Decoding Multicomponent Separations by the Autocovariance Function

The statistical model of peak overlap clearly explains that the number of observed peaks is much smaller than the number of components present in the sample. The Fourier analysis of multicomponent chromatograms can not only identify the ordered or disordered retention pattern but also estimate the average spot size, the number of detectable components present in the sample, the spot capacity, and the saturation factor (Felinger et al., 1990). Fourier analysis has been applied to estimate the number of detectable components in several complex mixtures. 

An adaptation of Fourier analysis to 2D separations can be established by calculating the autocovariance function (Marchetti et al., 2004). The theoretical background of that approach is that the power spectrum and the autocovariance function of a signal constitute a Fourier pair, that is, the power spectmm is obtained as the Fourier transform of the autocovariance function. 

A multicomponent 2D chromatogram is considered as a series of 2D peaks with random position and height. For the sake of simplicity, here we assume that the peaks are modeled with bidimensional Gaussian peaks, thus the signal is expressed as 

The 2D autocovariance function can also be calculated from the 2D chromatogram acquired in digitized form, 

A nonlinear curve fitting procedure of the experimental (Eq. 4.28) to the theoretical (Eq. 4.27) 2D autocovariance function can serve to perform some fundamental characterization of the 2D separation. The total volume (Vy) and the peak height dispersion (/a() can be readily measured in the chromatogram, thus the number of components (m) and the peak widths (a, and ay) can be estimated (Marchetti et al., 2004). 

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