Nonuniform sampling and spectral aliasing

The Nyquist theorem stipulates the largest sampling interval sufficient to avoid aliasing is the reciprocal of the spectral bandwidth. When data are not sampled uniformly, the Nyquist theorem no longer applies, and aliasing phenomena become more complex. For samples selected from an evenly spaced grid, signals that are within the nominal bandwidth of the grid can give rise to aliases. The effective bandwidth afforded by a set of nonuniformly sampled evolution times does not necessarily correspond to spacing of the grid from which the samples are selected, but instead depends on the actual distribution of sample times. For conventional uniform sampling there is no distinction between the grid spacing and the sampling interval. For nonuniform sampling, an effective bandwidth can be inferred from the greatest common divisor of the sample times, provided that none of the sample times are irrational. A simple way to increase the effective bandwidth for a set of nonuniformly spaced samples is to randomly select them from an oversampled grid. For a given grid spacing, "bursty" sampling helps to minimize aliasing artifacts. We show that some spectral artifacts arising from nonuniform sampling are aliases, and that increasing the effective bandwidth shifts these artifacts out of the spectral window and improves spectral quality. An advantage of nonuniform sampling is that some of the benefits of oversampling can be realized without incurring experiment time or resolution penalties. We illustrate the improvements that can be obtained with nonuniform sampling in the indirect dimension of a SOFAST-HMQC experiment.