Quantized sampling as sampling with uncertainty in time and amplitude

In classical sampled quantization, the signal is sampled at discrete times and at discrete values, resulting in uncertainty of the signal amplitude. However, the sampling times and the boundaries of the quantization intervals are still assumed to be known with infinite precision. The aim of this paper is to study quantization and sampling when these quantities are not known with infinite precision by considering quantized sampling in a general framework as sampling with uncertainty in time and amplitude. We define the concept of a valid quantized sample and consider a quantized sampling of a signal as a collection of valid quantized samples. We show that for continuous signals, a set of valid quantized samples generates a secondary set of valid quantized samples. We illustrate that oversampling can reduce reconstruction errors because oversampling can reduce the uncertainty in the secondary quantized samples. In particular, these secondary quantized samples have uncertainty approaching zero as oversampling increases, provided the sampling time and quantization thresholds are known with infinite precision. For a class of T-periodic bandlimited signals, this implies that the reconstruction error is a function of the oversampling ratio, the uncertainty in the sampling time, the stepsize of the quantizer, and the uncertainty in the quantization thresholds.