Fourier transform basic principles

Such a function exhibits peaks (Fig. 9C) that correspond to interatomic distances but are shifted to smaller values (recall the distance correction mentioned above). This finding was a major breakthrough in the analysis of EXAFS data since it allowed ready visualization. However, because of the shift to shorter distances and the effects of truncation, such an approach is generally not employed for accurate distance determination. This approach, however, allows for the use of Fourier filtering techniques which make possible the isolation of individual coordination shells (the dashed line in Fig. 9C represents a Fourier filtering window that isolates the first coordination shell). After Fourier filtering, the data is back-transformed to k space (Fig. 9D), where it is fitted for amplitude and phase. The basic principle behind the curve-fitting analysis is to employ a parameterized function that will model the... 

Considerable effort has gone into solving the difficult problem of deconvolution and curve fitting to a theoretical decay that is often a sum of exponentials. Many methods have been examined (O Connor et al., 1979) methods of least squares, moments, Fourier transforms, Laplace transforms, phase-plane plot, modulating functions, and more recently maximum entropy. The most widely used method is based on nonlinear least squares. The basic principle of this method is to minimize a quantity that expresses the mismatch between data and fitted function. This quantity /2 is defined as the weighted sum of the squares of the deviations of the experimental response R(ti) from the calculated ones Rc(ti) ... 

Figure 8.3 gives the basic layout of a continuous wave NMR spectrometer. These intruments were the original type of instrument and have largely been replaced by Fourier transform instruments. However, the principles of operation are broadly similar ... 

The foregoing has been a brief introductory discussion of NMR which has concentrated on some basic principles that are very useful in understanding the technique. The actual practice of NMR today is much more advanced. The incorporation of Fourier transform techniques has revolutionized NMR spectroscopy. All types of pulse sequences and two-dimensional (2D) techniques have been developed to provide even greater structural detail than has been discussed above. A discussion of such techniques belongs in a more specialized text, but it must be remembered that while these techniques are faster, more sensitive, and much more sophisticated, they are still largely based on the principles presented here, as is the interpretation of the results. 

A full discussion of the theory and performance of FTMS (Fourier transform mass spectrometry) is beyond the scope of this chapter however, books and review articles are available for the interested reader.1541 For the purpose of this discussion, an outline of the basic principles and performance characteristics of FTMS which are important for the measurement... 

Spectroscopy has become a powerful tool for the determination of polymer structures. The major part of the book is devoted to techniques that are the most frequently used for analysis of rubbery materials, i.e., various methods of nuclear magnetic resonance (NMR) and optical spectroscopy. One chapter is devoted to (multi) hyphenated thermograviometric analysis (TGA) techniques, i.e., TGA combined with Fourier transform infrared spectroscopy (FT-IR), mass spectroscopy, gas chromatography, differential scanning calorimetry and differential thermal analysis. There are already many excellent textbooks on the basic principles of these methods. Therefore, the main objective of the present book is to discuss a wide range of applications of the spectroscopic techniques for the analysis of rubbery materials. The contents of this book are of interest to chemists, physicists, material scientists and technologists who seek a better understanding of rubbery materials. 

This book intends to supply the basic information necessary to apply the methods of vibrational spectroscopy, to design experimental procedures, to perform and evaluate experiments. It does not intend to provide a market survey of the instruments which are available at present, because such information would very soon be outdated. However, the general principles of the instruments and their accessories, which remain valid, are discussed. Details concerning sample preparation and the recording of the spectra, which is the subject of introductory courses, are assumed to be known. Special procedures which are described in monographs, such as Fourier transformation or chemometric methods, are also not exhaustively described. This book has been written for graduate students as well as for experienced scientists who intend to update their knowledge. 

In Feature 25-7 we described the basic operating principles of the Michelson interferometer and the function of the Fourier transform to produce a frequency spectrum... 

For this reason, the reviewer proposes to introduce the reader to Fourier transform spectroscopy in the hope that he will make use of it. The basic physical principles of spectroscopy and the theory and practice of Fourier transform spectroscopy are described. Its advantages and disadvantages are discussed relative to spectroscopic problems and always with reference to the grating spectrometer as representing conventional spectroscopy. 

Several techniques have been used to investigate the reactivity of the metal carbide cluster ions formed in a laser vaporization source. The earliest investigations performed by Castleman s group relied on a preliminary mass selection of the desired cluster. The ion beam was then injected into a drift tube where the selected cluster encounters the reactant mixed with helium as a buffer gas. The FTICR (Fourier-transform ion cyclotron resonance) mass spectrometer studies reported by Byun, Freiser and co-workers basically rely on the same principle even though the total pressure of the reaction chamber is 10 torr, compared with 0.7 torr in Castleman s experiments. A new method of forming met-car ligand complexes was then reported by Castleman et al. this involved the direct interaction of the vaporized metal with mixtures of methane and selected reactant gases. ... 

A Light microscopy and X-ray crystallography share the same basic principle B X-Rays are scattered by electrons C The diffraction pattern corresponds to the Fourier transform of the crystal structure D The phase problem E Model building and refinement F Most-used types of electron-density maps... 

These considerations show how Fourier methods can be employed in analyzing systems with a number of resonance frequencies. As will be explained in the Section 3, nuclear magnetic systems are usually of this type, and the Fourier transform method discussed above is essentially that used in nuclear magnetic resonance. Therefore, the reader should bear in mind that a simple mechanical system was used to explain the basic principles but that all these considerations are already the first step of an introduction to Fourier transform nuclear magnetic resonance (FTNMR). 

Although the basic principles of ESR and NMR are similar, practical difficulties mean that pulse methods are less useful in ESR. This is because the pulse power required to produce the frequency span of a typical ESR spectrum would be several kilowatts and the pulse very short (nanoseconds). The pulse or FT (Fourier transform) ESR spectrometer is usually based on a standard CW instrument because it is often useful to record a standard ESR spectrum before carrying out pulse experiments. The pulse microwave source is usually a travelling wave tube other essentials... 

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