Fast Fourier transform techniques

The most notable advance in computational crystallography was the availability of methods for rehning protein structures by least-squares optimization. This developed in a number of laboratories and was made feasible by the implementation of fast Fourier transform techniques [32]. The most widely used system was PROLSQ from the Flendrickson lab [33]. 

Future development of spectroscopic structure-determination methods will depend on the availability of more powerful photon and particle sources as well as advances in photon and particle detectors. Impressive progress has been made in molecular structure determinations based on advances in computation power and in computational algorithms, such as fast Fourier-transform techniques, for nearly every form of spectroscopy and diffraction analysis. Hajdu and co-work-... 

In the present work, we monitor the laser-driven dynamics designed by the present formulation by numerically solving the time-dependent Schrodinger (5.2). It is solved by the split operator method [52] with the fast Fourier transform technique [53]. In order to prevent artificial reflections of the wavepacket at the edges, a negative imaginary absorption potential is placed at the ends of the grid [54]. The envelope of the pulses employed is taken as... 

All the three integrals in Equation (33) are simply convolutions of one-dimensional functions and can be sequentially evaluated by using the fast Fourier transform technique. 

Application of fast-Fourier transform techniques to the discrete-dipole approximation. Optics Letters 16(15) 1198-1200. 

An alternative method of calculating the power spectrum is available in the fast Fourier transform technique [23], because ... 

One solution to the problem is to increase the ionization probability. This can be done by choosing primary ions with heavy mass, for example, Bi+ or even Ccarbon atoms. The noise level can also be reduced by techniques of digital image processing. For example, a fast Fourier transform technique has been used to remove noise from the image. This technique transforms an image from a space domain to a reciprocal domain by sine and cosine functions. Noise can be readily filtered out in such domain. After a reverse Fourier transform, filtered data produces an image with much less noise. 

Using whatever propagation method, one has to evaluate the action of the Hamiltonian operator on the wavefunction P(r). This is normally carried out by expanding P(f) in a suitable basis set and then evaluates the operator action on basis functions. One can use the FFT (fast Fourier transform) techniques (7,14), discrete variable representation (DVR) (15,16) techniques, or simply calculate matrix elements of the operator in a given basis set. 

Other finite element methods are available, including ones which employ fast Fourier Transform techniques. However these have been applied more to time-dependent problems and to my knowledge these have yet to be applied to multi-dimensional vibrational calculations. 

The digital signal processor, using fast fourier transform techniques generates a power spectrum of the original signal. The power spectrum is what is used to determine the particle size distribution of the material being analyzed. 

Schmehl, R., Nebeker, B. M., and Hirleman, E. D. (1997) Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique, / Opt. Soc. Am. A, 14,3026-3036. 

Goodman, J. J., Draine, B. T, and Flatau, P. J. (1991) Application of fast-Fourier-transform techniques to the discrete-dipole approximation. Opt. Lett., 16,1198-1200. 

Prior to 1966, spectroscopists who measured spectra interferometrically used the same basic algorithm for their computations. This involved the use of what is now known as the classical, conventional, or discrete Fourier transform. Although it is true that few people today use this algorithm in view of the substantial time advantages to be gained by use of the fast Fourier transform technique (described in Section 4.2), an understanding of the conventional Fourier transform leads to a better comprehension of more advanced techniques. 

As the number of points to be calculated increases beyond about 10,000, the calculation time for a spectmm can become prohibitive, even for very fast present-day computers, for which that computation can take many hours. Prior to the development of fast, readily available computers, this problem was especially annoying and did not appear to be resolvable until about 1966. At that time, Forman [2] published a paper on the application of the fast Fourier transform technique to Fourier spectrometry. This technique had been described in the literature by Cooley and Tukey [3] one year earlier. This algorithm extended the use of Fourier transform spectrometry to encompass high-resolution data in all regions of the infrared spectmm. It is described in the next section. 

Impedance spectroscopy is frequently used to characterize unstable systems that change with time. One way to reduce the effects of system alteration during the measurement is to reduce the total measurement time by using a multisine technique, which is also frequently called time-domain or FFT (fast Fourier transform) technique [2]. 

Forman, M. L. (1966). Fast Fourier transform technique and its application to Fourier spectroscopy. Journal of Optical Society of America, 56, 978-9. 

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