Here one should think of the momentum vectors of the applied gradient pulse as the frequency tuples of the standard Fourier expression.
Hence the diffusion component for each frequency (or direction gradient) in a overall diffusion probability function is none other than the it's corresponding Fourier component.
Hence, inverting the Fourier components of the diffusion signal will give you back the total probability function of the diffusing molecules that are dancing according to Brown and Einstein .
P. S. In a nutshell, it is a Fourier mixing of the directional information with the Fourier modes.
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