Technical Report: A Unified Framework for Analysis of Path Selection Based Decode-and-Forward (DF) Cooperation in Wireless Systems

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1 Techncal Report: A Unfed Framework for Analyss of ath Selecton Based Decode-and-Forward DF Cooperaton n Wreless Systems Neeraj Varshney Student ember, IEEE and Adtya K. Jagannatham, ember, IEEE I. VALUES OF ARAETERS t AND a, {SR, SD, RD} FOR DIFFERENT FADING CHANNELS The values of parameters t and a, {SR, SD, RD} for varous fadng channels ncludng η µ and κ µ are shown n Table I. II. SILIFICATION OF THE ROBABILITY OF ERROR FOR THE EVENT ϕ IN 6 Usng the result for F γmn x = F γsr x + F γrd x F γsr xf γrd x from Eq5,, the resson for re ϕ n 6 n the man paper can be rewrtten as, re ϕ= = γsd re ϕ, γ SD F γmn f γsd γ SD dγ SD, α γsd γsd re ϕ, γ SD F γsr f γsd γ SD dγ SD + re ϕ, γ SD F γrd f γsd γ SD dγ SD α α γsd γsd re ϕ, γ SD F γsr F γrd f γsd γ SD dγ SD, α α Substtutng the ressons for F γ x, {SD, SR, RD}, f γsd x gven n, 3 respectvely n the man paper and usng re ϕ, γ SD = π dθ 3, yelds π sn π/γ SD sn θ DRAFT

2 Fadng Channel DF fβ t a Raylegh β δ δ δ Nakagam-m, m Generalzed Nakagam Nakagam-q, q Nakagam-n Webull, c Log-Normal, σ Shadowed-Rcan, b, m, Ω κ-µ dstrbuton η-µ dstrbuton Gamma-Gamma, µ, ν, η s β m s p m +n δ Γm c s β +q qδ n Γ+ c.5 b µ +κ µ κ m m β m mβ δ m Γm p, where p = +q β +n β δ 4q δ δ c/ β c bm bm+ω µ + β δ δ Γm Γm +s I q 4 β I n β m s, s >, m m s 4q δ +n β δ δ Γ + c 4.34 πσβ log βµ σ µ µ κ δ µ + πµ µ + h µ βµ Γµ H µ δ µ +, n c/ m β.5ω b F m, ; β b m+bω I µ µ +κ β δ µ h β δ I µ +η µ κ +κ β δ µ H β For format : < η <, h = +η, H 4 = η For format : < η <, h =, H η = η k= ζ k ν, µβ ν+k + k= ζ k µ, νβ µ+k where ζ k a, b = δ +n δ c c µ σ µ, µ η 4 η πab a+k η a+k snbaπk!γaγbγab+k+ f ν < µ then t gg = ν/ and t gg = ζ ν, µ f ν > µ then t gg = µ/ and t gg = ζ µ, ν TABLE I m m δ m Γm s p m Γm +q qδ n Γ+ c c δ 4.34 πσ µ σ m.5 b bm bm+ω µ +κ µ µ κ δ µ Γµ πµ µ h µ Γµ Γµ +.5δ µ, t gg a gg VALUES OF ARAETERS t AND a, {SR, SD, RD} FOR DIFFERENT FADING CHANNELS. the bound for re ϕ as, re ϕ π + π a RD a SD t RD + α t RD+ a SR a SD t SR +α t SR+ η tsd + η a SR a RD a η SD t SR +t RD +α t SR+t RD + tsr + η trd + η tsd + tsd + γ t RD+t SD + SD tsr + trd + η η γ t SR+t SD + SD sn π/γ SD sn θ sn π/γ SD sn θ γ τ SD sn π/γ SD sn θ dγ SD dγ SD dγ SD where τ = t SD + t SR + t RD +. The resson above can be smplfed usng the dentty x n µx = n!µ n 3.36-, 4 followed by gnorng the negatve term n the resultng resson to yeld the bound n 8 n the man paper as, re ϕ a SRa SD Γt SR + t SD + ζt SR + t SD + t SR + α t SR+ sn π/ t SR +t SD + η tsr + η tsd + dθ,

3 + a trd + RDa SD Γt RD + t SD + ζt RD + t SD + t RD + α t RD+ sn π/ η η tsd + t RD +t SD +. III. SILIFICATION OF THE ROBABILITY OF ERROR FOR THE EVENT ϕ IN Usng the result for f γmn x = f γsr x + f γrd x F γsr xf γrd x f γsr xf γrd x from Eq5,, the resson for re ϕ n n the man paper can be rewrtten as, re ϕ = π = π + π π sn π/γ mn γ mn = sn F γsd αγ mn f γmn γ mn dγ mn dθ, θ sn π/γ mn sn F γsd αγ mn f γsr γ mn dγ mn θ sn π/γ mn sn θ sn π/γ mn sn θ sn π/γ mn sn θ F γsd αγ mn f γrd γ mn dγ mn F γsd αγ mn F γsr γ mn f γrd γ mn dγ mn F γsd αγ mn f γsr γ mn F γrd γ mn dγ mn dθ. 3 Usng the ressons for F γ x, f γ x, {SD, SR, RD} gven n, 3 respectvely n the man paper, the above resson can be smplfed as, re ϕ π π + a SDa RD α t SD+ t SD + a SD a SR α t SD+ t SD + a SDa SR a RD α t SD+ η t SD + t SR + a SDa SR a RD α t SD+ t SD + t RD + η tsd + η η η tsd + η tsd + η tsd + η trd + tsr + tsr + trd + η γ t SD+t SR + mn γ t SD+t RD + mn tsr + trd + η sn π/γ mn sn θ sn π/γ mn sn θ dγ mn γmn τ sn π/γ mn sn dγ mn θ dγ mn γ τ mn sn π/γ mn sn θ The resson above can be further smplfed usng the dentty x n µx = n!µ n 3.36-, 4 followed by gnorng the negatve term n the resultng resson to yeld the dγ mn dθ.

4 bound n n the man paper as, re ϕ a SRa SD Γt SR + t SD + ζt SR + t SD + α t SD+ t SD + sn π/ t SR +t SD + η + a RDa SD Γt RD + t SD + ζt RD + t SD + α t SD+ t SD + sn π/ t RD +t SD + tsr + η η tsd + trd + η tsd +. 4 IV. IO-OSTBC BASED COOERATION WITH ATH SELECTION The ressons for the SER and dversty order n the IO-OSTBC based cooperatve system can be readly obtaned by substtutng Ns N Ns R c m d m SD SD a SD =, t N s N d m SD! δsd SD = N s N d m SD, Ns N Ns R c m r m SR SR a SR =, t SR = N s N r m SR, N s N r m SR! and δ SR Nr R c m Nr N RD d m RD, a RD = trd = N r N d m RD, N r N d m RD! δ RD n equatons 4, 5 gven n the man paper respectvely as, e Θ Ns N ζn s N d m SD +N s N r m SR r m SR η η NsNd m SD sn π/ NsN dm SD +N sn rm SR ΓN s N r m SR +N s N d m SD N s N r m SR α N sn r + αnsndmsd m SR N s N d m SD + Θ Nr N ζn s N d m SD +N r N d m RD d m RD η η Ns N d m SD sn π/ N sn d m SD +N r N d m RD ΓN r N d m RD +N s N d m SD + αnsndmsd, 5 N r N d m RD α N rn d m RD N s N d m SD d ath,ostbc =N s N d m SD + mn{n s N r m SR, N r N d m RD }, 6 where Θ = a SD a SR = N sn d m SD!N sn rm SR! Θ = a SD a RD = N s N d m SD!N r N d m RD! N s R c m SD δ SD N s R c m SD δsd Ns N d m SD Ns N d m SD N s R c m SR δsr Nr N d m N r R c m RD RD. δrd Ns N r m SR and Further, the optmal power for ths system can be obtaned by usng the polynomal equaton gven below C t SR+ C t RD+ =, 7

5 where C and C are gven as, C = ζn sn d m SD +N r N d m RD ΓN s N d m SD +N r N d m RD N r N d m RD η N rn d m RD sn π/ NrN dm RD Nr R c m Nr N RD d m RD, N r N d m RD! δ RD C = ζn sn d m SD + N s N r m SR ΓN s N d m SD + N s N r m SR N s N r m SR η N sn r m SR sn π/ N sn r m SR Ns N Ns R c m r m SR SR. N s N r m SR! δ SR V. GENERALIZED ANALYSIS FOR JOINT TRANSIT-RECEIVE ANTENNA AND ATH SELECTION JTRAS and Smlar to the IO-OSTBC based cooperaton, usng the values a SD = N sn d m SD /δsd m SDN s N d Γm SD + N sn d m SD!, t SD = m SD N d N s, a SR = N sn r m SR /δsr m SRN s N r Γm SR + NsNr m SR!, t SR = m SR N s N r, a RD = N rn d m RD /δrd m RDN r N d Γm RD + N rn d m RD!, t RD = m RD N d N r, derved n the man paper for the JTRAS based system, the closed form ressons for the asymptotc SER and dversty order n JTRAS based cooperaton are gven by, e Θ ζn s N d m SD +N s N r m SR sn π/ NsN dm SD +N sn rm SR ΓN s N r m SR +N s N d m SD η + Θ ζn s N d m SD +N r N d m RD sn π/ NsN dm SD +N rn d m RD ΓN r N d m RD +N s N d m SD NsNrm SR η Ns N d m SD N s N r m SR α NsNrm SR η + αn sn d m SD N s N d m SD Nr N d m RD η NsNd m SD N r N d m RD α N rn d m RD + αnsndmsd N s N d m SD, 8 d ath,jtras =N s N d m SD + mn{n s N r m SR, N r N d m RD }, 9

6 Ns where Θ = a SD a SR = N rn d m SD /δsd m SD NsN dm SR /δsr m SR N snr Γm SD + N sn d and Θ m SD!Γm SR + NsNr m SR! = a SD a RD = N s N r Nd m SD/δSD m SD N sn dmrd /δrd m RD N rn d Γm SD + N sn d m SD!Γm RD + N rn d. oreover, the optmal power for the JTRAS m RD! system can be obtaned by usng the equaton 7, where C and C are defned as, C = ζn sn d m SD +N r N d m RD ΓN s N d m SD +N r N d m RD N r N d m RD η NrN dm RD sn π/ N rn d m RD N r N d m RD /δrd m RDN r N d Γm RD + NrN d m RD!, C = ζn sn d m SD + N s N r m SR ΓN s N d m SD + N s N r m SR N s N r m SR η NsNrm SR sn π/ N sn r m SR N s N r m SR /δsr m SRN s N r Γm SR + NsNr m SR!. VI. ADDITIONAL SIULATION RESULTS In order to demonstrate the system performance for hgher order SK modulaton schemes, we consder a path selecton based SISO system n whch each lnk erence Nakagam-m fadng wth severty parameters m SR = m RD =, m SD = and average channel gans δsr =, δ SD = δ RD =.. From Fg., t can be observed that the monte-carlo results obtaned for the hgher order SK modulaton closely match wth the asymptotc SER approxmaton n 4 gven n man paper, whch clearly valdates that the analytcal framework developed n ths work s applcable for a general -SK modulaton. It can also be noted that the system performance sgnfcantly degrades as the order or the number of constellaton ponts ncreases. However, the system acheves the dentcal dversty order of t SD +mn{t SR +t RD }+ = m SD +mn{m SR, m RD } = 3 for each modulaton scheme. 6SK SER 3 4SK or QSK 8SK 4 5 SISO, =.773, =.87 Optmal SISO, =.5, =.5 Equal Asymptotc Bound Optmal, Analytcal Asymptotc Bound Equal, Analytcal SNRdB Fg.. SER erformance of path selecton based SISO system correspondng to the transmsson of QSK or 4-SK, 8-SK, and 6-SK modulated symbols.

7 On the other hand, Fg shows that the value of cooperaton threshold α also affects the system performance. It can be clearly seen n Fg. that the end-to-end system performance sgnfcantly mproves as the cooperaton threshold α ncreases. Therefore, one can note that n addton to power allocaton, the value of cooperaton threshold α also plays a key role, whch can also be optmzed to enhance the end-to-end performance of the path selecton scheme. SER 3 4 α=.,.5, 5 SISO, =.5, =.5 Equal Asymptotc Bound Equal, Analytcal SNRdB Fg.. SER erformance of path selecton based SISO system for dfferent values of cooperaton threshold α. REFERENCES. D. Yacoub, The κ-µ dstrbuton and the η-µ dstrbuton, IEEE Antennas and ropagaton agazne, vol. 49, no., pp. 68 8, 7. N. Varshney, V. Krshna, and A. Jagannatham, Capacty analyss for path selecton based DF IO-STBC cooperatve wreless systems, IEEE Communcatons Letters, vol. 8, no., pp , 4. 3 K. R. Lu, Cooperatve communcatons and networkng. Cambrdge Unversty ress, 9. 4 A. Jeffrey and D. Zwllnger, Table of ntegrals, seres, and products. Access Onlne va Elsever, 7.