Memoryless scalar quantization for random frames

Abstract

Memoryless scalar quantization (MSQ) is a common technique to quantize generalized linear samples of signals. The non-linear nature of quantization makes the analysis of the corresponding approximation error challenging, often resulting in the use of a simplifying assumption, called the “white noise hypothesis” (WNH) that is useful, yet also known to be not rigorous and, at least in certain cases, not valid. We obtain rigorous reconstruction error estimates without relying on the WNH in the setting where generalized samples of a fixed deterministic signal are obtained using (the analysis matrix of) a random frame with independent isotropic sub-Gaussian rows; quantized using MSQ; and reconstructed linearly. We establish non-asymptotic error bounds that explain the observed error decay rate as the number of measurements grows, which in the special case of Gaussian random frames show that the error approaches a (small) non-zero constant lower bound. We also extend our methodology to dithered and noisy settings as well as the compressed sensing setting where we obtain rigorous error bounds that agree with empirical observations, again, without resorting to the WNH.