arq by awais

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fails to detect the presence of errors or fails o determine theexact ocations of theerrors. neither case, an erroneousword sdelivered to theuser. Sinceno retransmission i s

required in an FEC error-control system, no feedback channelis needed. The throughput of the system s constant, and isequal to the rate of the code used by the system.

FEC error-control system s, however , do have some draw-backs . When a received word s detected in erro r, it mus t bedecoded, and the decoded word must be delivered to the userregardless of whether t scorrect or incorrec t. Since the

probability of adecodingerror smuch greater than theprobability of anundetected error, it isharder to achievehigh sys tem reliabil ity with FEC schemes. In order to attainhigh reliability, a ong, powerful error-correcting code mustbe used anda argecollectio n of errorpatternsmust becorrecte d. This makes ecoding ard to implement andexpensive. Forhese reasons, ARQ schemesre oftenpreferredver FEC schemes fo rrrorontrolnatacommunications ystems uch as packet-switching atanetwor ks and computer communications networks. However,incommunications ordatastorage)syst ems where eed-back hannelsarenotavailable or retransm issionsnotsuitable for some reason, FEC is the only choice.

This aper surveys a number of ARQ schemes. They

represent lternative olutions to the esign of retrans-missionrotocols,articularlyhe mode in hich thetransmitterstores,orders,and etransmits hecodewordswhich have been receive d n error. These different schemeshave arisen primarily n an attempt to combat he problemtha t, when the channel error rate ncreases, he hroughputof an ARQ error-control system may deteriorate very rapidly.This is caused by he imewastedn etransmitting hecodewor ds detected in error . This problem becomes part icu -larly severe f here s significant round-trip delay betweenthe ransmission of a codeword and he receipt of its errorstatus nformationbackat the transm itter. Long delay sinevitabl e when satellites or long errestria lchannels arebeing used.

Anotherapproach to errorcontrol s hrough theuse ofhyb rid ARQ schemes whic h i ncorpora te bo th FEC and retran s-mission.Hyb rid ARQ schemes offer hepotential orbetterperformance if appropriate ARQ and FEC schemes areproperly combined. Either block or convolu tional codes may

be used for FEC. This paper also discusses different class esof hybrid ARQ schemes and their performance.

Basic A R Q Schemes

Based on retransm ission strategies , there are three basictypes of ARQ schemes: stop-and- wait ARQ, go - b a c k 4 ARQ,andelective-repeat ARQ [3 , 16-21]. The s t o p - a n d - w a i t

scheme repres ents the simples t ARQ procedure nd wasimplemented in early error-control systems. The IBM Binary

SynchronousCommunica tion (BISYNC) procedure,or x-ample, was of the stop-and-wait type [22]. In a stop-and-waitARQ error-control sy stem, the transmi tter sends a codewordto the receiver and waits for an acknowledgment, as shownin Fig. 1. A positive acknowledgment (ACK) from the receiverindicates hat he ransmittedcodewordhas been success-fully received, and the transmitter sends the next codewordin the queue. Anegativeacknowl edgment (NAK) from hereceive r ndicates hat the transmitte dcodewordhas beendetected in error; the transmitte r then resends the codewordandagainwaits oranacknowledgment.Retransmissionscontinue until the transmitter receives an ACK.

This scheme i s simple but inheren tly inefficient because ofthe idle ime spent waiting or an acknowledgme nt o f each

transmitte d codewor d. One possible emedy s to make heblock (or code) length n extremely long. However, the use ofa very ong block ength does not really provide a solution,since he probability hat a block contains errors ncreaseswithhe lockength. Hence, usingong lockengthreduces thedleimeutncreasesherequency ofretransm issions or each codeword. Moreover, a ong blocklength may be impractical n many applications because ofrestrictions mposed by the data format.

By the 1970's, ARQ systems were in extensive use in

packet-s witched and other data networks . Higher data ratesand tilization of satellite hannelswithon gound-tripdelays stablishedhe need for ontin uousransmissionstrategies to replace hestop-and-waitprocedures. nter-

nationalstandardsorganizations suchas CCITT (the Inte r-national Telegraph and Telephone ConsultativeCommittee)began making ffortsor rotocol tandardization. Thisresulted n the high-leveldata ink.contr ol (HDLC) and theCCITT X.25 standards . These envisaged the use o f a go-back-

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N ARQ system on full duplex links. This remains the standardfor packet-switching networks.

The basicgo-back-N ARQ scheme is llust rated n Fig. 2.The transmitter continuously ransmits codewords n orderand henstores hempending eceipt of anACKlNAK or

each. The acknowledgment oracodewordarrivesafteraround-tripdelay,definedas he ime ntervalbetween hetransmission of a codeword and he receipt of an acknowl-edgment for that codeword. During this nterval, N - 1 othercodewordsarealso ransmitted. Whenever he trans mitte rreceivesa NAK indicating hataparticularcodeword,sa ycodeword i , was received n error, t stops transmitting newcodew ords. Then it goes back to codeword i and proceeds to

retransmitha todewordndhe N - 1 succeedingcodewordswhichwere ransmittedduring one round-tripdelay. Atheeceivingnd,heeceiveriscardsheerroneously eceivedword i and all N - 1 subsequentlyreceived words, whether they are error-free or not. Retrans-missioncontinuesuntilcodeword i isposit ivelyacknowl-

edged. In each retransmission for codeword i , the transmitterresendsheameequence fodewords. As soon scodeword i is ositively cknowledg ed,heransmitterproceeds to transmit new codewords.

The main draw back of go -back-N ARQ is tha t, whenever areceived word s detected n error, he eceiver also ejectsthe next N - 1 received words, even hough many of themmay be error ree. As a result, hey must be retransmitted.This represents a waste of transmissions, which can result insevere deterioration of throughputerformancefargeround-tripelaysnvolved. For example,onsidersatellitechannelwitha ound-tripdelay of approximately700 ms. If the codeword length n i s 1000 bits long and the bitrates 1 Mb/s,henn one round-trip elay, N = 700

codewordsare ransmitte d. Therefore,when one receivedwor d s detected in error , 700 received words are rejected. Iferrors occur often enough, he system hroughput may alloff very rapidly.

The go -b ac k4 ARQ scheme becomes quite ne ffective orcommunications ystemswithhighdata atesan dargeround-trip elays.Thisneffectivenesss caused byheretransmission of manyerror-freecodewords ollowingacodew ord detected in error. This can be overcome by usingthe selective-repeat ARQ protoco l. n a selec tive-re peat ARQerror-controlsystem,codewordsar ealso ransmittedcon-

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tinuously.However,heransmitter nly resendshosecodewords that are negatively acknowledged (NAK’ed). Afterresending NAK’ed codeword,heransmitter ontinuestransmittingnewcodewordsn he ransmitterbuffer asillustra ted n Fig. 3). With his scheme, abuffermust be

providedat he eceiver to store heerror-freecodewordsfollowingeceivedordetectednrror, because,ordinarily, codewo rds must be delivered to the end user incorrect order, for example, in point-to-point communications.When the firs t NAK’ed codeword is succe ssfully received, thereceiver then releases any error-free codewords n consecu-tive order from the receiver buffer until the next erroneo uslyreceivedwordsncountered. ufficienteceiverufferstorage mus t be provided in a selective-repeat AR Q system;otherwise, buffer overflow may occur and codewords may belost.

Reliability and Throughput Efficiencies of the Basic A RQ Schemes

The perfo rmance of an ARQ error-co ntrolystemsnormally measured by ts eliability and hroughput effici-ency. nan AR Q system, he eceivercommitsadecodingerror whenever it accepts a received word with undetectederrors. Such an event is calle d an e rror eve nt. Let P(E) denotethe probability o f an error event. Clearly, for an ARQ systemto be reliable,he P(E) should be made ery mall. Thethrough put efficienc y (simple through put) of an ARQ systemis defined as the ratio of the average number of info rmat ion

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bits success fully accepted by the receiver per unit t ime to thetotal number of bits that could be tran smi tted per unit time.Al lhree asic ARQ schemes chieve the same ystemreliabili ty, but they provide different throughpu t efficiencies.

Suppose that an ( n , k ) inear block code C is used for errordetection nan ARQ sys tem. Let usdefine he ollowingprobabilities:

PC = probability hata received n-bitwordcontains noerror;

Pd = probabil ity that a re ceived w ord conta ins a detectableerror pattern; andPe = probability hat a eceived word contains anunde-

tectable error pattern.

Obviously, PC+ Pd + Pe = 1 . The probabil ity PC depends on thechannelerror tatistics,andheprobabilities Pd and Pe

depend on both he channel statistic s and he choice of the(n ,k ) error-detectingode C. Pe is ormally alledhe

pro bab ili ty of undetected error of the code.A received word is accepted by the receiver only if it either

contains no errors or contains an undetectable error pattern.Since an undetectable error pattern can occur on the nitia ltransmission of awordoronany etransmission, he P ( E )that he eceivedword n an ARQ system sacceptedandthat a decoding error s made is give n by

P(E)= pe + PdPe + Pd2 Pe + . . .

= Pe (1 + Pd + Pd2 + . . .) ( 1 )

If theerror-detecting code C isproperly chosen, Pe can bemade very small relative to P C , and hence P ( E ) can be madevery small. If the receive d word contains a detectable errorpattern, it isnotacceptedby he eceiveranda etrans-mission s requested.

For a random error channel with bi t error rate t,

PC = (1 - ) " . (2)

It has been proved hat inearblock codes existwith theprobability o f undetectederror Pe satisfying he ollowingupper bound [6, 7 , 101:

PeI l - 1 - ) k ] 2 - ( " - k ' . (3)

Codes satisfying the above bound have been found and willbe discussed na atersection. If sucha code is used forerror etection and if the umber of parit ybits, n - k , issufficiently arge, P, can be made very small relative to PC

and hence P ( E ) << 1 . For example, let C be the (2047,2014)triple-error-correcting primitive BCH code. This code satisfiesthe bound given by (3 ) [ 1 4 ] . Suppose that his code is used

for rror etectionn n ARQ system . Let E = (a veryhigh bit error rate).Then PC- 1.25 X l o - ' , and Pe 5 lo-' ' . From( l ) , e have

P ( E ) I x

From this example we see that high system reliability can beachievedbyan ARQ err or-c ontr ol scheme usin gvery ittle

parity overhead.

Now we examine the throughpu t performan ce of the threebasic A R Q schemes. For simplicity, we assume that theforward channel is a random-error channel with bit error rate

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c andha theeedback hannelsnoiseless.First, weconsider the stop-and-wait ARQ scheme. Let A be the idle ti meof the trans mitter between two s uccess ive tr ansmis sions. Let

6 be the bitate of theransmitt er. Even thoughhetransmitter does not ransmitduring he dleperiod, he

effect of the idle period o n the thro ughput mu st be taken intoconsideration. n one round-trip delay ime, he ransmi ttercould ransmit n+hs bits f it didnot emain dle. For acodeword to be receivedcorrectly, heaverage numberofbits that the transmitter could have transmitted is

Ts w = (n + A6)Pc + 2( n + A S ) P c ( I - P C )

+ 3(n + AS) P , ( 1 -+ . . . + (n + As) PC (1 - PC) ' - ' + . . .

= (0+ AS)PC(1 4- 2(1 - PC) + 3 (1 - PC) + . . .)

Therefore, the throughput of a stop-and-wait ARQ system is

where k / n is the rate of the code used by he system. Fordata communications systems with high data rates and largeround-tripdelays, A 6 / n becomes very arge because it isimpractical to use a very large n. In this case, the th roughputperformance ofhe stop-and-wait A R Q scheme ecomesunacceptable. For example, consider a satellite communica-tions system with a data rate of 1 Mb ls . Assume he round-trip delay or a satellite channel to be 700 ms. Then ~6 =

0.7 X l o 6 . To make ASln small compared to one, n should bechosen in he range of one mill ion bits ! It is mpractical to

use sucha ong code.Suppose that we choose n =10000bits. Then A6/n = 70. In his case, the hroughpu t of thesystem becomes negligible.

Ina go -ba ck 4 (GBN)ARQ syste m, etransmi ssion of an

NAK'ed codewordalways nvolves esending N codewords.Consequently, oracodeword to be successfully eceived,the average number of transmissions s

TGBN = 1 ' PC + (N + 1)Pc(1 - C )+ (2N + 1)Pc(1 - P c ) ~

+ . . . + (QN+ l ) P c ( l - P c ) Q+ . . .

= 1 +N ( 1 - PC)

PC

and the throughput of a GBN ARQ system s

We see that he hrou ghpu t dependson both hechannel

error ateand he ound-tripdelay N. The term (1 - P c ) Nrepresents the effect of the hannel rrorate and theround-tripdelay. For communicationssystemswhere hedata rate is not to o high and the round-trip delay is small, Ncan be made small by choosing a reasonably ong code. Inthis case, theeffect of the ound-trip delay s nsignifi cantand he GBNARQ scheme provid es high h rough put perfo r-mance. However, or a communications system with a highdata rate and a ong round-trip delay, N may become verylarge. A s a esult, (1 - P c ) N becomes significantly arge,especially when thechannelbi terror ate E ishigh. This

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wouldmakehehroughputerforman ce of GBNRQinadequate. For example,considerasatellitecommunica-t ionssystemwithadata ate of 1 Mbls anda ound-tripdelay of 700 ms. If we choose n = 10000 bits, then N= 70.For E = lb4,we have PC= 0.886 and (1 - Pc)N 8. This givesa thro,ughput of less than 10% of the code rate k /n .

Figure 4 shows how vGBNvaries with bit error ate E forvarious values of code block length n. Throughput values arecomputed for tw o values of round-tri p delay, namely 30 ms,such as might be the case for a ong terrestrial ine, and 700

ms, which is typical for a satellite channel. In bothases, thebit rate is 64 kb/s . Each code block is assumed to contain 32bits ofoverhead, includingparityan dcontrolbi ts. We seethat, or such a ow data ate, he GBNARQ systemdoesprovideatisfactoryhroughputerforman ce when theround-tripdelay sno t to o large,suchas ina errestrialsystem. For a atellite ystem,however,hehroughputbecomes very low for a bit error rate of t >

Now consider elective-repeat (SR) ARQ systemorwhich he receiver has an nfinite buffer to store he error-freecodewords when a received wor d s detected inerror.We will call such a system an deal SR ARQ system. For acode word o be successfully accepted by he eceiver, heaverage number of transmissions needed is

r,, = 1 . PC + 2 . P,(1 - C )+ 3 . P c ( l - Pc)2 f . . .

+ P . Pc( l- c ) Q - '+ . .

1

P C '

---

Hence, the through put of an ideal SR ARQ system s

We see that he hrough put does not depend on the round-trip delay. As a result, SR ARQ o ffers significan t benefits forsatelliteand ong errestrialchannels.Figure 5 presentsacomparison of ideal SR and GBN ARQ's for a sate llite channe l

with a data rate of 1.5 Mb /s and a round-trip delay of 700

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ms. We see that SR ARQ sign ifica ntly outp erfo rms GBNARQ,and provides acceptable throughput even at a bit error rate

The highhroughput erformance of ideal SR A R Q i sachieve d at the expense of exten sive buffering (theoreticallyinfinite buffering) and more complex logic at both transmitterand eceiver. If a inite buffer is used at the receiver (as isthecase inpracticalsystems),bufferoverflowma yoccurwhich reduces the hroughp utperforman ce of the system.

However, fasufficiently ongbuffer s usedand ifbufferoverflows roperly andled , even witheductionnthroughput, SR ARQ still significan tly outperform s the othertw o ARQ schemes, especia lly orsystemswhere hedatarate s high and the round-trip delay s arg e.

There are two methods for handling buffer overflow n anSR ARQ system with a finite receiver buffer: one is to deviseretransmissionstrategies so thatbufferoverflowcan beprevented, and heother s to deviseamechanism or hetransmitter to detect theoccurrence of a eceiverbufferoverflow event so that he lost codewords can be properlyretransmitted. The firstmethod susuallysimpler,but thesecond provides better throughput performance.

Finite receiver buffer SR ARQ schemes which do no t allow

buffer overflow are reported n Metzner [23], Miller and Lin[24], and Weldon [25]. These schemes employmixed-moderetransmi ssion strategies. One such scheme is the selective-repeat plus go-back-N (SR+GBN) AR Q scheme [24]. When thetransmitter irst receives a NAK for a given codeword (saycodeword i ) , t retransmits that codew ord and then continuestransmitting other new codewords, as in the basic SR mode(see Fig. 6). If another NAK is eceived or odeword i ,indicating hat ts second transmission attempt was unsuc-cessful, he ransmitter switches o he GBN retransmis sionmode. That is, it sends no more new codewords but ba cks up

of = 10-4.

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to codeword i and resends thatcodeword and he N - 1succeeding codewordshat were transmittedfterheprevious transmission attempt of codeword i (see Fig. 6). The

transmitter taysnhe GBN mode unti l odew ord i ispositivelyacknowle dged. At the receiver, when hesecondtransmission attempt of codeword i is detected in error, thesubsequent N - 1 received words are discarded, regardlessof whether they were received error-free or not.

This scheme achievessuperior hroughputperformancecomparedwith GBNARQ, because of thebenefitsgainedfrom he use of the S R mode for he irst etransmissionattempt. The use of thesecondary mode (GBN) guaran teesthat buffer overflows cannot occur at the receiveras long asabuffer sprovided orstoring N codewords. The schemecan have an even highe r throughput if more b uffer storage isprovided at he receiver , and the ransmitter is designed topermitmore han one retransmissionattempt oragivenNAK’ed codeword n he SR mode beforeswitching to theGBN mode. If v retransmissions n the SR mode are allowedbeforeheransmitterwitchesohe GBN mode, thereceiver buffer must be able to store Y (N- 1) + 1 codewordsto prevent buffer overflow. The throughpu t for Y retransmis-sions n the S R mode is given by

Figure 7 showshow he hroughpu t of the SR+GBNARQscheme varies w ith bit e rror rate for several values f Y . As v

increases,hehroughputerformance of the SR+GBNapproaches the deal case.

Another mixed-mode scheme is called the selective-repeatplus stutt er (SR-tST) ARQ scheme [24]. This s he same asthe SR+GBN scheme except hat, nstead of using he GBNmode after v retransmissionattempts of agiven NAK’edcodeword, the transmitter switches to the ST mode in whichit repeatedly retransmits hat codeword until t receives anACK.TheSR+STARQ scheme issimplerbut less efficienttha n the SR+GBN ARQ scheme.

Avariati on of theabovemixed -mode ARQ scheme wassuggested by Weldon [25]. When the transmitter s n the SRmode followingheeceipt of a NAK, it retransmits theNAK’ed codeword 91 times stuttering) . Then it proceeds o

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transmit other codewords waiting in the transmitter buffer insequence from where it left of f. The number 91 can be chosento provide maximum hroughput or a given error rate and

delay.Typically, 91 = 3 provides,good esults. If all q1retransmissions of a codeword are received with detectableerrors, hen he ransmitter everts to the GBN mode. ForQI= 1, Weldon’s scheme becomes SR+GBN A R Q with Y = 1.Actually, Weldon’sscheme isaclevercombina tion of theSR+GBN and SR+ST ARQ schemes. It can be generalized to

have multilevel repetitions of a NAK’ed codeword before thetransmitterswitches to the GBN mode. As thenumber oflevels increases, the throughput performance approaches theideal case, and of cours e the size of the receiver buffer alsoincreases. Weldon’s scheme provides higher throughput thanthe SR+GBNARQ scheme for eryhighbi terror ates.However, it is also more complex.

A selective-repeat ARQ scheme with a finite receiver bufferwhich allows buffer overflo w has recently been reported by



Yu and Lin [26]. In this scheme, a mechanism is devised forthe transmitter to detect the occurrence of a receiver bufferoverflow event so that the lost words an be properlyretransmitted. This scheme providesbetter hroughput per-formance han hemixed- mode schemes forhighbi terrorrates; however, it is more complicated to implement.

Linear Block Codes or Error Detection

The reliability of an ARQ system depends on the choice ofthe ode sed forerrordetectio n. If the code forerrordetection is properly chosen, the probability that the receivercommits a decoding error can be very small.

Consider errordetectionwithinearblock codes ona

random error channel with bit error rate Korzhik [6] provedthat here exist ( n ,k ) linear codes with prob ability Pe of anundetected error satisfyi ng the bound

Pe 52- ( " - k '1 - 1 - ) k ) (8)

for all n and k and all t , 0I5 1/2. He proved this result byaveraging P e over all the systematic (n,k) linear block codes.It is an existence proof and no method has been found orconstructingodes to satisfyhe ound iven y (8).

However,a ewsmall classes of kno wn codeshave beenproved to satisfy a weaker bound,

p, 5 2-Wi (9)

fo r 0 5 t 5 1/2 . These are the inear perfect codes (such asHamming codes and the (23,12) Golay code), the ir ualcodes,and thedistance4pr im iti ve BCH codes [8, 91. If acode satisfyin g the bound of (9) is used for error de tection inan A RQ system, he P(E ) of thesystemcan be made very

small by using a moderate number of parity bits, say 32.Recently, Kasami,Klove,andLin [13, 141 cons idere d he

ensemble of alleven-weight ( n , k ) linear codes andprovedthat there exist codes with Pe satisfying the following bound,

P, 52-("-k' { 1 + (1 - 24 " - 2(1 - € y ) (10)

for 0 i 5 1/2. This bound is tighter than.the bounds givenby (8) and (9). They also e stablished sufficient conditio,ns forcodes to satisfyheiround. Based on theufficientconditions,hey wereble to provehat the follow ingclasses of codes satisfy the bou nd given by (10):

1) d istance-8 (24, 12) Golay code;2) dis tance-4 Hamm ing codes of lengths 2" - 1 and 2m;3) distance-6 primitive BCH codesof length 2m - 1 with

m L 5 ( X + 1 s a factor of the generator polynomial);4) distance-6 extended p rim itiv e BCH codes of length 2";5) distanc e-8 primitive BCH codes of length 2" - 1 with m

6) distance-8 extended primitive BCH codes of length 2m

Of course, these classes of codes alsosatisfy heboundsgive n by (8) and (9), and he y are good or error detection.For example, he CCITT recom mend ation X.25 forpacket-switcheddatanetw orks [27] adoptsadistance-4Hammingcode with 16 paritybits orerrordetection. The natura llength of this code is 32767 bits ong. In practice, the engthof a data packet s no more ha n a ew housand bits ong,which s much shorter han he natural ength of the code.Consequently, a shortened v ersion of the code is used.

odd and m L 5;

with m odd and m 1 5 .

1 1

Hybrid A RQ Error-Control Schemes

Drawbacks of the A R Q and FEC schemes can be overcomeif the two basicerror-co ntrol schemes areproperlycom-bined. Such acombination of the wobasicerror-controlschemes is referred to as hybrid A R Q [3, 19, 28, 291. A hybridAR Q system consists of a n FEC subsys tem contain ed n anAR Q system. The function of the FEC syste m is to reduce thefrequency of re transmi ssion by correctin g the error patternswhichoccurmostrequently. Thisncreaseshe system

throughputerformance.However, when a less-frequenterror pattern occurs and s detected, the receiver requests aretransmission ather hanpassing the unrelia bly decodedmessage to the user. This increases the system re liab ility. Asa esult,apropercombinatio n of FEC and AR Q provideshighereliabilityhan n FEC system alonend higherthroughput than a system with ARQ alone.

Hybrid A RQ schemes can be classified into two categories,nam ely type-l and type-I1 schemes [3, 291. A straightforwardtype-l hybrid ARQ scheme uses a code which s designed forsimultaneouserrordetectionandcorrection [l-51. When areceived word is detected in error, the receiver first attemptsto correct the errors. If the number of errors (or the length ofan errorurst)s ithin theesigned error-correcting

capa bility of the code, the errors will be corrected and hedecoded message will be delivered to the user or saved inthebufferuntil it is eady to be passed to the user. If an

uncorre ctable error pattern s detected, he receiver rejectsthe receivedwordndequestsetransmission. Theretransmissions he same codeword. When the retrans-mitted codew ord s received, he receiver again attempts to

correct the errors (if any). If the decoding s not successful,th e receiveragain ejects he received word and equestsanother retransmission. This continues until the codeword iseithersuccessfully eceivedorsuccessfully decoded. Errorcorrection may be included in an y typ e of ARQ scheme. Sincea code is used for both error correction and detection n atype-l hybrid AR Q system, t requires more parity-check bits

thanacode used only orerrordetection napure A R Qsystem. As a result, he overhead or each transmission sincreased. When the channel error rate is low, a type-l hybridAR Q systemhas ower hroughput han tscorrespondingAR Q system. However, when the channel error rate is high, atype-l hybrid ARQ system provides higher throughput than itscorresponding A RQ system, because its rror- corre ctioncapability reduces the etransmission requency, as illus-trated in Fig. 8.

Type-l hybrid A RQ schemes are best suited for com munica-tions systems n which a airly constant evel of noise andinterferencesanticipatedonhe hannel.nhis case,enough error correctio n can be designed into the system to

correct the vast majority of received words, thereby greatly

reducing he number of retransmissions and enhancing thesystem performance. However, or a nonstationary channelwhere the bit error rate changes, a type-l hybrid ARQ scheme

has some drawbacks. When the channel bit error rate s ow(for xample, atellite hanneln oodweather),hetransmission s smooth and no (or ittle) error correction sneeded. As a esult, the extraparity-checkbits orerrorcorrection ncluded n each transmission represent a waste.When thehannelseryoisy,he designed error-correctingcapabilitymay become inadequate. A s a esult,the frequency of retransm ission increases and hence reduces

December 1984-VO1.22, No . 12




the hroughput.Several ype-lhyb rid ARQ schemes usin g

either block or convolutional codes have been proposed andanalyzed [28, 30-361.For achannelwithanonstationarybi terror ate, one

would ike to design an adaptiv e hybrid ARQ system. When

the chann el is quiet, the sys tem beh aves jus t like a pure ARQsystem, with only parity-check bits for error detection beingincluded in each transmission. Therefore, thehroughputperformance s he sameas that of apure ARQ system.However,when he hannel becomes noisy ,extraparity-check bits are needed. This concep t orms he basis o f thetype-ll hybrid ARQ schemes. A message in its first transmis-

sion s coded with parity-check bits for error detection only,as napure ARQ scheme. When the eceive rdetects hepresence of errors in a received word, it saves the erroneous

word n a buffer, and at h e same time requests a retrans-mission. The retransmission is not the original codeword butablock of parity-checkbitswhich s ormed based on heoriginal message and an error-correcting 'code. When thisblock of parity-check bits is received, it is used to correct theerrors n the erroneous word stored n the receiver buffer. Iferrorcorrection sno tsuccessful, he eceiver equestsasecond retransmission of the NAK'ed word. The secondretransmissionmay be eithera epetition of theoriginalcodeword or another block of parity-check bits. This dependsonheetransmissiontrategyndheype of error-correcting code to be used.

The concept of parity retransmission or error correctionwas irst ntroducedbyMandelb aum [37]. The first ype -ll

hybrid ARQ usingarity-retransmission trategywasproposedbyMetz ner [38, 391. Metzner's scheme was aterextended andmodifiedbymanyothers [29, 40-491. Amongall the type -ll hybr id ARQ schemes that have been reported,the scheme proposed by L in and Yu [29] is the most an alyzedand fo rms the b asis for th e schemes repo rted in [41-491. TheLin-Yu scheme provides bo th high system reliability an d highthroughput performance.

WithLin-Yu's ype-llhybrid ARQ scheme, tw o codes areused; one is a high rate (n ,k ) code Co which s designed forerrordetectiononly, and theother s a half-rate nvertible( 2k ,k ) code C1 whichsdesigned orsimultaneouserror

December 1984-Vol. 22 , No. 1 2IEEE Communications Magazine 1 2

correction and error detection [l-51, for instance, Correcting tor fewer errors and simultan eously detectin g d (d > t) orfewer errors. A (2k,k) code is said to be invertible if, knowingonly the k parity-check its of a odeword,he orre-sponding k information bits can be uniquely determined by

an nversionoperati on [3, 291. That is osay, heparity-check section contains he same amount of informatio n asthe message section.

When a message D of k informationbits s eady ortransmission, it is encoded into a codewo rd (D,Q) of n bits

basedon he error-detecting code CO, here Q denotes then -k parity-check bits. The codeword (D,Q) is then transmitted.At he same time, he ransmitte rcomputes he k par i ty-check bits, denoted P(D) , based on he message D and hehalfatenvertible (2k,k)ode C1. Thus, (D,P(D)) iscodeword in CI. he k-bit parity block P(D) is not transmittedbut stored n the retransmission buffer of the transmitter forlater use.

Let (b.0) denote theeceivedwordorresponding to

(D,Q). When @.a) is eceived, he eceivercomputes hesyndrome of (0.4) basedon CO . If thesyndrome szero,then b is assumed to be error ree and will be accepted bythe eceiver. If thesyndrome snonze ro, he presenceoferrors n (b,d) s detected. The erroneous message b is thensaved in theeceiver uffer, nd NAK is sent to thetransmitter.Upon eceiving his NAK, the ransmitter en-codes he k-bit parity block P(D) into a codeword (P (D) ,Q( ' ) )of n bitsbasedon heerror-detecting code C O , where Q ( ' )denotes the n-k parity-check digits for P(D) . Then the p arityword ( P ( D ) , Q ( ' ) ) is ransmitted. Let (P (D) ,@l ) ) denotehereceived word corresponding o (P (D) ,Q( l ) ) ,When (P(D),@l)) s

received,ts yndromes omputed ased n C O . If thesyndrome s zero, then P(D) is assumed to be error free andthemessage D is ecovered rom P(D) by nversion . If thesyndrome s nonzero, then P(D) and the erroneous messageb (stored n he receiver buffer) ogether are used for errorcorrectionbasedon hehalf-ratecode CI. f theerrors n(b,p(D)) formorrectable rror attern,heywill becorrected. The decoded message D is hen accepted by hereceiver and an ACK is sent to the tran smitter. If the errors in(b,P(D)) form a detectable but noncorrectable error pattern,then s discarded, he erroneous parity block P(D) is storedin the receiver buffer, and a NAK i s sent to the transmitter.

Upon receiving he second NAK for he message D, thetransmitteresendsheodeword (D,Q). When (b,a) isreceived, it s syndrome is again computed based onCO.f thesyndrome szero, b isassumed to be error ree and_ isaccepted by the receiver, and the erroneous parity block (D )is then discarded. If the s ndrome is nonzero, then D and theerroneous parity block 8 0) (stor ed n the receiverbuffer)together are used for error correctio_n based on CI.f elrorcorrection sno tsuccessful, hen P(D) isdiscarded, D is

storednhe eceiverbuffer,anda NAK is sent to thetransm itter. The next retransmission will be the parity word(P(D).Qfl)). Therefore, theetransmissionsrelternaterepetitions of th e parity word ( P ( D ) , Q " ) ) and the nformation_codeword (D,Q) . The receiver stgres the received message Dand he received parity block P(D ) alternately. The retrans-missionscontinueuntil themessage D is inally ecoveredeither by nversion or by decoding. The throughput behaviorof the above type -ll hybridARQ scheme in the selective-repeatmode is llustrate d n Fig. 8.

The alternateparity-data etransmissionstrategycan beincorporatedwith ny of thehree asicypes of ARQschemes and their variations. It is particularly effective when



it is used in conjunction w ith SR A R Q . Type-ll SR hybrid A R Q

schemes using he parity-data ransmission strategy and afinite eceiverbufferhave been proposedandanalyzedbyLin and Yu [29] and Wang and Lin [43]. Lin and Yu showedthat, even with a receiver buffer of size N, their ype-ll SR

hybrid A R Q schemeanchieveheamehroughputperformanceas the ideal SR A R Q withan nfinite eceiverbuffer, y sing alf-ratenvertible ode C1 whichsdesigned for correcting only a few errors, say t = 4 or 5. fC1 is designed to correct more han 5 errors, heirscheme

can provide much higher hroughput han the ideal SR ARQwith an nfinite receiver buffer, as show n n Fig. 9. W e seethat there is a substantial trade-off between a small amountof error correction and a large amount of buffer storage. Forlargervalues of t, the throughputcurveswill ollow thegeneral characteristic for the type-ll hybrid ARQ illustrated inFig. 8. Linand Yu alsoshowed hat heirhybridschemeprovides the same order of reliability as a pure A R Q scheme.

The decoding complexity for the type-ll hybrid ARQ schemeusingalternateparity-data etransmissionan dahalf-rateinvertibleodes nlylightly reaterhanhat of acorrespondingype-lhybrid A R Q scheme wit h .the amedesigned error-correctingapability. The ext raircuitsneeded aren inversionircuit basednhe half-rate

invertible code C1 , which is simply a linear sequential circuit,andanerror-detectioncircuitbased on the error-detec tingcode CO.

Type-ll Hybrid ARO Schemes Using Convolutional Codes

The alternate arity-dataetransmissionype-ll ybridARQ schemecan be incor porat ed with a rate-112 convolu-tional codeusingeitherViterbiormajority-logic decoding[44, 45,47-49].The error-detectingcode CO is stil l an ( n ,k )

block code. However, the error-correcting code C1 is a half-rate (2,l,m) convolutional code with memory order m [3]. Inthe followin g, we will escribe cheme singViterbidecoding.

Let G1 X ) and G 2 ( X) be the wo generator polynomial s ofthe hal f-rate convoluti onal code C1. When a data sequence ofk bits s eady or ransmission, it is irst encoded ntoacodeword / ( X ) in Co . The information sequence / (X ) is thentransf ormed nto two sequences, P1 ( X) = I ( X ) G 1 ( X) and Pz(X)

= / ( X ) G 2 ( X ) , each n+m bits ong. Note that the2.(n+m)-bitsequence obtained by nterleaving PI (X ) and & ( X ) is a codesequence for / ( X ) based on the half-rate convoluti onal code

C1.The sequence P l ( X ) = I (%)Gl (X) is then transmitted to thereceiver and the sequence P2(%) is stored n the transmitterbuffer or possible retransmis sion at a ater ime. Let Pl(X)denote he received sequence corresponding to Pl(X). When

/?(X) is received, it is divided byGI(%) . Let ~ L ( X ) nd f i$t) bethe quotient and remainder, respectively. If R1(X) = 0, /1(X) is

checkedasednherror-detectingode CO. If it ssyndrome S l (X ) is zero, r l (X) is assumed to be error-free andidentical to the transmitted nformati on sequence / (X) . Thereceiver then accepts r1(X) (with n-k par ity bits deleted). f&(X) # 0 or S1(X) # 0, errors are detected n Pt(X). Pl(X) isthen saved n the receiver buffer for reprocessing at a atertime.

A t the ameime, the receiver ends N A K to thetransmi tter. Upon receiving his N A K , the transmitter sendstheequence PZ(X) = /(%)G2(X) to the receiverfirstretransmission or / ( X ) ) . Let &(X) be the received sequence

corresponding to P2(X). When A ( % ) is received, it is dividedb y G2(X). Let T2(X) and &(10 be the quotient and remainder,respectively. If /%(X) = 0, / z ( X ) is then checked based on theerror-detecting code Co. If itssyndrome &(X) iszero, henrz(X) is assumed to be eu or free nddentical to thetransmitted sequence / (X) . In this case, the receiver acceptsh ( X ) and discards the sequence A ( % ) . If /%(X) # 0 or SdX) #0, then &(X) togetherwith Pl(X) (which sstored n thereceiverbuffer)are hendecodedbased on thehalf-rateconvolutional code CI sing a Viterbi decoder. Let T*(X) be

thedecodedsequence. p(X) s hencheckedbased on theerror-detec ting code CO. If he syndrome Sf%) is zero, henthe receiver accepts p(X). f ts syndrome S ( X ) is not zero,then A ( X ) is discarded, &(X) is stored n the receiver buffer,and he receiver sends another N A K to the transmitter. Thenext etransmission will be PI(%). The alternate etrans-missionsof &(X) and P2(X) continueuntil / ( X ) is inallyrecovered.

For receiver buffer size N , the throughput efficiencies ofthe abo ve hybrid A R Q scheme for block ength n = 1024 andvarious memory orders m are shown n Fig. 10. We see thatthe chemeoffers ignificantlybetterhroughputperfor-mance than pure SR A R Q with nfinite receiver buffers whenthe channel bit error rate s high.

One disadvantage ofhe abovechemeshathethroughput efficiency drops rapidly to 0.5 when the channelis noisy enough to req uire retransmi ssions. A refinement ofthe above scheme, first ntroduced by Lugand and Costello[45], uses higher-rate onvolutiona l odes to achievehigher hroughput when the channel s noisy, at a cost ofsome ncreased receiver complexi ty.

13

December 1984-Vol. 22, No. 12




In the Lugand-Costello scheme, the error-de tecting code CO

issti l lan ( n , k ) block code. However, heerror-correctingcode isa (3,2,rn) con volu tion al code [3], denoted C3 withgenerator matrix

G(X) = pr ]631 ( 8

GE x ) 632 (X )

After a data sequence of k bits s encoded into a codewo rd/ ( X ) in C o , ( X ) is ransfo rmed nto wo sequences PI(%) =

I ( X ) G l ( X ) an d P31(X) = /(%)G31(%). Note that he sequenceobtained ynterleaving Pl(X) and is code sequencefo r / (X) based on hehalf-rateconvolutional code CI with

generator polynomials G1 X) and G31 (X ) . The sequence PI (X)is then transmitted to the receiver, while P31(X) is stored nthe transmitter buffer for possible use at a ater time. When? I ( %) is received, he same procedure s ollowed as in thehalf-ratecase. If errorsare detected in h ( X ) , then A ( X ) issaved in the receiver buffer, and the receiver sends a NAK lmessage to the tran smitter. Upon receiving a NAKl message,the ransmitter switches to transforming codewords J(X) inCO into he wo sequences &(X) = J(X)Gz(X) and P32 =

J(X)G32(X). These sequences, when nterleaved, orm a codesequence for J(X) based on the half-rate convolutional codeC2 withgeneratorpolynomials G2(X) and & ( X ) . The se-quence P2(X) is ransmitted to the eceiver,and & ( X ) isstored n he ransmitter buffer. When &(X) is received, he

same procedure s again ollowed. If errors are detecte d nh ( X ) , a NAK2 message is sent to the transmitter and & ( X ) isstored. Upon receivinga NAK2, the rans mitter sends thesequence P d X ) =. P31(X) + P3E(X) = /(%)G31(%) + J(X)G32(X) tothe receiver (the first retransmission for both / ( X ) and J(X)).Note that the sequence obtained by nterleaving &(X) , Pz(X) ,

and & ( X ) is a code sequence fo r / ( X ) and J(X) based on thetwo-thirds-rate onvolutional code C3. When &(X) ise-ceived, hen &(X) togetherwith & ( X ) and &(X) (whicharestored n the receiver buffer) are decoded based on the codeC3 using a Viterbi decoder. Let T* (X) and p(X) e the decodedsequences. 'They rehen othhecked based onhe

December 1984-Vol. 22,o. 12IEEE Communications Magazine 1 4

error-detecting code C O . If bothsyndromesare zero, thereceiveraccepts I*(%) and J * (% ) . If both yndromesarenonzero, & ( X ) is discarded, &(X) and &(X) are stored, andthe receiver sends a NAK3 message to the transmitte r. Thisinstructs the transmitter to resend PI(%) . Retransmissions ofP l ( X ) , & ( X ) , an d P3(X) continue until both / ( X ) and ,/(X) arefinally ecovered. If only one syndromes zero, severaloptions are available. These include subtracting the effect ofthe decoded sequence from A ( % ) , and then trying to decodethe other sequence based on one of the half-rate codes C I or

C E . A ull discussion and analysis of hese options can befound n [49].

Although the above scheme requiresmore omplexretransmissionprotocol,amorecomplex decoder (for thesame undetectederrorprobability),anda onger eceiverbuffer han hehalf-rate scheme, its hrough putefficiencydrops to only 0.66 when the channel becomes noisy enoughto require etransmissions.Asimilar scheme using hree-

fourths-ra te onvolutio nal codes maintain shroughputefficiency of at least 0.75 [49]. The throughput efficiencies ofthe three schemes discussed,al lusingamemoryorder3code, are compared in Fig. 11.

Type-l Hybrid A R O SchemesUsingConvolutional Codes

On channe ls hereairlyonstantoiseevels

anticipa ted, type-l hybrid ARQ schemes can offer a throug h-put advantage over type-ll schemes (see Fig. 8). s well as asimpler protocol. Type-l schemes using convolu tional codeshave been proposedbyseveralauthors [33, 34,50-521. Inthissection, we will eview wo ecen t ype-l schemes foruse inpacket-switchingne two rks [34, 521. Both of thesemake use of convolutional codes withsequentialdecoding

[2, 3,51.Inonventionalequentialecoding of convolutional

codes, ecoding roceeds ntilheeceived message iscomplet ely decoded or an erasure s declared . Erasures are

normallydeclaredwhendecodin g ime becomes excessive.



In type -I ybrid ARQ sequen tial ecoding,hisime-outcondi tion can be used a s a signal to request a retransmis-sion. Assume that a data sequence / ( X ) , ready for transmis-sion,s encoded intoodeword V(X) inn (n,k,m)convolutional code C I . (X ) is hen ransmittedover hechannel,and he eceived sequence P ( X ) is decoded byasequential decoder. If decoding is completed within the time-out limi t, he decoded sequence P(X) is accepted by thereceiver.Otherwise,a etransmission s equested.This scalled t h e t i m e - o u t a l g o r i t h m (TOA) [52]. he time required to

decode providesanaturalerror-detectingmechanismwithsequential decoding. When the received sequence is not ver ynoisy,ts decoded quick ly, and noetransmissionsnecessary. When the eceived sequence isnoisy,however,decoding takes a onger time due to the tree-searching rulesofheecoder. If theime-outimits exceeded, aretransmission s requested.

Since each decoded branch in the code resu lts n k databits being delivered to the user, the throughput efficiency ofthe TOA assuming an deal SR protocol s defined as

where CA is the average number of comp utations the decoder

performs per decoded branch, i n c l u d i n g e t r a n s m i s s i o n s . Inthenoiselesscase, Ca = 1 and qToA= k / n , thecode rate.Note that, since the number of computa tions performed by a

sequential decoder is a random variable which depends o nthe noisiness of the received sequence, CA must be computedas a statistical average. The existence of the optimum time-out limit which minimizes CA and hence maximizes TOA hasbeen sho wn naly tica lly , and has lso been ver ifie d ycomputer simulations [34].When the optimum time-out imitis used, CA increases nly lightlywithncreasing codememoryorder m. Since theundetectederrorprobability Pe

is n xponentially ecreasingunction of m [34], argememory orders can be used to achieve extremely small un-detected error probabilities without reducing the throughput.

A more sophisticated approach to detecting errors can beused to further improve the throughput efficiency. The metricof the best path n a sequential decoder is monit ored. Whentheslope of thismetric becomes to o negative, hat s, hemetric of the best path s decreasin g too rapidly, decodingstops and a retransmission s requested. As in the TOA, themetric of thebestpathprovidesanaturalerror-detectingcapabili ty. f the received sequence isno tverynoisy, hemetric along he best path will end o ncrease at a airlyuniform ate,withonlyoccasionaldips. The slope of thismetric,averagedoveranappropriatenumber of branches,will then remain positive. However, f the received sequenceisnoisy, he decoder may ollowan ncorrectpathwhosemetricdeclinesover everalbranches . The slope of the

metric will then turn negative, and may exceed a hreshold,thereby triggering a retransmission request.

This s called he s l o p e c o n t r o l a l g o r i t h m (SCA) [34]. headvantage fhe SCA ishatt anecognize oisyreceived sequence as soon as erroneous bits are processed.by the decoder, rather than waiting for the time-out imit tobe exceeded. This reduces C A , and hence increases hrough-put efficiency,compared to the T O A. The existence of anoptimumhresholdnhelopefhemetric, hichminimizes CA and hence maximizes the throughput efficiencySCA of the SCA, has been shown analy tical ly and verified by

computer simulation [34]. igure 12 compares the throughputefficiencyfhe TOA and SCA, bothperating underoptimumconditions,overa angeofbit-error atescorre-sponding oaverynoisychan nel. The code used in thecomparison was a (2,1,11) convolutional code with optimumdistance pro perties for sequ ential decodin g. Clearly, the SCAhasasmaller Ca , and hence a la rger hrough put , than theTOA. Therefore, he SCA provides a very effici ent means ofobtainingahighhroughput and aow undetected rrorprobability on noisy channels with relatively constant errorstatistics.

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10-14,1977.

1468-1496, NOV.1978.

Shu Lin was born n Nan king , China, on May 20, 1937. He receivedthe B.S.E.E. degree from National Taiwan University, Taipei, T aiwan,China, in 1959, andhe M S . and Ph.D. degrees in ElectricalEngineering from Rice University,Hous ton, TX, in 1964 and 1965,respectively.

From 1965 to 1981, he waswith heDepartm ent of ElectricalEngineering,University of Hawaii,Honolulu. n 1982 he joined he

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Depa rtment of Electrical Engineering, Texas A&M University, CollegeStation . n 1983, he rejoined he University of Hawaii where he is aprofessor. He spenthe 1978-1979 aca dem ic ear s isitin gscientis t at he BM Thom as J. Watson Research Center, Yorktow nHeights, NY, where he workedonerror-controlprotocols ordatacommunica tions systems. From 1976 to 1978 he serv ed as associateeditor or AlgebraicCodingTheory. He haspublishednumeroustechnicalpapers in IEEE Transactionsandother efereed ournals.He ishe uthor f Anntroductionorror-Correcting Codes(Englewood Cliffs, NJ: Prentice-Hall, 1970). He also coa utho red (withD. J. Costello) hebook, Error ControlCoding: undamentalsan dApplications (EnglewoodCliffs, NJ: Prentice-Hall, 1982). Hiscurrentresearch nterests ncludechannelcoding,error-controlprotocols,and satellite communications.

Dr. Linsmember of theBoard fGovern ors of the IEEE

of the IEEE.Infor matio n The ory Group, is a member of Sigma Xi, and is a Fellow

Daniel J. Costello, Jr.. was born n Seattle, WA, on August 9,1942.He receive d the B.S.E.E. degree fro m Seattle Unive rsity, Seattle, WA,in 1964, an d he M.S. and Ph.D. degrees in ElectricalEngineeringfrom the University of Notre Dame, Notre Dame, IN , in 1966 and 1969,respectively.

Inhe ummer f 1966, he served s n ssociateesearchengineer at the Boeing Aerospace Division, Seattle, WA. In 1969,hejoined the faculty of the Illinois Institute of Technolog y, Chicago, IL,as an assistant p rofessor of Electrical Engineering. He was promotedto associate profess or in 1973, and to full p rofe sso r in 1980. He sp entthe summer o f 1971 as a esearch associate at Cornell University,Ithaca, NY, and was a visiting professor at he University of Notre

Dame durin g the 1983-84 ac ademic year. In addition, he has servedas a professional consultant for Western Electric, Illinois Institute ofTechnology Research Institute,otorolaommunications,ndDigital Transmission Systems.

Dr. Costello has been a Member of the IEEE since 1969, and waselectedaSeniorMember in 1978. He belongs o he nformationTheory Group and he Commu nication s Society . Since 1983, he hasbeen a member o f the In formation Th eory Group Board of Governors,and in 1984 was elected second vice-president of he BOG. He hasserved as an associate editor of Communication Theory for the /€€E

associate editor for Coding Techniques for the /E€€ Transactions onTransactions nCommunications, and ince 1983 has been an

Information Theory.Dr. Costello'sesearchnterestsre in therea o f digital

communica tions, with special emphasis on coding heory, nforma-tionheory,multiuser ystems, ommunications etworks, rrorcontrol, nd pread-spec trum ommunica tions. He has uthored

over 50 technic al public ations n his ield, and n 1983 coauthored(with Shu Lin)a extbookentitled Error ControlCoding: unda-mentals and Applications. He has served as principal investigator ontwelve esearchgrants,andasanassociate nvestigatoron woothers. He has also supervised en Ph.D. dissertations at the IllinoisInstitute of Technology.

Michael J. Miller receive d he B.E. hon ors and the graduate Dip.Ed.degrees from the University of Adelaide, Adelaide, South Australia in1961, and he M.Sc.(Eng.) degree fro m Queen's Univers ity, King ston ,Ont., Canada in 1973. He received the Ph.D. degree fro m the Univ ersityof Hawa ii, Hono lulu, n 1981.

He was w ith the A ustralian Postm aster General's De partment from1956-1966, in iti al ly as a cadet engineer and subsequen tly as a radioengineeresponsibleoradioelayystemevelopmentndoperations.Since 1966, he has been on he aculty of theSouthAustra lian nstitute of Technology, Adelaide, Australia, where he is

currently an associate professor. n 1980 he spent six months as avisiting esearch colleague at he University of Hawaii. His currentresearch interests are i n the fields o f error control coding and digitalradio transmission.

Dr. Millers

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