Asymptotic entropy-constrained performance of tessellating and universal randomized lattice quantization

Two results are given. First, using a result of Csiszar (1973) the asymptotic (i.e., high-resolution/low distortion) performance for entropy-constrained tessellating vector quantization, heuristically derived by Gersho (1979), is proven for all sources with finite differential entropy. This implies, using Gersho´s conjecture and Zador´s formula, that tessellating vector quantizers are asymptotically optimal for this broad class of sources, and generalizes a rigorous result of Gish and Pierce (1968) from the scalar to the vector case. Second, the asymptotic performance is established for Zamir and Feder´s (1992) randomized lattice quantization. With the only assumption that the source has finite differential entropy, it is proven that the low-distortion performance of the Zamir-Feder universal vector quantizer is asympotically the same as that of the deterministic lattice quantizer