End-to-End HARQ in Cognitive Radio Networks

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End-to-End HARQ in Cognitive Radio Networks

Ao Weng Chon

Graduate Institute of Communication Engineering National Taiwan University

Taipei, Taiwan r97942044@ntu.edu.tw

Kwang-Cheng Chen

Graduate Institute of Communication Engineering National Taiwan University

Taipei, Taiwan chenkc@cc.ee.ntu.edu.tw

Abstract—Cognitive radio networks (CRN) may greatly enhance the throughput based on a given bandwidth. CRN has a unique feature, consisting of uni-directional opportunistic wireless links, and packets may be cooperatively relayed from source node to destination node through one or multiple paths, while each path consists of multi-hop opportunistic wireless fading links. Effective end-to-end error control is therefore a must to complete transportation of packets in CRN, though not being touched in open literatures. Traditional ARQ cannot generally function over uni-directional opportunistic links. We consequently develop a novel hybrid ARQ based on amplify- and-forward cooperative relay of CRN to reach the purpose of end-to-end error control at session level. Multiple coded sub- packets are sent through multiple paths, and the destination node decodes these received coded sub-packets from different paths. This HARQ for CRN works even missing some coded sub-packets. By the concept of outage, we also study allocation of coded sub-packets on multiple paths in an information theoretical view and develop a coding rate adaptation scheme for CRN such that end-to-end error control is possible.

I. INTRODUCTION

Facing the increasing demands of wireless services and the underutilization of licensed spectrum, cognitive radio (CR) to sense the spectrum and to opportunistically access the spectrum holes of primary (licensed) systems has emerged as a technology to enhance spectrum utilization. Networking CRs and primary system nodes to form a cognitive radio network (CRN) can use cooperative relay to greatly enhance network efficiency given a fixed bandwidth [1]. However, due to opportunistic access, CRN actually consists of uni-directional opportunistic wireless links [2]. A transmission opportunity may exist in one direction but no warranty in another direction.

Packets in CRN may be cooperatively relayed from source node to destination node through one or multiple paths, while each path consists of multi-hop uni-directional opportunistic wireless links. An effective end-to-end error control is thus necessary to facilitate reliable packet transmission in CRN.

Traditional link level ARQ/HARQ schemes based on bidi- rectional link assumption (the existence of feed-back channel) cannot generally function over uni-directional opportunistic links. ARQ is a link layer protocol used for packet error detection and retransmission. Error-detection bits (such as CRC bits) are attached and transmitted along with the data bits forming the message packet. Retransmission occurs when error is signaled. HARQ is a variation of ARQ and has two types in general. In Type-1 HARQ, FEC (forward-error cor-

rection) bits are additionally added. It reduces the number of retransmissions but a fixed through-put offset always exists due to the fixed FEC redundancy. Type-2/IR HARQ is proposed as a scheme to relax the fixed redundancy by introducing redundancy incrementally. In Type-2 HARQ, only data and error-detection bits are sent in the first transmission attempt.

Additional FEC parity bits are sent successively, combines with the corrupted packet and parity already received, then decodes, until the message packet is recovered or all the FEC parity bits are transmitted. An overview of the development of link level HARQ can be found in [3] and references therein.

In this paper, we consequently develop a novel end-to-end (session level) HARQ scheme to reach the purpose of end- to-end error control. The proposed scheme is as follows. We generate a coded packet consisted of data bits, error-detection bits and FEC bits at the source and divide the coded packet into multiple coded sub-packets. These coded sub-packets are sent through multiple forward paths. (A forward (backward) path from source (destination) to destination (source) consists of uni-directional and/or bi-directional links. Many different forward and backward paths may exist. We take the advantage of spatial diversity by using multiple paths between source and destination.) Each intermediate node within a path amplifies and forwards the coded sub-packets to next hop. (Within a path, link-by-link fully decode-and-forward is not possible due to the uni-directional characteristic of a link and amplify- and-forward is used instead.) Decoding is only performed at destination by combining coded sub-packets gathering from different paths. A session level ACK (NACK) is only generated by the destination passing through backward paths to the source provided that the origin message can (cannot) be suc- cessfully recovered. The idea of link level coded cooperation scheme is first introduced in [4]–[6]. Incremental redundancy is generated through cooperative relay in order to exploit spatial diversity. We here generalize it to session level in a network perspective. In [7] [8], performance analysis of other diversity and cooperative relay schemes are discussed.

Please note that the link level Type-2 HARQ as a rate adaption scheme generalizes Type-1 HARQ. Incremental redundancy is sent successively as needed. Decoding delay is multiple link delays. It will be transformed into multiple end- to-end path delays here if we simply imitate the concept of Type-2 HARQ. Such long decoding delay may not be suitable for some applications. Our proposed coding rate

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adaption scheme works in a “backward” sense. All redundancy is transmitted at first attempt. The coding rate will change successively according to the information piggybacked along the acknowledgement packet of the first attempt.

The remainder of the paper is organized as follows. In section II-A, link and path models are presented. It will be used for performance analysis of the coding scheme in section II-B. In section III, we will discuss how to decide the ratio of the coded packet allocated to each path (i.e. the size distribution of coded sub-packets) in an information-theoretic view. It is formulated as an optimization problem. In section IV, the impact of network delay is addressed. In section V, a coding rate adaption scheme is introduced. Section VI is the simulation results. Section VII is the conclusion.

II. END-TO-ENDHARQSCHEME

A. Link and path model

We assume that there exist L link-disjoint paths between the source and the destination as shown in figure 1. Each path i, 1 ≤ i ≤ L, has Mi− 1 intermediate nodes. Each link between a pair of node is modeled as an independent slow flat Rayleigh fading channel. The received signal at node Ri,j is yi,j = h_i,jx_i,j−1 + z_i,j. Channel gain hi,j is Rayleigh distributed. xi,j−1 is the signal generated from the previous hop R_i,j−1. z_i,j is independent zero-mean additive white Gaussian noise with variance N0. Each intermittent node amplifies and forwards the received signal subject to the same signal energy constraint Eb. The signal generated by node Ri,j

will be

xi,j= αi,jyi,j, α²_i,j= Eb

h²_i,jE_b+ N₀ (1) α_i,j is the amplifying coefficient. The per hop received SNR is defined as γi,j = h²_i,jEb/N0. It is exponential distributed with mean γi,j. The equivalent end-to-end path SNR γeq_i of path i is [9],

γ_eq_i =





Mi

Y

j=1

1 + 1

γi,j

− 1





−1

, 1 ≤ i ≤ L (2)

At high SNR, (2) reduces to

γeq_i =





M_i

X

j=1

1 γ_i,j





−1

, 1 ≤ i ≤ L (3)

It is more tractable and corresponds to the amplifying coefficient αi,j= 1/hi,j. The above equation relates the path SNR with the SNRs of its composed links.

B. Coding scheme

We assume that the rate of the FEC code is R. Message packet size is K bits. Coded packet size is N = K/R bits.

We divide the coded packet into L coded sub-packets. They are transmitted over L different paths. The L coded sub-packets are received, combined and decoded at the destination. The

Fig. 1. CRN network topology

pairwise error probability (PEP) between two codewords can be evaluated as

P (d|γeq) = Q

p2d₁γeq1+ 2d2γeq2+ ... + 2dLγeqL

(4) γeq = (γeq₁, γeq₂, ..., γeq_L). d is the Hamming distance between the transmitted codeword and the codeword obtained after decoding. di is the portion of Hamming distance con- tributed from the i’th coded sub-packet transmitted through path i, 1 ≤ i ≤ L, and d = d1+d2+...+dL. The unconditional pairwise error probability can be evaluated as

P (d) = Z

γ_eq

P (d|γ_eq)f (γ_eq)dγ_eq (5) f (γ_eq) = f (γ_eq₁)...f (γ_eq_L) is the probability density function of γeq. In high SNR, using the parameterization method introduced in [10], (5) can be evaluated as

P (d) ≈ QL

i=1(2i − 1) L2^L+1

L

Y

i=1



 1 d_i

M_i

X

j=1

1 γ_i,j



 (6)

We can see that diversity gain of order L is achieved. Also, the denominator of (6) has a term QL

i=1di corresponding to the coding gain. It is maximized when di ≈ d/L, i.e., the total Hamming distance is distributed evenly among each coded sub-packets.

The above result can be shown in an alternative way by noticing that in high SNR, the performance of the weakest link dominates the path,

γ_eq_i ≈ γ_b_i≡ min (γ_i,1, γ_i,2, ..., γ_i,M_i), 1 ≤ i ≤ L (7) γ_b_i is exponential distributed with mean

γ_b_i = 1

1 γi,1 +_γ¹

i,2 + ... + _γ¹

i,Mi

, 1 ≤ i ≤ L (8) Using the moment generating function method, (5) can be simplified and bounded as

P (d) = 1 π

Z ^π₂

0

1 + d1γb₁

sin²θ

−1

...

1 + dLγb_L

sin²θ

−1

dθ

≤ 1

2

1

1 + d1γb₁

...

1

1 + dLγb_L

< 1 2

L

Y

i=1



 1 di

M_i

X

j=1

1 γi,j



 (9)

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It is the same as (6) with a scaling factor.

For convolutional code, the bit error rate and packet error rate can be bounded as [11]

PBER≤ Z

γeq

min



 1 2,1

k

∞

X

d=d_f

∞

X

w=1

waw,dP (d|γeq)



f (γeq)

dγeq (10)

P_{P ER}≤ 1 − Z

γ_eq



1 − min



1,

∞

X

d=d_f

∞

X

w=1

a_w,dP (d|γ_eq)









B

f (γ_eq)dγ_eq (11)

where k is the number of input bits per trellis stage. df is the free distance of the code. aw,d is the number of codewords corresponding to input weight w and output weight d. B is the number of trellis stages in the codeword.

III. CODED PACKET DIVISION PROBLEM

In this section, we want to discuss how to decide the ratio of the coded packet allocated to each path, i.e. the size distribution of coded sub-packets. Our objective is to choose a division strategy to minimize the outage probability given a fixed source information rate. The strategy we should use depends on how much side information (path statistics) avail- able at the source. First, we consider the case that the source does not have any information about the statistics of paths. A reasonable strategy is to divide the coded packet evenly into the L paths. In other words, the source uses different paths in an equal time-sharing fashion. log₂(1 + γeq_i), 1 ≤ i ≤ L is the end-to-end capacity of path i. The source is assumed to have information rate R. In the high SNR regime, the outage probability P^out is

P^out = P

L

X

i=1

1

Llog₂(1 + γeq_i) < R

!

= P

L

X

i=1

log₂(1 + γeq_i) < LR

!

≥ P

L

\

i=1

{log₂(1 + γeqi) < R}

!

=

L

Y

i=1

P (log₂(1 + γ_eq_i) < R)

=

L

Y

i=1

P γ_eq_i < 2^R− 1

≈

L

Y

i=1



(2^R− 1)

M_i

X

j=1

1 γi,j



 (12)

In the low SNR regime, it becomes P^out = P

L

X

i=1

1

Llog₂(1 + γ_eq_i) < R

!

≈ P

L

X

i=1

γ_eq_i < LR log₂e

!

(13) Next, we consider the case that each source s has side information of mean statistics of paths between itself and its destination. Here, we present this exploration in a network perspective with multiple source-destination pairs using the network utility maximization framework [12]. Each source s has fixed information rate R^s and utility Us(ρs) which is a concave function in ρs. ρs is the reliability of s and equals 1 − P_s^out. w^s_i is the ratio of a coded packet of s admitted to its ith path (the time-sharing fraction of its ith path),P

iw^s_i = 1.

They are decision variables we want to find such that the sum utility of sources is maximized.

The network structure is described as follows. It has E links, 1 ≤ e ≤ E, and N source-destination pairs, 1 ≤ s ≤ N . Each source s has L^s paths, 1 ≤ i ≤ L^s, between itself and its destination. (We do not assume link-disjoint paths here.) A matrix H_ei^s is defined for each source s. H_ei^s = 1 when path i of source s uses link e, otherwise it equals 0.

Before presenting the optimization problem, we address that even in the limited SNR regime, the outage probability P_s^out is not jointly convex in w^s_i, 1 ≤ i ≤ L^s.

In the high SNR regime, P_s^out = P

L^s

X

i=1

w^s_ilog₂(1 + γ_eq^s_i) < R^s

!

≈ P

L^s

Y

i=1

(γ^s_eq

i)^w^sⁱ < 2^R^s

!

(14) In the low SNR regime,

P_s^out = P

L^s

X

i=1

w^s_ilog₂(1 + γ_eq^s

i) < R^s

!

≈ P

L^s

X

i=1

w^s_iγ^s_eq_i< R^s log₂e

!

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The optimization problem is,

maximize X

s

U_s(1 − P_s^out)

subject to P_s^out = P

L^s

X

i=1

w_i^slog₂(1 + γ_eq^s_i) < R^s

! , ∀s X

s

X

i

H_ei^sw^s_i ≤ 1, ∀e X

i

w_i^s= 1, ∀s 0 ≤ w^s_i ≤ 1, ∀s, i

1 ≤ s ≤ N, 1 ≤ i ≤ L^s, 1 ≤ e ≤ E (16)

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The second constraint is the capacity constraint. The optimization problem can be solved only numerically. Each source s will allocate ratio w^s_i^∗ to its ith path.

IV. NETWORK DELAY

In this section, we analyze the impact of network delay.

We model each multi-hop path as M/M/1 tandem queues.

Suppose that there exists an information flow with Poisson arrival rate λ between the source and the destination. The flow is divided into L sub-flows passing through L different paths with Poisson rate λi = wiλ, 1 ≤ i ≤ L,P

iwi = 1. (wi

can be chosen as described in section III.) Each node Ri,j

has its own background traffic with Poisson arrival rate bi,j

and the processing time is exponential distributed with mean 1/µ_i,j. The probability density function of link packet delay τ_i,j (queueing time plus processing time) of R_i,j is

f_τ_i,j(t) = µ_i,j(1 − ρ_i,j)e^−µ^i,j^(1−ρ^i,j^)t (17) where ρi,j = (λi+Pj

k=1bi,k)/µi,j. The probability density function of end-to-end packet delay τi of path i is the Mi-fold convolution of fτ_i,j(t), j = 1, ..., Mi

fτi(t) = fτi,1+...+τ_i,Mi(t) =

"_M_i Y

l=1

µi,l(1 − ρi,l)

#

×

M_i

X

j=1

e^−µ^i,j^(1−ρ^i,j^)t QMi

k=1,k6=j(µi,k(1 − ρi,k) − µi,j(1 − ρi,j))(18) The probability that the end-to-end packet delay of path i is smaller than t can be computed as P_i(t) ≡ P (τ_i ≤ t) = Rt

0fτ_i(x)dx. Suppose that there has an end-to-end packet delay threshold Tth, i.e. packets are considered to be loss when the end-to-end delay is greater than the threshold. For each path i, it happens with probability Pi(T_th). The performance degradation due to network delay can be characterized by the expected number of received coded sub-packets. It changes form L to PL

i=1P_i(T_th).

V. CODING SCHEME WITH RATE ADAPTION

As described in introduction, if we follow the design of typical link level Type-2 HARQ for rate adaptation, we may face a long decoding delay. It is because that only one out of L coded sub-packets is forwarded at a time and we wait an end-to-end round trip time to see if decoding succeeds.

Then we decide whether we should forward the next coded sub-packet. The decoding delay will be multiple end-to-end round trip times facilitating successful decoding. There exists a trade-off between decoding delay and ability of fine adaption of code rate.

Here we propose a two phase rate adaption scheme. In the following, the coded packet is assumed to be evenly divided into coded sub-packets. In “probe” phase, the source has no idea of the conditions of paths between itself and the destination. This may be due to the first attempt of transmission or after abrupt changes of conditions of paths indicated by previous decoding failure. The source transmits

all L coded sub-packets in this phase. (I.e. it uses the lowest code rate to probe the paths.) This initial rate setting leads to high probability of successful decoding and thus low decoding delay.

At the destination, we first consider the case that the receiver gathers, combines all the L coded sub-packets, then decodes. We want to deduce the required number of coded sub-packets actually facilitating the successful decoding. We first try to combine and decode the L − 1 coded sub-packets gathered from paths having highest path SNR. If decoding still succeeds, we throw away the least favor one and combine the L − 2 coded sub-packets having highest path SNR. This process continues until we fails to decode at combining the M − 1 best coded sub-packets. At this time, we know that combining the M best coded sub-packets is just enough (It corresponds to code rate R ∗ _M^L.) The remaining L − M coded sub-packets are not necessary. Note that each relay along a path needs to include the channel side information γi,j in its header to facilitate calculation of equivalent path SNR at the destination. The header of a coded sub-packet is assumed protected by a separated powerful low rate code and can be recovered individually. The destination will generate an acknowledgement of successful decoding to the source attaching the information of the combining ratio _M^L and the paths traversing by the M best coded sub-packets. The source can use this information to adapt the code rate and choose the appropriate paths for successive packet transmissions. That is, the source sends M out of L coded sub-packets through the reporting M paths. This comes the beginning of “settling”

phase. In “settling” phase, we continue trying to reduce the number of coded sub-packets transmitted at each successive transmission.

The “settling” phase ends when decoding failure occurs at the destination (due to abrupt changes of conditions of paths). A NACK is generated at the destination requesting transmission of all remaining coded sub-packets. After that, all L received coded sub-packets are combined to decode. We are now in “probe” phase again. In this rate adaptive scheme, there are regeneration points corresponding to the destination having all L coded sub-packets, i.e., beginning of “probe” phase. The regenerative cycle consists of the interlacing “probe” phase and “settling” phase. We assume that the regenerative cycle has average period W in unit end-to-end round trip time (RTT). The two phases interlace with each other of length 1 and W − 1 in unit RTT. Note that, if conditions of paths change abruptly within one round trip time, we are always in probing phase (W = 1), i.e. we always use the lowest code rate corresponding to total L coded sub-packets transmission.

A. Performance analysis

In the high SNR regime, we use the approximation (7) in the following derivation and assume that γb₁, .., γb_L have equal mean γb for simplicity. In the “probing” phase, the PEP can be derived as (9). In the “settling” phase, the PEP changes due to now choosing the M best paths from the L paths rather than M independent paths from the L paths. The

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joint probability density function of path statistics will become f γ_b₍₁₎, ..., γ_b_{(M )}, the upper M order statistic of γb1, ..., γ_b_L. Here γb₍₁₎ ≥ ... ≥ γb_(L). The PEP is

PM(d|γb₍₁₎, ..., γb_{(M )}) = Qq

2d1γb₍₁₎+ ... + 2dMγb_{(M )}

(19) with change of variables [13],

xi = γb_(i)− γb_(i+1), i = 1, ..., L − 1 xL = γb_(L)

or, γb_(i) = x_i+ ... + x_L, i = 1, ..., L (20) xi, i = 1, ..., L, are independent and distributed as exponential with mean ^γ_i^b. Substitute (20) in (19), it becomes

PM(d|γb₍₁₎, ..., γb_{(M )})

= Qp

2d1x1+ 2(d1+ d2)x2+ ... + 2(d1+ ... + dM)xL

The unconditional PEP can be computed as

PM(d)

= Z

P_M(d|γ_b₍₁₎, ..., γ_b_{(M )})f (γ_b₍₁₎, ..., γ_b_{(M )})dγ_b₍₁₎...dγ_b_{(M )}

= 1 π

Z ^π₂

0

1 + d1γb

sin²θ

−1

1 +(d1+ d2)^γ₂^b sin²θ

!−1

...

× 1 +(d₁+ d₂... + d_M)^γ_L^b sin²θ

!⁻¹ dθ

≤1 2

1

1 + d1γb

1

1 + (d₁+ d₂)^γ₂^b

! ...

× 1

1 + (d₁+ d₂... + d_M)^γ_L^b

!

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We can observe that diversity gain of order L is still achieved.

For convolutional code, the packet error probability after combining the M best coded sub-packets can be bounded as

P_PER(M ) ≤ 1

− Z



1 − min



1,

∞

X

d=d_f

∞

X

w=1

a_w,dP_M(d|γ_b₍₁₎, ..., γ_b_{(M )})









B

f (γb₍₁₎, ..., γb_{(M )})dγb₍₁₎, ..., dγb_{(M )} (22) B. Throughput analysis

We define the normalized throughput as T = K

lavg

(23) where lavg is the average number of total transmitted bits per K information bits. In the “probe” phase of the rate adaption scheme, we have l_avg^p = N . In the “settling” phase, the average number of transmitted bits depends on number of code sub-packets used to facilitate successful decoding. PPER(i) is

Fig. 2. Packet error rate with L = 2, 4, 6, 8, 10 corresponds to different order of path diversity.

Fig. 3. Packet error rate with coded sub-packet loss, L = 10.

the packet error probability combining i coded sub-packets defined in (22), so

l_avg^s ≈N

L(1 − PPER(1)) +

L

X

j=2

jN

L (1 − PPER(j))

j−1

Y

i=1

PPER(i)

+N

L

Y

i=1

PPER(i) (24)

The “probe” phase and “settling” phase interlacing with each other have length 1 and W-1 in unit RTT respectively, we have

lavg≈ 1

Wl^p_avg+W − 1

W l^s_avg (25)

VI. SIMULATIONRESULTS

We assume that message packet size is 120 bits. In figure 2, convolutional code with rate 1/2 is used and with rate 1/3 in other figures. Constraint length is 7. Modulation scheme is BPSK. Links are slow Rayleigh fading channels.

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Fig. 4. We only need to combine subset of received coded sub-packets from L = 10 paths to maintain fixed packet error rate 10⁻³.

Fig. 5. The number of coded sub-packets needed to achieve PER=10⁻³, L = 10.

Fig. 6. Throughput increases due to rate adaption.

In Figure 2, message packet is encoded, divided into L coded sub-packets and transmitted through L different paths,

L = 2, 4, 6, 8, 10. We can see that as L increases, packet error rate decreases due to higher order of path diversity. The number of hops of a path is uniformly distributed between 1 and 3. γ is the average SNR of a link and is same for all links.

In Figure 3, performance degradation due to coded sub-packet loss is shown with L = 10. γ_eq is the path SNR. In Figure 4, we can see that when γ_eq increases, we only need to combine subset of L = 10 received coded sub-packets to maintain a fixed packet error rate (10⁻³), e.g. we need 8 out of 10 coded sub-packets when γeq= 11(dB) , PER=10⁻³. In Figure 5, the exact number of coded sub-packets needed with different γeq

is shown. In Figure 6, throughput increases due to result of coding rate adaption.

VII. CONCLUSION

In this paper, we propose, in a unified fashion, an end- to-end HARQ scheme in cognitive radio network consisting of unidirectional and opportunistic links. End-to-end error control is developed based on coded cooperation among paths and amplify-and-forward within a path. Decoding is only performed at destination by combining coded sub-packets gathered from paths. Through simulations, we show that performance of coded cooperation can be improved by ex- ploiting path diversity. Facing the opportunistic and dynamic environment, performance degradation due to packet loss is also shown. The coded packet division problem is modeled as an optimization problem. A coding rate adaption scheme is provided to improve the end-to-end error control efficiency, such that reliable packet transmission in CRN is feasible.

REFERENCES

[1] C.-H. Huang, Y.-C. Lai, and K.-C. Chen, “Network capacity of cognitive radio relay network,” Physical Communication, vol. 1, no. 2, pp. 112 – 120, 2008.

[2] K.-C. Chen, B. K. Cetin, Y.-C. Peng, N. Prasad, J. Wang, and S. Lee,

“Routing for cognitive radio networks consisting of opportunistic links,”

Wireless Communications and Mobile Computing,Wiley, 2009.

[3] C. Lott, O. Milenkovic, and E. Soljanin, “Hybrid arq: Theory, state of the art and future directions,” in Information Theory for Wireless Networks, 2007 IEEE Information Theory Workshop on, July 2007, pp. 1–5.

[4] T. Hunter and A. Nosratinia, “Diversity through coded cooperation,”

IEEE Trans. Wireless Commun., vol. 5, no. 2, pp. 283–289, Feb. 2006.

[5] A. Nosratinia, T. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Commun. Mag., vol. 42, no. 10, pp. 74–80, Oct. 2004.

[6] M. Janani, A. Hedayat, T. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: space-time transmission and iterative decoding,” IEEE Trans. Signal Process., vol. 52, no. 2, pp. 362–371, Feb. 2004.

[7] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf.

Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

[8] D. Tse and P. Viswanath, Fundamentals of Wireless Communication.

Cambridge University Press, 2005.

[9] M. Hasna and M.-S. Alouini, “Outage probability of multihop transmission over nakagami fading channels,” IEEE Commun. Lett., vol. 7, no. 5, pp. 216–218, May 2003.

[10] A. Ribeiro, X. Cai, and G. Giannakis, “Symbol error probabilities for general cooperative links,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1264–1273, May 2005.

[11] E. Malkamaki and H. Leib, “Evaluating the performance of convolutional codes over block fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1643–1646, Jul 1999.

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[12] M. Chiang, S. Low, A. Calderbank, and J. Doyle, “Layering as optimization decomposition: A mathematical theory of network architectures,”

Proceedings of the IEEE, vol. 95, no. 1, pp. 255–312, Jan. 2007.

[13] M.-S. Alouini and M. Simon, “An mgf-based performance analysis of generalized selection combining over rayleigh fading channels,” IEEE Trans. Commun., vol. 48, no. 3, pp. 401–415, Mar 2000.