Double Fourier transformation

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

Although the idea of generating 2D correlation spectra was introduced several decades ago in the field of NMR [1008], extension to other areas of spectroscopy has been slow. This is essentially on account of the time-scale. Characteristic times associated with typical molecular vibrations probed by IR are of the order of picoseconds, which is many orders of magnitude shorter than the relaxation times in NMR. Consequently, the standard approach used successfully in 2D NMR, i.e. multiple-pulse excitations of a system, followed by detection and subsequent double Fourier transformation of a series of free-induction decay signals [1009], is not readily applicable to conventional IR experiments. A very different experimental approach is therefore required. The approach for generation of 2D IR spectra defined by two independent wavenumbers is based on the detection of various relaxation processes, which are much slower than vibrational relaxations but are closely associated with molecular-scale phenomena. These slower relaxation processes can be studied with a conventional... 

The second development that has revolutionized the practice of NMR was the introduction of multidimensional spectroscopy. This was initialized by Jeener [2], who showed that, by introducing a second pulse and varying the time between them, a second time-axis could be constructed. A double Fourier transformation yields the familiar two-dimensional spectrum, nowadays known by everyone as the COSY spectrum. Ernst, already involved in the development of FT-NMR, showed that the concept was more generally applicable [3], and paved... 

Fig. 6. Basic scheme of the NOESY sequence which provides essentially homo-nuclear cross-relaxation rates from a double Fourier transform of the signal S( 2)-...

Thus the required scattering function can be obtained from eqns (12.5) and (12.6) by a double Fourier transform. This is straightforward but somewhat lengthy to perform if the geometrical term is added back as unaffected by the crack, then the result is... 

The COSY spectrum is produced by a double Fourier transformation with respect to tl and t2, and its cross peaks indicate which H nuclei are mutually J-coupled. 

Example. As a model for two-stage diffusion take i = 1,2 and F as in (7.1). Then 7i,2 = 2 and 72,1 =7i- F°r computing the cross-section for neutron scattering one needs to know the probability density Gs(r, t) that a molecule that, at t = 0, was at r = 0 will, at time t, be at r. The differential cross-section is its double Fourier transform GS(A , co). It is convenient to apply the Fourier transformation in space right away to (7.4) so that both operators Ff reduce to factors,... 

The image formed by a convergent lens (Figure 2) is the double Fourier transform of the object 0. The electron diffraction... 

J/2) from the CH2 protons and the other at 5-94 Hz from the CH proton whose resonance in a normal spectrum is a 1 2 1 triplet with outer components which precess at twice the rate of the components of the CH2 doublet. (148) In fact each spin-echo normally contains a number of different frequency components which can be separated by electronic filtration or by Fourier transformation to yield a set of J-spectra which can then be conveniently displayed in a two-dimensional plot. In practice the second approach is normally used, and thus the set of spin-echoes is subjected to a double Fourier transformation according to equation (2) in which cafe must be taken to combine correctly the real and imaginary transforms. (26)... 

FIG. 15. A schematic diagram of the operation of a double Fourier transformation program for obtaining two-dimensional spectra. Sine and cosine transforms are represented by S and C, and the corresponding double transforms by CC, SC, CS, and SS. Operations in the sequence are represented by solid lines and information transfer by dotted lines. From ref. 22. 

It is important to realize that double Fourier transformation is not an essential part of two-dimensional NMR spectroscopy. This was made clear by Ernst in his early article (26) and when the spin system is simple there may be no particular advantage in using it. This approach was adopted in studies of C-enriched methyl formate in which various selective H and/or pulses were applied to individual transitions and it was found possible to deduce an accurate value for v( C) and draw conclusions about relaxation behaviour. (164-166) Detailed analysis (167) also shows that the sensitivity of two-dimensional Fourier transform spectroscopy can be as good as half that achieved in ordinary one-dimensional experiments. In this connection we should note that the time-saving gain of Fourier transformation... 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

Figure 6-4 (a) The result of the COSY experiment after double Fourier transformation for a single isolated nucleus such as that in Figure 6-3. (b) The result of the COSY experiment for two uncoupled nuclei. 

Since Eq. (29) is orientation independent, all crystallites are refocused at the same time t2e=R I,p)ti, giving rise to an isotropic echo. After a double Fourier transform in and tj.the NMR resonances appear along the anisotropic axis A with direction Vi=R(f,pjv2, where V2 and Vj are the single- and multiple-quantum dimensions frequencies, respectively. The projection of the spectrum onto an axis perpendicular to A yields an isotropic, highly resolved, spectrum. 

Divinylbenzene-hydrophilic methacrylate copolymer 944 DNOC 1350 Domesticine 1064 DOtz benzannulation 454-459 Double Fourier transform filtering 984, 985 DRD 953 Drinking water,... 

It is easy to discover by plots of (34) that the shape of this signal is not actually circular or elliptical, especially because of the presence of the sine term. The double Fourier transform of the signal (34) according the procedure (25) and... 

Figure 5.7. A generalised scheme for the collection of a two dimensional data set. The experiment is repeated many times with the t -period incremented at each stage and the resulting FIDs stored separately. Following double Fourier transformation with respect to first t2 and then ti, the two-dimensional spectrum results.

In a 2D experiment one or more scans are acquired with a delay tl that is incremented in subsequent acquisitions to generate a time domain tl. The time domain tl in conjunction with the acquisition time domain t2 generates a 2D data set that upon double Fourier transform gives a 2D spectrum. In a very simplified view all 2D experiments can be described as series of ID experiments but in practise the situation is rather more complicated because to achieve quadrature detection in both dimensions phase cycling or pulse field gradients must be used. Consequently the processing of 2D data sets depends upon the detection mode and the experimental setup. 

Data analysis of EXAFS spectra involves double-Fourier-transform procedures whose precision is highly dependent on the accessible energy range. At high pressures, the contributions from the first shell of neighbours are readily accessible, and, less frequently, so are those from the second and third shells. A detailed review of XAS under high pressures can be found in reference 2 and the references therein. 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

For quadmpolar nuclei, the dependence of the pulse response on Vq/Vj has led to the development of quadmpolar nutation, which is a two-dimensional (2D) NMR experiment. The principle of 2D experiments is that a series of FIDs are acquired as a function of a second time parameter (e.g. here the pulse length applied). A double Fourier transformation can then be carried out to give a 2D data set (FI, FI). For quadmpolar nuclei while the pulse is on the experiment is effectively being carried out at low field with the spin states determined by the quadmpolar interaction. In the limits Vq 4 Vj and Vq Vj the pulse response lies at Vj and (/ -I- )vj respectively so is not very discriminatory. However, for Vq Vj the pulse response is complex and... 

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