1 Bivariate Rician shadowed fading model J. Lopez-Fernandez, J.F. Paris Senior Member, IEEE, and E. Martos-Naya Abstract—In this paper 1 we present a bivariate Rician in the exploration of bi and multivariated distributions. Bi- shadowed fading model where the shadowing is assumed to variated distributions have been profusely explored in [5–11] follow a Nakagami-m distribution. We derive exact expressions including bivariate Rayleigh, Rician, Nakagami-m, Hoyt and involving a single integral for both the joint probability density κ µ distributions among others. To the best of the author’s function (PDF) and the joint cumulative distribution function − (CDF) and we also derive an exact closed-form expression for knowledge the correlated Rician shadowed scenario has not themomentgeneratingfunction(MGF).Asadirectconsequence yetbeenaddressedintheliterature.Inthispaperwepresenta we obtain a closed-form expression for the power correlation bivariate Rician shadowed distribution model inspired by the 7 coefficient between Rician shadowed variables as a function of approachpresentedin[12]forcorrelatedRayleigh,Ricianand 1 thecorrelationcoefficientbetweentheunderlyingvariablesofthe Nakagami-m fading channels. 0 model.Additionally,intheparticularcaseofintegermweshow 2 that the PDF can be expressed in closed-form in terms of a sum In our work we obtain exact expressions for the PDF and ofmMeijerG-functionsoftwovariables.Resultsareappliedto CDF with one single integral involving the Kummer conflu- n analyze the outage probability (OT) of a dual-branch selection ent hypergeometric function F (, ; ) and derive a simple a combiner (SC) in correlated Rician shadowed fading and the 1 1 · · · J algebraic closed-form expression for the MGF. Moreover, for evaluation of the level crossing rate (LCR) and average fade the case of integer m we reach a closed-form expression for 1 duration (AFD) of a sampled Rician shadowed fading envelope. 1 the PDF in terms of a finite sum of Meijer G-functions of IndexTerms—BivariateRicianshadowed,correlation,diversity two variables. The results are then used to analyze the outage ] reception, outage probability, level crossing rate, average fade probability (OT) of a dual-branch selection combiner (SC) in T duration. correlated Rician shadowed fading and to evaluate the level I . crossing rate (LCR) and average fade duration (AFD) of a s c I. INTRODUCTION sampled Rician shadowed fading envelope. [ The remainder of this paper is organized as follows: in The random fluctuation of the signal amplitude transmitted 1 Section II we introduce the statistical model which is then through a wireless channel has been extensively modeled in v analyzed in Section III. This section is subdivided in three the literature [1]. Besides classical models like Rayleigh and 1 subsections where the PDF, CDF and MGF are calculated 8 Rice, more sophisticated statistical distributions are recalled separately. Application results are presented in Section IV 9 to fit the behavior of the fading signal if accuracy is needed whereas the main conclusions are exposed in Section V. 2 in more intricate scenarios. In [2] a modification of Rice 0 model is presented where the amplitude of the Line-of-Sight 1. (LOS) component is assumed to randomly fluctuate following II. BIVARIATERICIANSHADOWEDMODEL 0 aNakagami-mdistribution.Theresultantdistributionisnamed The following notation will be used throughout the paper. 7 Rician shadowed and fits to the land mobile satellite (LMS) Wedenotetheabsolutevalueandtheexpectationofarandom 1 : channel experimental data [2] and has also proven to fit to variable X as X and E(X) respectively. The notation X Y v the underwater acoustic channel (UAC) fading behavior [3]. stands for X |con|ditioned to Y. We write X (µ,σ|2) i ∼ Nc X TwoparameterscharacterizetheRicianshadoweddistribution. to denote that random variable X distributes as complex r i.e. the Rician factor K which denotes the ratio between the Gaussian with complex mean µ and variance σ2. a average power in the LOS and the diffuse components and m WestartbymodelingasetofcorrelatedRayleighenvelopes that describes the level of fluctuation of the LOS and ranges (cid:112) G =σ 1 ρX +σ√ρX , k =1,2, (1) from 0.5 to , were m means no fluctuation (constant k − k 0 ∞ →∞ LOS component). The proposed model includes as particular where X ,k = 1,2 are independent random variables dis- k cases, the Rice (m ) and the Rayleigh (K =0) models. tributed as (0,1) and ρ is real number with 0 ρ 1 In [2] closed-form→e∞xpressions for the probability density [12]. It can∼beNecasily2checked that E(G ) = 0, E(G≤2) =≤σ2 k k function(PDF)andmomentgeneratingfunction(MGF)where and that ρ is the cross correlation coefficient between G and 1 presented while the cumulative density function (CDF) was G . Next we model the fluctuation of the LOS component by 2 derived in [4]. This (and any other) univariate random model incorporating a complex random variable Z whose envelope has applications on single input-single output systems but in is Nakagami distributed with real shaping factor m 0.5 and thecaseofdiversitycombiningsystemsthespatialdistribution E(Z 2) = Ω , which results in our proposed expr≥ession for N of reception antennas may cause the signals to present some mo|de|ling correlated Rician shadowed random variables degree of correlation which has motivated growing interest (cid:112) H =G +Z =σ 1 ρX +σ√ρX +Z, k =1,2. (2) k k k 0 − 1This work has been submitted to the IEEE for possible publication. In this model G and Z will account for the diffuse and Copyright may be transferred without notice, after wich this version may k nolongerbeaccessible LOS components respectively. Notice that the model assumes 2 that the fluctuation of the LOS component (modeled by Z) is ocowmTehdmeodnrisatntrodibobumotethdvavwraiiratibhalbeElse(s|HHHkk1|2a)anrd=eHiσn2d2i(2v1.id+uaKlly) Rfoircikan=sh1a,d2- fRk|V(rk,v)= Ωrk2 exp(cid:18)−rk22+Ω|2v|2(cid:19)I0(cid:18)|vΩ|r2k(cid:19), k =(71),2, | | where K = ΩN/σ2 is the Rician factor. The pair of random where variables H1 and H2 defined in (2) follow a bivariate Ω2 =σ2(1 ρ)/2. (8) | | | | − Rician shadowed distribution, a fact that will symbolically be Notice that it is v instead of the real or imaginary parts expressed as (σ,K,m,ρ). RS | | ∼B of v what appears in the conditioned PDF in (7) so we can assess that f (r ,v) = f (r ,r ) for k = 1,2. Next, III. STATISTICALANALYSIS sinceR V aRnkd|VR kV areindRekp|eRn3denkt,t3heirjointPDFcanbe 1 2 | | written as the product of the marginal PDFs resulting in In this section the PDF, CDF and MGF of the bivariate Rician shadowed model described in (2) are derived. r r (cid:18) r2+r2+2r2(cid:19) f (r ,r ,r )= 1 2 exp 1 2 3 (R1,R2)|R3 1 2 3 Ω4 − 2Ω2 A. Derivation of the PDF I (cid:16)r1r3(cid:17)I (cid:16)r2r3(cid:17). (9) × 0 Ω2 0 Ω2 Lemma 1: Let (R ,R ) (σ,K,m,ρ) with σ, K, m, 1 2 RS ∼B Taking now Z = z to be fixed, the random variable R = ρ real positive, m 0.5 and 0 ρ 1; then its joint PDF 3 ≥ ≤ ≤ V is also Rice distributed as can be seen from the definition fR1,R2(r1,r2) is given by i|n |(4) and its PDF takes the form ×f(cid:90)R01∞,Rx2e(xr1p,(cid:16)r2σ)−2ρ(=1(1+−8ρσ(ρ)6)mρxm(ρ1+2ρ−(cid:17)KρI))m02(cid:16)(cid:18)r1σr222(e1rx1−px(cid:18)ρ−)(cid:19)σrI2102(1(cid:18)+−σr2(cid:17)22ρ2()1r(cid:19)2−xρ)(cid:19) whfeRre3|Z(r3,z)= Ωr3(cid:48)2 exΩp(cid:48)2(cid:18)=−σr3222ρ+Ω/2(cid:48)|2z.|2(cid:19)I0(cid:18)|zΩ|(cid:48)r23(cid:19), ((1101)) F m;1; K x2 dx, (3) ×1 1 σ2ρ(ρm+K) Sfince o(nrly,r|z)| tsaokethsapta(r1t0i)nc(a1n0)beweexcparneswserditeasfR3|Z(r3,z)= where I0() denotes the modified Bessel function of the first R3|R4 3 4 kind and z·ero order and F (, ; ) is the Kummer confluent r (cid:18) r2+r2(cid:19) (cid:16)r r (cid:17) hypergeometric function [1131].· · · fR3|R4(r3,r4)= Ω3(cid:48)2 exp − 32Ω(cid:48)24 I0 Ω3(cid:48)24 . (12) Proof: Let define R = H , R = H , R = V and R = Z where V is defi1ned|as1| 2 | 2| 3 | | Finally, the determination of fR4(r4) is straightforward as 4 | | R4 = Z is Nakagami distributed with parameters m and | | Ω , whose PDF is given by [1] V =σλX +Z. (4) N 0 (cid:18) (cid:19) m TofhecojnodinittioPnDaFl PoDfFRs1asand R2 can be expressed as a function fR4(r4)= Γ(2mm)mΩmNr42m−1exp −ΩNr42 . (13) Substituting (9), (12) and (13) in (5) and reorganizing the fR1,R2(r1,r2)= double integral we get (cid:90) (cid:90) 0∞ 0∞f(R1,R2)|R3(r1,r2,r3)fR3|R4(r3,r4)fR4(r4)dr3dr4.(5) fR1,R2(r1,r2)= Γ(m16)(σK6mρσ(21)−mρ)2r1r2exp(cid:18)−r122+Ω2r22(cid:19) The procedure consists in finding the successive conditional ×(cid:90) ∞r3exp(cid:16)−σ2(ρ1(+1ρ)ρ)r32(cid:17)I0(cid:16)rΩ1r23(cid:17)I0(cid:16)rΩ2r23(cid:17)IR4(r3)dr3, PDFs and then performing double integration. Using the 0 − (14) definition of V in (4) we can rewrite H in (2) as k whereI (r )standsfortheintegralwithrespecttor which (cid:112) R4 3 4 H =σ 1 ρX +V, k =1,2. (6) is a function of r k k 3 − Consider now V =v to be fixed. Then, the random variables IR4(r3)=(cid:90) ∞r42m−1I0(cid:16)rΩ3r24(cid:17)exp(cid:16)−mρKρ+σK2 r42(cid:17)dr4. H and H become independent (and so do R and R ) as 0 (cid:48) 1 2 1 2 (15) they are a function of independent random variables X and (cid:16) (cid:17) 1 Using the identity I (ax) = F ;1;a2x2 [14] where X respectively. Moreover, both R and R turn into Rician 0 0 1 4 2 1 2 F (; ; ) is the confluent hypergeometric function and using variables with PDF [1] 0 1 · · the integral [13, eq. 5-7.522] a closed form solution can be 2In the practical application of this model (see section IV), |H1| and obtained for IR4(r3), namely |aHnt2en|nwaililnactwcooundtifffoerretnhtetsiimgnealinesntavnetlsopoerleinveltwreocediivfefedreenitthearntiennnoanseasitngthlee I (r )= Γ(m)(cid:16) Kσ2ρ (cid:17)m F (cid:16)m;1; K r2(cid:17). sametime.AssumingacommonvalueofZ meansthatthefluctuationofthe R4 3 2 ρm+K 1 1 σ2ρ(ρm+K) 3 (16) LOS component varies in a larger time-space scale than that of the diffuse components. Substituting(16)in(14)wegettheproposedexpression(3). 3 In case of integer m, expression (3) can be further manip- Substituting (12), (13) and (24) in (22) the integral with ulated so that a closed-form expression for the PDF can be respecttoR isfoundtobeidenticaltoI (r )in(15).Using 4 R4 3 obtained involving a sum of m Meijer G-functions of two the closed form result for I (r ) shown in (16) we get the R4 3 variables. proposed CDF. Corollary 1: Let (R ,R ) (σ,K,m,ρ) with σ, K, 1 2 RS ∼ B ρ real positive, 0 ρ 1 and positive integer m; then its ≤ ≤ C. Derivation of the MGF PDF can be expressed in closed form as Lemma 3: Let (R ,R ) (σ,K,m,ρ) with σ, K, m, 1 2 RS ∼B ρ real positive, m 0.5 and 0 ρ 1 and let P =R2 and fR1,R2(r1,r2)=c1r1r2exp(cid:18)−σr212(1+r22ρ)(cid:19)m(cid:88)−1(cid:0)mk2−k1!(cid:1)ck2 P2 =R22; then the≥MGF of P1≤and≤P2 is given by1 1 − k=0 (cid:18) (cid:12) (cid:12) (cid:12) (cid:19) ×G01,,10::11,,03::11,,03 −k(cid:12)(cid:12)(cid:12) 0,01,/12/2(cid:12)(cid:12)(cid:12) 0,10/,12/2(cid:12)(cid:12)(cid:12)c3r12,c3r22 , (17) MP1,P2(θ1,θ2)= σ2ρ(mKρ+K) − (a θ θ +a θ +a θ +a )m 1 whereG()istheMeijerG-functionoftwovariables(seeII.13 1 1 2 2 1 3 2 4 − , (25) in [15]) a·nd × (b1θ1θ2+b2θ1+b3θ2+b4)m (cid:16) (cid:17)m where 8π2 mρ mρ+K c1 =σ4ρ(1 ρ)(2K m(1 ρ)), (18) a1 =σ2(1+ρρ)(1−ρ), a2 =a3 = −ρ1, a4 = σ12ρ, (26) c2 =ρ(ρm+−KK)2(2(1K−−ρm) (1− ρ)), (19) b1 =σ2(1ρ−ρ)(cid:16)1+ρ− 2mKρ(1+−Kρ)(cid:17), (27) − − m+K mρ K b =b = , b = . (28) c3 = . (20) 2 3 −ρm+K 4 σ2ρ(mρ+K) 2(2K m(1 ρ)) − − Proof: See Appendix A. Proof: Following the same procedure used to proof Lemmas 1 and 2, we replace f (r ,r ,r ) by (θ ,θ ,r ) in (5) which(Rgi1v,Res2)|R3 1 2 3 B. Derivation of the CDF M(P1,P2)|R3 1 2 3 Lemma 2: Let (R ,R ) (σ,K,m,ρ) with σ, K, m, 1 2 RS ∼B ρ real positive, m 0.5 and 0 ρ 1; then its joint CDF (θ ,θ )= FR1,R2(r1,r2) is g≥iven by ≤ ≤ (cid:90)M∞(cid:90)P1∞,P2 1 2 (θ ,θ ,r )f (r ,r )f (r )dr dr . 2( mρ )m (cid:90) ∞ (cid:18) x2 (cid:19) 0 0 M(P1,P2)|R3 1 2 3 R3|R4 3 4 R4 4 3 4(29) F (r ,r )= mρ+K xexp R1,R2 1 2 σ2ρ −σ2ρ (cid:104) (cid:16)x r (cid:17)(cid:105)(cid:104)0 (cid:16)x r (cid:17)(cid:105) Since P1 R3 and P2 R3 are independent random variables 1 Q1 , 1 1 Q1 , 2 their joint M| GF can be| written as the product of individual × − Ω Ω − Ω Ω (cid:16) (cid:17) MGFs, this is F m;1; K x2 dx, (21) ×1 1 σ2ρ(ρm+K) (θ ,θ ,r )= (θ ,r ) (θ ,r ). where Q1(·,·) is the first order Marcum Q−function. M(P1,P2)|R3 1 2 3 MP1|R3 1 3 ·MP2|R3 2 (330) Proof: We can proof Lemma 2 using the same approach Moreover, both P R and P R are non-central chi-square employed for Lemma 1 replacing f (r ,r ,r ) by 1| 3 2| 3 F (r ,r ,r ) in (5) which g(iRv1e,sR2)|R3 1 2 3 randomvariableswhoseMGFisreadilyobtainedfrom[16,eq. (R1,R2)|R3 1 2 3 2-1-117].IfwesubstitutethisMGFin(30)andthensubstitute (12), (13) and (30) in the expression for the MGF (29) and reorganize the double integral we can write F (r ,r )= R1,R2 1 2 (cid:90) (cid:90) ∞ ∞ F (r ,r ,r )f (r ,r )f (r )dr dr . 0 0 (R1,R2)|R3 1 2 3 R3|R4 3 4 R4 4 3 4 4( m )m (cid:89)2 (cid:20) 1 (cid:21) (22) (θ ,θ )= σ2K MP1,P2 1 2 Γ(m)σ2ρ 1 σ2(1 ρ)θ k k=1 − − Both R1 R3 and R2 R3 are independent Rician random (cid:90) variables wh|ose CDF is|[1] ∞r exp(cid:0) d r2(cid:1)I (r )dr , (31) × 3 − 1 3 R4 3 3 0 F (r ,r )=(cid:104)1 Q (cid:16)r3,rk(cid:17)(cid:105), k =1,2, (23) where Rk|R3 k 3 − 1 Ω Ω 1 θ θ so we can write d1 = σ2ρ − 1 σ2(11 ρ)θ − 1 σ2(12 ρ)θ , (32) 1 2 − − − − F(R1,R2)|R3(cid:104)(1r1,rQ2,r(cid:16)3r)3=,r1(cid:17)(cid:105) (cid:104)1 Q (cid:16)r3,r2(cid:17)(cid:105). (24) tahnadt waphpeeraereIdR4in(r3th)eisdetrhievastiaomneoifntthegeraPlDwFitahndresCpDecFt wtohoRs4e − 1 Ω Ω · − 1 Ω Ω solution is shown in (16). Substituting (16) in (31) we get 4 1 m=0.5 (cid:16) (cid:17)m (cid:89)2 (cid:20) 1 (cid:21) m=1 (θ ,θ )= 2 ρm m=3 MP1,P2 1 2 σ2ρ ρm+K 1 σ2(1 ρ)θk 0.8 m→∞ k=1 − − (cid:90) ∞r exp(cid:0) d r2(cid:1) F (cid:16)m;1; K r2(cid:17)dr . (33) × 3 − 1 3 1 1 σ2ρ(ρm+K) 3 3 0 0.6 TheintegralwithrespecttoR canbesolvedinclosed-form 3 using again the integral [13, eq. 5-7.522] yielding S R ρ 0.4 (cid:16) (cid:17)m (cid:89)2 (cid:20) 1 (cid:21) (θ ,θ )= 1 ρm MP1,P2 1 2 σ2ρd1 ρm+K 1 σ2(1 ρ)θk k=1 − − (cid:16) (cid:17) 0.2 F m,1;1; K . (34) ×2 1 σ2ρ(ρm+K)d1 Finally, making use of the equivalence F (a,1;1;z) = 2 1 (1 z) a and after some algebraic simplification, (34) can 0 − 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 − be expressed in the final form presented in (25). ρ This result can be used to obtain a closed form expression that relates the correlation coefficient ρRS of the square Fig. 1. Correlation coefficient ρRS of square envelopes of the bivariate envelopesP andP ofthebivariateRicianshadowedrandom Ricianshadowedrandomvariablesversusthecorrelationcoefficientρofthe 1 2 underlyingmodelrandomvariablesforK=1anddifferentvaluesofm. variables to the correlation coefficient ρ of the underlying modelrandomvariables.Thisisstraightforwardtoverifysince the correlation coefficient ρ is defined in terms of the RS reduction in the value of K would also reduce the residual moments of P and P , namely 1 2 correlationforanyfixedm.InthissenseifwemakeK 0all → the curves on Fig. 1 will converge to the curve corresponding ρBS = (cid:112)E[P2E][P1EP22[]P−]E[(cid:112)P1E][·PE2[]P1]E2[P ], (35) to m→∞. 1 − 1 · 2 − 2 IV. APPLICATIONS and the moments in (35) can obtained as derivatives of (θ ,θ ) with the well known expressions Next we use the derived expression for the bivariate Rician MP1,P2 1 2 shadowed CDF to analyze some interesting scenarios in com- E[Pn]= ∂nMP1,P2(θ1,θ2)(cid:12)(cid:12)(cid:12) , k =1,2, (36) minudnuicaal-tibornasncshtasretilnegctiwonithcotmhebionuintagge(SpCr)obaanbdiliftoyllo(OwTin)gstwuidtyh k ∂θn (cid:12) k θk=0 second-order statistics of sampled Rician shadowed fading E[P1P2]= ∂2MP1,P2(θ1,θ2)(cid:12)(cid:12)(cid:12) . (37) channels. ∂θ ∂θ (cid:12) 1 2 θ1=θ2=0 A. Outage probability of dual-branch SC Using (25) in (36) and (37), and substituting the results in (35) we reach a closed-form expression for ρ which is In SC the receiver selects the branch with higher instanta- BS rather lengthy and is not shown here for simplicity. Fig. 1 neous SNR, γk, k =1,2 so the output SNR of the combiner depicts ρ versus ρ for different values of m. According to is RS themodelin(2),thecorrelationbetweenthesquareenvelopes γSC =max(γ1,γ2) (38) P and P of H and H respectively has two origins. One 1 2 1 2 In the Rician shadowed fading scenario under study, the is explicitly determined by the parameter ρ and the other is instantaneous SNR of the kth branch is a consequence of the common level of the LOS component R2E fluctuation that we assumed to affect both variables. This γ = k s, k =1,2, (39) k explains the behavior of the plots in Fig. 1. When ρ 1, N0 → H1 H2 which means P1 P2 and hence ρBS 1. while the average SNR is given by → → → Howeverwhenρ 0therestillremainsaresidualcorrelation → E[R2]E σ2(1+K)E between P1 and P2 due to the common level of fluctuation γ = k s = s, k =1,2, (40) N N of the LOS component which makes ρ tend to a non- 0 0 BS zero value. This value depends on the relative level of the where E is the symbol energy and N is the noise spectral s 0 LOS component fluctuation which is determined by both m density which is assumed to be the same in both branches. and K. i.e., see in Fig. 1 that ρ decreases as m grows Without loss of generality we will consider the normalization BS (less fluctuation of the LOS component) for any fixed ρ. For E /N = 1 in the forthcoming expressions. The outage s 0 the limiting case of m (no fluctuation of the LOS probability is defined as the probability that the instantaneous → ∞ component) see that ρ 0 when ρ 0. Similarly, a SNR γ falls below a certain threshold γ . Using the previous BS th → → 5 definitions the outage probability in SC can be expressed as a 100 function of the CDF in (21) as m=0.5 P (γ )=Pr(γ <γ ) (41) out th SC th 10−1 =Pr(γ1 <γth,γ2 <γth) (42) m=3 =Pr(R2 <γ ,R2 <γ ) (43) 1 th 2 th =Pr(R1 <√γth,R2 <√γth) (44) γ)th 10−2 m→∞ < =F (√γ ,√γ ). (45) R1,R2 th th SC γ Using (21) the expression for the outage probability is Pr( 10−3 Pout(γth)= 2(1+K)(γmρmρ+ρK)m (cid:90) ∞xexp(cid:18)−(1+γρK)x2(cid:19) 10−4 ρ=0.1 0 (cid:20) (cid:18) (cid:113) (cid:113) (cid:19)(cid:21)2 ρ=0.5 1 Q x 2(1+K), 2γth(1+K) ρ=0.9 × − 1 γ(1(cid:16)−ρ) γ(1−ρ) (cid:17) 10−55 10 15 20 25 30 ×1F1 m;1;γKρ((ρ1m++KK))x2 dx. (46) AverageSNR,γ (dB) In Fig. 2 the outage probability with SC is depicted using Fig.2. OutageProbabilityasafunctionoftheaverageSNRforK =10, numerical computation of (46) with MATLAB as a function anddifferentvaluesofmandρ.Linescorrespondtotheoreticalexpressions of the average SNR γ for γ = 10 dB, K = 10, and andmarkerscorrespondtosimulations th different values of m and ρ . Notice that as m grows, the effect of the fluctuation of the LOS component diminishes level u in the positive (or negative) direction will be denoted yielding a decrease in outage probability for all the values as LCR(u) and can be analytically expressed as of ρ considered. The extreme case of no fluctuation of the LOScomponent(m )correspondstoacorrelatedRician Pr R u fadingscenario.The→imp∞actofdifferentdegreesofcorrelation LCR(u)= { 1 2 }, (47) T S between the signals arriving at the two branches can be also examined from Fig. 2. Particularly, notice that the outage where R1 (cid:44)R[n+1] and R2 (cid:44)R[n]. Noting that Pr R1 < { probability increases as the two branches correlate, e.g. as u,R2 >u =Pr R1

Bivariate Rician shadowed fading model

Similar Bivariate Rician shadowed fading model