Asymptotic analysis of optimal fixed-rate uniform scalar quantization

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 ∆N for symmetric uniform scalar quantization decreases as 2σ N V −1 1 / 6N 2 ( ) , where N is the number of quantization levels, σ 2 is the source variance and V (⋅) is the inverse of V (y) = y P(σ X > x)dx y ∞ ∫ . Equivalently, the optimal support length N∆N increases as 2σ V −1 1 / 6N 2 ( ). Granular distortion is asymptotically well approximated by ∆N 2 / 12 , and the ratio of overload to granular distortion converges to a function of the limit τ ≡ lim y→∞ yE[X| X > y], provided, as usually happens, that τ 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 DN is obtained; and τ = 1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by ∆N 2 / 12 . Applying these results to the class of two-sided densities of the form b x β e x α , which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that τ = 1, that ∆N decreases as (ln N) 1/α / N , that DN is asymptotically well approximated by ∆N 2 / 12 and decreases as (ln N) /α / N 2 , and that more accurate approximations to ∆N are possible. The results also apply to densities with one-sided infinite support, such as Rayleigh and Weibull, and to densities whose tails are asymptotically similar to those previously mentioned.