434 ieee journal on selected areas in ...434 ieee journal on selected areas in communications, vol....

434 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 25, NO. 2, FEBRUARY 2007

Collaborative Decoding inBandwidth-Constrained Environments

Arun ’Nayagam, John M. Shea, and Tan F. Wong

Abstract— We present a cooperative communication schemein which a group of receivers can collaborate to decode amessage that none of the receivers can individually decode.The receivers act as a virtual antenna array in which thecombining must be performed over bandwidth-constrained links.The proposed approach is targeted at systems in which thecooperative information must be digitized, such as for wirelessor wired links that are constrained to use digital modulation.In such systems, previously proposed schemes such as amplify-and-forward would require that a large amount of informationbe exchanged when there are many collaborating nodes. Theapproach presented in this paper, called improved least-reliablebits (I-LRB) collaborative decoding, provides a higher level ofadaptation than previously proposed cooperative schemes. TheI-LRB scheme utilizes reliability information and informationabout competing paths in soft-input, soft-output (SISO) decodersto adaptively select the amount of information that is neededto correct a particular part of a message, as well as whichbits should be exchanged. Simulation results show that theproposed approach offers a significant performance advantageover a constrained-overhead, incremental form of maximal ratiocombining (MRC).

Index Terms— Cooperative communications, collaborative de-coding, distributed antenna array, SISO decoding, user cooper-ation, cooperative diversity.

I. INTRODUCTION

MOTIVATED by the information-theoretic study of therelay channel in [1], several network-based approaches

to achieve spatial diversity have been studied in recentyears [2]–[6]. In these schemes multiple nodes take advantageof spatial diversity in wireless communications by collabo-rating to efficiently relay and combine the different receivedcopies of a message. This type of collaboration is useful whenthe physical sizes of the radios do not permit the use ofmultiple antennas. The diversity achieved through collabo-ration has been termed cooperative diversity. Two relayingschemes for user cooperation, called decode-and forward(D-F) and amplify-and-forward (A-F) were considered forfading channels in [2], [7]. In D-F, the relay decodes thesource transmission, and if successful, the relay re-encodesthe information using the same code used at the source and

Manuscript received February 1, 2006; revised July 1, 2006. This workwas supported by the Office of Naval Research under Grant N00014-02-1-0554, by the National Science Foundation under Grant ANI- 0220287, andby the DoD Multidisciplinary University Research Initiative administered bythe Office of Naval Research under Grant N00014-00-1-0565.

Arun ’Nayagam is with Intellon Corp., Ocala, FL (e-mail:[email protected]).

John M. Shea and Tan F. Wong are with the Wireless In-formation Networking Group (WING), University of Florida (e-mail:{jshea,twong}@ece.ufl.edu).

Digital Object Identifier 10.1109/JSAC.2007.070218.

forwards it to the destination. Sendonaris et al. [3] studyrelaying-based user cooperation in a cellular CDMA systemto increase the sum capacity of the network. The schemesin [3] and D-F are based on simple repetition coding. Severalrelaying techniques that use better error correction codes havebeen studied recently in [8], [9], [10]. These coded cooperationschemes are based on D-F, and only differ from D-F in thecoding scheme used at the relay to re-encode the informationbits from the source. These coded cooperation schemes donot easily scale to cooperation with more than one relay. Alsonote that all the coded cooperation schemes rely on correctdecoding at the relay. In A-F, the relay amplifies the receivedanalog signal subject to a power constraint before forwardingit to the destination. A more general variation to the A-Fscheme is the compress-forward (C-F) scheme (cf. [6]), forwhich several practical implementations have been recentlyproposed in [11], [12], [13].

Collaborative decoding schemes were proposed in [14],[4], [5], [15] as techniques to allow a group of receivers tocollaborate to recover a message transmitted from a distanttransmitter. In this approach, nodes that receive the messagefrom the transmitter are modeled as elements of a virtualantenna array. Traditional antenna arrays use maximal-ratiocombining (MRC) [16] at a central combiner to combine thesignals received at different elements. This is made possibleby high-bandwidth cables that connect the combiner to thearray elements. We consider a scenario in which the nodesin the virtual array are connected via bandwidth-constrainedlinks. These links may be either wireless or wired. Either typeof link may need to be used efficiently because it carries othertraffic. For instance, the collaborating nodes may be part of alarger wireless ad hoc network, or a wired link may be usedto exchange cooperative information for many users amonga group of base stations or access points. As the number ofusers and their data rates go up, the amount of information tobe exchanged becomes important.

In many systems, the use of digital modulation does notpermit the transmission of arbitrary analog signals, and thusamplifying/scaling and forwarding the analog received signalsis not possible. For instance, many mobile devices operatetheir power amplifiers in the saturation region and utilizesignals with a low peak-to-average power ratio. The links be-tween base stations are often either optical or microwave linksthat employ digital modulation. In such systems, the receivedsymbols have to be quantized and transmitted as a bit stream.The messages exchanged by cooperating nodes contribute tooverhead in the system, and will henceforth be referred toas cooperation overhead. If the relays in A-F are required

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’NAYAGAM et al.: COLLABORATIVE DECODING IN BANDWIDTH-CONSTRAINED ENVIRONMENTS 435

to use digital modulation to relay the received symbols, thiswould result in high overhead since demodulator outputs mustbe quantized and broadcast for each received symbol at eachcollaborating node. Thus the overhead become particularlylarge as the number of collaborating nodes increases. Thishigh overhead will not be acceptable in systems that areconstrained in bandwidth or that require a certain minimumthroughput guarantee. The objective of collaborative decodingis to achieve performance that is close to MRC while requiringsignificantly lower cooperation overhead.

The problem of estimating the transmitted codeword undera communication constraint is similar to previous work ondistributed detection and estimation under communicationconstraints (cf. [17], [18], [19], [20], [21]). However, unlikepreviously considered detection problems, we are not consider-ing a simple binary hypothesis, and unlike previously consid-ered estimation problems, we are not considering parameterestimation in the presence of noise. Our work is, however,similar in some aspects to [18] and [19] in that the informationused in our distributed detection problem is limited basedon a measure of the quality of that information. In [18], theinformation at a sensor is “censored” (not transmitted) if thequality is too low. In [19], the information from a sensor iseither transmitted as a hard decision or log-likelihood ratiodepending on the quality of the information. In each of theseapproaches, the decision about what information should betransmitted is made at the sensor before transmission to adata fusion center. In the scenario that we consider, datafusion occurs within the group of sensors and the selection ofinformation to exchange is based on the quality of informationat the node that performs the fusion and the quality of theinformation at each of the other sensing nodes. Furthermore,our algorithm is an iterative fusion process that utilizes thestructure of the error-control-coded data to estimate the qualityof the decisions for different output bits.

The selection of which information to exchange based onreliability information that is generated in soft-input, soft-output decoding is one of the primary features that distin-guishes collaborative decoding from other cooperative com-munication schemes. In collaborative decoding, combiningis performed for only those trellis sections that are deemedunreliable at the output of the SISO decoder. This adapts theinformation exchange to each individual channel instantiationthat causes errors in the decoder. In this paper we present acollaborative decoding scheme that provides better utilizationof soft information in the soft-input, soft-output decoder toprovide good performance with low collaborative overheadfor convolutionally encoded communication. We show thatreliability information can be temporally correlated becauseit comes from the same set of competing paths in the codetrellis. We then develop a collaborative decoding scheme thatutilizes this fact. We show that competing paths in the codetrellis can be explicitly computed using calculations that arealready performed in the max-log-MAP implementation ofthe BCJR [22] algorithm. We also show how the amount ofinformation can be adapted for each unreliable trellis section.We then design a collaborative decoding scheme that makesuse of these results. Collaborative decoding may be considereda C-F scheme because it attempts to minimize the information

Fig. 1. System topology for collaborative decoding.

that must be exchanged among the nodes in order to achievecorrect decoding by exploiting the inherent redundancy inthe information. However, unlike the C-F schemes proposedin [11], [12], [13], we allow communication among all thereceiving nodes and do not utilize Wyner-Ziv or Slepian-Wolfsource coding. Rather, we iteratively refine the estimate of theinformation by utilizing a posteriori probabilities generated inthe decoding process.

Unlike previously proposed A-F, D-F, and C-F schemes,our scheme offers a higher level of adaptation to the qualityof the received information. A-F and D-F combine the sameinformation for all the bits in a codeword, and previouslyinvestigated C-F schemes provide a constant level of com-pression across the received symbols. Our scheme adapts thetrellis sections for which combining is performed based on thechannel realization, and the amount of information combinedis adapted to the reliability of that trellis section as determinedby the SISO decoder. The scheme we develop offers thefollowing advantages over previous cooperative schemes thatcan potentially be used in the virtual array:

• unlike schemes based on D-F, correct decoding is notrequired at any of the nodes,

• the cooperating overhead is smaller than schemes basedon A-F in which the signals to be relayed are quantizedand transmitted using digital modulation, and

• the schemes scale easily to multiple cooperating nodes.

II. SYSTEM MODEL

The system model studied in this work is shown in Fig. 1. Adistant transmitter broadcasts a packet to a cluster of receivingnodes. Typical scenarios could be military applications inwhich a battleship broadcasts a message to a platoon ofsoldiers on the mainland or commercial applications wherein abase station communicates with a cluster of mobile users. Themessage at the source is packetized and encoded with a codethat permits SISO decoding. The codeword is then broadcastto a cluster of receiving nodes that will attempt to decode themessage. The received message for symbol i at node j can bemodeled as

ri,j = ajxi + ni,j , (1)

where xi is the transmitted symbol at time i; aj is the channelcoefficient at receiving node j, which we assume is fixedover each packet; and ni,j is white Gaussian noise. In all that


follows, we consider rate R = 1/2 codes, but it is straight-forward to generalize the work to other code rates.

We consider two different scenarios for the packet destina-tion. In the anycast scenario, if any node in the cluster decodesthe message correctly, then we consider the message to besuccessfully received. For instance, if the cooperating nodesare a cluster in an ad hoc network, any of the nodes maybe able to forward the message on to its ultimate destination.We also consider a unicast scenario in which messages aretargeted to a particular radio in the cluster. If any node (otherthan the destination) decodes the message correctly, it canforward the message to the destination. In both the scenarios,collaborative decoding is initiated only if none of the nodesin the cluster is able to decode correctly. Thus, our proposedscheme is utilized only when the packet cannot be recoveredvia D-F (selection diversity).

In the schemes that we propose, the nodes use the outputsof the SISO decoders to select which information should beexchanged and which nodes should transmit that information.The a posteriori probability (APP) log-likelihood ratio (LLR)at the output of a SISO decoder is a real number and iscommonly referred as the soft output. The sign and magnitudeof the soft output for an information bit represent the harddecision and the reliability of that decision, respectively [23].The sample mean of the reliabilities at node i, µi, is a mea-surement of the overall reliability of the decoder’s decision.We assume that the nodes exchange the µis after the firstdecoder iteration and that combining occurs at the node withthe largest µi, which we refer to as the “best” node. The nodesthen broadcast information about a selected set of the receivedsymbols (as in A-F) to the best node. The cooperative processcan go through several iterations, each of which consists ofthree parts. In the first part of the iteration, the nodes identifyinformation to be exchanged. In the second part, a selectedgroup of nodes will transmit that information to the best node.In the final part of each iteration, the best node decodesthe message and checks whether it has decoded correctly.The process stops if the best node has decoded the messagecorrectly or if the limit on the number of iterations is reached.

In each iteration, we constrain the maximum number of bitsthat can be transmitted in the cooperative process. This maybe necessary in many systems to ensure that the cooperativeprocess does not conflict with the transmission of additionalpackets from the source or does not occupy channel resourcesthat are required for other traffic in the network. We specifythe constraint as a portion of the total information exchangedif MRC is to be performed at one of the nodes in the cluster.Let N be the information block size, R be the code rate,Nrx be the number of receivers, and q be the number ofbits used to quantize the channel observations1. Then theoverhead for MRC is θMRC = Nq(Nrx−1)/R bits. The largeθMRC will be not acceptable for many applications. Hence,we constrain the amount of information that can exchanged inthe cooperating cluster to be a fraction p of θMRC . Note thatthis places a limit on the maximum amount of information

1Note that for all of the schemes considered, we assume equal quantizationlevels at all of the nodes, which is not optimal. However, determining theoptimal quantization levels under a fixed bit constraint is known to be anNP-hard problem. The interested reader is referred to [20].

exchange in the cooperative process for a particular packet;however, the actual amount of information exchanged forany particular packet may be much less because we allowthe cooperative process to terminate whenever the packet isdecoded correctly.

We next describe the two main cooperative schemes thatwill be compared in this paper. The first, which we callconstrained-overhead incremental MRC (COI-MRC), is aniterative form of MRC in which the overhead is constrainedas explained above. The second scheme is a collaborativedecoding scheme called the improved least-reliable bits (I-LRB) scheme. Because of the complexity of this scheme, weonly provide an overview of it in this section. A detaileddescription of I-LRB is given in Section IV after we developsome necessary decoding techniques.

A. Constrained-overhead incremental MRC

Consider first an implementation of full MRC in a groupof collaborating radios. Each node (other than the best node)scales its received symbols by the fading gain, quantizes them,and transmits them to the best node. As mentioned above, thiswould result in a large overhead. A variant of this schemethat can offer even better performance than MRC with loweroverhead is incremental MRC (I-MRC). In incremental MRC,the cooperation is done over several iterations2. In iteration i,the node with the i + 1th largest µi transmits informationabout all of its received symbols to the best node3. Then thebest node combines that information with its own receivedsymbols and any previously received information, decodesthe message, and checks whether the message has decodedcorrectly. If the message decodes correctly, the cooperativeprocedure terminates, and thus the average overhead of I-MRC is typically much less than MRC. In addition, becausedecoding is performed after each information exchange, I-MRC can achieve a slightly lower error probability than MRCbecause combining information from nodes in deep fades isavoided on early termination of combining.

Although I-MRC has a lower average overhead than MRC,the overhead in each iteration consists of all of the receivedsymbols from one node, and the maximum overhead is thesame as MRC. As explained above, it may be necessaryto constrain the maximum overhead. Thus, we introduce aconstrained-overhead I-MRC (COI-MRC) scheme. In COI-MRC, the overhead is constrained to pNq(Nrx − 1)/R bits.We allow a total of Niter = Nrx − 1 iterations, so in eachiteration, pNq/R bits are exchanged. The information in eachiteration represents a set of pN/R received symbols from thebest node that has not previously transmitted all of its receivedsymbols. The set of symbols is uniformly selected from theremaining set of symbols at that node. Once all of the symbolsat a node have been transmitted, then the next best node (interms of µi) will transmit information for its received symbols.

2We thank an anonymous reviewer of a previous paper for proposing thiscooperative scheme.

3Note that for quasi-static fading channels the value of µi is generallydominated by the fading coefficient. If two nodes have similar fading coeffi-cients, this approach allows us to choose the one whose received informationprovides more confidence in decoding.


After each round of information exchange, the best nodeuses MRC to combine the new information with its previouslyreceived information. The best node then decodes the message.If the message decodes correctly or if the maximum numberof iterations has been reached, collaboration ends. Otherwise,another iteration of information exchange is performed.

B. Overview of Improved Least-Reliable Bits CollaborativeDecoding

The MRC-based schemes are effective approaches for co-operation. However, these schemes are “dumb” schemes inthe sense that they do not utilize information that is availablethat could improve the performance for the same constrainton the collaborative overhead. SISO decoders offer the abilityto assess which bit decisions are reliable and which areunreliable. By first exchanging information that can improvethe unreliable bit decisions, we may be able to achieve a bettertradeoff between overhead and performance.

The scheme that we propose is based on the least-reliablebits (LRB) schemes that were proposed in [14], [5]. In theseLRB schemes, each node identifies the set of bits with theleast reliabilities (i.e., smallest magnitude of the APP LLR)and requests information for these bits from every other node.Our technique improves on the prior LRB schemes in severalways:

1) We request information at only the best node, so thatthe overhead from the information requests is reduced.

2) We exchange information for the received symbolsrather than the decoded bits because this provides betterperformance on fading channels.

3) We utilize the fact that the set of LRBs is often cor-related, and we develop techniques to avoid requestingtoo much information because of this correlation.

4) The amount of information required to correct a bitdepends on its reliability, so we present a techniqueto adapt the amount of information based on a bit’sreliability.

5) Not all bits that surround an unreliable bit will neces-sarily help to correct that bit, so we present a techniqueto select the set of bits which are most likely to correctan unreliable bit.

We refer to the new approach as the improved LRB (I-LRB) scheme. In this paper, we demonstrate how the goalsof the I-LRB scheme can be achieved for convolutionallyencoded communications by utilizing information generatedin the max-log-MAP implementation of the BCJR decodingalgorithm. The details of I-LRB with convolutional codesare given in Section IV after we develop several decodertechniques in Section III.

III. THE DECODER

As previously mentioned, in this paper we consider the im-plementation of the I-LRB collaborative decoding scheme forconvolutionally encoded communications. We utilize featuresof the max-log-MAP implementation of the BCJR algorithmto identify which information to exchange and how much in-formation to exchange. We begin by defining the terminologyand notation used in this section.

A. Terminology and notation

The terminology and notation introduced here are specificto rate 1/2 convolutional codes. It is straight-forward togeneralize these to rate k/n codes.

• input and output labels: An input label is used to indicatethe input that causes a particular state transition in thecode-trellis, and an output label is used to indicate thecorresponding output caused by that state transition.

• path and event: A sequence of valid state transitionsin the trellis is called a path through the trellis. Notethat every codeword represents a path through the trellis.Because the code is linear, the difference between anytwo codewords is a path through the trellis. Such a pathis also called an event.

• valid state: A valid state lies on any path through thetrellis. Because the trellis starts and stops in the all-zerosstate, not every state is a valid state near the ends of thetrellis.

• metric: The Euclidean distance between the receivedvector r and any codeword c, ‖r− c‖2, is referred to asthe metric4. Note that the metric is a maximum-likelihood(ML) decision statistic for additive white Gaussian noise(AWGN) channels.

A summary of all of the notation used in this paper is providedin Table I.

B. Max-log-MAP decoding of convolutional codes

The BCJR algorithm is a bitwise maximum a posteriori(MAP) decoder [22], which minimizes the bit error proba-bility. When implemented in the log domain, the inputs toa BCJR MAP decoder are a priori probability LLRs andLLRs for the received symbols, and the output consists ofAPP LLRs. For each information bit ui, the log-MAP decodercomputes the APP LLR as

L(ui|r) = lnP(ui = 0|r)P(ui = 1|r) = ln

∑c∈Ci

+P(c|r)∑

c∈Ci−

P(c|r) , (2)

where Ci+ and Ci

− are defined in Table I. In what follows,we consider a nonfading (i.e., aj = 1 in (1)) additive whiteGaussian noise (AWGN) channel. The results extend easily tothe case of quasi-static fading by premultiplying the codewordby the channel coefficient aj . A suboptimal implementationof the log-MAP decoder called the max-log-MAP decoder isobtained by using the approximation ln(

∑xi) = max(ln(xi))

to evaluate the log-APP in (2). Using this approximation andassuming that all the codewords are equally likely, the softoutput for codewords transmitted on an AWGN channel withnoise variance σ2 can be written as [24]

L(ui|r) = minc∈Ci

+

(‖r − c‖2

2σ2

)− min

c∈Ci−

(‖r − c‖2

2σ2

). (3)

Note that the maximum-likelihood (ML) codeword/path cML isa codeword that is closest to the received vector r,

cML = argminc∈C

‖r− c‖2.

4Note that a metric is associated with a particular codeword. In other words,each codeword has a different metric.


TABLE I

NOTATION USED IN THIS PAPER

N Block-size (the number of sections in the code-trellis).

ui The input to the encoder at time i; i.e., the input label for trellis section i. For binary codes considered in thispaper ui ∈ {0, 1}. We will refer to ui as the information bit.

ci = [c0i , c1i ] The output of the encoder at time i. This is a two-dimensional vector consisting of the two parity bits outputby the encoder at each time. If BPSK is used for modulation then cj

i ∈ {−1, 1},∀j ∈ {0, 1}. Since every cicorresponds to a particular branch in the trellis , ci will used as the output labels for the branches in the trellisat time i. We will use parity bits or coded bits to refer to the output labels at any particular time in the trellis.

c = [c1, . . . , cN ] A valid codeword (output labels on a path through the trellis). Appropriate subscripts will be used to indicatethe codeword being considered.

ri = [r0i , r1

i ] The received vector corresponding to ci.

r The received vector corresponding to c.

cba [ca, ca+1, . . . , cb−1, cb]. rb

a is similarly defined.

ul(c) Input label at trellis section l in codeword c.

c(l) Component l in codeword c. Note that this refers to a particular bit in the corresponding output label.

Ci+ {c : ui(c) = 0} i.e., the set of all codewords with input label 0 at trellis section i.

Ci− {c : ui(c) = 1} i.e., the set of all codewords with input label 1 at trellis section i.

C The set of all valid codewords. C = Ci−

SCi

+.

S Set of states in the trellis. For the memory-two code considered in this paper, there are four states.

Therefore, S = {0, 1, 2, 3}.

sk State of the encoder at time k. Note sk ∈ S .

S(→ s) The set of valid states at time k − 1 that have branches leading into state s at time k.

S(s →) The set of valid states at time k + 1 that have branches emerging from state s at time k.

sk(c) The state that codeword c passes through at time k.

αi(s) log(P (si = s), ri1)

γi(s′, s) log(P (si = s, ri|si−1 = s′))

βi(s) log(P (rNi+1|si = s))

N (µ, σ2) Represents a Gaussian distribution with mean µ and variance σ2 .

It is possible that there is more than one ML codeword(although this occurs with probability zero for the unquantizedAWGN channel), in which case we arbitrarily choose one ofthe paths as the ML codeword.

♦ Definition 1. Competing codeword/path cicomp: The

competing path at trellis section i is the path that is closest tothe received vector among all paths that differ from the MLpath in the input label for trellis section i,

cicomp = argmin

{c∈C:ui(c) �=ui(cML)}‖r − c‖2. (4)

As in the case of the ML codeword, there may be more thanone codeword that satisfies (4), in which case the tie is brokenby choosing one of the codewords arbitrarily. Note that theremay be different ci

comp for different values of i.Then the reliability for bit i, which is the magnitude of the

soft information in (3), can be expressed as

Λi � |L(ui|r)| =1

2σ2

{‖r− ci

comp‖2 − ‖r− cML‖2

}. (5)

Since the distance between r and the ML codeword is smallerthan the distance between r and any other codeword, thedifference in (5) is always positive. A high value of reliabilityimplies that the metrics of the ML path and the next best pathwith the opposite input label for bit i are far apart, and hencethere is a lower probability that the decoder chose the wrongpath and made a bit error. Thus, reliability is a measure ofthe correctness of the bit decision. This has also been shownvia simulation results in [25], [26]. A bit with high reliability

is more likely to have decoded correctly than a bit with lowreliability.

The I-LRB scheme that is described in Section IV utilizesboth the bit reliabilities and knowledge of cML and ci

comp indetermining which information should be exchanged in thecollaborative decoding process. In the next section, we detailhow cML and ci

comp can be determined for a particular trellissection.

C. Obtaining the ML and competing path using the BCJRalgorithm

Following the development in [27], the soft informationin (3) can be expressed as

L(ui|r) = maxCi

+

(αi−1(s′) + γi(s′, s) + βi(s)

)

− maxCi

−

(αi−1(s′) + γi(s′, s) + βi(s)

), (6)

where αk(s), γk(s′, s), and βk(s) are defined in Table I.It can also be shown that (see [27])

αi(s) = maxs′∈S(→s)

(αi−1(s′) + γi(s′, s)) (7)

βi−1(s) = maxs′∈S(s→)

(βi(s′) + γi(s, s′)) (8)

γi(s′, s) ∝ −‖ri − ci‖2, (9)

where s′ ∈ S(→ s) and s′ ∈ S(s →) are defined in Table1, α0(0) = 0 and βN (0) = 0. Thus, it is seen from (9)


that γi(s′, s) is proportional to the branch metric (cf. [28]),P (ri|ci), used in the Viterbi algorithm (where the constant ofproportionality depends on only the channel coefficient andsignal-to-noise ratio).

Let the ordered pair of states (si−1, si) that maximizes thefirst term in (6) be (s+

i−1, s+i ). Let (s−i−1, s

−i ) be the ordered

pair of states that maximizes the second term. By comparing(3) and (6), it is seen that one of the ordered pairs of states(s+

i−1, s+i ) or (s−i−1, s

−i ) corresponds to cML, while the other

ordered pair corresponds to cicomp. For example, if

maxCi

+

(αi−1(s′) + γi(s′, s) + βi(s)

)

> maxCi

−

(αi−1(s′) + γi(s′, s) + βi(s)

),

then si−1(cML) = s+i−1, si(cML) = s+

i , and si−1(cicomp) =

s−i−1, si(cicomp) = s−i . Thus, when computing soft output for

trellis section i, it is possible to identify the branches throughthe trellis at time i that correspond to the ML path and thecompeting path.

We now introduce two theorems that will enable us toobtain cML and ci

comp in a straight-forward manner using thecomputations performed by the decoder.

Theorem 1: The branch selection theoremGiven the state in the code trellis at time k, sk = s′ andthe vector of received symbols r, the following statements aretrue:(a) Trace-back: The state transition s∗ → s′, where sk−1 =s∗ = argmax

s∈S(→s′){αk−1(s) + γk(s, s′))}, is a branch on a

codeword c∗ that satisfies c∗ = argmin{c∈C:sk(c)=s′}

‖rk1 − ck

1‖2.

(b) Trace-forward: The state transition s′ → s∗, where sk+1 =s∗ = argmax

s∈S(s′→)

{γk+1(s′, s) + βk+1(s)}, is a branch on a

codeword c∗ that satisfies c∗ = argmin{c∈C:sk(c)=s′}

‖rNk+1−cN

k+1‖2.

Proof: See Appendix.Theorem 2: The conditional path selection theorem.

Given a state transition at time i, i.e., si−1 = s′ and si =s∗, let C∗ represent the set of all paths through the trellis(codewords) passing through this transition at time i. That is,C∗ = {c ∈ C : si−1(c) = s′, si(c) = s∗}. Then the sequenceof state transitions{s′0, s′1, . . . , s′i−2, s

′, s∗, s∗i+1, . . . , s∗N} given by (10) and (11)

on the following page corresponds to a codeword c∗ that isclosest to the received vector r among all the codewords inC∗, c∗ = argmin

c∈C∗‖r− c‖2.

Proof: The proof follows by repeated application of the trace-back and trace-forward theorems.

Note that fixed-point implementations of the decoder couldlead to multiple codewords with the same metric. In sucha case, the ties are resolved randomly to obtain the desiredcodeword (cML or ci

comp). That is, whenever there are ties inselecting the next or previous state during branch selection,one of the states is chosen randomly to break the tie. Asmentioned earlier, the state transitions from time i − 1 to ithat correspond to a ML path and a competing path can be

obtained during the computation of the soft output for bit i.Given the states si−1(cML), and si(cML), a ML codeword cML

can be obtained using the conditional path selection theorem.The codeword output by the conditional path selection theoremis closest in Euclidean distance to the received vector amongall paths that pass through si−1(cML) and si(cML) and is thusa ML path. Similarly, a competing path can be obtained usingthe conditional path selection theorem given si−1(ci

comp), andsi(ci

comp). By recording information about the states that leadto the maximum values in (7) and (8) during the BCJRalgorithm, cML and ci

comp can be computed with no additionalcomputations.

During the trace-back (or trace-forward) procedure, ifsi−k(cML) = si−k(ci

comp) for some k, then the sequenceof state transitions obtained for any time before k will bethe same for cML and ci

comp. Similarly, if si+k(cML) =si+k(ci

comp), then the sequence of state transitions will bethe same for cML and ci

comp for any time after k. Thus,it is sufficient to execute the trace-back and trace-forwardprocedures until si±k(cML) = si±k(ci

comp).

IV. IMPROVED LEAST RELIABLE BITS

COLLABORATIVE DECODING

In this section we describe the Improved LRB (I-LRB)collaborative decoding scheme for convolutionally encodedcommunications. It is well known that errors at the outputof a convolutional code are bursty, and similarly the soft-output/reliabilities are temporally correlated [29], [26]. Onereason for this correlation is that bits that are close to eachother in the trellis may often share the same competingcodeword/path. For max-log-MAP decoding, such bits haveexactly the same reliability, as can be seen from (5). We haveverified this occurrence through simulation.

Recall that in I-LRB, the best receiver sorts the trellissections according to the reliabilities and requests informationfrom the other collaborating nodes to improve the decodingof some set of least reliable bits. The LRBs will often occurin groups because they are caused by the same error event,and thus it is only necessary to provide enough information tocorrect the error event in order to correct all of the bit errorscaused by that event. Moreover, we show that some of thereceived symbols corresponding to a LRB may not be usefulin resolving the most likely error event. In the rest of thissection, we first propose a simple analytical technique thatcan be used to determine how much information needs to betransmitted for each least reliable bit. We then describe howthe decoder can use information about the ML and competingpaths to decide which information can most efficiently correctany bit errors in the LRBs. Finally, we provide a detaileddescription of the I-LRB scheme for convolutionally encodedcommunications.

A. Estimation of request size

During the collaborative decoding process, the decoder mustact under the assumption that any LRB is in error, when in factthe error probability for even the least reliable bit is generallyless than 0.5 (otherwise, we would just invert that bit decision).Given the reliability of a LRB, the decoder needs to estimate


s′k−i = argmaxs∈S(→s′

k−i+1)

{αk−i(s) + γk−i+1(s, s′k−i+1)}, i = 2, 3, . . . , k (10)

s∗k+i = argmaxs∈S(s∗

k+i−1)

{βk+i(s) + γk+i(s∗k+i−1, s)}, i = 1, 2, . . . , N − k (11)

the amount of information that should be requested to correctthe bit. The most likely error event for bit i is the event thatseparates cML and ci

comp, i.e.,

ei = cML ⊕ cicomp,

where ⊕ represents the XOR (addition or subtraction ina binary field) operator. For linear convolutional codes, asconsidered in this paper, ei is a codeword.

The reliability in (5) can be further simplified as

Λi =1σ2

rT · (cML − cicomp). (12)

If the channel from the distant transmitter to the collaboratingcluster in Fig. 1 does not have unit channel gains, then thereliability at the jth receiver can be expressed as

Λi,j =1σ2

a∗jr

T · (cML − cicomp), (13)

where we have suppressed the dependence of cML and cicomp

on the receiver number, j.The decoder tries to estimate the amount of information

required to change the decision from the ML path to thecompeting path (assuming that this will correct the error). LetcML(k) and ci

comp(k) denote the kth parity bit on the MLpath and competing path for information bit i, respectively.If cML(k) = ci

comp(k), then that parity bit does not provideany distinction between the two paths in the trellis. Thus,requesting information about such parity bits from the othercollaborating nodes will not be helpful in resolving betweenthese two paths. In the most likely case, in which either cML

or cicomp is the correct codeword, the decoder will improve its

decision only if additional information is received for thoseparity bits that differ in value in the ML and competingcodeword.

♦ Definition 2. Candidate set of parity bits Si for trellissection i: The set of parity bits that differ in value in the MLand competing codewords,

Si = {k : cML(k) �= cicomp(k)} = {k : ei(k) = 1}. (14)

Once the candidate set of parity bits is obtained, the decodertries to estimate the number of parity bits from the candidateset Si that have to be requested from other nodes in order forthe decoder to decide in favor of ci

comp instead of cML.Let r∗ be the received vector at receiver 1 after requesting

κ coded symbols from another receiver5, say receiver 2. Thedecoder estimates the minimum number of additional codedsymbols (κ) that will change the decision from cML to ci

compwith probability greater than some threshold. That is, after re-ceiving the additional information, we desire a high probabilityof satisfying expressions (15), (16), and (17) on the following

5r∗ is obtained by combining the original received vector r and theadditional symbols using MRC.

page, where η is the subset of the candidate set that has beentransmitted in this iteration, and r

′corresponds to the symbols

received due to those transmissions; i.e., r∗ = a∗1r + a∗

2r′ (a∗

2

is the conjugate of the fading coefficient at receiver 2). Using(13), we obtain 2a∗

1rT · (cML − ci

comp) = 2σ2Λi, where Λi isthe reliability of trellis section i before combining. Note thatin the above equations, cML and ci

comp refer to the ML pathand competing path encountered in computing the soft outputfor trellis section i before receiving additional coded symbolsfrom receiver 2.

As previously mentioned, the decoder assumes that theML path has been incorrectly chosen over the competingpath. Then we can calculate the required value of κ underthe assumption that the all-zeros CW has been transmitted,in which case ci

comp(l) = 1 and cML(l) = −1, ∀l ∈ Si.Since the all-zeros CW is the true transmitted codeword,r′(l) ∼ N (a2, σ

2). Thus,

Xi �Xl∈η

η⊂Si,|η|=κ

2a∗2r

′(l)[cML(l)−ci

comp(l)] ∼ N (−4a22κ, 16a2

2κσ2).

Thus the decoder estimates that after the first retransmission,correct decoding is made if Xi < −2σ2Λi.

The decoder estimates the number of coded bits κ for whichinformation is required from another receiver as follows,

minκ

P (Xi < −2σ2Λi) ≥ Θ (18)

minκ

Q

(σ2Λi − 2a2

2κ

2√

a22κσ2

)≥ Θ, (19)

where κ is the number of parity bits retransmitted and Θis a predefined threshold. Thus, the decoder estimates thenumber of bits to be retransmitted as the minimum number thatwould cause the decoder to decide in favor of ci

comp insteadof cML with a probability that is at least Θ. This provides theminimum number of bits that is most likely to correct bit iif it is in error. P (Xi < −2σ2Λi) will be referred to as thecorrection probability after combining (Pc).

B. Estimation of the request set

After the decoder estimates κ from the candidate set, itneeds to select the subset of κ parity bits in Si for whichinformation will be requested from another receiver. Weestimate an instantaneous SNR for each trellis section involvedin the error event ei that separates cML and ci

comp to decide thecandidate set for collaborative exchange. The receiver sorts thetrellis sections in the error event according to the instantaneousSNRs and requests for κ parity bits from the trellis sectionswith low SNRs.

The concept of instantaneous SNR was proposed in [30]for use in selecting which symbols should be retransmitted in


‖r∗ − cicomp‖2 < ‖r∗ − cML‖2 (15)

=⇒ 2r∗T · (cML − cicomp) < 0 (16)

=⇒ 2rT · (cML − cicomp) +

∑l∈η

η⊂Si,|η|=κ

2r′(l)(cML(l) − ci

comp(l)) < 0 (17)

TABLE II

INSTANTANEOUS SNR ESTIMATION FOR TRELLIS SECTIONS BASED ON THE AVERAGE OF THE INSTANTANEOUS SNRS OF THE PARITY BITS IN THE

CANDIDATE SET

Output label on trellis section i Output label on trellis section i Estimate of the instantaneous SNRfor cML for ci

comp for trellis section i

1 1 −1 − 1 (|r0i | + |r1

i |)/2−1 1 1 − 1 ′′−1 1 1 1 |r0

i |−1 1 −1 − 1 |r1i |

an ARQ scenario. Several different schemes were consideredin [30], and the one described here was found to offer thebest performance. If for a particular trellis section i, cML andci

comp differ in only one parity bit, then the instantaneous SNRof that section is equal to the absolute value of the LLR ofthe received symbol corresponding to that parity bit. If fora particular trellis section i, cML and ci

comp differ in bothparity bits, then the instantaneous SNR of the trellis sectionis the average of the instantaneous SNRs of the two paritybits. The receiver selects κ parity bits corresponding to trellissections with the lowest SNRs from the candidate set. Theinstantaneous SNR of a particular trellis section for differentoutput labels on cML and ci

comp is given in Table II. Note thatall possible output labels can be obtained by interchanging theoutput labels on the ML and competing paths in each row ofTable II.

C. Detailed description of I-LRB collaborative decoding

With the above approaches to estimate the request size andthe request set, we can describe I-LRB collaborative decodingin detail. Upon initiation of collaboration, the nodes broadcasttheir µs to determine the best receiver. Starting with the bestreceiver, let the receivers be numbered RX1 to RXNrx . Thesecond best receiver RX2, transmits its fading coefficient a2

to RX1. RX1 needs the fading coefficient to estimate thenumber of coded symbols that have to be requested.

Let the number of iterations in collaborative decoding bedenoted by Niter . For the results presented in this paper, we setNiter = Nrx − 1. Given the overhead constraint, RX1 limitsthe number of bits that can be exchanged in each iteration topθMRC/Niter . In each iteration, RX1 sorts the informationbits according to the reliabilities, and obtains the competingpath for each LRB using the technique described in SectionIII-C. Then for each LRB,

1) RX1 estimates κ using (19).2) RX1 obtains the candidate set and the set of parities to

be requested based on the instantaneous SNRs.3) RX1 broadcasts κ and the indices of the parity bits that

need coded symbols from another node.4) For each bit index, the best node that has not previ-

ously transmitted information for that bit will transmit

Fig. 2. The code-trellis for the (5, 7) convolutional code with examples ofthe notation used in this paper.

information for that bit. Each node scales its receivedsymbols by the channel coefficient and broadcasts thatinformation for a bit. If κ > |Si| (the number of codedsymbols required is more than the size of the candidateset), then coded symbols are obtained from the next bestreceiver until a total of κ symbols are transmitted.

Consider an example to illustrate step 4 above. Assume thatthe codeword shown in bold in Fig. 2 is the competing path forbit i and that the ML path is the all-zeros path. Assume thatthis is the first iteration in which bits in this candidate set havebeen selected to receive information from the collaboratingnodes. For the sake of exposition, assume that trellis sectionsi− 1, i, and i + 1 have increasing instantaneous SNRs in thatorder. If κ = 2, information about ci−1 will be obtained fromRX2. If κ = 3, information about ci−1 and c1

i will be obtainedfrom RX2. If κ = 7, coded symbols for all the parity bits inthe candidate set are obtained from RX2, and coded symbolsfor ci−1 are obtained from RX3.

Once the appropriate number of coded symbols are com-bined for the LRB, RX1 requests for coded symbols for thenext LRB that has a different competing path. As previouslydescribed, coded symbols for a particular trellis section from aparticular collaborating nodes are only ever transmitted once.Using the previous example, assume that the branch fromstate 2 to state 1 has already received coded symbols fromRX2 because this branch was part of a different competing


Fig. 3. Probability of block error for different numbers of collaboratingnodes when the overhead constraint is fixed at 5% of the overhead for MRC.

path for some other bit that had a reliability less than thatof bit i. Then when κ = 3, information for ci−1 and c1

i+2

will be obtained from RX2 (assuming that coded symbols forthese bits have not been obtained from RX2 earlier). Also, ifcoded symbols for c1

i are required in the next iteration, theyshould be obtained from RX3, and not RX2. This procedureis repeated until a total of pθMRC/Niter bits are exchangedwithin the cluster. Note that this includes the bits required toindex the parity bits requested by RX1. In practice, all ofthe information requests can be performed at the beginningof an iteration, followed by each receiver’s response startingfrom RX2 to RXNrx . RX1 combines all of the receivedinformation with its previously received information usingMRC (on a bit-by-bit basis). If RX1 is able to decode correctlyor the maximum number of iterations has been reached,then the collaborative decoding process terminates. Otherwise,another iteration of collaborating decoding is performed.

V. RESULTS

In this section, we present the performance of our col-laborative decoding scheme. For all the results in this pa-per, a rate 1/2, memory-three, non-recursive, non-systematicconvolutional code with generator polynomials 1 + D2 and1 + D + D2 ((5, 7) in octal notation) is used for encoding atthe distant transmitter. The message consists of N = 900-bitpackets. We use 5-bit quantization, which has been shown toachieve performance close to no quantization [31], for the softinformation exchanged in the cluster. The channel betweenthe distant transmitter and the cluster of cooperating nodes isassumed to be a quasi-static Rayleigh fading channel, wherethe fading is constant over each packet. For all results, thenumber of collaborating iterations Niter = Nrx − 1. Thecorrection probability after combining is set to Pc = 0.95.

Results are first shown for systems in which the cooperationchannels are error-free. This prevents the results from beingconstrained to a particular coding or modulation scheme.For instance, in many scenarios, the cooperative links mayutilize a different physical layer than the link from the distanttransmitter. Results for noisy cooperation channels are shown

Fig. 4. Throughput for different numbers of collaborating nodes when theoverhead constraint is fixed at 5% of the overhead for MRC.

later, where we assume nonfading AWGN channels amongthe nodes in the cluster. The performance of various schemesare demonstrated in terms of the block error rate (PB),average overhead (Θavg), and the throughput. Here we definethe throughput as η = N/(N/R + Θavg), which assumesthat the channel occupancy of the forward transmission andcollaborative information is the same. The overhead consistsof the µ’s and the ai of the second-best receiver before thefirst iteration, along with the bit indices for the requestedinformation and the soft information in each iteration. Theµ’s and single ai require (Nrx + 1)q bits. The overhead forthe bit indices is computed by assuming that each bit indexis transmitted using �log2(N)� bits. Even with this naiverepresentation with no compression, we will show that I-LRB provides significant improvements in throughput whencompared to other schemes.

We first show results for the anycast scenario (in which thepacket is assumed to be received correctly when any of thenodes is able to decode correctly) with error-free cooperationchannels. The block error rate for I-LRB and COI-MRC isshown in Fig. 3 for different numbers of collaborating nodes.For these results, a 5% overhead constraint with respect tothe overhead required for MRC was imposed. It is observedthat I-LRB outperforms COI-MRC for all sizes of the coop-erating cluster shown. It is also seen that the gain offeredby COI-MRC increases as the number of collaborating nodesincreases. For example, with a target block error rate of 10−2,I-LRB outperforms COI-MRC by approximately 2 dB whenthere are 8 collaborating nodes. The corresponding throughputfor this scenario is shown in Fig. 4. It is seen that throughputfor I-LRB is larger than the throughput for COI-MRC forall the cases. At Eb/N0 = 2 dB and with eight collaboratingreceivers, I-LRB increases the throughput by almost 30% withrespect to COI-MRC. The performance of selection diversitywith eight nodes is also shown in Fig. 3 and Fig. 4. It is seenthat the block error rate of COI-MRC is only slightly lowerthan that of selection diversity with eight receivers, but thethroughput of COI-MRC is significantly lower. This clearlyshows that the additional bandwidth used for combining in


Fig. 5. Probability of block error overhead for COI-MRC and I-LRB with8 cooperating nodes and different constraints on the overhead.

Fig. 6. Average cooperation overhead for COI-MRC and I-LRB with 8cooperating nodes and different constraints on the overhead.

COI-MRC is not worth the negligible improvement in errorperformance. It is also seen that I-LRB can provide around2 dB improvement in block error rate when compared toselection diversity and simultaneously provide a 5%-20%improvement in throughput. The throughput advantage withrespect to selection diversity may be further improved byutilizing source coding for the bit indices in I-LRB.

The block error rates of COI-MRC and I-LRB are comparedin Fig. 5 for different overhead constraints when there are eightcollaborating nodes. The corresponding average cooperationoverhead is shown in Fig. 6. I-LRB achieves a lower blockerror rate with a lower cooperation overhead thereby leadingto an improvement in throughput as seen from Fig. 7. Thethroughput of I-MRC (COI-MRC with no overhead constraint)and that of a single receiver are also shown in Fig. 7. ThoughI-MRC has the best block error rate among all the schemes(see Fig. 5), it has a lower throughput when compared to I-LRB or COI-MRC. Thus, it is clear that I-MRC achieves goodblock error rate performance at the cost of higher overhead.It is also observed that the throughput of I-LRB decreases

Fig. 7. Throughput for COI-MRC and I-LRB with 8 cooperating nodes anddifferent constraints on the overhead.

Fig. 8. Average number of iterations per collaborative decoding attemptrequired by 8 receivers.

when the overhead constraint is relaxed. This implies that theimprovement in block error rate is not significant as morecombining is allowed in the cooperating cluster. The increasein overhead caused by relaxing the overhead constraint dom-inates the decrease in block error rate, leading to a lowerthroughput. Thus, the I-LRB scheme is capable of providing alarge increase in throughput with a very small overhead. Thisis because I-LRB targets the trellis-sections that are likely tobe in error and adapts the amount of information combinedfor these sections based on their reliabilities. Thus, among allthe schemes compared (I-MRC and COI-MRC are variantsof A-F, selection diversity is an instance of D-F in error-freecooperation channels), I-LRB offers the best trade-off betweenthroughput and reliability (block error rate). At 0 dB, I-LRBhas the potential to improve the throughput by over 75% withrespect to COI-MRC, and by over 160% with respect to MRC(or I-MRC).

The average number of iterations required by the COI-MRC and I-LRB schemes is shown in Fig. 8. Collaborativedecoding is terminated faster in I-LRB than in COI-MRC. For


Fig. 9. Throughput for COI-MRC and I-LRB for the unicast and anycastscenarios, eight cooperating nodes and 5% constraint on the overhead.

Fig. 10. Probability of block error for COI-MRC and I-LRB as a function ofthe signal-noise-ratio (EC/N0) in the cooperating cluster. Results are shownfor varying forward signal-to-noise ratios (EF /N0), eight cooperating nodes,and 15% constraint on the overhead.

example, at an SNR of 0 dB and 5% overhead constraint, theaverage number of iterations for I-LRB is 47% smaller thanfor COI-MRC. This frees the network for use by other trafficand reduces the amount of energy used in transmission anddecoding.

The throughputs for the anycast and unicast scenarios arecompared in Fig. 9 for eight receivers and a 5% overhead con-straint. In the unicast system, cooperation using COI-MRC andI-LRB is performed until one node decodes correctly. Then thesource message is relayed to the intended destination. Thus,relaying to the intended destination at the last step contributesto additional overhead when compared to the anycast system.It is seen that the unicast system performs relatively poorerwhen compared to the anycast scenario. However, for boththe anycast and unicast scenarios, I-LRB outperforms COI-MRC. Performance of the unicast scenario on cooperationchannels with errors show similar performance. Note that it ispossible to implement a pure D-F cooperation scheme for the

Fig. 11. Throughput for COI-MRC and I-LRB as a function of the signal-noise-ratio (EC/N0) in the cooperating cluster. Results are shown for varyingforward signal-to-noise ratios (EF /N0), eight cooperating nodes, and 15%constraint on the overhead.

unicast scenario. For error-free cooperation channels, the blockerror rate is the same as that of selection diversity, but thethroughput is a lower than selection diversity due to relayingof the source message to the intended destination in the laststep. The performance of D-F is also shown in Fig. 9. It is seenthat COI-MRC has a throughput lower than that of D-F, but I-LRB provides a higher throughput than D-F (by approximately15% at 0 dB).

We now consider the performance with noisy intra-clusterchannels. For these results, we assume that same physical layeris used as for transmission from the distant transmitter. Thus,BPSK modulation is used with the (5, 7) convolutional code.The probability of block error and throughput for eight nodescollaborating under a 15% overhead constraint in AWGNcooperation channels is shown in Figs. 10 and 11, respectively.Each curve is obtained for a fixed bit energy-to-noise ratio(EF /N0) on the fading forward channel (between the distanttransmitter and the cluster) and varying bit energy-to-noiseratios (EC/N0) in the collaborating cluster. Even with thissimple code, it is seen that I-LRB offers better block error ratesfor EC/N0 ≥ 2 dB. In addition, the throughput approximatelyachieves its maximum value when EC/N0 ≥ 4 dB. Thus,good performance in the I-LRB scheme can be achieved withmoderate intra-cluster SNRs. For EF /N0 = 0 dB, I-LRBimproves the throughput by up to 50% for moderate SNRsin the cooperation channels. Note that COI-MRC outperformsI-LRB at very low EC/N0. Most of the loss comes from theerrors in the bit indices. The other nodes may not transmitinformation for the trellis sections requested by the best node,due to errors in decoding the indices. With more-powerfulcodes of the same rate, performance of error-free collaborationcan be achieved at lower SNRs in the collaboration channels.

VI. CONCLUSIONS

In this paper, we present a novel cooperative commu-nication scheme called improved least-reliable-bits (I-LRB)


collaborative decoding for user cooperation in bandwidth-limited scenarios. We also present a new constrained-overheadincremental MRC (COI-MRC) scheme that offers good per-formance with lower overhead than full MRC. In the I-LRBcollaborative decoding scheme, the cooperating nodes iteratebetween a process of information exchange and decoding.The advantage of the I-LRB scheme over COI-MRC andother previously proposed cooperation strategies is that theinformation exchanged in the collaborative process is carefullychosen based on information generated in the SISO decoder.There are two levels of adaptation in I-LRB. First, I-LRBadapts the set of bits for which information is requested basedon the reliabilities of the different trellis sections. Second, foreach chosen trellis section, I-LRB adapts the number of codedsymbols for which information is requested based on the bitreliabilities.

Simulation results show that I-LRB achieves a lower prob-ability of block error with a lower average collaborativeinformation exchange than the COI-MRC scheme. The resultsshow that I-LRB can provide a 30%-60% improvement inthroughput over COI-MRC in several scenarios. The overheadrequired for this improvement is less than 5% of the overheadof traditional combining schemes like MRC. I-LRB alsoprovides a 2 dB improvement in the probability of blockerror over selection diversity while simultaneously improvingthe throughput by 5% − 20%. Results for noisy AWGNcollaboration channels show that only moderate SNRs arerequired for the intra-cluster channels in order to achieveperformance close to that on error-free channels. Thus, I-LRB offers an efficient approach for collaboration when themaximum collaborating overhead is constrained.

APPENDIX

To prove the Trace-back procedure in Theorem 1, we firstprove the following Lemma.Lemma: αk(s) ∝ min

c∈C:sk(c)=s‖rk

1 − ck1‖2 for any state s at

time k that is on the path of a valid codewordProof: By mathematical induction.Note that α0(0) = 0. Then α1(0) is computed using (7) as

α1(0) = 0 + γ1(0, 0) (20)

because there is only one valid state leading into state 0 attime 1. Similarly, α1(2) = 0 + γ1(0, 2). The lemma does notapply to the other states at time 1 because they are not validstates for a rate 1/2 convolutional code initialized to state 0 attime 0. So using (9), the lemma holds for k = 1.

Assume that the lemma holds for time k − 1. Then

αk(s∗) = maxs∈S(→s∗)

(αk−1(s) + γk (s, s∗)) (21)

∝ maxs∈S(→s∗)

„max

c∈C:sk−1(c)=s− ‖rn

1 − ck−11 ‖2 + γk(s, s∗)

«(22)

∝ minc∈C:sk(c)=s∗

‖rk1 − ck

1‖2, (23)

where (22) follows from the assumption about the claim, andthe last equation follows from (9). Thus, the claim is true fortime k. The principle of induction completes the proof.

Remark: From the lemma, αk(s) is proportional to thepartial-path metric (log P (rk

1 |ck1)) [28] of the surviving path

at state s at time k in the Viterbi algorithm when the branchmetric is the Euclidean distance.

Proof of the trace-back theorem:Compare the trace-back theorem and (7). The trace-backtheorem chooses the previous state (si−1) that correspondsto the branch involved in computing the alpha for the currentstate (si). Since αk(s) is proportional to the partial path metricof the surviving path leading to sk = s, the branch involved incomputing αk(s) is part of the corresponding surviving path.

Thus, conditioned on the current state, the trace-back theo-rem chooses the previous state as the state at time k−1 on thesurviving path at time k. The proof of the trace-back procedurefollows because the surviving path has the best partial-pathmetric (min ‖rk

1 − ck1‖2) among all paths c that pass through

sk = s.

REFERENCES

[1] T. M. Cover and A. A. El Gamal, “Capacity theorems for the relaychannel,” IEEE Trans. Inf. Theory, vol. 25, pp. 572–584, Sept. 1979.

[2] J. N. Laneman, D. Tse, and G. Wornell, “Cooperative diversity inwireless networks: efficient protocols and outage behavior,” IEEE Trans.Inf. Theory, vol. 50, pp. 3062–3080, Dec. 2004.

[3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity–part I: system description,” IEEE Trans. Commun, vol. 51, pp. 1927–1938, Nov. 2003.

[4] T. F. Wong, X. Li, and J. M. Shea, “Distributed decoding of rectangularparity-check code,” IEE Electronics Lett., vol. 38, pp. 1364–1365, Oct.2002.

[5] A. Avudainayagam, J. M. Shea, T. F. Wong, and X. Li, “Reliabilityexchange schemes for iterative packet combining in distributed arrays,”in Proc. 2003 Wireless Communications and Networking Conference,vol. 1, pp. 832–837.

[6] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies andcapacity theorems for relay networks,” IEEE Trans. Inf. Theory, vol. 51,no. 9, pp. 3037–3063, Sept. 2005.

[7] J. N. Laneman, “Cooperative diversity in wireless networks: algorithmsand architectures,” Ph.D. dissertation, M.I.T, Sept. 2002.

[8] T. E. Hunter and A. Nosratinia, “Cooperative diversity through coding,”in Proc. 2002 IEEE Int. Symp. Inform. Theory, p. 220.

[9] T. Hunter, S. Sanayei, and A. Nosratinia, “Outage analysis of codedcooperation,” IEEE Trans. Inf. Theory, pp. 375–391, Feb. 2006.

[10] B. Zhao and M. Valenti, “Distributed turbo coded diversity for the relaychannel,” IEE Electronics Lett., vol. 39, pp. 786–787, May 2003.

[11] Z. Liu, V. Stankovic, and Z. Xiong, “Wyner-Ziv coding for the half-duplex relay channel,” in Proc. 2005 IEEE Int. Conf. Acoustics, Speech,and Signal Proc.

[12] R. Hu and J. Li, “Exploiting Slepian-Wolf codes in wireless usercooperation,” in Proc. 2005 IEEE Workshop on Signal Proc. Advancesin Wireless Commun., pp. 275–279.

[13] R. Hu and J. T. Li, “Practical compress-forward in user cooperation:Wyner-Ziv cooperation,” in Proc. 2006 IEEE Int. Symp. Inform. Theory.

[14] T. F. Wong, X. Li, and J. M. Shea, “Iterative decoding in a two-node distributed array,” in Proc. 2002 IEEE Military CommunicationsConference (MILCOM), vol. 2, pp. 704.2.1–5.

[15] A. Avudainayagam, J. M. Shea, T. F. Wong, and Y. Fang, Designand Analysis of Wireless Networks. Nova Science Publishers, 2004, ch.Cooperative techniques in wireless communications.

[16] J. G. Proakis, Digital Communications, Fourth Edition. McGraw-Hill,2000.

[17] R. Viswanathan and P. Varshney, “Distributed detection with multiplesensors I: fundamentals,” Proc. IEEE, vol. 85, no. 1, pp. 54–63, Jan.1997.

[18] C. Rago, P. Willett, and Y. Bar-Shalom, “Censoring sensors: a low-communication-rate scheme for distributed detection,” IEEE Trans.Aerosp. Electron. Syst., vol. 32, no. 2, pp. 554–568, April 1996.

[19] C.-T. Yu and P. K. Varshney, “Paradigm for distributed detection undercommunication constraints,” Optical Eng., vol. 37, no. 2, pp. 417–426,1998.


[20] J. Hu and R. Blum, “On the optimality of finite-level quantizations fordistributed signal detection,” IEEE Trans. Inf. Theory, vol. 47, no. 4,pp. 1665–1671, May 2001.

[21] A. Ribeiro and G. Giannakis, “Bandwidth-constrained distributed esti-mation for wireless sensor networks–part I: Gaussian case,” IEEE Trans.Signal Process., vol. 54, no. 3, pp. 1131–1143, March 2006.

[22] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding oflinear codes for minimizing symbol error rates,” IEEE Trans. Inf. Theory,vol. 20, pp. 284–287, March 1974.

[23] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary blockand convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, pp. 429–445,Mar. 1996.

[24] M. P. C. Fossorier, F. Burkert, S. Lin, and J. Hagenauer, “On the equiv-alence between SOVA and max-log-MAP decodings,” IEEE Commun.Lett., vol. 2, no. 5, pp. 137–139, May 1998.

[25] J. M. Shea, “Reliability-based hybrid ARQ,” IEE Electron. Lett., vol. 38,no. 13, pp. 644–645, June 2002.

[26] A. Roongta and J. M. Shea, “Reliability based hybrid ARQ usingconvolutional codes,” in Proc. 2003 Int. Conf. Commun. (ICC), pp.2889–2893.

[27] W. E. Ryan, “Concatenated codes and iterative decoding,” in WileyEncyclopedia of Telecommunications (J. G. Proakis ed.) New York:Wiley and Sons, 2003.

[28] S. Lin and D. J. Costello, Error Control Coding: Fundamentals andApplications. Prentice Hall, 1983.

[29] L. Reggiani and G. Tartara, “Probability density functions of softinformation,” IEEE Commun. Lett., vol. 6, pp. 52–54, Feb. 2002.

[30] A. Avudainayagam, A. Roongta, and J. M. Shea, “Improving theefficiency of reliability-based hybrid-ARQ with convolutional codes,”in Proc. 2005 IEEE Military Communications Conference.

[31] G. Montorsi and S. Benedetto, “Design of fixed-point iterative decodersfor concatenated codes with interleavers,” IEEE J. Sel. Areas Commun.,vol. 19, pp. 871–882, May 2001.

Arun ’Nayagam received the B.E degree in elec-tronics and communication engineering in 2000from Anna University, India, and the M.S. andPh.D. degrees in electrical and computer engineeringfrom the University of Florida in 2001 and 2006,respectively. His research interests include wirelesscommunications, codes on graphs and belief prop-agation, iterative decoding techniques, cooperativecommunication, and applied error control coding.He is currently employed as a Senior PowerlineResearch Engineer at Intellon Corporation.

John M. Shea (S’92-M’99) received the B.S.(with highest honors) in Computer Engineering fromClemson University in 1993 and the M.S. and Ph.D. degrees in electrical engineering from ClemsonUniversity in 1995 and 1998, respectively.

Dr. Shea is currently an Associate Professor ofelectrical and computer engineering at the Univer-sity of Florida. Prior to that, he was an AssistantProfessor at the University of Florida from July 1999to August 2005 and a post-doctoral research fellowat Clemson University from January 1999 to August

1999. He was a research assistant in the Wireless Communications Program atClemson University from 1993 to 1998. He is currently engaged in researchon wireless communications with emphasis on error-control coding, cross-layer protocol design, cooperative diversity techniques, and hybrid ARQ.

Dr. Shea was selected as a Finalist for the 2004 Eta Kappa Nu OutstandingYoung Electrical Engineer Award. He received the Ellersick Award from theIEEE Communications Society in 1996. Dr. Shea was a National ScienceFoundation Fellow from 1994 to 1998. He is an Associate Editor for theIEEE Transactions on Vehicular Technology.

Tan F. Wong received the B.Sc. degree (1st classhonors) in electronic engineering from the ChineseUniversity of Hong Kong in 1991, and the M.S.E.E.and Ph. D. degrees in electrical engineering fromPurdue University in 1992 and 1997, respectively.He was a research engineer working on the highspeed wireless networks project in the Departmentof Electronics at Macquarie University, Sydney,Australia. He also served as a post-doctoral researchassociate in the School of Electrical and ComputerEngineering at Purdue University. Since August

1998 he has been with the University of Florida, where he is currentlyan associate professor of electrical and computer engineering. He serves asEditor for Wideband and Multiple Access Wireless Systems for the IEEETransactions on Communications and as the Editor for the IEEE Transactionson Vehicular Technology.

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