Optimization of Unequal Error Protection Rateless Codes for Multimedia Multicasting

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by  Yu Cao

JOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.XX,NO.YY,MONTH2015 1
Optimization of Unequal Error Protection Rateless
Codes for Multimedia Multicasting
Yu Cao, Steven D.Blosteinand Wai-YipChan
Abstract: Rateless codes have been shown to be able to provide A critical aspect of robust multimedia multicast is channel-
greaterflexibilityandefficiencythanfixed-ratecodesformulticast coding performance. Traditional fixed-rate FEC encounters
applications. In thefollowing, we optimizerateless codes for un- the problem of channel heterogeneity as in the case of Reed-
5 equalerrorprotection(UEP)formultimediamulticastingtoaset Solomon(RS) codesthataretargetedforonespecific loss rate
1 ofheterogeneoususers.Theproposeddesignshavetheobjectivesof [12]. Ratelessfountaincodes[13] areefficientandflexiblefor
0 providingeitherguaranteedorbest-effortqualityofservice(QoS).
broadcasting or multicasting over erasure channels. The rate-
2 Arandomlyinterleavedratelessencoderisproposedwherebyusers
less property enables (1) a transmitter to generate, as needed,
n onlyneedtodecodesymbolsuptotheirownQoSlevel. Thepro-
an unlimited number of encoded symbols, and (2) a receiver
a posedcoderisoptimizedbasedonmeasuredtransmissionproper-
to successfully recover any subset of the encoded symbols of
J tiesofstandardizedraptorcodesoverwirelesschannels.Itisshown
size slightly greater than the number of information symbols.
0 that a guaranteed QoS problem formulation can be transformed
1 intoaconvexoptimizationproblem,yieldingagloballyoptimalso- Raptorcodes[14]duetotheirhighperformanceandlowcom-
lution.Numericalresultsdemonstratethattheproposedoptimized plexity are fountain codes that have been incorporated into

] random interleaved UEP rateless coder’s performance compares the third generation partnership program (3GPP) Multimedia
T
favorablywiththatofotherrecentlyproposedUEPratelesscodes. Broadcast/Multicast Services (MBMS) standard [15]. In [16],
I
. raptorcodeshavebeenextensivelyevaluatedforMBMSdown-
s
c loaddelivery. Amorerecentversionappearsin[17],andback-
[ IndexTerms: multimedia,errorcontrolcoding,unequalerrorpro- ground can be found in [18]. Rateless codes have been ap-

tection,raptorcodes,videotransmission plied to SVC-based multi-sourcestreaming [19], adaptiveuni-
1
v cast streaming [8], and SVC streaming from multiple servers
7 I. Introduction [20]. A JSCC rateless coding framework for scalable video
0 broadcast appears in [21]. Applications to distributed video
3 Multimedia transmission is a main driver for explosive data
streaming for relay/cooperation based receiver-driven layered
2 traffic growth in wired and wireless networks. While decades
multicasting is foundin [22], while [23] and [24] use fountain
0
ofresearchhavebeenconductedindesigningreliablemultime-
. codesfordistributedvideocachingviausercooperation.
1 dia transmission over error-prone channels, multimedia multi-
0 cast over lossy packet networks is still challenging due highly While the raptorcode itself is not suited for progressivede-
5 coding, multimedia has a hierarchical source symbol priority
variable channel conditions among different users, QoS con-
1 structure necessitating unequal error protection (UEP), some-
straintsandmultimediadevices,e.g.,smartphones,tablets,lap-
:
v tops. Scalablevideocoding(SVC)[3]isusefulasitlayersthe timesreferredto aspriorityencodingtransmission(PET)[25].
i Numerous UEP approaches to multimedia transmission have
X source to enable efficient progressive reconstruction at the re-
beenproposed[4][26][27][28]. In[4],MohrproposesaPET-
ceiver.
r
a Protection against channel impairments can be achieved by basedpacketizationschemefortransmittingcompressedimages
overnoisychannels. In[26], theMohrschemeis optimizedto
usingcodesthatprovideforwarderrorcorrection(FEC):Reed-
minimizeend-to-enddistortion.Optimizationofreceiver-driven
Solomon [4], low density parity check (LDPC) [5], Turbo [6]
networkshasalsobeeninvestigated[29]. Rate-distortion-based
and fountain [7] [8], as well as joint source-and-channelcod-
optimization can be found in [30]. Rather than incorporate
ing (JSCC) [5] [6] [7]. Other approaches that exploit source
codeperformanceintotheoptimization,theseexistingoptimiza-
scalability toprovideUEPusehybridautomaticrepeatrequest
tionapproachesgenerallyemploymaximumdistanceseparable
orcross-layeroptimization[9][10][11].Theaboveapproaches,
(MDS)codes. Inthispaper,codeperformanceistakenintoac-
however,weremainlyenvisionedforpoint-to-pointlinksanddo
countintheUEPratelesscodeoptimization.
notconsiderheterogeneoususers’QoS. Asaresult, adaptation
ofJSCCtomultimediamulticastisofteninefficientinthatthey Notsurprisingly,UEPratelesscodedesignmethodshavere-
catertothelowestQoSuser. centlyappeared.In[31],messagesymbolsareencodedbynon-
uniform selection of source symbols and applied to MPEG-II
Manuscript received May 07, 2014; approved for publication by Prof. videotransmissionin[32]. In[33],expandingwindowfountain
EmanueleViterbo,Editor,December13,2014.
(EWF)codesorganizesourcesymbolsintoasequenceofnested
ThisresearchissupportedbytheNaturalSciencesEngineeringandResearch
windows.In[34],EWFcodesareappliedtoscalablevideomul-
CouncilofCanadaGrantSTPSC356826-07,andbyDiscoveryGrant41731.
Y.CaoiswithHuaweiTechnologiesCanada,Kanata,Ontario,CanadaK2K ticasting.Windowingapproachesforratelesscodesthatachieve
3J1,[email protected],S.D.BlosteinandW.-Y.ChanarewithDepartment equal error protection (EEP) have been proposed in [35] and
ofElectricalandComputerEngineering,Queen’sUniversity,Kingston,Ontario,
[36]. The sliding-window(SW) rateless code design proposed
Canada,K7L3N6,{steven.blostein,chan}@queensu.ca.
Theresultsofthepaperarepresentedinpartin[1]and[2]. in [36] is applied to wireless video broadcasting in [37]. Ah-
1229-2370/14/$10.00 (cid:13)c 2014KICS
2 JOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.XX,NO.YY,MONTH2015
madet. al[38]achieveUEPinvideomulticastusingtheLuby UsersubscribersareclassifiedintoJ classesaccordingtore-
Transform(LT)[39]viablockduplication. In[40],aUEPrate- ceptioncapability. ForClass j users, receptioncapabilityδ is
j
less code based on hierarchical graph coding is proposed for definedas the proportionof symbolsthat the receivercan suc-
media streaming. However, these previous UEP rateless code cessfully receive compared to the number of transmitted sym-
design approaches [31] [33] may compromise performance as bols, 1 ≤ j ≤ J. Therefore,in eachtransmissionsession, the
theyaltertheLTcode[39]degreedistributionunlessthedegree numberofsuccessfullyreceivedencodedsymbolsforeachuser
distributionisjointly-optimizedwithUEPparameters. inClassjisδ M,whereM isthenumberofsymbolstransmit-
j
Finally, previous approaches to UEP optimization for mul- ted1. Withoutlossofgenerality,weordertheclassesaccording
timedia have focused almost exclusively on providing best- toreceptionquality,i.e.,0<δ ≤δ ...≤δ ≤1.Forexample,
1 2 J
effortQoS,i.e.,maximizationofanaveragefidelitymeasureof aClass1usermayrepresentamobilecellphonewithlimitedre-
video/image quality of end users for a given transmission rate ceptionqualityduetosizeandpowerrestrictions,whileaClass
[4] [26] [30] or with rateless codes [21] [32] [34] [37]. As 2 user may represent an automobile equipped with larger an-
rateless codes have no pre-determined transmission rate, QoS tennaandhigherbandwidthservice. Receptioncapabilitiesare
maybeachievedbytransmittingenoughcodedsymbolstomeet determined by channel quality and bandwidth between server
users’ QoS demands. In contrast, our focus is on guaranteed and receiver, and no distinction is made between overhead in-
QoS optimization, i.e., minimizing resource usage under the curredbysymbolerasuresandlostsymboltransmissionoppor-
constraints of heterogeneous QoS guarantees. While [1] and tunitiesdue to clientbandwidthrestrictions. Users in different
[2] presented early versions of this approach, this paper pro- classes may also have different QoS requirements, and outage
videsmorecompletebackground,technicaldetail,amethodto QoSguaranteesareusedtoenableuserstorecoveragivenpor-
simplify constraints, as well as an example video multicasting tionofsourcedatawithanachievedtargetprobability. Without
application. loss of generality, the term peak signal-to-noiseratio (PSNR),
Ourmaincontributionsaresummarizedasfollows: acommonmeasureusedforvisualmediaquality,isusedtode-
1. a UEP scheme is proposed that uses random interleaving noteQoS.
of raptor coders that enables direct application of already op- Let K represent the number of information symbols in a
timized standardizedraptorcodesused for 3GPPMBMS [15]. raptor-codedsource block. Assume the server transmits M =
Whenappliedtomulticastingtoheterogenoususers,lowband- (1+ε)K encodedsymbolsinordertomeetallusers’QoSde-
widthclientsneednotreceiveencodedsymbolstargetedtohigh mands,whereεisthetotaltransmissionoverheadforalllayers
bandwidthclients,whichcansignificantlyreducereceivercom- neededto combatlosses of the heterogenoususers in the mul-
plexityandtimetodecode. ticast system. For scalability, the coded source block is parti-
2. the proposed design, optimized for multimedia multicast to tionedintoLlayersindecreasingorderofimportance: Layer1
heterogeneous users, contains QoS guarantees and factors in containsthemostimportantsymbolswhileLayerLcontainsthe
ratelesscodeperformance.Withstandardizedraptorcodes,this leastimportantsymbols. Forexample,invideoorimagecom-
guaranteed QoS optimization problem is shown to be convex pression terminology, Layer 1 might represent the base layer
with a simplified solution using Karush-Kuhn-Tucker (KKT) (BL), and Layer2 the first enhancementlayer (EL). The num-
optimalityconditions. berofsourcesymbolsinLayerl isdenotedbyS ,1 ≤ l ≤ L,
l
3. through a combination of simulation and analysis, perfor- andK = L S .
manceoftheproposedrandominterleavedUEPratelessdesign l=1 l
Successfuldecodingoflayerlrequireslayers1,2,...,L−1
iscomparedtootherEEPandUEPratelesscoders. P
tobedecodable. Ratherthanjointlyoptimizingthesourceand
The paper is organized as follows: Section II describes the
channelcoders,wefocusonoptimizingchannelcodingparam-
systemsetupandproposedUEPratelesscodedesign;SectionIII
eters for a given source coder. Therefore, we assume that the
presentstheproblemformulationsforguaranteedandbest-effort
valuesofS ,1 ≤ l ≤ Lareprovidedbyapre-determinedscal-
QoS;SectionIVprovidesthesolutionforguaranteedQoS.Sec- l
ablesourcecoder.
tionIVtransformstheoriginalproblemformulationforguaran-
teedQoSintoaconvexoptimizationproblemwhereoptimalse-
lectionprobabilitiesforinterleavingareobtainedinclosedform B. ProposedUEPratelesscode
forcertaincases orelse numerically. Comparisonswith recent
We propose a randomly interleaved UEP rateless encoder
UEPratelesscodingschemesareprovidedinSectionV.
structuretoprovideFECformultimediamulticastas shownin
Fig. 1. The encoder assumes that source symbols have been
II. Systemsetupandproposeddesign
allocated to the L layers prior to encoding. Encoding is per-
A. Systemsetup
formed by randomly selecting layer l with probability ρ for
l
Amultimediaserverthattransmitsmultimediacontentsimul- l=1,2,...,Lwhere L ρ =1.Encodedoutputsymbolsare
l=1 l
taneously to multiple users is considered, which may include generated by the raptor encoder for Layer l with code dimen-
P
streamingwithstrictdelayrequirements.Multimediacontentis sionS , degreedistributionΩ (x)andprecodeC . Theoverall
l l l
dividedintomultiplecodedblocks. Theserverfirstcompresses encodeddatastreamconsistsofinterleavedraptor-encodedsym-
each source block using a pre-defined source coder and then bolsfromtheLencoders. Fromtheabovedefinitions,εcanbe
addserrorprotectiontothesourceinformationusingarateless,
e.g.,raptororLTcode.Encodedsymbolsarethenmulticastover 1Foranalyticalsimplicity,thenumberofreceivedsymbolsforeachuserclass
awirelesslossypacketnetwork. ismodeledasδjmultipliedbythetotaltransmittedasin[34].
CAOetal.:OPTIMIZATIONOFUNEQUALERRORPROTECTIONRATELESS... 3
s.t. Prob(PSNR ≥γ )≥P , j =1,2,...,J, (3)
Layer 1 S1 RDiampetonrs eionnc=odeSr1 Sweithle cptr olabyaebri llity (cid:85)l Cuslaesrss 1 wherePSNR represejntsthjePSNjRofthesuccessfullyrecov-
Layer 2 S2 Raptor encoder (cid:71)1 eredsourcedatjaoftheClassjusergivenM =(1+ε)K trans-
.. Dimens..ion=S2 ... (1(cid:14)(cid:72))K (cid:71)2. Cuslaesr.ss 2 amnidtteodustaygmebporlos,baanbdiliγtyj,arnedsp1e−ctPivjedlye,nfootrettahregeCtlPasSsNjRutsherre.shTohlde
. . .
LayerL SL RDiampetonrs eionnc=odSerL Efsotnruecnaotamdiend s dyimgibtaoll (cid:71)J . atiiomnoisfttoheapllroocbaatebicliotideisngρlr,a1te≤s alc≤rosLs.layers through optimiza-
Class J The source (e.g., video, image) coder is assumed to be pro-
Transmitter/server Multicast users
channels gressive,sothatthereconstructionmediaqualityisdetermined
mainlybythesymbolerrorsinthelowestlayerencounteredin
Fig.1. TheproposedrandominterleavedUEPraptorcodingmethod.
therecoveryprocess. Letq ,l=1,2,...,L,representthePSNR
l
achievedwhenLayers1tol aresuccessfullyrecoveredbyrap-
tordecoding,whereq ≤ q ... ≤ q . Foragivensourcecoder,
1 2 L
lowerboundedby if the source PSNR is represented as non-decreasing function,
f(·), ofthetotalnumberofsourcesymbolsdecodedbythere-
1 J ceiver, then q = f( l S ). For each class 1 ≤ j ≤ J, let
ε = (Sel) /δ −1, (1) l 1 l
min K i=1 i i gj ∈{1,2,...,L}betPheminimumindexthatsatisfiesqgj ≥γj.
X Inorderto satisfyPSNR ≥ γ , usersinClassj requirethe
gj j
where (Sel) ∈ {S ,S ,··· ,S } denotes the code dimension raptordecodertosuccessfullydecode,atminimum,Layers1to
i 1 2 L
oftheselectedlayerforuserClassi. g .ForagivenUEPraptorcodedesign,letP (l,j)representthe
j e
The proposed rateless coded scheme uses the random inter- errorprobabilitythattheClassj decoderfailstodecodelayerl
leavingtoachieveUEP.Whileprobabilisticencodinghasbeen giventransmission overheadε and receptionqualityδ . In the
j
usedinEWFratelesscodesin[34],aswellasin[41]and[42], most stringent case when decodingerrorsacross layers are in-
anadvantageoftheproposedschemeinFig. 1isthatthediffer- dependent,QoSrequirementsofenduserscanbesimplifiedto
entlayerscanbeencodedanddecodedseparately. Inaddition,
gj
in[31]and[34],degreedistributionsandselectionprobabilities
(1−P (l,j))≥P j =1,2,...,J. (4)
e j
need to be optimized jointly, which is a complicated task. A
l=1
practicaladvantageofFig.1whenappliedtoamulticastsystem Y
B. Best-effortQoSformulations
for users with different bandwidth constraints, low bandwidth
clients need not receive symbols generated from source layers While the above formulation focuses on minimizing trans-
targetinghighbandwidthclients,whichreducesthecomplexity mission overhead subject to satisfying guaranteed user QoS,
andtime-to-decodeforlowBWclients. this subsection considers transmission overhead that is upper
Itisworthnotingthatonemayaltertheorderingoftheout- bounded due to delay constraints or cost. For this scenario,
putsymbolsfromtherandominterleavedUEPraptorcoderus- given a maximum transmission overhead, ε , the service
max
ingschedulingalgorithmswhilemaintainingthepriorityofeach providerattemptsto provideusers ofdifferentclasses with the
layer. Investigationsalong these lines have been recently pro- best possible QoS. The following best-effortQoS problemex-
posedin[43]and[44]. Unlike[45][46],theproposedneednot tends that in [34] by 1) considering both constrained and un-
specifyapacketizationstructure;theschememaybeappliedto constrained cases, 2) allowing for allocating different weight-
datapacketsratherthantosymbols. ingfactorstodifferentuserclassesaswellas3)possessingthe
previouslymentionedadvantagesoftheproposedrandominter-
III. ProblemformulationswithQoSconstraints
leavedUEPraptorcodes.
A. GuaranteedQoSformulation The expected PSNR of users in Class j, which serves
as a measure of the best-effort QoS, can be evaluated as
We consider users that require playback media at a quality
E(PSNR ) = L p q , where q is the PSNR achieved
no lower than their own QoS requirement. Since the transmit- j l=1 l,j l l
when Layers 1 to l are successfully recovered,where p rep-
ter has to provide guaranteed QoS for all user classes before l,j
P
resentstheprobabilitythataClassj usersuccessfullyrecovers
the start of transmission of the next source block, system de-
Layers1tolbutfailstorecoverLayerl+1. Theoptimization
lay and throughputfor each source block is determinedby the
balancesusers of differentclasses with differentchannelqual-
maximumnumberoftransmittedsymbolsrequiredtosatisfythe
ities by assigning weighting coefficient w for Class j where
QoSofeachindividualuserclass. Asdelayisacriticalissuein j
multimedia multicast, the objective is to provide differentlev- 0 ≤wj ≤1, Jj=1wj = 1. Thechoiceofwj dependsonboth
theuserclassimportanceaswellasthenumberofusersinthat
els of QoS guarantees according to users’ requirements while
P
class. The weighted average PSNR over all user classes then
minimizingtotaltransmissionoverheadε:
becomestheobjectivefunction:
Problem1.0(GuaranteedQoS):
Problem2:(Best-effortQoS)
min ε (2) J L
ρ1,...,ρL max w p ·q (5)
j l,j l
ρ1,ρ2,...,ρL !
j=1 l=1
X X
4 JOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.XX,NO.YY,MONTH2015
One way to improve code performance for layers with fewer
subjectto ε≤ε (6)
max symbolsistomergesourcelayerswithsimilaroptimizedselec-
where tion probabilitiesρj into largerlayers. However, for video the
conditionk <200isunlikelytooccur.
p = Pe(l+1,j) li=1(1−Pe(i,j)) l =1,2,...,L−1 We also remark that standardized raptor codes outperform
l,j ( Li=1(1−PQe(i,j)) l =L. the recently proposedSW-raptorcodes [37]. For example, ac-
cording to Fig. 2(b) of [37], the SW-raptor codes have a de-
(7)
Q codingfailureprobabilityofalmost100%withcodedimension
InProblem2,noguaranteedminimumQoSisprovided. For K = 5000 and overhead ε = 0.03 while standardized raptor
a given maximum transmission overhead, the service provider codes have negligible decoding failure probability at the same
may instead aim to provide best-effort QoS to multiple user codedimensionandloweroverheadε=0.01accordingto(11).
classes, butunderthe additionalconstraintof a minimumQoS WhenmoregeneralLTorraptorcodesusingiterativedecod-
guaranteeforeachuserclass: ingareemployed,thedecodingfailureprobabilityP (l,j)can
e
Problem 3: (Best-effortQoS with constraints on individual beapproximatedbyassumingthatsymbolerrorsiniterativede-
classes) codingaremutuallyindependent
J L Pe(l,j)=1−(1−el,j)Sl, (12)
max w p ·q (8)
j l,j l
ρ1,ρ2,...,ρL j=1 l=1 ! whereel,j is thesymbolerrorprobabilityofa Class j userde-
X X
codingLayerl (also see (3) of [34]) whichcan be analytically
subjectto ε≤ε (9)
max determinedbyand-ortreeanalysis[47]. Sinceeachlayerisen-
coded by a separate rateless code, evaluating the symbol error
gj probability of each layer can be consider as a special case of
and (1−Pe(l,j))≥Pj j =1,2,...,J (10) (6)and(7)in[31]whereuniformselectionisused(kM = 1in
l=1 [31]),andEq. (12)canbeapproximatedusing
Y
wherep isgivenby(7)andg ∈{1,2,...,L} (j =1,2,...J)
l,j j
1 n=0
aisstpheecmiailnciamsuemoflPayroebrlienmde3xwthiatthsoauttisufiseesrqQgojS≥coγnj.stPraroinbtlse.m2is enl,j =(exp(−tSlδljΩ′(1−eln,−j1)) n≥1 (13)
In the next section, we show that Problem 1.0 can be trans-
formed to an equivalent convex optimization problem when where Ω(.) is the LT code degree distribution, Ω′(x) denotes
standardized raptor codes are employed. Unfortunately, while derivativewithrespecttox,nisthenumberofdecodingitera-
Problems 2 and 3 cannot be similarly transformed due to the tionsandtl isthetotalnumberofencodedsymbolstransmitted
form of the p expressions, they can still be solved numeri- forLayerl ineachtransmissionblock. Theasymptoticsymbol
l,j
callybysearchingthe(L−1)-dimensionalparameterspaceof error probabilityel,j = limn→∞enl,j of iterative decodingcan
{ρ1,ρ2,...,ρL−1},checkingtheconstraints(10)andtheresult- beestimatedbychoosingalargevalueninEq.(13)(see[47]).
ing averagePSNR (8). When L = 2, the numericalmethodis
B. Convexityanalysis
significantlysimplifiedasonlyρ ∈ [0,1]thatgivesthemaxi-
1
mumaveragePSNRneedstobedetermined. Numericalresults Foragiventransmissionoverheadε,t = (1+ε)Kρ ,2 and
l l
andcomparisonsforProblems2and3areprovidedlater. satisfies L t = (1+ε)K. Whenstandardizedraptorcodes
l=1 l
areused,substitutingm = t δ andk = S intoEqs. (11)and
l j l
IV. SolvingtheguaranteedQoSproblem P
(4),andtakingthelogarithmoftheconstraintsdescribedby(4),
A. Evaluationofdecodingfailureprobability Problem1.0istransformedto:
In theproposeddesign,existinghigh-performancestandard- Problem1.1:
ized raptor codes can be directly applied, which enable low
encoding/decoding complexity and overhead. Details about min ΣL t (14)
i=1 i
the pre-code, degree distribution and code constructioncan be t1,...,tL
foundin [15], (AnnexB). When standardizedraptorcodes are
employedwithmaximumlikelihood(ML)decodingforcodedi- s.t.−Σgj log[1−c αtl]+logP ≤0, j =1,2,...,J, (15)
mensiongreaterthan200,thedecodingfailureprobability,i.e., l=1 l j j
failuretodecodeksourcesymbolsaftermsymbolsaresuccess- wherecl = ab−Sl, αj = bδj andgj ∈ {1,2,...,L}. Thecon-
fullyreceived,havebeenshown,throughextensiveexperimen-
straint that t is non-negative is implicitly guaranteed by the
l
tation,tobeaccuratelymodeledby[16],
log(.) function. To ensure an integer solution, we compute
t ,t ,...,t as if real-valued, then round to the nearest larger
1 if m≤k 1 2 L
Pr(m,k)= (11)
e (abm−k if m>k 2Strictly speaking, tl is aBinomial-distributed random variable with mean
(1+ε)Kρl. However,therandomizationoftlhaslittleeffectontheproblem
ofinterestwhenaveragedoveralargenumberofrealizations. Inaddition,one
whereconstantsa = 0.85, b = 0.567. Fork < 200, Eq. (11)
canalwaysscheduletheselectionoflayerstomakesurethattlisproportional
underestimates the error probability due to short block length. toρl.
CAOetal.:OPTIMIZATIONOFUNEQUALERRORPROTECTIONRATELESS... 5
integer. Although the above transformation uses the decoding used in the formulation of [34], and if all the inequality con-
failureprobabilityevaluationofstandardizedraptorcodesgiven straintsareactive. Usingtheaboveassumption,thesolutionto
by Eq. (11), a similar method can be applied to other decod- Problem1.1canbe obtainedbyfindingt usingthe constraint
1
ing failure probability models that can be approximatedby an for Class 1 in Eq. (15) and substituting the solution of t into
1
exponentialfunction. thenextconstraint,solvingfort withtheconstraintforClass2
2
To solve Problem 1.1, we first prove convexity. As the ob- inEq. (15)etc. untilallofthevariablest ,t ,...,t aredeter-
1 2 L
jective function is linear, we only need to prove that the con- mined. However,since this simplificationhasnotbeenproven
straintfunctionsareconvexwithrespecttot ,l = 1,2,...,L. It tobeequivalenttoProblem1.1ingeneral,thesolutionobtained
l
can be shownthatforl = 1,2,...,L,the secondderivativesof inthismannerhastobeverifiedusingtheKKToptimalitycon-
−log(1−c αtl)withrespecttot satisfy ditions.Ifalltheinequalityconstraintsareactive,Eqs.(17)and
l j l
(19)areautomaticallysatisfied. Therefore,ifweobtainasolu-
∂2[−log∂(1t2−clαtjl)] = c(lα1tj−l(lcogααtlj)2)2 >0 j =1,2,...,J. (16)wtioencatn∗ soufbPsrtiotbulteemthe1.v1albuyesooflvti∗nigntfoj(Etq∗.)(2=0)0a,njd=ob1ta,i2n,λ..∗.,.JIf,
l l j λ∗ satisfiesEq. (18),i.e.,λ∗ ≥ 0,j = 1,2,...,J,thenwehave
j
proventhatthevalueoft∗weobtainedisindeedanoptimalso-
According to the second order condition of convex functions
[48], −log[1 − c αtl] is a convex function of t . Since lution of Problem 1.1. If the KKT optimality condition is not
l j l
satisfied,thennumericalmethodscanstillbeusedtosolvethis
nonnegative weighted sums preserve convexity [48], the con-
convexoptimizationproblem.
straint functions (15) are convex functions of the vector t =
[t ,t ,...,t ]T. Problem 1.1 can therefore be solved numeri-
1 2 L
C. Class-to-layermappingalgorithm
callybyavailableconvexoptimizationalgorithms[48]. We re-
markthattheaboveconvexityholdsnotjustforvaluesofaand Inthefollowing,weproposeanalgorithmtotransformagen-
bintheexponentialmodelofEq.(11)from[16]butalsomore eral guaranteed QoS problem into a problem with one-to-one
generallyovertherange0 < a < 1and0 < b < 1whichrep- mappingbetweenuser classes and channelcodinglayers. The
resentawidefamilyofexponentialfountaincodefailureproba- ideaistoreducethedimensionalityoftheproblembyremoving
bilitymodels. redundant user constraints and merging source-coding layers.
Let t = [t1,t2,...,tL]T and λ = [λ1,λ2,...,λJ]T be Theprocessisexplainedinthefollowingalgorithm:
the variable vectors of the primal and dual problems of Prob-
lem 1.1, respectively. If t∗ = [t∗,t∗,...,t∗]T and λ∗ = Algorithm1:(Class-to-layermappingalgorithm)
1 2 L
[λ∗,λ∗,...,λ∗]T represent sets of primal and dual optimal Step1(Userclassamalgamation):Repeatthefollowingclass
1 2 J
points, theymustsatisfy the Karush-Kuhn-Tucker(KKT)opti- amalgamationoperationuntil gi < gk for every i < k, where
malityconditionsfortheobjectivefunctionf (.)andconstraint 1≤i≤J,1≤k ≤J: foranypairofuserclassindicesiandk
0
functionsfj(.): wherei < k (henceδi < δk),ifClassiusershavethesameor
highertargetPSNRthresholdthanClass k users(i.e., γ ≥ γ
i k
∗
f (t )≤0, j=1,2,...,J (17) org ≥g ),weabsorbClasskintoClassi.
j i k
Step 2 (Source layer merging): Repeat until for every layer
1 ≤ l ≤ L,thereexistsaclassj,1 ≤ j ≤ J suchthatg = l:
∗ j
λ ≥0, j =1,2,...,J (18)
j if there existsa source layerl wherethere is nocorresponding
userclass(i.e.,noj existssuchthatg = l),Layerslandl+1
j
λ∗f (t∗)=0, j=1,2,...,J (19) are mergedto form a new sourcelayer l′ with codedimension
j j Sl′ =Sl+Sl+1.
Step 1 finds a set of the most demanding user classes with
respectto their channelconditions; Step 2 reducesthe number
J
∇f (t∗)+ λ∗∇f (t∗)=0 (20) ofchannelcodinglayerstotheminimumwithoutcompromising
0 j j
the performance. After performingAlgorithm 1, we can show
j=1
X
thefollowingfact:
where here f0(t) = ΣLi=1ti and fj(t) = −Σgl=j1log[1 − Lemma1:AfterperformingAlgorithm1,L=J andgj =j
c αtl]+logP ,j=1,2,...,J.SincetheoriginalProblem1.1is forj = 1,2,...,J. IfforeveryClass k thathasbeenabsorbed
l j j
convexandsatisfiesSlater’scondition,theaboveKKToptimal- intoClassiinStep1,P ≥P isalsosatisfied,thenthenewop-
i k
ity conditions provide the necessary and sufficient conditions timization problem after performingAlgorithm 1 is equivalent
foroptimality[48].Ingeneral,solvingtheKKTconditionisnot to Problem 1.1. In addition, any further partitioning of layers
straightforward. However,ifwecanidentifyasetofinequality cannotreducethe minimumtransmission overheadrequiredto
constraintsthataremostlikelytobeactive,i.e.,achieveequality achievetheQoSrequirements.
attheoptimalsolution,thenwecanobtainacorrespondingset Proof: First we showthatanyQoSconstraintdroppedfrom
ofprimalanddualsolutionpointsandverifytheoptimalitywith Step1(userclassamalgamation)isirrelevant.SupposetheQoS
KKTcondition. constraintofClassiusersissatisfied,i.e., gi (1−P (l,i))≥
l=1 e
AsimplificationtoProblem1.1arisesifwehaveaone-to-one P . Sincei<k,wehaveδ <δ . Hence,Classkusersreceive
i i k
Q
mapping between user classes and channel coding layers, i.e., morecodedsymbolsthanClassiusers.Therefore,thedecoding
g = j forj = 1,2,..,J andL = J, whichisthe assumption failureprobabilityP (l,i) > P (l,k)forall1 ≤ l ≤ L. Then,
j e e
6 JOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.XX,NO.YY,MONTH2015
becauseg ≥ g , fromthe assumptionofLemma1, P ≥ P , operateatthesymbollevel. Weassumetherearefourclassesof
i k i k
and users with reception capabilities and QoS requirementsshown
inTable1.
gk gk
(1−Pe(l,k)) > (1−Pe(l,i)) Table1. ExampleofuserclassesandtheirQoSrequirements
l=1 l=1
Y Y Userclassindex(j) 1 2 3 4
gi Userreceptioncapabilityδj 0.4 0.5 0.6 1
≥ (1−P (l,i))≥P ≥P .(21)
e i k UserQoSreq.(PSNRthr.γj(dB)) 25.79 29 27.25 40.28
l=1 #DecodedsymbolstoachieveQoS 400 1155 700 3800
Y
#Decodedsourcelayersrequired(gj) 1 4 2 15
Hence,theQoSconstraintforClasskusersisalsosatisfied. ProbabilitythresholdPl 0.8 0.9 0.85 0.95
NextweshowthatafterperformingAlgorithm1,thenumber
of sourcelayersL andthe numberofuser classes J are equal. Using the previously described simplification strategy for
Theclassamalgamationprocedureensuresthatthesetgj, j = classandlayermapping,weobservethatg2 >g3whileδ2 <δ3,
1,2,...,J is monotonically increasing with j. This fact does whichmeansthatClass3usershavebothbetterreceptioncapa-
notchangeafterperformingthesourcelayermergingprocedure. bilitiesandlowerPSNRrequirementsthanClass2users.There-
Since gj ∈ {1,2,...,L}, we have L ≥ J. On the other hand, fore, the QoS constraint from Class 3 users can be dropped.
source layer merging ensures that for any l = 1,2,...L, there Then, since the numberof layers requiredby the three classes
existsanintegerj ∈ {1,2,...,J}suchthatgj = l. Therefore, are1,4and15,afterthelayer-mergingprocedureofAlgorithm
we also have L ≤ J. Thus, L = J. Together with the fact 1,weobtainanewsetofchannellayerswithLayer1compris-
thatgj ismonotonicallyincreasingwithj,wecanconcludethat ing the BL, Layer 2 consisting of the first 3 ELs, and Layer 3
gj =j forj =1,2,...,J. consistingofthefourthtofourteenthELs. SinceP2 >P3,from
Finally,tocompletetheproof,intheappendixweshowthat Lemma 1, the new problem after mapping is equivalentto the
any further partitioning of layers cannot reduce the required original problem. The parameters of the transformed problem
minimumtransmissionoverhead.QED. afterthemappingareshowninTable2.
Remark1:TheconditionthatforeveryClasskthathasbeen
absorbed into Class i, P ≥ P , is a sufficient condition for Table2. UserclassesandQoSrequirementsafterthemapping.
i k
Lemma1 butnotanecessarycondition. Evenif thiscondition
Combinedclass-layerindex(jorl) 1 2 3
is notsatisfied, it is possible that the transformedproblemdue Receptioncapabilityδj 0.4 0.5 1
to Algorithm 1 results in the optimal solution. In addition, if PSNRthresholdγj(dB) 25.79 29 40.28
NumberofdecodedsymbolstoachieveQoS 400 1155 3800
thisconditionisviolated,toensurethattheoptimalsolutionof
Numberofdecodedlayersrequiredgj =j 1 2 3
the transformedproblemis theoptimalsolutionofthe original
ProbabilitythresholdPj 0.8 0.9 0.95
problem,wecanalwaysverifyiftheobtainedsolutionsatisfies NumberofsymbolsSlineachlayer 400 755 2645
all the constraints of the user classes that have been amalga-
mated in Step 1. If not, the convexProblem1.1 can be solved To determine the interleaving probabilitiesfor the standard-
numerically.ThisisfurtherillustratedinSectionIV-D. izedraptorcodesforthethreenewlayers,ρ ,ρ andρ needto
1 2 3
Remark2:Forbest-effortQoSProblem3,thetransformation bedeterminedtominimizet +t +t suchthat
1 2 3
giveninAlgorithm1maynotapply,asanoptimalsolutionalso
dependsonthefidelitymeasureofthemultimediasource. (1−ab(t1δ1−S1))≥P
1
Remark3: IntheoriginalgeneralproblemLandJ arearbi- (1−ab(t1δ2−S1))(1−ab(t2δ2−S2))≥P (22)
 2
trary,whichmeansitispossiblethatauserwithworsechannel (1−ab(t1δ3−S1))(1−ab(t2δ3−S2))(1−ab(t3δ3−S3))≥P3.
qualitymayhaveahigherQoSrequirement. Lemma1andthe
mappingalgorithmtransformtheoriginalproblemtoaprogres- Assumingalltheinequalityconstraintsareactive,weobtain
sivetransmissionproblemwherethereisaone-to-onemapping
a minimum overheadε = 36.2%, which is achievedwhen
min
betweenuserclassesandchannelcodinglayers.
ρ = 0.1946, ρ = 0.2933 and ρ = 0.5121. The solution
1 2 3
Remark4: InthecaseofL=J,thetransmissionoverhead is then verified to be optimal using KKT conditions. In con-
canbelowerboundedbyEq.(1)whichisindependentofcode
trast, equal error protection (EEP) allocation requires a mini-
optimization. Minimizing ε, as in Problems 1 and 1.1, maxi- mumoverheadof152%,afactorofoverfourhigher.
mizescodeperformance.
With the optimal selection parameters, we find that Class 3
usersoftheoriginalproblem(Table1)cansuccessfullydecode
D. Videomulticastingnumericalexample
the base layer and one enhancement layer with a probability
We now illustrate the mapping process and solution to the higherthan99.9%.Thismeansthatevenifthetargetprobability
guaranteed QoS problem for multicasting a H.264 SVC [3] thresholdP = 99%inTable1,whichviolatestheassumption
3
video-codedstreamwhichcontains15layers:abaselayer(BL) ofLemma1, theproblemtransformedbyAlgorithm1stillhas
and 14 enhancementlayers (ELs). Since our focus is on opti- thesameoptimalsolutionastheoriginalproblem. Asafurther
mizing a channel coder for a given source coder, the number remark,letussupposethattheconditionsofLemma1werevi-
ofinformationsymbolsandthecorrespondingPSNRvaluesare olated,andweassumetheextremecaseofP = 99%andvary
3
takenfromTableIof[34].Asin[34],eachsourcesymbolrepre- thevalueofδ withintherange0.5=δ <δ ≤1=δ . Inthat
3 2 3 4
sents400sourcebits. TheUEPratelessencodersanddecoders case,onlywhen0.5 < δ < 0.503,ourobtainedsolutiondoes
3
CAOetal.:OPTIMIZATIONOFUNEQUALERRORPROTECTIONRATELESS... 7
1.4 3
UEP
EEP
1.2
2.5
Proposed UEP
Minimum transmission overhead000...1468 Minimum transmission overhead1.125 OEEptPim oaple orpaetinragt ipnogi nptoint
0.2 0.5
0 0
5 10 15 20 25 30 35 40 45 50 0.2 0.3 0.4 0.5 ρ 0.6 0.7 0.8 0.9
S/K × 100 1
1
Fig.2. UEPversusEEPforvaryingS1/S2,standardizedraptorcodes, Fig.4. Theeffectoflayerallocationprobabilityρ1,standardizedraptor
K=1155;δ=[0.5,0.9];P =[0.95,0.9]. codes,L=2;K=1155;S=[400,755]δ=[0.5,0.9];P =[0.95,0.9].
1.4 byassumingthatallinequalityconstraintsareactive.Allresults
UEP
EEP showninFigs.2to4wereverifiedtosatisfytheKKToptimality
1.2
conditions. To achieveEEP, the ratio ρ /ρ is fixedto S /S .
mission overhead0.18 Fonpuigtmi.mb2eizrseshdoofwUbsEitPsmiiannnitdmheEutmEwPotrrlaaanypsetmorsrisicssoivodanersioeavd1se.rtFhh2ieega.rda3,ticεoo,mbreepqtawurieerese1ndUtEfho2Per
m trans0.6 and EEP as a function of channelreception quality of the first
Minimu0.4 userclass, δ1. ItcanbeseenthatUEPhasa significantadvan-
tage over EEP wheneverthe channelreception qualities of the
0.2
twoclassesdifferappreciably. Fig. 4plotsminimumtransmis-
00. 5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 sionoverhead,ε,asafunctionofselectionprobabilityρ1where
δ1/δ2 itisobservedthatεisverysensitivetothechoiceofρ1. Inpar-
ticular,anon-optimizedallocationschememaybesignificantly
Fig.3. UEPversusEEPforvaryingδ1/δ2,standardizedraptorcodes, outperformedbyEEP.
S=[400,755];δ2=0.9;P =[0.95,0.9].
To enable comparison with the UEP raptor code from [31]
as well as with EWF codes from [34], rather than use raptor
codes,weemployiterativelydecodedLTcodesthathavedegree
notsatisfytheQoSconstraintofClass3users.Inpractice,how- distribution[14]
ever, distinct classes would have a greater reception capability
differencethan 0.003 ×100%=0.6%. Ωr(x)=0.007969x+0.493570x2+0.166622x3
0.500
+0.072646x4+0.082558x5+0.056058x8+0.037229x9
V. Numericalandsimulationresults
+0.055590x19+0.025023x65+0.003135x66 (23)
This section provides comparisons of the proposed random
interleavedUEPratelesscodedesigntoEEPcodesandtoother for all layers as used for UEP codes in [31] and for the EWF
recentUEPratelesscodes. Theparametersofthedifferentsce- code[34]. Thatis,foranalyticalsimplicity,nopre-codeisused
nariosare describedin the correspondingfigurecaptions. Per- in any ofthe schemes. The decodingfailureprobabilityP on
e
formanceof LT codesare evaluatedusing and-ortree analysis theleftsideoftheconstraintfunctionsinEq.(4)isevaluatedas
while standardized raptor codes are evaluated using Eq. (12). follows: thesymbolerrorprobabilitye ofLayerlfortheUEP
l
Simulationsarealsousedtoconfirmtheand-ortreeanalysis. ratelesscodesin[31],theEWFcode,andtheproposedrandom
Figs. 2to4comparetheproposedUEPdesigntoEEPdesign interleaved scheme are estimated by and-or tree analysis and
fortheguaranteedQoSproblemwhenstandardizedraptor obtained using Eqs. (6) and (7) in [31], Eq. (7) in [33], and
codesareemployed. Theminimumtransmissionoverheadis Eq. (13) in this paper, respectively. The failure probability of
evaluatedusingthemethoddescribedinSectionIV-D.Forsim- decodingeachlayerisestimatedasPe(l)=1−(1−el)Sl.
plicity, only two layers are considered. The dimension of the Parameter optimization of the other schemes can be found
standardizedraptorcodeusedinlayerlisS . Theinefficiencies in [2] and are not reproducedhere. Fig. 5 in [2] provides the
l
incurred by the standardized raptor codes are characterized by minimumtransmissionoverheadrequiredtosatisfyalltheuser
thedecodingfailureprobabilityPr(.)inEq. (11)andaresmall constraintsoftheproposedrandominterleavedschemeaswell
e
asexpected.Theoptimalselectionprobabilityρ andminimum as that in [31] using different values of k , a parameter that
1 M
overhead for the UEP scheme are obtained by the simplified governsthedegreeofnon-uniformityofinputsymbolselection.
method described in Section IV for solving Problem 1.1, i.e., Fig. 6in[2]showsasimilarcomparisonbetweentheproposed
8 JOURNALOFCOMMUNICATIONSANDNETWORKS,VOL.XX,NO.YY,MONTH2015
schemeandtheEWFcode. Thesize ofthefirstwindowinthe random-interleaveddesignwouldlikelybelowerduetoalower
EWF code is fixed to the number of symbols in Layer 1 (S ). dimensiondecodingmatrixobtainedfromseparate-layerdecod-
1
Parameter Γ is the probability of choosing the more impor- ing. In addition, the code structure and generating matrix of
1
tant first layer during encoding (see [34]). It can be observed systematic standardized raptor code implementation has been
that when all schemesare optimized, the proposedrandomin- highly optimized, includingthe decoding schedule in the code
terleavedratelesscodeperformancematchesthatof[31]aswell constraintprocessor[49]. Totheauthors’bestknowledge,such
as [33]. The existence of two local minima in Fig. 6 in [2] techniqueshavenotbeenappliedtoEWFcodes,whichmayalso
is due to the symbolerror rates of the more importantbits not becomplicatedbytheiroverlappingstructure.
decreasingmonotonicallyasΓ increases(seeFig. 1in[34]). Fig.6showssourcereconstructionquality,intermsofPSNR,
1
An advantageof EWF codes [33] overthose in [31] is flex- of the proposedrandom interleaved and EWF schemes for the
ibility in deploying different degree distributions for different best-effort QoS formulations of Problems 2 and 3. Transmis-
windows. Fig. 5 plots transmission overheadas a function of sion of H.264 SVC coded CIF Stefan video sequence [34] is
numbers of symbols in the first layer or window for the three performed in two layers, with the first (base) layer containing
UEPschemeseachusingLTcodesafteroptimizationovertheir S =400symbolsandallenhancementstreamscomprisingthe
1
respective parameters where different degree distributions are secondlayerwith S = 3400symbols. Successfullydecoding
2
applied to differentEWF codewindowsas well as to different
layersoftheproposedUEPscheme. Degreedistributions,cho-

40
Proposed UEP, LT codes,Ω (x)=Ω(x)=Ω(x)
2 EWF, Ω1(x)=Ω2(x)=Ωr(x) EWF code, LT codes, Ω1(x)=1Ω2(x)=2Ωr(x)r
EWF, Ω(x)=Ω(500, 0.03, 0.5), Ω(x)=Ω(x) Proposed UEP, standardized raptor codes
1.9 1 rs 2 r
on overhead11..78 PPNrrooonpp−ooussneeiddfo UUrmEE PPse,, l ΩΩec11t((ixxo))n== ΩΩU2rEs((Px5),0= Ω0Ω(,x r(0)x=.)0Ω3r,(0x.)5), Ω2(x)=Ωr(x) Average PSNR35
missi1.6 30
ns
a
m tr1.5
u
m
ni1.4
Mi 25
0 0.05 0.1 0.15 0.2 0.25 0.3
1.3 ρ (for proposed UEP) or Γ (for EWF code)
1 1

500 1000 S 1500 2000 Fig.6. AveragePSNRperformanceofUEPschemes.LTcodeswithiterative
1
decoding,L=2;S=[400,3400];δ=[0.55,1];P =[0.95,0.8];εmax=
Fig.5. PerformancecomparisonsusingLTcodeswithdifferentdegree 1;w=[0.5,0.5].
distributionswithL=2;K=9000;δ=[0.4,0.8];P =[0.95,0.8].Also
shownareEWFfrom[33]andnon-uniformselectionfrom[31].
thefirstlayerprovidesaPSNRof25.79dBwhiledecodingboth
layersprovidesaPSNRof40.28dB.Performanceisplottedas
sen for the more important bits (MIB) and less important bits averagePSNRversusselectionprobabilityρ fortheproposed
1
(LIB), are denoted as Ω (x) and Ω (x), respectively. Degree randominterleavedschemeandthefirstwindowselectionprob-
1 2
distribution Ω (x) described by Eq. (23) is used as well as a ability Γ of the EWF code. Given ρ or Γ , average PSNR
r 1 1 1
truncatedrobustsolitondistribution(RSD)Ω (k ,δ,c),where is obtained numerically by setting ε = ε and substituting
rs rs max
k isthemaximumdegree,isappliedtotheMIBfortheEWF thecorrespondingdecodingfailureprobabilitiesp (i,j)into(7)
rs e
andproposedrandominterleavedschemes. ThetruncatedRSD and (8). It should be noted that selection probabilities ρ and
1
has better error performancecomparedto Ω (x) at the cost of Γ for the two different schemes are not directly comparable.
r 1
higherdecodingcomplexity. ItcanbeseenfromFig. 5thatthe Forthecross-markedandstar-markedcurves,wehaveusedthe
truncatedRSD fortheMIBprovidesa significantperformance LTcodewithaniterativedecoderanddegreedistributionΩ (x)
r
boost for both schemes. When the same degree distributions applied to all windows and layers. For these parameters, both
areused,theproposedrandominterleavedschemematchesthe the proposed random interleaved and EWF schemes provide a
performanceofexistingschemes. maximum average PSNR of around 32.4 dB. For Problem 3,
WenotethatinFig. 4,ε =0.432,indicatingthatthecode thefeasibleregionsofselectionprobabilitiesρ andΓ areob-
min 1 1
performancenearlyachievesminimumoverheadwhilethecode tainedbycheckingconstraints(10). WenotethatforProblem3,
performanceshown in Fig. 5 does notcome close to the min- themaximumachievableaveragePSNRsremainthesamesince
imum overhead. This difference is mainly attributable to the bothoptimaloperatingpointsoftheproposedUEPschemeand
useofstandardizedraptorcodes,whichincludesahighperfor- the EWF code lie inside the feasible regions. The diamond-
mancepre-codeaswellas efficientmaximumlikelihood(ML) markedcurveshowstheresultswhenstandardizedraptorcodes
decoding in contrast to the iterative decoding used for Fig. 5. areemployedfortheproposedrandominterleavedUEPscheme.
It can be arguedthatthe performanceof existing UEP designs A maximum average PSNR of 40.28 dB can be achieved for
in [31] and [33] can similarly benefit from a precode and ML 0.11 ≤ ρ ≤ 0.18, which, as expected, is significantly higher
1
decoding. However,ML decodingcomplexityof the proposed thantheothertwoLTcodedcurves. Wecanalsoobservefrom
CAOetal.:OPTIMIZATIONOFUNEQUALERRORPROTECTIONRATELESS... 9
Fig. 6 that different choices of ρ result in significant differ- Appendix
1
encesinaveragePSNR,showingtheneedforoptimization. Fi-
We prove the last part of Lemma 1, i.e., after performing
nally,weobservethatthesteepperformancecurveofstandard-
Algorithm 1, transmission overhead cannot be further reduced
izedraptorcodesresultsinonlytwoobtainedPSNRvalues.
with additional layer partitioning and selection probability re-
TheaboveLTcoding/iterativedecodingresultsareobtained
assignment. Let Scheme A denote the source-to-channellayer
usingand-ortreeanalysiswhichassumesinfiniteblocklength.
mappingproducedbyAlgorithm1anddenoteSchemeBasone
As a check, simulation of LT codes with degree distribution
which further partitions Layer l into Layers m and n with di-
Ω (x)anditerativebeliefpropagation(BP)decoding[39][14]
r mensions S and S , respectively. Denote the resulting opti-
are provided in Fig 7. Layer selection parameter ρ = 0.19 m n
1 mal selection probabilities for Scheme B which minimize the
obtained from Eqs. (12) and (13) determine the constraints in
transmission overhead as ρ and ρ for Layers m and n, re-
Problem 1.0. The horizontalaxis depicting transmission over- m n
spectively. We now show that the minimum required trans-
head includes the minimum overhead achieved by and-or tree
mission overhead is no larger by using Scheme A with se-
analysis (εand−or = 1.475), as well as 5% (1.525) and 10% lection probability ρ = ρ + ρ assigned to Layer l. For
(1.575)greaterthantheminimum.TheresultingPSNRforeach l m n
the same number of total transmitted symbols M, the effec-
user class is computed for each realization. The vertical axis
tive average raptor code rates for Layer l in Scheme A, Layer
showstherelativefrequencythatthePSNRislargerthanthede-
scilraesdsetsh.reItshcaonldb(ePsreoebn(PfrSomNRthje≥leftγsji)d)efoorfeFaigch. 7ofthtahtethtweosiumsuer- mRmin=ScMhSeρmmmeaBndaRndnL=ayeMrSρnnn,inreSspchecetmiveelBy. aWreitRholut=losMsSρlolf,
lationresultscloselymatchtheand-ortreeanalysis. Theprob- generality,weassumeρm/Sm ≥ρn/Sn. Thenitcanbeshown
taobitlhiteydoefsirreeadcphrinogbatbairlgiteyttPhSreNsRhoγldjPin.siAmlsuola,tbioyniniscrveearsyincglothsee that Rl = MS(ρmm++Sρnn) ≤ SρMnnρ(ρmm++Sρρnnnρ)n = MSρnn = Rn. As
j thedecodingfailureprobabilityoftheraptorcodesismonoton-
overheadto 1.525, a higherprobabilityin reachingtargetQoS
icallyincreasingwithcoderateforthesameuserclass,wehave
canbeobtained.
(1−P (l,j))≥(1−P (n,j))>(1−P (m,j))(1−P (n,j))
e e e e
for any class index j, where P (.) is the same decoding fail-
e
ure probability function as defined in (4). This means that for
1
thesamenumberoftransmittedsymbols,theoriginalmapping
0.95 scheme (Scheme A) has higher probability of successfully de-
CCllaassss 12,, ssiimmuullaattiioonn rreessuullttss coding all the symbols in Layer l than Scheme B for all user
≥γProb(PSNR )jj00.8.95 CCllaassss 12,, QQooSS pprroobbaabbiilliittyy tthhrreesshhoolldd PP12 c(c4loa)m,sspSeascr.heedTmthoeerSAecfhorereemq,uefioBrre.stFhlieensasslalmym,eirnaQipmtoouSrmccootdnraesnstrswamiintithssslidaorengsecorrvidbeiermhdeebanyd-
sionhavebetterperformanceforthesamecoderate,whichalso
0.8
impliesnofurtherlayerpartitioning.
0.75 1.48 1.5 1.52 1.54 1.56 1.58
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Yu Cao received his B.S. degree from Department
Theory,vol.42,no.6,pp.1737-1744,Nov.1996.
ofElectronicEngineering,TsinghuaUniversity,Bei-
[26] P.A. Chou, H.J. Wang and V.N. Padmannabhan, “Layered multiple de-
jing, China and his M.S. and Ph.D. degrees from
scriptioncoding",Proc.PacketVideoWorkshop,Nantes,2003.
DepartmentofElectricalandComputerEngineering,
[27] R.Hamzaoui,V.Stankovic,Z.Xiong,“Optimizederrorprotectionofscal-
Queen’sUniversity,Kingston,Canada. From2011to
ableimagebitstreams,"IEEESignalProcessingMag.,vol.22,no.6,pp.
2012,hewasapost-doctoralfellow atQueen’sUni-
91-107,Nov.2005.
versity,Canada. Since2012,hehasbeenworkingas
[28] V.Stankovic, R.Hamzaoui,andZ.Xiong,“Robustlayeredmultiplede-
aresearchengineeratHuaweiCanadaResearchCen-
scriptioncodingofscalablemediadataformulticast,"IEEESignalPro-
ter, Ottawa, Ontario, Canada. His research interests
cessingLetters,vol.12,no.2,pp.154-157,Feb.2005.
are forward error correction codes and optimization
[29] P.A. Chou, A.E. Mohr, A. Wang, and S. Mehrotra, “Error control for
forwirelesscommunications.Hiscurrentresearchfo-
receiver-drivenlayeredmulticastofaudioandvideo",IEEETrans.Multi-
cusisairinterfacedesignfornextgenerationradioaccessnetworks.
media,vol.3,no.1,pp.108-122,2001.
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tizedmedia",IEEETrans.Multimedia,vol.8,no.2,pp.390-404,2006.
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efficientMPEGvideotransmission,"Proc.MILCOM,pp.1-7,2009.
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“Expanding windowfountain codesforunequalerrorprotection," IEEE
Trans.onCommunications,vol.57,no.9,pp.2510-2516,Sep.2009. StevenD.BlosteinreceivedhisB.S.degreeinElec-
[34] D.Vukobratovic,V.Stankovic,D.Sejdinovic,L.Stankovic,andZ.Xiong, trical Engineering from Cornell University, Ithaca,
“Scalable Video Multicast Using Expanding Window Fountain Codes," NYandtheM.S.andPh.D.degreesinElectricaland
IEEETrans.Multimedia,vol.11,no.6,pp.1094-1104,Oct.2009. ComputerEngineeringfromtheUniversityofIllinois,
[35] C. Studlholme and I. Blake, “Windowed erasure codes," in Proc. Intl. Urbana-Champaign.HehasbeenwiththeDepartment
Symp.Inform.Theory(ISIT),pp.509-513,July2006. ofElectricalandComputerEngineeringQueen’sUni-
[36] M.C.O. Bogino, P. Cataldi, M. Grangetto, E. Magli, and G. Olmo, versitysince1988. HewasMulti-RateWirelessData
“Sliding-window digital fountain codes for streaming multimedia con- AccessMajorProjectleaderintheCanadianInstitute
tents,"inProc.IEEEISCAS,pp.3467-3470,2007. forTelecommunicationsResearch,aconsultanttoin-
[37] P. Cataldi, M. Grangetto, T. Tillo, E. Magli, and G. Olmo, “Sliding- dustryandgovernmentinimagecompression, target
window Raptor codes for efficient scalable wireless video broadcasting tracking,radarimagingandwirelesscommunications,
withunequallossprotection,"IEEETrans.onImageProcessing,vol.19, andspentsabbaticalleavesatLockheedMartinElectronicSystemsandatCom-
no.6,pp.1491-1503,June2010. munications Research Centre ofIndustryCanada. Hisinterests lieintheap-
[38] S.Ahmad,R.Hamzaoui,andM.M.Al-Akaidi,“Unequalerrorprotection plication ofsignal processing towireless communications systems, including
using fountain codes with applications to video communication," IEEE synchronization, networkMIMOandphysicallayeroptimizationformultime-
Trans.Multimedia,vol.13,no.1,pp.92-101,Feb.2011. diatransmission. HehasbeenamemberoftheSamsung4GWirelessForum
[39] M.Luby,“LTcodes,"Proc.43rdAnnu.IEEESymp.FoundationsofCom- and an invited distinguished speaker. He served as Department Head, IEEE
puterScience,Vancouver,Canada,pp.271-282,Nov.2002. KingstonSectionChair, Biennial SymposiumonCommunications Chair, and
[40] K.YangandJ.Wang,“Unequalerrorprotectionforstreamingmediabased EditorforIEEETransactionsonImageProcessingandIEEETransactions on
onratelesscodes",IEEETran.Comput.,vol.61,no.5,pp.666-675,2012. WirelessCommunications.HeisaregisteredProfessionalEngineerinOntario.