Multiple-description vector quantization with lattice codebooks: design and analysis

The problem of designing a multiple-description vector quantizer with lattice codebook Λ is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance results are obtained for quantizers based on the lattices A₂ and Zⁱ, i=1, 2, 4, 8, that make use of this labeling algorithm. The high-rate squared-error distortions for this family of L-dimensional vector quantizers are then analyzed for a memoryless source with probability density function (PDF) p and differential entropy h(p)<∞. For any a ε (0, 1) and rate pair (R, R), it is shown that the two-channel distortion d¯_o and the channel 1 (or channel 2) distortion d¯_s satisfy lim_R→∞d¯_o2^2R(1+a) =¼G(Λ)2^2h(p) 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