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Asymptotic analysis of optimal fixed-rate uniform scalar quantization

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2 Author(s)
Hui, D. ; Dept. of Electr. Eng. & Comput. Sci., Michigan Univ., Ann Arbor, MI, USA ; Neuhoff, D.L.

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 ΔN for symmetric uniform scalar quantization decreases as 2σN-1-1(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-1y P(σ-1X>x) dx. Equivalently, the optimal support length NΔN increases as 2σV¯-1(1/6N2). Granular distortion is asymptotically well approximated by ΔN2/12, and the ratio of overload to granular distortion converges to a function of the limit τ≡limy→∞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 DN is obtained, and τ=1 is both necessary and sufficient for the overall distortion to be asymptotically well approximated by ΔN2/12. 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 ΔN decreases as (ln N)1α//N, that DN is asymptotically well approximated by ΔN2/12 and decreases as (ln N)2α//N2, and that more accurate approximations to ΔN 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

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Information Theory, IEEE Transactions on  (Volume:47 ,  Issue: 3 )