  In this paper, the optimal sampling strategies (uniform or nonuniform) and
distortion tradeoffs for Gaussian bandlimited periodic signals with additive
white Gaussian noise are studied. Our emphasis is on characterizing the optimal
sampling locations as well as the optimal pre-sampling filter to minimize the
reconstruction distortion. We first show that to achieve the optimal
distortion, no pre-sampling filter is necessary for any arbitrary sampling
rate. Then, we provide a complete characterization of optimal distortion for
low and high sampling rates (with respect to the signal bandwidth). We also
provide bounds on the reconstruction distortion for rates in the intermediate
region. It is shown that nonuniform sampling outperforms uniform sampling for
low sampling rates. In addition, the optimal nonuniform sampling set is robust
with respect to missing sampling values. On the other hand, for the sampling
rates above the Nyquist rate, the uniform sampling strategy is optimal. An
extension of the results for random discrete periodic signals is discussed with
simulation results indicating that the intuitions from the continuous domain
carry over to the discrete domain. Sparse signals are also considered, where it
is shown that uniform sampling is optimal above the Nyquist rate.
