ISSN 33-365 (Prnt, ISSN 848-6339 (Onlne https://do.org/.7559/tv-67847 Orgnal scentfc paper Outage Probablty of Macrodversty Recepton n the Presence of Fadng and Webull Co- Channel Interference Mloš PERIĆ, Branmr JAKŠIĆ, Danela ALEKSIĆ, Dragan RANDJELOVIĆ, Mhalo STEFANOVIĆ Abstract: In ths paper macrodversty recepton wth macrodversty selecton combnng (SC recever and two mcrodversty SC recevers operatng over Gamma shadowed Webull multpath fadng channel n the presence of co-channel nterference subected to Webull short term fadng s analysed. Closed form resson for cumulatve dstrbuton functon (CDF of sgnals at outputs of mcrodversty recevers s calculated. These results are used for CDF evaluaton of macrodversty SC recever output sgnal envelope. Seres of derved ressons for outage probablty rapdly convergence from 5 to terms should be summed to acheve accuracy at 5 th sgnfcant dgt for any Gamma long term fadng parameters and Webull short term fadng parameters. In ths paper, nfluence of Gamma long term fadng severty parameter, Gamma long term fadng correlaton coeffcent and Webull short term fadng nonlnearty parameter on outage probablty are analysed and dscussed. Keywords: co-channel nterference; cumulatve dstrbuton functon (CDF; macrodversty (MAD; mcrodversty (MID; outage probablty; Webull INTRODUCTION Macrodversty (MAD technque wth MAD selecton combnng (SC recever and two or more mcrodversty (MID recevers can be used to reduce long term fadng effects and short term fadng effects on system performance of wreless communcaton system. MAD SC recever selects MID SC recever wth hgher sgnal envelope average power at nputs to provde servce to user resultng n long term fadng reducton and MID SC recever selects and outputs branch wth the hghest sgnal to nterference envelopes rato reducton of short term fadng effects reducton [ - 4]. There are multple dstrbutons that can be used to descrbe small scale envelope varatons n multthreaded Webull dstrbuton faded channels descrbng the varaton of small scale envelope varatons n nonlnear and non-lne-of-sght multstage channels wth a sngle cluster. Ths dstrbuton has parameter whch s n relaton to nonlnearty of fadng channels. When parameter s two, Gamma shadowed Webull fadng channel becomes Gamma shadowed Raylegh multpath fadng channel [5, 6]. Long scale sgnal envelope varaton can be descrbed usng Log-normal dstrbuton or Gamma dstrbuton. When long term fadng s descrbed by usng Gamma dstrbuton, outage probablty can be derved n closed form. Gamma dstrbuton has severty fadng parameter c. When parameter c goes to nfnty, Gamma shadowed multpath fadng channel becomes multpath fadng channel. When correlaton coeffcent goes to one, the least sgnal occurs smultaneously on both MID recevers resultng n system performance degradaton [7, 8]. There are more works n open techncal lterature consderng performance of wreless communcaton system n the presence of short term fadng, long term fadng and co-channel nterference. In [9], average level crossng rate and average fade duraton of MAD system wth MAD SC recever and two MID maxmal rato combnng (MRC recevers operatng over Gamma shadowed Nakagam-m multpath fadng channel are calculated. Wreless communcaton system wth SC recever operatng over Nakagam-m multpath fadng channel n the presence of Nakagam-m co-channel nterference s consdered. In ths paper, probablty densty functon, cumulatve dstrbuton functon, moments, outage probablty and bt error probablty of consdered wreless system are evaluated. In [], the second order statstcs of MAD system are analysed. In ths paper, MAD system wth MAD SC recever and two MID SC recevers s studed. Receved sgnal propagates over correlated Gamma shadowed Webull multpath fadng channel and co-channel nterference. Closed form resson for cumulatve dstrbuton functon of MAD SC recever output sgnal envelope s calculated. To the best author s knowledge, MAD system wth two MID receptons operatng over Gamma shadowed Webull fadng channel n the presence of Webull co-channel nterference s not reported n open techncal lterature. SYSTEM MODEL MAD technque wth MAD SC recever and two MID recevers operatng over Gamma shadowed Webull multpath fadng channel n the presence of co-channel nterference subected to Webull short term fadng s consdered. Model MAD system s shown n Fg.. Receved sgnals envelopes and co-channel nterference envelope at nputs and outputs of MAD and MID recever are denoted n Fg.. Fgure MAD system Receved sgnal envelopes at nputs n the frst MID recever are denoted wth x and x and the nputs n the second MID recever wth x and x. The co-channel 376 Techncal Gazette 5, (8, 376-38
Mloš PERIĆ et al.: Outage Probablty of Macrodversty Recepton n the Presence Fadng and Webull Co-Channel Interference nterference envelopes at nputs of the frst MID SC recever are denoted wth y and y, at nputs n the second MID recever wth y and y. Ratos sgnal to nterference are denoted wth η, η, η and η. At outputs of MID SC recevers desred sgnal envelopes are denoted wth x and x, co-channel nterference envelope wth y and y and ratos of sgnal to nterference are denoted wth η and η. At output of MAD SC recever desred sgnal s denoted wth x, nterference wth y and η s sgnal to nterference rato. At nputs of MID SC recevers desred sgnal average power s denoted wth and and nterference sgnal average power wth S and S. Probablty densty functon (PDF of x, =,; =, s [3]: x px ( x = x, =,, =, ; x. ( F η dt ( η = ( S S = t = S = ( + Sη = S ( + Sη The CDF of η, =, s + S η η ( η = ( ( η η η η = F F F S ( + Sη (6 (7 s Probablty densty functon (PDF of y, =,; =, Random varables and have correlated Gamma dstrbuton y py ( y = y, =,, =, ; y. ( S S The rato of derved sgnal envelope end co-channel nterference sgnal envelope s p ( = c c+ Γ( c( ρ ρ + c ρ ( ρ =! Γ ( + c + c + c (8 x η =, x = η y. (3 y The PDF of η s p ( η = d y y p ( y p ( x = η y x η = dy y y S + = (4 l S = ( S η ( + Sη The cumulatve dstrbuton functon (CDF of η s η η ( (5 F ( η = d tp ( t = S dt ( + St η η After substtutng + St = zand n (5, CDF of η becomes t t dt = dz S where c s Gamma long term fadng severty parameter, ρ s Gamma long term fadng correlaton coeffcent and s average value of or. Random varable S and S follow also uncorrelated Gamma dstrbuton c ps S ( SS = S c S Γ( c c S S c Γ( c where c s Gamma parameter and s average value of S or S. 3 ANALYTICAL RESULTS MAD SC recever selects MID SC recevers wth hgher sgnal envelope average power at nputs to provde servce to user. Therefore, the CDF of η s η η F = d d F p ( + η η S, S η + d d F p ( = η, S η = d d F p ( = η, S (9 ( Tehnčk vesnk 5, (8, 376-38 377
Mloš PERIĆ et al.: Outage Probablty of Macrodversty Recepton n the Presence Fadng and Webull Co-Channel Interference = S c c+ Γ( c( ρ ρ + c ρ ( ρ =! Γ ( + c + c ( I I + I3 ( + c+ ( ( ( ρ = where (a n s denotng the Pochhammer symbol []. The ntegral I s ( ( + c+ η Γ ( I = S + c+ + η U + c+ +, + c+ +, S ( ρ The ntegral I 3 s ( + c+ η Γ ( I = S + c+ + 3 η U + c+ +,+ c+ +, S ( ρ (8 (9 + c + I = d = ( ρ + c+ ( ρ = Γ (+ c+ I The ntegral I s + c + + d = + ( ρ η ( S ( ( CDF of η can be calculated by averagng CDF of η/s reected to S Fη( η d Sp S ( S Fη S η = = = ( ( ( ρ c Γ( c 4 5 6 + c ρ = c ( ρ c+ = Γ( c( ρ ρ! Γ ( + c + c ( + c+ ( I + I + I ( After substtutng z S η =, = Sη z and d = Sη dz, the ntegral I becomes + c+ + + c + + ( η d ( Sη I = S zz Sη z ( ρ + ( z (3 The ntegral I 4 s c + 4 d I = SS I S = c + dss S ( = + c+ ( ρ Γ (+ c+ Prevously ntegral can be solved by usng formulas for confluent hypergeometrc functon U(a,b,z []: a (4 + + t U ( a, b, s = dtt ( st Γ ( a ( a b For consdered case, parameters a, b and s are = + c + + (5 b = + c + + + + = + c + (6 + The ntegral I 5 s c + 5 I = ds S I S = 4+ 4c+ ( c = η Γ + + + c + + + c+ dss S η U + c+ +, + c+ +, S ( ρ ( Sη s = ( ρ (7 After substtutng Eq. (4, Eq. (5, Eq. (6 and Eq. (7 n Eq. (3, the resson for I becomes η After substtutng ( S = ρ z, S ( ρ = z η and ( ρ ds = dz n Eq. (, the resson for I 5 η becomes 378 Techncal Gazette 5, (8, 376-38
Mloš PERIĆ et al.: Outage Probablty of Macrodversty Recepton n the Presence Fadng and Webull Co-Channel Interference ( 4+ 4c+ 5 I = η Γ + c+ + c+ + + c+ ( ρ η c + + + c+ ( ρ dzz z η U + c+ + + c+ + z (,, Prevously ntegral can be solved by usng formulas (3 b Γ( b Γ( b c+ dt t ( st U ( a, c, t = Γ ( a+ b c+ (4 F ( b, b c+, a+ b c+ ; s where F s hypergeometrc functon []. For consdered case s b = c c + (5 + + + b c+ = c+ + + c+ c + = c + a+ b c+ = + c+ + c+ a = = + c+ + c + 3a 3 (6 (7 After substtutng Eq. (4, Eq. (5, Eq. (6 and Eq. (7 n Eq. (3, the resson for I 5 becomes ( 4+ 4c+ 5 c+ + + c+ ( ρ I = η Γ + c+ + η Γ Γ ( c+ + + c+ ( c+ Γ ( + c+ + c+ 3 3 (,, F c + + + c+ c + ( ρ + c+ + c+ 3 3; η The ntegral I 6 s c + 6 = d 3 = 4+ 4c+ η Γ ( c c + + + c+ dss S I SS I S (8 = + + + (9 η U + c+ +,+ c+ +, S ( ρ η After substtutng ( S = ρ z, S ( ρ = z η and ( ρ ds = dz n (, the resson for I 6 η becomes ( 4+ 4c+ 6 c+ + + c+ ( ρ I = η Γ + c+ + η c + + + c+ ( ρ dzz z η U + c+ + + c+ + z = (,, 4+ 4c+ ( c = η Γ + + + c+ + + c+ ( ρ η Γ ( c+ + + c+ Γ ( c+ Γ ( + c+ + c+ 4 4 F c + + + c+, c +, ( ( ρ + c+ + c+ 4 4; η 4 NUMERICAL RESULTS (3 Cumulatve dstrbuton functon (CDF of consdered NAD system n terms of sgnal to nterference rato (SIR for dfferent values of Gamma long term fadng severty parameter c and Gamma long term fadng correlaton coeffcent ρ s shown n Fg.. CDF ncreases as SIR ncreases and outage probablty goes to one for hgh values of SIR or when SIR goes to nfnty. The nfluence of SIR on outage probablty s hgher for lower values of SIR. When severty parameter c ncreases, outage probablty decreases resultng n system performance mprovement. The nfluence of parameter c on outage probablty s hgher for lower values of SIR, also, the nfluence of severty parameter c on outage probablty s hgher for lower values of parameter c. When correlaton coeffcent decreases, CDF decreases and performance qualty s worse. The nfluence of correlaton coeffcent on outage probablty s hgher for lower values of SIR. The nfluence of correlaton coeffcent on outage probablty s smlar for lower and hgher values of parameter c. Cumulatve dstrbuton functon versus Gamma long term fadng severty parameter c for dfferent values of Gamma long term fadng correlaton coeffcent ρ s shown n Fg. 3. CDF decreases as parameter c ncreases and CDF fast decreases to lower values of correlaton coeffcents. In the numercal analyss, standard values of the system parameters are used [3]. Values of Gamma long term fadng severty parameter c go from to 3. Values of Gamma long term fadng correlaton coeffcent ρ go from to. Tehnčk vesnk 5, (8, 376-38 379
Mloš PERIĆ et al.: Outage Probablty of Macrodversty Recepton n the Presence Fadng and Webull Co-Channel Interference F(η.. c=, ρ=.75 c=, ρ=.5 c=, ρ=.5 c=, ρ=.75 c=, ρ=.5 c=, ρ=.5 =, =, =.5 correlaton coeffcent and Webull short term fadng nonlnearty parameter on outage probablty s analysed and dscussed. The system performance s better for lower values of outage probablty. Outage probablty decreases when Gamma shadowng severty parameter ncreases. Outage probablty has hgher values for lower values of Webull nonlnearty parameter. The nfluence of Webull parameter on outage probablty s hgher for lower values of Gamma fadng severty parameter. When correlaton coeffcent goes to one, the least sgnal occurs smultaneously on both MID SC recevers. In ths case, MAD system becomes MID system. 6 REFERENCES F(c.6.6.59.58.57.56.55.54-4 - η [db] Fgure Cumulatve dstrbuton functon ρ=. ρ=. ρ=.3 ρ=.4 =, η=, =, =.5.53..5.5.75..5.5.75 3. c Fgure 3 Cumulatve dstrbuton functon versus Gamma long term fadng severty parameter c 5 CONCLUSION The MAD technque wth the MAD SC recever and two MID SC recevers are operatng on the shaded multlayer pale channels n the presence of the co-channel nterference subected to short-term fadng. Receved derved sgnal and co-channel nterference erence Webull multpath fadng and correlated Gamma long term fadng. MAD SC recever reduces Gamma long term fadng effects and MID SC recevers reduce Webull short term effects fadng on system performance. MAD SC recever selects MID SC recever wth hgher sgnal envelope average power to provde servces to user and MID selects and outputs branch wth the hgher rato of sgnal envelope and nterference envelope resultng n system performance mprovement. Closed form resson for cumulatve dstrbuton functon for sgnals at outputs MID SC recevers s calculated. By usng these formulas, cumulatve dstrbuton functon of MAD SC recever output sgnal envelope s effcently calculated. Outage probablty of MAD system can be evaluated from cumulatve dstrbuton. The nfluence of Gamma long term fadng severty parameter, Gamma long term fadng [] Stuber, G. L. (3. Moble communcaton, nd ed. Kluwer Academc Publsher, Dordrecht. [] Smon, M. K. & Aloun, M. S. (. Dgtal Communcaton over Fadng Channels. John Wley & Sons, USA. https://do.org/./47697 [3] Panc, S., Stefanovc, M., Anastasov, J., & Spalevc, P. (3. Fadng and Interference Mtgaton n Wreless Communcatons. CRC Press, USA. https://do.org/./b675 [4] Mukheree, S. & Avdor, D. (3. Effect of Mcrodversty and Correlated Macrodversty on Outages n a Cellular System. IEEE Transactons on Wreless Technology, (, 5-59. https://do.org/.9/twc..86363 [5] Shankar, P. M. (8. Analyss of mcrodversty and dual channel macrodversty n shadowed fadng channels usng a compound fadng model. Internatonal Journal of Electroncs and Communcatons (AEUE, 6(6, 445-449. https://do.org/.6/.aeue.7.6.8 [6] Yacoub, M. D. (7. The η-μ dstrbuton and the κ-μ dstrbuton. IEEE Antennas and Propagaton Magazne, 49(, 68-8. https://do.org/.9/map.7.37983 [7] Malmgren, G. (997. On the performance of sngle frequency networks n correlated shadow fadng. IEEE Transactons on Broadcastng, 43(, 55-65. https://do.org/.9/.598364 [8] Goldsmth, A. (5. Wreless communcatons. Cambrdge Unversty press, Cambrdge, UK. https://do.org/.7/cbo9785844 [9] Sekulovc, N., Stefanovc, M., Mlovc, D., & Stanocc, S. (4. Second-Order Statstcs of System wth N-branch Mcrodversty and L-branch Macrodversty Operatng over Gamma Shadowed Nakagam-m Fadng Channels. Internatonal Journal of Communcaton Systems, 7(, 39-4. https://do.org/./dac.369 [] Stefanovc, D., Panc, S., & Spalevc, P. (. Secondorder statstcs of SC macrodversty system operatng over Gamma shadowed Nakagam-m fadng channels. AEU Internatonal Journal of Electroncs and Communcatons, 65(5, 43-48. https://do.org/.6/.aeue..5. [] Gradshteyn, I. S. & Ryzhk, I. M. (. Table of Integrals, Seres and Products. Academc Press, San Dego, USA. 38 Techncal Gazette 5, (8, 376-38
Contact nformaton: Mloš PERIĆ et al.: Outage Probablty of Macrodversty Recepton n the Presence Fadng and Webull Co-Channel Interference Mloš PERIĆ, Faculty of Electronc Engneerng, Unversty of Nš Aleksandra Medvedeva 4, 8 Nš, Serba E-mal: mlosperc@yahoo.com Branmr JAKŠIĆ, Faculty of Techncal Scences, Unversty of Prštna Knaza Mloša 7, 38 Kosovska Mtrovca, Serba E-mal: branmr.aksc@pr.ac.rs Danela ALEKSIĆ, College of Appled Techncal Scences Aleksandra Medvedeva, 8 Nš, Serba E-mal: danela.alekscnkolc@gmal.com Dragan RANDJELOVIĆ, Academy of Crmnalstc and Polce Studes Cara Dusana 96, 8 Belgrade, Serba E-mal: dragan.randelovc@kpa.edu.rs Mhalo STEFANOVIĆ, Faculty of Electronc Engneerng, Unversty of Nš Aleksandra Medvedeva 4, 8 Nš, Serba E-mal: msa.profesor@gmal.com Tehnčk vesnk 5, (8, 376-38 38