The Fast Fourier Transform Method

In order to evaluate the effect of the kinetic energy operator operating on the wavefunction it is convenient to use the fast Fourier transform (FFT) methodology [237]. The kinetic energy operator entering the DFT problem is 

Using a discrete representation of the wavefunction 4 (x, y, z ) at N = NxNyN grid points we can introduce the Fourier transform as 

The derivative of the wavefunction with respect to x, for example, is now given as 

Thus the operator working on the wavefunction can be evaluated by replacing the coefficients cimn with 

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Figure 8 Structures of docked salicylhydrazine inhibitors in the IN catalytic core. Salicylhydrazines positioned by the MOE program are shown as stick models, those positioned by the fast Fourier transform method are shown as ball and stick models. SHI, SH2, and SH30 are shown on the top left, top right, and bottom, respectively.

Figure 6 shows the Fourier transformed and phase-corrected spectrum in the frequency domain. The fast Fourier transform method yields in this case 512 points with a resolution of 1 KHz. The splitting of the lines occurs due to the nuclear quadrupole interaction. 

On the basis of the above method, the algorithm of the calculation of the Fourier transforms expressed by Eqs (2.85) and (2.100) is calculated by using the Fast Fourier Transform Method [68]. 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

For the experimental model in Fig. 13, white Gaussian noise was used as the input signal to exeite the dynamies of the aetuator, whose displaeement as the output signal was measured using a laser displaeement sensor foeused 0.5 mm from its tip. The fast Fourier transform method was employed to obtain the transfer funetion in Eq. 27. The validity of this model was realized under a dynamie signal shown in Fig. 14 (John et al. 2008b). It must be noted the model validation is the last step in a typieal system identifieation proeedure. 

The two most frequently used grid methods to solve the Schrodinger equation are the discrete variable representation [19-21] (DVR), and the fast Fourier transform method [6, 22] (FFT). [Please see Chapter 3 for a critical comparison between the two grid methods.] In this dissertation, we exclusively use DVR because it allows us to tailor the grid, in a simple fashion, to the shape of the physical and absorbing potentials. 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

The Fourier analyzer is a digital deviee based on the eonversion of time-domain data to a frequeney domain by the use of the fast Fourier transform. The fast Fourier transform (FFT) analyzers employ a minieomputer to solve a set of simultaneous equations by matrix methods. 

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

The fast Fourier transform can be carried out by rearranging the various terms in the summations involved in the discrete Fourier transform. It is, in effect, a special book-keeping scheme that results in a very important simplification of the numerical evaluation of a Fburier transform. It was introduced into the scientific community in the mid-sixties and has resulted in what is probably one of the few significant advances in numerical methods of analysis since the invention of the digital computer. 

The spectral method is used for direct numerical simulation (DNS) of turbulence. The Fourier transform is taken of the differential equation, and the resulting equation is solved. Then the inverse transformation gives the solution. When there are nonlinear terms, they are calculated at each node in physical space, and the Fourier transform is taken of the result. This technique is especially suited to time-dependent problems, and the major computational effort is in the fast Fourier transform. 

The first term governs smoothness through the second differences of dn. The second term imposes the consistency between the solution values dn and the data values im. The tradeoff is controlled by varying parameter jS. Frieden (1975) explores the method briefly. Hunt (1973) applies it to images in a computationally efficient way that uses the fast Fourier transform. 

When using the fast-Fourier-transform algorithm to calculate the DFT, inverse filtering can be very fast indeed. By keeping the most noise-free inverse-filtered spectral components, and adding to these an additional band of restored spectral components, it is usually found that only a small number of components are needed to produce a result that closely approximates the original function. This is an additional reason for the efficiency of the method developed in this research. 

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... 

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