Asymptotically optimal block quantization

for the mean-square quantizing error where N is the. number of level&p(x) is the probability density of the input, and E’(x) is the slope of the compressor curve. The formula, an approximation based on the assumption that the number of levels is large and overI& distortion is negligible, is a useful tool for analytical studies of quantfzation. This paper gives a bedstlc argument generallhg Bemett’s formula to block quantization wbere a vector of random variables is quantized. The approach is again based on the. asymptotic situation where N, tke number of quantized output vectors, is very large. Using the resulting heuristic formula, an optimhtlon is performed leading to an expression for the minimum quantizing noise attainable for any block quantizer of a given block size k. The results are consistent with Zador’s results and speciaiize to known results for tke oneand two-dimensional casea and for the case of White. block length (k+m). The same heuristic approach also gives an alternate derivation of a bound of Elias for multidimensional quantization. Our approach leads to a rigorous metkod for obtaining upper bounds on the minimum distortion for block quantizers. In particular, for k = 3 we give a tigkt upper bound that may in fact be exact. ‘Ihe idea of representing a block quantizer by a block “compressor” mapping followed with an optimal quantizer for uniformly distributed random vectors is also explored. It is not always possible to represent an optimal quautizer with tbis block companding model.