Covariance NMR spectroscopy by singular value decomposition

Covariance NMR is demonstrated for homonuclear 2D NMR data collected using the hypercomplex and TPPI methods. Absorption mode 2D spectra are obtained by application of the square-root operation to the covariance matrices. The resulting spectra closely resemble the 2D Fourier transformation spectra, except that they are fully symmetric with the spectral resolution along both dimensions determined by the favorable resolution achievable along omega2. An efficient method is introduced for the calculation of the square root of the covariance spectrum by applying a singular value decomposition (SVD) directly to the mixed time-frequency domain data matrix. Applications are shown for 2D NOESY and 2QF-COSY data sets and computational benchmarks are given for data matrix dimensions typically encountered in practice. The SVD implementation makes covariance NMR amenable to routine applications.