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

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 results are obtained for quantizers based on the lattices A(2) and Z(i), 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) < infinity. For any a is an element of (0, 1) and rate pair (R, R), it is shown that the two-channel distortion do and the channel 1 (or channel 2) distortion (d) over bar (s) satisfy lim(R --> infinity) (d) over bar (0)2(2R(1+a)) = (1)/(4) G(Lambda )2(2h(p)) and lim(R --> infinity) (d) over bar (s)2(2R(1-a)) = G(S-L)2(2h(p)) where G(Lambda) is the normalized second moment of a Voronoi cell of the lattice Lambda and G(S-L) is the normalized second moment of a sphere in L dimensions.