collision channel
Giuseppe Caire and DanielaTuninetti
January 22, 2001
Abstract
In next generation wireless communication systems, packet-oriented data transmission
will be implemented in addition to standard mobile telephony. We take an information-
theoreticviewofsomesimpleprotocolsforreliablepacketcommunicationbasedon\Hybrid-
ARQ",overaslottedmultipleaccessGaussianchannelwithfadingandstudytheirthrough-
put (totalbit/s/Hz)and average delayunder idealizedbut fairly generalassumptions. As
anapplication ofthe renewal-rewardtheorem, we obtainclosed-formthroughputformulas.
Then, we consider asymptotic behaviors with respect to various system parameters. The
throughput of ARQ protocols is compared to that of CDMA with conventional decoding.
Interestingly,theARQsystemsarenotinterference-limitedevenifnomultiuserdetectionor
jointdecodingisused, asopposed toconventionalCDMA.
Keywords. Packet radio,informationtheory,multiusersystems,wireless communications.
1 Introduction
In order to support new services (e.g., wirelessmobile access to the Internet), next generation
wireless communication systems will implement packet-oriented data transmission in addition
to standard mobile telephony [60 ]. This implies bursty sporadic communication from a large
populationofusers,thatmayrequireinstantaneouslargedataratesandverysmallerrorproba-
bilitiesforashorttime. Motivatedbytheaboveconsideration,wetakeaninformation-theoretic
Mobile Communications Department, Institut EURECOM, Sophia-Antipolis, France. E-mail:
view ofsome simpleprotocols forreliable packetcommunicationbased on \Hybrid-ARQ", i.e.,
oncombiningchannel coding and Automatic Retransmission reQuest(ARQ)[48 , 6].
Several types of Hybrid-ARQ protocols have been proposed (see [48 , 6] and references
therein). The throughput of ARQ schemes can be improved by packet combining, i.e., by
keepingtheerroneous received packets and usingthem fordetection. Packetcombiningcan be
based on hard decisions [36 , 16 , 43 , 51 , 50 ] oron soft channeloutputs [8, 3, 45 ]. In the latter
case, several noisy observations of the same packet obtained by retransmission are combined
by a suitable diversity technique (e.g., maximal-ratio, equal-gain or selection combining [29 ]).
Softdecodingofmaximal-ratiocombinedpackets can beseen asan elementaryformof Hybrid-
ARQ, basedon soft-decodingof arepetitioncodeof variable length. Thisideacan beextended
to more general classes of codes. In [22 , 45 , 46 ], Rate Compatible Punctured Convolutional
(RCPC) codes are used inan incremental redundancy ARQ scheme. Transmissionstarts with
the highest rate code of the RCPC code familyand additional redundancybits are requested
whenevernecessary. Morerecently,Turbo-codes[5 ]havebeensuggestedascandidatesforpacket
combining,exploitingthefactthattheyaresystematicandproduceincrementalredundancyby
puncturingthe paritybits[62 , 25 ].
AnalysisofHybrid-ARQprotocolsintermsofthroughput,errorratesanddelaycanbefound
in[34 , 33 , 31 ,30 , 47 , 38 ,37 , 58 , 55 ,41, 20 ]. Most works carryout a \separated analysis", i.e.,
consideracompletelysymmetricsystemwithrespecttoanyuser,andstudythebehaviorofthe
protocolforaparticularreferenceusermodeledasaMarkovchain. Ingeneral,analysisdepends
onthetypeofcodesanddecoding/errordetectiontechniqueemployed. Modellingthesystemby
a Markov chain might be complicated, sincein each state one must convey all the information
aboutthe memoryofthesystem. In[32 ], Zorziproposesthe useofrenewal theory [42 ] inorder
to analyzeARQprotocols.
As remarkably well illustrated by Ephremides in [1 ], information theoretic techniques are
notyetofwidespreaduseinthedomainof networking withrandomuseractivity. Steps inthis
direction are represented by the work of Shamai and Wyner on cellular systems [52, 53 ] and
of Telatar and Gallager [13 ]. In [13 ], the multiple access Gaussian channel is assimilated to a
processor-sharing system and is analyzed as a queue with single server and an innite buer.
The required service for each user is dened in terms of random coding bound on the error
applicationof Little's Theorem[7].
This paperis mainlyinspiredbythe work of [13 , 32 ]. We assume thatusers transmittheir
signalburstsat highinstantaneouspowerand inacompletelyuncoordinated way. Thereceiver
is formed by a bank of single-user decoders, and doesnot implement joint decoding, i.e., each
decoder treats the signals from other users as noise.
1
We refer to this model as the Gaussian
collision channel [17 ]. The transmission of each user is governed byan Hybrid-ARQ protocol,
designed to cope with background noise, fading and interference (or \collisions") from other
users.
We study the system performance in terms of throughput (total bit/s/Hz) and average
delay for three simpleidealized protocols: a coded version of Aloha, a repetition scheme with
maximal-ratio packet combining and an incremental redundancy scheme with general coding.
Byapplyingtherenewal-rewardthereom[42 ],weobtainaclosed-formthroughputformulaunder
a delay constraint (time-out) and code rate constraint. Since we consider random coding and
typical set decoding, our results are independent of the particular coding/decoding technique
andshould be regardedasa limitintheinformationtheoreticsense.
ThesystemthroughputiscomparedtothatofCDMAwithconventionalsingle-userdetection
and decoding. Interestingly, the ARQ system is not interference-limited even if no multiuser
detection or joint decoding is used, i.e., arbitrarily high throughput can be obtained simply
by increasing the transmit power of all users, as opposed to conventional CDMA where the
throughput tends to a nite limit as all users increase their transmit power [10, 56 ]. As a
byproductofthisanalysis,weprovideastrongeroperationalmeaningtotheinformationoutage
probabilityof block-fading channels and we obtain the closed form probability distribution of
signal-to-interference plus noise ratio (SINR) with Rayleigh fading and a Poisson-distributed
numberof interferers, extendingtheresult of[53 ].
The paper is organized as follows: in Section II we describe the system model; in Section
III we deal with typical set decoding and error detection; in Section IV we carry out system
throughputanalysis;inSectionVwepresentsomelimitingbehaviorsandinSectionVIwepoint
outourconclusions. The proofsofthe resultsareprovidedinthe Appendices.
1
Even though any point in the capacity region of multiple access channels can be implemented with low
complexity bysuccessive \stripping" [4 ], thisrequires agooddeal ofcoordinationamongthe userswhichmust
allocate theirrateandpowerinaproperway[9 ,44 ,27 ]andmaynotbesuitedtorandomuseractivity.
2 The slotted Gaussian collision channel with feedback
Inthe systemunderinvestigation,N
u
usersshare a commonradiochannelwithcomplexbase-
bandequivalent bandwidth[ W=2;W=2] inorder to transmit their informationmessages to a
single receiver. Users are provided with a common time reference. The time axis is divided
inslots of duration T and userstransmit signal bursts ofduration slightlylessthan T,aligned
withtheslots. Apartfrom theslotted transmissionmode,usersare completelyuncoordinated.
EachusercantransmitaboutL=bWTcindependentcomplexsymbolsoverone slot(assuming
WT 1[59 ]).
A discrete-time signal model is adopted and slots are denoted by their index s. Let y
s
=
(y
s;1
;:::;y
s;L ),x
k;s
=(x
k;s;1
;:::;x
k;s;L
) and
s
=(
s;1
;:::;
s;L
)denotethereceived signal,the
transmitted signal from user k and the background noise during slot s, respectively. Noise is
assumedcircularly-symmetriccomplexGaussian,withi.i.d. componentswithvarianceN
0 . User
k transmitswithconstant average energypersymbolE
k
=E[jx
k;s;`
j 2
].
Thepropagationchannelisassumedslowlytime-varyingandfrequency-atforeachuser. In
particular,thecomplexchannelgainc
k;s
of userk overslotsisassumed to beconstant on the
wholeslot(block-fading [26 ]). Thereceived signal over slotscan be written as
y
s
= X
k2K(s) c
k;s x
k;s +
s
(1)
whereK(s)f1;:::;N
u
gdenotesthe setof active usersoverslots.
User k encodes its information messages, of b bits each, independently of other users, by
usingachannelcodewithcodebookC
k C
LM
oflengthLM overthecomplexnumbers,where
M is a given integer. Code words are dividedinto M subblocks of length L, each of which is
modulatedinto a signal burstand is transmittedoverone slot. We let C
k;m
,form=1;:::;M,
denotethepuncturedcodeoflengthmLobtainedfromC
k
bydeletingthelastM msubblocks.
Each user selects the slots for transmission according to its own time-hopping (pseudo-
)randomsequence,independentlyoftheotherusers[24 ]. Time-hoppingsequencescanbeseenas
random\on-o"processes,whereausercantransmitonlywhen itis\on". Weassumethatthe
receiver knows a priori the time-hoppingruleof all usersinthe system [24 ].
2
Transmissionis
2
Thisassumptionisnotparticularlyrestrictive,andisanalogoustothestandardassumptionofCDMAwith
pseudo-random\long"spreading[2], wherethereceiverisassumedtoknowthespreadingsequencesof allusers
governed bythefollowingsimpleHybrid-ARQprotocol,runinadecentralizedwaybyeachuser
k. Whenanewcodewordisreadyfortransmission,userk sendstherstLsymbolsontherst
allowed slot, says
1
,according to its time-hopping rule. The receiver decodes thecode C
k;1 by
processingthereceivedsignal y
s1
. Ifdecodingissuccessful,apositiveacknowledgement (ACK)
is sent back to user k over an error free and delay free feedback channeland the transmission
of the current code word stops. On the contrary, if the receiver detects an error, a negative
acknowledgement (NACK) is sent. In thiscase, user k sendsthesecond blockof L symbols of
the same code word on the next allowed slot, says
2
. Now, thereceiver decodes the code C
k;2
by processing the received signal blocksfy
s
1
;y
s
2
g. Again, ifdecoding is successful an ACK is
sent and the transmissionof thecurrent code wordstops. On thecontrary,ifa decodingerror
is detected, a NACK is sent back and user k transmits the third block of L symbols of the
same codewordon thenext allowed slot. Theprocessgoeson thisway: afterthe transmission
of m bursts of the current code word, code C
k;m
is decoded by processing the received signal
fy
s
:s2S
k;m
g,where S
k;m
=fs
1
;:::;s
m
g denotesthesequence of slotswhere transmissionof
user k took place. If successful decoding occurs at them-th transmission, the eective coding
rateforthe currentcode wordisR =m bits/s/Hz, whereR
=b=L.
3
In general, the slots s 2 S
k;m
are non-adjacent. We let n denote the delay (expressed in
numberofslots)betweentheinstantwhereacodewordisgeneratedandthecurrenttime(time
ticks at theslotrate). Obviously,mn (see examplein Fig.1). In any practicalapplication,
an informationmessage must be delivered to the receiverwithin a maximumdelay of N slots,
whereforsimplicityN is assumedcommonto all usersandall messages. If successfuldecoding
does not occur within delay N, the message becomes useless. Moreover, since the code words
of C
k
have M subblocks, thesame message can be transmittedin at mostM signal bursts. If
successfuldecodingdoesnot occur withinM transmittedbursts,the message is lost. We shall
refer to N and M as the \delay" and \rate" constraints, respectively. The transmission of a
code wordcan stopin threecases: i)Successfuldecoding occurs at them-thtransmitted burst
itwishes todecode. Inpractice, we mightthinkofanaccessmechanism,runat amuchslowertime-scalethan
packettransmission,thatassignstonewusersenteringthesystematime-hoppingsequence.
3
Forlarge WT,a complexsymbol(ordimension)canbetransmittedapproximately inonesecond andone
Hz. Moreprecisely,thespectraleÆciencyexpressedinbit/s/Hzcanbeobtainedbymultiplyingthecodingrate
(bit/complexsymbol)by themodulationspectraleÆciency(expressedincomplexsymbols/s/Hz),thatdepends
onthemodulationexcessbandwidth[21 ].
n = 8 m = 4
Code word generation time
Current time
Figure 1: Example: m = 4 transmitted bursts (shadowed) over n = 8 slots since the current
code wordgeneration.
and innslots,withm M and nN;ii)No successfuldecoding occursafter M transmitted
burstsandnN slots;ii)No successfuldecodingoccursafterN slotsandmM transmitted
bursts.
There are several ways to handle transmission failures (cases (ii) and (iii) above). For
example, in the case of time-sensitive information, the current message is simply discarded.
Inotherapplications,delayisnotastrictrequirement(N isverylarge)andthecurrentmessage
may be kept in the transmission buer for a later attempt. Several practical ARQ protocols
have beenproposed to handle transmissionfailures (see references in Section1). The analysis
carriedoutintherest of thispaperconsidersthesimpliedscenario whereaninnite sequence
ofmessagesisavailableto allusersand,inthecaseof transmissionfailure,thecurrent message
is discarded and the next message is encoded and transmitted in exactly the same way. The
time-hoppingsequenceforslotselectionisnotmodiedbytransmissionfailures(e.g.,thereisno
idlestate, waiting forbetter channelconditions). It is important to noticethat each user runs
its own ARQprotocolindependentlyof theother users. The onlywayinwhichusersinuence
each others isthrough mutual interference (collisions)that occurswhen several users transmit
theirburstsoverthesame slot.
The single-userdecoderforuser k hasperfect knowledge ofthe channelgainc
k;s
and ofthe
SINR
k;s
=
k;s E
k
N
0 +
P
j2K(s):j6=k
j;s E
j
(2)
forallssuch thatk 2K(s), where
k;s
=jc
k;s j
2
denotesthechannelpowergain. Estimationof
thechannelgainand oftheSINRcanbeachieved inpractice byinsertingtrainingsymbolsinto
each signalburst,ascurrentlydone inmostCDMA and TDMA cellularstandards [61 ].
The Hybrid-ARQ protocol described above is a general incremental redundancy scheme
(denotedby \INR" forbrevity). We consideralso thefollowingparticular cases.
Generalized Slotted Aloha. The slotted Aloha protocol [7 ] (denoted by \ALO" for
brevity)is obtained byassuming foreach user k a suboptimal decoder thatconsiders onlythe
lastreceived signal block. In classicalAloha it isassumed that a decoding failure occurs(and
is detected) whenever a collision occurs. In mobile systems, users might be received at very
dierentpowerlevels becauseoffading,shadowinganddierent distancesfrom thereceiver. In
this case, a packed can be decoded successfully even if a collision occurs (capture eect) [20 ].
Here,weconsiderageneralizedALOwherechannelcodingisusedandmessagesmaybedecoded
correctlyeven inthepresence ofcollisions,dependingonthe SINR.
Repetitiontime-diversity. Asimpletime-diversityscheme(denotedby\RTD"forbrevity)
isobtainedbyrepeatingthesame burstofL symbols[8 , 3]. This isequivalent to constructthe
usercodeC
k
asaconcatenated code,whereC
k;1 C
L
istheoutercode anda simplerepetition
codeoflengthM istheinnercode. Afterthem-thtransmission,thereceiverperformsMaximal-
RatioCombining(MRC)[8 ]ofthesignalsfy
s
:s2S
k;m
ganddecodestheoutercodeC
k;1 based
on the combined signal y = P
s2S
k ;m c
k;s y
s
. Because of the analogy with a rake receiver that
performs MRC of thedierentmultipathcomponents ofthereceived signal [21 ],thisscheme is
sometimesreferredto as\repetition rake" [28 ].
3 Coding, decoding and error detection
We assume thatall code booksC
k
are generated randomlyand independently,with i.i.d. com-
ponents, accordingto a given pdf q(x) over C withmean zero and varianceE
k
. Foreach user,
anencodingfunction
k
:f1;:::;2 RL
g!C
k
isdened andrevealed to thereceiver.
A key point of the ARQ schemes described in Section 2 is that decoding errors should be
detected. Any complete decoding function, based on a partition of the channel output space
into 2 RL
regions (e.g., MAP decoding or ML decoding), is not suited to this purpose, unless
an explicit error-detection stage after channel decoding is introduced (e.g., in many cellular
systems a CRC is inserted into the informationmessage [6 , 61 ]). This however is undesirable
since it decreases the throughput by adding extra redundancy. An alternative is the use of
possiblysuboptimaldecodersintermsoferrorprobability,butfeaturingabuilt-inerrordetection
capability. Moreover, it is desirable to decode all punctured codes C
k;m
,for m =1;:::;M, by
thesame decoder.
In particular, we examine the following error correction/detection scheme. Consider de-
coding foruser k after m received blocks, and let x (w)
k
=
k
(w) be the transmitted code word
correspondingtoinformationmessagew. Thedecoderaddstothereceivedsignalfy
s
:s2S
k;m g
otherM mdummysignalblocksz
i
,generatedindependentlyof thereceivedsignal, 4
to form
4
Inpractice,indecodingofpuncturedconvolutionalcodesdummysymbolsaresettozero,butinthelimiting
caseconsideredhereitissuÆcientthattheyarestatisticallyindependentofthechannelinput.
theobservationY =(y
s
1
;:::;y
s
m
;z
1
;:::;z
M m
) oflengthLM,and thendecodesthe\mother
code" C
k
accordingto thetypical setrule
k :C
LM
!f1;:::;2 RL
;eg(see[59 ] andAppendixA
fordetails)denedasfollows:
LetE
w
betheevent thatx (w)
k
istheuniquecode word jointlytypical withY. Then,
k
(Y)=wb if, forsomewb2f1;:::;2 RL
g,theevent E
b w
occurs.
k
(Y)=einanyother case.
Since decoderk treats all other user signalsasadditivenoise, it \sees"a virtual additive noise
channelgiven by
y
s
=c
k;s x
k;s +v
k;s
(3)
where v
k;s
=
s +
P
j2K(s):j6=k c
j;s x
j;s
is the interference plus noise vector. We let p
k;s (yjx)
denote the single-letter transition pdf of the above channel, conditioned on the channelgains
fc
j;s
:j 2K(s)g and on theset of active users K(s), and we deneI(q(x);p
k;s
(yjx)) to be the
mutual information(per letter) of channel (3), expressed as a functionalof the pdfsq(x) and
p
k;s
(yjx). Obviously,I(q(x);p
k;s
(yjx)) variesrandomlyfromslotto slot,sinceitdependsonthe
randomsetK(s) andon therandomchannelgainsfc
j;s
:K(s)g.
We examinethebehaviorofcodesC
k;m
withdecoder
k
denedabove,foragiven sequence
of channeltransition pdfs P
= fp
k;s
(yjx) :s2 S
k;m
g. The average error probabilityis dened
by
Pr(errorjP;C
k )
=2 RL
2 RL
X
w=1 Pr(E
w
jw;P;C
k
) (4)
A decoding errorwhen message w is transmitted isnot detected if, forsome wb6=w, theevent
E
b w
occurs. Then, theaverage probabilityof undetectederroris denedby
Pr(undetected errorjP;C
k )
=2 RL
2 RL
X
w=1 Pr
0
@ [
b w 6=w
E
b w
w;P;C
k 1
A
(5)
Thefollowingresults,proved inAppendixA,showthatthetypicalsetdecoderdenedaboveis
asymptoticallyoptimalforbotherror andundetected errorprobabilities, forlargeburstlength
L:
Lemma 1 (achievability). Forall >0 there exist L and codesC
k 2C
LM
of size2 RL
with
Pr(errorjP;C )<forallm=1;:::;M and channelsequencesPsuchthat
P
s2S
k ;m
I(q(x);p
k;s
(yjx))>R . }
Lemma 2 (converse). For all channelsequences Psuch that P
s2S
k ;m
I(q(x);p
k;s
(yjx)) <R ,
Pr(errorjP;C
k;m
)!1 foranycodeC
k;m 2C
Lm
ofsize2 RL
asL!1. }
Lemma3 (error detection). Forall >0and channelsequencesPthere exists Lsuch that
any code C
k 2C
LM
of size2 RL
satises Pr(undetected errorjP;C
k
)<. }
Theoptimalinputdistributionq(x)oftheinterference channel(3)isnotknowningeneral[59 ].
Forthesakeofmathematicaltractability,weconsider(somewhatarbitrarily)circularly-symmetric
complexGaussianinputsforall users. Then,themutualinformation P
s2S
k ;m
I(q(x);p
k;s (yjx))
takes theform
I
k;m
= X
s2S
k ;m log
2 (1+
k;s
) (6)
From theabove resultswe havethat, byusingGaussiancodesandtypical setdecodingat each
stepm of theARQprotocols ofSection 2,theprobabilityofdecodingerroris arbitrarilysmall
if R < I
k;m
, very large if R > I
k;m
and decoding errors are detected with arbitrarily large
probability,forsuÆcientlylargeL. Practicalfuturesystemformobiledatatransmissionwillbe
characterized byavery largevalueof theproductWT,inorder to supportlargeinstantaneous
bit rates. This motivates a system analysis under the assumption of very large L.
5
In this
regime,we shall assumethat, forall k and m, Pr(errorjR<I
k;m
) =0,Pr(error jRI
k;m )=1
andPr(undetected error)=0.
Remark: Boundeddistanceanditerativedecoding. Obviously,thetypicalsetdecoder
consideredaboveisnotsuitedforpracticalimplementations. However, itisinterestingtonotice
thatsomenon-MLpracticaldecodingschemesshowabehaviorsimilartothetypical-setdecoder.
Forexample, bounded-distancedecoding [39 ] outputsthemessage w ifthe received signalfalls
inside a sphere centered on the code word corresponding to w, while if the received signal is
notinanysphere, anerrormessage eisdeclared. Anotherexampleisprovidedbytheiterative
decodingscheme [23 ]used todecodeTurbo-codes. Thecomponentcodesof theTurbo-codeare
individuallydecodedbysymbol-by-symbolsoft-in soft-outdecoders sharingandupdatingsome
common information about the reliabilityof the symbol-wise decisions. Typically, if the code
5
Forexample, inthe3rdgenerationUMTSstandardapacket-radiorandomaccessschemeissupportedwith
variable slot duration 0:625 T 10ms, bandwidthW = 5MHz [49 ] and modulationspectraleÆciency up
to0:2,obtainedbyusingdirect-sequencespread-spectrummodulationwithraised-cosinepulseswithroll-o0:22
andspreadinggain4. ThismeansthatL=b0:2WTcisbetween625and10000 complexsymbolsperslots.
wordis correctly decodedall component decoders agree on thesymbol-wise decisions,whilein
the presence of decoding errors the decoders keep on reversing the symbol decisions at each
iteration[11 ]. This illbehavior, aswell asthe lowreliabilityforsome symbols, can be usedas
errorindicators[14].
Remark: analogy with the block-fading channel. Under the assumptionof Gaussian
usercodemadehere,thechannelmodel(3)istotallyanalogoustotheblock-fadingAWGNchan-
nel withperfect channelstate informationat the receiver introducedin [26 ]. In [26 ], decoding
is always performed after M blocks and the probabilityof decoding failurefor largeL is given
byPr(I
k;M
R ), and isreferred to asinformation outage probability. Outageprobabilitynds
a very natural interpretation as the limiting error probability for large block length averaged
over the random coding ensemble and over the fading states [18 ]. A questionleft open in [26 ]
andinmanysubsequentworksis whetheritexistsa codesequence (forincreasingvaluesofthe
blocklengthL)witherrorprobabilityarbitrarilysmallforallfadingstatessuchthatI
k;M
>R .
Notice that this is not a trivial question, since if the choice of the code sequence depends on
the particular fading state, outage probabilitywould not be achievable (it would require side
informationat thetransmitter). Theexistenceof codesasymptotically good forall fadingstates
satisfyingI
k;M
>R is given byLemma 1 (see the detailsof theproof inAppendixA). In this
respect, information outage probability is not just an average probabilityof error over a code
ensemble,butitcan be approachedbya given (deterministic)sequence of codes.
ALO and RTD schemes. In analogy to what done above for the INR scheme, we can
dene random coding and typical-set decoding for ALO and RTD. For the sake of brevity,
we state without details that, as L ! 1, also for these schemes there exist codes for which
Pr(errorjR<I
k;m
)and Pr(undetected error)vanishand Pr(errorjRI
k;m
) goesto 1,provided
thatthecorrectexpressionforthemutualinformationisused. ALOtakesintoaccountonlythe
mostrecent received signal burst,therefore thecorrespondingI
k;m
is givenby
I
k;m
=log
2 (1+
k;sm
) (7)
In RTD, the SINR after MRC of m received bursts is given by the sum P
s2S
k ;m
k;s [21 ],
thereforethe corresponding I
k;m
isgiven by
I
k;m
=log
2 0
@
1+ X
s2S
k;s 1
A
(8)
4 Throughput analysis
In thissection we compute the throughput of the ARQ protocols of Section 2 with thecoding
and decoding scheme of Section 3, in the limit for large L. Our analysis is valid under the
followingidealizedassumptions:
1. An innitenumberof informationmessagesis availableforeach users. As soon asa user
stops the transmission of the current code word, it encodes the next packet and starts
its transmission in the next selected slot. As explained in Section 2, transmission of a
codewordcanstopeitherbecausesuccessfuldecodingoccurs,orbecausethedelayorrate
constraintsN and M are violated(decoding failure).
2. TheACK/NACKfeedbackchannelis delay-free and error-free.
3. Users select slots for transmission so that the number of slots between two consecutive
transmissionsofthesame userisi.i.d.,geometricallydistributedwithidenticalparameter
p
t
for all users. In order words, on each slot s each user transmits a signal burst with
probability p
t
and does not transmit with probability 1 p
t
. The expected number of
userstransmittingovera slotis G=p
t N
u
(average channelload).
4. Thechannelpower gains
k;s
are i.i.d. randomvariables(RVs), forallusersand slots.
5. Allusers have thesame transmit SNR
=E=N
0
(i.e.,E
k
=E 8k=1;:::;N
u ).
Let t count the numberof slots,b
k
(t) the number of information bitsfrom user k successfully
decoded up to slot t and R
k (t)
= b
k
(t)=L the corresponding number of bit/s/Hz. The overall
throughput measuredinbit/s/Hz isgiven by
= lim
t!1 1
tL Nu
X
k=1 b
k (t)
= N
u lim
t!1 1
t R
1
(t) (9)
wherethesecond linefollows fromthe symmetryof thesystemwithrespectto anyuser.
Consider user 1 transmission. Under the above assumptions, the event that user 1 stops
transmittingthecurrentcode wordisrecognizedtobea recurrent event [42 ]. Arandomreward
transmissionstopsbecausesuccessfuldecoding,andR=0bit/s/Hziftransmissionstopsbecause
delay/rate constraint violation. We can applytherenewal-rewardtheorem [42 ] andget
lim
t!1 1
t R
1 (t)=
E[R]
E[T]
with prob. 1 (10)
whereT is therandomtimebetween two consecutive occurrencesof therecurrent event (inter-
renewaltime). Thus,thedesired throughputgeneral expressionis
=N
u E[R]
E[T]
(11)
In order to evaluate E[R], the mean reward, and E[T], the mean inter-renewal time, we focus
on the transmission of a given code word of user 1 and we dene the auxiliary RV M to be
the number of transmitted bursts between the instant when the code word is generated and
the instant when its transmission is stopped (i.e., between two consecutive occurrences of the
recurrent event). We dene the event A
m
= fI
1;m
> R g, and the probability q(m) that the
randomsequence I
1;1
;I
1;2
;:::;I
1;m
;:::ofmutualinformationat theuser1decodercrosseslevel
R at the m-th step (and not before), or, in other words, the probability of having successful
decodingwith mtransmitted bursts. This isgiven by
q(m) = Pr(A
1
;:::;A
m 1
;A
m )
= Pr(A
1
;:::;A
m 1
) Pr(A
1
;:::;A
m )
= p(m 1) p(m) (12)
where
p(m)
=Pr(A
1
;:::;A
m )=1
m
X
`=1
q(`) (13)
Thejointprobabilitydistributionof T and M
f
T;M (n;m)
=Pr(T=n;M=m)
isobtainedexplicitlyasfollows(inthecaseM Notherwisetherateconstraintismeaningless):
f
T;M
(n;m)= 8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
: (1 p
t )
N
n=N;m=0
v(N;m)+ 0
B
@ N
m 1
C
A(1 p
t )
N m
p m
t
p(m) n=N;1mM 1
v(n;M)+ 0
B
@ n 1
M 1
1
C
A (1 p
t )
n M
p M
t
p(M) M nN;m=M
v(n;m) mnN 1;1mM 1
0 elsewhere
(14)
(we use theshort-handnotation v(n;m) for 0
@ n 1
m 1
1
A
(1 p
t )
n m
p m
t
q(m)). In AppendixBwe
show that (14) is a well-dened probability distribution for all 0 p
t
1, N M > 0 and
non-negative non-increasingsequence fp(m)g with p(0)=1.
Atthispoint,wearereadyto computeE[R]andE[T]. ArewardRisobtainedfor(T;M)=
(n;m) ifsuccessful decoding occursin nslots with m transmitted bursts. This correspondsto
placingm 1transmissions intherst n 1 slotswithoutsuccess, andthe m-th transmission
inthen-thslotwithsuccess, whichoccurswithprobabilityv(n;m). Therefore,
E[R] = R M
X
m=1 N
X
n=m
v(n;m)
= R
2
4
1 M 1
X
`=0 0
@ N
` 1
A
(1 p
t )
N `
p
`
t p(`)
N
X
`=M 0
@ N
` 1
A
(1 p
t )
N `
p
`
t p(M)
3
5
(15)
Theaverage inter-renewaltime issimplygiven by
E[T] = M
X
m=0 N
X
n=1 nf
T;M (n;m)
= M 1
X
m=0 1
p
t p(m)
2
4
1 m
X
`=0 0
@ N+1
` 1
A
(1 p
t )
N+1 `
p
`
t 0
@ N
m 1
A
(1 p
t )
N m
p m+1
t 3
5
(16)
Finally,the desiredclosed-formexpressionforthesystem throughputis given by
N;M
=R G 2
4
1 M 1
X
`=0 0
@ N
` 1
A
(1 p
t )
N `
p
`
t p(`)
N
X
`=M 0
@ N
` 1
A
(1 p
t )
N `
p
`
t p(M)
3
5
M 1
X
m=0 p(m)
2
4
1 m
X
`=0 0
@ N+1
` 1
A
(1 p
t )
N+1 `
p
`
t 0
@ N
m 1
A
(1 p
t )
N m
p m+1
t 3
5
(17)
(we write
N;M
inorder tostressthedependenceon N andM). Protocols INR,RTDand ALO
described before, for given parameters R , M, N, N
u
and G, dier in the probabilities p(m).
ConsiderrstINRandRTD.Theseschemeshavememory,sincethereceiveraccumulatesmutual
information, forINR, or SINR, forRTD, over the sequence of slots fs 2S
1;m
g. From (6) and
(8),since
1;s
isnon-negative,itis apparentthattherandomsequence fI
1;m
gisnon-decreasing
withprobability1. Then,A
` A
m
forall `m andwe canwrite
p(m)=Pr(A
m
)=Pr(I
1;m
R ) (18)
ForALO,I
1;m
given by(7)hasnoparticular monotonebehavior. However, thereceiverhasno
memoryof pastsignal burstsand theevents A
m
arei.i.d.. Then,we can write
p(m) = Pr(A
1
;:::;A
m )
= m
Y
i=1 Pr(A
i
)=Pr(A
1 )
m
(19)
Finally,for allprotocols examinedweobtaina compactexpressionforp(m) as
p(m)= 8
>
>
>
>
>
<
>
>
>
>
>
: Pr
P
s2S1;m log
2 (1+
1;s )R
INR
Pr
log
2 (1+
P
s2S
1;m
1;s )R
RTD
Pr(log
2 (1+
1;1 )R )
m
ALO
(20)
Some interesting properties of
N;M
can be derived immediatelyfrom (17) and (20). It can be
easilyshownthat
N;M
isa decreasingfunctionof p(m) andthat
Pr 0
@ X
s2S
1;m log
2 (1+
1;s )R
1
A
Pr 0
@
log
2 (1+
X
s2S
1;m
1;s )R
1
A
Pr(log
2 (1+
1;1 )R )
m
forall m=1;:::;M. Then,asexpected,thethree protocolsare relatedby
(INR)
(RTD)
(ALO)
(21)
ForALO,wehave that
(ALO)
N;M
=R G(1 p(1)) (22)
independentlyof N and M. This resultis expected,sinceALO hasnomemory andbothdelay
andrate constraintsareirrelevant. It iseasy toshowthatthesequencep(m) forbothINR and
RTD is\sub-geometric",i.e., thatp(m)p(1) m
form=1;2;:::,withequalityonlyform=1.
From this observation, it is prossible to show that for, bothINR and RTD, the throughput is
increasinginN and M,i.e., that
N+`;M+r
N;M
(23)
forall `;r 0,withequalityfor`=0;r =0 only. This result isintuitive,sinceit makessense
thatthethroughputisgoingto increasebyrelaxingthedelayortherateconstraints. However,
it is not completely trivial since both the numerator (average reward) and the denominator
(average inter-renewaltime) of (17) are increasingfunctions of N and M. Asa matter of fact,
both theINR and the RTD protocols have the nice feature that \thelonger we wait the more
we gain".
In classicalAloha analysis, itis customary to let N
u
! 1 whilekeepingG xed and nite
(innitepopulation[7 ]). Since p
t
=G=N
u
,thisisequivalentto letp
t
!0 in(17). Interestingly,
forall niteN,wehave
lim
pt!0
N;M
=R G(1 p(1))
Inother words,inthelimitforinnitepopulation,theINRand RTDprotocolswithnitedelay
constraint are equivalent to ALO. In fact, in this case a large number of users transmit with
very small probability, and the probability that a user transmit more than once in any nite
time N is negligible. Therefore, either the packet is successfully decodedat the rst attempt,
oritis discarded, like inALO.On thecontrary,forN !1 the limitforinnite population is
dierentforthe threeprotocols.
Inthenextsection,westudysomelimitingbehaviorsofthethroughputforanunconstrained
system, i.e., for N;M ! 1. We notice here that all three protocols without constraints yield
zero packet loss probability: the transmission of a code word ends only when it is correctly
decoded. Theunconstrained throughput (denotedsimplyby forthesake ofbrevity) iseasily
obtainedfrom (17)as
=
RG
1
X
m=0 p(m)
= RG
E[M]
(24)
where we used the fact that P
1
m=0
p(m) = P
1
m=1
mq(m) = E[M], the average number of
transmitted bursts needed for successful decoding. In passing, we notice that E[M]=p
t is the
mean delay (measured in slots) for the transmission of an information message (i.e., it is the
average numberof slotsbetweenthegenerationof a codewordand its successfuldecoding).
For ALO, can be computed in closed form since p(1) = Pr(log
2 (1+
1;1
) R ) can be
obtainedexplicitly(see AppendixD). Forthechannelwithoutfading we have
(ALO)
=RG
K(R;) 1
X
`=0 0
@ N
u 1
` 1
A
G
N
u
`
1 G
N
u
Nu 1 `
(25)
where
K(R ;)=
1
2 R
1 1
+1 (26)
is the maximum number of simultaneous users in a slot that can be correctly decoded (notice
that,dependingonRand,acollisiondoesnotcorrespondnecessarilytoanerror,sinceK(R ;)
might be largerthan 1). ForN
u
!1,(25) yields
(ALO)
=RG
K(R;) 1
X
`=0 e
G G
`
`!
(27)
that for K(R ;) =1 reduces to the well-known result of classicalslotted Aloha, =R Ge G
.
ForthechannelwithRayleigh fadingwe have
(ALO)
=RG Nu 1
X
`=0 0
@ N
u 1
` 1
A
G
N
u
`
1 G
N
u
Nu 1 `
e (2
R
1)=
2
`R
(28)
which forN
u
!1 yields
(ALO)
=RGe (2
R
1)= (1 2 R
)G
FortheINRandRTDitisnotpossibletondclosed-formexpressionsfortheprobabilitiesp(m).
However,thesecanbecalculatedeasilyforanymasfollows. LetZ =
1;1
andI =log
2 (1+
1;1 ).
Then, from denitions (20) we see that for INR, p(m) is the cdf of the sum of m i.i.d. RVs
distributed as I, evaluated in R , and for RTD, p(m) is the cdf of the sum of m i.i.d. RVs
distributedasZ,evaluatedin2 R
1. Forsmallm,p(m) canbeevaluatedfrom thedistribution
of
1;1
(e.g., byusingthe characteristic function). Since thisapproach involvesdiscreteFourier
transformswhoselengthincreaseswithm,itcannotbeappliedforlargem. Inthiscase,fromthe
centrallimittheorem[35 ]wehavethat 1
p
m P
s2S
1;m
1;s and
1
p
m P
s2S
1;m log
2 (1+
1;s
) areclose
to Gaussian RVs, for largem. Therefore, p(m) can be easilyevaluated from theGaussian cdf.
Forthe sake of brevity, we skipthe detailsof numerical calculations. However, itis interesting
to noticethatnoneof theresultsof thispaperareobtainedbyMonteCarlo simulation.
Figs.2and 3showvs. R ,fortheINR,RTDandALOprotocols,with =10dB,N
u
=50,
G = 1 and N = M ! 1 in AWGN and Rayleigh fading, respectively. For ALO on AWGN
channel,iszeroforR>log
2
(1+)=3:5sinceforhigherratestheSINRisnotenoughevenin
theabsenceof interferers(the systembecomes power-limited ratherthan interference-limited).
In thecase of Rayleigh fading, decreases with R but it is positive even for R >log
2
( 1+) ,
sincethere isa non zeroprobabilitythatthefading gain islargerthan one.
Fig.4showsvs. Rfor =10dB,N
u
=50andG=1,N =100 andincreasingvaluesofM
forINR on AWGN channel. As already pointedout, thecurve forM =1 coincideswithALO.
Thedierent curvesoverlapforsmallR sinceonetransmittedburstissuÆcient to decode. For
M >1 thethroughput is non zeroalso for R larger thanlog
2
( 1+) . Forexample, for M =2
themaximummutual informationthatcan be accumulated is2log
2
(1+)=6:92.
5 Limiting behaviors
In this section we study the three protocols in terms of their unconstrained throughput for
largeR ,G or.
5.1 Limits for large R
InAppendixCwe showthat
lim
R!1 =
8
>
>
>
>
>
<
>
>
>
>
>
: GE[log
2 (1+
1;1
)] INR
0 RTD
0 ALO
(29)
0 1 2 3 4 5 6 7 8 9 10 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
R
η
INR
RTD
ALO
Figure2: vs. R for =10dB, N
u
=50, G=1and N;M !1 on AWGNchannel.
0 1 2 3 4 5 6 7 8 9 10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
R
η
INR
RTD
ALO
Figure3: vs. R for=10dB, N =50, G=1and N;M !1 onRayleigh fadingchannel.
0 1 2 3 4 5 6 7 8 9 10 0
0.2 0.4 0.6 0.8 1 1.2 1.4
R
η
M=1
M=2
M=10
Figure 4: vs. R for=10dB, N
u
=50, G=1 and N =100 forINR on AWGN channel.
ALO and RTD schemes. ALO and RTD involve strongly suboptimalcoding schemes, for
whichE[M]growsfasterthanR . Thus,thelimitingiszero. Since =0forR=0andgoesto
0forlargeR , forbothprotocols thereexist an optimalchoice of 0<R<1 maximizing (see
Figs.2and3). ThisoptimalRdepends,ingeneral,onGand. Fromapracticalsystemdesign
point of view, thisshows that the burstspectral eÆciencyR shouldbe dimensionedaccording
to thechannelload Gand theSNR .
INR scheme. For INR, < GE[log
2 (1+
1;1
)] for all nite R . This fact is quite hard to
show directly by using (24), since the probabilities p(m) depend on R but a closed form is
notavailable. However, wecan provideasimpleindirectproofofthestatement asfollows. The
quantityC
=E[log
2 (1+
1;1
)]isthecapacityofthememorylessL-blockinterferencechannel[40 ]
given in (3), where the interference signal v
k;s
is circularly-symmetric complex Gaussian with
i.i.d. components and where
k;s
is the SINR for block s. A well-known result states that
feedback doesnot increasethecapacityof memorylesschannels [59 ]. Then,even iftheencoder
has available the sequence of past received vectors y
1
;:::;y
s 1
, the maximum transmissible
rate for channel (3) is C.
6
Hence, we conclude that = GC is actually the maximum
achievable throughput on this channel, irrespectivelyof thefeedback and forany choice of R .
From a practicalsystem designpoint of view,in theabsenceof rate and delayconstraintsit is
convenient to work with a very high burst spectral eÆciency R , irrespectively of the channel
loadGand the SNR.
The maximum throughput is achieved for innite delay. It is interesting to notice that,
with innite delay, the same maximum throughput (with zero packet loss probability) can be
achieved bya system withoutfeedback (just forwarderror correcting codes) [17 ]. It is natural
to ask why the ACK/NACK feedback channel should be implemented at all. The answer is
providedbycloser examinationoftheaverage delay: thesystemwithoutfeedbackneeds a very
large (innite) delay in order to transmit with arbitrarily small packet loss probability for all
valuesof[17 ]. Onthecontrary,theINRprotocolachieveszerotransmissionfailureprobability
withniteaverage delayforall strictlylessthan GC.
Fig. 5 shows the average number of transmitted bursts E[M] vs. for the ALO and INR
protocols inthe case of Rayleigh fading, for =10dB, N
u
= 50 and G=1, N;M ! 1. The
corresponding average delayis givenbyN
u
E[M]=G.
5.2 Limits for large G
We consideranoptimized systemwith respectto R andwelet =sup
R
. InAppendixC,we
showthat
lim
G!1 =
8
>
>
>
>
>
<
>
>
>
>
>
: log
2
(e) INR
log
2
(e) RTD
log
2 (e)
ALO
(30)
where = E[
1;1
], =sup
u0
u[1 F
(u)] and F
(u)
=Pr(
1;1
u) is the cdf of
1;1 . For
AWGN, F
(u) is a step function with jump in u = 1, therefore = = 1. For Rayleigh
6
Itis important tonotice that (3)is memorylessat the block level, butnot at the symbollevel. Feedback
doesnotprovideanycapacityincreaseifthefeedbackchannelworksattheslotrate,i.e.,itsendsbackthewhole
receivedvectorys atthe endof eachs-thslot. Thisispreciselythe waytheACK/NACKfeedbackworks. On
the contrary, capacity would be clearly increased by a feedback working at faster rate, which sends back the
componentsofy
s
assoonastheyarereceived,duringeachs-thslot.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0
1 2 3 4 5 6 7 8 9 10
η
E[M]
INR ALO
Figure 5: E[M] vs. for =10dB, N
u
=50 and G=1and N;M ! 1 forALO and INR on
Rayleigh fading channel.
fading, F
(u) =1 e u=
, therefore = =e and lim
G!1 =log
2
(e)=e. This shows that for
largechannelloadGall schemesareequivalentinAWGN,whileALOperformsworsethanINR
and RTD in Rayleigh fading. In fact, ALO considers only the most recent received block for
decoding. Hence, there is no \averaging eect" with respect to the fading aecting the useful
signal over a long sequence of slots. Fig. 6 shows vs. G for ALO, =10dB on AWGN and
Rayleigh fading channel.
In order to gain insight in the behavior of the ALO protocol, it is useful to take a closer
look at thethroughput curve in thecase of AWGN.
7
This curve is obtained by noticing that
thesupremumof (ALO)
given in(27) forxed Gand is always obtainedwhen R=log
2 (1+
=(1+K))forsomeintegerK,whereK+1isthemaximumnumberofusersthatcancollide
on the same slots without causing a decoding error. Therefore, maximizing with respect to
R (for given G) is equivalent to searching for the maximum of the expressionlog
2
(1+=(1+
K))G P
K
`=0 e
G
G
`
=`!overthenon-negativeintegersK. Inparticular,forsmallGthemaximum
isobtainedbyK =0. Inthiscase,thethroughputismaximizedbychoosingthelargestpossible
R , i.e., R = log
2
(1+), and by letting the protocol alone to take care of collisions, like in
conventional Aloha. As G increases, the maximum is obtained by larger and larger K. In
thiscase, thethroughput is maximizedby choosingR inorder to tolerate up to K interferers,
i.e., R = log
2
(1 +=(1 +K)) (a decoding error occurs only when there are more than K
interferers). In this way, thetask of coping withcollisions is sharedby channel coding and by
theretransmissionprotocol: channelcodingyieldsno errorsforup to K+1active usersinthe
slot,whileifthenumberofactive usersislarger thanK+1 retransmissionis needed.
As G becomes large, a very large number of users transmit in every slot. In the limit,the
systemisequivalenttoaCDMAsystemwithaninnitenumberofusersKandinnitespreading
gainN,such thatthe ratioK =N isequalto G [10 , 56]. In fact, thechannelload Gisprecisely
the(average) numberofusersperdimension(perchip). ThethroughputofsuchCDMAsystem
isgiven by [54 ,12 ]
(CDMA )
=GE
log
2
1+
1;1
1+G
(31)
Itsmaximumislog
2
(e)bit/s/Hz,obtainedforG!1. Interestingly,thethroughputofCDMA
7
ThethroughputofALOinAWGNconverges toitslimitingvaluelog
2
(e)veryslowly. Convergence canbe
seenonamuchlargerscaleofG. Wechosethisscaleinordertobetterillustratethebehaviorinapracticalrange
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0 5 10 15 20 25 30
η
G
AWGN Rayleigh AWGN limit Rayleigh limit
Figure6: vs. G forALO,=10dB inAWGN and Rayleigh fading.
islessthanlog
2
(e)forallniteG, whilefortheINR,RTDand ALOschemes theremight exist
arange ofG forwhich >log
2 (e).
In the case of INR, is thelimiting value forlarge R given by (29). Fig. 7 shows vs. G
for INR and CDMA, =10dB for AWGN and Rayleigh fading channel. Both the multiaccess
schemes have the same limiting throughput, equal to log
2
(e) , for G ! 1, however the INR
scheme tends to this limitfrom above whileCDMA from below. Also, notice that for largeG
the fading increases the throughput of the INR scheme, while it decreases the throughput of
CDMA. This can be interpretedin terms of the capture eect. With the bursty discontinuous
transmission of slotted INR, only a nite number of users are going to collide on every slot,
forevery niteG. Because of fading,some of the interferers are received with low power and,
apparently,thisprovideathroughputincreaseforlargeG. Onthecontrary,inCDMA allusers
transmitall the timeand interference converges quickly to a deterministicaverage. Because of
theconcavity ofthelogarithm, thisprovidesa throughputdecrease.
0 1 2 3 4 5 6 7 8 9 10 0
0.5 1 1.5 2
G
η
INR Rayleigh fading INR AWGN CDMA AWGN CDMA Rayleigh fading log 2 (e)
Figure7: vs. G forINR andCDMA, =10dBin AWGN and Rayleigh fading.
5.3 Limits for large
InAppendixCwe showthat
lim
!1
=1 (32)
forall protocols examined, as opposed to CDMA, forwhichthe limit of(31) for large yields
Glog
2
(1+1=G) forAWGN andGe G
Ei(1;G) inRayleigh fading(Ei(1;z)
= R
1
1 e
zt
=tdt).
This means that the ARQ system is not interference limited, even if no joint decoding is
implementedat thereceiver: arbitrarilyhighthroughput can beobtained by simplyincreasing
transmitSNR of all users, irrespectivelyofpower control, fading, etc... Intuitively,thisis due
to the fact that there is a non-zero probability that only one user is active on any given slot,
andcan transmitat very highinstantaneousrate.
6 Conclusions
Combinedchannelcodingandretransmissionprotocolsappeartobeaviableandsimplesolution
for reliable packet-radio communication requiringhigh instantaneous rates and very low error
probabilityand characterized byburstysporadictransmissionandbymild delaycontraints.
Inthispaper,wepresentaninformation-theoreticthroughputanalysisofsomeHybrid-ARQ
protocolsunderidealizedbutfairlygeneralconditions. Weshowedthattypicalsetdecodinghas
verydesirablepropertiesforHybrid-ARQ,inthelimitforlargeslotdimension. Fromarenewal-
rewardtheoryapproach,weobtainedclosed-formthroughputformulasforthreesimpleprotocols:
a generalization of slotted Aloha (ALO), a repetition time diversity scheme with maximal-
ratio packet combining (RTD) and an incremental redundancyscheme based on progressively
puncturedcodes(INR).We analyzedtheeect ofdelayandrateconstraintsonthethroughput,
aswellasthelimitingbehaviorwithrespectto theslotspectraleÆciency,thechannelloadand
the transmit SNR. Interestingly, all three protocols are not interference-limited, and achieve
arbitrarilylarge throughputbysimplyincreasing thetransmitpowerof allusers.
We conclude bypointing outsome future research directions inspiredbythisanalysis.
Practical coding and decoding schemes based on incremental redundancy and featuring
built-inerrordetectioncapabilityshouldbeusedwithHybrid-ARQ.Turbo-codes(orother
formsof concatenated coding)with iterativedecodingappearto be apromisingsolution.
However, the behavior of iterative decoders in the presence of decoding errors shouldbe
better characterized inorder to exploititforerrordetection.
The assumption that the receiver knows exactly the time-hopping sequences of all users
might not be realistic. If user activity is random and not known to the receiver, our
resultscan be seenas an upperboundon theachievablethroughput obtained by ageine-
aided receiver which knows a priori the active users in each slot. True random access,
wherethereceivermustalsodetect whichusersareactiveinordertomake theapproriate
packetcombining,mightbestudiedbyinsertinginourframeworkanactiveuserdetection
scheme.
Inthis work, we concentratedon a very simplereceiverthat does notattempt to decode
theusers jointly. A natural direction forfuture research is to considerjoint decoding at
thereceiver(e.g.,implementedbystripping). AtheoreticaldiÆcultyisrepresentedbythe
userrandomactivity[1 ]. Infact,becauseofrandomaccess,thecapacityregionvariesfrom
slot to slot and it is not known in advance, unless a complicated reservation/allocation
scheme is implemented. Also, the set of interfering usersmight be dierent from slotto
slot,anditisnotclearhowtocarryoutjoint decodingacrosstheslots. Firststepsinthis
directionaretaken in[19 , 27 ].
Inmostpracticalapplications,packet-radionetworksmustco-existwithother systems,as
forexampleaconnection-orientedCDMAsystemwherealargenumberof low-powerlow-
rate userstransmit continuously. Quite a lot of work hasbeen dedicated to the problem
of power control for burstytransmission, where closed-loop schemes are not eective, in
the fear that high-rate high-power bursty users might create too much interference to
an underlyingCDMA system. An appealing consequence of our study is the following:
insteadof trying to controlburstyusers, we can let them transmit at full-power. Thanks
totheARQprotocol, thesignal fromall burstyuserscan beeventuallydecodedcorrectly
and subtracted from the received signal, so that the underlyingCDMA system \sees" a
clean channel, as if the bursty users were not there. In this way, the two quite dierent
systemcould be layered one on top of the other. Obviously,in order to make this claim
rigorous several issues must be addressed in the details: perhapsthe most important of
whichisthedelay. Infact, CDMAuserscan bedecodedonlyafter thesignalfrombursty
users has been subtracted. Then, the variable decoding delay associated with the ARQ
protocol imposes a variable decoding delay also on the CDMA system. If CDMA users
have a strict delay constraint (e.g., due to real-time speech transmission, like in cellular
telephony), outages due to the occurrence of large decoding delay events must be taken
into account.
Acknowledgement
Theauthors would like to acknowledge thekindand fruitfulfeedback from Prof. SergioVerdu
andDr. MichaelMecking.
Appendices
A Proofs of Lemmas 1,2 and 3
Following standard continuity arguments [15 ], we consider a quantization of the input and a
partitionof theoutput of(3)andwe workon theresultingdiscretechannel. Theresultsforthe
continuous channel can be obtained by taking the supremumover all inputquantizations and
outputpartitions. Fixa sequence ofchanneltransitionprobabilitiesP=fp
k;s
(yjx):s2S
k;M g.
Let P(fx
k;s
;y
s
:s 2S
k;m
g), P(fx
k;s
: s2S
k;m
g) and P(fy
s
:s2 S
k;m
g) be the joint and the
marginal probability distributionsinduced by P and by the input distribution q(x). Since on
every slot s2S
k;m
thequantizedversion of channel(3)is a time-invariant DMC, forthe weak
lawof largenumbers[35 ] we have thefollowinglimitsinprobability:
lim
L!1 1
L log
2 P(fx
k;s
;y
s
:s2S
k;m g) =
X
s2S
k ;m H
k;s (X;Y)
lim
L!1 1
L log
2 P(fx
k;s :s2S
k;m
g) = mH
k (X)
lim
L!1 1
L log
2 P(fy
s
:s2S
k;m g) =
X
s2S
k ;m H
k;s
(Y) (33)
where
H
k;s (X;Y)
=
X
x;y q(x)p
k;s
(yjx)log
2 q(x)p
k;s (yjx)
H
k (X)
=
X
x
q(x)log
2 q(x)
H
k;s (Y)
=
X
x;y q(x)p
k;s
(yjx)log
2 X
x 0
q(x 0
)p
k;s (yjx
0
) (34)
arethejoint,inputand outputentropies perletterinslot s. The typical set A
k;m
isdenedas
thesetof all sequencesfx
k;s
;y
s
:s2S
k;m
gsatisfying
1
L log
2 P(fx
k;s
;y
s
:s2S
k;m g)+
X
s2S
k ;m H
k;s (X;Y)
1
L log
2 P(fx
k;s
:s2S
k;m
g)+mH
k (X)
1
L log
2 P(fy
s
:s2S
k;m g)+
X
s2S
k ;m H
k;s (Y)
(35)
Byletting I(q(x);p
k;s (yjx))
=H
k
(X)+H
k;s
(Y) H
k;s
(X;Y),and byfollowingthesame steps
in[59 , Th.8.7.1], we getthatanyrate lessthan 1
m P
s2S
I(q(x);p
k;s
(yjx)) is-achievable. In
particular, for m = M and given sequence of channels P, for suÆciently large L there exists
codes C
k
of length LM and rate R =M with error probability (with typical set decoding) less
thanif
R<
X
s2S
k ;M
I(q(x);p
k;s
(yjx)) (36)
In order to prove Lemma 1 we need to show that: i) there is a single code C
k
having error
probability uniformly less than over all sequences of channels P satisfying (36); ii) for all
1mM,ifR<
P
s2S
k ;m
I(q(x);p
k;s
(yjx)), then thepuncturedcode C
k;m
obtained from C
k
bytaking therst m subblocksof lengthLhasalso errorprobabilitylessthan.
From therandomcodingachievabilitypartand from thestrongconverse (it holdsforevery
sequenceof channelsasshown byLemma2) we have that
E
C
[Pr(errorjP;C)]!1
f P
s I(q(x);p
k ;s
(yjx))Rg
as L ! 1, where 1
fAg
denotes the indicator function of the event A and where E
C
denotes
expectationovertheensembleofallcodesofsize2 RL
andblocklengthLM generatedaccording
to the inputdistributionq(x). By averaging also with respectto thesequence of channels and
exchangingexpectationswithrespectto Candwithrespectto P(we canalwaysdoit, sincethe
integrandisnon-negative and boundedby1) we obtain
E
C [E
P
[Pr(errorjP;C)]] !Pr 0
@ X
s2S
k ;M
I(q(x);p
k;s
(yjx))R 1
A
Then,there existsa familyofcodesC
forincreasingL such that
E
P
[Pr(errorjP;C
)]Pr 0
@ X
s2S
k ;M
I(q(x);p
k;s
(yjx))R 1
A
(37)
forL suÆcientlylarge. Because of thestrong converse, Pr(errorjP;C
)! 1 forall P such that
P
s2S
k ;M
I(q(x);p
k;s
(yjx))R . Then,inordertosatisfy(37)itmustbePr(errorjP;C
)!0for
all channel sequencesP such that P
s2S
k ;M
I(q(x);p
k;s
(yjx)) >R . Thisshows that, asymptoti-
cally, thereexist codesC
such that
Pr(errorjP;C
)!1
f P
s I(q(x);p
k ;s
(yjx))Rg
Now, let C
k;M
=C
and assume thatfora given sequenceof channels
R<
X
s2S
k ;m
I(q(x);p
k;s
(yjx)) (38)
forsome 1mM. Then,we can extend thesequence of channels byadding to fp
k;s (yjx) :
s 2 S
k;m
g other M m dummy useless memoryless channels whose output is independent of
theinput. Since themutualinformationon thelastM m blocks iszero and because of(38),
theresultingsequence of M channelsP 0
satises Pr(errorjP 0
;C
k;M
)!0. Noticethat extending
thesequenceofchannelsisequivalentto appendingdummyoutputsignalblocksz
i
independent
of the channel input to the received signal fy
s
: s 2 S
k;m
g, as described in Section 3. This
concludesthe proofof Lemma1.
In orderto proveLemma 2 we usethelimitinprobability
lim
L!1 1
L log
2
P(fx
k;s
;y
s
:s2S
k;m g)
P(fx
k;s :s2S
k;m
g)P(fy
s
:s2S
k;m g)
= X
s2S
k ;m
I(q(x);p
k;s
(yjx)) (39)
wheretheLHSis thelimitingnormalized informationdensity overthem slotsandwhere, fora
xed sequence of channels,theRHS is a constant. Therefore, the inf-informationrate and the
sup-informationrate (seedenitionsin[57 ])coincide and,from[57 , Th. 7], thestrongconverse
holds,conditionallyon thesequence fp
k;s
(yjx):s2S
k;m g.
In orderto proveLemma 3 we usethesimplerelation
[
b w 6=w
E
b w
n
fx (w)
k;s
;y
s
:s2S
k;m g2=A
k;m o
8 w2f1;:::;2 RL
g andfor allm=1;:::;M. This impliesthat
Pr(undetected errorjw;P;C
k;m
) Pr
n
fx (w)
k;s
;y
s :s2S
k;m g2= A
k;m o
w
< (40)
wherethesecondinequalityholdsforarbitrary>0andsuÆcientlylargeL,sincetheprobability
thatthechannelinputandoutputsequencesarenotjointlytypical vanishesasL!1[59 , Th.
8.6.1]. Then,Lemma3 follows from averaging (40)overall transmittedmessages.
B Probability distribution of the inter-renewal time
Thejointpdf ofT and Mcan be expressedby
f
T;M
(n;m)= 8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
: (1 p
t )
N
n=N;m=0
v(N;m)+g(N;m) n=N;1mM 1
v(n;M)+r(n;M) M nN;m=M
v(n;m) mnN 1;1mM 1
0 elsewhere
(41)
wherewe dene
v(n;m) = 0
@ n 1
m 1
1
A
(1 p
t )
n m
p m
t q(m)
r(n;M) = 0
@ n 1
M 1
1
A
(1 p
t )
n M
p M
t p(M)
g(N;m) = 0
@ N
m 1
A
(1 p
t )
N m
p m
t p(m)
whereq(m)is denedin(12), p(m) in(13) and they arerelatedbyq(m)=p(m 1) p(m).
We show that(41) isa well-denedprobabilitydistributionforanyN M >0,0p
t 1
and non-negative decreasing sequence fp(m)g with p(0) =1. Since all terms in (41) are non-
negative, itis suÆcientto showthattheirsum is 1. We usetheidentity
N 1
X
n=k 0
@ n
k 1
A
a n k
(1 a) k+1
=1 k
X
`=0 0
@ N
` 1
A
a N `
(1 a)
`
(42)
(for0a1) andwrite
X
n;m f
T;M
(n;m)= M
X
m=1 N
X
n=m
v(n;m)+ M 1
X
m=0
g(N;m)+ N
X
n=M
r(n;M) (43)
Forthe sake of brevity,we lets(`)
= 0
@ N
` 1
A
(1 p
t )
N `
p
`
t
. The rst, second and thirdterms in
theRHS of(43) aregiven by
M
X N
X
n=m
v(n;m)=1 M 1
X
s(`)p(`) p(M) 1 M 1
X
s(`)
!
(44)
by
M 1
X
m=0
g(N;m)= M 1
X
m=0
s(m)p(m) (45)
andby
N
X
n=M
r(n;M)=p(M) 1 M 1
X
`=0 s(`)
!
(46)
wherewe usedthe factthat P
N
k=`+1
q(k) =p(`) p(N). The result followsbynoting thatthe
second and third term in the RHS of (44) are the opposite of the terms given in (45) and in
(46).
C Limits
C.1 Limits for large R
We want to establishthelimitingbehaviorofthesystemthroughput forlargeR ,inthecase of
N;M !1. Tothispurposewe considerlim
R!1
1=,where is given in(24).
We needthe followinglemmas:
Lemma C.1. LetX bea RV withcdf F
X
(x). Then,8y,
1
fxyg F
X
(y)F
X
(x)F
X
(y)+1
fxyg (1 F
X
(y)) (47)
}
Lemma C.2. Ifa
n
!aasn!1,then forany non-negative niteinteger k
b
n
= 1
n+k n
X
i=1 a
i
!a for n!1 (48)
Proof. It followsimmediatelyfrom theCesaro's mean theorem[59]. }
Lemma C.3. LetX
`
bei.i.d. zero-mean RVswithvariance 2
X
. Forall >0,wehave
lim
n!1 Pr
1
n n
X
`=1 X
`
<
!
= 1
lim
n!1 Pr
1
n n
X
`=1 X
`
<
!
= 0 (49)
Moreover, forsuÆcientlylargenwe have
Pr 1
p
n n
X
X
`
<
p
n
!
e n
2
=(2 2
X )
(50)
Proof. (49)follows immediatelyformtheweak lawoflargenumbersand (50)fromthecentral
limittheorem[35], byusingtheboundon theGaussiantailfunctionQ(x)exp( x 2
=2). }
INR protocol. We let X
i
=log
2 (1+
1;s
i ) for s
i 2S
1;m and
X
=E[X
i
]. Then, for
1
>0
andb=
X +
1
we can write
lim
R!1 1
= lim
R!1 1+
P
1
m=1 Pr(
P
m
i=1 X
i
<R )
R G
(a)
lim
R!1 1
R G 1
X
m=1 1
fRmbg Pr
1
m m
X
i=1 X
i b
!
(b)
1
Gb lim
R!1 1
bR =bc+1 bR=bc
X
m=1 Pr
1
m m
X
i=1 X
i
<b
!
(c)
=
1
G(
X +
1 )
(51)
where (a) follows by applying Lemma C.1 to the RV P
m
i=1 X
i
with x = R and y = mb; (b)
followsbynotingthatb=R1=(bR =bc+1);(c)followsfromLemmaC.2 andLemmaC.3,since
lim
m!1 Pr(
1
m P
m
i=1 X
i
X
<)=1. Similarly,
2
>0 andb=
X
2
wecan write
lim
R!1 1
= lim
R!1 1+
P
+1
m=1 Pr(
P
m
i=1 X
i
<R )
R G
lim
R!1 1
R G 1
X
m=1
"
Pr 1
m m
X
i=1 X
i
<b
!
+1
fRmbg
#
lim
R!1 1
GR 1
X
m=1 Pr
1
m m
X
i=1 X
i
<b
!
+ 1
Gb lim
R!1 1
bR =bc bR=bc
X
m=1 1
(a)
=
1
G(
X
2 )
(52)
Inorder to get (a), we usethe fact that, byLemma C.3, there exists nniteand independent
ofR suchthat
1
X
m=1 Pr
1
m m
X
i=1 (X
i
X )<
2
!
= n
X
m=1 Pr
1
m m
X
i=1 (X
i
X )<
2
!
+ 1
X
m=n+1 Pr
1
p
m m
X
i=1 (X
i
X )<
p
m
2
!
n
X
Pr 1
m m
X
(X
i
X )<
2
!
+ 1
X
e m
2
2
=(2 2
X )
(53)
