Incoherence of Partial-Component Sampling in multidimensional NMR

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In NMR spectroscopy, undersampling in the indirect dimensions causes reconstruction artifacts whose size can be bounded using the so-called coherence. In experiments with multiple indirect dimensions, new undersampling approaches were recently proposed: random phase detection (RPD) Maciejewski11 and its generalization, partial component sampling (PCS) Schuyler13. The new approaches are fully aware of the fact that high-dimensional experiments generate hypercomplex-valued free induction decays; they randomly acquire only certain low-dimensional components of each high-dimensional hypercomplex entry. We provide a classification of various hypercomplex-aware undersampling schemes, and define a hypercomplex-aware coherence appropriate for such undersampling schemes; we then use it to quantify undersampling artifacts of RPD and various PCS schemes.