  The sampling, quantization, and estimation of a bounded dynamic-range
bandlimited signal affected by additive independent Gaussian noise is studied
in this work. For bandlimited signals, the distortion due to additive
independent Gaussian noise can be reduced by oversampling (statistical
diversity). The pointwise expected mean-squared error is used as a distortion
metric for signal estimate in this work. Two extreme scenarios of quantizer
precision are considered: (i) infinite precision (real scalars); and (ii)
one-bit quantization (sign information). If $N$ is the oversampling ratio with
respect to the Nyquist rate, then the optimal law for distortion is $O(1/N)$.
We show that a distortion of $O(1/N)$ can be achieved irrespective of the
quantizer precision by considering the above-mentioned two extreme scenarios of
quantization. Thus, a quantization precision indifference principle is
discovered, where the reconstruction distortion law, up to a proportionality
constant, is unaffected by quantizer's accuracy.
