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

4PRELIMINARIES

mainreasonforintroducingαuisthatitsconstructionlogicallyprecedesthatofα

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Figure4:VariousedgesetsareillustratedforasublatticeofA2withN=31.Forclarityonlythesublatticepointsareshown.ThesetEu|α1=0isshownontheleft,Ed|α1=0ontheright.Theoriginislocatedatthecenterofthecircle.

Certainsetsassociatedwiththetwomapswillplayacentralroleinthedevelopment.The rstisEd(λ′)=α(V0(λ′)),thesetofalllabelsforpointsinV0(λ′)(thesubscriptdindicatesthattheedgesaredirected).ThecorrespondingsetofundirectededgesisdenotedbyEu(λ′).Morespeci cally,Eu(λ′)=αu(V0(λ′)).Thesetofall(directed)labelsisdenoted

byEd=λ′∈Λ′Ed(λ′)andthesetofallundirectedlabelsbyEu=λ′∈Λ′Eu(λ′).Itisalsousefultode netherestrictionEd|α1=0={(λ′1,λ′2)∈Ed,λ′1=0}.ThesetofundirectedFig.4.

edgesinEd|α1=0willbedenotedbyEu|α1=0.ThesetsEu|α1=0andEd|α1=0areillustratedinThereuseindexassociatedwithalabelλ′andchanneliisde nedtobeNi(λ′)=|{λ:αi(λ)=λ′}|,thenumberoflatticepointsλforwhichαi(λ)=λ′.followingproperties.

Property1:ThereuseindexNi(λ′)=N,forallλ′∈Λ′andi=1,2.InotherwordsInordertorendertheproblemtractable,weassumethatthelabelingfunctionhasthe