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

4PRELIMINARIES14

whenchannel2fails.NotethatforagivenΛ,onlytheexcessdistortiontermisa ectedbythelabelingfunctionα.From(12),itfollowsthat

¯s=d¯0+(1/2)d

¯0+(1/2)=d

( λ α1(λ) 2+ λ α2(λ) 2)P(λ)

λ∈Λ

λ′∈Λ′λ∈V0(λ′)

(13)(14)

→→Wealsointroducethefollowingnotation:d1(λ, e)= λ λ′1 2andd2(λ, e)= λ λ′2 2

→→→→→ ).andd(λ, e)=(d( e)+d( e))/2,where e=(λ′,λ′).Notethatd(λ, e)=d(λ,←e

s

1

2

1

2

s

s

( λ α1(λ) 2+ λ α2(λ) 2)P(λ).

Hencewhenappropriate,wewillwriteds(λ,e).Also,whentheedgeassociatedwiththe

→→latticepointisclearfromthecontext,wewillwrited(λ)ord( e)insteadofd(λ, e).It

s

s

s

isuseful(asadesignguideandfortheasymptoticswhichfollow)towritedownaslightlydi erentexpressionforthesidedistortionwherethesumistakenovertheedgesetEd=

′λ′∈Λ′Ed(λ):

¯s=d¯0+d

→e∈Ed

¯0+=d

→λ′∈Λ′ e∈Ed(λ′)

→→whereP( e)isequaltotheprobabilityofthelatticepointthattheedgelabels,i.e.,P( e)=Pr(Q(X)=λ).

→→ds( e)P( e)

→→ds( e)P( e)

(15)

4.2RateComputation

Expressionsfortherate(inbits/sample)willbederivednext.LetR0betheraterequiredtoaddressthetwo-channelcodebookforasinglechannelsystem1.Wewill rstderiveanexpressionforR0andthendeterminethe(per-channel)rateRofthemultipledescriptionsystem.