Asymptotic Gains of Generalized Selection Combining

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1 Asymptotic Gains of Generaized Seection Combining Yao a Z Wang S Pasupathy Dept of ECpE Dept of ECE Dept of ECE Iowa State Univ Iowa State Univ Univ of Toronto, Emai: mayao@iastateedu Emai: zhengdao@iastateedu Emai: pas@commutorontoca Abstract In this paper, empoying the poynomia approimation of the fading channe probabiity density function pdf for arge average signa-to-noise ratio ASNR, we derive genera asymptotic moment generating function GF epressions of the GSC output signa-to-noise ratio SNR for generaized fading channes Based on the GF resut, the asymptotic diversity and coding gains for GSC are derived Our anaytica resuts revea that the diversity gain of GSC is equivaent to that of maimum ratio combining RC, for different moduations and generaized fading channes incuding the case of correated branches We aso show that the difference in the coding gains for different moduations is manifested in terms of a moduation factor defined in this paper Asymptotic performance gap between GSC and RC, and the gap between non-coherent and coherent GSCs respectivey, are studied in terms of the coding gain I INTRODUCTION Generaized seection combining GSC is a practica and usefu diversity combining scheme for wireess communications, and has attracted a ot of research interest recenty [ [6 In GSC, N N L branches with the argest instantaneous SNRs are seected from a tota of L branches and combined, and we refer to it as GSC N,L Note that coherent GSC L, L and GSC,L correspond to the maimum ratio combining RC and the conventiona seection combining CSC, respectivey; whie non-coherent GSC L, L corresponds to the post-detection equa gain combining EGC However, some important issues about performance of GSC in generaized fading channes remain uncear: Performance gaps between CSC, RC and a genera GSC N,L for different N; Effect of different moduations on the GSC performance; 3 Effect of correated diversity branches Thus, an anaytica resut is much desired In this paper, we try to approach these issues for an asymptotic scenario of high ASNR Based on the GF resut of the GSC output SNR in [5, and the poynomia approimation of the fading channe pdf for sma instantaneous SNR [7, we derive diversity and coding gains for the error and outage probabiities for a GSC receiver with different moduations and in generaized fading channes, incuding independent but non-identicay distributed ind branches, and correated branches Our resuts demonstrate that GSC N,L achieves the same diversity gain as RC, for the error and outage probabiity performance We aso show that the difference in the coding gains for different moduations is manifested in terms of a moduation factor defined in this paper, and thus the performance gaps between different moduations can be anayticay predicted The performance oss of GSC with respect to wrt RC, and the performance improvement of GSC wrt CSC are anayticay quantized Aso, the noncoherent combining oss NCL, defined as the coding gain gap between coherent and non-coherent GSCs for differentia and non-coherent detection, is derived These resuts put new insight into the effect of different system and channe parameters on the GSC performance at high ASNRs, and aid in the design of an efficient GSC receiver II ASYPTOTIC GF OF THE GSC OUTPUT SNR A Background For a GSC N,L receiver, we define the SNR vector from the L receiving channes as γ =[γ n,γ n,,γ nl T, where γ n represents the instantaneous SNR in the n th diversity branch; n,,,l for L, and n,n,,n L is a permutation of,,,l We arrange the eements in γ in descending order as γ = [γ,γ,,γ L T,so γ is an ordered SNR set The GSC output SNR is given by γ s = N γ and its GF is defined as Φ γs s =E[ep γ s s From the resut in [5, a genera GF epression for the GSC output SNR on arbitrary ind fading channes is given by Φ γs s = e s f nn n,,n n N n <n <<n [ [ Φ n s, =N+ F n d where f n and F n = f n ydy are the pdf and the cumuative distribution function cdf epressions for the output SNR in the n th branch, respectivey, for n =,,L Φ n s, = f n ye sy dy is the incompete GF epression originay defined in [6, where f n is the pdf epression for the SNR γ n Note that Φ n s, = Φ n s, and Φ n, = F n, where F n is the cdf of the SNR γ n In, n,,n denotes the L n <n <<n possibe combinations of seecting the N argest SNR branches out of the L branches, and in n N the inde n N is chosen from the remaining L N + branches /3/$7 3 IEEE 349

2 However, does not provide the insight as to which parameters determine the performance of GSC at high ASNRs We study this issue in terms of the diversity order and coding gain, which is defined net For high ASNRs, if the error probabiity can be written as P Err =G c G d + o G d where is the input ASNR per branch, and o satisfies o the property that im = Then we ca G c as the coding gain which iustrates an improvement or degradation factor to the ASNR per branch, and G d as the diversity gain, which manifests the effective diversity order and determines the asymptotic sope of the error probabiity curves versus the input ASNR at high ASNRs, in a og-og scae, see eg [7 For outage performance of GSC with a predefined SNR threshod γ th, if the asymptotic outage probabiity [defined as P OT γ th =Prγ s <γ th can be epressed as Od P OT γ th = O c + o O d 3 γ th then we ca O c and O d as the outage coding gain and diversity gain, respectivey B GF for IND Channes For different fading channe types, the epressions for the poynomia approimation of the pdfs were originay given in [7 Here for competeness of this work we ist them in Tabe I, athough in a sighty different form due to the definition We epress the pdf as fγ =aγ t +oγ t for the high ASNR Aso, we find that for a the fading channe types studied in this paper, we can epress a as a = b/ t+, as shown in Tabe I For arge ASNRs, the incompete GF can be approimated as Φs, a e su u t du Furthermore, assuming s is a positive rea scaar, a condition which hods true for the anaysis we use ater, we can write Φs, aγt +,s/s t+, where Γt, = e u u t du is the compementary incompete Gamma function Aso, the cdf F is approimated as F a t+ /t + By substituting the resut above to the genera GF epression, we obtain [ L Φ GSC s a e s [ n,,n n N n <n <<n Γt n +,s s tn + L =N tn + =N+ t d 4 n + By using a change of variabe that y = s, we can obtain a genera asymptotic GF epression for GSC in generaized fading channes as a Φ GSC s s L t + n,,n n N n <n <<n Γt n +,y y L =N tn + L =N+ t n + e y dy 5 To proceed further, et g = /, where is the tota input TABLE I EXPRESSIONS FOR THE PDF AND OTHER PARAETERS FOR THE SNR IN A DIVERSITY BRANCH OVER RAYLEIGH, RICIAN, NAKAGAI-q AND NAKAGAI-m FADING CHANNELS IS THE INPUT ASNR Fading Types PDF epression fγ fγ aγ t, t, a, andb γ Rayeigh ep t =, a =/, b = +q Nakagami-q q ep +q γ 4q t =, a = +q q, q I γ 4q b = +q q +K Rician ep K +Kγ t =, a =+Ke K /, I K + Kγ/ b =+Ke K Nakagami-m γm mm m m ep Γm m γ t = m, a = Γm m, b = mm Γm ASNR divided by L Thus L k= g k = L Aso, we define two vectors b =[b,,b L and t =[t,,t L We obtain a genera GF resut as shown beow Proposition The asymptotic GF for the GSC N,L output SNR is reated to the ASNR by Φ GSC s F b, ts L t + where F b, t = L b g t + Γt n +, n,,n n <n <<n L =N tn + L =N+ t n + e n N 6 d 7 Using the fact that the GF for the SNR at the th branch for =,,L is approimatey given by Φ s a e su u t du Γt + = a s t = b g t+ Γt s t + we obtain the GF for GSC L, L or RC for generaized fading channes as Φ RC s b g t + Γt + s L 9 t + C Resuts for Correated Branches We study GSC N,L operating in a correated Nakagamim fading channe Let the power correation matri for the L branches be denoted by P =ρ ij, which is an L L matri whose i, jth entry is ρ ij Let us define a joint GF for γ = [γ,γ,,γ L as Φγs = Ee s γ, where s = [s,,s L Byusinga /3/$7 3 IEEE 35

3 resut in [8, we get Φγs = L + s /m m ρ + m s det ρl + m s L L ρl + m s L L ρl + m s m For the high ASNR case where, we get [ L det Φγs s /m m [ m ρij From we observe that the branch correation equivaent introduce an factor det [ ρij m to the GF, otherwise the resut is identica to that for the independent branches D Resuts for Specia Cases A Specia Case for IND Channes: Let us consider the case where the ASNRs in different branches can be non-identica Aso, the signa SNRs at the different branches can foow different famiies of distributions, incuding Rayeigh, Rician, and/or Nakagami-q distributions but ecuding Nakagami-m distribution with m For these channes, since t = for =,,L, we can write Γt n +,=Γ,=e Thus 6 can be simpified to Φ GSC s = b g t + s L b g t + t + N L e N d N L! s L N!N Further, assuming the ASNRs at a the branches are identica, we get b L L! Φ GSC s s L N!N III PERFORANCE ANALYSIS Based on the new asymptotic GF resuts, we derive the asymptotic coding and diversity gains for the error and outage probabiities of GSC in generaized fading channes A Average Error Probabiity The average symbo and bit error probabiities ASEP and ABEP for a arge cass of moduation are studied BPSK: By using the integra form of the BEP epression for BPSK [9, we get P BPSK π/ Φ GSC π sin dθ 3 θ where Φ GSC sin θ is obtained from 6 by repacing variabe s with sin θ Since s L t+ is the ony factor in 6 that is invoved in the integra 3, we can etract the factor in 3 which is directy reated with the BPSK moduation format as G,BPSK = π = π/ π sin θ L t+ dθ 4 Γ L t ++5 Γ L t ++ 5 We define G,BPSK as the moduation factor for BPSK The ABEP of BPSK can be rewritten as P BPSK = F b, t G,BPSK L t + By comparing P BPSK with, we get the diversity gain for GSC N,L as G d = L t +, and the coding gain as /Gd G c,bpsk = F b, t ΓG d +5 6 π ΓG d + When N = L, the ABEP for RC can be computed by using 9 as [ L b g t + Γt + G,BPSK P BPSK = L t + Thus the diversity gain of RC is aso given by G d = L t +, but the coding gain is given by G c = [ b g t + Γt + G /Gd,BPSK Proposition GSC N,L achieves the same diversity gain as the RC on both independent and correated fading channes, but the coding gains differ by a constant depending on N,L and the channe parameter {t } L We note that Proposition is true even for the case [ det ρij is rank deficient, in which case the diversity gain is smaer than L t + We aso note that for BPSK and a the other moduations studied ater in this paper, the foows proposition hods Proposition 3 The diversity gain of GSC N,L is given by G d = L t +, and the coding gain is given by G c = F b, t G /Gd 7 where G is the moduation factor, and b, t are the channe parameters defined in Tabe I Assume matri ρ ij has fu rank, thus G d = ml We observe that the power correation in Nakagami-fading channe simpy brings a coding gain oss of det [ /L ρij than independent diversity Due to the space imitations, we present ony the moduation factors for some popuar moduations beow, and more resut is given in [ PSK: The moduation factor is given by G,PSK = π/ Gd sin θ dθ π sinπ/ = B sin[ π/ G d +5, 5 π[sinπ/ G d /3/$7 3 IEEE 35

4 3 -ary Quadrature Ampitude oduation QA: G,QA = [ πg G d qam BG d +5, 5 B 5 G d +5, 5 4 BDPSK and BFSK: First, from the inverse Lapace transform of 6, we get the pdf of the GSC output SNR as f GSC γ F b, t G d γ G d /ΓG d 8 By using 8 and integrate the conditiona BEP over the fading channe pdf, we have G N = c k Γk + G d g G 9 d k! ΓG d k= where c k = k n= n, g =for BDPSK and g =5 for BFSK To summarize, the resuts above demonstrate that both coherent and non-coherent GSCs have the same diversity gain same as RC or post-detection EGC, and the differences in the coding gains for different moduations are manifested in the moduation factor G ony, which can be anayticay evauated by using the resuts given above B Outage Probabiity We can use 8 and obtain an asymptotic outage probabiity as P OT,GSCγ th F b, t ΓG d + /γ th G d The resut shows that for GSC the outage diversity gain is the same as the error probabiity diversity gain, that is O d = G d Aso, the outage coding gain is given by O c = F b,t ΓG d + /Gd IV DISCUSSION A Loss of Coding Gain of Coherent GSC The performance of coherent GSC N,L is upper-bounded by that of GSC L, L or RC, and in this subsection we study the performance gap in terms of the coding gain The oss of the coding gain ASNR oss per branch of GSC with respect to that of RC can be defined as β N,L = The genera resut is given by β N,L = Gc,RC G c,gscn,l F t k= Γt k + /Gd where G d is the diversity gain, and F t is given by F t = n,,n n N n <n <<n Γt n +, e L =N tn + L =N+ t n + d For the case of t = = t L = t =, but the ASNRs are not necessariy identica, can be reduced to /L L! β N,L = N!N We note that in [ it was proved that the ASNR penaty of GSC wrt the RC for PSK moduation at high ASNR in an independent and identicay distributed iid Rayeigh fading channe is given by L! /L N!N In comparison, our resut in is more genera because we proved that this hods true for a arge cass of moduations at high ASNR and in different ind channes where t =incuding mied Rayeigh, Rician and Nakagami-q fading distributions For a sma N and N>, we study the improvement factor of the coding gain wrt that of CSC, which is defined as α N,L = G c,gscn,l G c,gsc,l = β,l β N,L It is not difficut to show that /L N! α N,L = N, and N N L α N,L N When N = L/, the coding gain oss of GSC L/,L is given by by repacing N with L/ By appy the Stiring s formua we can get β L/,L L / e 3 It can be show that β L/,L decreases monotonicay as L increases and og β L/,L < db when L B Non-Coherent Combining Loss NCL We define the NCL as the coding gain oss of non-coherent GSC wrt coherent GSC for DPSK/NFSK For the coherent GSC, the moduation factor can be obtained by setting N = in 9, as given by G = By using 9 the NCL is g G d give by β N,L NC = G N,L c,coh /G N,L /Gd c,nc = GN / G /Gd c k Γk + G d = + N 4 k!γg d Since k= β NC N,L k= c k Γk+G d N k!γg d, we can concude that hods true for generaized fading channes studied in this paper It can be shown numericay that β N,L in 4 monotonicay decreases as G d increases Since G d increases when L increases, we observe that GSC N,L reduces the NCL wrt GSC N,N V NUERICAL RESULTS We first present eact and asymptotic ABEP resuts for GSC N,4 with BPSK moduation on an ind Rician fading channe in Fig We assume the ASNRs at different receiving branches foow an eponentiay decaying power deay profie PDP, where the ASNRs decrease by 5 db between the adjacent paths successivey The -ais is defined as the tota input ASNR divided by L We assume from the strongest to the weakest branches K = [5, 3,, db, respectivey The resuts in Fig show that the asymptotic ABEP curves become accurate at high ASNR for a N /3/$7 3 IEEE 35

5 Bit error probabiity Soid ines: eact Dashed ines: asymptotic N= L=4 Non coherent combining oss db N=5 N=6 N=7 N=8 N=9 N= Average SNR per branch db G d Fig Eact in soid ines and asymptotic in dashed ines ABEPs of GSC N, 4 on an ind Rician fading channe Fig 3 The NCL in db for DPSK/NFSK wrt G d in fading channes Coding gain oss db N= N=5 N=6 N=7 N=8 N=9 L= Soid ines: t= Dashed ines: t=3 N,N which is the post-detection EGC VI CONCLUSIONS We anayzed the asymptotic error and outage probabiities of GSC N,L, in terms of the diversity and coding gains, over generaized fading channes Our resuts demonstrate that the diversity gain of GSC N,L is the same as that of RC Aso the coding gain gaps between different moduations and GSC N,L with different N can be anayticay predicted In a future study, the performance of GSC in space-time processing and UWB communications wi be anayzed Number of branches L Fig Coding gain oss of GSC N, L wrt GSC L, L on fading channes, for t =soid ines and t =3dashed ines respectivey Net, we present the coding gain oss of GSC wrt RC in fading channes where parameter t is identica for a the L branches in Fig When L increases, the performance gap between GSC N,L for N<L and RC increases, especiay for CSC N = For L =and t =,the improvement of N =over N =is about 7 db, which is very cose to the 3 db upper bound When N = L/ =5, the coding gain oss is about 986 db, which is very cose to our predicted vaue 989 db by using 3 We aso note that when t =3, the coding gain oss of CSC becomes arger than that when t =; in the mean time, for a sma N N >, the performance improvement of GSC N,L wrt CSC N = is more significant Finay, we give the NCL of DPSK/NFSK in Fig 3 We observed that i when N increases, the NCL becomes arger; ii for a N, the NCL is a monotonicay decreasing function of G d The resuts demonstrate non-coherent GSC N,L for L>N has a smaer NCL than that for non-coherent GSC REFERENCES [ N Kong, T Eng, and L B istein, A seection combining scheme for Rake receivers, in the Proc ICUPC, Japan, 995, pp [ -S Aouini and K Simon, Appication of the Dirichet transformation to the performance evauation of generaized seection combining over Nakagami-m fading channes, Journa of Communications and Networks, vo, no, pp 5 3, arch 999 [3, An GF-based performance anaysis of generaized seection combining over Rayeigh fading channes, IEEE Trans Commun, vo 48, no 3, pp 4 45, arch [4 Y a and C C Chai, Unified error probabiity anaysis for generaized seection combining in Nakagami fading channes, IEEE J Se Areas Comms, vo 8, no, pp 98, Nov [5 Y a and S Pasupathy, Efficient performance evauation for generaized seection combining on generaized fading channes, IEEE Trans Wireess Commun, 3, accepted for pubication [6 A Annamaai and C Teambura, A new approach to performance evauation of generaized seection diversity receivers in wireess channes, in Proc IEEE Vehicuar Technoogy Conference, Fa, pp [7 Z Wang and G B Giannakis, A simpe and genera parameterization quantifying performance in fading channes, IEEE Trans Commun,, accepted for pubication [8 Y-C Ko, -S Aouini, and K Simon, Outage probabiity of diversity systems over generaized fading channes, IEEE Trans Commun, vo 48, no, pp , Nov [9 K Simon and D Divsaar, Some new twists to probems invoving the Gaussian probabiity integra, IEEE Trans Commun, vo 46, no, pp, Feb 998 [ Y a, Z Wang, and S Pasupathy, Asymptotic performance of generaized seection combining in fading channes, IEEE Trans Commun, arch 3, submitted for pubication [ Z Win, N C Beauieu, L A Shepp, B F Logan, and J H Winters, On the SNR penaty of PSK with hybrid seection/maima ratio combining over iid Rayeigh fading channes, IEEE Trans Commun, vo 5, no 6, pp 3, June /3/$7 3 IEEE 353