Skip to Main Content
A quantizer divides the range [0, 1] of a random variable into quantizing intervals the th such interval having length . We define the quantization error for a particular value of (unusually) as the length of the quantizing interval in which finds itself, and measure quantizer performance (unusually) by the th mean value of the quantizing interval lengths , averaging with respect to the distribution function of the random variable . is defined to be an optimum quantizer if for all . The unusual definitions restrict the results to bounded random variables, but lead to general and precise results. We define a class of quasi-optimum quantizers; is in if the different intervals make equal contributions to the mean th power of the interval size so that is constant for all . Theorems 1, 2, 3, and 4 prove that exists and is unique for given , and : that , where is the density of the absolutely continuous part of the distribution function of , and : that as ; and that if for finite , then .