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

5ALABELINGFUNCTIONFORA216

therateoverheadisR0 (2/L)log2(N).

multipledescriptionsystemusesatotalof2R=2R0 (2/L)log2(N)bits/sample,andso

5

ALabelingFunctionforA2

→e∈Ed(λ′)

Wenowlookforalabelingfunctionαforwhich

→ds( e)isminimizedandisinde-

pendentofλ′.Sincethedetailsarecomplicated,wewillworkoutthe rstexample–forthehexagonallatticeA2–quitecarefully.Wewillthenidentifycertaingeneralprinciplesandusethemtoconstructlabelingsforotherlattices.

ThelatticeA2maybeconsideredtobeasubsetofIR2orasasubsetofC

,where

√

ω= 1/2+i

3/2.AsublatticeΛ′ofalatticeΛissaidtobegeometricallysimilartoΛif

itcanbeobtainedbyscalingandrotatingand/orre ectingΛ[11].Tobemoreprecise,ifamatrixG′generatesΛ′andGgeneratesΛ,thenΛ′isgeometricallysimilartoΛifandonlyifG′=cUGB,forsomenonzeroscalarc,integermatrixUwithdeterminant±1,andrealorthogonalmatrixB.TheindexNisde nedastheratioofthefundamentalvolumesofΛ′andΛandisgivenintermsofthescalefactorcbyN=c2.Itcanbeshown[6]thatΛ′issimilartoΛifandonlyifNisoftheforma2 ab+b2,a,b∈

;

ifthisholdsthenΛ′isgeneratedbyu=a+bωandv=ω(a+bω).Inadditiontothis

K

restrictiononN,wewillrequire,forconvenienceonly,thatN=i=0Ai,whereAiisthenumberoflatticepointsatsquareddistanceifromtheorigin.Inotherwords,werequirethatNisthenumberofpointsinthe rstKshellsofthelattice,forsomeK=K(N).Thereareheuristicarguments,tobepresentedelsewhere,whichsuggestthattherearein nitelymanyvaluesofNwiththisproperty.Forexample,N=31hasthisproperty,sinceN=A0+A1+A2+A3+A4=1+6+6+6+12=31,and31isalsooftheforma2 ab+b2,