Wireless Pers Commun (2010) 55:441–455DOI 10.1007/s11277-009-9808-y

Performance Analysis of HARQ Transmissionin Cooperative DF Relaying Systems

Kan Zheng · Lijie Hu · Wenbo Wang · Lin Huang

Published online: 27 August 2009© Springer Science+Business Media, LLC. 2009

Abstract Hybrid automatic repeat request (HARQ) is the well-known technique toimprove the system throughput and link performance in wireless communication systems. Itcan also be applied together with the cooperative communication, which provides a new wayof exploiting the spatial diversity. Firstly, we present the analytical framework of decode-and-forward (DF) relaying systems with the general hop-by-hop HARQ mechanism in thispaper. Then, instead of capacity or outage performance which is only taken as the theoreticalguide, the block error rate bound of cooperative DF relaying systems with HARQ transmis-sion is analyzed by using weight enumerating functions. Numerical and simulation resultsdemonstrate the effectiveness of the proposed analytical method and show the gain of HARQtransmission in DF relaying systems.

Keywords Cooperative diversity · Decode-and-forward (DF) · Hybrid automaticrepeat request (HARQ) · Block error rate (BLER)

1 Introduction

To meet the explosive growth of demand for higher data bandwidth with greater reliability infuture wireless communication networks, the cooperative communication, which provides anew way of introducing spatial diversity by creating a virtual antenna array (VAA), becomesone of the most promising techniques [1]. Several repetition-based cooperative diversityalgorithms, such as amplify-and-forward (AF) and decode-and-forward (DF), are devel-oped to fully exploit the spectral diversity so that the outage probability can be reduced [2].Recently, standardization efforts of integrating cooperative relaying technologies to future

K. Zheng (B) · L. Hu · W. WangWireless Signal Processing and Network Lab, Key laboratory of Universal Wireless Communication,Ministry of Education, Beijing University of Posts & Telecommunications, 100088 Beijing, Chinae-mail: kzheng@ieee.org; zkan@buptnet.edu.cn

L. HuangOrange, France Telecom, R&D, Beijing, China

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wireless networks have commenced not only in the evolutional mobile Worldwide Inter-oper-ability for Microwave Access (WiMAX) but also in further advancements for third-generation(3G) long-term evolution (LTE) systems [3,4].

On the other hand, in wireless communication systems with appropriate feedback link,hybrid automatic repeat request (HARQ) mechanism can be incorporated into cooperativeprotocols where packet retransmissions are requested if previous transmissions are detectedin error. Various HARQ protocols with AF or DF relaying have been proposed for cooperativewireless networks and their long-term average throughput performances were evaluated byMonte Carlo simulations [5]. Depending on how to retransmit the received packets by therelay, there are two kinds of HARQ protocols in general as follows: (1) Upon reception ofa negative acknowledgement (NACK), the relay simply retransmits a copy of the receivedpackets from the source and the destination coherently combines the packets of the multipletransmissions, which is called as chase combining (CC) HARQ transmission [6]. (2) Afterthe source message is decoded and re-encoded into the codeword by the relay, the differ-ent part of the codeword is retransmitted from the relay to the destination, i.e. incrementalredundancy (IR) HARQ transmission [7].

To our knowledge, most of existed literatures have been focused on analyzing the outageor throughput of cooperative communication systems with HARQ transmission, which onlyprovides the theoretical performance bound for guide. For example, the outage probability ofan HARQ cooperative protocol has been derived in [8], under the hypothesis of nodes oper-ating on a single band in half-duplex mode with non-orthogonal transmission. Meanwhile,the throughput-delay performance of a half-duplex relay channel with HARQ was analyzedin [9], whose protocol uses a form of IR HARQ transmission with assistance from the relayvia space-time coding. Furthermore, a model was developed to analyze the performance interms of the probability of end-to-end delivery failure, delay distribution and throughput withthe method of discrete time Markov chain (DMTC) [10], where a number of relays supportthe transmission of each hop using a finite number of transmission rounds.

The HARQ transmission mechanism that we study in this paper is assumed to be appliedwith a time-slotted radio frame, where signals are transmitted in orthogonal channels so thattransmissions neither collide nor interfere. Moreover, the (re)transmission process follows astop-and-wait operation, which is commonly used such as in [11,12]. So far, no block errorrate (BLER) performance of cooperative communication systems with HARQ transmissionhas been investigated in detail, which is the exact measurement metric for performanceevaluation. Our main contributions in this paper are as follows:

– Firstly, we present the analytical framework for studying the BLER performances ofgeneral HARQ transmission in DF relaying systems under quasi-static Rayleigh fadingchannels.

– Secondly, the BLER union bounds of CC HARQ transmission in DF relaying systems arederived with the different number of (re)transmissions. Moreover, these analytical resultsare validated by the simulations.

In addition, the proposed analytical method can be conveniently extended to the scenarioswith different relaying schemes and HARQ protocols.

The reminder of this paper is organized as follows. After a brief description of the systemmodel and HARQ protocols in Sect. 2, the BLER performance is analyzed in Sect. 3. Thenumerical and simulation results are presented to demonstrate the effectiveness of HARQprotocols in Sect. 4. Finally, conclusions are drawn in Sect. 5.

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Performance Analysis of HARQ Transmission 443

S R DSR RDh h

S-R R-D S-R R-D t

Fig. 1 System diagram of a sample cooperative relaying system

2 System Overview

2.1 System Model

As shown in Fig. 1, let us consider a simple wireless cooperative communication systemin which a relay (R) node cooperates with a source (S) node to transmit information to adestination (D) node. We assume the half-duplex relay’s deployment which is motivatedfrom practical concern on the large difference between transmit and receive powers in manyapplications. Perfect channel state information (CSI) knowledge is assumed to be knownat the receiver. As in [8], all links are assumed to be long-term quasi-static wherein allHARQ rounds of a single packet experience a single channel realization. Subsequent packetsexperience independent channel realizations.

The signal transmitted by the source during the first time slot is denoted as x. We assumethat E{x} = 0 and E{|x |2} = 1 with the unified average transmitted power. Then, the signalsreceived by the relay in the ith transmission round through the channel between S to R isgiven by

y(i)S R = hS R x + z(i)

S R (1)

where hS R denotes the independent complex fading channel gain from S to R, modeled ashS R ∼ CN (

0, σ 2S R

)1 with σ 2

S R = E{|hS R |2}, the additive white gaussian noise plus interfer-

ence terms z(i)S R is modeled as z(i)

S R ∼ CN (0, σ 2

N

). In case of the retransmission at the S-to-R

link, i.e. i > 1, which is happened if the relay fails to decode the packet in the previous trans-missions, the relay combines the signals from the current and previous transmission rounds byusing maximal ratio combining (MRC) before decoding. Then, the resulting signal-to-noiseratio (SNR) at the S-to-R link can be expressed by

γ(i)S R = i |hS R |2/σ 2

N = iγ (1)S R , 1 ≤ i ≤ I (2)

with the mean of γ(i)S R = iσ 2

S R/σ 2N = i γ (1)

S R and I is the number of transmission rounds at theS-to-R link.

The relay detects and decodes the received signal. Only the packet passed cyclic redun-dancy check (CRC) will be re-encoded/re-modulated as xR , and transmitted at the second

1 A circularly symmetric complex Gaussian RV x with mean m and covariance R is denoted x ∼CN (m, R).

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time slot. Then, the signals received by the destination in the lth transmission round throughthe channel between R to D can be written as

y(l)RD = h RD xR + z(l)

RD (3)

where h RD denotes the fading channel gain from R to D, modeled as h RD ∼ CN (0, σ 2

RD

)

with σ 2RD = E{|h RD|2}, z(l)

RD ∼ CN (0, σ 2

N

)is additive noise plus interference received at the

destination. Similar to the operations at the relay, the MRC is applied to the signals receivedfrom the current and previous transmissions between R to D at the destination before decod-ing. Correspondingly, the instantaneous SNR can be written by

γ(l)RD = l|h RD|2/σ 2

N = lγ (1)RD, 1 ≤ l ≤ L (4)

with the mean of γ(l)RD = lσ 2

RD/σ 2N = lγ (1)

RD and L is the number of transmission rounds atthe link of R-to-D.

Different from the single-hop system, the retransmission of the two-hop system with DFrelaying may be happened in the hop either of S-to-R or of R-to-D. So, let us define themaximum number of HARQ retransmissions in DF relaying systems as M , which equals tothe sum of retransmission number in each individual link.

2.2 HARQ Protocol Description

Here we only focus on the hop-by-hop HARQ mechanism for cooperative DF relaying sys-tems, which has high transmission efficiency and short transmission delay. In this mechanism,the relay not only forwards the data and feedback information between the source and destina-tion but also generates its own feedback information and transmits it to the source. When theretransmission in one hop is happened, the receiver (relay or destination) combines the signalsfrom current and previous transmissions by maximal ratio combining (MRC) before decod-ing. At the end of transmission per hop, the receiver (i.e. the relay or destination) decodesthe packet and detects the error through the CRC. Then, the transmission result is feedbackthrough a 1-bit acknowledge (ACK) or non-acknowledge (NACK) message to the relay orsource. The NACK/ACK is assumed to be received error-free and with the negligible delay.

The HARQ mechanism considered in this paper is basically a stop-and-wait protocol. Aslong as NACK is received in each hop and the maximum number of HARQ retransmissionsis not reached, the source or relay successively retransmits the packet containing the sameinformation bits as before. Otherwise, the subsequent packets are transmitted. Not only chasecombining (CC) but also incremental redundancy (IR) can be used in this mechanism. Forthe sake of simplification, only CC is assumed in the following analysis.

3 BLER Performance Analysis

Before analyzing the BLER performance of CC HARQ transmission in DF relaying systems,we first have to calculate the error probability bound of the single-hop channel. The termi-nated convolutional codes with a finite uncoded block length K and coded block length Nare assumed to be used through this paper. Under the Rayleigh fading channel, the BLERupper bound of the received QPSK modulated signal conditioned on the instantaneous SNR

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Performance Analysis of HARQ Transmission 445

γ in single-hop transmission can be given by [13]

Pupper(γ ) = min

⎧⎨

⎩1,

N∑

d=d f

K∑

w=1

aw,d Q(√

dγ)⎫⎬

⎭(5)

where Q(x) = 1√2π

∫ ∞x exp(−y2/2)dy denotes the Gaussian Q-function, d f is the free

distance of the code, aw,d is the multiplicity of codewords corresponding to input weight w

and output weight d.Next, the end-to-end BLER can be calculated as the function of the average BLER of

different individual links. Then, the union bounds for the BLER are determined by usingweight enumerating functions before averaging.

3.1 Non-HARQ Transmission (M = 0)

To better understand HARQ transmission for DF relaying, let us first take a close look at theperformance for two-hop non-HARQ transmission, i.e. M =0.

Notice that no error at the destination is happened only when both of links from S to Rand from R to D are reliable. Therefore, on the assumption that the channels of differentlinks are independent, the end-to-end BLER of the DF relaying system conditioned on theinstantaneous γ

(1)S R and γ

(1)RD can be written as

�(M)(

e|γ (1)S R , γ

(1)RD

)= 1 −

[1 − P

(γ

(1)S R

)] [1 − P

(γ

(1)RD

)]

=[1 − P

(γ

(1)S R

)]P

(γ

(1)RD

)+ P

(γ

(1)S R

) , (6)

where P(γ

(1)S R

)and P

(γ

(1)RD

)are the BLERs of the S-to-R link and the R-to-D link con-

ditioned on the instantaneous γ(1)S R and γ

(1)RD , respectively. For sake of further analysis, we

define the function F0(x, y) = [1 − P(x)]P(y) + P(x).

Then, the corresponding conditional bound of �(0)(

e|γ (1)S R , γ

(1)RD

)is given by

�(0)upper

(e|γ (1)

S R , γ(1)RD

)=

[1 − Pupper

(γ

(1)S R

)]Pupper

(γ

(1)RD

)+ Pupper

(γ

(1)S R

)(7)

To obtain the average BLER, we must take the expected values of (7) over the distribution

of fading coefficients. So the unconditional bound of P(0)e

(γ

(1)S R , γ

(1)RD

)is calculated by

�(0)upper =

∞∫

0

∞∫

0

�(0)upper

(e|γ (1)

S R , γ(1)RD

)p

(γ

(1)S R

)p

(γ

(1)RD

)dγ

(1)S R dγ

(1)RD (8)

where p(x) is the probability density function (pdf) of random variable x. Under the Rayleighfading channel, the fading coefficients have an exponential distribution and their pdfs can bewritten as

p(γ

(1)S R

)= 1

γ(1)S R

exp

{

−γ(1)S R

γ(1)S R

}

p(γ

(1)RD

)= 1

γ(1)RD

exp

{

−γ(1)RD

γ(1)RD

} (9)

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446 K. Zheng et al.

3.2 HARQ Transmission (M = 1)

If the error is detected at the relay or destination by checking CRC, the packet will beretransmitted. When M is set to be one, the retransmission is happened once at the link eitherfrom S to R or from R to D. Errors at the destination occur only in three cases as shown inFig. 2.

On the assumption of the event �GS R,1 that the S-to-R transmission is received correctly in

its initial transmission round, the retransmission might be needed only at the link of R-to-D.The error event is happened only when the information bits still can’t be decoded correctlyafter the retransmission at the R-to-D link, i.e. (I, L)= (1, 2). In this case, the error probabilityand its bound conditioned on the event �G

S R can be calculated as

P{e|�G

S R

} = P(γ

(1)RD

)P

(γ

(2)RD

)

≤ PGupper = Pupper

(γ

(1)RD

)Pupper

(γ

(2)RD

) (10)

Otherwise, the original packet has to be retransmitted at the S-to-R link. Then, supposethat �E

S R is used to indicate the event that packet transmitted at the link of S-to-R is receivederroneously in its initial transmission round. The possibilities of the error event happenedare similar to the scenario of M =0, i.e.

RS D

MRC and CRC failed

MRC and CRC successful

NACK feedback

ACK feedback

Initial transmission

Retransmission

Fig. 2 Example on error events with HARQ transmission (M=1)

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Performance Analysis of HARQ Transmission 447

– (I, L)= (2,0): The retransmission at the S-to-R link fails.– (I, L)= (2,1): The retransmission at the S-to-R link is successful but the transmission at

the R-to-D link not.

So, the corresponding conditional error probability and its bound can be written as

P{

e|�ES R

}= F0

(γ

(2)S R , γ

(1)RD

)=

[1 − P

(γ

(2)S R

)]P

(γ

(1)RD

)+ P

(γ

(2)S R

)

≤ P Eupper =

[1 − Pupper

(γ

(2)S R

)]Pupper

(γ

(1)RD

)+ Pupper

(γ

(2)S R

). (11)

As shown in (2) and (4), γ(i)S R and γ

(l)RD have the linear relationship with γ

(1)S R and γ

(1)RD ,

respectively. Therefore, in case of M ≥ 1, the conditional BLER of DF relaying systemswith HARQ retransmission can be expressed by

�(1)(

e|γ (1)S R , γ

(1)RD

)=

[1 − P

(γ

(1)S R

)]P

{e|�G

S R

}+ P

(γ

(1)S R

)P

{e|�E

S R

}. (12)

Then, another function is defined for further study, i.e.

F1(x, y, u, v) = [1 − P(x)]P(u)P(v) + P(x)F0(y, u). (13)

Since P{e|�G

S R

} ≥ P{e|�E

S R

}, we can use Lemma 1 (See the proof in Appendix A) to

get the conditional upper bound of �(1)(

e|γ (1)S R , γ

(1)RD

)as

�(1)upper

(e|γ (1)

S R , γ(1)RD

)=

[1 − Pupper

(γ

(1)S R

)]PG

upper + Pupper

(γ

(1)S R

)P E

upper. (14)

Similarly, the bound of average BLER with HARQ retransmission is generated by takingthe expected value of (14) as

�(1)upper =

∞∫

0

∞∫

0

�(1)upper

(e|γ (1)

S R , γ(1)RD

)p

(γ

(1)S R

)p

(γ

(1)RD

)dγ

(1)S R dγ

(1)RD . (15)

3.3 HARQ Transmission (M = 2)

With the increased M, there are more options of error events when (re)transmission in eachhop. These error events can be first categorized according to transmission results of theS-to-R link. If the S-to-R transmission is received successfully in its initial transmissionround, the error event might be happened only after retransmission twice at the link of R-to-D. Otherwise, the cases are similar to the scenario of M= 1. As shown in Fig. 3, four errorevents in case of M= 2 are found, i.e.

– (I, L)= (1,3): The packet transmitted at the S-to-R link is received correctly in its initialtransmission round but in error at the S-to-R link even after retransmission twice.

– (I, L) =(2,2): The packet transmitted at the S-to-R link is received correctly afterretransmission once but in error at the S-to-R link even after retransmission once.

– (I, L) = (3,1): The packet transmitted at the S-to-R link is received correctly afterretransmission twice but in error at the S-to-R link without retransmission.

– (I, L) = (3,0): The retransmission at the S-to-R link fails even after retransmission twice.

To better understand the behavior of HARQ mechanism, the corresponding operation ateach link in case of error events is also summarized in Table 1. With the help of this table,

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448 K. Zheng et al.

RS D

MRC and CRC failed

MRC and CRC successful

NACK feedback

ACK feedback

Initial transmission

Retransmission

Fig. 3 Example on error events with HARQ transmission (M = 2)

the end-to-end BLER of HARQ transmission with M= 2 conditioned on the instantaneousγ

(1)S R and γ

(1)RD in DF relaying systems can be expressed by

�(2)(

e|γ (1)S R , γ

(1)RD

)

=[1 − P

(γ

(1)S R

)] 3∏

l=1

P(γ

(l)RD

)+ P

(γ

(1)S R

)F1

(γ

(2)S R , γ

(3)S R , γ

(1)RD, γ

(2)RD

)

=[1 − P

(γ

(1)S R

)] 3∏

l=1

P(γ

(l)RD

)+ P

(γ

(1)S R

) [1 − P

(γ

(2)S R

)] 2∏

l=1

P(γ

(l)RD

)

+2∏

k=1

P(γ

(k)S R

) [P

(γ

(3)S R

)+ P

(γ

(1)RD

)− P

(γ

(3)S R

)P

(γ

(1)RD

)], (16)

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Performance Analysis of HARQ Transmission 449

Table 1 Summary on operation at each link in case of error events (M = 2)

(I, L) = (1, 3) (I, L) = (2, 2) (I, L) = (3, 1) (I, L) = (3, 0)

S → R ♦ � � �R → D � ∅ ∅ ∅

S → R ∅ ♦ � �R → D � � ∅ ∅

S → R ∅ ∅ ♦ �R → D � � � ∅

Notes: ♦ = MRC and CRC successful; � = MRC and CRC failed; ∅ = not available

Firstly, the bound of F1

(γ

(2)S R , γ

(3)S R , γ

(1)RD, γ

(2)RD

)can be generated by the method used in

the case of M = 1. Then, by applying Lemma 1 into (16) recursively, the conditional BLER

bound of HARQ transmission with M = 2, i.e. �(2)upper

(e|γ (1)

S R , γ(1)RD

), can be calculated

while replacing all the P(γ ) terms in (16) with the corresponding Pupper(γ ). Finally, theunconditional BLER bound of HARQ transmission with M = 2 in DF relaying systems iscalculated by

�(2)upper =

∞∫

0

∞∫

0

�(2)upper

(e|γ (1)

S R , γ(1)RD

)p

(γ

(1)S R

)p

(γ

(1)RD

)dγ

(1)S R dγ

(1)RD . (17)

Note that this analysis can be easily extended to the case of M > 2. However, too largenumber of retransmission causes low transmission efficiency and large transmission delay,which can’t meet the quality of service requirements in the practical systems. Therefore, ouranalysis is limited to the case of M ≤ 2 in this paper. In addition, this analytical method canalso be applied in the multipath channel only when the joint probability density function ofthe received instantaneous SNR can be given in some special cases.

In principle, the BLER performance is improved after HARQ transmission due to theeffects of diversity gain so that the system throughput is also increased. However, since theretransmission costs more radio resources and the diversity gain becomes saturated with morenumber of retransmission, the throughput can not always be enhanced when the number ofretransmission is further increased.

4 Numerical and Simulation Results

The performances of HARQ transmission with DF relaying are evaluated by numerical anal-ysis and discussed in this section. We also give the performances of the DF relaying systemswithout HARQ transmission as reference, i.e. M = 0. QPSK modulation is used in DF relay-ing transmission. The family of rate-compatible punctured convolutional (RCPC) codes withgenerator polynomials G(171,133) is employed with rate Rc = 1/2, K= 200 and N= 400.The distance spectra aw,d is computed via computer enumeration for our analysis. Gener-ally, since the error event with minimum distance dominates the error performance, we onlysearch the distance spectra of the first nine Hamming weights and get the distance spectravalue as shown in Table 2.

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450 K. Zheng et al.

Table 2 Distance spectra value of RCPC codes with generator polynomials G(171,133)

d 10 11 12 13 14 15 16 17 18

∑Kw=1 aw,d 2090 0 7057 0 33738 0 202406 0 853102

Fig. 4 BLER performance in DF relaying system with/without HARQ (γS R , γRD) = (γ + 10 dB, γ )

Fig. 5 BLER performance in DF relaying system with/without HARQ (γS R , γRD) = (γ , γ + 10 dB)

Firstly, we consider representative scenarios that corresponding to those in which R islocated either close to S, close to D, or in the middle of S and D; the corresponding averageoutput SNRs (γS R, γRD) in logarithmic-scale are (γ + 10 dB, γ ), (γ , γ + 10 dB) or (γ , γ ),respectively. The block error rate (BLER) performances of DF relaying systems with or with-out HARQ transmission under these three different scenarios are shown in Figs. 4, 5 and 6,respectively, where both analysis and simulation results are given. Clearly, the analyticalresults match the simulation results well in all the cases including single transmission andHARQ transmission. From these results, we can find that the significant performance gain in

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Performance Analysis of HARQ Transmission 451

Fig. 6 BLER performance in DF relaying system with/without HARQ (γS R , γRD) = (γ , γ )

Fig. 7 BLER performance in DF relaying system with the fixed γRD = 20 dB

term of SNR can be achieved by using HARQ due to diversity gain. For example, as shownin Fig. 4, the SNR gain achieved retransmission once (i.e. M = 1) is about 3 dB if the targetBLER is 10−2 compared with the case of non-HARQ. With the increased M , (i.e. M = 2),the gain becomes more, (i.e. 4.7 dB). The similar gains can be found in Figs. 5 and 6.

With the fixed γRD = 20 dB, the BLER performances with/without HARQ in DF relay-ing systems are given in Fig. 7. When the SNR of the S-to-R link is low or medium (i.e. nomore than 20 dB), the BLER performances with/without HARQ in DF relaying systems areimproved by the increase of γS R . However, if the SNR of the S-to-R link is further increased(i.e. larger than 20 dB) with the fixed SNR of the R-to-D link (i.e. 20 dB), the BLER flooris happened because the performance of DF relaying systems is limited by any link of twohops.

Next, Fig. 8 shows the BLER performances of systems under the scenario, where the sumof the averaged SNR in dB of two hops is kept same, i.e. γS R + γRD = 30 dB with varying

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452 K. Zheng et al.

Fig. 8 BLER performance in DF relaying system with changing location of R (γS R + γRD = 30 dB)

Fig. 9 BLER difference between the analytical upper bound and the simulation results in DF relaying systemwith/without HARQ (γS R , γRD) = (γ + 10 dB, γ )

γS R . It can be seen that the BLER performance is best in case of γS R = γRD because theoverall performance is determined by the link with the poorer channel quality in DF relayingsystems. The gain due to HARQ transmission is almost same with the different γS R , whichmeans that the channel quality of any hop in DF relaying systems has the same effects on theHARQ gain. These analysis bounds also match well to the simulation results, proving theeffectiveness of our analysis method.

Furthermore, in order to show the tightness of the analytical bound, let us define the uni-fied difference between the analytical upper bound and the simulation results as ��(M) =�

(M)upper − �(M). Then, we compare the performances of ��(M) in Fig. 9, where the average

SNR γ is varying from 5 dB to 20 dB. With the increase of M , the BLER difference becomessmaller because the absolute value of BLER is decreased. The similar conclusion can bedrawn under the different average SNR γ .

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Performance Analysis of HARQ Transmission 453

5 Conclusion

In this paper, we presented a hop-by-hop HARQ mechanism in cooperative DF relayingsystem and analyzed its performance in term of BLER. In different scenarios, numericaland simulation results demonstrated the correctness and effectiveness of our proposed anal-ysis method. The HARQ in DF relaying systems can significantly improve performance ofDF relaying systems because it is capable of exploiting the diversity gain. Meanwhile, it isshown that the channel quality of any hop of DF relaying systems has the same effects on theHARQ gain. Furthermore, our future work will focus on extending the proposed analyticalframework to other HARQ mechanisms in more complex cooperative relaying systems.

Acknowledgments This work was supported in part by China NSFC under Grant 60802082 and ResearchFund for the Doctoral Program of Higher Education under Grant 200800131023.

Appendix

A Proof of Lemma 1

Lemma 1 Let X, Y and Z be three independent RVs with the upper bound of X , Y and Z ,

respectively, i.e. X ≤ X , Y ≤ Y and Z ≤ Z . Also, assume that 0 ≤ X, X , Y, Y , Z , Z ≤ 1.If Y ≥ Z, then

XY + (1 − X)Z ≤ X Y + (1 − X)Z (18)

Assume that

1 ≥ X = X + �X ≥ 01 ≥ Y = Y + �Y ≥ 01 ≥ Z = Z + �Z ≥ 0

(19)

with �X,�Y,�Z ≥ 0.Then, let us integrate (19) into (18) and get

XY + (1 − X)Z

≤ (X + �X)(Y + �Y ) + (1 − X − �X)(Z + �Z) (20)

Next, rewrite (20) as the follows:

�X (Y − Z) + �Y (X + �X) + �Z(1 − X − �X) ≥ 0 (21)

Now it is clear that equation (21), which is the equivalent of (18), exists only in case ofY ≥ Z.

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Author Biographies

Kan Zheng received the B.S., M.S. and Ph.D. degree from BeijingUniversity of Posts & Telecommunications (BUPT), China, in 1996,2000 and 2005, respectively, where he is currently associate professorin Universal Wireless Communication, Ministry of Education, BUPT.From April 2000 to October 2001, he was a system development engi-neer at TD-SCDMA R&D centre of Siemens (Ltd) at Beijing, China.His current research interests lie in the field of signal processing fordigital communications, with emphasis on PHY/MAC algorithms inwireless communication systems.

Lijie Hu received the B.S. degree from Beijing University of Posts &Telecommunications (BUPT), China, in 2007, where she is workingfor M.S. degree. Her current research interests include link adaptionalgorithms in wireless cooperative systems.

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Wenbo Wang received his B.S., M.S. and Ph.D. degree from BeijingUniversity of Posts & Telecommunications (BUPT), China, in 1986,1989 and 1992, respectively. He is currently a professor and dean ofschool of Telecommunication Engineering of BUPT. His research inter-ests include signal processing, mobile communications and wirelessnetwork.

Lin Huang received the B.S and M.S. from Beijing University of Posts& Telecommunications, China, in 2002 and 2005, respectively. Nowshe works on Orange Labs, France Telecom R&D, Beijing, China.Her current research interests lie in the field of cognitive/cooperativecommunication schemes.

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