Asymptotic performance of multiple description lattice quantizers

The high-rate squared-error distortions of a balanced multiple description lattice vector quantizer are analyzed for a memoryless source with probability density function p, differential entropy h(p)<∞, and lattice codebook Λ. For any a∈(0,1) and rate pair (R,R), it is shown that the two-channel distortion d¯₀ and the channel 1 (or channel 2) distortion d¯ _s satisfy lim_R→∞d¯₀2 ^2R(1+a)=G(Λ)2^2h(p)/4 and lim_R→∞d¯_s2^2R(1-a)=G(S _L)2^2h(p) where G(Λ) is the normalized second moment of a Voronoi cell of the lattice Λ and G(S_L ) is the normalized second moment of a sphere in L dimensions