Abstract

This paper studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean squared error. It is shown that when a symmetric source density with infinite support is sufficiently well behaved, the optimal step size D N for symmetric uniform scalar quantization decreases as 2s N -1 V -1 1/6N 2 () , where N is the number of quantization levels, s 2 is the source variance and V -1 () is the inverse of V (y) = y -1 P(s -1 X > x)dx y . Equivalently, the optimal support length ND N increases as 2s V -1 1/6N 2 () . Granular distortion is asymptotically well approximated by D N 2 /12, and the ratio of overload to granular distortion converges to a function of the limit t lim y y -1 E[X| X > y], provided, as usually happens, that t exists. When it does, its value is related to the number of finite moments of the source density; an asymptotic formula for the overall distortion D N is obtained; and t = 1 is both necessary...

Citations