Multiple Description Vector Quantization with Lattice Codebo(3)
发布时间:2021-06-07
发布时间:2021-06-07
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)=
122
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