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700 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005 Outage Probability at Arbitrary SNR with Cooperative Diversity Yi Zhao, Raviraj Adve, and Teng Joon Lim Abstract— Cooperative diversity improves the performance of nodes. Denote the set of all cooperating nodes as {c}. To guar- wireless networks by having several nodes transmit the same antee orthogonal transmissions, we consider a Time Division information. We present an outage probability analysis for a Multiple Access (TDMA) arrangement with m + 1 time slots. decode-and-forward system, valid at all signal to noise ratios (SNR). A closed form solution is obtained for independent The ﬁrst slot is used for the source to transmit its signal to and identically distributed (i.i.d.) channels, and two tight lower the destination as well as share it with the cooperative nodes. bounds are presented for correlated channels. If the channel between the source and a node is good Index Terms— Outage probability, cooperative diversity, wire- enough, this node becomes an active cooperating node, and less cooperative networks. decodes and forwards the source information. Denote the set of active cooperative nodes as C ⊂ {c}. In the following m slots, the active cooperative nodes repeat the source message I. I NTRODUCTION in a predetermined order [1]. OOPERATION helps create spatial diversity in wireless C networks, even if individual nodes do not use antenna arrays for transmission or reception [1], [2]. While asymptotic Assuming that the destination d has exact channel state information (CSI), maximum-likelihood combining of the signals received from all |C| + 1 nodes can be employed. Fur- (in SNR) performance analysis highlights the diversity order thermore, we assume that both the source and the cooperative achievable by various techniques, it is also important to study nodes do not have access to any transmit channel information. performance in the non-asymptotic or ﬁnite SNR regime so as to compare various schemes in practical settings. In III. O UTAGE P ROBABILITY: F ORMULAS AND B OUNDS this letter, we derive the exact outage probability of the decode-and-forward scheme1 for arbitrary signal-to-noise ratio A. Outage Probability for i.i.d. Channels (SNR) by using two powerful mathematical tools: Moment The mutual information between the source and cooperative Generating Function (MGF) and Order Statistics. We derive an nodes c = 1, . . . , m is [4]: exact closed-form expression for i.i.d. channels, and introduce 1 simple lower bounds for non-i.i.d. channels. Simulation results Ic = log (1 + SNRˆc ) , x (1) m+1 show that those bounds are very tight. As an example of the utility of our results, consider that where xc = |hs,c |2 and hs,c is the complex channel gain ˆ Laneman and Wornell proved in [1] that both the decode-and- between source s and node c, modelled as a zero-mean circu- forward and the space-time-coded cooperation can provide ˆ larly symmetric Gaussian random variable. xc is exponentially “full diversity” in the sense that the diversity order (using ˆ distributed with parameter λc . SNR is the transmit signal-to- 1 outage probability as a performance measure) is the total noise ratio, and the factor m+1 captures the TDMA nature number of cooperating nodes in the network. However, our of the scheme, in which node s is allowed to transmit its results will show that at ﬁnite SNR levels, a higher diversity information only a fraction 1/(m + 1) of the time. order does not necessarily translate into better performance. If the instantaneous mutual information Ic is higher than the This implies that the optimal number of cooperating nodes transmission rate R, we can assume that the cooperative node is in fact a complex function of the operating SNR and the successfully decodes the source bits, and thus belongs to the cooperative diversity scheme in use. active cooperative set C, or C = {c : Ic > R, c = 1, . . . , m}. As presented in [1], the mutual information of the decode- II. S YSTEM M ODEL and-forward transmission is We consider a system with a source node, s, communicating 1 I= log 1 + SNR x0 + xc , (2) with a destination, d, with the help of m other cooperating m+1 c∈C (or relay) nodes, which are called cooperative nodes or relay where x0 = |hs,d |2 and xc = |hc,d |2 are exponentially Manuscript received January 18, 2005. The associate editor coordinating distributed with parameter λ0 and λc , respectively. the review of this letter and approving it for publication was Prof. Carla- Fabiana Chiasserini. The outage probability is deﬁned as Pout = P I < R The authors are with the Department of Electrical and Computer En- where R is the required transmission rate for source s. Using gineering, of Toronto, Toronto, Ontario, Canada (e-mail: {zhaoyi, rsadve, the total probability law, we can write the outage probability limt}@comm.utoronto.ca). Digital Object Identiﬁer 10.1109/LCOMM.2005.08025. as: 1 Where the cooperative nodes attempt to decode the source’s bits, and P I<R = P I<R|C P C . (3) retransmit those bits if the decoding succeeds [3], C 1089-7798/05$20.00 c 2005 IEEE ZHAO et al.: OUTAGE PROBABILITY AT ARBITRARY SNR WITH COOPERATIVE DIVERSITY 701 Generally, this probability is very difﬁcult to compute since B. Convenient Lower Bounds the summation is over all the possible active cooperative sets, which has 2m items. However, if we assume independent and In Theorem 1 we derive the exact outage probability for- identically distributed (i.i.d.) fading, the closed-form outage mula for i.i.d. fading. Unfortunately, Pout does not have a nice probability can be derived. and simple form for general non-i.i.d. cases. However, we can Theorem 1: Under the assumption that the Rayleigh fading still obtain convenient bounds to avoid the complex numerical from the source to the cooperative nodes are i.i.d. (λc = ˆ summation in (3). We present two convenient lower bounds ˆ c = 1, . . . , m), and those from all nodes to the destination λ, as the following theorems. are also i.i.d. (λ0 = λc = λ, c = 1, . . . , m), the outage Theorem 2: One lower bound for the outage probability of probability of the system is a decode-and-forward system is m k m−k m ˆ ˆ m Pout = e−λγ 1 − e−λγ ˆ k Pout ≥ 1 − e−λc γ 1 − e−λ0 γ + (11) k=0 c=1 k (λγ)i m m 1 − e−λγ , (4) 1− 1 − e−λc γ ˆ 1 − e−λc γ/(m+1) . i=0 i! c=1 c=0 2(m+1)R −1 where γ = SN R . Proof: Based on the fact that Proof: First consider the conditional probability x0 + xc ≤ (m + 1)xmax , xmax = max xc , k c=0,...,m (m+1)R 2 −1 c∈C P I < R | C = {1, . . . , k} = P xc < SN R ﬁrst we have c=0 (5) Pout = P x0 + xc < γ ≥ P x0 < γ P k = 0 + We now use Moment Generating Function (MGF) to ﬁnd c∈C k the distribution of xsum = c=0 xc . Since each xc is expo- P (m + 1)xmax < γ P k = 0 , (12) nentially distributed with parameter λ, its MGF is Mc (s) = λ where k again represents the size of the active cooperative set, s+λ . Furthermore, the xc ’s are independent, so the MGF of xsum is [5] or k = |C|. k+1 λ Using a result in order statistics, we can obtain the CDF of Msum (s) = . (6) s+λ xmax as [6] m m Applying the inverse Laplace Transform, we can get the Fmax (x) = Fc (x) = 1 − e−λc x , pdf, and then the CDF of xsum as c=0 c=0 k i (λx) and therefore Fsum (x) = 1 − e−λx , (7) m i! i=0 P (m + 1)xmax < γ = 1 − e−λc γ/(m+1) . (13) therefore (5) becomes c=0 k (λγ)i For the other items in (12), it is simple to obtain Fsum (γ) = 1 − e−λγ . (8) i! m i=0 ˆ P k=0 = 1 − e−λc γ (14) Notice that this conditional distribution is only determined c=1 m by k, the size of the active set, and not by the identity of the ˆ k nodes in the set. As a result, we can re-write (3) as P k=0 = 1− 1 − e−λc γ (15) c=1 m −λ0 γ P x0 < γ = 1−e . (16) P I<R = P I < R | |C| = k P |C| = k (9) k=0 Substituting (13), (14), (15) and (16) into (12) completes From the mutual information formula (1), we have the proof. ˆ Theorem 3: Another lower bound for the outage probability P c ∈ C = P xc > γ = e−λγ , ˆ of a decode-and-forward system is ˆ ˆ since xc is exponentially distributed with parameter λ. It is m ˆ straightforward then to arrive at Pout ≥ 1 − e−λc γ 1 − e−λ0 γ + (17) k m−k c=1 m ˆ ˆ P |C| = k = e−λγ 1 − e−λγ (10) m m k ˆ 1− 1 − e−λc γ ac 1 − e−λc γ , c=1 c=0 Finally, substituting (8) and (10) into (9), we get (4), which λi completes the proof. where ac = i=c λi −λc . 702 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005 0 0 10 10 m=8 −2 −2 10 10 m=0 m=7 m=2 −4 −4 10 10 Outage Probability outage probability −6 10 −6 10 −8 10 −8 10 −10 10 −10 10 numerical result lower bound 1 −12 10 −12 lower bound 2 10 −14 10 −14 0 5 10 15 20 25 30 35 40 10 0 5 10 15 20 25 30 35 40 SNR (dB) SNR (dB) Fig. 1. Outage probability for the i.i.d. cases. Fig. 2. Outage probability and lower bounds for the non-i.i.d. cases. Proof: Similar to (12), we have Plots like Fig. 1 help in system design not only because m outage probability is an important QoS parameter in itself, but Pout ≥ P x0 < γ P k = 0 + P xc < γ P k = 0 , also because they allow us to determine the optimal number of c=0 cooperative nodes. For example, from Fig. 1 we can see that (18) when SNR= 20 dB, the optimal number of cooperative nodes m The MGF of c=0 xc can be written as is m = 2. When SNR increases to 30 dB, the optimal size m m changes to m = 4. An asymptotic analysis based only upon λc ai λc Msum (s) = = , (19) diversity order, on the other hand, will always point to m = 8 c=0 s + λc c=0 s + λc as the best setting because it yields the largest diversity order. λi Fig. 2 compares the two lower bounds with the actual Pout where ac = λi − λc obtained from numerical computations. In this simulation we i=c ˆ set R = 1 bit/sec/Hz, λc , λ0 and λc are i.i.d. uniformly m Therefore the CDF of c=0 xc is distributed in [0, 2]. From the ﬁgure we can see that the two m bounds are almost identical and very tight, providing very Fsum (x) = ac 1 − e−λc x (20) good approximations to the outage probability. c=0 Finally, substituting (20), (14), (15) and (16) into (18) R EFERENCES completes the proof. [1] J. N. Laneman and G. W. Wornell, “Distributed space-time-coded pro- m Since x0 + c∈C xc ≤ c=0 xc ≤ (m + 1)xmax , the tocols for exploiting cooperative diversity in wireless networks,” IEEE second lower bound is tighter, but requires slightly heavier Trans. Inform. Theory, vol. 49, pp. 2415–2425, Nov. 2003. [2] H. El Gamal and D. Aktas, “Distributed space-time ﬁltering for cooper- computation load. ative wireless networks,” Proc. IEEE Globecom, pp. 1826–1830, 2003. [3] A. Sendonaris, E. Erkip, and B. Aazhang, “Increasing uplink capacity via IV. S IMULATION R ESULTS user cooperative diversity,” Proc. IEEE Int. Symp. Info. Theory, p. 156, August 2002. Fig. 1 shows the outage probability of a decode-and-forward [4] T. M. Cover and J. A. Thomas, Element of Information Theory. system with the number of cooperative nodes ranging from 0 [5] K. Knight, Mathematical Statistics. Chapman and Hall, 1999. [6] H. A. David and H. N. Nagaraja, Order Statistics: Third Edition. John to 8. In the simulation we set R = 1bit/sec/Hz, and λc =ˆ Wiley, 2003. λ0 = λc = 1, c = 1, . . . , m.