For generalized gamma probability densities, this paper studies the monotonicity of step sizes of optimal symmetric uniform scalar quantizers with respect to mean squared-error distortion. The principal results are that for the special cases of Gaussian, Laplacian, two-sided Rayleigh, and gamma densities, optimal step size monotonically decreases when the number of levels N increases by two, and that for any generalized gamma density and all sufficiently large N, optimal step size again decreases when N increases by two. Also, it is shown that for a Laplacian density and sufficiently large N, optimal step size decreases when N increases by just one.

This paper investigates the convexity of the mean squared-error distortion of symmetric uniform scalar quantization with respect to step size. The principal results include proofs for odd numbers of levels that distortion is not convex for any symmetric density and that it is convex for even numbers of levels for densities, such as Gaussian, Laplacian, and gamma, but is not, in general for two-sided Rayleigh. For the latter case, an interval is derived that includes the optimal step size and over which the distortion is convex. The proofs of convexity use the Euler-Maclaurin formula applied to the second derivative of distortion, with upper bounds on the remainder term. These results imply that a zero of the derivative of the distortion for these densities, which has been previously conjectured optimal, is indeed the optimal step size, because the distortion is convex either globally or locally over a sufficiently wide interval to ensure a global minimizer.

Asymptotic formulas are derived for the mean-squared error (MSE) distortion ofN-level uniform scalar quantizers designed to be MSE optimal for one density function, but applied to another, asN→ ∞. These formulas, which are based on the Euler-Maclaurin formula, are then applied with generalized gamma, Bucklew-Gallagher, and Hui-Neuhoff density functions as the designed-for and applied-to densities. It is found that the mismatch between the designed-for and applied-to densities can disturb the delicate balance between granular and overload distortions in optimal quantization, with the result that, generally speaking, the granular or overload distortion dominates, respectively, depending on whether the applied-to density function has a lighter or heavier tail than the designed-for density. Specifically, in the case of generalized gamma densities, a variance mismatch makes overload distortion dominate for an applied-to source with a slightly larger variance, whereas a shape mismatch can tolerate a wider variance difference while retaining the dominance of the granular distortion. In addition, for the studied density functions, the Euler-Maclaurin approach is used to derive asymptotic formulas for the optimal quantizer step size in a simpler, more direct, way than previous approaches.

The paper investigates the mean-squared error (MSE) distortion of a variance-mismatched fixed-rate scalar quantizer that is optimal or asymptotically optimal in the minimum MSE sense for a Laplacian density with standard deviation σqbut is applied to another with standard deviation σp. A Nitadori-like formula is discovered for an optimal quantizer when ρ(△= σp/σq) = 1/2. Also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1/ N3/ρfor the heavy mismatch of ρ >; 3/2, as lnN/ N2for the critical mismatch of ρ = 3/2, and as 1/N2for the mild mismatch of ρ2is confirmed rigorously. In addition, optimal uniform quantizers are found to be heavily mismatched when ρ >; 1 and the resulting distortion decreases as (In N/N2)1/p.

This letter derives asymptotic formulas for the mean square error (MSE) distortion and the signal-to-noise ratio of a mismatched fixed-rate minimum MSE Laplacian quantizer. These closed-form formulas are expressed in terms of the number N of quantization points, the mean displacement, and the ratio p of the standard deviation of the source to that for which the quantizer is optimally designed. Numerical results show that the principal formula is accurate in that, for rate R = log2N ges 6, it predicts signal-to-noise ratios within 1 % of the true values for a wide range of variance and mean mismatch. The new findings herein include the fact that, for heavy variance mismatch of p > 3/2, the signal-to-noise ratio increases at the rate of 9/p dB/bit, which is slower than the usual 6 dB/bit, and the fact that an optimal uniform quantizer, though optimally designed, is slightly more than critically mismatched to the source. The derived formulas can be useful in quantization of speech or music signals, which are modeled well as Laplacian sources and have changing short-term variances.

This correspondence shows that the support growth of a fixed-rate optimum (minimum mean-squared error) scalar quantizer for a Laplacian density is logarithmic with the number of quantization points. Specifically, it is shown that, for a unit-variance Laplacian density, the ratio of the support-determining threshold of an optimum quantizer to 3//spl radic/2lnN/2 converges to 1, as the number N of quantization points grows. Also derived is a limiting upper bound that says that the support-determining threshold cannot exceed the logarithmic growth by more than a small constant, e.g., 0.0669. These results confirm the logarithmic growth of the optimum support that has previously been derived heuristically.

This paper determines how the support regions of optimal and asymptotically optimal fixed-rate scalar quantizers (with respect to mean-squared error) depend on the number of quantization points N and the probability density of the variable being quantized. It shows that for asymptotic optimality it is necessary and sufficient that the support region grow fast enough that the outer (or overload) distortion decreases as o(1/N/sup 2/). Formulas are derived for the minimal support of asymptotically optimal quantizers for generalized gamma densities, including Gaussian and Laplacian. Interestingly, these turn out to be essentially the same as for the support of optimal fixed-rate uniform scalar quantizers. Heuristic arguments are then used to find closed-form estimates for the support of truly optimal quantizers for generalized gamma densities. These are found to be more accurate than the best prior estimates, as computed by numerical algorithms. They demonstrate that the support of an optimal quantizer is larger than the minimal asymptotically optimal support by a factor depending on the density but not N, and that the outer distortion of optimal quantizers decreases as 1/N/sup 3/.

This paper extends Bennett's (1948) integral from scalar to vector quantizers, giving a simple formula that expresses the rth-power distortion of a many-point vector quantizer in terms of the number of points, point density function, inertial profile, and the distribution of the source. The inertial profile specifies the normalized moment of inertia of quantization cells as a function of location. The extension is formulated in terms of a sequence of quantizers whose point density and inertial profile approach known functions as the number of points increase. Precise conditions are given for the convergence of distortion (suitably normalized) to Bennett's integral. Previous extensions did not include the inertial profile and, consequently, provided only bounds or applied only to quantizers with congruent cells, such as lattice and optimal quantizers. The new version of Bennett's integral provides a framework for the analysis of suboptimal structured vector quantizers. It is shown how the loss in performance of such quantizers, relative to optimal unstructured ones, can be decomposed into point density and cell shape losses. As examples, these losses are computed for product quantizers and used to gain further understanding of the performance of scalar quantizers applied to stationary, memoryless sources and of transform codes applied to Gaussian sources with memory. It is shown that the short-coming of such quantizers is that they must compromise between point density and cell shapes.>

A recursively indexed scalar quantizer that performs as well as high-dimensional vector quantizers for several important sources, without the attendant complexity, is presented. The recursively indexed quantizer provides a simple technique, both in terms of design and operation, for use with entropy coding. It also provides a simple quantization technique for use in noisy channel conditions where a variable length code would be inappropriate but performance greater than that provided by Lloyd-Max quantizers is desired.>