Uniform scalar quantizers-distortion and the additive noise model

High-resolution infinite level uniform scalar quantizers are considered for the two cases when cell midpoints and cell centroids are used as reconstruction levels. First, it is shown that asymptotically, as the cell size goes to zero, midpoints induce the same MSE as centroids. Hence, they are asymptotically MSE optimal. Secondly, sufficient conditions are found under which the commonly used additive noise model, when midpoints are used, is asymptotically correct. It is also shown that for input probability density functions (pdf´s) that have a single jump discontinuity not at the origin, such a model is not well defined and consequently incorrect.