Quantization of Bandlimited Functions Using Random Samples

We investigate the compatibility of distributed noise-shaping quantization with random samples of bandlimited functions. Let f be a real-valued π-bandlimited function. Suppose R > 1 is a real number, and assume that $\left\{ {{x_i}} \right\}_{i = 1}^m$ is a sequence of i.i.d random variables uniformly distributed on $\left[ { - \tilde R,\tilde R} \right]$, where $\tilde R > R$ is appropriately chosen. We show that on using a distributed noise-shaping quantizer to quantize the values of f at $\left\{ {{x_i}} \right\}_{i = 1}^m$, a function f ^♯ can be reconstructed from these quantized values such that ${\left\| {f - {f^\sharp }} \right\|_{{L^2}[ - R,R]}}$ decays with high probability as m and $\tilde R$ increase.