Fourier transform mathematics

Fourier Transformation. Mathematical process of converting the interference free induction decay into a spectrum. 

The final step in obtaining the spectrum by the FTIR method is turning back the data obtained as a result of the repetitive interference action of the moving mirror into an intensity wavelength line. It is here that Fourier Transform mathematics is utilized. It is the signal intensity that is stored in a digital representation of the Interferogram. This information is then Fourier transformed by the computer into the frequency spectrum. 

Additionally, with the inclusion of computers as part of an instrument, mathematical manipulation of data was possible. Not only could retention times be recorded automatically in chromatograms but areas under curves could also be calculated and data deconvoluted. In addition, computers made the development of Fourier transform instrumentation, of all kinds, practical. This type of instrument acquires data in one pass of the sample beam. The data are in what is termed the time domain, and application of the Fourier transform mathematical operation converts this data into the frequency domain, producing a frequency spectrum. The value of this methodology is that because it is rapid, multiple scans can be added together to reduce noise and interference, and the data are in a form that can easily be added to reports. 

Figure 3.4 Steps in X-ray structure determination. X-ray scattering by the crystal gives rise to a diffraction pattern. From the diffraction pattern, the molecular structure can be determined using Fourier transformation mathematical calculations. Source For diffraction photograph, Nicholls Ft. Double hehx photo not taken by Franklin, BioMedNet News and Comments, 2003. http //news.bmn.com/news/story day=030425 story=l caption name [accessed April 28, 2003].)...

Inverse Fourier Transform Mathematical transformation from the... 

In 1946, both Purcell and Bloch and their coworkers independently reported the first NMR spectra of paraffin and water, respectively. They were awarded the Nobel Prize for physics in 1952. Twenty years later, Ernst and Anderson applied Fourier-transform mathematics to this technique, increasing instrument sensitivity and spectral resolution and opening the door to many possible applications. Today, NMR analysis of compounds not only reveals chemical structure and conformation, but also molecular mobility and internal dynamics of systems. 

Fourier transformation Mathematical transformation of time domain functions into frequency domain. 

Interferometers utilize the interference phenomenon to produce an interferogram at a specific field of view. The interferogram is subjected to Fourier transformation mathematics to create a measurement spectrum of a sample specimen. The basic components of a classical Michelson interferometer are demonstrated by Chandler (1951) and Steel (1983). 

Infrared spectroscopy is a family of techniques that can be used to identify chemical bonds. When improved by Fourier transform mathematical techniques, the resulting test is known as FTIR. An FTIR scan can be used to identify compounds rather in the same way as fingerprints are used to identify humans an FTIR scan of the sample is compared to the FTIR scans of known compounds. If a positive match is found, the sample has been identified an example is shown in Figure 8.8. Not surprisingly, FTIR results are sometimes called fingerprints by analytical chemists. 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

Fourier transform Mathematical process that can be used to analyze a function of time into the individual frequency components that it contains. 

Mathematical manipulation (Fourier transform) of the data to plot a spectrum... 

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. 

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. 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

Fourier transformation A mathematical operation by which the FIDs are converted from time-domain data to the equivalent frequency-domain spectrum. 

The essence of analyzing an EXAFS spectrum is to recognize all sine contributions in x(k)- The obvious mathematical tool with which to achieve this is Fourier analysis. The argument of each sine contribution in Eq. (8) depends on k (which is known), on r (to be determined), and on the phase shift

characteristic property of the scattering atom in a certain environment, and is best derived from the EXAFS spectrum of a reference compound for which all distances are known. The EXAFS information becomes accessible, if we convert it into a radial distribution function, 0 (r), by means of Fourier transformation ... 

We note that the wave packet (x, t) is the inverse Fourier transform of A k). The mathematical development and properties of Fourier transforms are presented in Appendix B. Equation (1.11) has the form of equation (B.19). According to equation (B.20), the Fourier transform A k) is related to (x, t) by... 

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