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Asymptotic quantization error of continuous signals and the quantization dimension

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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.

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Information Theory, IEEE Transactions on  (Volume:28 ,  Issue: 2 )