A Simple Probabilistic Relay Selection Protocol for ...

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

A Simple Probabilistic Relay Selection Protocol for Asynchronous Multi-Relay Networks Employing Rateless Codes Xijun Wang, Wei Chen, Member, IEEE, and Zhigang Cao, Senior Member, IEEE State Key Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing 100084, China Email: [email protected] {wchen,czg-dee}@tsinghua.edu.cn

Abstract—Cooperative communication with rateless codes has attracted much attention recently because it can provide both spatial diversity and high bandwidth efficiency. However, relay selection for asynchronous relaying, which shows great potential for practical applications, has not been carefully studied yet. In this paper, based on rateless codes, we propose a simple probabilistic relay selection protocol for asynchronous multi-relay networks, where no symbol-level inter-node synchronization or multiuser detection is needed. In such asynchronous networks, a decoding relay can help the other relays which have not decoded the message yet. Moreover, the tradeoff between the first hop and the second hop can be controlled by setting the decoding relay threshold. We analyze the average end-to-end throughput and find an optimal decoding relay threshold to maximize the throughput. Simulation results show the superiority of the proposed protocol at low Signal-to-Noise Ratios (SNRs) when the relays are close to the source.

I. I NTRODUCTION In cooperative communication systems, a number of nodes can be utilized as relays to assist a source in delivering data to its destination, thereby extending communication range and reducing energy expenses [1], [2]. The transmission reliability can also be improved by exploiting the broadcast nature of wireless channel and the inherent spatial diversity of multiple relays [3], [4]. Most recently, rateless codes have been shown to be well suited for cooperative relay transmissions because they not only achieve reliable communication under channel uncertainty but also enable flexible transmission durations of broadcast phase and multiple access phase [5]–[10]. Rateless codes [11], [12], as one of erasure coding schemes, neither impose a fixed code rate nor need the channel knowledge at the transmitter. A rateless encoder can generate everincreasing coded packets from a finite-length message until a positive acknowledgment (ACK) is received. These coded packets are accumulated in an arbitrary manner at the receiver. A rateless decoder can then recover the message as long as it accumulates enough coded packets that in aggregate are slightly longer in length than the message. This work was supported by NSFC project under Grant No. 60832008 and No. 60902001.

The application of rateless codes to single relay networks and multi-relay networks has been studied in [5], [6] and [7]– [9], respectively. Different from the previous studies in fixed rate networks [3], [4], the overall outage is virtually never experienced in rateless coded networks as the transmission duration could be indefinitely long. In [7], Molisch et al. proposed two cooperative relay transmission protocols, namely, quasisynchronous protocol: all decoding relays begin to forward the message simultaneously and relay does not help each other, and asynchronous protocol: the decoding relay forwards the message immediately and helps the remaining relays to decode the message. In both protocols, after the number of decoding relays reaches a preset value, all decoding relays will transmit collaboratively to the destination, which requires inter-relay synchronizations at the symbol level or multiuser detection at the destination. In order to dispense these complexities, relay selection has been investigated in [8], [9]. The relay with the best source-relay (s − r) link was selected to forward the message in [8]. On the other hand, a Signal-to-Noise Ratio (SNR) threshold based relay selection was proposed for quasisynchronous relaying systems in [9], where the node with the best link to the destination was selected from the source and the decoding relays. These previous work, however, paid less attention to the relay selection in asynchronous relaying networks, wherein rateless codes have a unique advantage. In this paper, we propose a rateless coding based probabilistic relay selection protocol for asynchronous relaying in ad-hoc networks. By utilizing rateless codes, an additional degree of freedom can be provided to set the number of decoding relays. In this protocol, the source transmits a potentially infinite number of coded packets until the number of decoding relays reaches a threshold. It is worth noting that the threshold is determined randomly by the source when a new message is generated. On the other hand, a relay joins the decoding relay set as soon as it decodes the message, and the destination selects the decoding relay with the best relay-destination (r−d) link from the set to forwards the message. It is also worth noting that the overheard information at the remaining relays accelerates their decoding process. The average end-to-end

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

throughput is derived by analyzing the average transmission time for the first hop and the second hop. We formulate an optimization problem that maximizes the average throughput, and further propose a probabilistic relay selection algorithm to obtain the optimal decoding relay threshold. We also evaluate the performance by simulations and compare our proposed protocol with two baseline protocols. The remainder of this paper is organized as follows: Section II introduces the system model. In Section III, we elaborate the probabilistic relay selection protocol and analyze the average end-to-end throughput. Section IV presents the simulation results, and finally, conclusions are drawn in Section V. II. S YSTEM M ODEL We consider a wireless relay network consisting of a source, s, a destination, d, and a set of decode-and-forward (DF) relays, Q = {1, 2, . . . , Q}. Each node is equipped with a single antenna, and all relays are FDD capable, i.e., they can transmit and receive simultaneously on channel 1 and channel 2, respectively. In addition, the direct link between the source and the destination is ignored due to the large distance. The wireless channels between different nodes are assumed to be frequency-flat, block-fading Rayleigh channels, with the block length being equal to the duration of one time slot. The channel fading coefficient between any two node, hij , which is assumed to be known only to the receivers, is modeled as a circularly symmetric complex Gaussian random variable with zero-mean and variance γ ij . These fading coefficients capture the effects of both path loss and Rayleigh fading. As a result, the channel power gain γij  |hij |2 is exponentially −α , where α distributed with the mean value given by γ ij = κlij is the path loss exponent and lij is the distance between node i and node j. The constant κ, which accounts for all of the other attenuation factors, is set to be 1 without loss of generality. We consider the case where the relay nodes are placed close to each other, and assume γ ij = 1 for i, j ∈ Q. It is also assumed in this paper that the distance between the source and different relays are the same, and so are the distance between the destination and different relays, i.e., ls1 = · · · = lsQ = ls , l1d = · · · = lQd = ld . All of nodes are assumed to employ rateless codes for encoding a message and the relay nodes use the same rateless codebook as the source. Each message consists of K data packets, with H bits/packet. At the transmitter side, the message is first encoded at packet-level using Luby-Transform (LT) rateless code [11]. A potentially infinite number of coded packets can be generated on the fly and every output coded packet is a binary addition of a random subset of K data packets. After appended with a Cyclic Redundancy Check (CRC) sequence, each LT-coded packet is encoded by a physical-layer channel code of rate R bit/channel-use (bpcu), and transmitted over the wireless channels in a time slot. The receiver try to decoded the packets at the physical layer first. If the CRC is incorrect, the corresponding packet will be discarded. Here, we assume that CRC bits are sufficient that any error can be detected. Then, the correctly received packets are used by the

LT decoder to recover the source message. When the receiver  accumulates any K = (1 + )K LT-coded packets, where  is the decoding overhead, it can completely recover the original message using belief propagation (BP) algorithm, and inform the transmitter via an ACK packet. For practical LT codes,  both K and  are discrete random variables [13]. Since  is a discrete random variable with non-negative value, the distribution of  can be modeled with a Poisson mixture density as in J (λ ) e−λj ,  ≥ 0, where [14]. Specifically, f () = j=1 ωj j ! J is the number of Poisson distribution components and ωj J is the combination weight (ωj ≥ 0 and j ωj = 1). These parameters can be estimated from the code realizations using the expectation-maximization (EM) algorithm. III. P ROBABILISTIC R ELAY S ELECTION P ROTOCOL In this section, we first describe the probabilistic relay selection protocol in detail. Then, we analyze the average end-to-end throughput. And finally, an optimization problem is formulated to find the optimal decoding relay threshold. A. Protocol Description The relay selection protocol is described as follows. Source: Assume that the source has a continuous stream of messages. Once it starts transmitting a message, limitless coded packets are broadcast on channel 1. The source ceases the current transmission and starts to transmit the next message as soon as there are L decoding relays. Thus L is referred to as decoding relay threshold, which could be set in a probabilistic way. In other words, the source sets L to a preset value with a given probability before transmitting a new message. Thus, the threshold is not fixed and may be different between the transmissions of different messages. We shall later show that setting the threshold in a probabilistic way can achieve the maximum throughput. In addition, we define the transmission time for the first hop, TLsr , as the time duration from the beginning of a transmission to the time when there are L decoding relays. Relays: All the relay nodes accumulate coded packets in order to decode the message. As soon as any relay has acquired  enough coded packets, say K , to reliably recover the message, it immediately informs the source its decoding success via an ACK packet and joins in the decoding relay set D. It is worth noting that other relays and the destination can also overhear the ACK packet. In each slot, the best decoding relay, i.e., the relay with the highest channel gain of r−d link, is chosen from D, and then it re-encodes the message with the same codebook as the source and forwards it to the destination on channel 2. The remaining relays in Q \ D can also hear the transmission of the best decoding relay and use maximum-ratio-combining (MRC) to combine the signal from the source on channel 1 and that from the best decoding relay on channel 2. Destination: Since the destination has the channel state information (CSI) from the decoding relays, it can request the best relay in D to transmit. Note that the best relay may be different in each time slot. Once the destination decodes the message, it sends back an ACK packet. And if the next

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message has already been decoded by a relay, it is forwarded to the destination immediately. The duration from the time when there are L decoding relays to the time when the destination decodes the message is defined as the transmission time for the second hop, denoted by TLrd . B. Performance Analysis Since the relays can receive and transmit simultaneously on different channels, the average end-to-end throughput of the wireless relay networks can be obtained as K

 rd  , (1) max TL   where E [TLsr ] and E TLrd will be given by Theorems 1 and 2, respectively, in the sequel. For any s − r link, the erroneous packets will be discarded at the receiver due to deep fading. By using the channel code with suitably long blocklength, the packet error probability can be accurately approximated by the outage probability, which 

η=

E [TLsr ] , E

R −1

−2

is given by Po,sj = Pr {log (1 + γsj ρ) < R} = 1 − e γ sj ρ , where j ∈ Q and ρ is the transmit SNR, which is assumed to be the same for all transmitters. Since the channel fading in each slot is independent, the sr , required for the relay j to recover transmission time1 , Tdec,j the source message, given the decoding overhead of the LT code, follows the conditional negative binomial distribution    sr = t | Kj = K(1 + j ) Pr Tdec,j

t−1 K(1+j ) t−K(1+j ) (1 − Po,sj ) Po,sj . (2) = K(1 + j ) − 1 Note that the packet erasure is independent for different s − r links. The decoding overhead j for different relays are also independent and identically distributed. Thus, the cumulative sr can be given by distribution function (cdf) of Tdec,j  sr sr (t) = Pr Tdec,j ≤t FTdec,j ∞     

   sr = Pr Tdec,j ≤ t | Kj = k Pr Kj = k =

k =K ∞

k =K

 IPo,sj

 1 f k ,t + 1 − k K 





  k −1 , K

(3)

where Po,sj = 1 − Po,sj and

Theorem 1: Let Δsr j denote the remaining time required for the relay j(j ∈ Q \ D) to decode the message when |D| = 1. The average transmission time for the first hop is given by E [TLsr ] = ⎧∞  Q  ⎪ ⎪ sr 1 − FTdec,j (t) , ⎨ t=0  ∞   ⎪ ⎪ 1 − FΔsr (x) , ⎩E [T1sr ] + L−1:Q−1

L = 1, 2 ≤ L ≤ Q.

(5)

x=0

where FΔsr (x) is the cdf of the (L−1)-th order statistic L−1:Q−1 of Q − 1 independent random variables Δsr j . Proof: For L = 1, the source transmits the next message as soon as there is any decoding relay. The transmission time T1sr is the first order statistic of Q independent random sr . By utilizing the variables, i.e., T1sr = Tdec,1:Q  results in [15], sr the cdf of T1sr , FT1sr (t)  Pr Tdec,1:Q ≤ t , is given by  Q sr (t) , (6) FT1sr (t) = 1 − 1 − FTdec,j

and hence the expected value of T1sr can be given by ∞ ∞  Q

  sr 1 − FT1sr (t) = 1 − FTdec,j (t) . (7) E [T1sr ] = t=0

t=0

For L ≥ 2, the source keeps transmitting the current message until there are L decoding relays. Once there is any decoding relay, other relays can combine the signals from the source and the best decoding relay. Therefore, the outage probability decreases and is given by Po,s˜rj = Pr {log (1 + γsj ρ + γr˜j ρ) < R}

R γ r˜j 2 −1 − =1 − exp − × γ sj ρ γ sj − γ r˜j  R

R

 2 −1 2 −1 exp − − exp − , γ sj ρ γ r˜j ρ

(8)

where γr˜j is the channel gain between the best decoding relay, r˜, chosen by the destination and the remaining relay j in Q \ D. In addition, when there is any decoding relay, the remaining relays have already accumulated some number of  j < K  ). The conditional probability  j (K coded packets, say K j  j given T sr and K  is mass function (pmf) of K 1 j      j = k | T1sr = t, Kj = k Pr K t k t−k (1 − Po,sj ) Po,sj  , k = 0, . . . , k − 1. (9) =  k k −1   x t−x t x (1 − Po,sj ) Po,sj x=0



Ix (a, b) =

a+b−1

j=a

(a + b − 1)! xj (1 − x)a+b−1−j (4) j!(a + b − 1 − j)!

is the regularized incomplete beta function. Then, we present E [TLsr ] in the following theorem. 1 Without loss of generality, for the rest of this paper, the length of a slot is set to be 1. Hence, the transmission time is equal to the number of slots.

Therefore, the remaining relays only have to accumulate K −  j more coded packets to recover the message. Similar to (2), K Δsr j follows conditional negative binomial distribution, and the   conditional pmf of Δsr j given Kj and Kj is      = x | K = k, K = k Pr Δsr j j j

  x−1 k −k x−k +k (1 − Po,s˜rj ) (Po,s˜rj ) . (10) =  k −k−1

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Since the decoding overhead for the remaining relays are independent with each other and independent with T1sr , the sr conditional cdf of Δsr j given T1 can be derived according to Eqns. (9) and (10). Specifically,  sr sr sr (x | t) = Pr Δ FΔsr j ≤ x | T1 = t j |T1 ∞     

   sr Pr Kj = k Pr Δsr = j ≤ x | T1 = t, Kj = k k =K 

∞ k −1

=



sr  Pr Δsr j ≤ x | Kj = k, T1 = t, Kj = k 

k =K k=0

   j = k | T sr = t, K  = k  1 f Pr K 1 j K



 

×

  k − 1 . (11) K

Therefore, according to Eqns. (6) and (11), the cdf of the order statistics Δsr L−1:Q−1 can be given by  FΔsr (x) = Pr Δsr L−1:Q−1 ≤ x L−1:Q−1 ∞

 sr sr Pr Δsr = L−1:Q−1 ≤ x | T1 = t Pr {T1 = t} =

t=K ∞

Since TLrd begins when |D| = L, the conditional pmf of   d,L is given Kd and K      d,L = k Pr TLrd = t | Kd = k , K

  t−1 k −k t−k +k (1 − Po,˜rd (L)) Po,˜rd (L). (17) =  k −k−1

TLrd

Therefore, the expected value of TLrd is given by

      K(1 + ) − E K d,L     d,L = . E TLrd = E E TLrd | Kd , K 1 − Po,˜rd (L) (18) By substituting (16) into (18), we get Eqn. (15). C. Throughput Optimization The decoding relay threshold has an important impact on the system performance. In order to achieve the maximum throughput, we can obtain the optimal decoding relay threshold, L∗ , via the following optimization problem    min max E [TLsr ] , E TLrd L

s.t.

IFΔsr |T sr (x|t) (L − 1, Q − L + 1) Pr {T1sr = t} ,

t=K

j

1

(12) TLsr

and hence the expected value of is given by ∞  

1 − FΔsr (x) . E [TLsr ] = E [T1sr ] + L−1:Q−1

(13)

x=0

By summarizing Eqns. (7) and (13), we get Eqn. (5). For the second hop, the destination chooses the best relay, denoted by r˜, from the decoding relay set D in each time slot. Therefore, the outage probability can be given by 

 Po,˜rd (|D|) = Pr log 1 + max hid ρ < R i∈D

 = Pr max hid i∈D

2 −1 < ρ R





=



2R − 1 1 − exp − γ id ρ

|D| .

(14)  rd  Based on the results of Theorem 1, we can present E TL in the following theorem. Theorem 2: The average transmission time for the second hop is given by   sr    sr  rd  K(1 + ) − L−1 r d (i) i=1 E Ti+1 − E [Ti ] Po,˜ , E TL = 1 − Po,˜rd (L) (15) where  is the average decoding overhead. Proof: When |D| ≥ 1, the destination can accumulate coded packets from the best relay r˜, and will experience different outage probability for different value of |D|. As a result, the average number of coded packets accumulated at the destination before |D| = L can be given by  L−1 

    sr  d,L = E Ti+1 − E [Tisr ] Po,˜rd (i) . E K i=1

(16)

1 ≤ L ≤ Q.

(19)

An analytical solution for L∗ is hard  to be obtained. Therefore, we analyze how E [TLsr ] and E TLrd vary with L first. It can be seen from Theorem 1 and 2 that E [TLsr ] monotonically  rd decreases increases with L, whereas E TL monotonically   with L. When E [TLsr ] intersects E TLrd , we can employ probabilistic decoding relay threshold. Specifically, Pr {L∗ = x} = 1 − Pr {L∗ = x + 1} = p,

(20)

where

sr rd − Tx+1 Tx+1 , (21) sr − T rd + T rd − T sr Tx+1 x x x+1   sr   rd    > E Ti+1 . and x = i | i ∈ Q, E [Tisr ] < E Tird , E Ti+1 Hence, the maximum throughput can be given by

p=

η∗ = K

sr rd Tx+1 − Tx+1 + Txrd − Txsr . sr T rd − T sr T rd Tx+1 x x x+1

(22)

  On the other hand, for disjoint E [TLsr ] and E TLrd , the optimal L∗ is either 1 or Q, i.e., Pr {L∗ = 1} = 1 or Pr {L∗ = Q} = 1. Correspondingly,  maximum throughput  the is η ∗ = K/E [T1sr ] or η ∗ = K/E TQrd . IV. S IMULATION R ESULTS In this section, we perform extensive simulations to evaluate the performance of the proposed protocol. We also compare the probabilistic relay selection (PRS) protocol with two baseline references, namely multi-relay cooperation (MC): all relays in the decoding relay set transmit collaboratively under an aggregate power constraint, and random relay selection (RRS): the destination randomly selects a relay from the decoding relay set. LT codes, as in [11], are employed in the simulations with parameters c = 0.1 and δ = 0.5. We also set Q = 10, R = 1 bpcu, α = 2, and K = 50.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

0.55

6

0.5 5 0.45 PRS MC RRS

0.4 0.35 10

15

20

SNR (dB)

25

4

30

3 10

0.65 2.5 0.6 0.55

2 0.5 0.45

1.5

15

20

25

30

SNR (dB)

Fig. 1. The maximum throughput η ∗ and the expected value of L∗ versus SNR ρ with the relays near the source (ls = 1, ld = 3).

0.35 10

PRS MC RRS

PRS MC RRS

0.4

15

20

25

30

SNR (dB)

1 10

15

20

25

30

SNR (dB)

Fig. 2. The maximum throughput η ∗ and the expected value of L∗ versus SNR ρ with the relays near the destination (ls = 3, ld = 1).

Fig. 1 shows the maximum throughput η ∗ and the expected value of the probabilistic decoding relay threshold, E [L∗ ], for a multi-relay network where the relays are close to the source. In this case, the second hop may become the bottleneck and more decoding relays are needed to provide the spatial diversity. It is shown that the throughput of PRS is larger than that of MC and RRS, and their throughput increase with SNR and merge at high SNRs. Since the relays are close to the source, the transmission time for the first hop decreases marginally as SNR increases, however, the transmission time for the second hop decreases evidently. As a result, E [L∗ ] decreases with the increasing of SNR. In addition, since PRS has the lowest outage probability, it needs less decoding relays to combat fading than MC and RRS. Both η ∗ and E [L∗ ] are shown in Fig. 2, where the relays are close to the destination. When ρ ≤ 16 dB, the first hop is the bottleneck, and hence the optimal decoding relay threshold L∗ is equal to 1, which implies that the source transmits a new message as soon as possible. Since no relay selection or cooperation is taken on the second hop, the throughput of three schemes are exactly the same. When ρ > 16 dB, the transmission time so that E [T1sr ]  the first hop decreases  rdfor ∗ is less than E T1 , and thus E [L ] > 1. Since the relays are close to the destination, the outage probabilities of three schemes are all small enough, and hence the differences of their impacts are negligible. Thus, E [L∗ ] are almost the same for different schemes, so are the maximum throughput η ∗ . In Fig. 3, the maximum throughput η ∗ and E [L∗ ] are illustrated with the relays located in the middle between the source and the destination. It is shown that PRS is superior to RRS, and the performance of PRS is almost the same to that of MC. Moreover, L∗ is a result of the balance between the first hop and the second hop. For PRS and MC, as the   ρ increases, decrease of E [TLsr ] is larger than that of E TLrd , and hence E [L∗] increases with ρ. However, for RRS, a larger decrease  of E TLrd is observed, leading to a decrease in E [L∗ ]. V. C ONCLUSIONS Based on the merits of rateless coding for relay transmission in ad-hoc networks, we proposed a simple probabilistic relay selection protocol for asynchronous relaying in this paper.

0.8

4

0.75 3.5 0.7

*

7

Maximum throughput (packets/slot)

0.65

3

0.7

*

8

4.5

0.85

E[L ]

0.7

0.6

3.5

0.75

E[L*]

Maximum throughput (packets/slot)

0.8 PRS MC RRS

9

E[L ]

10

Maximum throughput (packets/slot)

0.8 0.75

3

0.65 2.5 0.6

0.5 10

2

PRS MC RRS

0.55

15

20

25

SNR (dB)

30

1.5 10

PRS MC RRS 15

20

25

30

SNR (dB)

Fig. 3. The maximum throughput η ∗ and the expected value of L∗ versus SNR ρ with the relays in the middle (ls = ld = 2).

The average end-to-end throughput was analyzed by using order statistics. We also formulated an optimization problem to maximize the system throughput and obtained the optimal decoding relay threshold with the probabilistic relay selection algorithm. It was shown that PRS outperforms MC and RRS, especially when the relays are close to the source. In addition, for PRS, the maximum throughput can be achieved without requiring all of the relays to decode the message. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation divesity part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [2] ——, “User cooperation divesity part II: Implementation aspects and performance analysis,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1939– 1948, Nov. 2003. [3] J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415–2525, Oct. 2003. [4] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, “A simple cooperative diversity method based on network path selection,” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 659–672, March 2006. [5] J. Castura and Y. Mao, “Rateless coding for wireless relay channels,” IEEE Trans. Wireless Commun., vol. 6, no. 5, pp. 1638–1642, May 2007. [6] Y. Liu, “A low complexity protocol for relay channels employing rateless codes and acknowledgement,” in Proc. IEEE ISIT, Seattle, USA, July 2006, pp. 1244–1248. [7] A. F. Molisch, N. B. Mehta, J. S. Yedidia, and J. Zhang, “Performance of fountain codes in collaborative relay networks,” IEEE Trans. Wireless Commun., vol. 6, no. 11, pp. 4108–4119, Nov. 2007. [8] N. B. Mehta, V. Sharma, and G. Bansal, “Queued cooperative wireless networks with rateless codes,” in Proc. IEEE Globecom, New Orleans, USA, Dec. 2008. [9] R. Nikjah and N. C. Beaulieu, “Novel rateless coded selection cooperation in dual-hop relaying systems,” in Proc. IEEE Globecom, New Orleans, USA, Dec. 2008. [10] X. Wang, W. Chen, and Z. Cao, “Rateless coded chain cooperation in linear multi-hop wireless networks,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010. [11] M. Luby, “LT codes,” in Proc. 43rd Annual IEEE Symp. Foundations Computer Science (FOCS), Washington, DC, USA, Nov. 2002, pp. 271– 280. [12] A. Shokrollahi, “Raptor codes,” IEEE Trans. Inform. Theory, vol. 52, no. 6, pp. 2551–2567, June 2006. [13] G. Yue and X. Wang, “Anti-jamming coding techniques with application to cognitive radio,” IEEE Trans. Wireless Commun., vol. 8, no. 12, pp. 5996–6007, Dec. 2009. [14] X. Wang, W. Chen, and Z. Cao, “ARCOR: Agile rateless coded relaying for cognitive radios,” submitted to IEEE Trans. Veh. Technol. [15] H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. Hoboken, New Jersey, USA: John Wiley and Sons, Inc., 2003.