Suppose the original dataset
D contains the
n spectra in rows. The signals of the original dataset are generally preprocessed. The original spectra are compared to a reference spectrum. By subtracting a reference spectrum, often the average spectrum of the dataset, so called dynamic spectra are calculated which form the corresponding dynamic dataset
E. The presence and interpretation may be dependent on the choice of reference spectrum. The equations below are valid for equally spaced measurements of the perturbation.
Calculation of the synchronous spectrum A 2D synchronous spectrum expresses the similarity between spectral of the data in the original dataset. In generalized 2D correlation spectroscopy this is mathematically expressed as
covariance (or
correlation). :\phi(\nu_1 , \nu_2) = \frac{1}{n-1} y^T(\nu_1) . y(\nu_2) where: •
Φ is the 2D synchronous spectrum •
ν1 and
ν2 are two spectral channels •
yν is the vector composed of the signal intensities in
E in column
ν •
n the number of signals in the original dataset
Calculation of the asynchronous spectrum Orthogonal spectra to the dynamic dataset
E are obtained with the Hilbert-transform: :\psi(\nu_1 , \nu_2) = \frac{1}{n-1} y^T(\nu_1) . N . y(\nu_2) where: •
Ψ is the 2D asynchronous spectrum •
ν1 en
ν2 are two spectral channels •
yν is the vector composed of the signal intensities in
E in column
ν •
n the number of signals in the original dataset •
N the Noda-Hilbert transform matrix The values of
N,
Nj, k are determined as follows: • 0 if j = k • \frac{1}{\pi (k - j)} if j ≠ k where: •
j the row number •
k the column number ==Interpretation==