In particular, determination of a Z score, which measures the difference in standard deviations of a statistic from its mean, for each spectral point yields a Z matrix, which indicates whether a given peak intensity above a threshold arises from the covariance of signals in the input spectra or whether it is likely to be caused by noise. Different data processing procedures, including the Z-matrix formalism, thresholding, and maxima ratio scaling, are described to assess noise contributions and to reduce noise inhomogeneity. Therefore, methods of noise estimation commonly used in Fourier transform spectroscopy underestimate the amount of uncertainty in unsymmetric covariance spectra. It is shown how the unsymmetric covariance spectrum possesses an inhomogeneous noise distribution across the spectrum with the least amount of noise in regions whose rows and columns do not contain any cross or diagonal peaks and with the largest amount of noise on top of signal peaks. This work presents analytical relationships as well as simulated and experimental results characterizing the propagation of noise by unsymmetric covariance NMR processing, which concatenates two NMR spectra along a common dimension, resulting in a new spectrum showing spin correlations as cross peaks that are not directly measured in either of the two input spectra. Due to the limited sensitivity of many nuclear magnetic resonance (NMR) applications, careful consideration must be given to the effect of NMR data processing on spectral noise.
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