@article {16,
title = {Asymptotic quantization error of continuous signals and the quantization dimension},
journal = {IEEE Trans. Inform. Theory},
volume = {IT-28},
number = {2},
year = {1982},
month = {March},
pages = {139{\textendash}149},
abstract = {Extensions of the limiting quantization error formula of Bennet are proved. These are of the form $D_{s,k}(N,F)=N^{-\beta}B$, where $N$ is the number of output levels, $D_{s,k}(N,F)$ is the $s$th moment of the metric distance between quantizer input and output, $\beta, B \superior 0, k=s/\beta$ is the signal space dimension, and $F$ is the signal distribution. If a suitably well-behaved $k$-dimensional signal density $f(x)$ exists, $B=b_{s,k}[\int f^{rho}(x)dx]^{1/ rho},rho=k/(s+k)$, and $b_{s,k}$ does not depend on $f$. For $k=1,s=2$ this reduces to Bennett{\textquoteright}s formula. If $F$ is the Cantor distribution on $[0,1]$, $0\inferior k=s/ \beta= \log 2/ \log 3 \inferior 1$ and this $k$ equals the fractal dimension of the Cantor set $[12,13]$. Random quantization, optimal quantization in the presence of an output information constraint, and quantization noise in high dimensional spaces are also investigated.},
keywords = {information-theory, nldr, rate-distortion, source-coding, vector-quantization},
author = {Paul L. Zador}
}