Transformation covariant

ACC transforms = Auto-Cross-Covariance transforms —> autocorrelation descriptors... 

Covariance NMR mostly refers to any NMR experiment whose resulting data are subjected at some point to covariance analysis, covariance transformation or covariance treatment. Covariance NMR processing describes the steps that compute the covariance from a matrix of NMR data and yields the covariance map. The covariance map is equivalent to a NMR spectrum obtained after Fourier transformation, if the covariance was calculated obeying certain mathematical constraints, cf further below. In other words. 

FromEqs. (5.1) and (5.2), two conclusions can quickly be drawn. Firstly, Eq. (5.1) can be extended to Eq. (5.13), su esting directly that covariance transformations can be amended to three and higher dimension. 

According to Eq. (5.14), covariance transformations can be applied to any type of spectral data sets that are connected to each other by a common history or domain. The spectra thus generated represent heterospectral correlation maps. In NMR spectroscopy, this concept was taken up as unsymmetrical indirect covariance (UIC) NMR, relating, for example. 

Doubly indirect covariance (DIG) yields a symmetric correlation map [13]. It can further be considered as a bridge to the generalization of indirect covariance. It implies that correlation spectra, that are used for indirect covariance, themselves may be a creation of a previous covariance transformation. Scheme 5.1 serves as an example to elucidate this perspective. In Scheme 5.1A and B, indirect covariance generates homonuclear spectra. Part (C) leads to the creation of a heteronuclear correlation, which itself can be taken as component spectrum for transformation (A) or (B). It is apparent that Scheme 5.1 represents an oversimplification of the types of indirect covariance treatment. A wider variety of spectra that can be obtained by UIC processing is collected in Table 5.1. 

The matrix representation of the matrix formalism, in general, is more illustrative than the notation using sums and multiple indices. The equivalence between a 2-dimensional data matrix and a 2D spectrum or a mixed time-frequency data set is evident. While the covariance literature provides numerous examples of spectra obtained from covariance transformations of experimental data sets, which will be discussed further below, few sim-phfied models have been designed and applied for instruction purposes [14,17,18,32—34]. Thoroughly elaborated cases are described in Refs. [6,11,12,14,35-39]. 

For the purpose of adding a simple model to the following discussion. Figs. 5.1 and 5.2 present an overview of the covariance transformation types 1—4 on a selected set of matrices. The arrays used are of very small dimensions as compared to experimentally recorded NMR data. Still they reflect the essential features of 2D NMR spectra such as a H-H COSY or a H—TOCSY with a short mixing time, cf Fig. 5.1A, and an H-C HSQC, cf. Fig. 5.IB. The values of the matrix elements were arbitrarily chosen and indicate a signal or a correlation at that position. The relative signs should be interpreted as signal phases, cf. Fig. 5.IE as an example for a multiphcity-edited HSQC. 

GIC of the simplified models of the COSY and the multiplicity-edited HSQC. The spectrum resulting for A=1 resembles the UIC spectrum. Yet, the signal intensities are different when the square root operation >1 = 0.5 is applied, see Fig. 5.1 G. The positive effect of the GIC concerning the minimization of artefacts will be summarized in due course. It is not visible in this model. Fig. 5.1H symboHzes the transformation of a multiplicity-edited HSQC with an 1,1-ADEQUATE to yield a C—C correlation map, whose interpretation scheme will be given in Fig. 5.14. The construction of a 3R cube from two 2D spectra is illustrated in Fig. 5.11. It is the only covariance transformation among the examples that does not result from matrix multiphcation but from reconstruction according to Eq. (5.24). 

The two large-scale cormnercial vendors of spectrometer-independent NMR software currently offer covariance processing as a feature of their spectral manipulation and interpretation suites. The Spectrus platform by ADC/Labs and the MNova by Mestrelab Research provide stand-alone solutions that allow the import of all spectrometer data formats without the necessity of pre-processing. Covariance processing can be performed according to the direct, indirect, and generalized indirect formalism. The doubly indirect covariance transformation is formd in MNova. While... 

Instead of a multipHcity-edited HSQC as in the above experiments, a 1/cc inverted 1, -ADEQUATE was covariance transformed with a non-edited HSQC. A Jcc-edited HSQC-1, -ADEQUATE resulted from the GIC formahsm [31]. For the inversion of the Jcc correlations, 19 h of experiment time divided on 160 increments in the 0 domain were spent on a 600 MHz NMR instrument equipped with a 1.7 mm cryoprobe. The sample tube contained 500 pg of strychnine. A schematic representation of the Jcc-edited HSQC-1,1-ADEQUATE is given in Fig. 5.14. 

GB, but did not cause any problem to modem computer equipment. In combination with covariance transformation, spectral filtering based on signal mean positions and their line widths was performed. As a result, 22 of the 25 overlapping proton resonances of antamanide could be successfully disambiguated. 

Covariance NMR has established itself as a valuable tool in the ranks ofNMR methodologies. As direct and indirect covariance of 2-dimensional NMR data sets, it can replace the second Fourier transformation, whereas as unsymmetrical and GIC it follows Fourier transformation. Based on matrix algebra and statistical mathematics, covariance transformations were extended to doubly indirect and 3- and 4-dimensional covariance NMR. Since the parent or component data arrays can originate from any type of 2D NMR experiment, 2D NMR spectra were co-processed to yield C-N and C-P correlations of non-isotopicaUy labelled small molecules. Further heterospectroscopic covariance was used to concatenate NMR and MS data allowing to allocate the information of both to a compound. 

Common to the covariance transformations, sensitivity enhancement or experiment time reduction were observed, since in the case of combination spectra the sensitivity of the resulting correlation map would approach the sensitivity ofthe component spectra. For the HSQC-1,1-ADEQUATE, the signal-to-noise ratio was reported to be nearly as high as for the HSQC itself The HSQC might hence be used to enhance the signal-to-noise ratio ofthe... 

Again we note that the present complex symmetric ansatz has generated a covariant transformation, compatible with the classical Schwartzschild gauge but, as has been stated recurrently, we have not yet proven or explained the origin of time irreversibility. 

When U in (10.2.4) is a general non-singular mxm matrix the infinite set U forms a matrix group, the full linear group in m dimensions, denoted by GL(m). If the matrices are chosen to be unitary (thus leaving invariant any Hermitian scalar product, as we know from Section 2.2) then we obtain the unitary group U(m) and in this case the matrices of the covariant transformation in (10.2.4b) are... 

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