Determination of Two-Dimensional Correlation Spectra Using the Hilbert Transform

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Abstract
A computationally efficient numerical procedure to generate two-dimensional (2D) correlation spectra from a set of spectral data collected at certain discrete intervals of an external physical variable, such as time, temperature, pressure, etc., is proposed. The method is based on the use of a discrete Hilbert transform algorithm which carries out the time-domain orthogonal transformation of dynamic spectra. The direct computation of a discrete Hilbert transform provides a definite computational advantage over the more traditional fast Fourier transform route, as long as the total number of discrete spectral data traces does not significantly exceed 40. Furthermore, the mathematical equivalence between the Hilbert transform approach and the original formal definition based on the Fourier transform offers an additional useful insight into the true nature of the asynchronous 2D spectrum, which may be regarded as a time-domain cross-correlation function between orthogonally transformed dynamic spectral intensity variations.