Fourier transform theoretical background

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. 

Fig. 4. Nonlinear conductance oscillations at low field from a 6 /mi junction, (a) shows the oscillations as a function of both B and V. (A smoothed background has been subtracted to emphasize the oscillations.) The brightest (and darkest) lines, corresponding to tunneling between the lowest modes, break the V-B plain into regions I, II, and III. Additional positively-sloped bright and dark lines in II arise from other ID channels in the wires and are disregarded in our theoretical analysis. Also present is a slow modulation of the strength of the oscillations along the abscissa, (b) Absolute value of the peak of the Fourier transform of the conductance at a fixed V in region II as a function of V. Its slow modulation as a function of V is easily discerned.

The theoretical background for these four nuclei has already been presented in Chapters 3 and 4. Our treatment of spin, coupling, NOE, Fourier transformation, etc. can be applied to these nuclei without modification. The concept of chemical shift we also use without modification, but we must avoid exercising the predictive skills that we have developed for H and 13C chemical shifts to these nuclei (with some exceptions as noted). 

In this subsection, the theoretical background for SANS and neutron spin-echo measurements carried out with o/w- and w/o-droplet microemulsions will be presented. According to Milner, Safran and others, shape fluctuations in droplet microemulsions can be described in terms of spherical harmonics [42-44]. This offers the possibility to calculate a dynamic structure factor S(q,w) or its Fourier transform, i.e. the intermediate scattering function I(q,t) for the problem, which can be used to analyse dynamical measurements by neutron spin-echo spectroscopy [45]. For the scattering from thin shells I(q,t) was calculated [43]... 

In this Chapter the theoretical background that led to the Double Fourier Modulation technique has been presented, this is Fourier Transform Spectroscopy and Stellar Interferometry. 

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