The κ-µ Shadowed Fading Model with Integer Fading Parameters

1 The κ-µShadowed Fading Model with Integer Fading Parameters F. Javier Lopez-Martinez, Jose F. Paris and Juan M. Romero-Jerez Abstract— We show that the popular and general κ-µshadowed fading model with integer fading parameters µand m can be represented as a mixture of squared Nakagamiˆm(or Gamma) distributions. Thus, its PDF and CDF can be expressed in closed-form in terms of a finite number of elementary functions (powers and exponentials). The main implications arising from such connection are then discussed, which can be summarized as: (1) the performance evaluation of communication systems operating in κ-µshadowed fading becomes as simple as if a Nakagamiˆmfading channel was assumed; (2) the κ-µshadowed distribution can be used to approximate the κ-µdistribution using a closed-form representation in terms of elementary functions, by choosing a sufficiently large value of m; and (3) restricting the parameters µand mto take integer values has limited impact in practice when fitting the κ-µshadowed fading model to field measurements. As an application example, the average channel capacity of communication systems operating under κ-µ shadowed fading is obtained in closed-form. Index Terms— Wireless channel modeling, κ-µshadowed fading, Nakagami fading, probability density function, cumulative distribution function, Gamma distribution. I. INTRODUCTION The κ-µshadowed fading model was introduced in [1], and right after in [2] in an independent work, as a generalization of the popular κ-µmodel proposed by Yacoub [3]. Other generalizations of the κ-µdistribution using the inverse Gamma distribution [4], the inverse Gaussian distribution [5], and a multiplicative composition with the Gamma distribution [6], are also available in the literature. Recently, it was shown that the apparently unrelated η-µdistribution was also a particular case of the κ-µshadowed distribution [7], thus encapsulating this set of popular and general fading distributions in the literature in a single model. Ever since its inception, the κµshadowed fading model has gained a remarkable attention in the literature due to its versatility on modeling propagation conditions ranging from very favorable to worse-than-Rayleigh fading. It also provides an good fit to field measurements in diverse scenarios like device-to-device communications [2], or underwater acoustic channels [1, 8]. Unlike other general fading models [9–11], its chief probability functions (PDF and CDF) are given in closed-form. That This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accesible. F. J. Lopez-Martinez and J. F. Paris are with Departmento de Ingenier´ ıa de Comunicaciones, Universidad de Malaga - Campus de Excelencia Internacional Andaluc´ ıa Tech., Malaga 29071, Spain. Contact e-mail: [email protected]. J. M. Romero-Jerez is with Departmento de Tecnolog´ ıa Eectr´ onica, Universidad de Malaga - Campus de Excelencia Internacional Andaluc´ ıa Tech., Malaga 29071, Spain. being said, the computation of its CDF still requires for the evaluation of the confluent bivariate hypergeometric function Φ2[12, 9.262.2], which is not yet included in commercial mathematical software packages. This fact has not prevented the widespread use of the κ-µshadowed distribution in a number of practical scenarios of interest [13–15]. However, these results have a much more complicated form than their counterparts when assuming, for instance, the simpler and extremely popular Nakagamiˆm(or simply Nakagami) fading model1. In this paper, we show that the κ-µshadowed PDF and CDF can be expressed in terms of a finite number of elementary functions for a proper choice of the fading severity parameter values. Specifically, we show that the κ-µshadowed distribution can be expressed as a mixture of squared Nakagamiˆmdistributions, when the parameters µand mtake integer values. As we will later see, such restriction has little effect in practice when fitting field measurements to the κ-µ shadowed distribution, while being extremely convenient from a computational perspective. This connection considerably facilitates the performance evaluation of communication systems operating in κ-µshadowed fading channels. In fact, we show that any performance metric that is calculated by averaging over the distribution of the SNR in κ-µshadowed fading (e.g. bit error rate, capacity, outage probability...) can be readily and directly obtained as a linear combination of the results obtained when assuming Nakagamiˆmfading; the values of the weights for this linear combination are the coefficients of the mixture, which are given in closed-form. We also show that the κ-µshadowed fading model can be used to approximate the κ-µdistribution with arbitrary precision, by simply choosing a sufficiently large value of m. Thus, the computational benefits of the new representation of the κ-µshadowed fading model in terms of elementary functions can be extended to the κ-µdistribution (and also to the Rician distribution as a special case for µ= 1). As a direct application, we exemplify the usefulness of the results here unveiled to obtain exact expressions for the average capacity of the κ-µshadowed fading channel, which are considerably simpler than those originally obtained in [16]. The remainder of this paper is structured as follows: in Section II, the new expressions for the PDF and the CDF of the κ-µshadowed fading model are given in terms of a finite sum of elementary functions. Then, a number of relevant implications and useful properties arising from these 1In order to avoid any confusion between the mparameter of the κ-µ shadowed fading model and the homonymous parameter of the Nakagamiˆm fading model, we use a mwith a superscript to denote the latter. 2 results are discussed in Section III. An application example is presented in Section IV, whereas the main conclusions are outlined in Section V. II. NEW STATISTICS FOR THE κ-µSHADOWED DISTRIBUTION. Throughout this paper, we will characterize the distribution of the received power envelope in κ-µshadowed fading channels, or equivalently, the instantaneous SNR γat the receiver. Note that characterizing the distribution of the amplitude envelope ris straightforward by a simple change of variables r=√γ. Definition 1: A random variable γfollowing a κ-µshadowed distribution will be denoted as γ∼ S(¯γ;κ, µ, m), and its PDF will be given by fS(¯γ;κ, µ, m;x) = µµmm(1 + κ)µ Γ(µ)(m+µκ)mγx γµ−1 ·e−µ(1+κ) γx1F1m, µ;µ2κ(1 + κ) µκ +m x γ, (1) where 1F1(·)is the confluent hypergeometric function of the first kind [12, eq. (9.210.1)]. Definition 2: A random variable γfollowing a κ-µdistribution will be denoted as γ∼ KM(¯γ;κ, µ), and its PDF will be given by fKM(γ) =µ(1 + κ)µ+1 2 ¯γκµ−1 2eµκ γ ¯γµ−1 2 ×e−µ(1+κ)γ ¯γIµ−1 2µsκ(1 + κ)γ ¯γ!, (2) where Iν(·)is the ν-th order modified Bessel function of first kind. Definition 3: A random variable γfollowing a squared Nakagami distribution will be denoted as γ∼ K(¯γ; ˆm), and its PDF, assuming ˆm∈N, will be given by fK(¯γ; ˆm;x) = ˆm ¯γˆmxˆm−1 ( ˆm−1)!e−xˆm/¯γ.(3) Note that the squared Nakagami distribution is equivalent to a Gamma distribution with shape parameter ˆmand scale parameter ¯γ/ ˆm. After these preliminary definitions, we now provide new expressions for the PDF and CDF of the κ-µshadowed distribution for positive integer values of the fading severity parameters µand m. Theorem 1: Let γbe a random variable such that γ∼ S(¯γ;κ, µ, m)and let µ, m ∈N. Then, γis a mixture of squared Nakagami distributions, which PDF is given as follows: •If m<µ fS(¯γ;κ, µ, m;x) = µ−m X j=1 A1jfK(ωA1;µ−m−j+ 1; x) + m X j=1 A2jfK(ωA2;m−j+ 1; x). (4) •If m≥µ fS(¯γ;κ, µ, m;x) = m−µ X j=0 BjfK(ωB;m−j;x),(5) where A1j= (−1)mm+j−2 j−1 ×m µκ +mmµκ µκ +m−m−j+1 , A2j= (−1)j−1µ−m+j−2 j−1 ×m µκ +mj−1µκ µκ +mm−µ−j+1 , Bj=m−µ j m µκ +mjµκ µκ +mm−µ−j , (6) and where we have defined ωA1,∆1(µ−m−j+ 1) , ωA2,∆2(m−j+ 1) , ωB,∆2(m−j), (7) with ∆1,¯γ µ(1 + κ), ∆2,µκ +m m ¯γ µ(1 + κ). (8) Proof: See Appendix I. Expressions (4) and (5) give an exact representation of the κ-µshadowed distribution in terms of a finite mixture of squared Nakagami distributions. With this representation of the PDF, which is new in the literature to the best of our knowledge, a similar result for the κ-µshadowed CDF is now obtained. Corollary 1: Let γbe a random variable such that γ∼ S(¯γ;κ, µ, m)and let µ, m ∈N, then, the CDF of γis given as follows: •If m<µ FS(¯γ;κ, µ, m;x) = 1 − µ−m X j=1 A1je−x/∆1 µ−m−j X r=0 1 r!x ∆1r − m X j=1 A2je−x/∆2 m−j X r=0 1 r!x ∆2r . (9) 3 TABLE I PARAMETER VALUES FOR THE κ-µSHADOWED DISTRIBUTION WITH INTEGER µAND m, Case µ>m Case µ≤m M=µ M =m−µ Ci=       0i= 0 (−1)mm+i−2 i−1×hm µκ+mimhµκ µκ+mi−m−i+1 0< i ≤µ−m (−1)i−µ+m−1i−2 i−µ+m−1×hm µκ+mii−µ+m−1hµκ µκ+mi−i+1 µ−m<i≤µ Ci=m−µ ihm µκ+miihµκ µκ+mim−µ−i mi=(µ−m−i+ 1,0≤i≤µ−m µ−i+ 1 µ−m<i≤µmi=m−i Ωi=(¯γ µ(1+κ),0≤i≤µ−m µκ+m m ¯γ µ(1+κ)µ−m<i≤µΩi=µκ+m m ¯γ µ(1+κ) •If m≥µ FS(¯γ;κ, µ, m;x)=1− m−µ X t=0 Bje−x/∆2 m−j−1 X r=0 1 r!x ∆2r . (10) Proof: Using Theorem 1 and considering that the CDF of a squared Nakagami random variable is given by FK(¯γ; ˆm;x)=1−e−x/∆ ˆm−1 X r=0 1 r!x ∆r,(11) where we have defined ∆,¯γ/ ˆm, the proof is completed. A more compact and unified form for the PDF and the CDF of the κ-µshadowed fading distribution with integer fading parameters can be obtained after some manipulation, yielding fS(¯γ;κ, µ, m;x) = M X i=0 Ci xmi−1 (mi−1)! 1 Ωmi i e−x Ωi,(12) and FS(¯γ;κ, µ, m;x)=1− M X i=0 Cie−x Ωi mi−1 X r=0 1 r!x Ωir ,(13) where the parameters mi,Mand Ωiare expressed in Table I in terms of the parameters of the κ-µshadowed distribution, namely κ,µ,mand ¯γ. III. DISCUSSION After presenting the new results for the statistics of the κ-µ shadowed distribution with integer fading parameters, we now discuss about the main implications and insights that can be obtained from these results. A. Finite mixture representation Taking a deeper look at Theorem 1, one can observe that the expression for the mixture of squared Nakagami distributions has different form depending on whether mis larger or smaller than µ. When m≥µthe κ-µshadowed distribution is expressed as a proper mixture2of squared Nakagami distributions, on which the values of the weights Bjin (6) correspond to those of the binomial distribution, whose probability mass function is given by f(k;n, p) = Pr{X=k}=n kpk(1 −p)n−k,(14) with n=m−µand p=m µκ+m. In this situation, the κ-µ shadowed fading model can be regarded as a superposition of m−µparallel channels affected by Nakagami fading with different fading severities, being each of such channels used with a given probability3p. Conversely, for m<µthe PDF in (4) is given in terms of an improper mixture; i.e., (4) is expressed as a linear combination of two different sets of squared Nakagami distributions with coefficients A1jand A2j, on which the mixture coefficients are not necessarily non-negative. Another important insight arises by inspecting eq. (28) in the Appendix. We note that such MGF is expressed as the product of the MGFs of two Gamma distributions with scale parameters ∆1and ∆2, and shape parameters µ−mand m, respectively. Thus, a κ-µshadowed RV can be generated as the sum of two independent gamma RVs with such scale and shape parameters, provided that m < µ. Note that this holds for any {m, µ} ∈ R+. For the specific case on which the κµshadowed distribution reduces to the η-ˆµdistribution, i.e. µ= 2ˆµand m= ˆµwith κ= (1 −η)/2η[7], this observation coincides with the one given in [17]. 2We use the term proper mixture to denote any mixture distribution which can be expressed as a convex combination (i.e. a weighted sum with nonnegative coefficients that sum to 1) of other distributions. We also use the term improper mixture to denote any mixture distribution on which the mixture coefficients are not restricted to be non-negative. 3We must here note that this phenomenon is purely mathematical, as it arises from the observation of the new form of the PDF here derived. To the best of our knowledge, such observation does not have any connection with the physical model of the κ-µshadowed fading distribution (note that the physical models that originate this distribution [7] can be regarded as coherent combinations, in a maximal ratio combining form, of µRician shadowed variates). 4 01234 0 0.2 0.4 0.6 0.8 1 1.2 γ fγ(γ) m= 1 m= 2 m= 5 m= 10 m= 20 m= 100 m= 200 m→ ∞ Fig. 1. Convergence in distribution between the κ-µshadowed distribution and the κ-µdistribution as m→ ∞. Parameter values κ= 5,µ= 3 and ¯γ= 1. B. Convergence to the κ-µdistribution Intuitively, if we let m→ ∞ in the κ-µshadowed fading model, then the Nakagami-mPDF used to model the random fluctuation of the line-of-sight component degenerates to a deterministic distribution, being its PDF the Dirac delta function. Thus, this model reduces to the original κ-µdistribution in [3]. This implies that by virtue of Theorem 1, the κ-µshadowed distribution with integer fading parameters and sufficiently large mcan be used to approximate the κ-µdistribution. This is observed in Fig. 1, where the evolution of the κ-µshadowed distribution as mgrows is represented. However, a rigorous proof for the convergence in distribution between the κ-µshadowed and the κ-µfading models is not that evident. The following Lemmas formally establish the connections between the κ-µshadowed distribution and the κ-µdistribution in terms of weak convergence of probability measures. Lemma 1: Let {γm}∞ m=1 be a sequence of random variables with γm∼ S(¯γ;κ, µ, m)and the corresponding sequence of CDFs given by {FS(¯γ;κ, µ, m;γ)}∞ m=1. Then, this sequence of random variables weakly converges to the κ-µdistribution with mean ¯γand shaping parameters κand µ, i.e. lim m→∞ FS(¯γ;κ, µ, m;γ) = FKM(¯γ;κ, µ;γ),(15) where FKM represents the CDF of the corresponding κ-µ distribution. Proof: It is a direct consequence of the L´ evy’s continuity theorem [18]. Lemma 1 establishes that the κ-µshadowed distribution converges to the κ-µdistribution for sufficiently large m. Thus, it is possible to approximate the κ-µdistribution with integer µby a finite mixture of squared Nakagami distributions, by using the κ-µshadowed fading distribution with integer fading parameters and choosing an arbitrarily large parameter m∈N. These connections between the κ-µshadowed distribution and the κ-µdistribution can be extended to any probability measure obtained by expectation over the desired distribution, as stated in the following Lemma. Lemma 2: Let {γm}∞ m=1 be a sequence of random variables with γm∼ S(¯γ;κ, µ, m)and the corresponding sequence of CDFs given by {FS(¯γ;κ, µ, m;γ)}∞ m=1. Let Φ(γ)be any probability measure conditioned to γ, which is continuous and bounded as a function of γ. Then, the sequence of the expectations of Φover the κ-µshadowed random variables converges to the expectation of Φover a κ-µrandom variable with mean ¯γand shaping parameters κand µ, i.e. lim m→∞ Z∞ 0 Φ(γ)dFS(¯γ;κ, µ, m;γ) = Z∞ 0 Φ(γ)dFKM(¯γ;κ, µ;γ), (16) where FKM represents the CDF of the corresponding κ-µ distribution. Proof: It is a direct consequence of Lemma 1 and the Helly-Bray theorem [19]. C. A new gamma approximation to the Rician distribution In the milestone paper by Nakagami [20], a simple equivalence between the Nakagamiˆmdistribution and the Rician distribution was proposed. The connection between these distributions is established by a simple parameter transformation, setting ˆm= (1 + K)2/(1 + 2K). This approximation, also included in the reference textbook by Simon and Alouini [21, eq. (2.26)], has been widely employed in wireless communications in order to approximate the Rician distribution by the more tractable Nakagamiˆmdistribution. However, as argued by many authors [22, 23] such approximation has a severe flaw that has important impact when analyzing the system performance: the diversity order of Rician fading equals one, whereas the diversity order of Nakagamiˆmfading is ˆm. The diversity order is related to the behavior of the PDF around the origin, or equivalently to the asymptotic behavior of the MGF as s→ ∞. As consequences of Lemma 1, and setting µ= 1, we propose to approximate the Rician distribution by a finite mixture of squared Nakagami distributions, by using the Rician shadowed fading distribution and choosing an arbitrarily large parameter m∈N. The main benefit of this approximation relies on the fact that the diversity order of the Rician shadowed distribution is also 1. In Figs. 3 and 2, the behavior of the classical approximation to the Rician distribution proposed by Nakagami, and the one here proposed based on the Rician shadowed distribution with integer mis illustrated, representing the corresponding PDFs for different values of Kand m. Log-log scale was used in order to better observe the effect of increasing m. We can see that for low values of K, the approximation to the Rician distribution based on the Rician shadowed distribution is good even for moderate values of m. As Kgrows, a larger number of terms is required for the mixture approximation (i.e., a larger m) in order to converge to the Rician distribution. In both cases, the smoothness of the PDFs in the proximity of zero for the Rician and Rician shadowed distributions have similar shape, whereas the original approximation in [20] exhibits a very different behavior. 5 10−210−1100 10−2 10−1 100 γ fγ(γ) Rice m= 5 m= 20 m= 100 Nakagamiˆm Fig. 2. Gamma approximation to the Rician distribution with parameter Kusing the Rician shadowed distribution with parameter Kand integer m. Parameter values K= 3 and ¯γ= 1. Nakagami approximation [20] uses ˆm= (1 + K)2/(1 + 2K)=2.28. 10−210−1100 10−5 10−4 10−3 10−2 10−1 100 γ fγ(γ) Rice m= 5 m= 20 m= 100 Nakagamiˆm Fig. 3. Gamma approximation to the Rician distribution with parameter Kusing the Rician shadowed distribution with parameter Kand integer m. Parameter values K= 10 and ¯γ= 1. Nakagami approximation [20] uses ˆm= (1 + K)2/(1 + 2K)=5.76. D. Effect of considering integer fading parameters The parameter µ∈Nwas originally introduced by Yacoub [3] as the number of clusters of multipath waves that propagate in a non-homogeneous environment. As argued in [3], the restriction of the parameter µto take integer values is inherently linked to the underlying physical model for the κ-µdistribution. Thus, the η-µand κ-µfading models with integer µare usually referred to as physical models, and are often used to evaluate the performance of communication systems operating over generalized fading channels [17, 24– 28]. The consideration of a real-valued µindeed yields a larger flexibility to the model; however, the impact of considering an integer µdecreases as µgrows as observed in Fig. 4. 01234 0 0.2 0.4 0.6 0.8 1 µ= 1 µ= 5 γ fγ(γ) µ= 1.5 µ= 2.5 µ= 3.5 µ= 4.5 Fig. 4. Evolution of the κ-µshadowed fading PDF for different values of µ. Solid lines correspond to real-valued µ∈ {1.5,2.5,3.5,4.5}, whereas dashed lines correspond to the largest previous and the smallest following integer values µ∈ {1:5}. Parameter values κ= 6,m= 2 and ¯γ= 1. 01234 0 0.2 0.4 0.6 0.8 1 m= 1 m= 6 γ fγ(γ) m= 1.5 m= 2.5 m= 3.5 m= 4.5 m= 5.5 Fig. 5. Evolution of the κ-µshadowed fading PDF for different values of m. Solid lines correspond to real-valued m∈ {1.5,2.5,3.5,4.5,5.5}, whereas dashed lines correspond to the largest previous and the smallest following integer values m∈ {1:6}. Parameter values κ= 6,µ= 2 and ¯γ= 1. The restriction of the parameter mto take integer values only has a non-negligible impact in heavy shadowing environments (i.e. low values of m). However, as mgrows the PDFs corresponding to the real-valued mand its closest integer (i.e., largest previous or smallest following integer) counterpart tend to be indistinguishable; this is observed in Fig. 5. Thus, from the observation of Figs. 4 and 5 we see that the effect of restricting the fading severity parameters to take integer values is limited, unless mor µtake low values (i.e., severe LOS fluctuation or severe multipath, respectively). In fact, one may wonder whether there’s a lower value of m(or equivalently µ) under which the use of the κ-µshadowed fading distribution with integer fading parameters to approximate 6 the κ-µshadowed fading distribution with arbitrary fading parameters is not recommended. In our view, this will strongly depend on the application. Sometimes only the tail of the distributions is needed for further calculations, or even the performance metrics of interest end up being rather similar when some fading severity parameters vary. Thus, it may occur that the approximation error is still negligible for low values of mand µin some cases; this will be later exemplified in Section IV. We will now study the impact of restricting the fading severity parameters µand mto take integer values on the goodness of fit to field measurements. We use the empirical results presented in [8] for some underwater acoustic channels (UAC), for which the κ-µshadowed fading model showed the best fit. These measurements were conducted in the Mediterranean Sea near Cartagena (Spain), in shallow waters with depths in the range of 14-30m. Details on the specific measurement configuration, including a block diagram of the measurement equipment set-up, can be found in [8, Sect. 3.1]. We used a modified version of the Kolmogorov-Smirnov (KS) statistic in order define the error factor that is used to quantify the goodness of fit between the empirical and theoretical CDFs, which are denoted by ˆ Fr(·)and Fr(·) respectively, i.e, ,max x|log10 ˆ Fr(x)−log10 Fr(x)|.(17) As in [8], the CDF is used in log-scale with the aim of outweighing the fit in those values of received power closer to zero, i.e. those corresponding to a more severe fading. In Figs 6 and 7 we compare the set of measurements corresponding to the channels C9-32 and C9-64 with three different distributions: Rician, κ-µshadowed with integer fading parameters, and κ-µshadowed. For the C9-32 channel, we observe that the error factor increases from = 0.026 to = 0.083 when constraining µand mto take integer values. However, the fit is still improved when comparing to Rician fading, which yields = 0.111. With regard to the C9-64 channel, we now observe that the error factor is barely increased (i.e. = 0.060 instead of = 0.053) when forcing µand mto take integer values, compared to the general κ-µshadowed fading. For the case of Rician fading, we have = 0.104, which is clearly outperformed by the κ-µshadowed model with integer fading parameters. E. Performance analysis The calculation of performance metrics of interest such as error probability or channel capacity requires to integrate over the PDF or CDF of the SNR. Using the general representation of the κ-µshadowed fading distribution implies integrating over the hypergeometric functions that define its PDF and CDF. However, with the new representation for the κ-µ shadowed distribution functions, the performance analysis is greatly simplified. In the following Lemma, we show that analyzing the performance in κ-µshadowed fading has the same complexity as analyzing the much simpler Nakagamiˆm case. −30 −25 −20 −15 −10 −5 0 10−4 10−3 10−2 10−1 100 Normalized power envelope (dB) CDF Measured Rician κ-µshad (int) κ-µshad Fig. 6. Empirical vs theoretical CDFs of the received signal power for the UAC channel C9-32 [8]. Parameter values: Rician {K= 2.64; = 0.111}, κ-µshadowed int {κ= 12.84, µ = 1, m = 2; = 0.083};κ-µshadowed {κ= 4.06, µ = 1.13, m = 2.45; = 0.026}. −30 −25 −20 −15 −10 −5 0 10−3 10−2 10−1 100 Normalized power envelope (dB) CDF Measured Rician κ-µshad (int) κ-µshad Fig. 7. Empirical vs theoretical CDFs of the received signal power for the UAC channel C9-64 [8]. Parameter values: Rician {K= 0.03; = 0.104}, κ-µshadowed int {κ= 0.6, µ = 1, m = 6; = 0.060};κ-µshadowed {κ= 0.03, µ = 1.02, m = 6.32; = 0.053}. Lemma 3: Let h(γ)be a performance metric depending on the instantaneous SNR γ, and let hK(γ;m)be the metric in Nakagami fading with average SNR γobtained by averaging over an interval of the PDF of the SNR, i.e., hK(γ;m) = Zb a h(x)fK(¯γ;m;x)dx, (18) with 0≤a < b ≤ ∞. Then, the average performance metric in κ-µshadowed fading channels with average SNR γ, denoted 7 as hS(¯γ;κ, µ, m), can be calculated, given that µ, m ∈N, as hS(¯γ;κ, µ, m) = M X i=0 CihK(Ωi/mi;mi),(19) Proof: The average performance metric hS(¯γ;κ, µ, m) is calculated as hS(¯γ;κ, µ, m) = Zb a h(x)fS(¯γ;κ, µ, m;x)dx. (20) We can express fS(¯γ;κ, µ, m;x)as a mixture of squared Nakagami distributions, given in (12). Thus, combining (3) with (12) we have fS(¯γ;κ, µ, m;x) = M X i=0 CifK(Ωi/mi;mi;x).(21) Introducing (21) into (20), after some simple manipulation, (19) is obtained. The implications of this lemma are of great importance, as it means that any performance metric for which existing results are available for the Nakagamiˆmcase can be directly generalized to the κ-µshadowed case by means of a finite linear combination as described in (18). We will exemplify the usefulness of this Lemma in the next Section. IV. APPLICATION: AVERAGE CHANNEL CAPACITY The characterization of the average channel capacity in fading channels, defined as ¯ C[bps/Hz],Z+∞ 0 log2(1 + γ)fγ(γ)dγ, (22) where γis the instantaneous SNR at the receiver side, is a classical problem in communication theory [29–31]. The average channel capacity assuming κ-µshadowed fading was derived in [16], in terms of the unwieldy bivariate Meijer-G function. However, a direct application of Lemma 3 using the average channel capacity expression for Nakagamiˆmfading channels [32, eq. (23)] yields the following simple closed-form expression: C= log2(e) M X i=0 Cie1/Ωi mi−1 X k=0 Γ (−k, 1/Ωi) Ωk i ,(23) where Γ(·)is the upper incomplete Gamma function, which can be computed, when the first parameter is a negative integer, as [12, eq. (8.352.3)] Γ(−n, x) = (−1)n n!"−Ei (−x)−e−x n−1 X r=0 (−1)rr! xr+1 r#, (24) and Γ(0, x) = −Ei (−x),(25) where Ei(·)is the exponential integral function [12, eq. (8.211.1)]. In the next set of figures, the effect of the fading severity parameters µand mon the average capacity is investigated. Firstly, Figs. 8 and 9 illustrate the effect of min different 0 5 10 15 20 25 30 0 2 4 6 8 10 ¯γ(dB) ¯ C m= 1 m= 3 m= 5 m= 10 AWGN Fig. 8. Average channel capacity vs. average SNR, for different values of m. Parameter values are κ= 10 and µ= 3. The AWGN case is included as a reference. 0 5 10 15 20 25 30 0 2 4 6 8 10 ¯γ(dB) ¯ C m= 1 m= 3 m= 5 m= 10 AWGN Fig. 9. Average channel capacity vs. average SNR, for different values of m. Parameter values are κ= 1 and µ= 3. The AWGN case is included as a reference. conditions: strong LOS (κ= 10) and weak LOS (κ= 1), assuming a fixed value of µ= 3. In general terms, a larger m is translated into a larger capacity for a given SNR. However, the effect of mis much more pronounced in the strong LOS scenario, leading to a more severe performance degradation for lower m(i.e. heavy shadowing in the LOS component.). Conversely, in the weak LOS scenario the effect of mis barely noticeable. In Figs. 10 and 11, the effect of µis investigated in the same conditions as in the previous case: strong LOS (κ= 10) and weak LOS (κ= 1), assuming a fixed value of m= 3. We see that in the strong LOS scenario, increasing µhas little effect as the performance is dominated by the LOS component. Conversely, increasing the number of clusters µin the weak LOS scenario is translated into a better performance. 8 0 5 10 15 20 25 30 0 2 4 6 8 10 ¯γ(dB) ¯ C µ= 1 µ= 3 µ= 5 µ= 10 AWGN Fig. 10. Average channel capacity vs. average SNR, for different values of µ. Parameter values are κ= 10 and m= 3. The AWGN case is included as a reference. 0 5 10 15 20 25 30 0 2 4 6 8 10 ¯γ(dB) ¯ C µ= 1 µ= 3 µ= 5 µ= 10 AWGN Fig. 11. Average channel capacity vs. average SNR, for different values of µ. Parameter values are κ= 1 and m= 3. The AWGN case is included as a reference. V. CONCLUSIONS The statistical characterization of the κ-µshadowed model with integer fading parameters was here presented. Remarkably, the PDF and CDF of this very general model can be expressed in closed-form in terms of a finite sum of powers and exponentials, in the form of a mixture of Gamma distributions. The inherent mathematical complexity of the κµshadowed fading model is greatly reduced at the expense of a limited restriction in terms of flexibility. Most notably, the performance evaluation of communication systems operating under this fading channel model can be directly evaluated if existing results are available for the simpler Nakagamiˆmfading model. Thus, considering the κ-µshadowed fading model implies no additional complexity while allowing for a much larger flexibility in terms of propagation conditions, naturally including both LOS and NLOS scenarios. VI. ACKNOWLEDGEMENT This work has been funded by the Consejer´ ıa de Econom´ ıa, Innovaci´ on, Ciencia y Empleo of the Junta de Andaluc´ ıa, the Spanish Government and the European Fund for Regional Development FEDER (projects P2011-TIC-7109, P2011-TIC8238, TEC2013-42711-R, TEC2013-44442-P and TEC201457901-R). The first author would like to thank Marco di Renzo for enlightening discussion about the diversity order of the approximations to the Rician distribution. APPENDIX I PROOF OF THEOREM I The MGF of a κ-µshadowed random variable is given by [1] M(s) = (−µ)µmm(1 + κ)µ ¯γµ(µκ +m)ms−µ(1+κ) ¯γm−µ s−µ(1+κ) ¯γ m µκ+mm,(26) which can be written, in terms of ∆1and ∆2defined in (8), as M(s) = (1 −∆1s)m−µ (1 −∆2s)m.(27) The PDF is related to the MGF by the inverse Laplace transform L−1{M(−s); s;x}, therefore, it is clear that its analytical expression will depend on the exponent of the numerator being positive (or zero), i.e. m≥µ, or negative, i.e. m<µ. Let us consider m < µ. The MGF can be rewritten in a more convenient way for this case as M(s) = 1 (1 −∆1s)µ−m 1 (1 −∆2s)m,(28) and performing a partial fraction expansion we obtain M(s) = µ−m X j=1 A1j (1 −∆1s)µ−m−j+1 + m X j=1 A2j (1 −∆2s)m−j+1 , (29) where A1j= (−1)j−1m+j−2 j−1∆j−1 2∆m 1 (∆1−∆2)m+j−1, A2j= (−1)j−1µ−m+j−2 j−1∆j−1 1∆µ−m 2 (∆2−∆1)µ−m+j−1. (30) By plugging the defined ∆1and ∆2given in (8) into (30), the expressions of coefficients A1jand A2jgiven in (6) are obtained after some algebraic manipulation. Performing now an inverse Laplace transformation to the MGF as given in (29), (4) is obtained. Let us now consider m≥µand let us rewrite the MGF expression given in (27) as M(s) = ∆m−µ 1 ∆m 21 ∆1−sm−µ 1 ∆2−sm.(31) 9 The PDF can be obtained from the MGF given in (31) and by employing the derivative and the modulation properties of the Laplace transformation as follows: fS(¯γ;κ, µ, m;x) = L−1{M(−s) ; s.x} =∆m−µ 1 ∆m 2L−1     1 ∆1+sm−µ 1 ∆2+sm;s.x     =e−x/∆1∆m−µ 1 ∆m 2L−1   sm−µ 1 ∆2−1 ∆1+sm;s.x   =e−x/∆1∆m−µ 1 ∆m 2 ∆m 3 dm−µ dxm−µL−11 (1 −∆3s)m;s.x =e−x/∆1∆m−µ 1 ∆m 2(m−1)! dm−µ dxm−µxm−1ex/∆3, (32) where we have defined ∆3,∆1∆2 ∆2−∆1 .(33) From the Leibniz derivative rule we can write fS(¯γ;κ, µ, m;x) = e−x/∆1∆m−µ 1 ∆m 2(m−1)! m−µ X r=0 m−µ r ×dr dxrxm−1dm−µ−r dxm−µ−rex/∆3 =e−x/∆1∆m−µ 1 ∆m 2(m−1)! m−µ X t=0 m−µ r ×(m−1)! (m−1−r)!xm−1−r1 ∆m−µ−r 3 ex/∆3. (34) where we have considered that r≤m−1to calculate the rth order derivative of xm−1, thus fS(¯γ;κ, µ, m;x) = m−µ X t=0 m−µ r∆m−µ 1 ∆r 2∆m−µ−r 3 ×1 (m−1−r)!∆m−r 2 xm−1−re−x/∆2. (35) From the definitions of ∆1,∆2and ∆3given in (8) and (33), and noting that Bj=m−µ r∆m−µ 1 ∆r 2∆m−µ−r 3 ,(36) after some algebraic manipulation, (5) is finally obtained. REFERENCES [1] J. F. Paris, “Statistical Characterization of κ-µShadowed Fading,” IEEE Trans. Veh. Technol., vol. 63, no. 2, pp. 518–526, Feb 2014. [2] S. L. Cotton, “Human Body Shadowing in Cellular Device-to-Device Communications: Channel Modeling Using the Shadowed κ-µFading Model,” IEEE J. Sel. Areas Commun., vol. 33, no. 1, pp. 111–119, Jan 2015. [3] M. Yacoub, “The κ-µdistribution and the η-µdistribution,” IEEE Antennas Propag. Mag., vol. 49, no. 1, pp. 68–81, Feb 2007. [4] S. K. Yoo, S. L. Cotton, P. C. Sofotasios, M. Matthaiou, M. Valkama, and G. K. Karagiannidis, “The κ-µ/inverse gamma fading model,” in 2015 IEEE 26th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Aug 2015, pp. 425–429. [5] P. C. Sofotasios, T. A. Tsiftsis, K. H. Van, S. Freear, L. R. Wilhelmsson, and M. Valkama, “The κ-µ/Ig Composite Statistical Distribution in RF and FSO Wireless Channels,” in 2013 IEEE 78th Vehicular Technology Conference (VTC Fall), Sept 2013, pp. 1–5. [6] S. K. Yoo, S. L. Cotton, P. C. Sofotasios, and S. Freear, “Shadowed Fading in Indoor Off-Body Communication Channels: A Statistical Characterization Using the κ-µ/Gamma Composite Fading Model,” IEEE Trans. Wireless Commun., vol. 15, no. 8, pp. 5231–5244, Aug 2016. [7] L. Moreno-Pozas, F. J. Lopez-Martinez, J. F. Paris, and E. MartosNaya, “The κ-µshadowed fading model: Unifying the κ-µand η-µ distributions,” IEEE Trans. Veh. Technol., vol. 65, no. 12, pp. 9630– 9641, Dec 2016. [8] F. J. Ca˜ nete, J. L´ opez-Fern´ andez, C. Garc´ ıa-Corrales, A. S´ anchez, E. Robles, F. J. Rodrigo, and J. F. Paris, “Measurement and Modeling of Narrowband Channels for Ultrasonic Underwater Communications,” Sensors, vol. 16, no. 2, p. 256, 2016. [9] G. D. Durgin, T. S. Rappaport, and D. A. de Wolf, “New analytical models and probability density functions for fading in wireless communications,” IEEE Trans. Comm., vol. 50, no. 6, pp. 1005–1015, June 2002. [10] J. Salo, H. M. El-Sallabi, and P. Vainikainen, “Statistical analysis of the multiple scattering radio channel,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3114–3124, Nov 2006. [11] M. Rao, F. J. Lopez-Martinez, M.-S. Alouini, and A. Goldsmith, “MGF Approach to the Analysis of Generalized Two-Ray Fading Models,” IEEE Trans. Wireless Commun., vol. 14, no. 5, pp. 2548–2561, May 2015. [12] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 7th ed. Academic Press Inc, 2007. [Online]. Available: http://www.worldcat.org/isbn/012294755X [13] S. Kumar, “Approximate Outage Probability and Capacity for κ-µ Shadowed Fading,” IEEE Wireless Commun. Lett., vol. 4, no. 3, pp. 301–304, June 2015. [14] G. Chandrasekaran and S. Kalyani, “Performance Analysis of Cooperative Spectrum Sensing Over κ-µShadowed Fading,” IEEE Wireless Commun. Lett., vol. 4, no. 5, pp. 553–556, Oct 2015. [15] M. R. Bhatnagar, “On the Sum of Correlated Squared κ-µShadowed Random Variables and Its Application to Performance Analysis of MRC,” IEEE Trans. Veh. Technol., vol. 64, no. 6, pp. 2678–2684, June 2015. [16] C. Garc´ ıa-Corrales, F. J. Ca˜ nete, and J. F. Paris, “Capacity of κ-µ shadowed fading channels,” Int J Antennas Propag., vol. 2014, 2014. [17] N. Y. Ermolova and O. Tirkkonen, “The η-µFading Distribution with Integer Values of µ,” IEEE Trans. Wireless Commun., vol. 10, no. 6, pp. 1976–1982, June 2011. [18] B. E. Fristedt and L. F. Gray, A modern approach to probability theory. Springer Science & Business Media, 1996. [19] P. Billingsley, Convergence of probability measures. John Wiley & Sons, 1999. [20] M. Nakagami, “The m-distribution: A general formula of intensity distribution of rapid fading,” Statistical Method of Radio Propagation, 1960. [21] M. K. Simon and M.-S. Alouini, Digital communication over fading channels. Wiley-IEEE Press, 2005. [Online]. Available: http://www.worldcat.org/isbn/0471649538 [22] Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, pp. 1389–1398, Aug 2003. [23] N. C. Beaulieu and S. A. Saberali, “A generalized diffuse scatter plus line-of-sight fading channel model,” in 2014 IEEE International Conference on Communications (ICC), June 2014, pp. 5849–5853. [24] D. Morales-Jimenez and J. F. Paris, “Outage probability analysis for ηµfading channels,” IEEE Commun. Lett., vol. 14, no. 6, pp. 521–523, June 2010. [25] K. P. Peppas, G. C. Alexandropoulos, and P. T. Mathiopoulos, “Performance Analysis of Dual-Hop AF Relaying Systems over Mixed η-µand κ-µFading Channels,” IEEE Trans. Veh. Technol., vol. 62, no. 7, pp. 3149–3163, Sept 2013. [26] N. Y. Ermolova and O. Tirkkonen, “Outage Probability Analysis in Generalized Fading Channels with Co-Channel Interference and Background Noise: η-µ/η-µ,η-µ/κ-µand κ-µ/η-µScenarios,” IEEE Trans. Wireless Commun., vol. 13, no. 1, pp. 291–297, January 2014. [27] T. A. Tsiftsis, F. Foukalas, G. K. Karagiannidis, and T. Khattab, “On the Higher Order Statistics of the Channel Capacity in Dispersed Spectrum