Mathematical methods Fourier transformations

Resolution and sensitivity are essential to the collection of analytical chemical data with accuracy and precision. It is well known that mathematical transformation techniques enhance the resolution and sensitivity of spectroscopic methods. Fourier transform (FT), cross correlation (CC), and Fladamard transform (FIT) techniques allow for high resolution and high sensitivity of infrared spectroscopy (IR), fiuorometry, nuclear magnetic resonance... 

References Brown, J. W., and R. V. Churchill, Fourier Series and Boundary Value Problems, 6th ed., McGraw-Hill, New York (2000) Churchill, R. V, Operational Mathematics, 3d ed., McGraw-Hill, New York (1972) Davies, B., Integral Transforms and Their Applications, 3d ed., Springer (2002) Duffy, D. G., Transform Methods for Solving Partial Differential Equations, Chapman Hall/CRC, New York (2004) Varma, A., and M. Morbidelli, Mathematical Methods in Chemical Engineering, Oxford, New York (1997). 

There is significant debate about the relative merits of frequency and time domain. In principle, they are related via the Fourier transformation and have been experimentally verified to be equivalent [9], For some applications, frequency domain instrumentation is easier to implement since ultrashort light pulses are not required, nor is deconvolution of the instrument response function, however, signal to noise ratio has recently been shown to be theoretically higher for time domain. The key advantage of time domain is that multiple decay components can, at least in principle, be extracted with ease from the decay profile by fitting with a multiexponential function, using relatively simple mathematical methods. 

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. 

The basic mathematical method for power spectrum analysis is the Fourier transformation. By the way. transient fluctuation can be expressed as the sum of the number of simple harmonic waves, which is helpful for understanding fluctuation. A frequency spectrum analysis for pressure signals can yield a profile of the frequencies and that of the amplitude along the frequencies. The basic equation of Fourier transformation can be expressed as... 

Variations in organic structure of vitrinite concentrates were determined with Fourier transform infrared spectroscopy (FTIR). FTIR is a relatively new method for obtaining quantitative data from the organic constituents of coal and provides spectra of greater quality than conventional infrared spectrometers. The system employs an on-line minicomputer which enables the user to analyze data and perform a variety of mathematical manipulations. 

FT-ICR, first developed more than a decade ago (Comisarow and Marshall, 1974a,b), has become very popular in recent years for both analytical and ion/molecule reaction studies. In the literature this method is also frequently termed Fourier transform mass spectrometry (FTMS). The term FT-ICR, however, indicates the physical principles of the method more precisely and is less confusing the mathematical operation of Fourier transformation can also be applied to some other forms of mass spectrometry such as time-of-flight mass spectrometry as has been demonstrated recently (Knorr et al., 1986). 

Refs. [i] Macdonald JR (1987) Impedance spectroscopy, Wiley, New York [ii] Arfken G (1985) In Mathematical methods for physicists. Academic Press, Orlando, p 760 [iii] Butz T (2006) Fourier transformations for pedestrians. Springer, Berlin... 

The Fourier transform (FT) has revolutionised spectroscopy such as NMR and IR over the past two decades. The raw data are not obtained as a comprehensible spectrum but as a time series, where all spectroscopic information is muddled up and a mathematical transformation is required to obtain a comprehensible spectrum. One reason for performing FT spectroscopy is that a spectrum of acceptable signal to noise ratio is recorded much more rapidly then via conventional spectrometers, often 100 times more rapidly. This has allowed the development of, for example, 13C NMR as a routine analytical tool, because the low abundance of 13 C is compensated by faster data acquisition. However, special methods are required to convert this time domain ... 

Fourier transform A mathematical method of breaking a signal (function or sequence) into component parts (for example, any curve can be approximated by the summation of a finite number of sinusoidal curves). In genome informatics, the Fourier transform of a sequence is used as a means of extracting information about the sequence into a more tractable, smaller number of features. 

Reducing oxide based isobaric interferences in the ICP mass spectrum via gas flow modulation was proposed by Wetzel and Hieftje. After a careful manipulation of the central chaimel gas flow to impact distinguishable frequency specific behaviour of analyte and oxide ion species and application of a Fourier transform (FT) correction method, contributions from an analyte and oxide species superimposed at a given mass can be mathematically umavelled with a degree of success. Through application of this correction method, a greater than ten-fold error at m/z 156 caused by tlie interference of Ce 0 on Gd has been effectively eliminated. ... 

Beddow [42] showed how a number of particle silhouette shapes could be analyzed and reproduced by Fourier transforms. Gotoh and Finney [52] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid having the same volume, surface area and projected area as the original body. 

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