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 (MSE). It is shown that, when a symmetric source density with infinite support is sufficiently well behaved, the optimal step size Delta (N) for symmetric uniform scalar quantization decreases as 2 sigmaN(-1)(V) over bar (-1)(1/6N(2)), where N is the number of quantization levels, sigma (2) is the source variance and (V) over bar (-1)((.)) is the inverse of (V) over bar (y) = y(-1) integral (infinity)(y) P(sigma (-1) X > x) dx. Equivalently, the optimal support length N Delta (N) increases as 2 sigma(V) over bar (-1) (1/6N(2)), Granular distortion is asymptotically well approximated by Delta (2)(N)/12, and the ratio of overload to granular distortion converges to a function of the limit tau equivalent to lim(y --> infinity) y(-1) E[X]X > y], provided, as usually happens, that tau 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 tau = 1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by Delta (2)(N)/12. Applying these results to the class of two-sided densities of the form b \x \ (beta)e(-a \x \ alpha), which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that tau = 1, that Delta (N) decreases as (ln N)(1/alpha)/N, that D-N is asymptotically well approximated by Delta (2)(N)/12 and decreases as (ln N)(2/alpha)/N-2, and that more accurate approximations to Delta (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.
uniform scalar quantization optimal step size granular distortion generalized gamma density high-resolution quantization theory optimal support region overload distortion