Asymptotic quantization error of continuous signals and the quantization dimension
P. Zádor · IEEE Transactions on Information Theory · 1982
Extensions of the limiting qnanfizafion error formula of Bennet are proved. These are of the formD_{s,k}(N,F)=N^{-\beta}B, whereNis the number of output levels,D_{s,k}(N,F)is thesth moment of the metric distance between quantizer input and output,\beta,B>0,k=s/\betais the signal space dimension, andFis the signal distribution. If a suitably well-behavedk-dimensional signal densityf(x)exists,B=b_{s,k}[\int f^{\rho}(x)dx]^{1/ \rho},\rho=k/(s+k), andb_{s,k}does not depend onf. Fork=1,s=2this reduces to Bennett's formula. IfFis the Cantor distribution on[0,1],0<k=s/ \beta=\log 2/ \log 3<1and thiskequals 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.