384 IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 5, MAY 2006

Modeling Conditional FER for Hybrid ARQJian Gu, Yi Zhang, and Dacheng Yang

Abstract— This letter investigates the conditional frame errorrate (FER) of hybrid automatic repeat request (HARQ) retrans-mission conditioned on its previous erroneous transmission. Forsimplification, it is called ’conditional FER’ in this letter. On thebasis of theoretical analysis, this letter validates the necessity ofmodeling conditional FER in wireless communication systemswith HARQ and proposes a low-complexity implementationmethod. Simulation results prove the method’s accuracy.

Index Terms— Channel coding, conditional frame error rate,hybrid automatic repeat request (HARQ), wireless communica-tions.

I. INTRODUCTION

H IGH-SPEED data services, such as HTTP, FTP and real-time video, become more and more attractive in wireless

communication systems. In order to offer high-speed dataservices, some third generation (3G) wireless communicationsystems (cdma2000 1xEV and WCDMA HSDPA/HSUPA)adopt hybrid automatic repeat request (HARQ) mechanisms.Hybrid ARQ combines forward error correction (FEC) andconventional ARQ techniques. It greatly improves systemthroughput despite additionally required associated signal-ing, including acknowledgments/ negative acknowledgments(ACK/NACK) or HARQ information indicators.

Modeling Hybrid ARQ is very important for an accurateperformance evaluation of the aforementioned systems. In con-ventional system level simulations [1-3] of wireless commu-nication systems, the frame error rates (FER) of initial HARQtransmissions and subsequent retransmissions are predicted onbasis of the same FER curve. These conventional methods aresimple. However, their results are too optimistic [4,5], becauseHARQ retransmission, regardless of ACK feedback error,usually occurs on the condition that previous transmission isnot correctly received. For instance, FER achieved in the firsttransmission attempt is fer. Without modeling conditional FERin the conventional methods, the second transmission attempttransmitted with zero power achieves the residual FER offer2, which leads to overoptimistic system designs. Therefore,separate conditional FERs of HARQ retransmissions should beused for determining the actual error rate of data transmissionsin wireless communication systems. However, it is difficult toobtain conditional FERs for HARQ retransmissions throughlink level simulations with reasonable confidence intervals inthat too many conditions, including different coding rates and

Manuscript received October 20, 2005. The associate editor coordinatingthe review of this letter and approving it for publication was Dr. Cha’o-MingChang.

The authors are now with the School of Telecommunication Engineer-ing, Beijing University of Posts and Telecommunications, P.O.Box 93, No10 Xitucheng Road Beijing 100876 P.R. China (e-mail: jian gu@163.com;zhangy@buptnet.edu.cn; yangdc@bupt.edu.cn).

Digital Object Identifier 10.1109/LCOMM.2006.05006.

different Eb/N0 at different transmission attempts, have tobe considered. This letter proposes a low-complexity methodto model conditional FER in simulations and validates theproposed method through numerical results.

II. CONDITIONAL FER FOR RETRANSMISSIONS

A. Problem

This letter first of all analyzes the case for at most oneretransmission in subsections A-C and then generalizes it tocases with more than one retransmission in subsection D.

The performance of HARQ depends on the effective Eb/N0

after soft combining and the effective code rate1. The effectiveEb/N0 can be obtained by the equivalent SNR method (ESM)[6] for the reverse link and by the quasi-static method [1]or the extended actual value interface (AVI) method [2] forthe forward link. For example, according to [6], the effectiveEb/N0 in reverse link with pilot weighted combining receivercan be expressed by

2M(∑I

i=1

∑Pp=1 Eiα

2i,p

)2

F∑I

i=1

∑Pp=1

Eiα2i,p

Γi,p

(α2

i,pΓi,p + Λi,p + 1) (1)

where M is the number of modulation symbols per slot, F isthe number of information bits per slot, Ei is the signal poweron the ith slot, and αi,p, Γi,p and Λi,p are the fading amplitude,SNR of the channel estimator and received modulation symbolSNR on the pth path and ith slot, respectively.

Tn denotes the nth subpacket of the data packet, wheren = 1, ..., N . Tn = 0 or 1, respectively, denotes success orfailure in the nth HARQ transmission attempt.

Then, the FER of the initial transmission attempt can beexpressed as

p

(T1 = 1|

( Eb

N0

)1

=(eb/n0

)1, R1 = r1

)(2)

where(

Eb

N0

)1

is the effective Eb/N0 and R1 is the code rate.After soft combining after the first retransmission, the FER is

p

(T1 = 1, T2 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)(3)

where(

Eb

N0

)e

is the effective Eb/N0 and Re is the effectivecode rate after soft combining of two transmissions.

1Besides the effective Eb/N0 and the effective coding rate, more factorsare used to model HARQ, such as Doppler penalty and demapping penalty in[1], combination loss and incremental redundancy gain in [2], and equivalentstandard deviation and demapping penalty in [3]. It is straightforward to adaptthe analysis in this letter to a specific one.

1089-7798/06$20.00 c© 2006 IEEE

GU et al.: MODELING CONDITIONAL FER FOR HYBRID ARQ 385

It is impractical to obtain all the required

curves of the conditional FER, p

(T2 =

1|T1 = 1,

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

), for different

values of(

Eb

N0

)1, R1,

(Eb

N0

)e, Re through simulations with

reasonable confidence intervals. The number of requiredFER values is n1nem1me, where n1, m1, ne and me are,respectively, the sample number of

(Eb

N0

)1, R1,

(Eb

N0

)e, and

Re .It is even more impractical to get the conditional FER curves

for wireless communication systems in which the allowablemaximum number of HARQ transmissions, N , is larger than2.

B. Underlying Assumption

Assume that the CRC bits, the ACK feedback and theassociated control channel are error free. In this case, if theinitial transmission is correct, retransmissions do not happenat all, i.e.

p (T1 = 0, T2 = 1) = 0 (4)

In actual systems with possible transmission errors ofACKs, the retransmitted data of a correctly received packetis discarded before it is fed to the decoder. At the same time,the missing detection probability of CRC bits and the errorrate of the associated control channel are both very low inactual systems. Therefore, the given assumption is reasonablein all cases.

C. Proposed Solution

Because the HARQ procedure is causal, it can be obtainedthat

p

(T1 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

= p

(T1 = 1|

( Eb

N0

)1

=(eb/n0

)1, R1 = r1

)(5)

From (5), the conditional FER conditioned on erroneousreception of the 1st transmission is given by

p

(T2 = 1|

T1 = 1,(

Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

=

p

(T1 = 1, T2 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

p

(T1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1

) (6)

From (4), the residual FER after at most two Hybrid ARQtransmissions, i.e., the numerator in (6), can be expressed as

p

(T2 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

).

Coded symbols of HARQ transmissions are soft combinedbefore the channel decoder in a HARQ receiver. Therefore, theperformance after the two HARQ transmissions only dependson the effective Eb/N0 and coding rate after soft combining oftwo transmissions. Therefore, the residual FER after at mosttwo Hybrid ARQ transmissions can be rewritten as

p

(T2 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

= p

(T2 = 1|

( Eb

N0

)e

=(eb/n0

)e, Re = re

)(7)

By replacing the numerator of (6) with (7), we get

p

(T2 = 1|

T1 = 1,(

Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

=

p

(T2 = 1|

( Eb

N0

)e

=(eb/n0

)e, Re = re

)

p

(T1 = 1|

( Eb

N0

)1

=(eb/n0

)1, R1 = r1

) (8)

Therefore, we can obtain the FER curves at the initialtransmission and at the first retransmission through link levelsimulations. With (8), the number of required FER values isreduced to n1m1 + neme.

D. Generalization

Assumep (Tn−1 = 0, Tn = 1) = 0 (9)

In case of at most N transmissions, it is straightforwardto obtain the FER conditioned on the (n − 1)th transmissionattempt failure by

p

(Tn = 1|

Tn−1 = 1,(

Eb

N0

)e,n

=(eb/n0

)e,n

, Rn = rn,(Eb

N0

)e,n−1

=(eb/n0

)e,n−1

, Re,n−1 = re,n−1

)

=

p

(Tn=1|

(EbN0

)e,n

=(eb/n0

)e,n

,Re=re

)

p

(Tn−1=1|

(EbN0

)e,n−1

=(eb/n0

)e,n−1

,Rn−1=rn−1

)

(10)where N ≥ n > 1,

(Eb

N0

)e,n

and(

Eb

N0

)e,n−1

, respectively,

denote the effective Eb/N0 after n and n-1 transmissions, andRe,n and Re,n−1 denote the corresponding effective codingrates after these transmission attempts.

Therefore, the number of required FER values is reducedby the proposed method for all conditional FERs.

III. VERIFICATION THROUGH NUMERICAL RESULTS

Simulations were conducted for the turbo coded R-PDCHof the cdma2000 1xEV-DV system [7]. The link data rate is40.8 kbps with a BPSK modulation scheme. (1) is adopted forthe calculation of the effective Eb/N0. Single-path Rayleighfading channel with a velocity of 3km/h is modeled. Two

386 IEEE COMMUNICATIONS LETTERS, VOL. 10, NO. 5, MAY 2006

cases, in which n=2 and 3, are simulated. FER results areobtained after at least 100 residual erroneous frames arecollected. ACK feedback error rate is 1% in the simulations.It is shown that the conditional FER curve based on theerroneous (n − 1)th transmission is quite different from the

single transmission FER curve at large(

Eb

N0

)e,n−1

. This can

be explained as follows: If(

Eb

N0

)e,n−1

is large, the FER after

n − 1 transmissions is much smaller than one. Therefore, thenecessity of applying conditional FER to HARQ transmissionsimulations is also validated here.

From Fig. 1, both actual conditional FERs for n=2 and 3are similar to the conditional FER predicted by the proposedmethod, which proves the accuracy of our method. Noticethat the former is slightly smaller than the latter. It is becauseequation in the assumption of (4) should be approximateequation due to non-zero ACK feedback error rate. In thecase of maximum one retransmission, the exact solution to(8) should have a complement that can be expressed as

−p

(T1 = 0, T2 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1,(

Eb

N0

)e

=(eb/n0

)e, Re = re

)

p

(T1 = 1|

(Eb

N0

)1

=(eb/n0

)1, R1 = r1

)

(11)The numerator of (11) is usually small. Because ACK feed-back error rate is constant and p(T1 = 0) is nearly 1 inmedium and high (eb/n0)1 region, the numerator of (11)is nearly independent of (eb/n0)1. At the same time, thedenominator of (11) decreases with (eb/n0)1. Therefore, thecomplement is more visible in high (eb/n0)1 region than inlow (eb/n0)1 region.

The shown actual conditional FERs for n=2 and 3 aresimilar, because all the effective coding rates are 0.2 for the1st, 2nd and 3rd transmissions and power control works verywell in the simulations.

It can be obtained from the simulation results that theconditional FER at retransmission depends on the value ofthe effective Eb/N0 after the previous transmission. The largerthe effective Eb/N0 of the previous transmission, the largerthe conditional FER conditioned on the previous erroneoustransmission.

IV. CONCLUSION

Conditional Frame Error Rate (FER) is necessary for ac-curate modeling in simulations of wireless communicationsystems with HARQ, because the modeling without condi-tional FER delivers too optimistic results. An accurate andlow-complexity method to obtain the conditional FER withthe FER curve of the initial transmission has been introducedin this letter. As was shown, the conditional FER highlydepends on the quality of the previously received subpackets.The conditional FER is much worse than the FER at initialtransmissions, especially if the link quality at the previoustransmission is good.

0.5 1 1.5 2 2.510−3

10−2

10−1

100

(Eb/N0)e,n(dB)

FER (Eb/N0)e,n−1=0.5dB

(Eb/N0)e,n−1=1dB

(Eb/N0)e,n−1=1.3dB

(Eb/N0)e,n−1=1.5dB

FER at the 1st transmission

actual conditional FER, n=2

predicted conditional FER, n=2

actual conditional FER, n=3

predicted conditional FER, n=3

Fig. 1. The FER at the 1st transmission, the predicted conditional FER andthe actual conditional FER for the 2nd and 3rd HARQ transmissions.

ACKNOWLEDGMENT

The authors would like to thank Dr. Jack M. Holtzman ofQualcomm Inc for his valuable suggestions and comments onthis work, the editor and the anonymous reviews for theirinsightful comments, and Andreas Mueller of University ofStuttgart for his work improving the language.

REFERENCES

[1] R. Ratasuk, A. Ghosh, and B. Classon, “Quasi-static method for predict-ing link-level performance,” in Proc. 55th IEEE Vehicular TechnologyConference, Spring 2002, vol. 3, pp. 1298-1302.

[2] F. Frederiksen and T. E. Kolding, “Performance and modeling ofWCDMA/HSDPA transmission/HARQ-schemes,” in Proc. 56th IEEEVehicular Technology Conference, Fall 2002, vol. 1, pp. 472-476.

[3] J. Gu, “Link layer simulation modeling methodology for cdma20001xEV-DV,” 3GPP2 TSG-C, May 23, 2001, available at ftp.3gpp2.org.

[4] A. Saifuddin and A. Jain, “Reverse link soft-combining of re-transmittedframes,” 3GPP2 TSG-C, Jan. 7, 2002, available at ftp.3gpp2.org.

[5] A. Jain and T. Chen, “Reverse link system simulation results for Qual-comm RL DV proposal with soft-combining retransmissions,” 3GPP2TSG-C, Jan. 7, 2002, available at ftp.3gpp2.org.

[6] A. Das and A. Sampath, “Link error prediction for wireless systemsimulations,” in Proc. IEEE Wireless Communications and NetworkingConference 2004, vol. 1, pp. 507-512.

[7] Physical layer standard for cdma2000 spread spectrum systems, 3GPP2standard, C.S0002-D, Version 2.0, Sept 13, 2005.