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

eachchannelilabelisreusedexactlyNtimes.

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Property2:Theshiftproperty:αu(λ+λ′)=αu(λ)+λ′,forallλ∈Λandλ′∈Λ′. Property3:Eachundirectededgee={λ′1,λ′2},λ′1=λ′2,labelstwopoints,λaandλb,andλa+λb=λ′1+λ′2.

The rstassumptionmakesiteasytoparametrizethetradeo betweenthesideandcen-traldistortions.Thesecondassumptionreducesthelabelingproblemtooneoflabelinga niteset.Thethirdassumptionisasimplewaytoachieveexactbalancebetweenthetwodescriptions(itsimplicationswillbecomeclearerinSection5.2.3).Itwillproveusefultohavethefollowingde nitionsofequivalence.

De nition1:Twolatticepointsλ1andλ2aresaidtobeequivalentifeitherλ1andλ2orλ1and λ2lieinthesamecosetofΛrelativetothesublatticeΛ′.

De nition2:Twosublatticeedgese1ande2aresaidtobeequivalentiftheyareparallelTheequivalenceclassofanobjectwillbeindicatedbysquarebrackets.Thus[λ]istheequivalenceclassofλand[e]istheequivalenceclassofe,orequivalently,[λ]∈Λ/Λ′and[e]∈Eu/Λ′.

andofequallength,orequivalentlyife1+λ′=e2forsomeλ′∈Λ′.

4.1DistortionComputation

¯0isgivenbyTheaveragetwo-channeldistortiond ¯0= x λ 2pL(x)dx.d

λ∈Λ

V(λ)

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Sincethecodebookofthequantizerisalattice,alltheVoronoiregionsarecongruent.Furthermore,uponassumingthateachVoronoiregionissmallsothatpL(x)≈pL(λ)forx∈V(λ)anduponlettingνdenotetheL-dimensionalvolumeofaVoronoiregion,weobtain