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Asymptotic quantization error of continuous signals and the quantization dimension

TitleAsymptotic quantization error of continuous signals and the quantization dimension
Publication TypeJournal Article
Year of Publication1982
AuthorsPaul L. Zador
JournalIEEE Trans. Inform. Theory
VolumeIT-28
Pagination139–149
Date PublishedMarch
Keywordsinformation-theory, nldr, rate-distortion, source-coding, vector-quantization
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'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.