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。其工作原理是"整形"量化噪声，使其落入重建算子的零空间附近。本文研究了两种此类方案——具体而言即$\Sigma\Delta$量化与分布式噪声整形量化——与带限函数随机样本的兼容性。设$f$为实值$\pi$-带限函数，假设$R>1$为实数，且$\{x_i\}_{i=1}^m$是均匀分布在$[-\tilde{R},\tilde{R}]$上的独立同分布随机变量序列（其中$\tilde{R}>R$为适当选取的常数）。我们证明：通过使用噪声整形量化器对$f$在$\{x_i\}_{i=1}^m$处的函数值进行量化，可从这些量化值重建出函数$f^{\sharp}$，使得$\|f-f^{\sharp}\|_{L^2[-R, R]}$随着$m$与$\tilde{R}$的增大而以高概率衰减。需要强调的是，采样点$\{x_i\}_{i=1}^m$是完全随机的，即不具备任何预定义结构，这使得我们的研究成果具有开创性意义。