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Packet Combining Error Control for CDMA SystemsSlim Souissi and Stephen B.Wicker

School of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332-0250

Ahtract- Two examples of the application of p d e t combiningto CDMA are presented and analyzed: an uynduonolu DS/CDMAsystem supported by an averwed divcni ty combining scheme and asimilarsystcmuaingcodecombining. The ADC schemeuses a Viterbidecoder combined with a CRC code to generate retr wmissi on r equ"(., while the CC &e employsa aequcntid decoder operatingwith the time-out dgoritbm.Multiple received w e t s ue combinedto forma single.mom nliable &st. T h e error correcting decoderoperata on the combined packet, U opposed to the mcat ruenllyreceived individual packet (e.g. u in a type-I hybrid ARQ proto-col), substant idy inc-ng the probsbilityofacceptana with eachadditional transmission. We &o w through numerical andysis thatboth technigua provide a significant increase in the CDMA systemcapacity, rhroughput and reliability.

I. I N T R O D U C T ~ O NCode division multiple access (CDMA) systems are veryattractive for mobile communication applications becauseof their efficient use of the channel, their inherent resis-tance to fading, and their allowance for simultaneous dig-ital transmissions by a large community of relatively un-coordinated users. Unlike FDMA and TDMA, which areprimarily bandwidth l imited, CDMA is only interferencelimited. Any increase in the number of simultaneous trans-missions converts directly and proportionally into an in-crease in the effective channel noise level at the input toeach receiver. One app ropriate way for com batting channeldegradation in CDMA systems is the use of forward errorcorrecting (FE C) techniques. Despite their good through-put performance, FE C schemes have l imited reliabil i ty per-formance. Such a handicap is surmounted by properlycombining an FEC scheme with an ARQ scheme into atype-I hybrid ARQ protocol. Th e ARQ portion providesvery high reliability while the FEC portion, by correctingthe most likely error patterns, reduces the frequency of re-transmission requests [6].When type-I HARQ protocols are used in very noisy en-vironments, a large number of unreliable data packets arerejected. In order t o make use of th e information providedby these noise-corrupted packets, Chase [l ] has suggestedthat the packets be combined t o form a single, more reliablepacket. Such a scheme is adaptive in that only the mini-mum num ber of packets necessary to form a reliable packetis transmitted; the average number of retransmissions isminimized, allowing greater throughp ut while maintainingthe reliability of the accepted data.

In this paper, we analyze the performance of adaptiverat,e coding techniques in direct sequence CDMA systems.This work w w supported by National Science Foundation Grant SCR-9016276

Two methods of packet combining are considered: Av-eraged diversity combining (ADC) and Code Combining(CC). Th e comparison of the two schemes is based on ana-lytically derived performance bounds. It is shown th at theADC approach outperforms the CC method.11. DESCRIPTIONOF THE S Y S T E MThe network under consideration consists of numerous sub-scribers communicating via a single channel with one ormultiple base stations. We assume t hat during each packettransmission, the number of users sharing the channel isfixed, In pract ice th i s can be ach ieved by se t t ing t i m 4 o tintervals during which transmissions are allowed. An asyn-chronous direct sequence CD MA system model similar to

that described in [4] is used to characterize th e multiple-access channel. Gold sequences (21 with good correlationproperties are used to spread the d ata '.As opposed to conventional ARQ implementations wherethe decoder operates on each received codeword withoutregard to earlier received erroneous words, this paper as-sumes th at previously received copies of each d at a packetare available in the decoder buffer. Th e buffer is assumedto have sufficient length to prevent overflow. In the re-mainder of the section, we describe in more detail the di-rect sequence CDMA system aswell as the two m ethods ofpacket combining.A . Th e C DMA sysiem model

We consider the Spread Spectrum Multiple Aceeas sys-tem model that was proposed by Pursley [4]. It is one ofthe most frequently used models for the analysis of asyn-chronous direct sequence CDMA systems. Th e resultingtransmitted signal fo r th e kth user is given by

s& ) = d G a k ( t ) d t ( t ) c o s ( w , t + eb) , (1)where & ( t ) an d ak(1) represent, respectively, the transmit-ted sequence of dat a bits and t he spreading code sequence( d i ( t ) an d a t ( t ) E {-1, l ) ) , is the phase angle for thek" user, P is the common signal power, and fc = 2nw ,is the common carrier frequency. st. (t) is passed throughthe channel where it is combined with the other signals inthe system. Assuming IC active transm itters, the received

'These sequences arc required to have very low cross-correlationfunc-Th e autocorrelation funct ion must be very sharp to gumanteetions.proper acquisition

0-7803-1750-5/94/$3.00 8 1994 IEEE

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Figure 1: A CDMA system receiver

X(')= (zy', z!), ...,z$)): i f h transmitted copy of thesame da ta packet .Y ( * )= (y:), y;), ...,~2) ) :i f hreceived copy of the samedat a packet.E'') = (zy),$),...,$)): combined packet formedby L copies of Y ( ; ) .

T h e transmision/retransmissionprotocol adopted for thisscheme uses two codes: a CR C code COfor error detectionand a convolutional code C1 for error correction. A noise-less feedback channel is available 80 t ha t the t r ans mi t te -is reliably informed of the s tat us of each decoded packet.When a message I of k bits is ready for transmission itis first encoded for error detection using CO,and then en-coded for error correction using C1 . Th e resul t ing N-bitcodeword X(') is t ransmit ted. Th e receiver at temp ts todecode this first copy Y ( l ) . If the packet is recognized tohave a correctable error pattern then it is acknowledged;otherwise, a retransmission request is sent to th e transmit-ter. Th e transm itter responds to retransmission requestsby sending another copy ~ ( 2 ) .The receiver then attemptsto decode the new received packet ~ ( 2 ) .If the packet isdecoded correctly, then it is acknowledged, the decodedpacket is sent to the da ta s ink, and al l copies of the re-ceived packet are discarded. Otherwise t he receiver com-bines the packets in the buffer along with the most recentlyreceived copy and the decoder a tte mp ts to decode the com-bined packet. If the combined packet cannot be decodedreliably, then a retransmission is requested and the trans-mit ter sends another copy X ( 3 ) .T he receiver continues torequest and com bine packets until a decoding operation issuccessful.

Here 7 4 t ) is a white Gaussian Process with a tw*sidedspectral density % an d rt is the t im e delay of th e k f huser.B i an d r k are two independent, uniformly distributed ran-dom variables. The re is no loss in generality in consideringonly the first user's receiver. As shown in figure 1, th edecision variable at the output of the in tegrate and dumpfilter is given by

z (T)= ITr ( t ) a l ( t ) cos (w , t ) . (3)makes a decision as to whether +1 or -1 was

z (T)can be expressed(4 )

Th esent by examining the sign of[41 z(T)= D ( T )+ M ( T )+ I (T )

Th e first term is the desired signal. T he second term repre-sents the contribution of the additive white Gaussian noiseto z(T);it is a Gaussian random variable with mean zeroand variance equal to N 0 / 4 . Th e 1 s t t erm r ep res en t s theinterference from the other (IC - 1) users. Th e signal tonoise ratio at the input of the first user's receiver, aver-

Assume that L copies, {Y ( i ) } ,have been received andrecognized to be unreliable. Thes e packets are combinedinto a single packet, $'), by taking the average of the softdecision values of each repeated packet symbol,

(8)l Laged over all possible pha se shifts, tim e delays and d atasymbols, is defined as i= l&L) = - $),-SNR= ~ D(TI2 (5 ) where yj'" = z r ) + TI:), zy)= f l and TI?) is a Gaussian

random variable of variance U: . If we assume that the vari-v a r ( z ( T ) ) ' ance is constant for all i and equal to u2,then v a r ( e ) ) =0 2 / L , ( j =A Gaussian approximation on the probability of bit erroris given by [4] ,,.

PE = Q(&%), (6 ) C. Code combining with sequential decodingwhere the function Q( . ) is the Gaussian in tegral functiondefined as

In sequential decoding, th e most PoPu~ ar algorithm isdu e to Fano [6]. When the channel is d iscrete and mem-oryless (DMC), the Fano branch metr ic at node i is given

(9)(7 ) by1 "Q(+ )= /i e-('/')y'dy.

B. The Averaged Diversity Comb iner (AD C)We first introduce the following notation: where R is the code rate, t i is the i l h channel input symboland r; is the corresponding received symbol. Th e path

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metric through node m is simplym

i =OLe t C denote the num ber of com putations required to de-code the first j information symbols. Under the t ime-outcondition, a packet is assumed t o be in error if the decodingtime exceeds a certain threshold T,,,. Since the decod-ing t ime is proportional to the number of computations,a packet is declared unreliable if the number of computa-tions C exceeds a certain threshold Cm,,,.Th e probabili tyof decoding failure (erasure) of a packet is therefore

P ( F )= P ( C L G a s ) CPCL , (11)where k is the number of information bits per packet, 0 isa constant less than or equal to one and a is the Paretoexponent. Th e Pareto exponent is given by the implicitsolution of

(12)E& ) - R,- -PE& ) is the Gallager function given by

where j an d i are the inpu t and o u tput channel symbols ,respectively, and P ( i ) is the probability assignment of in-put letter k. For a system using FEC/ARQ protocols, theprobability of retransmission for a given packet is equal toIn [3], the m ethod of packet combining was applied to asystem using convolutional encoding and sequential decod-ing. When L copies of the sa me d at a packet are receivedand declared unreliable, they a re combined and the result-ing packet decoded. At this po int, the encoder is viewedfrom the receiver as double-staged. The first stage is theordinary convolutional encoder and the second stage is anL-repetition encoder. A k-bit information packet I=(i l ,iz_. ik ) is viewed as first encoded into an N-bit packet

X = ( z l , z 2 .. z , ~ )and then in to an L N-bi t packet X ( L )=( z ~ ) , z ~ . . z ~ ) ;...;zj), z j2 ) . . z jL) ;...;zc),z$)..zE)),basedon the L-repetition code. T he resulting packet is transm it-ted over the channel. T h e received packet is

P ( F ) .

where yjil denotes the j l h bit of the ithreceived copy-. Thecliannel may also be viewed as a DMC apparent channelwliicli produces for each input repetition-symbol (z j,zj...zj) L output letters (y,!),~,!)_ _ _ yjL) ) . In order to applythe fundamental sequential decoding results to the sequen-tial decoder w ith L-r epetition-co de com bining, we begin bydefining the L-repetition Fano metric as

The num ber of compu tations per decoded bit st i l l followsthe Pa reto distribution given in equation (11).The appar-ent Pare to exponent QL relative to the L-repetition channelis given by --E Y ( Q L ) - R ,QL

where E L L ) ( a ~ )is the Gallager function of the apparentchannel given by

E i L ) ( n r )=

(16)L is the number of symbol repetit ions and m an d j re p rasent the channel input and o utpu t symbols respectively.

111. APPLICATIONTO C D M AWe consider the two meth ods of packet combining (ADCand CC ) in conjunction with a direct sequence CDMA sys-tem . For the case of the BS C channel, th e channel crossoverprobability is given by equation ( 6 ) an d for th e c a ~ eof theAWGN channel, the multiple access noise is assumed t o beadditive Gaussian with noise variance given by u a r ( r ( T ) ) .A . CDMA with Averaged Divers i ty CombiningTh e input t o the averaged diversity combiner is the de-cision variable* z given by equation (3). Le t Y(i ) = ( y f ) ,y f ) , __ ., be the it* received copy. Each bit yy isassociated with its individual decision variable 27. T h ecorresponding set of decision variables Z()= ( z y ) ,z t , ..,&I)is formed. The averaged diversity combining processconsists of taking the average over all the sets of decisionvariables Z ( ) t o fo rm a single set of updated decision vari-ables

The combined packet = ( c y , c,, ...,.E) is nowformed by the action of the decision device (see figure 1)on the decision variables 5.Wi t h o u t loss of generalitywe consider the first bit of the combined packet. Since theCDMA system is asynchronous, the random variables zli)( i = 1 ...L ) are statist ically independent. A ssuming thatthey all have the sam e mean and variance (i.e. given byvar($)) ) then the variance ofe)isThe effective signal-to-noise ratio after L packets are com-bined is S NR(L)= L .SNR,

We omit th e bit period T ~n the notation z ( T ) .

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where SNR represents the signal-to-noise ratio withoutcombining. The averaged diversity combining of L packetsis equivalent to an increase of the effective received SNRby a factor of L.Throughput Performanc e: Th e sys tem is a concatenatedconvolut ional encoder lvi terbi decoder pair and an outercliannel uses CRC error detection. T o determine the per-stream at the out put of the Viterbi decoder (i.e. t he in-

upper and lower bounds developed earlier for the BER a tthe outp ut of the Viterbi decoder, we can bound P(RiM).

P(R$YJw)= 1- (1- p())NUP-< P(R$)) 5 P(R& ; j ) = 1 - (1 -pf,,).

coding system in which an inner channel is formed by a (26)Tr,& is then bounded by [3]

formance of this system , we mu st first characterize th e bit m i m1+ n P(RYjo,,,)5 Tr,& 5 1 + EP(R$Lp ) . (27)p u t of the CRC decoder) . Let Ri, dMand RiM de- i = l j = 1 i= liiote, respectively, the events Viterbi-decoded sequencecontains no errors, Viterbi-decoded sequence containserrors tha t are undetectable by the CR C decoder andViterbi-decoded sequence contains errors tha t are detectable The system throughput is defined as th e ratio th e num-ber of illformation bits accepted by th e receiver to th e totalnumber of bits transmitted, It is given byby the CR C decode;. Note th at this last event is equiv-alent to the event a retransmission request is generated.The superscript M denotes t he num ber of combined copieswhich form the sequence at the input to the Viterbi de-coder. It is clear that

P ( R i M ) )+P(RiM))+ P ( R y ) )= 1. (20)I f we denote by p ( M )the effective channel BER at the out-put of the Viterbi decoder given the combination of Mreceived copies of a packet, then

P(R!)) = ( 1 - P ( ~ ) ) ~ , (21)where N is the number of bi ts per t ransmit ted sequence.Since P ( d L ) is quite small for most CRC codes, thisquantity is neglected and P(R&L)is written as

p() is now characterized using upper and lower boundstha t are a function of the convolutional code in use. Whensoft decision Viterbi dec oding is used, p ( M ) is bounded by

(24)where (%)(M) is the effective bit energy to noise ratioalter L packets are combined,

(25)a nd T ( S ,1) is the weight enum erato r for the convolutionalcode i n use. Let Tr,adcbe the expected number of transmis-sions before a packet is accepte d by t he receiver. Using t.lie

(28)RcRCrcBode = -,Ti.adcwhere R, is the rate of the convolutional code (the asymp-totic rate loss due to th e encoder memory is assumed to benegligible) and R,,, i s the r a t e o f the C RC code .B. CC in Conjunci ion wi th CDMAIn order to derive upper and lower bounds on the aver-age number of transmissions (Tr,cc),we use equation (27)in which P(RY) is replaced by P ( F ( L ) )(probability of de-coding failure at the LIh a t t emp t ) . P ( F ( L ) )is determinedbywhere k i s the number of information bits, CmOzis themaximum number of computat ions al lowed to decode thepacket in the time-out alg orith m and IYL is the effectivePareto exponent corresponding t o the combination of Lpackets. a~ depends on the convolutional code rate, thenumbers of users in the CDMA sys tem and the numberof combined packets. In order to evaluate P ( F ( L ) ) ,weneed to determine th e explicit formul a for computing therepet i tion P areto exponent QL. This can b e done bysolving (15). Once T,,,, determined, the data throughputof the system is given by e,, = R/T,+.

P ( F ( L ) )k PLkC;;:, (29)

I V . NUMERICALRESULTSA DS/CDMA system using packet combining has been nu-merically analyzed. Sprea ding is impleme nted through aset of Gold codes of period G5. When averaged diversitycombining is used, error detection is provided by a CRC-12code while error correction is based on a (2,1,3) convolu-tional code with generator polynomials (15J7). At thereceiver a soft decision Viterbi decoder is employed. Th esystem supported by code combining uses the same con-volutional code. T he decoding is based on the time-outalgorithm for which Cma== 4094 computat ions and thepacket size N is equal t o 2047 bits. For this case, upperand lower bounds are very tight and are therefore repre-sented by the same curve.Figure 2 and figure 3 show upper and lower bounds onthe probability of retransmission request by the receiver for

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different values of the num ber of combined packets ( L ) . AsL increases we observe a significant decrease in the proba-bility of packet rejection.Figure 4 shows the throughput of the CDMA sys tem(w ith CC and A D C) as a function of the numb er of activeusers. We note that ADC has a bet ter throughp ut perfor-mance than CC . Actual ly , th is is due the fact that the sof tdecision Viterbi decoder h as bett er error correction perfor-mance than the sequential decoder. Furthermore, when er-ro r detection is based on th e time-out al gorith m, a correctpacket can be declared in error if the maximum number ofcomputations CmOzis reached, while for the ADC schemea packet is first completely decoded and then it is checkedfor the presence of errors. Only erroneous data packetsare retrans mitte d, thus allowing a lower average number oftransmissions and therefore a higher throughput.

VI. CONCLUSIONAn analysis of the performance of two methods of packetcombining in a DS/CDMA sys tem over the AWGN chan-nel has been presented. It has been shown that accept-able levels of throughput and reliability performance areprovided well beyond the point at which a conventionalCDMA system would have collapsed. Th e ADC schemeusing a CR C error detect ing code has been shown to havebetter throughput performance than the CC scheme o gcrating under the time-out algorithm. By employing anadditional error detecting code, ADC restricts retransmis-s ions to the erroneous da ta while the C C sys tem, l imited bythe time-out condition, occasionally requests the retrans-mission of packets that could have been correctly decoded.

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Figure 2: CC: Probability of Decoding Failure per Packet( k 1 0 Users an d Cmnz/L= 2).

\\J l l l

E W ( U 1

Figure 3: ADC: Probabil i ty of Packet Error (K=10 Usersand Packet Size = 2047 bits).(I ] D. Chase, Code Combining- A Maximum-Likelihood Decod-

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1 0 . 6j 0.4~ ~ . ~ ~ ~ . . .o.21 1 $7:-, , , 1. . ing Techniques to a DSfCDMASystem Using ConvolutionalEncoding and Scq uentid De codin g, Infernational Symposiumon Comm~nicalionTheory an d Applicaiions, Ambleside, Ul