Multidimensional NMR spectroscopy typically employs phase-sensitive detection, which results in hypercomplex data (and spectra) when utilized in more than one dimension. Nonuniform sampling approaches have become commonplace in multidimensional NMR, enabling dramatic reductions in experiment time, increases in sensitivity and/or increases in resolution. In order to utilize nonuniform sampling optimally, it is necessary to characterize the relationship between the spectrum of a uniformly sampled data set and the spectrum of a subsampled data set. In this work we construct an algebra of hypercomplex numbers suitable for representing multidimensional NMR data along with partial-component nonuniform sampling (i.e. the hypercomplex components of data points are subsampled). This formalism leads to a modified DFT–convolution relationship involving a partial-component, hypercomplex point-spread function set. The framework presented here is essential for the continued development and appropriate characterization of partial-component nonuniform sampling.
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