ₂1−𝑚,𝑚;2;−1+𝐾𝛾𝛾,−1+𝐾𝛾𝑚𝛾,𝐾+𝑚 (6) where 𝛾≜𝐸[𝛾], 𝐾 is the classical Rice parameter, 𝑚 is the parameter corresponding to the line-of-sight (LOS) fluctuations, and Φ₂(⋅) is the bivariate confluent hypergeometric function [8].

4. New Fading Models

In this section the author has selected five new fading models which are not listed in [1]. The models presented in this section do not exhaust the list of new fading models recently proposed and not included in [1].

4.1. The 𝜂-𝜇 Fading Model

This model was first proposed by Yacoub in 2000 [10] and later enhanced in [11]. From a physical point of view, this model corresponds to a non-line-of-sight (NLOS) scenario where different clusters of multipath waves can propagate simultaneously. The PDF of the 𝜂-𝜇 distribution in format 1 (see [11] for the details) is given by 𝑓_𝛾𝛾=2√𝜋𝜇^𝜇+1/2ℎ^𝜇𝛾Γ𝜇𝐻^𝜇−1/2𝛾𝛾^𝜇−1/2×exp−2𝜇ℎ𝛾𝛾𝐼_𝜇−1/22𝜇𝐻𝛾𝛾,ℎ≜2+𝜂⁻¹+𝜂4,𝜂𝐻≜⁻¹−𝜂4, (7) where 𝛾≜𝐸[𝛾], 𝐼_𝑝(⋅) is the modified Bessel function of the first kind [8], 𝜂 is a positive real number representing the scattered-wave power ratio between the in-phase and quadrature components of each cluster of multipath, and 𝜇 is a positive real number representing the effective number of propagation clusters. This model contains the Rayleigh, Nakagami-𝑞 (Hoyt), and Nakagami-𝑚 models as particular cases by a suitable combination of the parameters 𝜂 and 𝜇. In [11] a physical model was given for the 𝜂-𝜇 model when 𝜇 is a natural number; this model is useful for simulation of 𝜂-𝜇 random variables since these can be expressed in terms of Gaussian random variables. The MGF corresponding to the PDF given in (7) was expressed in elementary terms in 2008 [12]: ℳ_𝛾(𝑠)=4𝜇²ℎ2(ℎ−𝐻)𝜇+𝑠𝛾2(ℎ+𝐻)𝜇+𝑠𝛾^𝜇. (8) In 2010, an exact closed-form expression for the CDF of the 𝜂-𝜇 distribution was derived in terms of the bivariate confluent function [13]: 𝐹_𝛾𝛾=1−(𝐻/ℎ)^2𝜇Γ1+2𝜇2ℎ𝜇𝛾𝛾^2𝜇×Φ₂𝐻𝜇,𝜇;1+2𝜇;−1+ℎ2ℎ𝜇𝛾𝛾𝐻,−1−ℎ2ℎ𝜇𝛾𝛾. (9) Expressions (7), (8), and (9) complete the fundamental characterization of the 𝜂-𝜇 fading model by closed-form expressions.

4.2. The 𝜅-𝜇 Fading Model

This model was also proposed by Yacoub in 2000 [14] and later enhanced in [11]. From a physical point of view, this model corresponds to a line-of-sight (LOS) scenario where different clusters of multipath waves can propagate simultaneously. The PDF of the 𝜅-𝜇 distribution is given by 𝑓_𝛾𝛾=𝜇(1+𝜅)^(𝜇+1)/2𝜅^(𝜇−1)/2𝛾𝛾exp𝜇𝜅𝛾^(𝜇−1)/2−𝜇()𝛾×exp1+𝜅𝛾𝐼_𝜇−12𝜇𝜅(1+𝜅)𝛾𝛾, (10) where 𝛾≜𝐸[𝛾], 𝜅 is a real positive number representing the power ratio between dominant and scattered-wave components, and 𝜇 is a positive real number representing the effective number of propagation clusters. This model contains the Rayleigh, Nakagami-𝑛 (Rice), and Nakagami-𝑚 models as particular cases by a suitable combination of the parameters 𝜅 and 𝜇. In [11] a physical model was also given for the 𝜅-𝜇 model when 𝜇 is a natural number; this model is useful for simulation of 𝜅-𝜇 random variables since these can be expressed in terms of Gaussian random variables. The CDF of the 𝜅-𝜇 distribution was directly given in [11] in terms of the generalized Marcum 𝑄 function [1]: 𝐹_𝛾𝛾=1−𝑄_𝜇2𝜅𝜇,2()𝜇𝛾1+𝜅𝛾, (11) while the MGF was expressed in exact closed-form in 2008 [12]: ℳ_𝛾(𝑠)=𝜇(1+𝜅)𝜇(1+𝜅)+𝑠𝛾^𝜇𝜇exp²𝜅(1+𝜅)𝜇(1+𝜅)+𝑠𝛾−𝜇𝜅. (12) Therefore, expressions (10), (11), and (12) provide a complete set of alternative characterizations for the 𝜅-𝜇 fading model.

4.3. The 𝛼-𝜇 Fading Model

Again, this model was proposed by Yacoub in 2002 [15] and improved in 2007 [16]. From a physical point of view, this model is appropriate to consider nonlinear propagation phenomena. In fact, the 𝛼-𝜇 distribution is a rewritten form of the Stacy (generalized gamma) distribution. Defining Ω_𝛼=𝐸[𝑅^𝛼], the PDF of 𝛾=(𝑅²/Ω₂)𝛾 in this model is given by 𝑓_𝛾𝛾=𝛼𝜇^𝜇𝛾^{(𝛼𝜇/2)−1}2Γ𝜇𝛾^𝛼𝜇/2𝛾exp−𝜇𝛾^𝛼/2, (13) where 𝛾≜𝐸[𝛾] and 𝛾≜(Ω_𝛼^2/𝛼/Ω₂)𝛾; otherwise, 𝛼 and 𝜇 are shaping parameters. The CDF of the 𝛼-𝜇 fading model is expressed as 𝐹_𝛾𝛾=Γ𝜇,𝜇𝛾/𝛾^𝛼/2Γ𝜇, (14) where Γ(⋅,⋅) is the incomplete gamma function [8]. The MGF of this distribution was derived in exact closed-form in 2009 (see [17]) by the Meijer 𝐺 function [8]: ℳ_𝛾(𝑠)=𝛼𝜇^𝜇2Γ𝜇𝛾^𝛼𝜇/2𝑘^1/2𝑙^{(𝛼𝜇−1)/2}(2𝜋)^{(𝑙+𝑘−2)/2}𝑠^𝛼𝜇/2×𝐺^𝑘,𝑙_𝑙,𝑘𝜇𝛾^𝛼/2𝑘𝑙^𝑙𝑠^𝑙𝑘^𝑘∣𝑖𝑙,1−𝛼𝜇/2𝑖(),𝑘,0 (15) where 𝑖(𝑛,𝜉)=𝜉/𝑛,(𝜉+1)/𝑛,…,(𝜉+𝑛−1)/𝑛 and 𝛼/2=𝑙/𝑘 such that gcd(𝑙,𝑘)=1 (great common divisor) in order to include noninteger values of 𝛼. As happens with the two previous models, the PDF, CDF, and MGF of the 𝛼-𝜇 distribution are known in exact closed-form by expressions (13), (14), and (15).

4.4. The 𝒢 Fading Model

The 𝒢 distribution was first proposed by Frery et al. [18], in the context of synthetic aperture radar (SAR) image modeling. In 2008 Laourine et al. proposed the application of this model to wireless fading channels [19]. The 𝒢 distribution can be seen as a composite Nakagami-𝑚-inverse Gaussian distribution. The PDF of the 𝒢 distribution is expressed in closed-form as [19] 𝑓_𝛾𝛾𝛾=𝐴^𝑚−1𝛼+𝛽𝛾^𝑚+1/2𝐾_𝑚+1/2𝑏,𝜆𝛼+𝛽𝛾𝐴≜𝛾^(1+2𝑚)/4Γ(𝑚)2𝜆𝜆𝜋𝜃exp𝜃𝑚𝛾^𝑚,1𝑏≜𝜃𝜆𝛾,𝛼≜𝜆𝛾,𝛽≜2𝑚𝜃, (16) where 𝛾≜𝐸[𝛾], 𝐾_𝑝(⋅) is the modified Bessel function of the second kind [8], the parameter 𝑚 comes from the underlying Nakagami-𝑚 distribution, and the parameters 𝜆 and 𝜃 correspond to the parameters of the underlying inverse Gaussian distribution. The CDF was also derived in [19]: 𝐹_𝛾𝛾(𝑚)×=1−𝐴Γ_𝑚𝑘=12^𝑘𝛾^𝑚−𝑘𝛽𝑏^𝑘()!𝐾𝑚−𝑘^{𝑚−𝑘+1/2}𝑏𝛼+𝛽𝛾𝛼+𝛽𝛾^{𝑚−𝑘+1/2}, (17) where 𝑚 is restricted to be a natural number. Otherwise, the MGF is given by [19] ℳ_𝛾(𝑠)=1+𝑚^𝑚−1_𝑘=1(−1)^𝑘+1𝐶^𝑘_𝑚−1()!×𝑘+1_𝑘𝑝=0𝐶^𝑝_𝑘2𝛼𝑠𝛽^{𝑘+1−𝑝}Γ𝑘+𝑝+1×𝐻_{−(𝑘+𝑝+1)}𝑏2𝛽𝑠+𝑠𝛼𝛽, (18) where 𝐶^𝑘_𝑝 is the binomial coefficient and 𝐻_𝑝(⋅) is the Hermite function [8]. To the best knowledge of the author, finding exact closed-form expressions for the CDF and MGF of the 𝒢 distribution, for arbitrary values of 𝑚, is an open problem.

4.5. The 𝜅-𝜇 Shadowed Fading Model

This model has been proposed by the author in 2014 [20]. It is a generalization of both the Rician shadowed model and the 𝜅-𝜇 model. The PDF of the 𝜅-𝜇 shadowed distribution is given by 𝑓_𝛾𝛾=𝜇^𝜇𝑚^𝑚(1+𝜅)^𝜇Γ𝜇𝑚+𝜇𝜅^𝑚𝛾𝛾𝛾^𝜇−1×𝑒^{−𝜇(1+𝜅)𝛾/𝛾}₁𝐹₁𝜇𝑚,𝜇;²𝜅(1+𝜅)𝛾𝜇𝜅+𝑚𝛾, (19) where 𝛾≜𝐸[𝛾], ₁𝐹₁(⋅) is the confluent hypergeometric function [8], 𝜅 is a real positive number representing the power ratio between dominant and scattered-wave components, 𝜇 is a positive real number representing the effective number of propagation clusters, and 𝑚 is a parameter associated with the LOS fluctuation. This model contains the Rician shadowed when 𝜇=1 and the 𝜅-𝜇 when 𝑚→∞. In [20] a physical model was also given for the 𝜅-𝜇 shadowed model when 𝜇 is a natural number; this model is useful for simulation of 𝜅-𝜇 shadowed random variables since these can be expressed in terms of Gaussian and gamma random variables.

The shaping parameters of this distribution have a clear influence on the amount of fading (AoF) of 𝛾 [1]. In absence of LOS fluctuation (𝑚→∞) the AoF is due to multipath and depends on 𝜅 and 𝜇 as follows [11]: 𝛾=𝑉𝛾AoF𝐸𝛾²=1𝜇1+2𝜅(1+𝜅)², (20) where 𝑉[⋅] is the variance operator. Note that the AoF in (20) is inversely proportional to 𝜇. Otherwise, the AoF of the LOS component is inversely proportional to 𝑚.

The CDF of the 𝜅-𝜇 shadowed distribution was also given in [20] in terms of the bivariate confluent hypergeometric function: 𝐹_𝛾𝛾=𝜇^𝜇−1𝑚^𝑚(1+𝜅)^𝜇Γ𝜇𝑚+𝜇𝜅^𝑚𝛾𝛾^𝜇×Φ₂𝜇()𝜇−𝑚,𝑚;𝜇+1;−1+𝜅𝛾−𝜇()𝛾,1+𝜅𝛾𝑚𝛾,𝜇𝜅+𝑚 (21) while the MGF is expressed as [20] ℳ_𝛾(𝑠)=−𝜇^𝜇𝑚^𝑚(1+𝜅)^𝜇𝛾^𝜇𝑚+𝜇𝜅^𝑚×𝑠−𝜇(1+𝜅)/𝛾^𝑚−𝜇𝑠−𝜇(1+𝜅)/𝛾𝑚/𝜇𝜅+𝑚^𝑚. (22) Therefore, expressions (19), (21), and (22) provide a complete set of alternative characterizations for the 𝜅-𝜇 shadowed fading model.

5. Conclusions

This review paper presents some important results concerning fading statistics from 2005 to present. The new results on the most relevant classical fading distributions have been described. A selection of novel proposals for fading distributions is also included. A summary of relevant features of these novel distributions is shown in Table 1. Several open problems concerning the fading distributions have been also stated.

Table 1 

Relevant statistical information on the novel fading distributions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by the Spanish Government and FEDER under Project TEC2011-25473.