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 $ΣΔ$ 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 $π$-bandlimited function $f$ 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. 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.

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。其工作原理是通过"整形"量化噪声，使其接近重构算子零空间。我们研究了两种此类方案——具体为ΣΔ量化与分布式噪声整形量化——与带限函数随机样本的兼容性。设R>1为实数，假设{x_i}_{i=1}^m是[-R̃,R̃]上均匀独立同分布的随机变量序列，其中R̃>R适当选取。我们证明：通过使用噪声整形量化器对实值π-带限函数f在{x_i}_{i=1}^m处的（随机符号翻转）值进行量化，可从这些量化值重构出函数f♯，使得当m和R̃增加时，高概率下∥f-f♯∥_{L²[-R, R]}衰减。该衰减对全体带限函数f一致成立。我们强调样本点{x_i}_{i=1}^m是完全随机的，即它们没有预定义结构，这使得我们的结果是该类问题的首项成果。