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 , where is the number of output levels, is the th moment of the metric distance between quantizer input and output, is the signal space dimension, and is the signal distribution. If a suitably well-behaved -dimensional signal density exists, , and does not depend on . For this reduces to Bennett\´s formula. If is the Cantor distribution on and this equals the fractal dimension of the Cantor set . Random quantization, optimal quantization in the presence of an output information constraint, and quantization noise in high dimensional spaces are also investigated.