Asymptotic analysis of optimal fixed-rate uniform scalar quantization

Studies the asymptotic characteristics of uniform scalar quantizers that are optimal with respect to mean-squared error (MSE). When a symmetric source density with infinite support is sufficiently well behaved, the optimal step size Δ_N for symmetric uniform scalar quantization decreases as 2σN^-1V¯ ^-1(1/6N²), where N is the number of quantization levels, σ² is the source variance and V¯^-1 (·) is the inverse of V¯(y)=y^-1 ∫_y ^∞ P(σ^-1X>x) dx. Equivalently, the optimal support length NΔ_N increases as 2σV¯^-1(1/6N²). Granular distortion is asymptotically well approximated by Δ_N²/12, and the ratio of overload to granular distortion converges to a function of the limit τ≡lim_y→∞y^-1E[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 D_N is obtained, and τ=1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by Δ_N²/12. Applying these results to the class of two-sided densities of the form b|x|^βe(-α|x| ^α), which includes Gaussian, Laplacian, Gamma, and generalized Gaussian, it is found that τ=1, that Δ_N decreases as (ln N)¹α//N, that D_N is asymptotically well approximated by Δ_N²/12 and decreases as (ln N)²α//N², and that more accurate approximations to Δ_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