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# Peak Retention Time Using Discrete Fourier Transform

Consider a noise-free signal (e.g. a chromatogram of two chemical species), for instance, the sum of two Gaussian functions. This signal is given by , where the user can set the values of parameters , , , and . These two Gaussian functions can show partial or even complete overlap.
The blue curve shows the original signal. In this plot, there are either three extrema (two peaks shown in red and one valley shown in green) or only one extremum (one peak shown in red), when the two Gaussian functions overlap sufficiently.
This Demonstration applies the discrete Fourier transform to compute the derivative of the signal . This derivative is shown by the red curve. In addition, the positions of the extrema of are indicated by the blue dots in the derivative plot. A list of these extrema is given using the "peaks" tab. Clearly, for a valley, and , and for peaks, and .
Similar calculations are also possible for real signals, which may have white noise. Choose the "noisy chromatogram" tab to see the results for one particular case.

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