New insight into the mathematical background of generalized two-dimensional correlation spectroscopy and the influence of mean normalization pretreatment on two-dimensional correlation spectra

Generalized two-dimensional (2D) correlation spectroscopy is considered from the point of view of linear algebra. It is shown that a synchronous spectrum is the same as the cross-product matrix of the experimental data. An asynchronous spectrum is also treated as the ordered scalar products of the dynamic vectors of the experimental matrix and its Hilbert transformation. This approach connects the theory of generalized 2D correlation spectroscopy with the well-known conceptions of classic correlation analysis. The importance of spectral normalization in the 2D correlation analysis and its influence on the 2D correlation spectra is also investigated. All calculations were done on a synthetic spectral model consisting of two components. The synchronous spectra obtained from the model matrix were compared with those obtained after spectral mean normalization. It was found that the results strongly depend on the pretreatment. We plotted the Hilbert transformation of the mean-centered model and found that the normalization leads to the disappearance of the asynchronous spectra in the two-component system. Also, it has been concluded that the influence of normalization is important just for the systems with large intensity variations. All the results presented here are quite general and can be applied irrespective of the nature of the perturbation.