FRAME ERROR RATE EVALUATION OF A C-ARQ PROTOCOL

WITH MAXIMUM-LIKELIHOOD FRAME COMBINING

Julián David Morillo Pozo and Jorge García Vidal

Computer Architecture Department (DAC), Technical University of Catalonia (UPC), Barcelona, Spain

Keywords: Automatic repeat request, Cooperative systems, Diversity methods, Land mobile radio diversity systems,

Protocols.

Abstract: In this paper a Cooperative-ARQ (C-ARQ) with a maximum-likelihood frame combiner (ML-FC) protocol is

studied. C-ARQ is well suited for wireless transmission in either infrastructure or ad-hoc networks, as it

exploits some of the unique characteristics of wireless media, such as the natural broadcast of wireless

transmission and receiver diversity. The frame combiner can help in the case in which any received frame is

correct, exploiting the same characteristics of wireless transmission but at a bit level. The paper studies the

Frame Error Rate for this kind of system, showing that significant improvements can be obtained.

1 INTRODUCTION

This paper deals with a recently proposed variant of

the well known ARQ (Automatic Repeat reQuest)

protocol, called Cooperative-ARQ (C-ARQ) (Miu,

2005), (Monti, 2005), (Morillo, 2005), (Zhao, 2005).

C-ARQ is well suited for wireless transmission in

either infrastructure or ad-hoc networks, as it

exploits the natural broadcast characteristic of

wireless transmission and receiver diversity

(Goldsmith, 2005). As in any ARQ protocol, when a

node receives a frame with erroneous bits, it will ask

for a frame retransmission. In C-ARQ, however, the

node has previously designated a subset of nearby

nodes as cooperator nodes. In case that a cooperator

node has correctly received the frame, it will

perform the retransmission instead of the transmitter

node. In case no cooperator node has the correct

copy of the frame, a retransmission from the

transmitter node will be asked.

In this document, a variant of this basic operation

scheme is studied: when all the cooperator nodes

have erroneous copies of the frame, instead of

asking a retransmission from the transmitting node,

the receiver uses the erroneous frames from them in

an attempt to reconstruct the original one. This

frame reconstruction will be performed by means of

a so-called frame-combiner (FC). The frame-

combining technique studied in this paper is called

maximum-likelihood (ML).

Assume that a node have M-1 cooperators. The

frame-combiner will take bit by bit the M frames

(frames from the receiving node plus frames from

the M-1 cooperator nodes) and will try to produce a

correct copy of the original frame. Although a frame

combiner is a Layer 2 module, -it deals with bits and

not with signals-, it shares some commonalities with

techniques used at physical level, such as the well

known Maximal Ratio Combining (MRC)

technique, that combines different signals received

on different antennas on the same device; see

(Goldsmith, 2005).

The paper presents a numerical evaluation of the

Frame Error Rate (FER) of a C-ARQ protocol with a

maximum-likelihood frame combiner (ML-FC) in

Rayleigh channels. The idea behind the frame

combiner is give to the system a second opportunity

when C-ARQ fails: instead of asking a

retransmission from the source we retransmit from

neighboring nodes of the destination.

Using C-ARQ with a frame combiner can

introduce degradation in throughput, as in some

occasions M-1 cooperator nodes must re-transmit

their incorrect frames to the receiving node. On the

other hand using higher order constellations in the

frame retransmissions from the cooperators could

alleviate this. The study of this, however, will be left

out of the scope of this paper.

The studied scheme does not present major

implementation challenges, as it only involves

changes in the driver software together with the

David Morillo Pozo J. and García Vidal J. (2008).

FRAME ERROR RATE EVALUATION OF A C-ARQ PROTOCOL WITH MAXIMUM-LIKELIHOOD FRAME COMBINING.

In Proceedings of the International Conference on Wireless Information Networks and Systems, pages 15-20

DOI: 10.5220/0002022700150020

 SciTePress

inclusion of new signalling packets (for example for

establishing the relation of cooperation and for

requesting frame copies). The ML-FC operation

requires that each node knows the average SNR

during frame reception. This information is readily

available, with different levels of accuracy, from

many of the wireless NICs used in current networks.

Previous studies on C-ARQ (e.g. (Miu, 2005),

(Monti, 2005), (Zhao, 2005)) report significant

improvements in terms of transmission power,

transmission range or throughput. Reference

(Morillo, 2005) studies a majority voting frame

combiner for AWGN and Rayleigh channels. An

ARQ variant (that the authors call Memory ARQ)

that uses frame combining of erroneous frames was

previously introduced in (Lau, 1986). This work,

however, is not in the framework of cooperative

ARQ, as all the copies come from the original

transmitter. Moreover, (Lau, 1986) analyzes the

performance of such system in an AWGN channel

while our work is focused on Rayleigh fading

channels. The work reported in (Eaves, 1977),

studies the probability of block error (i.e. FER) for

slow Rayleigh fading channels. Although it is not a

work on cooperative techniques nor on ARQ

protocol, it gives to us a good base line scenario to

compare with our proposal.

Next, Section II gives a brief description of the

C-ARQ variant that is studied in this paper. Section

III presents the Maximum-Likelihood (ML) frame

combiner (FC) integrated in the studied C-ARQ

scheme. Section IV presents the evaluation of the

FER for such kind of system.

2 COOPERATIVE ARQ

In this section we give a brief description of the C-

ARQ variant studied in this paper: Let us assume a

wireless ad-hoc network. For a given node x of the

network, we define R

as the set of nodes which

receive the signal from x with some minimal quality

parameters. Let y be a node of R

, to which node x

wants to send a frame, and let d be the distance

between x and y. From the set R

, we form a subset

of cooperating nodes, that we call C

, which

includes all nodes from that y receives signals with

an excellent quality (including y itself), and that are

willing to cooperate with y. We assume a perfect

channel between y and nodes in C

, due to this

excellent signal quality. Although this is a strong

assumption, it can be justified by the proximity

between y and nodes in C

. In the rest of the paper

we assume |C

| = M. Usually, we will have that

distance between y and nodes of C

will be bounded

to d’, with d’<<d. We can thus approximate the

distance between x and nodes of C

to the value d.

We assume that when x transmits a frame

addressed to y, nodes of C

observe different values

of SNR, following a Rayleigh distribution:

)(

≥=

−

where the average SNR, γ*, is the same for all nodes

and constant with time. We also assume that for

each node, the SNR value is constant during a frame

reception, but it can be different for each frame.

Let x transmit a frame to y, which is received by

nodes of C

. We assume that these nodes can

identify that the final destination of the frame is y

even in presence of transmission errors (e.g. using a

strong error correction code for the corresponding

header fields). After receiving the frame, every node

checks for its correctness using for instance a CRC.

For simplicity, it is assumed throughout this paper

that the error detection code used will detect all

errors introduced by the channel. In practice, this is a

reasonable assumption since the probability of an

undetected error can be made very small.

In a C-ARQ system without frame combining,

nodes in C

that correctly receive the frame will

keep a copy of it. If node y detects that its reception

is erroneous, it will ask one of these nodes for a

retransmission of a correct copy. Only in the case

that any node in C

has correctly received the frame,

y would ask for a retransmission to node x.

In a C-ARQ with FC, in contrast, even if the

frame was received with errors, nodes in C

will

keep a temporary copy of it. In case node y finds

that the frame has suffered errors, it will request

their cooperators for a correct copy of the frame. In

case there is no correct frame received by any

cooperator, it will send a signaling packet to its

cooperators requesting a retransmission of their

erroneous frame copies (Figure 1). Cooperators of y

will send, in turns, their copies of the frame,

attaching the measured value of SNR during the

frame reception, γ

, until y is able to correctly decode

the frame by performing the frame combining.

Recall that we assume a slow fading channel and,

therefore, that γ

is constant during frame reception,

but in general different for each cooperator. At each

reception of the information sent by each cooperator

- and assuming that this received copy of the frame

has not a correct CRC-, terminal y uses a Maximum-

Likelihood (ML) decision rule for constructing a

“hypothetically correct” received frame (Figure 1).

That is, y uses a statistically optimal fusion rule in

WINSYS 2008 - International Conference on Wireless Information Networks and Systems

terms of minimum detection error probability. Only

in the case that this reconstructed frame is still

incorrect in the last step (y has received all copies

from all its cooperators), a retransmission is required

from node x (Figure 1) if the maximum number of

such retransmissions has not been exceeded (this

number is normally set to 11 in 802.11 but could be

set below this value for this system with greater

resilience).

Received

frame

End

Correct?

yes

Ask for

correct copy

Correct

copy?

yes

End

Ask for

incorrect copy

Maximum

Likelihood

Correct?

yes

End

# of copies

= M ?

yes

Ask for retransmission

to transmitter

Figure 1: Cooperative ARQ scheme with maximum-

likelihood frame combiner.

Note that we do not focus on the order in which

cooperators send their respective frames to y, and we

assume some pre-established order. How this order

is set up is left for future work and is out of the

scope of this document.

This paper studies the impact of C-ARQ and ML

Frame-Combining on the FER.

3 THE ML FRAME COMBINING

TECHNIQUE

The ML decision rule for obtaining a possible

correct copy of the frame is the following: Let

BER

(γ

)

be the i-th node BER (directly derived from

its SNR, γ

). Let S

and S

be the sets of cooperator

nodes of y that have detected a given bit as 1 or 0

respectively. Assuming that the a-priori probabilities

of “1” or “0” are identical, the ML decision rule for

this bit would be:

Decide “1” if

() ()

()

(1 ( )) ( )

ii ii

iS iS

ii ii

iS iS

BER BER

γγ

∈∈

−<

−

∏∏

(1)

and decide “0” otherwise

Note that when SNR is the same for all cooperating

nodes, then the proposed decision rule is equivalent

to the Majority Voting (MV) scheme proposed in

(Morillo, 2005), due to the fact that all frames have

the same weight. Each cooperator node coherently

detects its own information with a given BER

depending on the modulation scheme used and the

channel model considered. In the case of Rayleigh

channels and for some modulation schemes, this ML

decision rule can be simplified.

4 FER PERFORMANCE

EVALUATION

The exact evaluation of the FER for C-ARQ+FC is

involved, even for simple modulation techniques.

Even the evaluation of FER for C-ARQ leads to

non-closed expressions; see (Eaves, 1977). On the

other hand, FC mechanism only enters into play

when all frames received by a node and its

cooperators are incorrect, meaning that we should

introduce this condition into the probability

expressions.

We take thus the next approach: Firstly, we study

the ML-FC in isolation. Secondly, an analysis of the

performance of C-ARQ without FC is done. It is

easy to show that each of these mechanisms in

isolation would lead always to worse cases than the

C-ARQ+FC mechanism in conjunction. Finally, we

study FER for the combined C-ARQ-FC scheme

using Monte-Carlo simulations. We assume that the

modulation method is BPSK, without loss of

generality.

The expression for the FER for a Rayleigh fading

channel in the case in which no cooperation is

exploited is given by:

()()

[

]

γγ

dBEReFER

∫

∞−

−−=

)(

(2)

where γ* is the average SNR, and L is the frame

length; see (Eaves, 1977). In the case of BPSK

modulation,

(

)

(

)

γγ

2QBER =

. Figure 2 presents

the calculated FER versus frame length L for an

average SNR of 24 dB, which corresponds to a 10

-3

BER.

FRAME ERROR RATE EVALUATION OF A C-ARQ PROTOCOL WITH MAXIMUM-LIKELIHOOD FRAME

COMBINING

Figure 2: FER vs Frame Length in Rayleigh fading

channels. SNR=24dB. BPSK modulation.

4.1 FER of ML-FC

We focus first on the maximum-likelihood FC

mechanism and study it in isolation. We have

performed a Monte-Carlo simulation calculating the

FER that can be obtained for different values of

average SNR, and for different values of M, the

number of cooperators. Simulations have been done

using OCTAVE for frame lengths (L) equal to 1,000

and 10,000 bits. The results for L=1,000 bits are

presented in Figure 3.

Figure 3: FER of ML-FC vs SNR in Rayleigh fading

channels for different values of M and for L=1000 bits.

BPSK Modulation.

From Figure 3 we can see the great

improvements that can be obtained with the ML-FC

technique studied in this paper. For example, for an

SNR=33 (~15dB) the FER decreases from 0.15 for

the case M=1 -no cooperation- to 0.025 for the case

of just combining two frames, meaning that we have

a reduction of FER of an order of magnitude. In

fact, note that for the case M=2, the ML-FC does not

perform any frame combining and simply chooses

the frame with higher SNR. For M=3 the FER for

the same SNR is about 0.0026. Increasing M, on the

other hand, increases the number of frame

retransmissions.

Figure 4 presents similar results for the case

L=10,000 bits.

Figure 4: FER of ML-FC vs SNR in Rayleigh fading

channels for different values of M and for L=10000 bits.

BPSK Modulation.

4.2 FER of C-ARQ Protocol

Let us focus now on the C-ARQ protocol and study

it in isolation, i.e. without ML-FC, as we have done

for the FC.

An analytical expression can be easily derived

from (2) for the case of C-ARQ. If we have M

cooperating nodes (including the destination node

itself) and considering that the γ

are independent

and identically distributed random variables, the

FER of C-ARQ will be:

(3)

This expression represents the FER from the

transmitter point of view, i.e., the probability that a

retransmission from x in the model presented in

Section II would be necessary.

The goodness of the C-ARQ protocol can be

observed in Figure 5, together with the impact of the

number of cooperators in the FER. It can be seen

how FER decays orders of magnitude with just 2 or

3 cooperators.

()()

[

]

∏

∫

∞

−

−−=

iARQC

dBEReFER

γγ

ARQC

FERFER )]([

−

WINSYS 2008 - International Conference on Wireless Information Networks and Systems

Figure 5: FER of C-ARQ vs SNR in Rayleigh fading

channels for different values of M and for L=1000 bits.

BPSK Modulation.

4.3 FER of Combined C-ARQ +

ML-FC

Once presented the results for each component of

the system in isolation, let us focus on the

performance of such system as a whole.

We have performed simulations for L=1,000 and

10,000 bits, and for different number of cooperators

(M). Figure 6 shows the results for M=2 and M=1. It

is clear that the use of C-ARQ-FC considerably

reduces the overall FER values. Figure 6 is also

interesting for a particularity of the M=2 case:

normally ML-FC in isolation lead to a lower FER

than the C-ARQ in isolation. For M=2, however, this

is not true, as in this case ML-FC in fact does not

perform any frame combining: it will simply choose

the frame with higher SNR (that could be

erroneous). C-ARQ, on the other hand, will choose a

correct frame if it exists. Sometimes it can happen

that the correct frame is the one with lower SNR. In

this case ML-FC will fail while C-ARQ will not.

Figure 6: FER of the whole system vs SNR in Rayleigh

fading channels for M=2 and L=1000. BPSK Modulation.

Figure 7 presents the case M=3 and L=1,000. As

stated previously, ML-FC in isolation performs

better than C-ARQ in isolation, while the

combination of both mechanisms performs only

slightly better than ML-FC.

Figure 7: FER of the whole system vs SNR in Rayleigh

fading channels for M=3 and L=1000. BPSK Modulation.

Figure 8 shows the obtained values for the case

M=4, L=1,000. As expected, similar conclusions can

be drawn, although now the difference between the

C-ARQ in isolation and the combined C-ARQ-FC

scheme is larger.

The question of whether C-ARQ in isolation or

the combined C-ARQ-FC schemes are feasible

alternatives, is very much system dependent, and is

left out of the scope of this paper. The evaluation of

this question depends on factors like the distance

between source and receiver, between receiver and

cooperators, etc. The idea of the FC is to give a

second opportunity in trying to avoid

retransmissions from a source (possibly far away) by

substituting it with transmissions of nearby nodes

that could spend less power. Another open issue

could be the system performance in the case of using

Hybrid-ARQ techniques integrated in the system.

Figure 8: FER of the whole system vs SNR in Rayleigh

fading channels for M=4 and L=1000. BPSK Modulation.

FRAME ERROR RATE EVALUATION OF A C-ARQ PROTOCOL WITH MAXIMUM-LIKELIHOOD FRAME

COMBINING

5 CONCLUSIONS

In this paper we study and evaluate the FER for a

cooperative ARQ scheme with a maximum-

likelihood frame combiner integrated in it that

exploits space diversity and cooperation between

neighboring nodes. This paper shows how the

benefits of space diversity and node cooperation

have also a great impact on the FER performance

(maybe more interesting at the Link Layer than the

BER). The hardware complexity of the system is

clearly reduced with respect to MRC or to

cooperative transmission techniques, although new

protocol signaling is needed.

ACKNOWLEDGEMENTS

This work has been supported by Spanish Ministery

of Science and Technology under grant TSI2007-

66869-C02-01 and by the NoE EuroFGI of the VI

FP of EU.

REFERENCES

Goldsmith, A., 2005. Wireless Communications,

Cambridge University Press.

Miu, A., Balakrishnan, H., Koksal, C. E., 2005. Improving

Loss Resilience with Multi-Radio Diversity in

Wireless Networks. In ACM Mobicom 2005.

Monti, P., Tacca, M., Fumagalli, A., 2005. Optimized

Transmission Power Levels in a Cooperative ARQ

Protocol for Microwave Recharged Wireless Sensors.

In IEEE ICC’05.

Morillo-Pozo, J., García-Vidal, J., Pérez-Neira, A., 2005.

Collaborative ARQ in Wireless Energy-Constrained

Networks. In ACM-SIGMOBILE DIAL-M-POMC

International Workshop on Foundation of Mobile

Computing.

Nosratinia, A., Hunter, T.E., Hedayat, A., 2004.

Cooperative communication in wireless networks. In

IEEE Communications Magazine, vol. 42, no. 10, pp.

68-73.

Zhao, B., Valenti, M. C., 2005. Practical Relay Networks:

A Generalization of Hybrid-ARQ. In IEEE JSAC, Vol.

23, No. 1.

Cover, T. M., El Gamal, A. A., 1979. Capacity theorems

for the relay channel. In IEEE Trans. on Information

Theory, pp. 572-584, Sept. 1979.

Laneman, J. N., Tse, D. N. C., Wornell, G. W., 2004.

Cooperative diversity in wireless networks: Efficient

protocols and outage behaviour. In IEEE Trans. on

Information Theory, pp. 3062-3080, Dec. 2004.

Lau, C., Leung, C., 1986. Performance Analysis of a

memory ARQ scheme with soft decision detectors. In

IEEE Trans on Commun., Aug. 1986.

Ishioka, K., Takanashi, H., Tanaka, T., 1995. Improved

Throughput Characteristics by ARQ with weighted

Majority Decision. In Proc. IEEE ICUPC’95, pp. 467-

471.

Eaves, Reuben E., Levesque, Allen H., 1977. Probability

of Block Error for Very Slow Rayleigh Fading in

Gaussian Noise. In IEEE Transactions on

Communications, March 1977, pp. 368-374.

Tacca, M., Monti, P., Fumagalli, A., 2005. Cooperative

and Non-Cooperative ARQ Protocols for Microwave

Recharged Sensor Nodes. In Proc. EWSN’05, pp. 45-

56.

Chen, B., et. al, 2004. Channel aware decision fusion in

wireless sensor networks. In IEEE Transactions on

Signal Processing, Dec. 2004.

Cai, X., et. al, 2003. Performance of CDMA random

access systems with packet combining in fading

channels. In IEEE Trans. on Wireless Commun., May

2003.

Cui, S., et. al, 2004. Energy-efficiency of MIMO and

cooperative MIMO in sensor networks. In IEEE JSAC,

vol. 22, no. 6.

WINSYS 2008 - International Conference on Wireless Information Networks and Systems