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

5ALABELINGFUNCTIONFORA2

2.DeterminethediscreteVoronoisetV0(0).

3.Determineanundirectedlabelforeverypointλ∈V0(0).

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4.Extendthelabelingtotheentirelatticeusingtheshiftpropertyoftheundirectedlabels.

→5.Givenλanditsundirectedlabele,determinethecorrectdirectedlabel e(i.e.,deter-minewhichendpointofeistobesentonchannel1andwhichendpointonchannel2).

AsublatticeΛ′ofindexequalto31maybeobtainedbyconsideringallpointsoftheformau+bv,withu=5 ωandv=ω(5 ω)tticepointshavebeenlabeledwithlower-caselettersa,b,c,...andsublatticepointswithupper-caselettersO,A,B,C,....Inorderto xthecoordinatesystem,notethatthesublatticepointOistheoriginofthecomplexplaneandthelatticepointsaandchaverepresentations1andωrespectively.Relativetothisbasis,therepresentationsofsublatticepointsAandCare5 ωandω(5 ω),respectively.ThediscreteVoronoisetV0(0)={O,a,b,c,...,y,z,aa,ab,ac,ad}isalsoshowninFig.5.Notethat|V0(0)|=31.PointsinV0(0)willbelabeledusingdirectededgesobtainedfromthefollowingsetof28undirectededges:

Eu(0)={{O,O},

{O,A},{O,B},{O,C},{O,D},{O,E},{O,F},{A,C},{B,D},{C,E},{D,F},{E,A},{F,B},{A,D},{B,E},{C,F},

{G,D},{G,E},{H,E},{H,F},{I,F},{I,A},{J,B},{J,A},{K,C},{K,B},{L,C},{L,D}},

(20)(19)