The Performance of PCM Quantization Under Tight Frame Representations

TL;DR

This paper analyzes the PCM quantization error for finite unit-norm tight frames in real space, deriving bounds without the White Noise Hypothesis and showing the error's behavior in different dimensions and frame constructions.

Contribution

It provides new bounds on PCM quantization error for tight frames, extending previous results and demonstrating the error's non-vanishing nature with increasing redundancy.

Findings

Error bound of O(δ^{3/2}) for certain 2D frames

Error bound of O(δ^{(d+1)/2}) in higher dimensions

Error does not diminish to zero with increased redundancy

Abstract

In this paper, we study the performance of the PCM scheme with linear quantization rule for quantizing finite unit-norm tight frame expansions for $\R^d$ and derive the PCM quantization error without the White Noise Hypothesis. We prove that for the class of unit norm tight frames derived from uniform frame paths the quantization error has an upper bound of $O(\delta^{3/2})$ regardless of the frame redundancy. This is achieved using some of the techniques developed by G\"{u}nt\"{u}rk in his study of Sigma-Delta quantization. Using tools of harmonic analysis we show that this upper bound is sharp for $d=2$. A consequence of this result is that, unlike with Sigma-Delta quantization, the error for PCM quantization in general does not diminish to zero as one increases the frame redundancy. We extend the result to high dimension and show that the PCM quantization error has an upper bound $O(\delta^{(d+1)/2})$ for asymptopitcally equidistributed unit-norm tight frame of $\R^{d}$.