On the reconstruction of bandlimited signals from random samples quantized via noise-shaping

Abstract

Noise-shaping quantization techniques are widely used for converting bandlimited signals from the analog to the digital domain. They work by ``shaping" the quantization noise so that it falls close to the reconstruction operator's null space. We investigate the compatibility of two such schemes, specifically $\hat{I}\pounds \hat{I}”$ quantization and distributed noise-shaping quantization, with random samples of bandlimited functions. Suppose $R>1$ is a real number and assume that $\{x_i{\}}_{i=1}^m$ is a sequence of i.i.d random variables uniformly distributed on $[-\tilde{R},\tilde{R}]$, where $\tilde{R}>R$ is appropriately chosen. We show that by using a noise-shaping quantizer to quantize the (randomly sign flipped) values of a real-valued $\ddot{I}€$-bandlimited function $f$ at $\{x_i{\}}_{i=1}^m$, a function $f^{♯}$ can be reconstructed from these quantized values such that $\Vert f-f^{♯}{\Vert }_{L^2[-R,R]}$ decays with high probability as $m$ and $\tilde{R}$ increase. This decay holds uniformly over all bandlimited $f$. We emphasize that the sample points $\{x_i{\}}_{i=1}^m$ are completely random, that is, they have no predefined structure, which makes our findings the first of their kind.