The η−μ/IG Distribution: A Novel PhysicalMultipath/Shadowing Fading Model

Paschalis C. Sofotasios1, Theodoros A. Tsiftsis2, Mounir Ghogho1,3, Leif. R. Wilhelmsson4 and Mikko Valkama5

1School of Electronic and Electrical Engineering, University of Leeds, LS2 9JT, Leeds, United Kingdom.e-mail: {p.sofotasios;m.ghogho}@leeds.ac.uk

2Department of Electrical Engineering, Technological Educational Institute of Lamia, 35100 Lamia, Greece.e-mail: tsiftsis@teilam.gr

3International University of Rabat, Rabat-Sale-Zemmour-Zaer, Morocco.4Ericsson Research, Scheelevagen 19, SE-221 83 Lund, Sweden.

e-mail: leif.r.wilhelmsson@ericsson.com5Department of Communications Engineering, Tampere University of Technology, FI-33101, Tampere, Finland.

e-mail: mikko.e.valkama@tut.fi

Abstract—The aim of this work is the formulation and deriva-tion of the η−μ/Inverse Gaussian composite distribution whichcorresponds to a physical fading model. The η−μ distribution isa generalized small-scale fading model which accounts effectivelyfor non-line-of-sight scenarios and includes as special casesthe widely known Nakagami-m, Rayleigh, Hoyt and one sidedGaussian distributions. Similarly, the inverse Gaussian (IG)distribution is a convenient model which was recently shownto characterize shadowing more efficiently than the widely usedgamma distribution. To this effect, the proposed η−μ/IG modelprovides an overall efficient characterization of multipath andshadowing effects which typically occur simultaneously. Theoffered modelling accuracy is achieved thanks to the remarkableflexibility of its parameters. This is also verified by the fact thatthe proposed model is capable of providing good fittings to exper-imental data that correspond to realistic wireless communicationscenarios while they include as special cases the widely knownNakagami-m/IG, Rayleigh/IG and Hoyt/IG composite fadingmodels. Novel analytic expressions are derived for the envelopeand power probability density function (pdf) of the η−μ/IGmodel. The derived expressions can be utilized in various studiesin radio communications, free space optical communicationsand ultrasound imaging, among others. Indicatively, an analyticexpression is derived for the outage probability (OP) of η−μ/IGfading channels.

Index Terms—η−μ distribution, inverse Gaussian distribution,generalized multipath fading, shadowing, outage probability1.

I. INTRODUCTION

Fading is a physical phenomenon that significantly degradescommunication signals during wireless propagation. A com-mon approach for accounting for this effect has been feasiblethrough utilization of suitable statistical distributions, so calledfading distributions. Based on this, popular models such asRayleigh, Nagakami-m, Weibull and Hoyt have been shown

1This work was supported by the Academy of Finland under the project259915 “In-band Full-Duplex MIMO Transmission” and the Research Pro-gram DGRES (MIS 380360) within the Research Activity Archimedes III,funded by the NSRF 2007-2013, Greece.

to be capable of modelling efficiently small-scale fading inNon-Line-of-Sight (NLOS) communication scenarios whereasRice distribution has been typically employed in characteriz-ing multipath fading in Line-of-Sight (LOS) communicationscenarios, [1]–[3] and the references therein. Capitalizing onthese models, M. D. Yacoub proposed three generalised fadingmodels, namely, the α−μ, the κ−μ, the η−μ and subsequentlythe λ−μ and the κ−μ Extreme models, [4]–[10]. Likewise, theα−κ−μ Extreme distribution was recently proposed in [11].These models have been extensively shown to be particularlyuseful due to the remarkable flexibility offered by their namedparameters which render them capable of providing adequatefittings to results obtained from field measurements. This isalso evident by the fact that they include as special casesthe aforementioned small-scale fading distributions which havebeen widely used during the past decade.

However, it is fundamentally known that multipath andshadowing effects typically occur simultaneously. As a re-sult, in spite of the usefulness of the aforementioned fadingmodels, their capabilities are limited to the characterizationof either the one or the other effect. In other words, theyfail to account for both multipath and shadowing. There-fore, the necessity for composite statistical models that canprovide adequate characterization of the fading effect as awhole became evident. Motivated by this, the authors in [12]proposed the Rayleigh/gamma fading model, which is alsoknown as K−distibution, (K) while Shankar in [13] exploitedthe flexibility of Nakagami-m distribution, which includesRayleigh distribution as a special case, for introducing theNakagami-m/gamma composite distribution. This model isalso known as generalised K−distribution, (KG). In the samecontext, the Weibull/gamma distribution was proposed in [14]while the κ−μ/gamma, the η−μ/gamma, λ−μ/gamma andα−κ−μ/gamma models were recently reported in [15]–[21].

It is recalled that gamma distribution has been largelyused as a substitute of log-normal distribution. The reason isthat the log-normal distribution often appears intractable dueto its rather inconvenient algebraic representation. However,it is noted here that the inverse Gaussian (IG), or Wald,distribution has been shown to be closer to log-normal dis-tribution than the gamma distribution and hence it constitutes

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IEEE ICC 2013 - Wireless Communications Symposium

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a more effective substitute in characterising the shadowingeffect [22]. This was also justified by means of Kullback-Leibler (KL) Divergence which determines the closenessof two continuous statistical distributions in terms of thevalue of the involved relative entropy, in bits/message [23].The performance of digital communications systems overIG shadowing and Nakagami−m/IG multipath/shadowingenvironments was recently analyzed in [24], [25], respectively.

Motivated by the above, this work is devoted to the in-troduction, formulation and derivation of the η−μ/IG com-posite distribution which constitutes a physical fading model.Specifically, after formulating each model mathematically,novel analytic expressions are derived for the correspondingprobability density functions (pdf). The validity of the offeredexpressions is verified through comparisons with numericalresults while their behaviour is assessed under different para-metric scenarios. It is also shown that the proposed distributionincludes as special cases the Nakagami-m/IG, Rayleigh/IGand Hoyt/IG composite fading models. Importantly, owingto their convenient algebraic representation, the offered ex-pressions are useful mathematical tools that can be efficientlyutilized in analytical studies related to the performance eval-uation of digital communications over η−μ/IG generalizedmultipath/shadowing channels. Indicatively, a novel analyticexpression for the corresponding outage probability (OP) isderived in the present work.

II. THE η−μ, GAMMA AND INVERSE GAUSSIAN FADING

MODELS

A. The η−μ ModelThe η−μ distribution is a general fading distribution that

accounts for the small-scale variation of fading signals inNLOS communication scenarios. It is expressed in terms oftwo physical parameters, η and μ and it holds for two formats,namely Format-1 and Format-2. In the former the η parameterdenotes the ratio of the powers between the multipath waves inthe in-phase and quadrature components, whereas in the latterit denotes the correlation coefficient between the scatteredwave in-phase and quadrature components of each clusterof multipath2. Likewise, the μ parameter denotes - in bothformats - the inverse of the normalised variance and relates tothe number of multipath clusters in the environment [4], [8].

For a fading signal with envelope R and r =√E(R2), the

η−μ envelope pdf is mathematically represented as [8]

pR(r) =

4√πμμ+ 1

2hμ

Γ(μ)Hμ− 12

r2μ

r2μ+1e−2μh r2

r2 Iμ− 12

(2μH

r2

r2

)(1)

where E(.) and r denote expectation and the root-mean-square(rms) value of R, respectively. Furthermore, the parameter μis positive and is expressed as,

μ =E2(R2)

2Var(R2)

[1 +

H

h

](2)

where Var(.) denotes variance. The parameters h and H aredefined as,

h =2 + η−1 + η

4, H =

η−1 − η

4(3)

in Format-1 with 0 < η <∞ and as,

h =1

1− η2, H =

η

1− η2(4)

2The Format-2 of the η−μ distribution is also known as λ−μ distribution.

in Format-2 with −1 < η < 1 [8].The η−μ fading model includes as special cases other

widely known small-scale fading distributions. Specifically,the Hoyt distribution is obtained for μ = 0.5 and b =(1 − η)/(1 + η), where b denotes the Nakagami−q or Hoytparameter. Likewise, the one sided Gaussian distribution isattained for η → ∞ or η → 0 whereas the Nakagami−mdistribution is obtained for μ = m with η → 0 or η → ∞in Format-1 and η = ±1 in Format-2. Finally, the Rayleighdistribution is obtained for μ = 0.5 with η = 1 in Format-1and η = 0 in Format-2 [9].

B. The Gamma Fading ModelThe log-normal distribution has been traditionally regarded

as the optimum statistical model for characterising the shad-owing effect, [1]–[3]. Nevertheless, in spite of its usefulness ithas been largely shown that when involved in integrands alongwith other elementary and/or special function, its algebraicrepresentation often becomes intractable. This is particularlythe case in analyses related to the derivation of importantperformance measures in the field of digital communications.A solution to this issue was reported in [12] where the gammadistribution was proposed as an accurate substitute to log-normal distribution. The envelope pdf of gamma distributionis given by [12, eq. (4)], namely,

pY(y) =

yb−1e−yΩ

Γ(b)Ωb, y ≥ 0 (5)

where the term b ∈ R+ is its shaping parameter and

Ω = E(Y 2). Evidently, the algebraic form of the gammadistribution is undoubtedly more convenient and thus it istractable both analytically and numerically. Thanks to this, thegamma distribution has been used in myriads wireless commu-nication applications and has been additionally exploited in thederivation of the K and KG composite multipath/shadowingmodels which have been also widely used in the context ofdigital communications over fading channels [12], [13].

C. The Inverse Gaussian Fading ModelIt was mentioned above that the gamma distribution has

been largely used as an adequate substitute of log-normaldistribution for modelling shadowing effect. Based on the samefoundations the authors in [22] proposed the Inverse Gaussiandistribution as a better substitute of the log-normal distribution.More specifically, with the aid of the Kullback-Leibler (KL)divergence, which is a measure that determines the distancebetween two continuous statistical distributions by means ofthe involved relative entropy, in bits/message [23], the authorsdemonstrated that the relative entropy between log-normaland IG distributions is significantly smaller compared to therelative entropy between log-normal and gamma distributions.To this effect, it was shown that the IG distribution representsmore closely the log-normal distribution compared to thegamma distribution and therefore, it constitutes a more effec-tive substitute. Mathematically, the pdf of the IG distributionis expressed according to [26, eq. (27.1.1)], namely,

pY(y) =

√λ

2πy3e−λ(y−θ)2

2yθ2 =

√λ

2π

1

y√ye−

λy

2θ2 eλθ e−

λ2y .

(6)The above expression is valid for y > 0, while λ ∈ R

+

is the shape parameter of the distribution and θ ∈ R+

relates to the mean value of the corresponding fluctuations.Capitalizing on the capability of the IG distribution in mod-elling the shadowing effect, the authors in [22] and [25]

5716

proposed the Rayleigh/IG and the Nakagami−m/IG, or G,multipath/shadowing fading models and provided detailedperformance evaluations for various communication scenarios.

III. THE η−μ/IG FADING DISTRIBUTION

A. Model Formulation

According to the basic principles of the theory of statistics,the pdf of a composite statistical distribution is constituted bythe superimpose of two or more statistical distributions. In thepresent case, this is realized by superimposing one multipathand one shadowing distribution, namely,

p(r) �

∫ ∞

0

pR|Y

(r | y)pY(y)dy (7)

where pW |Y

(γ | y) denotes the corresponding multipathdistribution with mode y. Evidently, the η−μ/IG compositefading model is formulated by firstly setting r2 = y in (1) andsubstituting in (7) along with (6) yielding,

p(r) = A∫ ∞

0

r2μ

yμ+2e−2μh r2

y e−λy

2θ2 e−λ2y Iμ− 1

2

(2μH

r2

y

)dy

(8)where

A =

√2μμ+ 1

2hμ√λ

Γ(μ)Hμ− 12 e−

λθ

.

Evidently, the derivation of an analytic expression for thepower pdf of the η−μ/IG is subject to evaluating the integralin (8).

B. The Special Case μ ∈ N

A closed-form expression for equation (8) for the specialcase that μ ∈ N can be derived with the aid of the finiteseries representation for the In(x) function. To this end, bymaking the necessary change of variables in [27, eq. (8.467)],substituting in (8) and carrying out some basic algebraicmanipulations yields,

p(r) =

μ−1∑i=0

μμ−i−1 hμ (μ)i√λ r2μ−2i−1

i!√πHμ+i Γ(μ− 1)22i+

12 e−

λθ

×⎧⎪⎪⎪⎨⎪⎪⎪⎩(−1)

i

∫ ∞

0

yi−μ− 32e−

λy

2θ2

eB1y

dy︸ ︷︷ ︸I1

(9)

+(−1)μ∫ ∞

0

yi−μ− 32e−

λy

2θ2

eB2y

dy︸ ︷︷ ︸I2

⎫⎪⎪⎪⎬⎪⎪⎪⎭

where (a)n � Γ(a+n)/Γ(a) denotes the Pochhammer symbol[27] and

B1 = 2μr2(h−H) +λ

2,

B2 = 2μr2(h+H) +λ

2

Importantly, the integrals I1 and I2 have the same algebraicform and can be expressed in closed-form according to [27,

eq. (3.471.9)]. To this end, by making the necessary variabletransformation, it follows that:

I1 = 2

(2B1θ

2

λ

) i−μ2 − 1

4

Ki−μ− 12

(√B1λ

2θ2

)(10)

and

I2 = 2

(2B2θ

2

λ

) i−μ2 − 1

4

Ki−μ− 12

(√B2λ

2θ2

). (11)

where Kν(x) denotes the modified Bessel function of thesecond kind, [27]. Therefore, by inserting (10) and (11) in(9), recalling that Kν(x) � K−ν(x) and performing longbut simple algebraic manipulations by using B1 and B2, thefollowing closed-form expression is deduced for the envelopepdf of the η−μ/IG fading distribution,

p(r) =

μ−1∑i=0

μμ−i−1 hμ (μ)i eλθ θi−μ− 1

2 r2μ−2i−1

i!√πHμ+i Γ(μ− i) 22i−

32 λ

i−μ2 − 3

4

×

⎧⎪⎪⎨⎪⎪⎩(−1)

i

Kμ−i+ 12

(√4μλ (h−H) r2+λ2

θ

)[4μ (h−H) r2 + λ]

μ−i2 + 1

4

+ (−1)μKμ−i+ 1

2

(√4μλ (h+H) r2+λ2

θ

)[4μ (h+H) r2 + λ]

μ−i2 + 1

4

⎫⎪⎪⎬⎪⎪⎭ .

(12)

Having derived equation (12), the corresponding SNR pdfof the η−μ/IG fading model is obtained with the aid of [3,eq. (2.3)]. To this end, it immediately follows that,

p(γ) =

μ−1∑i=0

μμ−i−1 hμ (μ)i eλθ λ

μ−i2 + 3

4 γμ−i−1

i!√θ πHμ+i Γ(μ− i) 22i−

12 γμ−i

×

⎧⎪⎪⎨⎪⎪⎩(−1)

i

Kμ−i+ 12

(1θ

√4μλ θ (h−H) γ

γ+ λ2

)[4μ θ (h+H) γ

γ+ λ

] μ−i2 + 1

4

+ (−1)μKμ−i+ 1

2

(1θ

√4μλ θ (h−H) γ

γ+ λ2

)[4μ θ (h+H) γ

γ+ λ

] μ−i2 + 1

4

⎫⎪⎪⎬⎪⎪⎭

(13)

which is also valid for μ ∈ N.

C. The general case μ ∈ R

In spite of the usefulness of the above closed-form expres-sion for integer values of μ, an analytic expression for the pdfof η−μ/IG for the more general case of non-integer valuesof μ is also necessary. Such an expression can be derived byexpressing the Iν(x) function in (8) according to [28, eq. (19)].

5717

To this end, by making the necessary variable transformationequation (8) can be re-written as follows:

p(r) �√2λhμ

Γ(μ) e−λθ

n∑j=0

Γ(n+ j)n1−2jμ2(μ+j)H2j

j! Γ(n− j + 1)Γ(μ+ j + 1

2

)× r4(μ+j)−1

∫ ∞

0

1

y2(μ+j)+ 32

e−1y (2μhr

2+λ2 ) e−

λy

2θ2 dy︸ ︷︷ ︸I3

.

(14)

Notably, the I3 integral has the same algebraic representationas the I1 and I2 integrals and hence, it can be also expressedin closed-form in terms of [27, eq. (3.471.9)]. To this effect,by making the necessary change of variables it follows that,

I3 =λμ+j+ 1

4 K2 (μ+j)+ 12

(√λθ

√4μhr2 + λ

)(2μhr2 + λ

2

)μ+j+ 14 θ2μ+2j+ 1

2 2μ+j− 34

. (15)

Therefore, by substituting (15) into (14) yields the followinganalytic expression,

p(r) �n∑

j=0

Γ(n+ j)n1−2jμ2(μ+j)H2jλμ+j+ 34 hμe

λθ

j!(n− j)!Γ(μ)Γ(μ+ j + 1

2

)2μ+j− 5

4 θ2(μ+j)+ 12

× r4(μ+j)−1(2μhr2 + λ

2

)μ+j+ 14

K2(μ+j)+ 12

(√λ

θ

√4μhr2 + λ

)(16)

which holds for μ ∈ R.The corresponding SNR pdf can be straightforwardly de-

duced by applying again [3, eq. (2.3)] in (16) yielding,

p(γ) �n∑

j=0

Γ(n+ j)n1−2jμ2(μ+j)H2jλμ+j+ 34hμ 2

54−μ−j e

λθ

j! (n− j)!Γ(μ)Γ(μ+ j + 1

2

)√θ γ 2 (μ+j)

× γ2(μ+j)−1(2μ θhγ

γ+ λ

2

)μ+j+ 14

K2(μ+j)+ 12

(√λ

θ

√4μhθγ

γ+ λ

)

(17)

It is noted here that as n → ∞, the above equations reduceto the following exact infinite series representations,

p(r) =

∞∑j=0

μ2(μ+j)H2jλμ+j+ 34 hμe

λθ

j! Γ(μ)Γ(μ+ j + 1

2

)2μ+j− 5

4 θ2(μ+j)+ 12

× r4(μ+j)−1(2μhr2 + λ

2

)μ+j+ 14

K2(μ+j)+ 12

(√λ

θ

√4μhr2 + λ

)(18)

and

p(γ) =∞∑j=0

μ2(μ+j)H2jλμ+j+ 34hμ 2

54−μ−j e

λθ

j! Γ(μ)Γ(μ+ j + 1

2

)√θ γ 2 (μ+j)

× γ2(μ+j)−1(2μ θhγ

γ+ λ

2

)μ+j+ 14

K2(μ+j)+ 12

(√λ

θ

√4μhθγ

γ+ λ

)

(19)

respectively. To the best of the authors’ knowledge, equations(12)−(13) and (16)−(19) are novel.

IV. APPLICATIONS IN WIRELESS COMMUNICATIONS

It is widely known that the convenience of the algebraicrepresentation of important performance measures is crucial instudies related to analytical performance evaluation of digitalcommunications. The reason is that when the algebraic formof a corresponding measure is tractable, it becomes morelikely to ultimately derive the desired expressions in closed-form. Based on this, the derived expressions can be efficientlyutilized in various analytic studies relating to the performanceevaluation of digital communications over composite multi-path/shadowing fading channels. Among others, they can bemeaningfully employed in deriving explicit expressions forperformance measures such as, error probability, probabilityof outage, channel capacity under different constraints andhigher order statistics. Such expressions can be derived inthe context of classical and emerging technologies such assingle channel and multichannel communications, cognitiveradio, cooperative systems and free-optical communications.

A. Outage ProbabilityOutage probability is a critical performance measure of

communication links as it denotes the probability that thesignal strength (SNR) fails to exceed a predetermined suf-ficient threshold, γth. Mathematically, this is expressed asPout =

∫ γth

0pγ(γ)dγ [25]. Therefore, for the case of η−μ/IG

for all values of μ, we utilize the SNR pdf in (17) yielding,

p(γ) �n∑

j=0

Γ(n+ j)n1−2jμ2(μ+j)H2jλμ+j+ 34 hμ 2

54−μ−j e

λθ

j! (n− j)!Γ(μ)Γ(μ+ j + 1

2

)√θ γ 2 (μ+j)

×∫ γth

0

γ2(μ+j)−1 K2(μ+j)+ 12

(√λθ

√4μhθγ

γ+ λ

)(

2μ θhγγ

+ λ2

)μ+j+ 14

dγ.

(20)

Notably, the above integral has the same algebraic repre-sentation as the integral in [25, eq. (32)] which is given inclosed-form in [25, eq. (35)]. Hence, by setting accordingly:p = 2μ + 2j; q = 2μ + 2j; x = γ; y = 2μhθ/γ; z = λ/2,and carrying out some algebraic manipulations, one obtainsthe following analytic expression for the outage probability3,

p(γ) �n∑

j=0

Γ(n+ j)n1−2jμ2(μ+j)H2jλμ+j+ 34 hμ 2

54−μ−j

j! (n− j)!Γ(μ)Γ(μ+ j + 1

2

)√θ γ 2 (μ+j)e−

λθ

×

⎡⎢⎢⎣1−

2μ+2j∑l=1

γl γ2μ+2j−lth K2μ+2j−l+ 1

2

(√2λθ

√2μhθγ

γth +λ2

)(μh

√2λ)l(2μ+ 2j)1−l

(2μhθγth

γ+ λ

2

)μ+j+ 1−2l4

⎤⎥⎥⎦

(21)which is valid for μ ∈ R.

V. NUMERICAL RESULTS

This Section is devoted to the analysis of the generalbehaviour of the offered expressions. Fig. 1 illustrates theenvelope pdf of the η−μ/IG distribution for θ = 1.1,λ = 0.8 and different values of η and μ. One can notice thecorresponding variations even at slight changes of η or μ. Thisflexibility demonstrates the overall flexibility of the proposedmodel and its capability to match accurately various resultsobtained from field measurements.

3As n→∞, the terms Γ(n+j)n1−2j/(n−j)! vanish and (21) becomesan exact infinite series.

5718

0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

PD

F

η = 0.1, μ = 0.3 η = 0.4, μ = 0.3 η = 0.4, μ = 0.7 η = 0.9, μ = 1.6

Fig. 1. Envelope pdf of η−μ/IG distribution for θ = 1.1, λ = 0.8 anddifferent values of η and μ.

0 5 10 15 20 25 30 3510

−12

10−10

10−8

10−6

10−4

10−2

100

Average SNR (dB)

Out

age

Pro

babi

lity

η = 0.3, μ = 0.3 η = 1.1, μ = 0.8 η = 1.9, μ = 1.3 η = 2.7, μ = 2.8

Fig. 2. Outage probability of η−μ/IG fading channels for λ = 0.8, θ = 1.1and different values of η and μ.

In the same context, Fig. 2 depicts the behaviour of the OPalso for θ = 1.1, λ = 0.8 and different values of η and μ. Asexpected, the OP is inverse proportional to the value of η andμ. This rapid change is clear since for γ = 20dB, the valueof the OP for η = μ = 0.3 is four orders of magnitude higherthan the OP value for η = 1.9 and μ = 1.3.

VI. CLOSING REMARKS

This work was devoted to the introduction, formulationand derivation of the η−μ/IG multipath/shadowing fadingdistribution. This model emerged from the superimpose ofthe η−μ generalised small-scale fading model and the re-cently proposed IG shadowing model and it shown to beparticularly flexible in accounting for both multipath andshadowing effects. Novel analytic expressions were derivedfor the corresponding envelope and power probability densityfunction which were utilized in the derivation of a novelanalytic expression for the corresponding outage probability.Importantly, the proposed model includes as special cases theHoyt/IG, Nakagami-m/IG and Rayleigh/IG fading modelsand therefore, it can be considered a useful mathematical toolin applications related to the analytical performance evaluationof wireless communications over composite fading channels.

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