Asymptotic analysis of optimal fixed-rate uniform scalarquantization
Hui, D.; Neuhoff, D.L.
Information Theory, IEEE Transactions on
Volume 47, Issue 3, Mar 2001 Page(s):957 - 977
Digital Object Identifier 10.1109/18.915652
Summary: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-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 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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