Sampling and Distortion Tradeoffs for Bandlimited Periodic Signals

In this paper, the optimal sampling strategies (uniform or nonuniform) and distortion tradeoffs for stationary Gaussian bandlimited periodic signals with additive white Gaussian noise are studied. Unlike the previous works that commonly consider the average distortion as the performance criterion, we justify and use both the average and variance of distortion as the performance criteria. To compute the optimal distortion, one needs to find the optimal sampling locations, as well as the optimal pre-sampling filter. A complete characterization of optimal distortion for the rates lower than half the Landau rate is provided. It is shown that nonuniform sampling outperforms uniform sampling. In addition, this nonuniform sampling is robust with respect to missing sampling values. Next, for the rates higher than half the Landau rate, we find bounds that are shown to be tight for some special cases. 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. Finally, we consider a sampling/quantization scheme for compressing the signal. Here, we show that the total distortion can be expressed as the sum of sampling and quantization distortions. This implies a lower bound on the distortion via Shannon's rate distortion theorem.