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

4PRELIMINARIES15

InordertoderiveexpressionsforR0andR,weusethefactthateachquantizerbinhasidenticalvolumeνandthatpL(x)isapproximatelyconstantoverVoronoiregionsofthesublatticeV(λ′).ThesecondassumptionisvalidinthelimitastheVoronoiregionsbecomesmallandisstandardinasymptoticquantizationtheory.TherateR0=H(Q(X))isgivenby[18]

pL(x)dxlog2R0= (1/L)

≈ (1/L)

≈h(p) (1/L)log2(ν).

isroughlyconstantovereachVoronoiregionofΛ′toget

R= (1/L)pL(x)dxlog2

λ′∈Λ′

pL(x)dx

λ

λ

V(λ)V(λ)

V(λ)

pL(x)dxlog2pL(λ)ν

(16)

ForR,weevaluatetheentropyH(α1(Q(X)))andthenusetheapproximationthatpL(x)

≈ (1/L)≈ (1/L)

λ′∈Λ′

1′

λ∈α 1(λ)

V(λ)

λ′∈Λ′

≈h(p) (1/L)log2(Nν).

1′λ∈α 1(λ)

V(λ)

1′λ∈α 1(λ)

pL(x)dx log2(pL(λ′)Nν)

pL(x)log2(pL(λ′))dx (1/L)log2(Nν)

1′

λ∈α 1(λ)

V(λ)

pL(x)dx

V(λ)

(17)

Observethatintheaboveequation,thetermNνissimplythevolumeofafundamentalregionforthesublatticeΛ′(sinceithasindexNinΛ).Uponwriting(17)intermsofR0weobtain

R=R0 (1/L)log2(N).

(18)

¯0.InsteadaAsinglechannelsystemwouldhaveusedR0bits/sampletoachievethesamed