Low-resolution scalar quantization for Gaussian sources and squared error

This correspondence analyzes the low-resolution performance of entropy-constrained scalar quantization. It focuses mostly on Gaussian sources, for which it is shown that for both binary quantizers and infinite-level uniform threshold quantizers, as D approaches the source variance σ², the least entropy of such quantizers with mean-squared error D or less approaches zero with slope -log₂e/2σ². As the Shannon rate-distortion function approaches zero with the same slope, this shows that in the low-resolution region, scalar quantization with entropy coding is asymptotically as good as any coding technique.