Asymptotic quantization error of continuous signals and the quantization dimension

Extensions of the limiting qnanfizafion error formula of Bennet are proved. These are of the form Dₒ, ₊ (N, F) =N^-B, where N is the number of output levels, Dₒ, ₊ (N, F) is the s th moment of the metric distance between quantizer input and output, , B>0, k=s/ is the signal space dimension, and F is the signal distribution. If a suitably well-behaved k -dimensional signal density f (x) exists, B=bₒ, ₊ f^ (x) dx^1/, =k/ (s+k), and bₒ, ₊ 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 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.