Fast Fourier techniques

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. 

The average residence time is the sum of the diffusional value and a reversible reaction time component of retardation. The results in the time domain are no longer obtainable from a simple shift in s the roots of the denominator follow from a quadratic equation in s (see Cleaves et al., 1988 for details). The Fast Fourier technique remains though. 

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

The frequency-domain format eliminates the manual effort required to isolate the components that make up a time trace. Frequency-domain techniques convert time-domain data into discrete frequency components using a mathematical process called Fast Fourier Transform (FFT). Simply stated, FFT mathematically converts a time-based trace into a series of discrete frequency components (see Figure 43.19). In a frequency-domain plot, the X-axis is frequency and the Y-axis is the amplitude of displacement, velocity, or acceleration. 

Noticing the fact that the formula for determining surface deformation takes the form of convolution, the fast Fourier transform (FFT) technique has been applied in recent years to the calculations of deformation [35,36]. The FFT-based approach would give exact results if the convolution functions, i.e., pressure and surface topography take periodic form. For the concentrated contact problems, however. 

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]. 

Particle shapes influence properties such as surface area, bulk density, flow, and so on. A number of methods are available for describing shape from simpler qualitative descriptions, through property ratios, to techniques that employ fast Fourier transformations to describe the projected perimeter of the particle. The measurement of the shape and the relevance of the data obtained are generally the two difficulties associated with particle shape. Fortunately, in the processing of materials physically unlike those in chemical processing, shape is perhaps is less significant and is more often than not inherently accounted for in the nominal diameter. 

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. 

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-... 

The m/z values of peptide ions are mathematically derived from the sine wave profile by the performance of a fast Fourier transform operation. Thus, the detection of ions by FTICR is distinct from results from other MS approaches because the peptide ions are detected by their oscillation near the detection plate rather than by collision with a detector. Consequently, masses are resolved only by cyclotron frequency and not in space (sector instruments) or time (TOF analyzers). The magnetic field strength measured in Tesla correlates with the performance properties of FTICR. The instruments are very powerful and provide exquisitely high mass accuracy, mass resolution, and sensitivity—desirable properties in the analysis of complex protein mixtures. FTICR instruments are especially compatible with ESI29 but may also be used with MALDI as an ionization source.30 FTICR requires sophisticated expertise. Nevertheless, this technique is increasingly employed successfully in proteomics studies. 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by... 

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... 

In order to better understand the experimental results, we performed quantum mechanical calculations using the fast Fourier transform (FFT) split-operator technique, which was previously employed by Meier and Engel [38]... 

---