The κ-µ Shadowed Fading Model with Integer Fading Parameters

Transcription

1 The κ-µ Shadowed Fading Model with Integer Fading Paraeters F. Javier Lopez-Martinez, Jose F. Paris and Juan M. Roero-Jerez arxiv:69.37v [cs.it] Jan 7 Abstract We show that the popular and general κ-µ shadowed fading odel with integer fading paraeters µ and can be represented as a ixture of squared Nakagai- ˆ or Gaa distributions. Thus, its PDF and CDF can be expressed in closed-for in ters of a finite nuber of eleentary functions powers and exponentials. The ain iplications arising fro such connection are then discussed, which can be suarized as: the perforance evaluation of counication systes operating in κ-µ shadowed fading becoes as siple as if a Nakagai- ˆ fading channel was assued; the κ-µ shadowed distribution can be used to approxiate the κ-µ distribution using a closed-for representation in ters of eleentary functions, by choosing a sufficiently large value of ; and 3 restricting the paraeters µ and to take integer values has liited ipact in practice when fitting the κ-µ shadowed fading odel to field easureents. As an application exaple, the average channel capacity of counication systes operating under κ-µ shadowed fading is obtained in closed-for. Index Ters Wireless channel odeling, κ-µ shadowed fading, Nakagai fading, probability density function, cuulative distribution function, Gaa distribution. I. INTRODUCTION The κ-µ shadowed fading odel was introduced in [], and right after in [] in an independent work, as a generalization of the popular κ-µ odel proposed by Yacoub [3]. Other generalizations of the κ-µ distribution using the inverse Gaa distribution [4], the inverse Gaussian distribution [5], and a ultiplicative coposition with the Gaa distribution [6], are also available in the literature. Recently, it was shown that the apparently unrelated η-µ distribution was also a particular case of the κ-µ shadowed distribution [7], thus encapsulating this set of popular and general fading distributions in the literature in a single odel. Ever since its inception, the κ- µ shadowed fading odel has gained a rearkable attention in the literature due to its versatility on odeling propagation conditions ranging fro very favorable to worse-than-rayleigh fading. It also provides an good fit to field easureents in diverse scenarios like device-to-device counications [], or underwater acoustic channels [, 8]. Unlike other general fading odels [9 ], its chief probability functions PDF and CDF are given in closed-for. That This work has been subitted to the IEEE for possible publication. Copyright ay be transferred without notice, after which this version ay no longer be accesible. F. J. Lopez-Martinez and J. F. Paris are with Departento de Ingeniería de Counicaciones, Universidad de Malaga - Capus de Excelencia Internacional Andalucía Tech., Malaga 97, Spain. Contact e-ail: fjlopez@ic.ua.es. J. M. Roero-Jerez is with Departento de Tecnología Eectrónica, Universidad de Malaga - Capus de Excelencia Internacional Andalucía Tech., Malaga 97, Spain. being said, the coputation of its CDF still requires for the evaluation of the confluent bivariate hypergeoetric function Φ [, 9.6.], which is not yet included in coercial atheatical software packages. This fact has not prevented the widespread use of the κ-µ shadowed distribution in a nuber of practical scenarios of interest [3 5]. However, these results have a uch ore coplicated for than their counterparts when assuing, for instance, the sipler and extreely popular Nakagai- ˆ or siply Nakagai fading odel. In this paper, we show that the κ-µ shadowed PDF and CDF can be expressed in ters of a finite nuber of eleentary functions for a proper choice of the fading severity paraeter values. Specifically, we show that the κ-µ shadowed distribution can be expressed as a ixture of squared Nakagai- ˆ distributions, when the paraeters µ and take integer values. As we will later see, such restriction has little effect in practice when fitting field easureents to the κ-µ shadowed distribution, while being extreely convenient fro a coputational perspective. This connection considerably facilitates the perforance evaluation of counication systes operating in κ-µ shadowed fading channels. In fact, we show that any perforance etric that is calculated by averaging over the distribution of the SNR in κ-µ shadowed fading e.g. bit error rate, capacity, outage probability... can be readily and directly obtained as a linear cobination of the results obtained when assuing Nakagai- ˆ fading; the values of the weights for this linear cobination are the coefficients of the ixture, which are given in closed-for. We also show that the κ-µ shadowed fading odel can be used to approxiate the κ-µ distribution with arbitrary precision, by siply choosing a sufficiently large value of. Thus, the coputational benefits of the new representation of the κ-µ shadowed fading odel in ters of eleentary functions can be extended to the κ-µ distribution and also to the Rician distribution as a special case for µ =. As a direct application, we exeplify the usefulness of the results here unveiled to obtain exact expressions for the average capacity of the κ-µ shadowed fading channel, which are considerably sipler than those originally obtained in [6]. The reainder of this paper is structured as follows: in Section II, the new expressions for the PDF and the CDF of the κ-µ shadowed fading odel are given in ters of a finite su of eleentary functions. Then, a nuber of relevant iplications and useful properties arising fro these In order to avoid any confusion between the paraeter of the κ-µ shadowed fading odel and the hoonyous paraeter of the Nakagai- ˆ fading odel, we use a with a superscript to denote the latter.

2 results are discussed in Section III. An application exaple is presented in Section IV, whereas the ain conclusions are outlined in Section V. II. NEW STATISTICS FOR THE κ-µ SHADOWED DISTRIBUTION. Throughout this paper, we will characterize the distribution of the received power envelope in κ-µ shadowed fading channels, or equivalently, the instantaneous SNR at the receiver. Note that characterizing the distribution of the aplitude envelope r is straightforward by a siple change of variables r =. Definition : A rando variable following a κ-µ shadowed distribution will be denoted as S ; κ, µ,, and its PDF will be given by f S ; κ, µ, ; x = µµ + κ µ Γµ + µκ e µ+κ x F µ x, µ; µ κ + κ µκ + x, where F is the confluent hypergeoetric function of the first kind [, eq. 9..]. Definition : A rando variable following a κ-µ distribution will be denoted as KM ; κ, µ, and its PDF will be given by f KM = µ+ µ + κ κ µ e µκ µ e µ+κ I µ µ κ + κ, where I ν is the ν-th order odified Bessel function of first kind. Definition 3: A rando variable following a squared Nakagai distribution will be denoted as K ; ˆ, and its PDF, assuing ˆ N, will be given by f K ; ˆ; x = ˆ ˆ x ˆ ˆ! e x ˆ/. 3 Note that the squared Nakagai distribution is equivalent to a Gaa distribution with shape paraeter ˆ and scale paraeter / ˆ. After these preliinary definitions, we now provide new expressions for the PDF and CDF of the κ-µ shadowed distribution for positive integer values of the fading severity paraeters µ and. Theore : Let be a rando variable such that S ; κ, µ, and let µ, N. Then, is a ixture of squared Nakagai distributions, which PDF is given as follows: If < µ f S ; κ, µ, ; x = + If µ µ j= A j f K ω A ; µ j + ; x A j f K ω A ; j + ; x. j= f S ; κ, µ, ; x = µ j= 4 B j f K ω B ; j; x, 5 where A j = + j j [ ] [ ] j+ µκ, µκ + µκ + A j = j µ + j j [ ] j [ ] µ j+ µκ, µκ + µκ + [ ] j [ ] µ j µ µκ B j =, j µκ + µκ + and where we have defined with ω A µ j +, ω A j +, ω B j, µ + κ, µκ + Proof: See Appendix A µ + κ. Expressions 4 and 5 give an exact representation of the κ-µ shadowed distribution in ters of a finite ixture of squared Nakagai distributions. With this representation of the PDF, which is new in the literature to the best of our knowledge, a siilar result for the κ-µ shadowed CDF is now obtained. Corollary : Let be a rando variable such that S ; κ, µ, and let µ, N, then, the CDF of is given as follows: If < µ µ F S ; κ, µ, ; x = j= j= j A j e x/ r= µ j A j e x/ r x. r! r= r x r! 9

3 3 TABLE I PARAMETER VALUES FOR THE κ-µ SHADOWED DISTRIBUTION WITH INTEGER µ AND, Case µ > Case µ M = µ M = µ i = C i = +i [ ] [ ] i+ i µκ < i µ µκ+ µκ+ i µ+ C i = µ [ ] i [ ] µ i µκ i [ ] i µ+ [ ] i µκ+ µκ+ i+ i µ+ µκ µ < i µ µκ+ µκ+ { µ i +, i µ i = i = i µ i + µ < i µ { µ+κ Ω i =, i µ µκ+ µ < i µ µ+κ Ω i = µκ+ µ+κ If µ µ F S ; κ, µ, ; x = t= j B j e x/ r= r x. r! Proof: Using Theore and considering that the CDF of a squared Nakagai rando variable is given by ˆ F K ; ˆ; x = e x/ r= x r, r! where we have defined / ˆ, the proof is copleted. A ore copact and unified for for the PDF and the CDF of the κ-µ shadowed fading distribution with integer fading paraeters can be obtained after soe anipulation, yielding and f S ; κ, µ, ; x = F S ; κ, µ, ; x = M i= C i x i i! M C i e x Ω i i= i r= Ω i i e x Ω i, r x, 3 r! Ω i where the paraeters i, M and Ω i are expressed in Table I in ters of the paraeters of the κ-µ shadowed distribution, naely κ, µ, and. III. DISCUSSION After presenting the new results for the statistics of the κ-µ shadowed distribution with integer fading paraeters, we now discuss about the ain iplications and insights that can be obtained fro these results. A. Finite ixture representation Taking a deeper look at Theore, one can observe that the expression for the ixture of squared Nakagai distributions has different for depending on whether is larger or saller than µ. When µ the κ-µ shadowed distribution is expressed as a proper ixture of squared Nakagai distributions, on which the values of the weights B j in 6 correspond to those of the binoial distribution, whose probability ass function is given by fk; n, p = Pr{X = k} = n p k p n k, 4 k with n = µ and p = µκ+. In this situation, the κ-µ shadowed fading odel can be regarded as a superposition of µ parallel channels affected by Nakagai fading with different fading severities, being each of such channels used with a given probability 3 p. Conversely, for < µ the PDF in 4 is given in ters of an iproper ixture; i.e., 4 is expressed as a linear cobination of two different sets of squared Nakagai distributions with coefficients A j and A j, on which the ixture coefficients are not necessarily non-negative. Another iportant insight arises by inspecting eq. 8 in the Appendix. We note that such MGF is expressed as the product of the MGFs of two Gaa distributions with scale paraeters and, and shape paraeters µ and, respectively. Thus, a κ-µ shadowed RV can be generated as the su of two independent gaa RVs with such scale and shape paraeters, provided that < µ. Note that this holds for any {, µ} R +. For the specific case on which the κ- µ shadowed distribution reduces to the η-ˆµ distribution, i.e. µ = ˆµ and = ˆµ with κ = η/η [7], this observation coincides with the one given in [7]. We use the ter proper ixture to denote any ixture distribution which can be expressed as a convex cobination i.e. a weighted su with nonnegative coefficients that su to of other distributions. We also use the ter iproper ixture to denote any ixture distribution on which the ixture coefficients are not restricted to be non-negative. 3 We ust here note that this phenoenon is purely atheatical, as it arises fro the observation of the new for of the PDF here derived. To the best of our knowledge, such observation does not have any connection with the physical odel of the κ-µ shadowed fading distribution note that the physical odels that originate this distribution [7] can be regarded as coherent cobinations, in a axial ratio cobining for, of µ Rician shadowed variates.

4 4 f = = = 5 = = = = 3 4 Fig.. Convergence in distribution between the κ-µ shadowed distribution and the κ-µ distribution as. Paraeter values κ = 5, µ = 3 and =. B. Convergence to the κ-µ distribution Intuitively, if we let in the κ-µ shadowed fading odel, then the Nakagai- PDF used to odel the rando fluctuation of the line-of-sight coponent degenerates to a deterinistic distribution, being its PDF the Dirac delta function. Thus, this odel reduces to the original κ-µ distribution in [3]. This iplies that by virtue of Theore, the κ-µ shadowed distribution with integer fading paraeters and sufficiently large can be used to approxiate the κ-µ distribution. This is observed in Fig., where the evolution of the κ-µ shadowed distribution as grows is represented. However, a rigorous proof for the convergence in distribution between the κ-µ shadowed and the κ-µ fading odels is not that evident. The following Leas forally establish the connections between the κ-µ shadowed distribution and the κ-µ distribution in ters of weak convergence of probability easures. Lea : Let { } = be a sequence of rando variables with S ; κ, µ, and the corresponding sequence of CDFs given by {F S ; κ, µ, ; } =. Then, this sequence of rando variables weakly converges to the κ-µ distribution with ean and shaping paraeters κ and µ, i.e. li F S ; κ, µ, ; = F KM ; κ, µ;, 5 where F KM represents the CDF of the corresponding κ-µ distribution. Proof: It is a direct consequence of the Lévy s continuity theore [8]. Lea establishes that the κ-µ shadowed distribution converges to the κ-µ distribution for sufficiently large. Thus, it is possible to approxiate the κ-µ distribution with integer µ by a finite ixture of squared Nakagai distributions, by using the κ-µ shadowed fading distribution with integer fading paraeters and choosing an arbitrarily large paraeter N. These connections between the κ-µ shadowed distribution and the κ-µ distribution can be extended to any probability easure obtained by expectation over the desired distribution, as stated in the following Lea. Lea : Let { } = be a sequence of rando variables with S ; κ, µ, and the corresponding sequence of CDFs given by {F S ; κ, µ, ; } =. Let Φ be any probability easure conditioned to, which is continuous and bounded as a function of. Then, the sequence of the expectations of Φ over the κ-µ shadowed rando variables converges to the expectation of Φ over a κ-µ rando variable with ean and shaping paraeters κ and µ, i.e. li ΦdF S ; κ, µ, ; = ΦdF KM ; κ, µ;, 6 where F KM represents the CDF of the corresponding κ-µ distribution. Proof: It is a direct consequence of Lea and the Helly-Bray theore [9]. C. A new gaa approxiation to the Rician distribution In the ilestone paper by Nakagai [], a siple equivalence between the Nakagai- ˆ distribution and the Rician distribution was proposed. The connection between these distributions is established by a siple paraeter transforation, setting ˆ = + K / + K. This approxiation, also included in the reference textbook by Sion and Alouini [, eq..6], has been widely eployed in wireless counications in order to approxiate the Rician distribution by the ore tractable Nakagai- ˆ distribution. However, as argued by any authors [, 3] such approxiation has a severe flaw that has iportant ipact when analyzing the syste perforance: the diversity order of Rician fading equals one, whereas the diversity order of Nakagai- ˆ fading is ˆ. The diversity order is related to the behavior of the PDF around the origin, or equivalently to the asyptotic behavior of the MGF as s. As consequences of Lea, and setting µ =, we propose to approxiate the Rician distribution by a finite ixture of squared Nakagai distributions, by using the Rician shadowed fading distribution and choosing an arbitrarily large paraeter N. The ain benefit of this approxiation relies on the fact that the diversity order of the Rician shadowed distribution is also. In Figs. 3 and, the behavior of the classical approxiation to the Rician distribution proposed by Nakagai, and the one here proposed based on the Rician shadowed distribution with integer is illustrated, representing the corresponding PDFs for different values of K and. Log-log scale was used in order to better observe the effect of increasing. We can see that for low values of K, the approxiation to the Rician distribution based on the Rician shadowed distribution is good even for oderate values of. As K grows, a larger nuber of ters is required for the ixture approxiation i.e., a larger in order to converge to the Rician distribution. In both cases, the soothness of the PDFs in the proxiity of zero for the Rician and Rician shadowed distributions have siilar shape, whereas the original approxiation in [] exhibits a very different behavior.

5 5 f Rice = 5 = = Nakagai- ˆ f µ = µ = 5 µ =.5 µ =.5 µ = 3.5 µ = Fig.. Gaa approxiation to the Rician distribution with paraeter K using the Rician shadowed distribution with paraeter K and integer. Paraeter values K = 3 and =. Nakagai approxiation [] uses ˆ = + K / + K =.8. Fig. 4. Evolution of the κ-µ shadowed fading PDF for different values of µ. Solid lines correspond to real-valued µ {.5,.5, 3.5, 4.5}, whereas dashed lines correspond to the largest previous and the sallest following integer values µ { : 5}. Paraeter values κ = 6, = and =. f 3 4 Rice = 5 = = Nakagai- ˆ 5 f = = 6 =.5 =.5 = 3.5 = 4.5 = Fig. 3. Gaa approxiation to the Rician distribution with paraeter K using the Rician shadowed distribution with paraeter K and integer. Paraeter values K = and =. Nakagai approxiation [] uses ˆ = + K / + K = Fig. 5. Evolution of the κ-µ shadowed fading PDF for different values of. Solid lines correspond to real-valued {.5,.5, 3.5, 4.5, 5.5}, whereas dashed lines correspond to the largest previous and the sallest following integer values { : 6}. Paraeter values κ = 6, µ = and =. D. Effect of considering integer fading paraeters The paraeter µ N was originally introduced by Yacoub [3] as the nuber of clusters of ultipath waves that propagate in a non-hoogeneous environent. As argued in [3], the restriction of the paraeter µ to take integer values is inherently linked to the underlying physical odel for the κ-µ distribution. Thus, the η-µ and κ-µ fading odels with integer µ are usually referred to as physical odels, and are often used to evaluate the perforance of counication systes operating over generalized fading channels [7, 4 8]. The consideration of a real-valued µ indeed yields a larger flexibility to the odel; however, the ipact of considering an integer µ decreases as µ grows as observed in Fig. 4. The restriction of the paraeter to take integer values only has a non-negligible ipact in heavy shadowing environents i.e. low values of. However, as grows the PDFs corresponding to the real-valued and its closest integer i.e., largest previous or sallest following integer counterpart tend to be indistinguishable; this is observed in Fig. 5. Thus, fro the observation of Figs. 4 and 5 we see that the effect of restricting the fading severity paraeters to take integer values is liited, unless or µ take low values i.e., severe LOS fluctuation or severe ultipath, respectively. In fact, one ay wonder whether there s a lower value of or equivalently µ under which the use of the κ-µ shadowed fading distribution with integer fading paraeters to approxiate

6 6 the κ-µ shadowed fading distribution with arbitrary fading paraeters is not recoended. In our view, this will strongly depend on the application. Soeties only the tail of the distributions is needed for further calculations, or even the perforance etrics of interest end up being rather siilar when soe fading severity paraeters vary. Thus, it ay occur that the approxiation error is still negligible for low values of and µ in soe cases; this will be later exeplified in Section IV. We will now study the ipact of restricting the fading severity paraeters µ and to take integer values on the goodness of fit to field easureents. We use the epirical results presented in [8] for soe underwater acoustic channels UAC, for which the κ-µ shadowed fading odel showed the best fit. These easureents were conducted in the Mediterranean Sea near Cartagena Spain, in shallow waters with depths in the range of 4-3. Details on the specific easureent configuration, including a block diagra of the easureent equipent set-up, can be found in [8, Sect. 3.]. We used a odified version of the Kologorov-Sirnov KS statistic in order define the error factor ɛ that is used to quantify the goodness of fit between the epirical and theoretical CDFs, which are denoted by ˆF r and F r respectively, i.e, ɛ ax x log ˆF r x log F r x. 7 As in [8], the CDF is used in log-scale with the ai of outweighing the fit in those values of received power closer to zero, i.e. those corresponding to a ore severe fading. In Figs 6 and 7 we copare the set of easureents corresponding to the channels C9-3 and C9-64 with three different distributions: Rician, κ-µ shadowed with integer fading paraeters, and κ-µ shadowed. For the C9-3 channel, we observe that the error factor increases fro ɛ =.6 to ɛ =.83 when constraining µ and to take integer values. However, the fit is still iproved when coparing to Rician fading, which yields ɛ =.. With regard to the C9-64 channel, we now observe that the error factor is barely increased i.e. ɛ =.6 instead of ɛ =.53 when forcing µ and to take integer values, copared to the general κ-µ shadowed fading. For the case of Rician fading, we have ɛ =.4, which is clearly outperfored by the κ-µ shadowed odel with integer fading paraeters. E. Perforance analysis The calculation of perforance etrics 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-µ shadowed fading distribution iplies integrating over the hypergeoetric functions that define its PDF and CDF. However, with the new representation for the κ-µ shadowed distribution functions, the perforance analysis is greatly siplified. In the following Lea, we show that analyzing the perforance in κ-µ shadowed fading has the sae coplexity as analyzing the uch sipler Nakagai- ˆ case. CDF 3 Measured Rician κ-µ shad int κ-µ shad Noralized power envelope db Fig. 6. Epirical vs theoretical CDFs of the received signal power for the UAC channel C9-3 [8]. Paraeter values: Rician {K =.64; ɛ =.}, κ-µ shadowed int {κ =.84, µ =, = ; ɛ =.83}; κ-µ shadowed {κ = 4.6, µ =.3, =.45; ɛ =.6}. CDF Measured Rician κ-µ shad int κ-µ shad Noralized power envelope db Fig. 7. Epirical vs theoretical CDFs of the received signal power for the UAC channel C9-64 [8]. Paraeter values: Rician {K =.3; ɛ =.4}, κ-µ shadowed int {κ =.6, µ =, = 6; ɛ =.6}; κ-µ shadowed {κ =.3, µ =., = 6.3; ɛ =.53}. Lea 3: Let h be a perforance etric depending on the instantaneous SNR, and let h K ; be the etric in Nakagai fading with average SNR obtained by averaging over an interval of the PDF of the SNR, i.e., h K ; = b a hxf K ; ; x dx, 8 with a < b. Then, the average perforance etric in κ-µ shadowed fading channels with average SNR, denoted

7 7 as h S ; κ, µ,, can be calculated, given that µ, N, as h S ; κ, µ, = M C i h K Ω i / i ; i, 9 i= Proof: The average perforance etric h S ; κ, µ, is calculated as h S ; κ, µ, = b a hxf S ; κ, µ, ; x dx. We can express f S ; κ, µ, ; x as a ixture of squared Nakagai distributions, given in. Thus, cobining 3 with we have M f S ; κ, µ, ; x = C i f K Ω i / i ; i ; x. i= Introducing into, after soe siple anipulation, 9 is obtained. The iplications of this lea are of great iportance, as it eans that any perforance etric for which existing results are available for the Nakagai- ˆ case can be directly generalized to the κ-µ shadowed case by eans of a finite linear cobination as described in 8. We will exeplify the usefulness of this Lea in the next Section. IV. APPLICATION: AVERAGE CHANNEL CAPACITY The characterization of the average channel capacity in fading channels, defined as C[bps/Hz] + log + f d, where is the instantaneous SNR at the receiver side, is a classical proble in counication theory [9 3]. The average channel capacity assuing κ-µ shadowed fading was derived in [6], in ters of the unwieldy bivariate Meijer-G function. However, a direct application of Lea 3 using the average channel capacity expression for Nakagai- ˆ fading channels [3, eq. 3] yields the following siple closed-for expression: C = log e M i= i C i e /Ωi k= Γ k, /Ω i Ω k, 3 i where Γ is the upper incoplete Gaa function, which can be coputed, when the first paraeter is a negative integer, as [, eq ] Γ n, x = n n! and [ n Ei x e x r= ] r r r! x r+, 4 Γ, x = Ei x, 5 where Ei is the exponential integral function [, eq. 8..]. In the next set of figures, the effect of the fading severity paraeters µ and on the average capacity is investigated. C = = 3 = 5 = AWGN db Fig. 8. Average channel capacity vs. average SNR, for different values of. Paraeter values are κ = and µ = 3. The AWGN case is included as a reference. C = = 3 = 5 = AWGN db Fig. 9. Average channel capacity vs. average SNR, for different values of. Paraeter values are κ = and µ = 3. The AWGN case is included as a reference. Firstly, Figs. 8 and 9 illustrate the effect of in different conditions: strong LOS κ = and weak LOS κ =, assuing a fixed value of µ = 3. In general ters, a larger is translated into a larger capacity for a given SNR. However, the effect of is uch ore pronounced in the strong LOS scenario, leading to a ore severe perforance degradation for lower i.e. heavy shadowing in the LOS coponent.. Conversely, in the weak LOS scenario the effect of is barely noticeable. In Figs. and, the effect of µ is investigated in the sae conditions as in the previous case: strong LOS κ = and weak LOS κ =, assuing a fixed value of = 3. We see that in the strong LOS scenario, increasing µ has little effect as the perforance is doinated by the LOS coponent. Conversely, increasing the nuber of clusters µ in the weak

8 8 C µ = µ = 3 µ = 5 µ = AWGN db Fig.. Average channel capacity vs. average SNR, for different values of µ. Paraeter values are κ = and = 3. The AWGN case is included as a reference. C µ = µ = 3 µ = 5 µ = AWGN db Fig.. Average channel capacity vs. average SNR, for different values of µ. Paraeter values are κ = and = 3. The AWGN case is included as a reference. LOS scenario is translated into a better perforance. V. CONCLUSIONS The statistical characterization of the κ-µ shadowed odel with integer fading paraeters was here presented. Rearkably, the PDF and CDF of this very general odel can be expressed in closed-for in ters of a finite su of powers and exponentials, in the for of a ixture of Gaa distributions. The inherent atheatical coplexity of the κ- µ shadowed fading odel is greatly reduced at the expense of a liited restriction in ters of flexibility. Most notably, the perforance evaluation of counication systes operating under this fading channel odel can be directly evaluated if existing results are available for the sipler Nakagai- ˆ fading odel. Thus, considering the κ-µ shadowed fading odel iplies no additional coplexity while allowing for a uch larger flexibility in ters of propagation conditions, naturally including both LOS and NLOS scenarios. VI. ACKNOWLEDGEMENT This work has been funded by the Consejería de Econoía, Innovación, Ciencia y Epleo of the Junta de Andalucía, the Spanish Governent and the European Fund for Regional Developent FEDER projects P-TIC-79, P-TIC- 838, TEC3-47-R, TEC P and TEC4-579-R. The first author would like to thank Marco di Renzo for enlightening discussion about the diversity order of the approxiations to the Rician distribution. APPENDIX A PROOF OF THEOREM I The MGF of a κ-µ shadowed rando variable is given by [] µ M s = µµ + κ µ s µ+κ µ µκ +, 6 s µ+κ µκ+ which can be written, in ters of and defined in 8, as M s = s µ s. 7 The PDF is related to the MGF by the inverse Laplace transfor L {M s; s; x}, therefore, it is clear that its analytical expression will depend on the exponent of the nuerator being positive or zero, i.e. µ, or negative, i.e. < µ. Let us consider < µ. The MGF can be rewritten in a ore convenient way for this case as M s = s µ s, 8 and perforing a partial fraction expansion we obtain M s = where µ j= A j s µ j+ + A j = j + j j A j = j µ + j j j= A j s j+, j +j, 9 j µ µ +j. 3 By plugging the defined and given in 8 into 3, the expressions of coefficients A j and A j given in 6 are obtained after soe algebraic anipulation. Perforing now an inverse Laplace transforation to the MGF as given in 9, 4 is obtained.

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