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

4PRELIMINARIES

thefollowingexpressionforthetwo-channeldistortion[15]

x 2dxV(0)¯0≈d

13

ν1+2/L

isgivenby

¯0≈G(Λ)ν2/L.d

,(9)

(10)

¯1andd¯2andd¯sdef¯1+d¯2)/2.Wenowderiveexpressionsfortheaveragedistortionsd=(dWhenonlydescription1isavailable,thedistortionisgivenby

¯1= x α1(λ) 2pL(x)dxd

==

λ∈Λλ∈Λλ∈Λ

V(λ)

2

λ∈Λ

V(λ)

x λ+λ α1(λ) 2pL(x)dx x λ 2pL(x)dx+

λ∈Λ

V(λ)

¯0+=d

(a)

V(λ)

λ α1(λ) 2pL(x)dx+

(11)

V(λ)

x λ,λ α1(λ) pL(x)dx

¯0+=d

λ∈Λλ∈Λ

λ α1(λ) 2P(λ)+2 λ α1(λ) 2P(λ),

λ∈Λ

xpL(x)dx

V(λ)

λpL(x)dx,λ α1(λ)

V(λ)

(12)

whereP(λ)=Pr(Q(X)=λ),and(a)followsbyassumingthatλisthecentroidofitsVoronoiregion.Thisistruefortheuniformdensity.Fornonuniformdensities,thereisanerrortermwhichgoestozerowiththesizeoftheVoronoiregion.The rsttermin(12)isthetwo-channeldistortionandthesecondtermistheexcessdistortionwhichisincurred