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 $\Sigma\Delta$ quantization and distributed noise-shaping quantization, with random samples of bandlimited functions. Let $f$ be a real-valued $\pi$-bandlimited function. 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 values of $f$ at $\{x_i\}_{i=1}^m$, a function $f^{\sharp}$ can be reconstructed from these quantized values such that $\|f-f^{\sharp}\|_{L^2[-R, R]}$ decays with high probability as $m$ and $\tilde{R}$ increase. We emphasize that the sample points $\{x_i\}_{i=1}^m$ are completely random, i.e., they have no predefined structure, which makes our findings the first of their kind.

翻译：噪声整形量化技术广泛应用于将带限信号从模拟域转换到数字域。其原理是通过“整形”量化噪声，使其接近重构算子的零空间。本文研究两种此类方案——具体为ΣΔ量化与分布式噪声整形量化——与带限函数随机采样的兼容性。设f为实值π带限函数，取实数R>1，假设{x_i}_{i=1}^m为独立同分布于[-\tilde{R},\tilde{R}]的随机变量序列，其中\tilde{R}>R适当选取。我们证明：利用噪声整形量化器对函数f在{x_i}_{i=1}^m处的值进行量化后，可从这些量化值重构出函数f^#，使得\|f-f^#\|_{L^2[-R, R]}随m和\tilde{R}增大而高概率衰减。需强调的是，采样点{x_i}_{i=1}^m完全随机，即无预定义结构，这使得本研究成为该类问题的首项成果。