Fourier transformation computer processing

Because of this, S(k) is measured in the time domain. We obtain a one-dimensional imaging in the frequency domain using the Fourier transform. This process is carried out by a computer. 

Step 2 Fourier Transform. Compute the correlation functions hots h) ( ) = 91 (Hobs h) - S(h)))) with being the center of the peak actually processed. Only few discrete points of each correlation function are actually chosen for the evaluation, namely the points located at... 

This process is continued until there is only one component. For this reason, the number N is taken as a power of 2. The vector [yj] is filled with zeroes, if need be, to make N = 2 for some p. For the computer program, see Ref. 26. The standard Fourier transform takes N operations to calculation, whereas the fast Fourier transform takes only N log2 N. For large N, the difference is significant at N = 100 it is a factor of 15, but for N = 1000 it is a factor of 100. 

Radiofrequency spectroscopy (NMR) was introduced in 1946 [158,159]. The development of the NMR method over the last 30 years has been characterised by evolution in magnet design and cryotechnology, the introduction of computer-based operating systems and pulsed Fourier transform methods, which permit the performance of new types of experiment that control production, acquisition and processing of the experimental data. New pulse sequences, double-resonance techniques and gradient spectroscopy allow different experiments and have opened up the area of multidimensional NMR and NMRI. 

A Fourier transform infrared spectroscopy spectrometer consists of an infrared source, an interference modulator (usually a scanning Michelson interferometer), a sample chamber and an infrared detector. Interference signals measured at the detector are usually amplified and then digitized. A digital computer initially records and then processes the interferogram and also allows the spectral data that results to be manipulated. Permanent records of spectral data are created using a plotter or other peripheral device. 

For all of the cases considered earlier, a C(t) function is subjected to Fourier transformation to obtain a spectral lineshape function 1(G)), which then provides the essential ingredient for computing the net rate of photon absorption. In this Fourier transform process, the variable 0) is assumed to be the frequency of the electromagnetic field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to 1(G)) is ... 

This Fourier transform process was well known to Michelson and his peers but the computational difficulty of making the transformation prevented the application of this powerful interferometric technique to spectroscopy. An important advance was made with the discovery of the fast Fourier transform algorithm by Cooley and Tukey 29) which revived the field of spectroscopy using interferometers by allowing the calculation of the Fourier transform to be carried out rapidly. The fast Fourier transform (FFT) has been discussed in several places 30,31). The essence of the technique is the reduction in the number of computer multiplications and additions. The normal computer evaluation requires n(n — 1) additions and multiplications whereas the FFT method only requires (n logj n) additions and multiplications. If we have a 4096-point array to Fourier transform, it would require (4096) (4095) or 16.7 million multiplications. The FFT allows us to reduce this to... 

Model correlation functions. Certain model correlation functions have been found that model the intracollisional process fairly closely. These satisfy a number of physical and mathematical requirements and their Fourier transforms provide a simple analytical model of the spectral profile. The model functions depend on the choice of two or three parameters which may be related to the physics (i.e., the spectral moments) of the system. Sears [363, 362] expanded the classical correlation function as a series in powers of time squared, assuming an exponential overlap-induced dipole moment as in Eq. 4.1. The series was truncated at the second term and the parameters of the dipole model were related to the spectral moments [79]. The spectral model profile was obtained by Fourier transform. Levine and Birnbaum [232] developed a classical line shape, assuming straight trajectories and a Gaussian dipole function. The model was successful in reproducing measured He-Ar [232] and other [189, 245] spectra. Moreover, the quantum effect associated with the straight path approximation could also be estimated. We will be interested in such three-parameter model correlation functions below whose Fourier transforms fit measured spectra and the computed quantum profiles closely see Section 5.10. Intracollisional model correlation functions were discussed by Birnbaum et a/., (1982). 

---