Inner-product NMR spectroscopy: A variant of covariance NMR spectroscopy

We propose a variant of covariance NMR spectroscopy, namely, inner-product NMR spectroscopy, originally suggested in Takeda (2015). The mathematical operation of inner-product NMR is the same as that of covariance NMR, except that subtraction of the average value of the variable is intentionally omitted, so that the correspondence of the spectrum with that of conventional two-dimensional Fourier-transformation is established without having to request the average to become vanishingly small. We demonstrate inner-product NMR for ¹³C DARR correlation experiments in a polycrystalline sample of ¹³C-labeled l-alanine. In covariance, we show that the mixing-time dependence of the peaks is influenced considerably by the choice of the carrier frequency and thereby the center of the spectrum, whereas the inner-product approach is free from such an undesirable effect, while keeping the merit of the covariance NMR.