Multiple Description Vector Quantization with Lattice Codebo(3)

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

1INTRODUCTION

3

whereasifanL-dimensionallatticeΛisusedasacodebook,thedistortionsatis es

limd

¯RR→∞

(R)22=G(Λ)22h(p),(2)

whereG(Λ)isthenormalizedsecondmomentofaVoronoicellofthelattice.Indimensionsgreaterthanone,latticesexistforwhichG(Λ)isstrictlysmallerthan1/12.Forexample,in8dimensions,itispossibletogain0.66dBbyusingthelatticeE8ascomparedtouniformscalarquantization[11].Itisalsoknownthrougharandomquantizingargument[31]thatquantizersexistforwhichtheproductd

¯(R)22Rapproaches22h(p)/(2πe)astherateincreases.Furthermore,itfollowsfromratedistortiontheory[5]thatnosmallervaluecanbeachievedfortheaboveproductinthelimitofin niterate.Themaximumgainpossibleoverentropy-codedscalarquantizationis1.53dBandlatticesprovideausefulmethodforclosingthisgap.

Fromnowonwewillrestrictourattentiontothecaseoftwochannels.ConsideramultipledescriptionquantizerwhichsendsinformationacrosseachchannelatarateofRbits/sample.Theperformanceofthissystemismeasuredintermsofthreedistortions:the

two-channeldistortiond

¯0,whenbothdescriptionsareavailabletothedecoder;thechannel1distortiond

¯1,whenonlythe rstdescriptionisavailableandthechannel2distortiond¯2,whenonlytheseconddescriptionisavailable.Wewillfurtherassumethatd

¯1=d¯2=d¯sandwillrefertothiscommonvalueasthesidedistortion.Theobjectiveistodesignvector

quantizersthatminimized

¯0undertheconstraintd¯sagivenboundD≤Ds,foragivenratepair(R,R)andsontheside-channeldistortion.

Ithasbeenshown[27]thatforauniformentropy-codedmultipledescriptionquantizer,andanya∈(0,1),thedistortionssatisfy

limd

¯1R→∞

0(R)22R(1+a)=¯2h(p)

,

Rlim→∞

d

s(R)22R(1 a)=

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