Abstract The problem of designing a multiple description vector quantizer with lattice codebook \Lambda is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance resu

4PRELIMINARIES8

thequantizermappingbyλ=Q(x).Informationabouttheselectedcodevectorλisthensentacrossthetwochannels,subjecttorateconstraintsimposedbytheindividualchannels.Thisisdonethroughalabelingfunctionαfollowedbyentropycoding.Thelabelingfunctionαmapsλ∈Λtoapair(λ′1,λ′2)∈Λ′×Λ′,whereΛ′isasublatticeofΛwithindexN.Thecomponentfunctionsofαaredenotedbyα1andα2,whereα1(λ)=λ′1andα2(λ)=λ′2.

ForsimplicityweassumethatΛ′isgeometricallysimilartoΛ,i.e.,Λ′canbeobtainedbyscaling,rotatingandpossiblyre ectingΛ.NotethatpointsinthelatticeΛaredenotedbyλ,possiblywithsubscripts,whereassublatticepointswillbedenotedbyλ′orλ′′,possiblywithsubscripts.

InFig.3,aportionofthehexagonallatticeA2isillustrated,alongwithageometricallysimilarsublatticeofindex31.Thelatticepointslieattheintersectionofthestraightlinesinthehexagonalgrid(onlysomeofthepointsareshown).Thesublatticepointsaremarkedwithupper-caseletters.Observethatthelatticeis31timesasdenseasthesublattice,i.e.,thereare31latticepointsforeverysublatticepoint.

Atthedecoder,ifonlychannel1works,thereceivedinformationisusedtodecodeλ′1,andifonlychannel2works,theinformationreceivedoverchannel2isusedtodecodeλ′2.Themappingαisassumedtobeone-to-onesothatifbothchannelsworkλcanberecoveredfrom(λ′1,λ′2).(Inpractice,ifonlyonechannelisworkingitmaybebettertodecodethereceivedvectortosomefunctionofλ′1orλ′2ratherthantoλ′1orλ′2itself.Ifλ′1isreceivedbutλ′2isnot,forinstance,wewoulddecodeλ′1asthecenterofmassofallpointsλ∈Λsuchthatthe rstcomponentofα(λ)isλ′1.Wewillignorethiscomplicationinordertosimplifytheanalysis.)

GivenΛ,Λ′andα,therearethreedistortionsandtworatesassociatedwithanMDVQ.Foragivenxmappedtothetriple(λ,λ′1,λ′2)bytheMDVQ,thetwo-channeldistortiond0isgivenby x λ 2,thechannel1distortiond1by x λ′1 2andthechannel2distortiond2by x λ′2 2(weassumethattheinnerproductofL-dimensionalvectorsx=(x1,x2,...,xL)