Fourier transform infrared mathematical methods

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. 

Fourier Transform Infrared (FT-IR) Spectroscopy With the introduction of commercial FT-IR spectrometers, the application of oil analysis by IR became relatively commonplace for production oil analysis laboratories. The mathematically intensive infrared data analysis techniques that were difficult or impossible to perform on the earlier IR systems became easy on these systems. In addition, quantitative analysis measurement techniques such as peak height, peak area, local baselines and more sophisticated matrix methods could be easily employed in the analysis, and the automation of lubricant analysis became commercially viable. 

In recent years, Fourier-transform infrared (FT-IR) spectroscopy has found increasing favour in laboratories. This more recent method is based on the old idea of the interference of radiation between two beams to yield an interferogram. An interferogram is a signal produced as a function of the change of pathlength between the two beams. The itwo domains of distance and frequency are interconvertible by the mathematical method of Fourier transformation. 

Medicine and Pharmaceuticals. Applied mathematics, particularly the use of Fourier analysis and wavelet theory, has sparked explosive growth in the fields of medicine and pharmaceutical development. The enhanced analytical methods available to chemists and bioresearchers through NMR and Fourier transform infrared (FTIR) spectroscopy feciUtate the identification and investigation of new compounds that have potential pharmaceutical applications. In addition, advanced statistical methods, computer modeling, and epidemiological studies provide the foundation for unprecedented levels of research. 

C. P. Schultz, The potential role of Fourier transform infrared spectroscopy and imaging in cancer diagnosis incorporating complex mathematical methods, Technol. Cancer Res. Treat., 2002, 1, 95-104. 

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... 

The major advance towards routine use in the mid-infrared region -j j- g. came with a new mathematical method (or algorithm) devised by Cooley and Tukey in 1965 for fast Fourier transformation. This was ... 

At last, it should be mentioned that, as always applies to Fourier methods , the major advantage of FTNMR is the multiplex advantage (see also (Chapter 1). Ernst and Anderson have shown that the gain in the signal-to-noise ratio between FTNMR and conventional NMR is proportional to the square root of the number of spectral elementsS . This result is the same as that obtained for infrared Fourier spectroscopy. In this context, it should be noted that a complete analogon to infrared Fourier spectroscopy is the nuclear magnetic Fourier-transfoim resonance with an incoherent rf-field (stochastic resonance) . Of course, Fourier methods depend on electronic computers to perform the Fourier transform of the measured data and were widely used when sufficiently cheap computers became available. The use of the compute- and of the mathematical treatment of the experimental... 

Two methods of recovering spectra by use of complex mathematical methods are known as Hadamard transform and Fourier transform spectroscopy. Both have been applied successfully to infrared spectroscopy. 

Since FT-IR spectrometry is based on the interference of waves of light (or radiation), first an account of this phenomenon is briefly given, before explaining the Fourier transform method by which an infrared spectrum is obtained from a measured interferogram. Some characteristics of FT-IR spectrometry, namely, wavenumber resolution, measurable wavenumber region, and accurate determination of wavenumbers are discussed. To facilitate the understanding of the description, which inevitably requires some mathematical formulations, many illustrations are provided. 

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