Fourier transform principles

The rigid-body superposition method RigFit[133] in FlexS is based on some of the above Fourier transform principles. Other examples of the usefulness of Fourier transformation in virtual screening and modeling can be found in [134-138]. 

NMR experiments are not restricted to solutions but can also be conducted directly in the solid state. NMR was first observed in solids in 1945. Solid-state NMR has gained momentum since the introduction of the Fourier transform principle (after 1975). Recently, 750 MHz s-NMR instruments have been introduced. In order to obtain high-resolution s-NMR spectra, special techniques and spectrometer designs are employed. Although it is possible to use the same spectrometer for both solution and solid-state studies (and manufacturers are developing systems which can be modulated for any technique like 1-NMR, s-NMR, NMR Imaging, NMR Microscopy, or Localised NMR spectroscopy), usually each customer configures a particular spectrometer for only one experimental technique. 

Following the discovery of the NMR phenomenon in 1945 by two groups of physicists [1,2] NMR was applied to a wide range of systems, both solids and solutions. However, it rapidly became obvious that solution-state NMR was exceptionally useful to chemists because the high resolution achieved (with linewidths for H less than 1 Hz) aUowed small but important effects (i.e. chemical shifts and splittings due to coupling constants) to be observed. Solid-state NMR became for a while the esoteric preserve of a few hardy physicists and physical chemists. This situation became even more apparent after the introduction of the Fourier transform principle made high-quality spectra obtainable. 

From the Heisenberg uncertainty principle as stated in Equation (1.16) estimate, in cm and Hz, the wavenumber and frequency spread of pulsed radiation with a pulse length of 30 fs, typical of a very short pulse from a visible laser, and of 6 ps, typical of pulsed radiofrequency radiation used in a pulsed Fourier transform NMR experiment. 

Conceptually, the problem of going from the time domain spectra in Figures 3.7(a)-3.9(a) to the frequency domain spectra in Figures 3.7(b)-3.9(b) is straightforward, at least in these cases because we knew the result before we started. Nevertheless, we can still visualize the breaking down of any time domain spectrum, however complex and irregular in appearance, into its component waves, each with its characteristic frequency and amplitude. Although we can visualize it, the process of Fourier transformation which actually carries it out is a mathematically complex operation. The mathematical principles will be discussed only briefly here. 

Attenuated Total Reflectance Fourier Transform Infrared Spectroscopy. Attenuated total redectance (atr) ftir spectroscopy is based on the principle of total internal redection (40). Methods based on internal redection in the uv and visible regions of the spectmm are also common in addition to those in the ir region. The implementation of internal redection in the ir region of the spectmm provides a means of obtaining ir spectra of surfaces or interfaces, thus providing moleculady-specific vibrational information. 

Interferometry is difficult in the uv because of much greater demands on optical alignment and mechanical stabiUty imposed by the shorter wavelength of the radiation (148). In principle any fts interferometer can be operated in the uv when the proper choice of source, beam spHtter, and detector is made, but in practice good performance at wavelengths much shorter than the visible has proved difficult to obtain. Some manufacturers have claimed operating limits of 185 nm, and Fourier transform laboratory instmments have reached 140 nm (145). 

Such effects principally cannot be observed in multi band detectors such as a UV diode array detector or a Fourier transform infrared (FTIR) detector because all wavelengths are measured under the same geometry. For all other types of detectors, in principle, it is not possible to totally remove these effects of the laminar flow. Experiments and theoretical calculations show (8) that these disturbances can only be diminished by lowering the concentration gradient per volume unit in the effluent, which means that larger column diameters are essential for multiple detection or that narrow-bore columns are unsuitable for detector combinations. Disregarding these limitations can lead to serious misinterpretations of GPC results of multiple detector measurements. Such effects are a justification for thick columns of 8-10 mm diameter. 

Figure 1.33 The underlying principle of the Redfield technique. Complex Fourier transformation and single-channel detection gives spectrum (a), which contains both positive and negative frequencies. These are shown separately in (b), corresponding to the positive and negative single-quantum coherences. The overlap disappears when the receiver rotates at a frequency that corresponds to half the sweep width (SW) in the rotating frame, as shown in (c). After a real Fourier transformation (involving folding about n = 0), the spectrum (d) obtained contains only the positive frequencies.

A number of 2D NMR experiments, such as NOESY, have a mixing period incorporated in their pulse sequence. In principle, precession of j magnetization also occurs during the mixing period. Why do we not need to have a third Fourier transformation to monitor the precession frequencies that occur during the mixing period ... 

Shifting the origin in the Fourier space by uci, we obtain the wave-function FT[0(r)]e > , from which the lens aberration term can be eliminated in principle by multiplication with the inverse of the aberration phase factor e . The inverse Fourier transform gives finally the amplitude and phase of the true object wave 0 (f). 

There are many specific ways to generate equally spaced tags but they are all based on the same principle of manipulating the rf pulses to generate equally spaced bands of rf radiation in the frequency domain. It is well known that under ordinary conditions, meaning normal levels of nuclear spin excitation, the frequency spectrum of the rf excitation pulse(s) is approximately the Fourier transform of the pulses in the time domain. Thus, a single slice can be generated in the... 

Principles and Characteristics Fourier-transform infrared detection in SFC is attractive because it can offer structural information about the analytes [372]. The coupling was introduced in 1983 [373]. Various approaches have been advanced ... 

Principles and Characteristics Both mid-IR (2.5-50 p.m) and near-IR (0.8-2.5 p.m) may be used in combination to TLC, but both with lower sensitivity than UV/VIS measurements. The infrared region of the spectrum was largely ignored when the only spectrometers available were the dispersive types. Fourier-transform instruments have changed all that. Combination of TLC and FTIR is commonly approached in two modes ... 

Figure 3.4 shows (i) a line spectrum (one-dimensional dispersive spec-trographic record), (ii) a spectrometric record, (iii) an interferogram obtained by a Fourier transform spectrometer, and (iv, v) two- and three-dimensional double dispersive spectra recorded e.g. by Echelle spectrometers. In principle, all forms may be obtained by OES. 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

In the mid-IR, routine infrared spectroscopy nowadays almost exclusively uses Fourier-transform (FT) spectrometers. This principle is a standard method in modem analytical chemistry45. Although some efforts have been made to design ultra-compact FT-IR spectrometers for use under real-world conditions, standard systems are still too bulky for many applications. A new approach is the use of micro-fabrication techniques. As an example for this technology, a miniature single-pass Fourier transform spectrometer integrated on a 10 x 5 cm optical bench has been demonstrated to be feasible. Based upon a classical Michelson interferometer design, all... 

The blue crab (Callinectes sapidus) catches in the Chesapeake Bay have been measured [64] for the years 1922-1976 A.D., see figure 20. We have made a Fourier transform of the yield in millions of pounds per year (1 million pounds = 454 metric tonnes) versus years into amplitude versus period in years, see figure 21 and find principle cyclic periods of 8.6, 10.7, and 18.0 years. 

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