Fourier transform , generally

By using the integration theorem of two-sided Fourier transformation generalized to fractional calculus [32], namely,... 

In addition to covering Raman microscopy, this book has a wealth of information on Raman instrumentation in general. Elving P J and Winefordner J D (eds) 1986 Fourier Transform Infrared Spectroscopy (New York Wiley)... 

Other techniques for mass measurement are available, but they are not as popular as those outlined above. These other methods include mass measurements on a standard substance to calibrate the instrument. The standard is then withdrawn, and the unknown is let into the instrument to obtain a new spectrum that is compared with that of the standard. It is assumed that there are no instrumental variations during this changeover. Generally, this technique is less reliable than when the standard and unknown are in the instrument together. Fourier-transform techniques are used with ion cyclotron mass spectrometers and give excellent mass accuracy at lower mass but not at higher. 

The crystalline mineral silicates have been well characterized and their diversity of stmcture thoroughly presented (2). The stmctures of siHcate glasses and solutions can be investigated through potentiometric and dye adsorption studies, chemical derivatization and gas chromatography, and laser Raman, infrared (ftir), and Si Fourier transform nuclear magnetic resonance ( Si ft-nmr) spectroscopy. References 3—6 contain reviews of the general chemical and physical properties of siHcate materials. 

In the infrared spectral range in general Fourier transform (FT) interferometers are used. In comparison with dispersive spectrometers FTIR enables higher optical throughput and the multiplex advantage at equivalent high spectral resolution. In... 

The isotope has a nuclear spin quantum number I and so is potentially useful in nmr experiments (receptivity to nmr detection 17 X 10 that of the proton). The resonance was first observed in 1951 but the low natural abundance i>i S(0.75%) and the quadrupolar broadening of many of the signals has so far restricted the amount of chemically significant work appearing on this rcsonance, However, more results are expected now that pulsed fourier-transform techniques have become generally available. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

The transform option is selected from the plot menu bar. It displays a box which allows the user to select an operation to be performed on an entire axis of data. These can be any of three general types. The first are algebraic series of operations called "scripts". The second are unit transformations. The third are higher operations such as integration or Fourier Transform. 

Often the actions of the radial parts of the kinetic energy (see Section IIIA) on a wave packet are accomplished with fast Fourier transforms (FFTs) in the case of evenly spaced grid representations [24] or with other types of discrete variable representations (DVRs) [26, 27]. Since four-atom and larger reaction dynamics problems are computationally challenging and can sometimes benefit from implementation within parallel computing environments, it is also worthwhile to consider simpler finite difference (FD) approaches [25, 28, 29], which are more amenable to parallelization. The FD approach we describe here is a relatively simple one developed by us [25]. We were motivated by earlier work by Mazziotti [28] and we note that later work by the same author provides alternative FD methods and a different, more general perspective [29]. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

Two-dimensional NMR spectroscopy may be defined as a spectral method in which the data are collected in two different time domains acquisition of the FID tz), and a successively incremented delay (tj). The resulting FID (data matrix) is accordingly subjected to two successive sets of Fourier transformations to furnish a two-dimensional NMR spectrum in the two frequency axes. The time sequence of a typical 2D NMR experiment is given in Fig. 3.1. The major difference between one- and two-dimensional NMR methods is therefore the insertion of an evolution time, t, that is systematically incremented within a sequence of pulse cycles. Many experiments are generally performed with variable /], which is incremented by a constant Atj. The resulting signals (FIDs) from this experiment depend... 

There are actually two independent time periods involved, t and t. The time period ti after the application of the first pulse is incremented systematically, and separate FIDs are obtained at each value of t. The second time period, represents the detection period and it is kept constant. The first set of Fourier transformations (of rows) yields frequency-domain spectra, as in the ID experiment. When these frequency-domain spectra are stacked together (data transposition), a new data matrix, or pseudo-FID, is obtained, S(absorption-mode signals are modulated in amplitude as a function of t. It is therefore necessary to carry out second Fourier transformation to convert this pseudo FID to frequency domain spectra. The second set of Fourier transformations (across columns) on S (/j, F. produces a two-dimensional spectrum S F, F ). This represents a general procedure for obtaining 2D spectra. 

Dc correction is a process by which the contribution of the receiver dc is omitted from the FID. The dc level is generally determined by examining the last (one-fourth) portion of the FID (tail), which is more likely to have the maximum dc contribution of the receiver. The level is then subtracted automatically from each FID of the data set before Fourier transformation. 

One of the main points in the papers by Mermin and his collaborators [9 - 18] is the insistence on the primacy of reciprocal space. The properties of the Fourier transform of the density rather than the density itself determine those properties which are of importance for "generalized" crystallography. As pointed out by Mermin that point view was stressed in a paper by Bienenstock and Ewald already in 1962 [26]. 

By means of numerical convolution one can obtain Xg t) directly from sampled values of G t) and Xj(t) at regular intervals of time t. Similarly, numerical deconvolution yields Xj(t) from sampled values of G(t) and Xg(t). The numerical method of convolution and deconvolution has been worked out in detail by Rescigno and Segre [1]. These procedures are discussed more generally in Chapter 40 on signal processing in the context of the Fourier transform. 

The Fourier transform may be considered as a special case of the general data transform... 

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