The validity of the additive noise model for uniform scalar quantizers

A uniform scalar quantizer with small step size, large support, and midpoint reconstruction levels is frequently modeled as adding orthogonal noise to the quantizer input. This paper rigorously demonstrates the asymptotic validity of this model when the input probability density function (pdf) is continuous and satisfies several other mild conditions. Specifically, as step size decreases, the correlation between input and quantization error becomes negligible relative to the mean-squared error (MSE). The model is even valid when the input density is discontinuous at the origin, but discontinuities elsewhere can prevent the correlation from being negligible. Though this invalidates the additive model, an asymptotic formula for the correlation is found in terms of the step size and the heights and positions of the discontinuities. For a finite support input density, such as uniform, it is shown that the support of the uniform quantizer can be matched to that of the density in ways that make the correlation approach a variety of limits. The derivations in this paper are based on an analysis of the asymptotic convergence of cell centroids to cell midpoints. This convergence is fast enough that the centroids and midpoints induce the same asymptotic MSE, but not fast enough to induce the same correlations.