DF vs. AF
In this section, we characterize the effective throughput for the cooperative ARQ schemes presented in the preceding sections. The throughput is defined as the average number of successfully transmitted data divided by the average number of transmission time. By definition, it can be formulated as
T hroughput = Tp× Rrate× [1 − Pout,N(Rrate)]
Tp× E{N } = Rrate× [1 − Pout,N(Rrate)]
E{N } . (5.1)
Two phase:
T hroughput = 2R × [1 − Pout,N(2R)]
E{N } . (5.2)
One phase:
T hroughput = R × [1 − Pout,N(R)]
E{N } . (5.3)
The outage probability Pout,N(R) for each cooperative ARQ scheme has been given in the previous chapter. It remains to characterize the expected time of ARQ transmissions, E{N}, for each scheme. To simplify the following derivation, we define the outage event for cooperative phase as E0 and the outage event of following ARQs phase as Ei. For the cooperative ARQ , we have
Two phase:
E{N} = 2P {E0} + 3P {E0∩ E1} + 4P {E0∩ E1∩ E2} + · · ·
+ (N + 2)P {E0∩ E1∩ E2∩ · · · ∩ EN} (5.4)
One phase:
E{N } = P {E0} + 2P {E0∩ E1} + 3P {E0∩ E1∩ E2} + · · ·
+ (N)P {E0∩ E1∩ E2∩ · · · ∩ EN} (5.5)
DF protocol:
Pout,0 = P {E0}, Pout,1 = P {E0∩E1} · · · Pout,N = P {E0∩E1∩E2+· · ·+EN} Because P {E0∩E1}+P {E0∩E1} = P {E0}, we can obtain the P {E0∩E1} = Pout,0−Pout,1. By using the similar method to analyze the average transmission, we can get
[Two phase]
E{N } = 2(1 − Pout,0) + 3(Pout,0− Pout,1) + · · · +
(N + 1)(Pout,N −2− Pout,N −1) + (N + 2)Pout,N −1
= 2 +
N −1X
i=0
Pout,i. (5.6)
[One phase]
E{N} = (1 − Pout,0) + 2(Pout,0− Pout,1) + · · · + (N − 1)(Pout,N −2− Pout,N −1) + (N )Pout,N −1
= 1 +
N −1X
i=0
Pout,i. (5.7)
AF protocol:
Because AF protocol without decoding set problem, the outage event in each ARQ phase is independent event. Pout,0= P {E0}, Pout,1= P {E0∩ E1} = P {E0}P {E1}
· · · Pout,N = P {E0∩E1∩E2∩· · ·∩EN} = P {E0}P {E1} · · · P {EN}. we can obtain the P {E0∩ E1} = P {E0}P {E1} = Pout,0− Pout,1. Thus, the average transmission of cooperative ARQ for AF protocol can be shown as
[Two phase]
E{N } = 2(1 − Pout,0) + 3(Pout,0− Pout,1) + · · · +
(N + 1)(Pout,N −2− Pout,N −1) + (N + 2)Pout,N −1
= 2 +
N −1X
i=0
Pout,i. (5.8)
[One phase]
E{N} = (1 − Pout,0) + 2(Pout,0− Pout,1) + · · · + (N − 1)(Pout,N −2− Pout,N −1) + (N )Pout,N −1
= 1 +
N −1X
i=0
Pout,i. (5.9)
From the simulation result, he effective way to improve the throughput is using type-IV ARQ scheme for DF OC-BF. For AF OC-BF, the throughput can be more effectively
improved by using more best relays.
0 5 10 15 20 25 30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
P/N0dB
Effective throughput
type-I OCBF2 ARQ1 type-I OCBF2 ARQ2 type-I OCBF2 ARQ3
Figure 5.1: Effective throughput for TP DF type-I in different ARQ times [M = 5, R = 2, α = 0.5]
0 5 10 15 20 25 30
Figure 5.2: Effective throughput for TP DF type-II and type-III in different ARQ times [M = 5, R = 2, α = 0.5]
Figure 5.3: Effective throughput for TP DF type-III and type-IV in different ARQ times [M = 5, R = 2, α = 0.5]
0 5 10 15 20 25 30
Figure 5.4: Effective throughput for OP DF type-III and type-IV in different ARQ times [M = 5, R = 2, α = 0.5]
Figure 5.5: Effective throughput for TP AF type-III and OP AF type-III in different ARQ times [M = 5, R = 2, α = 0.5]
0 5 10 15 20 25 30 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
P/N0 dB
Throughput
TP type-III OC-BF-1 ARQ3 TP type-III OC-BF-2 ARQ3 TP type-III OC-BF-3 ARQ3 TP type-III OC-BF-4 ARQ3 TP type-III OC-BF-5 ARQ3 OP type-III OC-BF-1 ARQ3 OP type-III OC-BF-2 ARQ3 OP type-III OC-BF-3 ARQ3 OP type-III OC-BF-4 ARQ3 OP type-III OC-BF-5 ARQ3
Figure 5.6: Effective throughput for TP AF type-III and OP AF type-III OC-BF-i i = 1, 2, · · · M [M = 5, R = 2, α = 0.5]
Chapter 6 Conclusions
We investigated the outage probabilities and characterized the effective throughput of several types cooperative ARQ scheme for DF and AF protocols by using opportunistic cooperative beamforming. For DF OC-BF, the outage probability of OC-BF-M can be achieved by OC-BF-2 and the outage performance is constrained by decoding set. For AF OC-BF, the outage probability becomes better by choosing more relays and the SNR offset can be reduced by using one more relay. In the effective throughput, the effective way to improve the throughput is using type-IV ARQ scheme for DF OC-BF. For AF OC-BF, the throughput can be more effectively improved by using more best relays.
Appendix A
Z T
n
0
Z T −(n)tn
n−1
0
Z T −(n)tn−(n−1)tn−1 n−2
0
· · ·
Z T −(n)tn−(n−1)tn−1···(2)t2 1
0
tL−nn dt1dt2· · · dtn= FN(T, L)
= TL(L − n)!
n!nL−nL! (A.1)
proof:
n=2 : the above equation can be given as Z T
2
0
Z T −2t2
0
tL−22 dt1dt2 = Z T
2
0
(T − 2t2)tL−22 dt2 = F2(T, L) (A.2)
By using the formula that is given by Z u
0
xv−1(u − x)p−1dx = u(p+v−1)(p − 1)!(v − 1)!
(p + v − 1)! (A.3)
We can obtain the function: F2(T, L) = 2!2TL(L−2)(L−2)!L!
n=3: assume s = t − 3t3
Bibliography
[1] E. Telatar, “Capacity of multi-antenna gaussian channels,” European Trans. on Telecommunications, vol. 10, no. 6, pp. 585–595, 1999.
[2] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. on Information Theory, vol. 59, no. 10, pp. 2415–2525, 2003.
[3] David N.C.Tse J.Nicholas Laneman, “Coopeative diversity in wireless networks:
Efficient protocols and outage behavior,” IEEE Trans. on Information Theory, vol.
50, pp. 3062–3080, 2004.
[4] Y. Zhao, R. Adve, and T. J. Lim, “Improving amplify-and-forward relay networks:
optimal power allocation versus selection,” in Proc. IEEE ISIT. Seattle, WA, July 2006.
[5] Aggelos Bletssa Hyundong Shin and Moe Z., “Cooperative communicaions with outage-optimal opportunistic relaying,” IEEE Trans. on Wireless Communications, vol. 6, no. 9, pp. 3450–3460, 2007.
[6] Jung-Bin Kim and Dongwoo Kim., “Coopeative system with distributed beam-forming and its outage probability,” pp. 1638–1641, 2007.
[7] Y. Zhao, R. Adve, and T. J. Lim, “Beamforming with limited feedback in amplify-and-forward cooperative networks,” in Proc. IEEE Globecom. Washington, D.C., Nov. 2007.
[8] T. W. Ban, B. C. Jung, D. K. Sung, and W. Choi, “Performance analysis of two relay selection schemes for cooperative diversity,” in Proc. IEEE PIMRC. Athens, Greece, Sep. 2007.
[9] I. Stanojev, O. Simeone, and Y. Bar-Ness, “Performance of multi-relay collaborative hybrid-ARQ protocols over fading channels,” IEEE Communications Letters, vol.
10, no. 7, pp. 522–524, 2006.
[10] J.-B. Kim and D. Kim, “Cooperative system with distributed beamforming and its outage probability,” in Proc. IEEE VTC-Spring. Dublin, Ireland, April 2007.
[11] M.O. Hasna and M.S. Alouini, “End-to-end performance of tranmission systems with relays over rayleigh-fading channels,” IEEE Trans. on Communications, vol.
2, no. 6, pp. 1126–1131, 2003.
[12] Karim G. Seddik Ahmed K. Sadek K.J. Ray Liu, “Outage analysis and optimum power allocation for multinode relay networks,” Signal Processing Letters, vol. 14, no. 6, pp. 1126–1131, 2007.
[13] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Elsevier, 7rd edition, 2007.
[14] SAEED GHAHRAMANI, Fundamentals of probability with stochastic processes, Prentice Hall, 3rd edition, 2005.
[15] H.N. Nagarja H.A.David, Order statistics, Wiley, 3rd edition, 2003.