On the Calculation of the Incomplete MGF with ... - CORE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. YY, XXXX 2016 1

On the Calculation of the Incomplete MGFwith Applications to Wireless Communications

F.J. Lopez-Martinez, J.M. Romero-Jerez and J.F. Paris

Abstract— The incomplete moment generating function(IMGF) has paramount relevance in communication theory,since it appears in a plethora of scenarios when analyzingthe performance of communication systems. We here present ageneral method for calculating the IMGF of any arbitrary fadingdistribution. Then, we provide exact closed-form expressions forthe IMGF of the very general κ-µ shadowed fading model,which includes the popular κ-µ, η-µ, Rician shadowed and otherclassical models as particular cases. We illustrate the practicalapplicability of this result by analyzing several scenarios ofinterest in wireless communications: (1) physical layer security inthe presence of an eavesdropper, (2) outage probability analysiswith interference and background noise, (3) channel capacitywith side information at the transmitter and the receiver, and (4)average bit-error rate with adaptive modulation, when the fadingon the desired link can be modeled by any of the aforementioneddistributions.

Index Terms—Fading channels, κ-µ shadowed fading, momentgenerating function, physical layer security, secrecy capacity,outage probability, channel capacity, bit error rate.

I. INTRODUCTION

THE moment generating function (MGF) has played apivotal role in communication theory for decades, as a

tool for evaluating the performance of communication systemsin very different scenarios [1–4]. The MGF of the signal-to-noise ratio (SNR) γ, defined as the Laplace transform of theProbability Density Function (PDF) of γ, is well-known formost popular fading distributions [5–7] and hence enables for asimple characterization of the performance metrics of interestin closed-form.

A more general function extensively used in communicationtheory is the so-called incomplete MGF (IMGF), also referredto as truncated MGF, or interval MGF. This function has anadditional degree of freedom by allowing the lower (or equiva-lently, upper) limit of the integral in the Laplace transform, sayζ, to be greater than zero (or equivalently, lower than∞), andalso appears when characterizing the performance in a numberof scenarios of interest: order statistics [8–11], symbol and bit

This work was presented in part at European Wireless 2015. F. J. Lopez-Martinez and J. F. Paris are with Departamento de Ingenierıa de Comu-nicaciones, Universidad de Malaga - Campus de Excelencia InternacionalAndalucıa Tech., Malaga 29071, Spain. J. M. Romero-Jerez is with De-partamento de Tecnologıa Eectronica, Universidad de Malaga - Campus deExcelencia Internacional Andalucıa Tech., Malaga 29071, Spain. Contact e-mail: [email protected] work has been funded by the Consejerıa de Economıa, Innovacion,Ciencia y Empleo of the Junta de Andalucıa, the Spanish Governmentand the European Fund for Regional Development FEDER (projectsP2011-TIC-7109, P2011-TIC-8238, TEC2013-42711-R, TEC2013-44442-Pand TEC2014-57901-R).

error rate calculation with multiantenna reception [12, 13],capacity analysis in fading channels [2], outage probabilityanalysis in cellular systems [14, 15], adaptive schedulingtechniques [16], cognitive relay networks [17] or physicallayer security [18]. Despite its usefulness, to the best of ourknowledge the expressions of the IMGF are largely unknownfor fading distributions other than the classical ones: Rayleigh,Rice, Nakagami-m and Hoyt. In fact, a general and systematicway to find analytical expressions for the IMGF does notyet exist, thus requiring the use of state-of-the-art numericaltechniques for its evaluation [2].

Thence, there is a twofold motivation for this paper from apurely communication-theoretic perspective: first, we presenta general theory for deriving the IMGF of the SNR for anarbitrary distribution. Specifically, we show that the IMGF ofany fading distribution is given in terms of an inverse Laplacetransform of a shifted version of the MGF scaled by theLaplace domain variable. This implies that the IMGF shouldhave a similar functional form as the cumulative distributionfunction (CDF). Secondly, we exemplify the usefulness of thisresult by deriving a closed-form expression for the IMGF ofthe κ-µ shadowed distribution [7, 19], which includes popularfading distributions such as κ-µ, η-µ [20, 21] or Rician-shadowed [22] as particular cases, and for all of which theIMGF had not been previously reported.

In order to illustrate its practical applicability, we introduceseveral scenarios of interest in wireless communications: Asa main application, we focus on a physical-layer securityset-up on which two legitimate peers (Alice and Bob) wishto communicate in the presence of an external eavesdropper(Eve). The characterization of the maximum rate at whicha secure communication can be attained, i.e. the secrecycapacity CS , is a classical problem [23, 24] in communicationtheory. Remarkably, the research on physical-layer security ofcommunication systems operating in the presence of fadinghas been boosted in the last years ever since the original worksin [25, 26]. The fact that Eve and Bob observe independentfading realizations adds an additional layer of security tocommunication compared to the conventional set-up for theGaussian wiretap channel [24]. Hence, a secure communica-tion is feasible even when the average SNR at Bob is lowerthan the average SNR at Eve.

In the literature, there is a great interest on understandinghow the consideration of more sophisticated fading modelsthan Rayleigh may impact the secrecy performance attendingto different metrics: the outage probability of secrecy capacity(OPSC), the probability of strictly-positive secrecy capacity(SPSC) or the average secrecy capacity (ASC). Specifically,

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2 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. XX, NO. YY, XXXX 2016

fading models such as Nakagami-m [27], Rician [28], Weibull[29], two-wave with diffuse power [30], Nakagami-q [31]or κ-µ [32] have been considered. However, the statisticalcharacterization of the OPSC in a tractable form is oftenunfeasible due to the involved mathematical derivations. Thus,approximations are usually required for evaluating the OPSC[30, 32], being only possible the calculation of the SPSC inclosed-form [28, 29, 32].

Very recent works have proposed novel approaches toderiving the secrecy performance metrics in a general way: in[33], a duality between the OPSC and the outage probabilityanalysis in the presence of interference and background noisewas presented, which greatly simplifies the analysis of theOPSC calculation for an arbitrary choice of fading distribu-tions for the desired and eavesdropper links. In [34], a unifiedMGF approach to the analysis of physical-layer security inwireless systems was introduced, allowing for a numericalevaluation of the secrecy performance metrics for arbitraryfading distributions.

As we will later see, the practical application of our resultsallows that the OPSC can be evaluated directly by specializingthe IMGF of the SNR at Bob at some specific values. Thus,the OPSC (and hence the SPSC as a special case) can beevaluated in closed-form provided that such IMGF is given inclosed-form. This is exemplified for the very general case ofthe κ-µ shadowed distribution [7].

We also illustrate how the IMGF can be used to obtain otherperformance metrics in relevant scenarios in communicationtheory which, despite looking rather dissimilar at a first glance,they all require for the computation of the IMGF. The first oneof these additional scenarios is the outage probability analysisof wireless communications systems affected by interferenceand background noise. As previously stated, this problem wasrecently shown to be mathematically equivalent to the OPSCin [33]; thus, the outage probability in this scenario will alsobe expressed in terms of the IMGF under the same conditionsassumed when computing the OPSC. The second additionalscenario is related to the analysis of the channel capacity whenside information is available at both the transmitter and thereceiver sides [35]. According to the framework introduced in[2], the capacity in this scenario can be expressed in termsof the IMGF. Finally, we also analyze a classical performancemetric in communication theory, which is the average bit-errorrate (ABER) with adaptive modulation [36]. The ABER in thisscenario under arbitrary fading is also given in terms of theIMGF of the fading distribution.

The remainder of this paper is structured as follows: inSection II, the main mathematical contributions of this paperare presented, within the most relevant is a general wayto deriving the IMGF of any arbitrary fading distribution.Closed-form expressions for the IMGF of the κ-µ shadoweddistribution are also given, as well as for all the specialcases included therein (κ-µ, η-µ, Rician shadowed, Rician,Nakagami-m, Nakagami-q, Rayleigh and one-sided Gaussian).Then, in Section III these mathematical results are used topresent an IMGF-based approach to the physical-layer securityanalysis in wireless systems, as well as to illustrate howadditional scenarios of interest in wireless communications

can also be analyzed by using the IMGF. Numerical resultsare given in Section IV, whereas the main conclusions areoutlined in Section V.

II. MATHEMATICAL RESULTS

Definition 1 (Lower IMGF): Let X be a non-negativerandom variable and ζ a non-negative real number, the lowerIMGF of X is defined as

MlX(s, ζ) ,

∫ ζ

0

esxfX (x) dx. (1)

Definition 2 (Upper IMGF): Let X be a non-negativerandom variable and ζ a non-negative real number, the upperIMGF of X is defined as

MuX(s, ζ) ,

∫ ∞ζ

esxfX (x) dx. (2)

Obviously, the MGF of X is obtained from these IMGFs asMX(s) =Ml

X(s,∞) =MuX(s, 0). Both incomplete IMGFs

are easily related through the following equation

MuX (s, ζ) =MX (s)−Ml

X (s, ζ) , (3)

In the following Lemma, we present a general expression forthe IMGF of an arbitrarily distributed non-negative randomvariable.

Lemma 1: Let X be a non-negative random variable withMGFMγ(s). Its lower IMGFs can be computed by the inverseLaplace transform of the scaled-shifted MGF, i.e.

MlX (s, ζ) = L−1

1

pMX (s− p) ; p, z = ζ

, (4)

where Lh (z) ; z, p ,∫∞

0e−pzh (z) dz represents the

Laplace transform from the z-domain to the p-domain, andL−1 H (z) ; p, z the inverse Laplace transform from thep-domain to the z-domain.

Proof: Let us consider the upper IMGF in (1) as afunction of the upper integration limit, i.e.

Λ(z) =

∫ z

0

esxfX (x) dx. (5)

The Laplace transform of Λ(z) in the p-domain can beexpressed as

LΛ (z) ; z, p = L∫ z

0

esxfX (x) dx; z, p

=

1

pLeszfX (z) ; z, p

=1

pLfX (x) ; z, p− s =

1

pMX (s− p) ,

(6)

where both the definite integral property and the modulationproperty of the Laplace transform have been applied in orderto complete the proof.

This Lemma provides a general way of computing theIMGF of any fading distribution in terms of the conventionalMGF. Interestingly, (4) involves an inverse Laplace transform

LOPEZ-MARTINEZ et. al: ON THE CALCULATION OF THE INCOMPLETE MGF WITH APPLICATIONS TO WIRELESS COMMUNICATIONS 3

of a shifted version of the MGF, scaled by the Laplacedomain variable p. Thus, it is expectable that the IMGF has afunctional form similar to the CDF, since the CDF arises as aparticular case of the lower IMGF when evaluated in s = 0,i.e.

FX(ζ) =MlX(0, ζ). (7)

This result also suggests that the IMGF of any distributionfor which either the CDF or the MGF are not availablein closed-form, will not be likely to have a closed-formexpression. Otherwise, we would be finding an expression forsuch CDF or MGF as a special case. This is the case of somerelevant fading distributions in the literature like Durgin andRappaport’s Two-Wave with Diffuse Power (TWDP) fadingmodel [37], or the Beckmann distribution [38, 39]. The CDFfor these distributions is only available in integral form, buttheir MGFs have a closed-form expression [4, 5]. Anothervalid example is the lognormal distribution, for which theCDF has a closed-form expression in terms of the GaussianQ-function, but its MGF has been largely unknown [40].

As consequences of Lemma 1, the IMGF of any non-negative RV can be obtained in a very general form by aninverse Laplace transform operation over the MGF. As wewill now show, this Laplace transform can be computed inclosed-form for a number of fading distributions of interest.In the following corollaries (and summarized in Table I),we derive closed-form expressions for the IMGF of the κ-µshadowed distribution and the special cases included therein,namely the Rician shadowed, κ-µ and η-µ distributions. Allthese expressions are new in the literature to the best of ourknowledge.

Corollary 1: Let γ be a κ-µ shadowed random variable withE γ = γ, and non-negative real shape parameters κ, µ andm, i.e, γ ∼ Sκµm(γ;κ, µ,m)1 [7]. Then, its lower IMGF isgiven in the first entry of Table I, where Φ2(·) is the bivariateconfluent hypergeometric function defined in [41, eq. 9.261.2].

Proof: For the sake of clarity let us write the MGF of theκ-µ shadowed distribution as follows

Mγ (s) = (−1)µA(s− a)m−µ(s− b)−m

A =µµmm (1 + κ)

µ

γµ (µκ+m)m

a =µ (1 + κ)

γ

b =µ (1 + κ)

γ

m

µκ+m

. (8)

In order to apply Lemma 1, the scaled-shifted MGF of γ is

1The symbol ∼ reads as statistically distributed as.

first expressed as follows

1

pMγ (s− p) =

A(−1)µ

p(s− p− a)m−µ(s− p− b)−m

=A

p(p− s+ a)m−µ(p− s+ b)−m

=A

p1+µ

(1−

(s− ap

))m−µ(1−

(s− bp

))−m=

A

p1+µ

(1−

(s− ap

))−(µ−m)(1−

(s− bp

))−m.

(9)To perform the inverse Laplace transform the following pair

can be considered [42, pp. 290]

Lxγ−1Φ

(n)2 (β1, ..., βn; γ;λ1x, ..., λnx) ;x, p

=

=Γ (γ)

pγ

(1− λ1

p

)−β1

...

(1− λn

p

)−βn,

Re γ > 0,Re p > max 0,Re λ1 , ...,Re λn .(10)

Joining (9), (10) and Lemma 1 the proof is completed.

It must be noted that the IMGF here obtained has the samefunctional form as the CDF given in [7, eq. 6]; thus, evaluatingthe IMGF has the same complexity as evaluating the CDF.

Corollary 2: Let γ be a Rician shadowed random variablewith E γ = γ, and non-negative real shape parameters Kand m, i.e, γ ∼ SKm(γ;K,m) [22]. Then, its lower IMGF isgiven in the second entry of Table I.

Proof: Specializing the results for the κ-µ shadoweddistribution for µ = 1, the IMGF for the Rician-shadowedcase is obtained with K = κ.

Again, the IMGF of the Rician Shadowed distribution hasthe same functional form as its CDF [43]. We must note thatfor the case of m ∈ Z, we have that Φ2(1 − m,m; 2; ·, ·)function reduces to a finite sum of Laguerre polynomials[43, eq. 4]; similarly, for m being a positive half integer,then Φ2(1 −m,m; 2; ·, ·) function reduces to a finite sum ofKummer hypergeometric functions, modified Bessel functionsand Marcum-Q functions [43, eq. 7].

Corollary 3: Let γ be a κ-µ random variable withE γ = γ and non-negative real shape parameters κ and µ,i.e, γ ∼ Sκµ(γ;κ, µ) [20]. Then, its lower IMGF is given inthe third entry of Table I, where Qµ is the generalized MarcumQ function of µ-th order [5].

Proof: Using the MGF of a κ-µ distributed RV γ in [6],and according to Lemma 1, the lower IMGF can be expressedas

Mlγ (s, z) = L−1

1pMγ(s− p); p, z

= L−1

1p

µµ(1+κ)µ

(µ(1+κ)−γs+γp)µ exp(

µκγs−µκγpµ(1+κ)−γs+γp

); p, z

,

(11)

After some algebra, the terms in (11) can be convenientlyrearranged so the inverse Laplace transform has the following


TABLE ILOWER IMGFS FOR THE κ-µ SHADOWED FADING MODEL AND PARTICULAR CASES INCLUDED THEREIN. UPPER IMGFS CAN BE OBTAINED USING THE

MGFS GIVEN IN [6, 7, 22] AND RESTATED IN THE TABLE, AND THEN USING (3).

Fading model IMGF Mlγ(s, z) and MGF Mγ(s)

κ-µ shadowed Mlγ(s, z) =

µµ−1mm(1+κ)µ

Γ(µ)γµ(µκ+m)mzµ × Φ2

(µ−m,m; 1 + µ;

(s− µ(1+κ)

γ

)z,(s− µ(1+κ)

γm

µκ+m

)z)

Mγ(s) =(−µ)µmm(1+κ)µ

γµ(µκ+m)m

(s−µ(1+κ)

γ

)m−µ(s−µ(1+κ)

γm

µκ+m

)mRician shadowed Ml

γ(s, z) =mm(1+K)γ(K+m)m

z × Φ2

(1−m,m; 2;

(s− (1+K)

γ

)z,(s− (1+K)

γm

K+m

)z)

Mγ(s) = −mm(1+K)

γ(K+m)m

(s− (1+K)

γ

)m−1(s− (1+K)

γm

K+m

)mκ-µ Ml

γ(s, z) =µµ(1+κ)µ

(µ(1+κ)−γs)µ exp(

µκγsµ(1+κ)−γs

)×[1−Qµ

(√2µκ

µ(1+κ)µ(1+κ)−γs ,

√2(µ(1+κ)γ

− s)z

)]

Mγ(s) =µµ(1+κ)µ

(µ(1+κ)−γs)µ exp(

µκγsµ(1+κ)−γs

)η-µ Ml

γ(s, z) = µ2µ−1

2Γ(2µ)γ2µ

((1+η)2

η

)µz2µ × Φ2

(µ, µ; 2µ+ 1;

(s− µ(1+η)

ηγ

)z,(s− µ(1+η)

γ

)z)

Mγ(s) =

(µ2(2+η−1+η)

[(1+η)µ−γs][(1+η−1)µ−γs]

)µ

form:

Mlγ (s, z) = exp (−µκ)

(µ(1+κ)

γ

)µexp

((s− µ(1+κ)

γ

)z)×

L−1

1

pµ+11

1−(µ(1+κ)γ −s)/p

exp(

1pµ2κ(1+κ)

γ

); p, z

,

(12)

This inverse Laplace transform can be calculated in terms ofthe bivariate confluent hypergeometric function Φ3(·) definedin [41, eq. 9.261.3] by using the transform pair in [44, 4.24.9].This yields

Mlγ (s, z) = µµ(1+κ)µzµ

Γ(µ+1)γµ exp (κµ) exp((s− µ(1+κ)

γ

)z)

×Φ3

(1, µ+ 1;

(µ(1+κ)

γ − s)z, µ

2κ(1+κ)γ z

),

(13)

Finally, we make use of the existing connection betweenthe Φ3(1, µ + 1; a; b) function and the generalized MarcumQ-function, which holds for µ ∈ R. Adapting [45, eq. 34] tothe specific case here addressed, we have

Φ3 (1, µ+ 1; a, b)

Γ(µ+ 1)= exp

(a+ b

a

)a−µ

[1−Qµ

(√2 ba ,√

2a

)].

(14)

Combining this equation with (13) yields the final expressiongiven in the third entry of Table I. This completes the proof.

Note that the IMGF is given in terms of the well-knownMarcum Q-function, just like the CDF of the κ-µ distributionoriginally derived by Yacoub [20].

Corollary 4: Let γ be a η-µ random variable withE γ = γ and non-negative real shape parameters η and µ,

i.e, γ ∼ Sηµ(γ; η, µ) [20]. Then, its lower IMGF is given inthe fourth entry of Table I.

Proof: Leveraging the recent connection between the κ-µshadowed distribution and the η-µ distribution [21], the IMGFof the η-µ power envelope in format 1 is obtained2 by settingthe parameters of the κ-µ shadowed distribution to µ = 2µ,κ = 1−η

2η and m = µ.

This expression is coincident with the one obtained in [45],and also has the same complexity as the original CDF for theη-µ distribution derived in [46] .

III. APPLICATIONS

A. Physical Layer Security: single-antenna scenario

Aided by the previous mathematical results, we will nowshow how the IMGF can be used to directly analyze theperformance in wireless scenarios. Specifically, we aim to de-termine the physical layer security of a wireless link betweentwo legitimate peers (Alice and Bob) in the presence of anexternal eavesdropper (Eve) [25, 26]. We first consider thatBob, Eve and Alice are equipped with single-antenna devices;the inclusion of multiple antennas at Bob or Eve will be laterdiscussed in this section. Depending on the characteristics ofthe propagation environment, random fluctuations affecting thedesired and wiretap links need to be modeled with a specificdistribution.

The performance in this scenario can be characterized by

2According to the original definition in [20], this format implies that η ∈[0,∞).


the secrecy capacity CS , defined as

CS , Cb − Ce = log2

(1 + γb1 + γe

)> 0, (15)

where γb and γe are the instantaneous SNRs at Bob andEve, respectively, and Cb and Ce denote the capacities of thecommunication links between Alice and Bob, and betweenAlice and Eve, respectively.

In many cases, Eve’s channel state information (CSI) isnot available at Alice and hence information-theoretic securitycannot be guaranteed. This may be the case on which Eveis a passive eavesdropper, and hence Alice has no way tohave access to Eve’s CSI. Conversely, perfect knowledge ofBob’s CSI by Alice can be assumed. Thus, Alice selects aconstant secrecy rate RS for transmission; in this situation,the outage probability of the secrecy capacity (OPSC) givesthe probability that communication at a certain secrecy rateRS > 0 cannot be securely attained. This metric is computedas follows

PRS , Pr CS 6 RS (16)

= Pr

γb 6

(2RS − 1

)( 2RS

2RS − 1γe + 1

)(17)

= 1− Pr CS > RS . (18)

The probability of strictly positive secrecy capacity can beobtained as a particular case of (16) by setting RS = 0.

Finally, another secrecy performance metric of interest isthe ε-outage secrecy capacity Cε. This is defined as the largestsecrecy rate RS for which the OPSC satisfies (PRS≤ ε), with0 ≤ ε ≤ 1. For a certain ε, this metric is computed as

Cε , supRS :PRS≤ε

RS , (19)

We will first assume that the fading experienced by theeavesdropper can be modeled by the κ-µ shadowed distribu-tion, i.e. γe ∼ Sκµm(γe;κ, µ,m). This distribution [7] is wellsuited to model both line-of sight (LOS) and non-LOS (NLOS)scenarios, and also includes most popular fading distributionsin the literature as special cases. We will also consider thatthe fading severity parameters µ and m take integer values3,which allows for a simpler mathematical tractability.

With this only restriction, and for any arbitrary fadingdistribution at the legitimate link between Alice and Bob, theOPSC can be computed using the following Lemma.

Lemma 2: Let us consider the communication between twolegitimate peers A and B in the presence of an externaleavesdropper E. Let γb and γe be the instantaneous SNRsat B and E, respectively, and γb and γe the average SNRsat B and E, respectively. If γe ∼ Sκµm(γe;κe, µe,me) with

3This can be justified as follows: the parameter µ in the κ-µ distributionintroduced by Yacoub [20] was defined as the number of clusters of multipathwaves propagating in a certain environment; thus, according to this definitionthe consideration of integer µ is related to the physical model for the κ-µdistribution. Equivalently, the restriction of m to take integer values doesnot have a major impact unless the LOS component is affected by heavyshadowing (i.e. very low values of m). In practice, this restriction has anegligible effect, and specially when the κ-µ shadowed distribution is usedto approximate the κ-µ distribution in a more tractable form [47].

µ,m ∈ Z+, then for any finite rate RS ≥ 0 the OPSC canbe expressed in terms of the IMGF of γb as

PRS =

M∑i=0

Cieαβi

mi−1∑r=0

r∑k=0

(−αβi)rk!(r − k)!

∂kMuγb

(s, z)

∂sk

∣∣∣∣s=−βiz=α

+Mlγb

(0, 2RS − 1)

=

M∑i=0

Cieαβi

mi−1∑r=0

r∑k=0


∂kMuγb

(s, z)

∂sk


+ Fγb (α) , (20)

where Fγb(·) is the CDF of γb, α = 2RS−1, βi = 1/(2RSΩi),and the parameters M , mi, Ωi and Ci are related to κ, µ, mand γe as described in Table II in Appendix B.

Proof: See Appendix A.

Expression (20) yields the OPSC in any scenario on whichthe eavesdropper’s fading channel can be modeled with theκ-µ shadowed distribution, for any arbitrary choice of thefading distribution for the legitimate channel. Thence, we canuse the results in Table I combined with (3) to derive analyticalexpressions for the secrecy performance in those scenarios onwhich the fading at the desired link can be modeled by thegeneral κ-µ shadowed distribution, or any of the particularcases included therein. Note that there is no need to restrict theparameters µ and m of the legitimate channel to take integervalues, as the IMGF derived in Table I holds for any µ,m ∈R.

In some scenarios, Eve may only have access to signalsarriving from NLOS paths [48, 49]. Under this premise, wecan assume that the eavesdropper link can be modeled bythe Rayleigh distribution. This leads γe to be exponentiallydistributed with average SNR γe, i.e. γe ∼ Exp(γe). Thus, thefollowing corollary arises as a special case of Lemma 2.

Corollary 5: Let us consider the communication betweentwo legitimate peers A and B in the presence of an externaleavesdropper E. Let γb and γe be the instantaneous SNRs atB and E, respectively, and γb and γe the average SNRs at Band E, respectively. If γe ∼ Exp(γe), then for any finite rateRS ≥ 0 the OPSC can be expressed in terms of the IMGF ofγb as

PRS =Mlγb

(0, 2RS − 1) + e2RS−1

2RS γeMuγb

(− 1

2RS γe, 2RS − 1

)=Fγb

(2RS − 1

)+ e

2RS−1

2RS γeMuγb

(− 1

2RS γe, 2RS − 1

),

(21)

where Fγb(·) is the CDF of γb.

Proof: Following the same derivation in Appendix A andsetting κ = 0 and µ = 1 yields the desired result.

Note that these results provide a systematic way to derivethe OPSC for any arbitrary fading distribution in the legitimatelink, provided that the IMGF of the SNR at Bob is known.We must also note the OPSC for γb γe does not dependon the distribution of γe, but only on the distribution of γb


and the average SNR at Eve γe [31]. Thus, the considerationof Rayleigh fading for the eavesdropper link as performedin Corollary 5 has a negligible effect in practice, whilesimplifying the analysis.

If now assuming Rayleigh fading also for the legitimatechannel, the OPSC expression in (21) reduces to the one orig-inally calculated in [25]. This can be checked by setting κ = 0and µ = 1 in the IMGF of the κ-µ distribution in the thirdentry of Table I. Using the equivalence Q1(0,

√2x) = e−x

given in [5, eq. 4.45], and after some manipulations we obtain

PRS = 1− e−2RS−1γb

γbγb + 2RS γe

. (22)

B. Physical Layer Security: multi-antenna scenario

In the previous analysis, we have explicitly assumed thatall the agents in the system are equipped with single-antennadevices. We here show that the extension for the case on whichEve and Bob are multi-antenna devices can be straightfor-wardly carried out.

First, note that in the derivation carried out in Appendix Ain order to prove Lemma 2, there is neither any restrictionrelated to the number of antennas used by Bob, nor relatedto the combining strategy carried out. Hence, Lemma 2 canbe applied as is for a multi-antenna configuration at Bob.In this case, the only requirement to derive the OPSC is todetermine the IMGF of the SNR after combining. For instance,if maximal ratio combining (MRC) is used by Bob we havethat γb =

∑NBi=1 γbi and

Muγb

(s, z) =

NB∏i=1

Muγbi

(si, zi) , (23)

where NB is the number of receive antennas at Bob, γbidenote the per-branch instantaneous SNRs, and independencebetween receive branches has been considered.

When assuming multiple antennas at Eve with i.i.d.branches under κ-µ shadowed fading, the extension is alsostraightforward when considering that Eve performs MRCreception in order to maximize the receive SNR; since thesum of NE i.i.d. κ-µ shadowed random variables is also κ-µshadowed-distributed with µeq = NE · µ, meq = NE ·m andγe = NE · γe, the OPSC in the multiantenna scenario can beexpressed as in Lemma 2.

C. Outage probability analysis with interference and back-ground noise

The performance characterization of wireless communica-tion systems in the presence of interference is a very importantproblem in communication theory, ever since the advent ofdigital cellular systems, whose performance is known tobe limited by the interference received from nearby cells.Let us denote as γd the instantaneous SNR at the intendedreceiver, and let us denote as γi the aggregate instantaneousinterference-to-noise ratio corresponding to the set of interfer-ing signals affecting the receiver.

The outage probability in this scenario, defined as theprobability that the signal-to-noise plus interference ratio isbelow a given threshold γth, can be calculated as

OPNI = Pr γd 6 γth (γi + 1) . (24)

As pointed out in [33], expressions (24) and (16) are for-mally equivalent, up to some scaling of the random variablesand setting γth = 2RS − 1. This means that both OPNI andPRS will have the same functional form for the same choiceof distributions.

The derivation of the outage probability of communicationsystems in the presence of interference and background noisein a κ-µ shadowed/κ-µ shadowed scenario, which is an openproblem in the literature, is directly obtained by using theIMGF given in the first entry of Table I, under the same con-ditions assumed in Section III. Recent results in the literaturearise as special cases [50].

D. Channel capacity with side information at the transmitterand the receiver

Let us now consider the scenario on which a transmitter,subject to an average transmit power constraint, communicateswith a receiver through a fading channel. Assuming perfectchannel knowledge at both the transmitter and receiver sides,the transmitter can optimally adapt its power and rate. TheShannon capacity in this scenario is known to be given by thefollowing expression [35],

C =

∫ ∞0

log

(γ

γ0

)fγ(γ)dγ, (25)

where a normalized bandwidth B = 1 was assumed forsimplicity. In (25), γ is the instantaneous SNR and γ0 is acut-off SNR determined by the average power constraint. Analternative expression for this capacity was proposed in [2] interms of the Ei-transform, which makes use of the exponentialintegral function Ei(·) as integration kernel, yielding

C =1

log(2)

∫ ∞0

Ei(−x) exp(x)Ψ(x, γ0)dx, (26)

where the ancillary function Ψ(x, γ0) is defined as

Ψ(x, γ0) ,Muγ

(x

γ0, x

)+

1

γ0

∂Muγ (s, z)

∂s

∣∣∣∣s= xγ0

z=x

. (27)

Thus, (26) provides an alternative way of computing thecapacity in this scenario for an arbitrary distribution of theSNR, in terms of the IMGF and its first derivative. Using theexpressions for the IMGF derived in Table I, capacity resultsare obtained for the κ-µ shadowed fading channels, and allthe special cases included therein. These results are also newin the literature.

E. Average bit-error rate with adaptive modulation

Adaptive modulation makes use of channel knowledge at thetransmitter side in order to optimally design system parameterssuch as constellation size, transmit power, coding rates andschemes, and many others [36]. One extended alternative is the


design of the constellation size and power in order to maximizethe average throughput, for a certain instantaneous bit-errorrate (BER) constraint. In this scenario, the average BER Pbof adaptive modulation with M -QAM is well approximatedusing [36, eq. 9.7] and [36, eq. 9.72], as

Pb ≈

N−1∑j=1

kj∫ γjγj−1

0.2 exp(−1.5 γ

2kj−1

)fγ (γ) dγ

N−1∑j=1

kj∫ γjγj−1

fγ (γ) dγ

, (28)

where γ represents the instantaneous SNR, γ is the averageSNR, N is the number of fading regions, γj are the SNRswitching thresholds and kj is the number of bits per complexsymbol employed when γj−1 ≤ γ < γj . Note that thedenominator in (28) represents the exact average spectralefficiency and, for convenience, γN−1 ,∞. Using the IMGFof γ, the following closed-form expression is obtained as

Pb ≈0.2

N−1∑j=1

kj

Ml

γ(− 1.5

2kj−1, γj−1)−Ml

γ(− 1.5

2kj−1, γj)

N−1∑j=1

kjMl

γ(0, γj−1)−Mlγ(0, γj)

≈0.2

N−1∑j=1

kj

Ml

γ(− 1.5

2kj−1, γj−1)−Ml

γ(− 1.5

2kj−1, γj)

N−1∑j=1

kj Fγ(γj−1)− Fγ(γj).

(29)

Therefore, the average BER in this scenario for any fadingdistribution can be easily obtained by evaluating a finite num-ber of terms involving the IMGF. More specifically, closed-form results new in the literature can be obtained for the caseof the κ-µ shadowed fading distribution and special cases.

IV. NUMERICAL RESULTS

In this section we provide numerical results for some of thepractical scenarios previously analyzed. Specifically, we focuson the outage probability of the secrecy capacity studied inSection III under different fading scenarios. We assume thatBob and Eve are only equipped with one antenna, and for theeavesdropper’s channel we set κ = 0 and µ = 1. The effectof system parameters on the ε-outage secrecy capacity is alsoinvestigated. All the results shown here have been analyticallyobtained by the direct evaluation of the expressions developedin this paper: Additionally, Monte Carlo simulations have beenperformed to validate the derived expressions, and are alsopresented in all figures, showing an excellent agreement withthe analytical results. Details on how to compute the confluentbivariate function Φ2 are given in [7, App. E].

In Figs. 1-5, the OPSC is represented considering differentfading models as a function of the average SNR at Bob γb, fordifferent sets of values of the fading parameters. We assumein these figures that the normalized rate threshold value usedto declare an outage is RS = 0.1, and an average SNR at Eveγe = 15 dB.

Figs. 1 and 2, show results for the κ-µ shadowed fadingconsidering, respectively, small (κ = 1.5) and large (κ = 10)

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

γb(dB)

PrC

s≤

Rs

m=0.5

m=12

Simulation

µ= 1, 2, 3, 4

Fig. 1. Outage probability of secrecy capacity under κ-µ shadowed fading asa function of γb, for different values of m and µ. Parameter values: κ = 1.5,γe = 15 dB and RS = 0.1.

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

γb(dB)

PrC

s≤

Rs

m=0.5

m=12

Simulation

µ= 1, 2, 3, 4

Fig. 2. Outage probability of secrecy capacity under κ-µ shadowed fading asa function of γb, for different values of m and µ. Parameter values: κ = 10,γe = 15 dB and RS = 0.1.

LOS components in the received wave clusters for differentvalues of the µ parameter and also considering light (m = 12)or heavy (m = 0.5) shadowing for the LOS components. Asexpected, as the fading parameter µ increases, the diversitygain increases too, resulting in a higher slope of the curvesin the high SNR regime, and with diminishing returns asµ increases. Note that µ represents the number of receivedwave clusters when it takes an integer value. It can also beobserved that the performance is always better when the LOScomponents are lightly shadowed, and this improvement ismuch more noticeable for large LOS components.

The impact of shadowed LOS components on performancecan be observed in Fig. 3, where the outage probability of thesecrecy capacity under Rician shadowed fading is presentedfor different values of the m and K parameters. It can beobserved that it is more beneficial for the performance to havesmall LOS components (K = 1.5) if they are affected byheavy shadowing. Conversely, if the shadowing is mild, large


0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

γb (dB)

PrC

s≤

Rs

K=1.5

K=10

Simulation

m= 0.5, 4 ,12

Fig. 3. Outage probability of secrecy capacity under Rician shadowed fadingas a function of γb, for different values of K and m. Parameter values:γe = 15 dB and RS = 0.1.

0 5 10 15 20 25 30 35 40

10−4

10−3

10−2

10−1

100

γb(dB)

PrC

s≤

Rs

κ = 1.5

κ = 10

Simulation

µ= 1, 2, 3

Fig. 4. Outage probability of secrecy capacity under κ-µ fading as a functionof γb, for different values of κ and µ. Parameter values: γe = 15 dB andRS = 0.1.

LOS components always yield a lower outage probability. Thisappreciation can be confirmed by observing the results in Fig.4, which depicts the outage probability under κ-µ fading, i.e.,when the LOS components do not experience any shadowing.

Fig. 5 presents results for the η-µ fading, i.e.. in a NLOSscenario on which the in-phase and quadrature components ofthe scattered waves are not necessarily equally distributed. Weconsider format 1 of this distribution, for which η representsthe scattered-wave power ratio between the in-phase andquadrature components of each cluster of multipath, and thenumber of multipath clusters, when µ is a semi-integer, isrepresented by 2µ . It can be observed that, when the in-phaseand quadrature components are highly imbalanced (η = 0.04),the performance is poorer. On the other hand, increasing thenumber of multipath clusters have always a beneficial impacton performance, as the instantaneous received signal power issmoothed.

Fig. 6 shows the normalized ε-outage secrecy capacity Cε

0 5 10 15 20 25 30 35 4010

−4

10−3

10−2

10−1

100

γb(dB)

PrC

s≤

Rs

η=0.04

η=0.8

Simulation

µ= 0.5, 1, 3

Fig. 5. Outage probability of secrecy capacity under η-µ fading as a functionof γb, for different values of η and µ. Parameter values: γe = 15 dB andRS = 0.1.

under κ-µ shadowed fading as a function of γb, where thecapacity normalization has been computed with respect to thecapacity of an AWGN channel with SNR equal to γb. We haveassumed m = 2 and an average SNR at Eve of γe = −10 dB.The corresponding results for η-µ fading are presented in Fig.7, also for γe = −10 dB. In both figures it can be observed thatwhen the outage probability is set to a high value (ε = 0.8),better channel conditions (i.e. µ = 6, κ = 10 for κ-µ shadowedfading and µ = 4, η = 0.9 for η-µ fading) yield a lower ε-outage capacity. Conversely, better channel conditions resultsin a higher capacity for lower values of ε. Further insight canbe obtained from Fig. 8, where it is shown the normalizedε-outage secrecy capacity Cε under κ-µ fading as a functionof the outage probability ε and for different average SNRs atEve, assuming an average SNR of 10 dB at the desired link.We observe that higher outage probability ε leads to higherCε, having an important influence the average SNR of theeavesdropper’s channel. It can also be observed that, as thechannel conditions improve, the normalized ε-outage secrecycapacity tends to one for all values of the outage probabilityfor low values of γe.

Note that wireless communication over fading channelsdoes not require necessarily the average SNR of the channelbetween Alice and Bob to be greater than the average SNRof the channel between Alice and Eve, since there is certainprobability that the instantaneous SNR of the main channelbeing higher than the instantaneous SNR of the eavesdropper’schannel (γb > γe) even when γb < γe. In fact, there is atrade off between the outage probability of the secrecy capacityPr CS ≤ RS and the ε-outage secrecy capacity Cε, wherea higher Cε corresponds to a higher outage probability ε, andviceversa. For that reason, results in Figs. 6, 7 and 8 showthat the normalized outage secrecy capacity may have non-zero values even when γb ≤ γe (for high values of ε).

V. CONCLUSIONS

A fundamental connection between the incomplete MGF ofa positive random variable and its complete MGF has been


−20 −10 0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

γb

Norm

alizedC

ǫ

µ=1, κ=1.5

µ=1, κ=10

µ=6, κ=10

Simulation

ǫ= 0,2, 0.4, 0.6, 0.8

Fig. 6. Normalized ε-outage secrecy capacity Cε under κ-µ shadowed fadingas a function of γb. Parameter values; m = 2, γe = −10 dB.

−20 −10 0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

γb(dB)

Norm

alizedC

ǫ

η=0.2, µ=1

η=0.9, µ=1

η=0.9, µ=4

Simulation

ǫ= 0.2, 0.4, 0.6, 0.8

Fig. 7. Normalized ε-outage secrecy capacity Cε under η-µ fading as afunction of γb. Parameter value γe = −10 dB.

presented. The main takeaway is that the IMGF is expectedto have a similar functional form as the CDF, and henceits evaluation should not require any additional complexity.Using this novel connection, closed-form expressions for theIMGF of the κ-µ shadowed distribution (and all the specialcases included therein) have been derived for the first timein the literature. This has enabled us to introduce a newframework for the analysis of the physical layer security inscenarios on which the desired link is affected by any arbitraryfading distribution, and the eavesdropper’s link undergoes κ-µshadowed fading.

We hope that the results in this paper may facilitate theperformance evaluation for more practical setups related tophysical layer security, such as those using artificial noisetransmission or the collaboration of a friendly jammer, and,in general for all the situations and scenarios on which theIMGF makes appearances.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

ǫ

Norm

alizedC

ǫ

µ=1, κ=1.5

µ=1, κ=10

µ=6, κ=10

Simulation

γe= 10, 0, -10 dB

Fig. 8. Normalized ε-outage secrecy capacity Cε under κ-µ fading as afunction of ε. Parameter value γb = 10 dB.

ACKNOWLEDGMENT

The authors would like to thank Unai Fernandez-Plazaolafor checking out the expressions for the κ-µ shadowed fadingdistribution in Appendix B.

APPENDIX APROOF OF LEMMA 2

From the definition of OPSC in (16), the probability ofachieving a successful secure communication is given by

Pr CS > RS = Pr

γe <

1

2RS(1 + γb)− 1

. (30)

Note that γe only takes non-negative values; hence, for thiscondition to occur in (30), then the inequality γb ≥ 2RS − 1must be satisfied, i.e.

Pr

γe <

1

2RS(1 + γb)− 1|γb < 2RS − 1

= 0. (31)

Therefore we can write

Pr CS > RS =

∫ ∞2RS−1

fγb(x)

(∫ 1

2RS(1+x)−1

0

fγe(y)dy

)dx

=

∫ ∞2RS−1

fγb(x)Fγe

(1

2RS(1 + x)− 1

)dx,

(32)

where FX(·) and fX(·) are the CDF and PDF of the randomvariable X , respectively.

Let us first assume that the fading experienced by theeavesdropper’s is modeled by th κ-µ shadowed distribution[7]. For integer values of µ and m, the CDF of the κ-µshadowed distribution can be expressed as a mixture of gammadistributions as described in Appendix B, as follows.

Fγe(γe) = 1−M∑i=0

Cie− γ

Ωi

mi−1∑r=0

1

r!

(γ

Ωi

)r. (33)


TABLE IIPARAMETER VALUES FOR THE κ-µ SHADOWED DISTRIBUTION WITH INTEGER µ AND m,

Case µ > m Case µ ≤ m

M = µ M = m− µ

Ci =

0 i = 0

(−1)m(m+i−2

i−1

)×[

mµκ+m

]m [µκ

µκ+m

]−m−i+10 < i ≤ µ−m

(−1)i−µ+m−1 ( i−2i−µ+m−1

)×[

mµκ+m

]i−µ+m−1 [µκ

µκ+m

]−i+1µ−m < i ≤ µ

Ci =(m−µ

i

) [m

µκ+m

]i [µκ

µκ+m

]m−µ−i

mi =

µ−m− i+ 1, 0 ≤ i ≤ µ−mµ− i+ 1 µ−m < i ≤ µ

mi = m− i

Ωi =

γ

µ(1+κ), 0 ≤ i ≤ µ−m

µκ+mm

γµ(1+κ)

µ−m < i ≤ µΩi = µκ+m

mγ

µ(1+κ)

Plugging (33) in (32) yields

Pr CS > RS = 1− Fγb(2RS − 1)−M∑i=0

Ci exp(− 1−2RS

2RS Ωi

)×∫ ∞

2RS−1

fγb(x)e− x

2RS Ωi

mi−1∑r=0

1

r!

(1 + x− 2RS

2RSΩi

)rdx︸︷︷︸

I

,

(34)

where the parameters M , mi, Ci and Ωi are defined in TableII in the Appendix B in terms of the parameters κ, µ, m and γeof the eavesdropper’s fading distribution. Using the binomialexpansion, and defining α = 2RS − 1, βi = 1/(2RSΩi), theintegral term I can be reexpressed as

I =

mi−1∑r=0

r∑k=0

(−α)r−k

k!(r − k)!βri

∫ ∞α

xkfγb(x)e−βixdx,

=

mi−1∑i=0

r∑k=0

αr−k

k!(r − k)!βri∂kMu

γb(s, z)

∂sk


(35)

where the general derivative property in the transform domainwas used, in order to identify the kth derivative of the IMGF.Finally, using (35) in (34) and (16) yields

PRS =

M∑i=0

Cieαβi

mi−1∑r=0

r∑k=0


∂kMuγb

(s, z)

∂sk


+ Fγb (α) . (36)

This completes the proof.

APPENDIX BCDF OF THE κ-µ SHADOWED DISTRIBUTION FOR INTEGER

FADING PARAMETERS

The CDF of the κ-µ shadowed fading model was originallygiven in [7] as

Fγ(γ) =µµ−1mm (1 + κ)

µ

Γ(µ)γµ (µκ+m)m z

µ (37)

× Φ2

(µ−m,m; 1 + µ;−µ(1+κ)

γ γ,−µ(1+κ)γ

mµκ+mγ

).

If the fading parameters µ and m take integer values, thisCDF can be expressed as a finite mixture of squared Nakagamidistributions, i.e. as a finite sum of exponentials and powers[47]. Manipulating the expressions in [47], we can compactlyexpress the CDF as

Fγ(γ) = 1−M∑i=0

Cie− γ

Ωi

mi−1∑r=0

1

r!

(γ

Ωi

)r, (38)

where the parameters mi, M and Ωi are expressed in TableII in terms of the parameters of the κ-µ shadowed distribution,namely κ, µ, m and γ.

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F. Javier Lopez-Martinez received the M.Sc. andPh.D. degrees in Telecommunication Engineering in2005 and 2010, respectively, from the Universityof Malaga (Spain). He joined the CommunicationEngineering Department at the University of Malagain 2005 as an associate researcher. In 2010 he stayedfor 3 months as a visitor researcher at UniversityCollege London. He was a Marie Curie postdoctoralfellow in the Wireless Systems Lab at StanfordUniversity (2012-2014), and at University of Malaga(2014-2015). Since 2015, he is an Assistant Profes-

sor at the Communication Engineering Department, University of Malaga.He has received several research awards, including the best paper award in

the Communication Theory symposium at IEEE Globecom 2013, the IEEECommunications Letters Exemplary Reviewer certificate in 2014, and theIEEE Transactions on Communications Exemplary Reviewer certificate in2015. His research interests span a diverse set of topics in the wide areas ofCommunication Theory and Wireless Communications: stochastic processes,wireless channel modeling, random matrix theory, physical layer security,massive MIMO and mmWave for 5G.


Juan M. Romero-Jerez holds MSc degrees both inTelecommunication Engineering and in Mathematicsfrom the University of Malaga, Spain, where healso obtained a PhD degree in TelecommunicationEngineering in 2001. In 1996 he joined the Elec-tronic Technology Department at the University ofMalaga, where he is an Associate Professor since2002. He was a Visiting Associate Professor inthe Electrical Engineering Department at StanfordUniversity from September 2005 to February 2006,and also from September 2007 to February 2008,

and from February 2016 to March 2016.He has participated in several research projects in the areas of packet

radio transmission, multiple antennas, interference management and cellularnetworks. His current research interest is in the area of wireless com-munications and, more specifically: wireless communications performanceanalysis, multipath fading, wireless channel modeling, diversity systems, smartantennas, MIMO performance and interference management. He is a SeniorMember of the IEEE, and an Associate Editor of the IEEE Transactions onWireless Communications.

Jose F. Paris received the M.Sc. and Ph.D. degreesin Telecommunication Engineering from the Univer-sity of Malaga, Spain, in 1996 and 2004, respec-tively. From 1994 to 1996 he worked at Alcatel,mainly in the development of wireless telephones.In 1997, he joined the University of Malaga whereis now an associate prodessor in the Communica-tion Engineering Department. His teaching activitiesinclude several courses on digital communications,signal processing and acoustic engineering. His re-search interests are related to wireless communica-

tions, especially channel modeling and performance analysis.In 2005, he spent five months as a Visiting Associate Professor at Stanford

University with Prof. Andrea J. Goldsmith. In 2014, he was credited as Fullprofessor (Catedratico de Universidad) by the Spanish Ministry of Education.He received the 2016 Neil Shepherd Memorial Best Propagation PaperAward by the IEEE Vehicular Technology Society. Currently, he serves asan Associate Editor at IEEE Communications Letters.

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