Fourier transform mathematical techniques

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. 

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. 

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. 

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. 

Fourier transform spectroscopy spect A spectroscopic technique in which all pertinent wavelengths simultaneously Irradiate the sample for a short period of time, and the absorption spectrum is found by mathematical manipulation of the Fourier transform so obtained. fur e a tranz,form spek tras-ko-pe fp See freezing point. 

What this equation indicates is that the intensity of the signal corning out of the beam splitter, I, can be written as the intensity in terms of wavenumbers, lv. This variable, lv, is the one the spectroscopist is interested in obtaining. Flowever, this intensity is inside an integral therefore, in order to obtain it, it is necessary to use a standard technique in mathematics called the Fourier transformation,... 

Because of this mathematical step, the technique is usually called Fourier transform infrared spectroscopy or FTIR spectroscopy. The Fourier transformation is a mathematical procedure that enables one to convert from the results of an interfero-gram back to intensities of a given wavelength. It is performed in a computer connected to the spectrometer. The result is the absorption spectrum of the sample, that is, the intensity of the absorbance as a function of the wavenumbers. 

Spectral Manipulation Techniques. Many sophisticated software packages are now available for the manipulation of digitized spectra with both dedicated spectrometer minicomputers, as well as larger main - frame machines. Application of various mathematical techniques to FT-IR spectra is usually driven by the large widths of many bands of interest. Fourier self - deconvolution of bands, sometimes referred to as "resolution enhancement", has been found to be a valuable aid in the determination of peak location, at the expense of exact peak shape, in FT-IR spectra. This technique involves the application of a suitable apodization weighting function to the cosine Fourier transform of an absorption spectrum, and then recomputing the "deconvolved" spectrum, in which the widths of the individual bands are now narrowed to an extent which depends on the nature of the apodization function applied. Such manipulation does not truly change the "resolution" of the spectrum, which is a consequence of instrumental parameters, but can provide improved visual presentations of the spectra for study. 

A Fourier transform is the mathematical technique used to compute the spectrum from the free induction decay, and this technique of using pulses and collecting transients is called Fourier transform spectroscopy. A Fourier transform spectrometer requires sophisticated electronics capable of generating precise pulses and accurately receiving the complicated transients. A good 13C NMR instrument usually has the capability to do H NMR spectra as well. When used with proton spectroscopy, the Fourier transform technique produces good spectra with very small amounts (less than a milligram) of sample. 

Equation 9 represents the IR spectmm (intensity versus wavenumber), which can be derived from expression (8) using a mathematical technique known as Fourier transformation. Needless to say, this requires spectrometer-interfaced computing power, which additionally provides the capacity for spectral manipulation such as deconvolution, smoothing, and subtraction. 

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