Fourier spectrum extension

Because no unique extension of the Fourier spectrum exists in the absence of additional information, the impetus of the research described in this chapter has been to discover as much prior information as possible about the data and the influences on them, and to find ways to formulate this information as constraints to produce increasingly more probable values for the... 

Two-dimensional FFT filtering is an extension of the one-dimensional case and both are based on the same FFT algorithms. The M noisy ac polarograms (N points each) can be represented as a M N matrix (M rows and N columns). First the FFT is performed on each row to yield an intermediate matrix and then the columns of the intermediate matrix are transformed. The result is a two-dimensional Fourier spectrum (also an M N matrix) which is multiplied by a two dimensional filter and inverse transformed to yield the smoothed ac polarographic data. In this study 8 ac polarograms of 256 points each were smoothed simultaneously using the two-dimensional FFT filtering technique. 

This effect induces a free induction decay (FID) signal in the detection circuit. The FID can be measured, and the normal absorption spectrum can be obtained by means of an inverse Fourier transform. A variety of experimental extensions have been developed for this approach. By means of particular pulse sequences it is possible to detect spin resonances selectively on the basis of a broad ensemble of properties such as spatial proximity and dipolar coupling strengths. The central fundamental quantity of interest is, however, still the energy spectrum of the nuclear spin,... 

Let the discrete spectrum, which consists of the coefficients of u(k) and v(k), be denoted by U(n) and V(n), respectively. The low-frequency spectral components U(n) are most often given by the most noise-free Fourier spectral components that have undergone inverse filtering. For these cases V(n) would then be the restored spectrum. However, for Fourier transform spectroscopy data, U(n) would be the finite number of samples that make up the interferogram. For these cases V(n) would then represent the interferogram extension. 

An obvious extension to 3D spectroscopy from 2D spectroscopy is the homonu-clear NOESY-NOESY [34]. There are two t variable times and one tj, which after Fourier transform provide three frequency domains. The 3D NOESY-NOESY spectrum of met-myoglobin cyanide, which contains low spin iron(III) in a heme moiety (see Fig. 5.7), has been successfully measured [35]. In Fig. 8.22 a slice of the 3D spectrum is shown at the I2-CH3 height. On the diagonal it shows all the dipolar connectivities between I2-CH3 and other protons off-diagonal... 

These considerations have to be applied to phenomena in which the external field has its origin in the solute (or, better, in the response of the solute to some stimulus). The characteristics of this field (behaviour in time, shape, intensity) strongly depend on the nature of the stimulus and on the properties of the solute. The analysis we have reported of the behaviour of the solvent under the action of a sinusoidal field can here be applied to the Fourier development of the field under examination. It may happen that the Fourier decomposition will reveal a range of frequencies at which experimental determinations are not available to have a detailed description of the phenomena an extension of the s(w) spectrum via simulations should be made. It may also happen that the approximation of a linear response fails in such cases the theory has to be revisited. It is a problem similar to the one we considered in Section 1.1.2 for the description of static nonlinear solvation of highly charged solutes. 

A third general issue regards the dynamic coupling between solute and solvent. To accurately model excited states formation and relaxation of molecules in solution, the electronic states have to be coupled with a description of the dynamics of the solvent relaxation toward an equilibrium solvation regime. The formulations of continuum models which allow to include a time dependent solvation response can be formulated as a proper extension of the time-independent solvation problem (of equilibrium or of nonequilibrium). In the most general case, such an extension is based on the formulation of the electrostatic problem in terms of Fourier components and on the use of the whole spectrum of the frequency dependent permittivity, as it contains all the informations on the dynamic of the solvent response [10-17],... 

Long before FT methods were applied to NMR, physicists and engineers had made extensive use of Fourier transformation to analyze signals that vary with time in order to extract the fundamental frequencies that are present. The basic relation between a time response s(t) and its corresponding frequency spectrum is... 

The next milestone, in the history of NMR [Frel], was the extension of the NMR spectrum to more than one frequency coordinate. It is called multi-dimensional spectroscopy and is a form of nonlinear spectroscopy. The technique was introduced by Jean Jeener in 1971 [Jeel] with two-dimensional (2D) NMR. It was subsequently explored systematically by the research group of Richard Ernst [Em 1 ] who also introduced Fourier imaging [Kuml]. Today such techniques are valuable tools, for instance, in the structure elucidation of biological macromolecules in solution in competition with X-ray analysis of crystallized molecules as well as in solid state NMR of polymers (cf. Fig. 3.2.7) [Sch2]. 

A theoretical model to relate the Wiener spectrum to the toner deposit parameters is difficult to construct because the mathematical difficulties of dealing with projections of transforms of probability distributions quickly "hide" any simple relationships. Models have been constructed however for a crowded monolayer photographic emulsion (11), and for multilayers of emulsion (12). Although the analysis was done for one-dimensional geometry, extension to two dimensions was outlined. A different approach will be used here, which relies on the linearity property of the Fourier transform, and assumes that the location of the toner particles is independent of neighbors. 

Speetroscopic techniqnes are among the most widely used to study the effects of penetration enhancers on the SC. Infrared (IR) spectroscopy, especially Fourier transform IR (FTIR), has been used extensively to study the structure of SC in vitro. IR spectroscopy deals with the interaction between a molecule and radiation from the IR region of the electromagnetic spectrum (IR region = 10,0(X) to 400 cm , with the most useful region for functional analysis 4000 to 400 cm ). IR radiation causes the excitation of the vibrations of covalent bonds within that molecule. These vibrations include the stretching and bending modes. FTIR spectroscopy provides... 

Horlick and Yuen make extensive use of aliasing or folding of different spectral regions into the final spectrum. This topic is discussed in many references on Fourier transform spectroscopy (c.f. reference 19), and therefore will not be developed here. It is because of their use of aliasing that several spectra in Figure 9 have multiple frequency scales. 

The science of spectrum analysis is extensive and comprehensive. Therefore, the analyst who is interested in skillfully applying these techniques is advised to consult the complete list of references at the end of this chapter. Several interesting topics not covered here, such as Fourier transformation and resolution enhancement, are treated by several of the references. 

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