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

1INTRODUCTION2

1Introduction

Weconsidertheproblemofdesigningamultipledescriptionvectorquantizerforamem-orylesssourcewithprobabilitydensityfunctionp,di erentialentropyh(p)<∞andthesquared-errordistortionmeasure.Amultipledescriptionvectorquantizerencodesvectorsfortransmissionoveratwo-channelcommunicationsystem.Theobjectiveistosendinfor-mationaboutthesourcevectorovereachchannelinsuchawaythatgoodperformanceisobtainedwhenbothchannelsworkandthedegradationissmallwheneitherchannelfails.Itisassumedthattheencoderhasnoknowledgeaboutthestateofachannel,i.e.,itdoesnotknowwhetherachannelhasfailedorisworking.

Therecentinterestinthemultipledescriptionproblemislargelybecauseoftheapplica-tiontoimage,videoandvoicecommunicationsoverpacketnetworkswithnonzeroprobabilityofpacketloss.Thelossofapacketcouldbesigni cantifitresultsinalossofalargeblockofsourcesamples,suchasalargeimageblockoralargeblockofspeech.Onewaytoimproveperformanceistoplacedi erentencodedversionsofagivenblockofsourcesamplesintoseveralpacketsinsuchawaythatifsomeofthesepacketsarereceived,adegradedversionofthesourceblockmayberecovered.Thisleadstotheformulationofthemultipledescriptionsourcecodingproblem.

Forthesingle-descriptionproblem,oneofthebene tsofvectorquantizationoverscalarquantizationisareductioningranulardistortion.ThisisbecauseinhigherdimensionsitispossibletoconstructVoronoicellsthataremore“spherical”thanthehypercube.Tobemorespeci c,uniformscalarquantizationcoupledwithentropycodingisknowntohave

¯(R)atentropyRbits/samplesatisfying[16]meansquarederror(mse)d

22h(p)2R¯limd(R)2=

R→∞