Nonuniform sampling: Bandwidth and aliasing

For spectroscopic measurements, there are good reasons why one might consider using nonuniformly nonsimultaneously sampled complex data. The primary one is that the effective bandwidth, the largest spectral window free of aliases, can be much wider than with uniformly sampled data. In this article, we discuss nonuniformly nonsimultaneously sampled data, describe how these data are traditionally analyzed, analyze them using probability theory, and show how probability theory generalizes the discrete Fourier transform: first for uniformly sampled data, then for nonuniformly sampled data, and finally for nonuniformly nonsimultaneously sampled data. These generalizations demonstrate that aliases are not so much removed by nonuniform nonsimultaneous sampling as they are moved to much higher frequencies.